analele univ buc 2001

ANALELE UNIVERSITATIIBUCURESTI

MATEMATICA, ANO L, 2001

Annual National Conference ”Caius Iacob” of Fluid Mechanics and Technical Ap-plications 3

Dumitru-Ion ARSENIE and Ichinur OMER, Determination of the pressure ina pipeline with uniformly distribution of the discharge 5

Florin BALTARETU, Numerical prediction of air flow pattern in a ventilatedroom 11

Florin BALTARETU, Cornel MIHAILA, Numerical simulation of a horizontalbuoyant jet deflected by the Coanda effect 17

Galina CAMENSCHI, The effects of the temperature - dependent viscousity onflow in cooled channel 23

Claude CARASSO and Ruxandra STAVRE, Numerical simulation of a jet ofink 31

Adrian CARABINEANU, Numerical and qualitative study of the problem ofincompressible jets with curvilinear walls 37

Mircea Dimitrie CAZACU, On partial differential equations of the viscous liquidrelative flow through the turbomachine blade channel 45

Mircea Dimitrie CAZACU and Loredana NISTOR , Numerical solving of thebidimensional unsteady flow of a viscous liquid, generated by displacement of a flatplate 53

Eduard - Marius CRACIUN, Behaviour of the−42m piezoelectric crystal con-

taining a crack in antiplane state 61

Liviu Florin DINU , An example of interaction between two gasdynamic objects:a piecewise constant solution and a model of turbulence 67

Alexandru DUMITRACHE , An Analytical Model of the Glass Flow during thePressing Process 79

C. FALUP-Precurariu, D. MINEA, Oana FALUP-PRECURARIU, LauraDRACEA, The dynamic of mechanical forces on lung properties 87

Constantin FETECAU , Nonsteady shearing flow of a fluid of Maxwellian type 93

2 Sumar Sommaire Contents

Florin FRUNZULICA and Lucian IORGA, An adaptive method for structuredmeshes in CFD problems 99

Stelian ION, A Numerical Method for Richards’ Equation 109

Mirela KOHR, Boundary Integral Method for an Oscillatory Stokes Flow Prob-lem 115

Mircea LUPU, Adrian POSTELNICU, Ernest SCHEIBER, Analytical Methodfor maximal drag airfoils optimization in cavity flows 123

Dorin MARINESCU, On a Boltzmann Model of Fermions 141

Anca Marina MARINOV, A Two Dimensional Mathematical Model for Simu-lating Water and Chemical Transport in an Unsaturated Soil 149

Alexandru M. MOREGA, A Numerical Analisys of Laminar Transport Processesin Ducts with Cross-Sectional Periodicity 159

Sebastian MUNTEAN, Romeo F. SUSAN-RESIGA, Ioan ANTON and Vic-tor ANCUSA, Domain Decomposition Approach for 3D Flow Computation inHydraulic Francis Turbine 169

Elena PELICAN, Constantin POPA, Approximate orthogonalization of lin-early independent functions with applications to Galerkin-like discretization tech-niques 179

Dumitru POPESCU, Stelian ION and Maria Luiza FLONTA,Appearance of pores through black lipid membranes due to collective thermic move-ment of lipid molecules 185

Mihai POPESCU, Optimality and Non-Optimality Criteria for Singular Control193

Lucica ROSU, Liliana SERBAN, Dan PASCALE, Cornel CIUREA andCarmen MAFTEI, The study of the water stability in canals with rectangularcross section by means of frequency analysis 199

Valeriu Al. SAVA, Translation flows of non-local memory-dependent micropolarfluids 205

Romeo F. SUSAN-RESIGA and Hafiz M. ATASSI, Nonreflecting Far-FieldConditions for Unsteady Aerodynamics and Aeroacoustics 211

Victor TIGOIU, On the Uniqueness of the Solution of the Initial and BoundaryValue Problem for Third Grade Fluids 223

Annual National Conference ”Caius Iacob” ofFluid Mechanics and Technical Applications

anniversary session

Institute of Applied Mathematics ”Caius Iacob” –ten years

In the Romanian scientific life it is acknowledged as a tradition the factthat the researchers in the Fluid Mechanics field participate at a scientificmeeting every year in October.

This scientific event, also known as National Colloquium on Fluid Me-chanics and its Technical Applications for a long time, initiated since 1959 bythe Society of Mathematical Science has become, thanks to all fluid mechan-ics researchers’ effort, an annual meeting. It take place in various economicor universitary centers all over the country.

In addition to the usual significance of this event, the meeting that willghater us this year has a special meaning: it celebrate ten years since theInstitute of Applied Mathematics ”Caius Iacob” of Romanian Academy wasfounded. The process of setting up such an institute has begun after thebreaking up of the scientific institutes of Romanian Academy and has beencarried on by the Colloquies.

As for this year’s Conference, we do consider it necessary that this tra-dition should be preserved and that an old goal should be achieved: ”that alarge amount of productive units, as well as the youth in Universities shouldbe involved in such scientific events: [..] that the Fluid Mechanics courseshould be introduced in every University” (Caius Iacob and C. Ciobanu,preface of the Proceedings of National Colloquium on Fluid Mechanics, Iasi,1978)

3

Analele Universitatii Bucuresti, MatematicaAnul L(2001), pp. 5–10

Determination of the pressure in a pipeline withuniformly distribution of the discharge

Dumitru Ion ARSENIE and Ichinur OMER

November 2, 2001

Abstract - In the literature, the most studied is the samples pipelines in the steady and

unsteady conditions. We propose to study the waterhammer phenomenon in the pipeline

with uniformly distribution of the discharge. By using the impulse variation theorem, we

obtain the variation of the medium velocity, the variation of the velocity of elastic wave

propagation and the variation of the pressure along the pipeline. We have determined

these when in the final section occurs a quick totally close manoevre of the valve. We

applie the results into a numerical example, using an original program.

Key words and phrases : pipeline with uniformly distribution of the discharge,

pressure, discharge, velocity of the elastic wave propagation, waterhammer.

Mathematics Subject Classification (2000) : 76A05

1 Introduction

In the literature the most studied pipeline is the simple pipeline, in thesteady and unsteady flow, less treated being specials pipelines.

In this paper we propose to study the waterhammer in the pipeline withuniformly distribution of the discharge in the case of the quick totally closemanoeuvre of the valve which is in the final section of the pipeline. Thispipe represent one of the simplified models of water supply pipes, irrigationsetc.

We shall obtain the variation of the medium velocity, the variation of thevelocity of elastic wave propagation and the variation of the pressure alongthe pipeline. In the end of the paper we present a numerical example .

Useful notations : p - the pressure, β - ununiform coefficient of the rate(Boussinesq), ρ - water density, v - flow velocity, A - area of the transversalsection, Q - discharge, t - the time, V - the volume, ∆p-the medium pressurevariation on the segment, vx,t - the velocity in the x section and at t time,

cj=∆x

∆t- the medium velocity of elastic wave propagation on the segment

5

6 D.-I. Arsenie , I. Omer

j, j = 1, n− 1,the segment j being delimited by the sections i and i+1,

B =q0∆x

2A, νi+1 =

vi+1,i

B. vx,0 = v1,0 +

q0

A(L− x) - the velocity in thex

section at initial time before the stop manoeuvre of the valve. For the sectionx of the segment j and after the moment when the waterhammer wave hasarrived in this section, the velocity is:

vx,t = vi,t +q0

A

√1 +

∆pj

p0[L− (i− 1) ∆x− x] .

In the numerical example, the pipeline with the uniformly distribution of thedischarge was divided into equals lengths segments ∆x. In the x - coordinatesections which delimit these segments, we applied the precedents formulas,considering: x = i∆x; i = 1, n.

The first section is the final section (going out of the pipeline with theuniformly distribution of the discharge) and the last section is the initialsection (entrance of the water in the pipeline with the uniformly distributionof the discharge), the sections order being in the same direction like thewaterhammer propagation direction.

One of the used hypothesis refers to the way of the discharge variationwhich leaves the pipe and which has an uniformly distribution in the initialsteady flow. We considered that the specifically discharge q0, correspondingto the pressure p0, is changed when the steady condition, respective when thepressure in the pipeline become p0 + ∆p, by the orifice-nozzle type relation:

q0 = k√

p0 , qx = k√

p0 + ∆px

=⇒ qx = q0

√1 +

∆px

p0(1)

2 Mathematical model

To determine the proposed relation, we applie the variation of impulse the-orem on the control volume which delimit the liquid comprised into thesections i and i+1 (the segment j).

In the projection on the water flow direction (the horizontally axis), thescalar relation is.

−∆piA = βρviQi − βρvi+1Qi+1+∫

V

ρ∂vx

∂tdV (2)

Determination of the preassure 7

Figure 1: The calculus scheme

Because the pipeline’s axis is horizontal, the weight of the liquid (−→G )

and the reaction of the pipeline’s wall (−→R ) don’t appear in this equation.

Replacing the expression of the velocity’s derivative by a finit difference:

∂vx

∂t∼= vx,t − vx,0

∆t(3)

and solving the integral, the relation (2)becomes:

∆pi+1 = ρcj

[v1,0 +

q0

A

(i− 1

2

)∆x− vi,i−1

](4)

−ρcjq0∆x

2A

√1 +

∆pj

p0

Making the follow notation yi+1 =ci+1

cj, the relation (4) can be write

thus:


√1 +

∆pj

p0=

v1,0

B+ 2i− 1− vi+1,0

B+

yi+1∆vi+1

B(5)

Rasing to square the relation (5) and solving the equation in yi+1, weobtain the calculus relation for this parameter:

yi+1 =1

νi+1

v1,0

B+ 2i− 1− vi+1,0

B+

√1 +

∆pj

p0

(6)

Because the segments have small length, we admitted that between the

medium value√

1 + ∆pj

p0and the values of the segments extremities is one

relation corresponding to the linear variation:√

1 +∆pj

p0=

12

(√1 +

∆pi

p0+

√1 +

∆pi+1

p0

)(7)

In the relation (6) result that have appeared two unknown variablesyi+1and ∆pi+1. Between these two variables exists the Jukovski relation:

∆pi+1 = −ρci+1∆vi+1 = −ρcjyi+1∆vi+1 (8)

For solve this problem we used an iterative procedure, considering theinitial approach ∆pj

∼= ∆pi, which permits to calculate yi+1, respective ci+1.Then with the relation (8) we determine ∆pi+1 and thus we may calculateagain ∆pj in the next approach. In the numerical example treated, theiterative process stopped when the relative error of the value ∆pj lowersunder 1%.

The velocity vi+1is:

vi+1,i = B(

√1 +

∆p1

p0+ 2

√1 +

∆p2

p0+ . . .

+2

√1 +

∆pi

p0+

√1 +

∆pi+1

p0) (9)

and the variation of velocity:

∆vi+1 = vi+1,i − vi,0 (10)

For the final segment 1, the previous relations become

Determination of the preassure 9

y1 = 1 +1ν2

+1ν2

√1 +

∆p1

p0(11)

∆v2 = B

(√1 +

∆p1

p0+

√1 +

∆p2

p0− 2

)− v1,0 (12)

Figure 2: The distribution of velocity’s variation, pressure variation andvelocity of elastic wave propagation along the pipeline.

3 Numerical example

We consider a pipeline with the uniformly distribution of the discharge forwhich we know the follows characteristically elements: d = 250 mm, l =200m, Qi = 80 l/s, q0 = 0,0002m3/ms, c1 = 1000m/s, p1=4 bari (400000N/m2).We divided the pipeline into equals lengths segments ∆x = 10m and


we determined the variation of the medium velocity, the variation of thevelocity of elastic wave propagation and the variation of the pressure alongthe pipeline (for four segments).

4 Conclusions

We observe in the Figure (2) that the medium velocity, the velocity of elasticwave propagation and the pressure diminish along the pipeline, strating fromthe final section of the pipeline where is the valve. It is possible that insome situations the pressure’s variations becomes so small, that they maybe neglected. The quality aspect (the attenuation of the waterhammer alongthe pipeline) doesn’t meet in the simple pipeline.

References

[1] Trofin, P., Water Supply. E.D.P., Bucuresti, 1983.

[2] Jeager, Ch., Fluid Transients in Hydro-Electric Engineering PracticeBlackie, London, 1977.

[3] Cioc, D. Hydraulic. E.D.P., Bucuresti, 1975.

Dumitru Ion ArsenieOvidius University Constantza, B-dul Mamaia nr.124,8700-Constanta, ROMANIAE-mail: [email protected]

Ichinur OmerOvidius University Constantza, B-dul Mamaia nr.124,8700-Constanta, ROMANIAE-mail: [email protected]


Numerical prediction of air flow pattern in aventilated room

Florin BALTARETU

November 2, 2001

Abstract - This paper is concerned with the numerical simulation of turbulent air

flow in a ventilated room with ceiling slot air supply and return. The room is operated

isothermally with the walls and supply air at the same temperature. In order to avoid

the shortcut of the flow, the distance between the supply slot and the vertical wall is

chosen as to produce the attachment of the inlet jet on the vertical wall by the Coanda

effect. Turbulence k−ε model with finite volume method and the SIMPLE computational

algorithm are used to study the air flow in the room. Numerical results and comparison

with experimental data of time averaged velocity and turbulence characteristics in the

room are presented.

Key words and phrases : finite volume method, k − ε model, indoor air flow


1 Introduction

Indoor air quality and thermal comfort are a major concern of modern times,because most of the time are spent in interior activities. It is therefore nec-essary to study the air flow patterns in the room, which has a great influenceon the spreading of contaminants and on the temperature distribution withinthe room. This paper describes a numerical investigation, using turbulencek − ε model with finite volume method and the SIMPLE computational al-gorithm, in order to validate the capability of a personal code to predict theair flow pattern inside a ventilated room.

2 Experimental measurements

Xu et al. [7] conducted experimental measurements at the University ofMinnesota. Interior measurements were made using a computer controlledthree-axis positioning system, with a position accuracy within 0.4 mm in

11

12 F. Baltaretu

all three directions. The chamber used has inside dimensions of 1.95 mwide, 1.45 m high and 1.95 m in depth, in order to simulate a single roomwith a scalin factor of 2.0. The air was supplied downward through a slot1.97 m long and 5 cm wide, designed to provide fully developed turbulentchannel flow at the exit. The return was of the same design, and both supplyand return slots were located 0.36 m from the nearest wall. The inlet airvelocity was 200 ft/min (1.02 m/s). All the other experimental conditionsare specified in [7].

3 Model equations

We considered the general governing differential equations:

∂Uj

∂xj= 0 (1)

ρ∂Ui

∂t+ ρ

∂

∂xj(UiUj) = − ∂p

∂xi+

∂

∂xj(τij − ρuiuj) (2)

τij = µ

[(∂Ui

∂xj+

∂Uj

∂xi

)− 2

3δij

∂Uk

∂xk

](3)

with Boussinesq approximation:

−ρuiuj = 2µtSij − 23δij

(µt

∂Uk

∂xk+ ρk

)(4)

and the standard k− ε turbulence model of Launder & Spalding (1974) [6]:

∂ (ρk)∂t

+∂

∂xi

(ρUik − µt

σk

∂k

∂xi

)= ρ (Pk − ε) (5)

∂ (ρε)∂t

+∂

∂xi

(ρUiε− µt

σε

∂ε

∂xi

)= ρ

ε

k(Cε1Pk − Cε2ε) (6)

µt = ρCµCDk2

ε(7)

Pk = νt

(∂Ui

∂xj+

∂Uj

∂xi

)∂Ui

∂xj(8)

The turbulence model constants are:

Cµ = 0.5478, CD = 0.1643 ⇒ CµCD = 0.09

σk = 1.0, σε = 1.314, Cε1 = 1.44, Cε2 = 1.92 (9)

Air flow pattern 13

Because of the known limitations of the standard k− ε model, which uses asingle time scale, poor predictions are obtained in a certain number of casesand especially in the case of free jets and in the case of reattachment flows.For this reason, we considered the Chen & Kim (1987) [2] turbulence modelmodification, which improves the results for non-equilibrium turbulence. Byconsidering an additional time scale, a new term is added into (6):

Sε = ρCε4P 2

k

k(10)

The modified turbulence model constants are:

σk = 0.75, σε = 1.15, Cε1 = 1.15, Cε2 = 1.9 (11)

In order to take into account the effect of relaminarization, the turbulentviscosity is multiplied by a damping funtion of local turbulent Reynoldsnumber:

νt = fRetCµCDk2

ε, Ret =

k2

νε(12)

We used the damping function of Jones & Launder (1972) [3]:

fRet = exp

[−3.4(

1 + Ret50

)2

](13)

We considered the non-dimensional variables:

x∗i =xi

Lref, t∗ =

t

Lref/Uref, U∗

i =Ui

Uref, p∗ =

p− pref

ρU2ref

k∗ =k

U2ref

, ε∗ =ε

U3ref/Lref

, Γ∗ =Γ

LrefUref(14)

The general transport equation is:

∂φ∗

∂t+

∂

∂x∗j

(U∗

j φ− Γ∗∂φ∗

∂x∗j

)= S∗φ (15)

4 Boundary conditions

The boundary conditions at the wall are considered by placing the firstinternal node within the fully turbulent region, and using the so-called wallfunctions (semi-empirical laws of the wall):

τw

ρ=

(C

1/2D kP

)1/2

U+P

UP , kP =U2

P

C1/2D

(U+

P

)2, εP =

(C

1/2D kP

)3/2

κyP (16)

14 F. Baltaretu

where:

U+P =

1κ

ln y+P + B , y+

P =

(C

1/2D kP

)1/2

νyP (17)

and the constants take the values κ = 0.41, B = 5.0.For inlet section, the values for k and ε are taken:

k = I2U2 , ε = (CµCD)3/4 k3/2

lm(18)

where I stands for turbulence intensity, and lm ∼= 0.1W , with W the slot’swidth.

5 Numerical method

The numerical method is explained in detail in [1]. We choosed the finitevolume method [4], using total implicit scheme for convection-diffusion termdiscretization. The source term was liniarized, in the form:

S = SC + SP φP (19)

with SP ≤ 0 condition.The method ”coefficient and value” is used for implementing boundary

conditions, as fictive source terms.For solving the velocity-pressure coupling, the SIMPLE pressure correc-

tion procedure [4] is chosen, using hybrid differencing scheme [5]:

aE = max (aE ,−Fe, 0) , aW = max (aW , Fw, 0) ,

aN = max (aN ,−Fn, 0) , aS = max (aS , Fs, 0) (20)

The solution of discretized equations is obtained with the TDMA line-by-line relaxation method, by rewriting the discret analog as:

−aEφE + aP φP − aW φW = [aNφN + aSφS + b] (21)

For all the variables, under-relaxation is used:

φP = αφn + (1− α) φo (22)

with the under-relaxation factors listed below:

Variable U V p k ε

α 0.3 . . . 0.7 0.3 . . . 0.7 0.2 . . . 0.3 0.5 . . . 0.7 0.5 . . . 0.7

In order to avoid physically incorrect negative values of k and ε, we useda limitation procedure, which consist in imposing positive minimum values.

Air flow pattern 15

6 Numerical results

The numerical results are presented in figure 1 and figure 2.

(a) Data of Xu et al. [7] (b) Present simulation

Figure 1: Comparision of velocity distribution

Figure 2: Turbulence kinetic energy distribution

7 Conclusions

The general pattern of air flow obtained using a personal code is very sim-ilar with the pattern presented by Xu et al. [7]. One can observe that theinlet jet attaches the right side vertical wall at a distance of approximatively0.83 m from the floor, in very good agreement with experimental data. Theturbulence kinetic energy distribution is also very similar with the exper-imental one, excepting the outlet region. This fact occurs because of thelack of k − ε model to produce an accurate evaluation of the kinetic energyproduction term for the regions with large velocity gradients.

16 F. Baltaretu

References

[1] Baltaretu, F., Mathematical and numerical modelling of transport phe-nomena in buoyant jets. Ph.D. Thesis, U.T.C.B., 2001.

[2] Chen, Y.S., Kim, S. W., Computation of turbulent flows using an ex-tended k − ε model, NASA CR-179204, 1987.

[3] Jones, W.P., Launder, B. E., The prediction of laminarization with atwo-equation model of turbulence, Int. J. Heat Mass Transfer, 15(1972),301-314.

[4] Patankar, S. V., Numerical Heat Transfer and Fluid Flow. Hemisphere,New York, 1980.

[5] Spalding, D. B., A novel finite difference formulation for differential ex-pressions involving both first and second derivatives, Int. J. Num. Meth.Engng., 4(1972), 551-559.

[6] Wilcox, D. C., Turbulence modeling for CFD. DCW Industries, LaCanada, CA, 1994.

[7] Xu, J., Liang, H., Kuehn, T. H., Comparison of numerical predictionsand experimental measurements of ventilation in a room, ProceedingsROOMVENT’94 (1994), 2, 213-227.

Florin BaltaretuU.T.C.B. (Civil Engineering Technical University of Bucharest)Thermodynamics and Heat Transfer DepartmentBd. Pache Protopopescu 66, sect.2, RO-73232, Bucharest-39, ROMANIAE-mail: [email protected]


Numerical simulation of a horizontal buoyant jetdeflected by the Coanda effect

Florin BALTARETU and Cornel MIHAILA

November 2, 2001

Abstract - In the present paper, the reattachment of a horizontal buoyant jet on a plane

wall - by the Coanda effect - is investigated numerically using a finite volume method for

the solution of time averaged continuity, Navier-Stokes, and energy equations. The critical

condition of the Coanda effect breaking the buoyancy effect is reported. The numerical

method is based on the SIMPLE approach for velocity and pressure coupling, and the

k−ε turbulence model including buoyancy correction is used. The results are expected to

be of importance for indoor air distribution and heating and air-conditioning of industrial

spaces.

Key words and phrases : finite volume method, k−ε model, buoyancy, Coanda effect


1 Introduction

An important problem relating the heating or cooling of a large interiorspace using air jets is how to control the buoyancy effect. Considering theheating case, it is essential to limit the action of the warm air in the occupiedzone, otherwise the air rises, and the local heating effect is lost. A veryeffective method for ”capturing” the jet is the Coanda effect (attachment ofthe jet in the near presence of a solid surface), and such a flow is often called”an offset jet” [5], [6]. However, critical conditions occur in the case of anon-isothermal jet, because the buoyancy effect and the Coanda effect act inopposite directions. This paper deals with numerical simulation of describedphenomena, having as an objective to find out the critical situation.

2 Experimental measurements

A series of experiments was conducted by Yamada et al. [9], by using amodel chamber, whose upper side and opposite end were opened. The jet

17

18 F. BAltaretu, C. Mihaila

considered is plane and horizontal, with initial velocity of 3,4 and 5 m/s .The distance between the inlet slot and the floor was considered 0.6 m to0.8 m.

3 Key factors on flow development

The key factors on flow development are:- the distance between the floor and the inlet slot, Hs;- the initial temperature difference, ∆T0, between the warm jet and the am-biant air;- the initial jet velocity, U0;- the width of the inlet slot, h;- the ambiant temperature, Ta;- the pressure values in the regions above and below the jet.Most of the factors mentioned above are present in the Archimedes number:

Ar =gh

U20

β∆T0 (1)

which is often used to characterize the evolution of a free non-isothermaljet. As shown by Yamada et al. [9], an more complete number for a non-isothermal offset jet is:

K = Ar

(Hs

h

)3/2

(2)

4 Model equations

The governing equations for a non-isothermal flow are:

∂Uj

∂xj= 0 (3)

∂Ui

∂t+

∂

∂xj

(UjUi − νe

∂Ui

∂xj

)= −1

ρ

∂p

∂xi+

∂

∂xj

(νe

∂Uj

∂xi

)+ giβ (T − Tr) (4)

∂T

∂t+

∂

∂xj

(UjT − αe

∂T

∂xj

)=

Sh

ρcp(5)

where νe represents effective kinematic viscosity and αe represents effectivethermal diffusivity:

νe = ν + νt , αe = α + αt =ν

Pr+

νt

σT(6)

Numerical simulation 19

The standard k − ε model for a non-isothermal flow is:

∂ (ρk)∂t

+∂

∂xi

(ρUik − µt

σk

∂k

∂xi

)= ρ (Pk + GB − ε) (7)

∂ (ρε)∂t

+∂

∂xi

(ρUiε− µt

σε

∂ε

∂xi

)= ρ

ε

k(Cε1Pk + Cε3GB − Cε2ε) (8)

where GB is the buoyancy term:

GB = −gνt

σh

1ρ

∂ρ

∂xi≈ g

νt

σTβ

∂T

∂xi(9)

In order to improve the performance of the k − ε turbulence model forreattaching flows, we used the Chen & Kim modified k − ε model.We also considered a damping function which includes corrections for theeffect of relaminarization and for buoyancy effects:

νt = fµCµCDk2

ε, fµ = fBfRet (10)

For relaminarization effects we used the Jones & Launder correction func-tion:

fRet = exp

[−3.4(

1 + Ret50

)2

], Ret =

k2

νε(11)

We chosed as buoyancy measure parameter the term B = GB/ε, and theChikamoto et al. correction function [4]:.

fB =

0 for B ≤ 1/a0

1− a0B for 1/a0 < B < 0 , where a0 = −1.1 for B ≥ 0

(12)

For this model:

Cε3 =

1.44 for GB > 00 for GB ≤ 0

(13)

5 Numerical results

We used the finite volume method for the discretization of the flow equationsand the SIMPLE procedure for pressure-velocity coupling. The numericalapproach is explained in [1], [2].Some numerical results are presented in figures 1 and 2. Numerical resultsshow that the attachment occur for K ≤ 0.2. In order to have a stableattachment, the K ≤ 0.15 condition is proposed.


6 Conclusions

The reattachment of a horizontal buoyant jet on a plane wall - by the Coandaeffect - is investigated numerically. The critical condition of the Coandaeffect breaking the buoyancy effect is reported, as a measure of the non-dimensional factor K.

(a) Velocity distribution (b) Temperature distribution

(c) Velocity distribution (d) Temperature distribution

Figure 1: U = 3.5 m/s: ∆T = 17.5C and ∆T = 15C

Numerical simulation 21

(a) Velocity distribution (b) Temperature distribution

(c) Velocity distribution (d) Temperature distribution

(e) Velocity distribution (f) Temperature distribution

Figure 2: U = 5 m/s: ∆T = 35C, ∆T = 33.5C, ∆T = 30C


References

[1] Baltaretu, F., Mathematical and numerical modelling of transport phe-nomena in buoyant jets. Ph.D. Thesis, U.T.C.B., 2001.

[2] Baltaretu, F., Numerical prediction of air flow pattern in a ventilatedroom, An. Univ. Bucuresti, Mat., 50 (2001), 11–16.

[3] Chen, Y.S., Kim, S. W., Computation of turbulent flows using an ex-tended k − ε model, NASA CR-179204, 1987.

[4] Chikamoto, T., Murakami, S., Kato, S., Numerical simulation of velocityand temperature fields within atrium based on modified k − ε modelincorporating damping effect due to thermal stratification, ProceedingsISRACVE (1992), 501-510.

[5] Hoch, J., Jiji, L. M., Two-dimensional turbulent offset jet-boundary in-teraction, ASME J. Fluid Eng., 103(1981), 1, 154-161.

[6] Holland, J.T., Liburdy, J. A., Measurements of the thermal characteris-tics of heated offset jets, Int. J. Heat Mass Transfer, 33(1990), 1, 69-78.

[7] Patankar, S. V., Numerical Heat Transfer and Fluid Flow. Hemisphere,New York, 1980.

[8] Wilcox, D. C., Turbulence modeling for CFD. DCW Industries, LaCanada, CA, 1994.

[9] Yamada, N., Kubota, H., Kurosawa, K., Yoshida, Y., Hanaoka, Y.,Local space heating by covering with a warm plane jet, ProceedingsROOMVENT’94 (1994), 2, 299-308.

Florin Baltaretu, Cornel MihailaU.T.C.B. (Civil Engineering Technical University of Bucharest)Thermodynamics and Heat Transfer DepartmentBd. Pache Protopopescu 66, sect.2, RO-73232, Bucharest-39, ROMANIAE-mail: [email protected]


The effects of the temperature - dependentviscousity on flow in cooled channel

Galina CAMENSCHI

November 2, 2001

Abstract - The mathematical model of an incompressible linear viscous fluid motion with

the dynamical viscousity coefficient µ depending only on the temperature in a thin cooled

channel with small curved walls is presented assuming a relation between the Reynolds

number and the ratio of middle breadth and the length of the channel. The governing

system of equations leads, following the mentioned assumptions, to differential equations

for the pressure and the temperature. The velocity field equations depends on the pressure,

the function µ(T ) and the geometry of the channel. The boundary conditions refers to

the adherence of the fluid to the channel walls, the prescription of the temperature on

them and then pressure difference at the exit and imput in the channel. The geometry

of the channel walls and the function µ(T ) being given the pressure and the temperature

can be determined. Particulary, the problem can be used as a model for basaltic fissure

eruptions.

Key words and phrases : cooled channel, viscous fluid, temperature-dependent vis-

cosity

Mathematics Subject Classification (2000) : 76D99

1 Introduction

Let S0 and S1 be two fixed rigid vertical walls with small curvature. In theOx1x2x3 coordinate system, where Ox1x3 belongs to the tangent plane inO to S0, the S1 surface equation is x2 = h(x1, x3) (fig.1). In the actualconfiguration

Bt = (x1, x2, x3)/x1 ∈ (−∞, +∞), x2 ∈ h(x1, x3), x3 ∈ (0, l)

the motion is governed by Cauchy’s equation, the continuity equation, theconstitutive equation and the internal energy equation in spatial description

23

24 Galina Camenschi

ρ~a = ρ~b + divTT ,

div~v = 0,

T = −pI + 2µ(T)D, in Bt

ρe = T ·D + ρr − div~q,

(1)

where ~a is the accelaration vector, T and D the stress tensor and thedeformation rate tensor.

S S0 1

x

x

xx

3

3

21

= l

B t

Figure 1: The geometry of the problem

We neglect the body forces ( ~b = ~0 )and the heat sources (r = 0) as wellas the scalar product T ·D for the incompressible fluid motion. The internalenergy density is considered by e = c T where c is the specific heat which isa constant. The heat flux vector ~q is given by the Fourier law

~q = −κ∇T, (2)

where T is the absolute temperature and κ is the constant heat conductivitycoefficient.

The previous hypothesis lead to

∂vl

∂t+ vm

∂vl

∂xm= −1

ρ

∂p

∂xl+

1ρ

∂

∂xm

(µ(T )

∂vl

∂xm+ µ(T )

∂vm

∂xl

), l = 1, 3 (3)

∂vm

∂xm= 0, (4)

∂T

∂t+ ~v · ∇T = a∆T, (5)

Temperature - dependent viscousity 25

wherea =

κ

ρc. (6)

The boundary conditions corresponding to the adherence of the fluid tothe walls, to the constant temperatures T1 and T0 at the walls as well as theimput in Bt, are expressed in the form

v1(x1, 0, x3) = v2(x1, 0, x3) = v3(x1, 0, x3) = 0,

v1(x1, h(x1, x3), x3) = v2(x1, h(x1, x3), x3) = v3(x1, h(x1, x3), x3) = 0,

T (x1, 0, x3) = T (x1, h(x1, x3), x3) = T1, T (x1, x2, 0) = T0.(7)

Let l be the characteristic length for the xi, (i = 1, 3) directions andd (the mean thicknes of the channel) the characteristic length for the x2

direction. Nondimensionalising the equations system (3)-(5) using the valuesof the characteristic time, speed and pressure given by

t0 =d2

a, v0 =

a l

d2, p0 =

µ0al2

d4, T∗ = T0 − T1 > 0, (8)

where µ0 = µ(T0) is a constant, one obtains

t = t0t0, xi = lx0

i , i = 1, 3, x2 = dx02

vi = v0v0i , i = 1, 3, v2 = v∗0v

02, p = p0p

0, µ = µ0µ0, T = T∗T 0.

(9)

Introducing (9) in (4) it results

v∗0 =d

lv0 =

a

d. (10)

The relation (6) gives

κ

ρca= 1. (11)

We denote the Reynolds, Prandtl and Peckle numbers by

Re =ρv0l

µ0, P r =

µ0c

κ, Pe = Re · Pr =

v0l

a. (12)

26 Galina Camenschi

Following [1] for the numerical data

d ≈ 0, 3m, l ≈ 10km, ρ ≈ 3000kg m−3 a ≈ 10−6ms−1,

T∗ = T0 − T1,≈ 100K, Re ≈ 10, P r ≈ 3 · 104 À 1,(13)

it results

ε =d

l¿ 1, Re · ε2 =

1Pr

¿ 1, P e =1ε2À 1, (14)

usually Pe ≈ 3 105.Introducing (9) and (10) in (4) one obtains the dimensionless continuity

equation

∂v0m

∂x0m

= 0, in Bt (15)

and the equations of motion (3), via(9) and (10), in the form

Re · ε2(

∂v01

∂t0+ v0

m

∂v01

∂x0m

)= −∂p0

∂x01

+ ε2∂

∂x0l

(2µ0 ∂v0

1

∂x01

)+

+∂

∂x02

(µ0 ∂v0

1

∂x02

)+ ε2

∂

∂x02

(µ0 ∂v0

2

∂x01

)+ ε2

∂

∂x03

(µ0 ∂v0

1

∂x03

+ µ0 ∂v03

∂x01

),

Re · ε4(

∂v02

∂t0+ v0

m

∂v02

∂x0m

)= −∂p0

∂x02

+ ε4∂

∂x01

(µ0 ∂v0

2

∂x01

)+

+ε2∂

∂x01

(µ0 ∂v0

1

∂x02

)+ ε2

∂

∂x02

(2µ0 ∂v0

2

∂x02

)+

+ε4∂

∂x03

(µ0 ∂v0

2

∂x03

)+ ε2

∂

∂x03

(µ0 ∂v0

3

∂x02

),

Re · ε2(

∂v03

∂t0+ v0

m

∂v03

∂x0m

)= −∂p0

∂x03

+ ε2∂

∂x0l

(µ0 ∂v0

3

∂x01

+ µ0 ∂v01

∂x03

)+

+∂

∂x02

(µ0 ∂v0

3

∂x02

)+ ε2

∂

∂x02

(µ0 ∂v0

2

∂x03

)+ ε2

∂

∂x03

(2µ0 ∂v0

3

∂x03

).

(16)


The equation for the temperature results from (5), via (9) and (10) as

∂T 0

∂t0+ v0

m

∂T 0

∂x0m

= ε2(

∂2T 0

∂x01

2+

∂2T 0

∂x03

2

)+

∂2T 0

∂x02

2. (17)

2 The velocity field and the presure equation

Using the estimations (14) in (16) and (17) one can write the dimensionalsystem equations for the linear viscous incompressible fluid motion havingthe dynamical viscousity coefficient µ(T ) in the cooled channel in the form

∂p

∂x1=

∂

∂x2

(µ(T )

∂v1

∂x2

),

∂p

∂x2= 0,

∂p

∂x3=

∂

∂x2

(µ(T )

∂v3

∂x2

),

∂vm

∂xm= 0, in Bt

∂T

∂t+ ~v · ∇T = a

∂2 T

∂x22

.

(18)

The equation (18)1,3, via (18)2 and the boundary conditions (7) lead to thev1 and v3 velocity components determination in the form

v1(x1, x2, x3) =∂p

∂x1

[ x2∫

0

x2

µ(T )dx2 −

h(x1,x3)∫

0

x2

µ(T )dx2 ·

x2∫

0

dx2

µ(T )

h(x1,x3)∫

0

dx2

µ(T )

],

(19)

v3(x1, x2, x3) =∂p

∂x3

[ x2∫

0

x2

µ(T )dx2 −

h(x1,x3)∫

0

x2

µ(T )dx2 ·

x2∫

0

dx2

µ(T )

h(x1,x3)∫

0

dx2

µ(T )

].

(20)If we denote h3

1 = h(x1, x3), the continuity equation, the equations (19),(20) and the adherence condition on S0 lead to determination of the v2

velocity component in the form

28 Galina Camenschi

v2(x1, x2, x3) = −(

∂2p

∂x21

+∂2p

∂x23

) [ x2∫

0

η∫

0

ξ

µ(T )dξ

dη+

+

h31∫

0

x2

µ(T )dx2

h31∫

0

dx2

µ(T )

x2∫

0

η∫

0

dξ

µ(T )

dη

]+

∂p

∂x1

x2∫

0

η∫

0

ξ

µ2(T )∂µ

∂x1dξ

dη +

+

−

h31∫

0

x2

µ2(T )∂µ

∂x1dx2 +

∂h

∂x1

h31

µ(x1, h31, x3)

x2∫

0

η∫

0

dξ

µ(T )

dη

h31∫

0

dx2

µ(T )

−

−

h31∫

0

x2

µ(T )dx2

h31∫

0

dx2

µ(T )

2

[(

h31∫

0

dx2

µ(T )

x2∫

0

η∫

0

1µ2(T )

∂µ

∂x1dξ

dη

+

+

−

h31∫

0

1µ2(T )

∂µ

∂x1dx2 +

∂h

∂x1

1µ(x1, h3

1, x3)

x2∫

0

η∫

0

dξ

µ(T )

dη

]+

+∂p

∂x3

x2∫

0

η∫

0

ξ

µ2(T )∂µ

∂x3dξ

dη +

[−

h31∫

0

x2

µ2(T )∂µ

∂x3dx2+

+∂h

∂x3

h31

µ(x1, h31, x3)

]×

x2∫

0

η∫

0

dξ

µ(T )

dη :

h31∫

0

dx2

µ(T )−

−h31∫

0

x2

µ(T )dx2 :

h31∫

0

dx2

µ(T )

2[

h31∫

0

dx2

µ(T )

x2∫

0

η∫

0

1µ2(T )

∂µ

∂x3dξ

dη

−

h31∫

0

1µ2(T )

∂µ

∂x3dx2 − ∂h

∂x3

1µ(x1, h3

1, x3)

x2∫

0

η∫

0

dξ

µ(T )

dη

.

(21)


The boundary condition on the wall S1 : v2(x1, h(x1, x3), x3) = 0, allowsus the determination of the pressure equation

(∂2p

∂x21

+∂2p

∂x23

)[ h31∫

0

x2∫

0

ξ

µ(T )dξ

dx2 −

h31∫

0

x2∫

0

dξ

µ(T )

dx2×

×h31∫

0

x2

µ(T )dx2 :

h31∫

0

dx2

µ(T )

]− ∂p

∂x1

h31∫

0

x2∫

0

ξ

µ2(T )∂µ

∂x1dξ

dx2 −

−

h31∫

0

x2

µ2(T )∂µ

∂x1dx2 − ∂h

∂x1

h31

µ(x1, h31, x3)

h31∫

0

x2∫

0

dξ

µ(T )dx2 :

h31∫

0

dx2

µ(T )

−

−

h31∫

0

x2

µ(T )dx2 :

h31∫

0

dx2

µ(T )

2

h31∫

0

dx2

µ(T )

h31∫

0

x2∫

0

1µ2(T )

∂µ

∂x1dξ

dx2 −

−

h31∫

0

1µ2(T )

∂µ

∂x1dx2 − ∂h

∂x1

1µ(x1, h3

1, x3)

h31∫

0

x2∫

0

dξ

µ(T )

dx2

−

− ∂p

∂x3

h31∫

0

x2∫

0

ξ

µ2(T )∂µ

∂x3dξ

dx2 +

[−

h31∫

0

x2

µ2(T )∂µ

∂x3dx2 +

− ∂h

∂x3

h31

µ(x1, h31, x3)

] h31∫

0

x2∫

0

dξ

µ(T )

dx2 :

h31∫

0

dx2

µ(T )−

−h31∫

0

x2

µ(T )dx2 :

h31∫

0

dx2

µ(T )

2

h31∫

0

dx2

µ(T )

h31∫

0

x2∫

0

1µ2(T )

∂µ

∂x3dξ

dx2

−

−

h31∫

0

1µ2(T )

∂µ

∂x3dx2 − ∂h

∂x3

1µ(x1, h3

1, x3)

×

h31∫

0

x2∫

0

dξ

µ(T )

dx2

= 0.

3 Comments

It easy to see that if the h(x1, x3) function for the surface S1 is given andµ(T ) is also given (as in [1] for example), the pressure can be determinedfrom the last equation.

30 Galina Camenschi

Then introducing the functions h(x1, x3) and µ(T ) in (19) - (21) thevelocity field can be also obtained.

From the equation (18)5 the absolute temperature can be determinedwith the initial condition T (0, x1, x2, x3) = T ∗(x1, x2, x3).

For the integration of the pressure equation we assume as usual thatpressure difference ∆p = p(x1, 0)− p(x1,l) is given and is a constant.

References

[1] J.J. Wylie, J. R. Lister - The effects of temperature dependent viscousityon flow in a cooled channel with application to basaltic fissure eruptions,J.Fluid Mech., 305, (1995), 239-261.

[2] K. R. Hefrich - Thermo-viscous fingering of flow in a thin gap: a modelmagma flow in dikes and fissures, J.Fluid Mech., 305, (1995), 219-238.

[3] D. Berscovici - A theoretical model of cooling viscous gravity currentswith temperature dependent viscousity, Geophys. Res. Lett., 21, (1994),1177-1180.

Galina CamenschiDepartment of Mechanics, University of Bucharest ,14 Academiei Str., 70109 Bucharest, Romania


Numerical simulation of a jet of ink

Claude CARASSO and Ruxandra STAVRE

November 2, 2001

Abstract - The hydrodynamics of a jet of ink, exiting from a printer is considered. A

one-dimensional model is obtained from the Navier-Stokes equations, by using a rational

asymptotic expansion. A partial differential system for the first order approximation is

obtained and some numerical computations are performed.

Key words and phrases : breakup of the jet, similar solutions

Mathematics Subject Classification (2000) : 76B10, 76D05, 76M45

1 Introduction

This paper deals with a one-dimensional model evolution equation, whichdescribes the hydrodynamics of a jet of ink, exiting from a printer. Themathematical model is obtained from the Navier-Stokes equations by us-ing a rational asymptotic expansion of the unknown functions. This is aproblem with applications in the industrial process of conception of printerswith jet of ink. By giving the vibrations of the liquid at the nozzles of thereservoir, we study, both from the theoretical and the numerical points ofview the breakup of the viscous jet. This problem was previously studiedin [1]. In this paper, the authors used the Cosserat model for obtaining,under some hypotheses, a nonlinear partial differential system. Some nu-merical computations were also performed. Experimental observations of jetbreakup phenomena have been carried out in [2], [3], [4], [6]. The study of aninfinite jet breakup is performed in [5]. The model, derived from the Stokesequations, is employed in extensive simulations to compute breakup timesfor different initial conditions; satellite drop formation is also supported bythe model and the dependence of satellite drop volumes on initial conditionsis studied. In Section 2, by using a cylindrical coordinate system, we givethe dimensionless equations, the boundary and the initial conditions whichdescribe the unsteady flow of a semi-infinite, incompressible Navier-Stokesjet. By assuming that the ratio between the radius of the nozzles to the

31

32 R. Stavre

dimensional axial length scale is an asymptotically small parameter whichcan be used in an asymptotic expansion of the unknown functions, we de-rive, in Section 3, the system of partial differential equations for the firstorder approximation. Our interest is in the description of the breakup phe-nomenon. In Section 4 we show that the system describing the leading orderapproximation has a singularity with the jet radius vanishing and the fluidvelocity becoming infinite after a finite time, at some axial location. Thenext section deals with the numerical study of the problem. The system forthe first order approximation is solved by using a finite difference method.The numerical experiments show that the breakup of the jet depends on theinitial conditions.

2 The mathematical model

We consider the evolution of a semi-infinite jet of incompressible viscousfluid (ink), exiting from the nozzles of a printer. We suppose that the flowis axisymmetric. Initially the jet is a semi-infinite cylinder of radius r0

(the radius of a nozzle) and constant velocity. By giving the vibrations ofthe fluid at the nozzles, we obtain a free boundary problem, in cylindricalcoordinates, (r, z). If ρ > 0 is the constant density of the fluid, µ > 0 theconstant viscosity of the fluid, α > 0 the constant surface tension coefficient,u, v and p the radial velocity, the axial velocity and the pressure of the fluid,respectively, R the unknown radius of the jet and v0, v1, f, r0 positive givenconstants,we obtain for the nondimensional variables:

r =r

r0, z =

z

L, (u, v) =

1vs

(u, v), p =r0

µvsp, R =

R

r0, t =

vs

r0t,

the following system:

kvz +1r(ru)r = 0,

Re(ut + uur + kvuz) = −pr + urr +1rur + k2uzz − u

r2, in Ω(t),

Re(vt + uvr + kvvz) = −kpz + vrr +1rvr + k2vzz,

(1)

u = Rt + kvRz,

2kRz(kvz − ur)− (1− k2R2z)(vr + kuz) = 0,

p(1 + k2R2z)− 2(ur + k3R2

zvz) + 2kRz(vr + kuz) on r(t) = R(z, t),

= A1 + k2(−RRzz + R2

z)R(1 + k2R2

z)1/2,

(2)

Numerical simulation of a jet of ink 33

u(r, 0, t) = 0,v(r, 0, t) = V0 + V1sinFt,R(0, t) = 1,

(3)

u(r, z, 0) = 0,v(r, z, 0) = V0,R(z, 0) = 1,

(4)

where r0 and L are the length scales in the radial and axial directions re-spectively, vs the velocity scale,

µvs

r0the pression scale,

r0

vsthe time scale,

k =r0

L,A =

α

µvs, V0 =

v0

vs, V1 =

v1

vs, F = 2πf

r0

vs. The unknown flow

region, Ω(t) is defined by:

Ω(t) = (r, z)/z > 0, 0 < r < R(z, t), ∀ t ≥ 0. (5)

3 The asymptotic expansions

Since the lenght scale L does not have a physical meaning, we suppose thatr0 << L, so that k can be considered as an asymptotically small parameter.We use as asymptotic parameter the coefficient of the highest-order deriva-tive of (1). We seek the solution (u, v, p, R) of the problem (1)-(4), as anexpansion in powers of k2, of the form:

u(r, z, t) = u0(r, z, t) + k2u1(r, z, t) + ..., (6)

v(r, z, t) = k−1(v0(r, z, t) + k2v1(r, z, t) + ...), (7)

p(r, z, t) = p0(r, z, t) + k2p1(r, z, t) + ..., (8)

R(z, t) = R0(z, t) + k2R1(z, t) + .... (9)

The boundary problem satisfied by the first order approximation (v0, R0) isthe following:

R0t + v0R0z +12R0v0z = 0,

BR0v0t−3R0v0zz−6R0zv0z+BR0v0v0z−AR0z

R0=0, in IR+ × IR+,

(10)

v0(0, t) = V0 + V1sinFt,v0z(0, t) = 0, ∀ t ≥ 0,R0(0, t) = 1,

(11)

v0(z, 0) = V0,R0(z, 0) = 1, ∀ z ≥ 0.

(12)

By solving (10)-(12) we obtain v0, R0.

34 R. Stavre

4 The breakup of the jet

In the sequel we shall prove that the system (10) has a solution with thefollowing property: the jet radius vanishes and the fluid velocity becomesinfinite after a finite time, at some axial location. The next theorem provesthis property.

Theorem 1. There exists (zb, tb) ∈ IR∗+ × IR∗

+ so that

limzzb

( limttb

R0(z, t)) = 0,

limzzb

( limttb

v0(z, t)) = ∞.(13)

Proof. Let (z0, t0) be an arbitrary point in IR∗+×IR∗

+. For (z, t) ∈ [0, z0[×[0, t0[,we define the new variables ζ = z0 − z and τ = t0 − t and we introduce the

similarity variable φ =ζ

τ δ. We seek a solution (R0, v0) of the system (10) of

the form: R0 = ταf(φ),v0 = τβg(φ).

(14)

Introducing (14) into (10) we obtain:

2(δφf ′ − af)− τ b−δ+1(2f ′g + fg′) = 0,3τ2a+b−2δf2g′′ + 6τ2a+b−2δff ′g′ −Aτa−δf ′+B(bτ2a+b−1f2g − δτ2a+b−1φf2g′ + τ2a+2b−δf2gg′) = 0.

(15)

Choosing a, b and δ so that τ does not appear explicitly in (15) (i. e.a = 1, b = −1/2, δ = 1/2) and introducing the new variable f1 = Af,denoted also by f, the system (15) becomes:

f ′(φ− 2g)− f(2 + g′) = 0,

3(f2g′)′ − f ′ +B

2f2(g2 − φg)′ = 0.

(16)

We seek a solution of (16) of the form:

f(φ) = a1φ

p + a2φp−1 + ...,

g(φ) = b1φn + b2φ

n−1 + ...,(17)

Numerical simulation of a jet of ink 35

with a1, b1 ∈ IR∗ and p, n ∈ Z. Introducing the expansions (17) in (16) itfollows:

(p− 2)a1φp + (p− 3)a2φ

p−1 + O(φp−2)− (2p + n)a1b1φp+n−1−

((2p + n− 2)a2b1 + (2p + n− 1)a1b2)φp+n−2 + O(φp+n−3) = 0,nBa2

1b21φ

2p+2n−1+Ba1b1((2n− 1)a1b2 + 2na2b1)φ2p+2n−2+O(φ2p+2n−3)−B

2(n + 1)a2

1b1φ2p+n−B

2a1(2(n + 1)a2b1 + na1b2)φ2p+n−1+O(φ2p+n−2)−

pa1φp−1 − (p− 1)a2φ

p−2 −O(φp−3) = 0.(18)

We study in the sequel the following cases:i) p < p + n− 1. In this case we obtain from (18) a contradiction.ii) p = p + n− 1. (18)1 becomes:

a1((p−2)−(2p+1)b1)φp+((p−3)a2−(2p−1)a2b1−2pa1b2)φp−1+O(φp−2)=0. (19)

In this case max(2p + 2n− 1, 2p + n, p− 1) =max(2p + 1, p− 1).ii1) 2p + 1 > p − 1. Then p > −2 and max(2p + 1, p − 1) = 2p + 1.

The coefficient of φ2p+1 in (18)2 is Ba21b

21 − Ba2

1b1, which yields b1 = 1.Introducing this value in (19) we get p = −3, in contradiction with p > −2.

ii2) 2p + 1 < p− 1. Then p < −2 and max(2p + 1, p− 1) = p− 1. From(18)2 we obtain, since a1 6= 0, p = 0, again a contradiction with p < −2.

ii3) 2p + 1 = p − 1. In this case p = −2 and the identification of thecoefficients (for φ−2, φ−3 in (4.6)1 and for φ−3, φ−4, φ−5 in (18)2 leads usto a contradiction. The last case is

iii) p > p + n − 1. In this case n < 1 and (18)1 yield p = 2; hencemax(2p + 2n− 1, 2p + n, p− 1) = max(2n + 3, n + 4, 1) =max(n + 4, 1).

iii1) n + 4 < 1. Then max(n + 4, 1) = 1 and from (18)2 we obtain−pa1 = 0, which is impossible for p = 2.

iii2) n + 4 ≥ 1. Then max(n + 4, 1) = n + 4 and (18)2 gives −B(n +1)a2

1b1 = 0 which yields n = −1. Hence the only possible values for p and nare 2 and −1, respectively. For the above determined values of a, b, δ, p, n,the relations (14) become:

R0 = (z0 − z)2(a1 + O(φ−1)),v0 = (z0 − z)−1(b1 + O(φ−1)),

(20)

and, hence, the assertion of the theorem follows.

5 Numerical results.

In the previous section it was proved that there exist solutions of (10) char-acterized by the breakup phenomenon. In this section we shall study the

36 R. Stavre

influence of the boundary conditions on the breakup. For solving the prob-lem (10)-(12) we use a finite difference method. Since for nonlinear problemsthe stability depends on the structure of the finite difference procedure, wechose an implicite scheme. We denote by ∆z and ∆t the mesh size on Ozand Ot, respectively and by fn

j the value f(j∆z, n∆t) for any function f.The values for the physical data, used for the computations, are: ρ = 800kg/m3, µ = 5 · 10−3 Pa·s, α = 10−3 N/m, r0 = 3 · 10−6 m, v0 = 5 m/s,v1 = 1 m/s. For the parameters L and vs, without a physical meaning, wechose the values 4 · 10−5 m and 0.05 m/s, respectively; ∆z = 0.4, ∆t = 0.1.

Experimental observations show that sometimes the jet radius does notvanish. The numerical computations, performed for different values of thefrequence of the vibrations, f, are in agreement with the experiments. Forsmaller values of f (f < 34 · 103 Hz), we obtain after 150 iterations in timea stabilization of the oscillations of the jet radius around its initial value. Ifwe increase the number of iterations in time, we obtain a similar behaviourof the jet radius. The breakup phenomenon appears for f > 34000 Hz.

References

[1] Carasso, C.,Largillier, A.,Regal M-C.,Formation des gouttes dans un flu-ide non newtonien soumis a un champs electrique, 2eme colloque franco-chilien de mathematiques appliquees, C. Carasso, C. Conca, R. Coreea,J. P. Puel ed., CEPADUES Editions,(1991), 133-144.

[2] Chaudhary, K. C.,Maxworthy, T., The nonlinear capillary instability of aliquid jet. Part 2-3. Experiments on jet behavior before droplet formation,J. Fluid Mech., 96, (1980), 275-297.

[3] Donelly, R. J.,Glaberson, W., Experiments on the capillary instability ofa liquid jet, Proc. Roy. Soc. Lond. A 290, (1966), 547-556.

[4] Goedde, E. F.,Yuen, M. C., Experiments on liquid jet instability, J. FluidMech., 40, (1970), 495-511.

[5] Papageorgiu, D. T., On the breakup of viscous liquid threads, Institute forComputer Applications in Science and Engineering Report 95-1, (1995).

[6] Peregine, D. H., Shoker, G.,Symon, A., The bifurcation of liquid bridges,J. Fluid Mech., 212, (1990), 25-39.

Ruxandra StavreInstitute of Mathematics ”Simion Stoilow”, Romanian Academy,P. O. Box 1-764 RO-70700 Bucharest, RomaniaE-mail: [email protected]


Numerical and qualitative study of the problem ofincompressible jets with curvilinear walls

Adrian CARABINEANU

November 2, 2001

Abstract - The jet flow problem concerning the discharge of a fluid (from an orifice in

a container) into the atmosphere is studied herein in the framework of the Helmholtz-

Kirchhoff model. The problem is reduced to the study of a system of nonlinear equations.

Using Leray-Schauder’s fixed point theorem we prove that the system of functional equa-

tions has at least one solution. Then we present a semi-inverse method which gives us the

possibility to calculate numerically the unknown free lines for jets whose walls consist of

semi-infinite straight lines and arcs of circle

Key words and phrases : incompressible flow, free-lines, jet, topological-degree

Mathematics Subject Classification (2000) : 76B10, 35J25

1 Introduction

The free - boundary streamline flow is still an open research field. Recentpapers dedicated to this subject are dealing either with numerical [2] or ana-litical [4] methods of investigation. The jet flow problem is concerned withthe discharge of a fluid from an orifice (in a fixed vessel or container) intoan atmosphere at constant pressure. For purposes of theory the convenientidealization assumes that the vessel has two semi-infinite walls $1 and $2

(extending to infinity upstream) each of them consisting of a semi-infinitestraight portion and a finite curvilinear portion nearby the orifice (figure1). In this paper we assume that the walls $1 and $2 are simmetric withrespect to the Ox - axis.

We assume that the wall $1 consists of an arc of circle having the radiusR and the length νπR and a semi-infinite straight line, streching to infinityupstream (x → −∞) and making with the Ox - axis the angle (1− µ)π

where 0 ≤ ν ≤ µ ≤ 12. Let A and B be the edges of the orifice of the jet (i.e.

the endpoints of the walls $1 and $2) and let L = |zA − zB| be the lengthof the orifice of the jet

37

38 A. Carabineanu

Two free lines λ1 and λ2 detach from the edges of the orifice (namedthe detachement points) and extend to infinity downsteam. The domainbounded by the walls of the vessel and by the free lines is the flow domain.We neglect the gravity and we consider that the jet emerges because ofthe difference of the pressures inside and outside the vessel. We considerthat the fluid is ideal, incompressible and the fluid flow is plane, steady andirrotational. Denoting by v = (u, v) the velocity of the fluid we have fromthe conditions of irrotationality and mass conservation

∂u

∂y− ∂v

∂x= 0,

∂u

∂x+

∂v

∂y= 0, (1)

u =∂ϕ

∂x=

∂ψ

∂y, v =

∂ϕ

∂y= −∂ψ

∂x. (2)

Figure 1: Flow domain

From the Cauchy - Riemann condition (2) we deduce that the functionf(z) = ϕ (x, y) + iψ (x, y) (named the complex potential) is holomorphic,and denoting by w (z) = u (x, y) − iv (x, y) the complex velocity, we havedf

dz= w.

The walls of the vessel and the free lines are stream-lines i.e.

ψ |$1∪λ1=h

2, ψ |$2∪λ2= −h

2. (3)

Incompressible jets 39

Outside the flow domain the fluid is at rest. From Bernoulli’s law wededuce that ∣∣∣∣

df

dz

∣∣∣∣ |λ1∪λ2= V0 = const. (4)

2 Levi-Civita’s method. The functional ecuation

The function

f =h

πln

(ζ +

1ζ

)+ i

h

2, ζ = ξ + iη, (5)

is the conformal mapping of the unit half-disk from the ζ - plane onto theinfinite horizontal strip of width h in the f−plane.

We introduce T. Levi - Civita’s function ω (z) = θ (ξ, η) + iτ (ξ, η)

w (ζ) = w (z (ζ)) =df

dz= V0 exp (−iω (ζ)) , (6)

From (6) it follows

τ (ξ, η) = ln|w (ζ)|

V0, θ (ξ, η) = − arg w (ζ) . (7)

θ (ξ, η) is the angle of the velocity with the Ox - axis . From relation (4)we deduce that

τ (ξ, 0) = 0, ξ ∈ [−1, 1] , (8)

On the unit half - circle the function θ (cos s, sin s) , s ∈ [0, π] is discon-tinuous in s =

π

2because

limsπ

2

θ (cos s, sin s) = −µπ, limsπ

2

θ (cos s, sin s) = µπ.

We shall introduce therefore the continuous function Ω (ζ) = Θ (ξ, η) +iT (ξ, η) , such that

Θ (cos s, sin s) = −µπ − θ (cos s, sin s) , s ∈[0,

π

2

), (9)

Θ (cos s, sin s) = µπ − θ (cos s, sin s) , s ∈(π

2, π

], (10)

T (ξ, 0) = 0, ξ ∈ [−1, 1] . (11)

40 A. Carabineanu

From (9) - (11) it follows

Re [ω (ζ) + Ω (ζ)] = −µπ, ζ = exp (is) , s ∈[0,

π

2

), (12)

Re [ω (ζ) + Ω (ζ)] = µπ, ζ = exp (is) , s ∈(π

2, π

], (13)

Im [ω (ζ) + Ω (ζ)] = 0, ζ = ξ ∈ [−1, 1] . (14)

From (12) - (14) we deduce that

ω (ζ) = 2µi lnζ − i

ζ + i+ 2µπ − Ω(ζ) (15)

From (6) and (15) it follows

dz

dζ=

1V0

exp(−2µ ln

ζ − i

ζ + i+ 2µiπ − iΩ(ζ)

)df

dζ. (16)

Denoting by l (s) the length of the arc from $1 having the endpointsz (exp (is)) and z (0) we deduce from (5) and (16) that

dl

ds=

h

πV0exp (T (s)) cot 2µ

(π

4− s

2

)tan s, s ∈

[0,

π

2

). (17)

(In the sequel we shall use the notations T (s) = T (cos s, sin s) , Θ(s) =Θ (cos s, sin s) , θ (s) = θ (cos s, sin s).)

The function z (ζ) mapps the unit half - circle onto a curve consisting ofhalf - lines and arcs of circle. According to Schwarz’s principle concerningthe analytic continuation the function z (ζ) can be extended in a vicinity ofthe half - circle exp (is) ; s ∈ [0, π] . Taking into account (16) one deducesthat the function Ω(ζ) can also be extended in a vicinity of the half - circleexp (is) ; s ∈ [0, π] . The conjugate harmonic functions T and Θ satisfythe relation

∂T

∂n=

∂Θ∂s

, (18)

where∂

∂nis the inward normal derivative and

∂

∂sis the tangential derivative.

Using U. Dini’s formula and seeking for Ω(ζ) such that Ω(0) = 0 we get

−iΩ(ζ) =1π

∫ 2π

0

∂T (s)∂n

ln (exp (is)− ζ) ds. (19)


From (11) it follows that the function Ω (ζ) can be extended to thewhole unit disk, acording to Schwarz’s continuation principle, by means ofthe relations

T (ξ, η) = −T (ξ,−η) , Θ(ξ, η) = Θ (ξ,−η) . (20)

On the other hand, because of the symmetry of the flow domain we have

T (ξ, η) = T (−ξ, η) , Θ(ξ, η) = −Θ(−ξ, η) . (21)

From (20) and (21) we deduce that

∂T

∂n(2π − s) = −∂T

∂n(s) , s ∈ [0, π] ,

∂T

∂n(π − s) =

∂T

∂n(s) , s ∈

[0,

π

2

].

(22)From (19) and (22) it follows :

−iΩ(ζ) =1π

∫ π2

0

∂T

∂n(s) ln

(exp (is)− ζ) (exp (−is) + ζ)(exp (−is)− ζ) (exp (is) + ζ)

ds. (23)

From (23), putting ζ = exp (iσ) and separating the real parts we obtain

T (σ) = − 12π

∫ π2

0

∂T

∂n(s) ln

(sin s + sin σ

sin s− sin σ

)2

ds, σ ∈[0,

π

2

). (24)

From (18) and (24) it follows :

T (σ) = − 12π

∫ π2

0

∂Θ∂s

(s) ln(

sin s + sin σ

sin s− sin σ

)2

ds, σ ∈[0,

π

2

). (25)

Let z (exp (is0)) ∈ $1 reprezent the point where the rectilinear and thecircular portions are matching. Obviously we have

dθ

dl(s) = 0, s ∈

(s0,

π

2

),

dθ

dl(s) = − 1

R, s ∈ (0, s0) . (26)

From (9), (17) and (26) it follows :

∂Θ∂s

(s) = 0, s ∈(s0,

π

2

), (27)

∂Θ∂s

(s) =h

πRV0exp (T (s)) cot 2µ

(π

4− s

2

)tan s, s ∈ (0, s0) . (28)

From (25), (27) and (28) we deduce :

T (σ) = −∫ s0

0

h exp (T (s))2π2RV0

cot 2µ(π

4− s

2

)tan s ln

(sin s + sin σ

sin s− sin σ

)2

ds

(29)

42 A. Carabineanu

3 The existence of the solution

We shall consider the operator F (T (s) , s0, h, k) : D× [0, 1] → C[0,

π

2

]×R2

given by the right hand side of the system of equations

T (σ) = −∫ s0

0


cot 2µ(π

4− s

2

)tan s ln

(sin s + sin σ

sin s− sin σ

)2

ds,

(30)

s0 = s0 + kνπR− h

πRV0

∫ s0

0exp (T (s)) cot 2µ

(π

4− s

2

)tan s ds, (31)

h =V0L + 2RV0 [cos (µ− ν) π − cos (µ− kν) π]

1 +2π

∫ 1

0sin θ (ξ, 0)

ξ − ξ−1

ξ + ξ−1

dξ

ξ

. (32)

From (30) - (32) we easily check that F (T (s) , s0, h, k) is continuouswith respect to T (s) , s0, h and uniformly continuous with respect to k.

In [1] one demonstrates that there is a constant s∗0 ∈(0,

π

2

)such that for

every solution(Tk (s) , s0k, hk)of the system (30) - (32) we have 0 ≤ s0k < s∗0and hk < V0L0, L0 = L + 2R [cos (µ− ν) π − cosµπ]. We denote by

M = maxσ∈[0, π

2 ]

L0

2π2R

∫ s∗0

0cot 2µ

(π

4− s

2

)tan s ln

(sin s + sin σ

sin s− sin σ

)2

ds

We consider the domain D =

T (s) ∈ C[0,

π

2

]; |T (s)| < M + 1

×

[0, V0L0]× [0, s∗0] . For any solution (Tk (s) , s0k, hk) of (30) - (32) we have

0 < maxs∈[0, π

2 ]|T (s)|+ |s0k|+ |hk| < M + V0L0 + s∗0 + 1,

whence it follows that F (T (s) , s0, h, k) has no fixed point on ∂D.Taking into account the expression of the kernel of the integral equa-

tion (30) it follows that F maps the cartesian product of an arbitrarybounded set of continuous functions with an arbitrary bounded set fromR2 onto the product of a bounded set of equi-continuous functions with abounded set from R2. Taking into account Arzela’s theorem we deduce thatF (T (s) , s0, h, k) is a compact operator.

For k = 0, the operator F (T (s) , s0, h, k) is constant and it is given bythe right hand side of the following relations

T (s) = 0, s0 = 0, h =V0L + 2RV0 [cos (µ− ν) π − cosµπ]

1 +2π

∫ 1

0sin

(2µ arcsin

2ξ

ξ2 − 1

)ξ − ξ−1

ξ + ξ−1

dξ

ξ

. (33)


Hence the topological degree of the operator I − F (T (s) , s0, h, 0) in0 ∈ C

[0,

π

2

]× R2 is 1. From Leray - Schauder [3] fixed point theorem it

follows that F (T (s) , s0, h, k) has at least one fixed point, ∀k ∈ [0, 1] .

4 The semi - inverse method. Numerical results

Figure 2: Example

Let us consider the operator L : C[0,

π

2

]→ C

[0,

π

2

]where

L (T ) (σ) = −∫ s0

0


cot 2µ(π

4− s

2

)tan s ln

(sin s + sin σ

sin s− sin σ

)2

ds.

(34)Considering T0 (σ) = 0, σ ∈

[0,

π

2

]we can establish the relations

L (T0) (σ) ≤ L3 (T0) (σ) ... ≤ L2n+1 (T0) (σ) ≤ ... ≤ T (σ) ≤

... ≤ L2n (T0) (σ) ≤ ... ≤ L2 (T0) (σ) ≤ 0, ∀σ ∈[0,

π

2

](35)

whence we deduce the lower and upper bounds for T :

limn→∞L

2n+1 (T0) (σ) ≤ T (σ) ≤ limn→∞L

2n (T0) (σ) , ∀σ ∈[0,

π

2

]. (36)

For h and s0 small enough the two bounds conicide because L is acontraction. In the sequel for various values of h and s0 we shall ver-ify numerically that the lower and upper bounds coincide. For calculating

44 A. Carabineanu

T = limn→∞L

2n+1 (T0) (σ) we consider an equidistant grid on[0,

π

2

]consist-

ing of the nodes 0 = σ0, σ1, ..., σm =π

2and another grid on [0, s0] consisting

of the nodes 0 = s0, s1, ..., sp = s0. Using Simpson’s quadrature formulaone calculates L (T0) (σi) and then, iteratively L2 (T0) (σi) ,L3 (T0) (σi) ....We stop the culculus when

∣∣Ln (T0) (σi)− Ln+1 (T0) (σi)∣∣ < ε where ε is

an apriori given small number. Using Cauchy’s formula we calculate thenΘ (cosσi, sinσi) . We calculate also Θ (ξi, 0) , ξi ∈ [0, 1] by means of SchwarzVillat formula. From (31) and (32) we then calculate ν and L.

Then by means of (16) one may calculate numerically the positions ofthe points belonging to the free lines. In figure 2, we give an example offlow domain and calculated free lines.

Acknowledgement. This work has been supported by the Ministry ofEducation and Research, Romania, under CNCSIS Grant 35262/2001.

References

[1] Carabineanu A., Asupra problemei jeturilor incompresibile cu pereticurbilinii. I, Stud. Cerc. Mat., 37, 5 (1985), 377-398.

[2] Hureau J., Brunon E., Legallais Ph., Ideal free streamline flow over acurved obstacle, J. Comput. Appl. Math, 72(1996) 193-214.

[3] Leray J., Schauder, Topologie et equations fonctionelles, Ann. Sci. Ec.Norm. Sup., 51(1934), 45-78.

[4] Lupu M., Scheiber E., Studiul unor probleme la limita inverse in cazuljeturilor fluide incompresibile, Stud. Cerc. Mat., 49, 3-4(1997), 197–209.

Adrian CarabineanuUniversity of Bucharest, Faculty of Mathematics,Str. Academiei 14, sector 1, Bucharest,RO - 70109, RomaniaE-mail: [email protected]


On partial differential equations of the viscousliquid relative flow through the turbomachine

blade channel

Mircea Dimitrie CAZACU

November 2, 2001

Abstract - Taking into consideration the reality of the relative flow into the rotor blade

field and the great advantage presented by the boundary conditions in the study of the

fluid relative flow through the rotational blade channel of a turbomachine, we present the

transformation of Navier-Stokes partial differential equation system, written in cylindrical

coordinates, from the absolute to the relative coordinate system. In the same time one

gives two characteristic examples of flow patterns, considered from the fixed and moving

coordinate systems.

Key words and phrases : viscous liquid, relative whirl, centrifugal turbomachines

Mathematics Subject Classification (2000) : 76D05, 76M99

1 Introduction

Leonhard Euler was the first researcher, who paid attention to the rela-tive flow of the inviscid liquid through the rotating impeller channels ofa hydraulic reaction turbine [10]. Although his experimental laboratorymodel has small sizes, he had foreseen the possibility of the cavitationalphenomenon appearance at the upper impeller inlet. To avoid this undesir-able phenomenon the engineers have realized the bigger hydraulic turbineswith a larger size at the inlet of the impeller channels, diminishing by thisway the entrance velocities and consequently increasing the pressures.

Therefore, by the enlarging of impeller channels, simultaneously with theemphasizing of the matter inertia property (maybe by the relevant experi-ence of Foucault’s pendulum) one arrives at the beginning of 20th century[11] to consider, but only for the centrifugal turbomachines, the dislocationflow so called relative whirl, whose negative effect was appreciated espe-cially at the outlet from the rotor (its effect at the inlet of the rotor being

45

46 M. D. CAZACU

reduced due to the pass flow) and only by modification of the velocity trian-gles, consequently with effect on the real realized pump head by introductionof so called head diminishing factor ε = Hth/Hth∞ < 1, due to the finitenumber of the blades and inertia of the fluid [15].

While the problem was solved for the moment, it could not explain thepositive effect of the relative whirl in the centripetal flows [1] as:in modern reaction hydraulic turbines, in centripetal pumps with greaterefficiency, in Ljungstrom’s steam turbine for the first pass flow, or thatworking with water of Donat Banki [14], as well as in the case of transversal(cross) fan [2].

With this occasion we observe that the energy losses may be consideredonly in the relative flow, which is only real flow, while the absolute flowis fictitious, although we can seen it from the fixed coordinate system [2].By the viscous liquid relative flow study, introduced by us in the last years[4][5][3][6][12][8][16][7], we emphasize (put into the evidence for the first time)the real aspect of the relative flow through the rotating blades of an impellerand the fiction of the absolute flow, even it can be calculated [4] and observedfrom the fixed coordinate system [2]. At the elaboration of this paper mycolleague Prof. Dr. Eng. Mihai Exarhu [9] contributed with a valuableremark, for which I thank him.

2 Partial differential equation system for viscous liquid ab-solute flow in cylindrical coordinates

The Navier-Stokes system, for unsteady flow in general, of nonlinear withpartial differential equations consisting from Newton’s action principle andmass conservation equation [13], written in cylindrical coordinates, conve-nient in general to the unsteady and three-dimensional motion of the viscousliquid in the turbomachines and reported to the fixed system of reference(Oa, Ra, Φ, Za) [4][5][6] with the notation from figure 1, are:

∂VR

∂Ta+ VR

∂VR

∂Ra+

VΦ

Ra

∂VR

∂Φ+ VZ

∂VR

∂Za− V 2

Φ

Ra+

1ρ

∂P

Ra=

= ν

(∂2VR

∂R2a

+1

R2a

∂2VR

∂Φ2+

∂2VR

∂Z2a

+1

Ra

∂VR

∂Ra− 2

R2a

∂VΦ

∂Φ− VR

R2a

) (1)

∂VΦ

∂Ta+ VR

∂VΦ

∂Ra+

VΦ

Ra

∂VΦ

∂Φ+ VZ

∂VΦ

∂Za+

VΦVR

Ra+

1ρRa

∂P

∂Φ=

= ν

(∂2VΦ

∂R2a

+1

R2a

∂2VΦ

∂Φ2+

∂2VΦ

∂Z2a

+1

Ra

∂VΦ

∂Ra+

2R2

a

∂VR

∂Φ− VΦ

R2a

),

(2)

On pde of the viscous liquid flow 47

∂VZ

∂Ta+ VR

∂VZ

∂Ra+

VΦ

Ra

∂VZ

∂Φ+ VZ

∂VZ

∂Za+

1ρ

∂P

∂Za=

= ν

(∂2VZ

∂R2a

+1

Ra

∂VZ

∂Ra+

1R2

a

∂2VZ

∂Φ2+

∂2VZ

∂Z2a

) (3)

∂VR

∂Ra+

VR

RA+

1Ra

∂VΦ

∂Φ+

∂VZ

∂Za= 0, (4)

in which, after Euler’s concept of the motion,−→V (VR, VΦ, VZ) =

−→V [Ra(Ta), Φa(Ta), Za(Ta), Ta] the absolute velocity of a fluid particle, rela-tive to the fixed coordinate system and P (Ra,Φa, Za, Ta) the static pressure,as functions of absolute variables, are the unknown functions, ρ and ν be-ing the liquid density and the cinematic viscosity respectively, considered asconstants.

Fig. 1 Two reference systems and decompositionof the velocity triangle

In this conditions and taking into consideration Euler’s conception of themotion, that permitted us to study the flow of a dense ensemble of particles,which fill whole the space, interacting by the surface forces of pressure andfriction, we can write more:

VR =dRa

dTa, VΦ = Ra

dΦdTa

, VZ =dZa

dTaand also

WR =dRr

dTr,Wθ = Rr

dθ

dTr,WZ =

dZr

dTr,

(5)

while the partial derivatives have no sense∂Ra

∂Ta=

∂Φ∂Ta

=∂Za

∂Ta=

∂Rr

∂Tr=

∂θ

∂Tr=

∂Zr

∂Tr≡ 0, excepting

∂Rr

∂Ra=

dRr

dRa= 1,

∂θ

∂Φ=

dθ

dΦ= 1,

∂Zr

∂Za=

dZr

dZa= 1,

∂Tr

∂Ta=

dTr

dTa= 1. (6)

48 M. D. CAZACU

With these considerations we can calculate the partial derivatives to trans-form the Navier-Stokes equation system for the relative flow:

∂Fa

∂Ra=

∂Fr

∂Rr

∂Rr

∂Ra︸︷︷︸1

+∂Fr

∂θ

∂θ

∂Ra︸︷︷︸0

+∂Fr

∂Zr

∂Zr

∂Ra︸︷︷︸0

+∂Fr

∂Tr

∂Tr

∂Ra︸︷︷︸0

and also

∂Fa

∂Φ=

∂Fr

∂Rr0 +

∂Fr

∂θ1 +

∂Fr

∂Zr0 +

∂Fr

∂Tr0,

∂Fa

∂Za=

∂Fr

∂Rr0 +

∂Fr

∂θ0 +

∂Fr

∂Zr1 +

∂Fr

∂Tr0,

∂Fa

∂Ta=

∂Fr

∂Rr0− ω

∂Fr

∂θ+

∂Fr

∂Zr0 +

∂Fr

∂Tr1.

(7)

3 Viscous fluid relative flow in rotational blade channel

To transform the Navier-Stokes equations, written in cylindrical coordinatesfor the fixed inertial system, in a non-inertial moving system, having thesame origin but a rotational motion ϕ = ωTa, with ω = const. (fig.1), weshall utilize the vectorial relation between the absolute V , relative W andtransport U = Rω velocities, in virtue of the three-entity property of thephysical space and projecting the velocity triangle on the three rectangulardirections R, θ or Φ and Z,

−→V a =

−→W r +

−→U t,⇒ VR = WR, VΦ = Wθ + U = Wθ + Rω, VZ = WZ . (8)

The last three relations can be obtained also by time derivation of the rela-tions between the absolute and relative variables, the second relation beingmultiplied by the radius R

Ra(Ta) = Rr(Tr) = R(T ), Za(Ta) = Zr(Tr) = Z(T ),Φ(Ta) = θ(Tr) + φ(Ta) or θ(Ta) = Φ(Ta)− ωTa,

(9)

at which we add

Ta = Tr = T 6= T (Ra, Rr, Φ, θ, Za, Zr), (10)

the time passing in the same manner for the all particles, found in anyspatial position and independently of the spatial variables in both referencesystems.

Though our interest in practice is the permanent motion of the turboma-chines, the fluid particle being in a steady flow between the moving blades inrelative trihedron, constituting a z-connected domain (z being the number of


impeller blades), in that case the streamlines coinciding with the particle tra-jectories, we shall consider the general study of the relative flow, as unsteady

flow−→W (WR,Wθ,WZ) =

−→W [Rr(Tr), θr(Tr), Zr(Tr), Tr], in the same Euler’s

conception, relative only to the moving system (Or, Rr, θr, Zr), interdepen-dent with the impeller blades, driven with the angular velocity ω = const.Taking into consideration the variable dependence of the absolute and rel-ative functions (7) we can calculate the partial derivatives obtaining thepartial differential equations, from the relations (5) to (7) observing that

∂θ

∂Ta=

∂Φ∂Ta

− ω = −ω. (11)

With these considerations the partial differential equation system (1-4) be-comes for the relative flow case

∂WR

∂T+ WR

∂WR

∂R+

Wθ

R

∂WR

∂θ+ WZ

∂WR

∂Z− (Wθ + Rω)2

R+

1ρP ′

R =

= ν

(∂2WR

∂R2+

1R2

∂2WR

∂θ2+

∂2WR

∂Z2+

1R

∂WR

∂R− 2

R2

∂Wθ

∂θ− WR

R2

),

(12)

∂Wθ

∂T+ WR

∂Wθ

∂R+

Wθ

R

∂Wθ

∂θ+ 2ωWR + WZ

∂Wθ

∂Z+

WθWR

R+

1ρR

P ′θ =

= ν

(∂2Wθ

∂R2+

1R2

∂2Wθ

∂θ2+

∂2Wθ

∂Z2+

1R

∂Wθ

∂R+

2R2

∂WR

∂θ− Wθ

R2

),

(13)∂WZ

∂T+ WR

∂WZ

∂R+

Wθ

R

∂WZ

∂θ+ WZ

∂WZ

∂Z+

1ρP ′

Z =

= ν

(∂2WZ

∂R2+

1R

∂WZ

∂R+

1R2

∂2WZ

∂θ2+

∂2WZ

∂Z2

) (14)

the mass conservation equation having the same form

∂WR

∂R+

WR

R+

1R

∂Wθ

∂θ+

∂WZ

∂Z= 0. (15)

4 Demonstrative cases

For example we analyse two bidimensional cases of the inviscid or viscousfluid steady flows:

4.1 Absolute water source, springing from the origin of the sys-tems in a steady flow.

A similar image is shown in [2] and represents what one see at large distancesfrom the relative system (fig.2), with which the camera is interdependent

50 M. D. CAZACU

and moves with a constant angular velocity ω = constant. In this case, whenan absolute of ideal or viscous fluid source springs from the origin, we haveVΦ = VZ ≡ 0 and from the mass conservation equation (4) with (9)

dVR

dR+

VR

R= 0, ⇒ VR =

K

R= WR, (4′)

the absolute equation system reduces to the first equation (1), which givesthe radial pressure distribution

VRdVR

dR+

1ρ

dP

dR= ν

(d2VR

dR2+

1R

dVR

dR−VR

R2

)= 0, ⇒ P (R) = P0−ρK2

2R2, (16)

the pressure distribution being function of the intensity of the flow, measuredby the constant K. For the relative motion (12) we see in addition a counter-flow with Wθ = −Rw, proportional to the radius R, as in figure 2.

Fig. 2 Relativ flow pattern in the trasversal fan

WRdVR

dR+

1ρ

dP

dR= ν

(d2VR

dR2+

1R

dVR

dR− WR

R2

)= 0,⇒ P (R)= P0 − ρK2

2R2,

(12′)with the same distribution of the pressure only on the radius, the pressurebeing a scalar, consequently independent of the referential system. In figure3, a we can see the variations with the radius of velocity radial componentand the trajectory θ angle

dR

WR=

Rdθ

Wθ, ⇒ θ = θ0 − ω

2KR2. (17)


4.2 Absolute potential whirl

In this case when VR = WR = VZ = WZ ≡ 0 and VΦ = VΦ(R) = Γ/2πR , forthe bidimensional flow and for the potential whirl with a constant circulation

Γ =

2π∫

0

VΦRdΦ = const. ⇒ VΦ =Γ

2πR=

γ

R= Wθ + Rω, (18)

we obtain from the first equation (1) or (12) the same pressure distribution(12’) and from equation (18) the velocity variation with the radius Wθ(R) =γR − ωR as in figure 3,b.

Fig. 3

References

[1] M. D. Cazacu, Pompe, ventilatoare, suflante, compresoare. Note decurs, Univ. Polit., Bucuresti,1954.

[2] M. D. Cazacu, Flow visualization at the liquid free surface. R.M.A.,Tome 34(1989), nr.6, 617–628. or The 5th International Symposiumon Flow Visualization, Prague, 21-25 August 1989, Journal of FlowVisualization and Image Processing, Vol. 1, Nr. 3, July-September 1993,181–188.

[3] M. D. Cazacu, A. Ciocanea, The influence of a duct vibration on the in-ternal flow, The 4th Conf. on Hydraulic Machinery and Hydrodynamics,Timisoara, 26-30 Sept.1994, Vol. I, Sect.I-Hydrodynamics, 3-10.

[4] M. D. Cazacu, R. M. R. Neacsu, Miscarea absoluta si relativa a lichidu-lui vascos dintre paletele de descarcare a presetupei, Conf. de Masinihidraulice si Hidrodinamica, Timisoara, 15-17 noiembrie, 1990, vol. IV- Pompe, echipamente, actionari si automatizari hidraulice, 83-88.

52 M. D. CAZACU

[5] M. D. Cazacu, R. M. R. Neacsu. Relative and absolute motion of theviscous liquid through rotating vanes of an impeller with and withoutpass-flow, The 4th Conference on Hydraulic Machinery and Hydrody-namics, Timisoara, 26-30 September 1994, Vol. I, Section III - Pumps,189–196.

[6] M. D. Cazacu, R. M. R. Neacsu, S. M. Tomeh, Miscarea relativa si planaa lichidului vascos ıntre paletele unei turbomasini centrifuge, Bul.St.alUniv.Polit., Timisoara, Tom 44 (58) Mecanica 1999, 595-602.

[7] M. D. Cazacu, S. M. Tomeh, On determination of the cavitational term∆h′ at a centrifugal pump, Workshop on Numerical Simulation for FluidMechanics and Magnetic Liquids, 24-25 mai 2001, 160–167.

[8] A.Ciocanea-Teodorescu, Influenta vibratiilor asupra curgerii interioareunei tevi. Teza de doctorat, Univ. ,,Politehnica”, Bucuresti, 1997.

[9] M. Exarhu, Consideratii privind schimbul de energie ın rotoareleturbomasinilor stationare si mobile, Col. de Mecanica Fluidelor siAplicatiile ei Tehnice, Galati, 1979.

[10] N. N. Kovalev, B. S. Kviatkovskii, Ghidroturbostroenie v S.S.S.R.GOSENERGOIZDAT, Moskva-Leningrad, 1957, p. 13.

[11] Kucharski. Stromungen im rotierenden Kanal, Zeitschrift Turbinenwe-sen, 1917, 201.

[12] R. M. R. Neacsu, Contributii la studiul schimbului de energie ın ro-torul turbomasinilor, Teza de doctorat, Universitatea ,,Politehnica” ,Bucuresti, 1997.

[13] T. Oroveanu, Mecanica fluidelor vascoase. Ed. Acad., Bucuresti, 1967.

[14] D. Pavel, Masini hidraulice, Vol.I, Ed. Energetica de Stat, Bucuresti1954.

[15] C. Pfleiderer, Stromungsmaschinen, Springer-Verlag, Berlin/GottingenHeidelberg, 1957.

[16] S. M. Tomeh, Contributii privind fenomenul de cavitatie la pompelecentrifuge, Teza de doctorat, Univ. ,,Politehnica”, Bucuresti, 2001.

Mircea Dimitrie CazacuUniversity “Politehnica” of Bucharest, Splaiul Independentei 313RO-772061 Bucharest, Romania,E-mail : [email protected]


Numerical solving of the bidimensional unsteadyflow of a viscous liquid, generated by displacement

of a flat plate

Mircea Dimitrie CAZACU and Loredana NISTOR

November 2, 2001

Abstract - This paper presents the numerical solving for the two-dimensional unsteady

flow of a viscous liquid contained in a channel, and produced by the displacement with

a constant velocity of a flat plate, which is disposed in his axis and perpendicularly on

the channel walls. It consists in the sudden starting of the plate, which moves afterwards

with a constant velocity. In this case we are interested on the frequency of the sepa-

rate alternate vortices, this frequency being present in the partial differential stream-line

equation. The theoretical results are compared with the experimental ones, obtained in

the POLITEHNICA University Laboratory, on the calibrating stands of flow meters with

separate alternate vortices and on the installation for visualization of viscous liquid flows.

Key words and phrases : viscous vortex flow, numerical integration, alternate vortices

separation.

Mathematics Subject Classification (2000) : 76D05, 75D17, 76M25

1 Introduction

The studied problem presents both theoretical and practical importance.The theoretical importance is presented by the numerical integration of theNavier and Stokes equation system in the special boundary conditions forunsteady macroturbulent flow. The practical importance consists in thestudy of the flow-meter with alternate separation vortices [1, 2, 3].

53

54 M.D. CAZACU, Loredana NISTOR

2 Equation system written in dimension anddimensionless form

The partial differential equation system consists in two motion equations:

U ′T + UU ′

X + V U ′Y + 1

ρP ′X = ν(U ′′

X2 + U ′′Y 2),

Sh u′t + uu′x + vu′y + Eu p′x =1

Re(u′′x2 + u′′y2),

(1)

V ′T + UV ′

X + V V ′Y +

1ρP ′

Y = ν(V ′′X2 + V ′′

Y 2),

Sh v′t + uv′x + vv′y + Eu p′y =1

Re(v′′x2 + v′′y2),

(2)

and mass conservation equation

U ′X + V ′

Y = 0, u′x + v′y = 0, (3)

for which we take as characteristic parameters: the length of the plate B, themoving velocity of the plate Up and the time period T of vortices separation.

3 Equation transformation for numerical solving

To eliminate the mass conservation equation, unstable in iterative numericalsolving [6], we introduce the stream function, by the relations:

U = Ψ′Y , or u =

U

Up=

Ψ′Y

Up= ψ′y,

V = −Ψ′X , or v =

V

Up= −Ψ′

X

Up= −ψ′x.

(4)

Considering the Schwarz relation, that refers to the commutativity propertyof mixed partial differentials of a 2nd order function, the mass conservationequation is verified identically.

To eliminate in the two motion equations the unknown function P (X,Y, T )on the boundary using the same Schwarz relation that refers to the equalityof the mixed partial differentials of 2nd order, we differentiate the formula(1) with respect to Y and formula (2) with respect to X, and replace thecomponents of the velocity with the streamline function components by theformula (4) substitutions, we shall have by substraction

Ψ′′′tX2 + Ψ′′′

tY 2 + Ψ′Y

(∂∆Ψ∂X

)−Ψ′

X

(∂∆Ψ∂Y

)= ν∆∆Ψ. (5)

Numerical solving 55

4 Writing of a dimensionless equation system

The writing of dimensionless equation system is important for generality ofthe numerical solution. It can be obtained choosing as work domain a rect-angle having the length L and the width C, in which the plate with lengthB is moving with the constant velocity Up, and we write the dimensionlessequation using the following notations:

x =X

B, y =

Y

B, u =

U

Up, v =

V

Up, ψ =

ΨL · Up

, τ =t

T, (6)

and Sh−1 = UpTB the Strouhal and Re = UpB

ν the Reynolds number of theflow. Dimensionless equation thus obtained will be

(ψ′′′τy2 + ψ′′′τx2

)Re Sh + Re

(ψ′y

∂∆ψ

∂x− ψ′x

∂∆ψ

∂y

)= ∆∆ψ. (7)

5 Method of stream function development in Taylor’s series

This method is based on the splitting of the motion area by a square grid(fig. 1) and the calculating of the unknown function ψ at a certain momentin every node of the grid. This calculus will be made using an iterativemethod, by determining the repetitive values in the grid from node to node,after a certain direction and way, until is obtained a convergent solution.

To calculate the unknown function in every node, we shall expand theunknown function using the finite Taylor series, until the term with the samederivative order as that of the differential equation verified by the unknownfunction ψ(x, y, τ). The partial differential equation is a 4th order equation,therefore we shall write the series until the 4th order, inclusive. The gridused for the calculus is a square integration grid with step χ for both di-rection of the distance in the purpose of not advantaging any of the flowingdirections. Anyway this grid will have a different (smaller) step on the timedirection, because of the sufficiently fast motion.


JJ+1J+2 J−1 J−2

I

I−1

I−2

I+1

I+2

x

y

τ

0

2

31

5 6

78 4

10

119

13 14

1516 12

0

0

δ

δ

x

y

δ τ

Fig. 1 Knot marking in space bidimensional grid

Using the function developments in finite Taylor’s series in space [6],at the moment t, for a single step, will have the values for the stream-linefunction in the knots of the grid, from 1 to 16, the only difference betweenthe values calculated in the points 1 to 8 and those calculated in the points9 to 16 being that, instead of the single step χ, will have double step 2χ.Using these formulas we can calculate the values for every partial differentialof the dimensionless streamline function, for ordinary points like 0, takinginto consideration the values of the dimensionless stream-line function in allthe other sixteen nodes around 0. The calculus is based on the convenientadding and substracting of different Taylor’s series developments [6]. Besidethe known terms of the partial differentials [6] we will have a mixt term

ψ′′xy =1χ2

[13(ψ5 + ψ7 − ψ6 − ψ8) +

148

(ψ14 + ψ16 − ψ13 − ψ15)].

Identically, we shall determine the other 3rd and 4th order partial differen-


tials. As far as it concerns the non-permanent part of the motion, the valuesof the partial differential with respect to time t will be

∂

∂τ∆x,y

ψ = ψ′′′τx2 + ψ′′′τy2 =∂

∂τ

[ψ1 + ψ3 − 2ψ0

χ2+

ψ2 + ψ4 − 2ψ0

χ2

]

=1

χ2=[ 4∑

i=1

ψi − 4ψ0 −4∑

i=1

ψ−i − 4ψ−0

].

(8)

6 Algebraic relation associated to partial differentialequation

The algebraic equation will be determined replacing the expressions of the di-mensionless partial differentials in the partial differential equation by the un-known function ψ(x, y, τ), expliciting its value in the ordinary point 0(x0, y0, τ0)Simplifying at most the above equation, the final form of the equation (7)

ψ0 =Re=

96Re Shχ2 + 480=[8(ψ2 − ψ4) + ψ12 − ψ10]··[ψ5 + ψ8 + ψ9 − ψ6 − ψ7 − ψ11 + 4(ψ3 − ψ1)]+

+[8(ψ3 − ψ1) + ψ9 − ψ11]··[ψ5 + ψ6 + ψ10 − ψ7 − ψ8 − ψ12 + 4(ψ4 − ψ2)]+

+Re Shχ2

Re Shχ2 + 5= ·[14

4∑

1

(ψi − ψ−i ) + ψ−0

]+

+10=

3Re Shχ2

(25

4∑

1

ψi − 110

8∑

5

ψi − 120

12∑

9

ψi

)

(9)

is the algebraic solution associated to the stream-line function, whichallow us to determine the stream-line function in every ordinary point 0of the grid. To solve rigorously this problem there are necessary severalboundary conditions.

7 Stability of numerical solution

Taking into consideration the hyperbolic type of stream line equation andthe impossibility to know the end values of the function ψ for the final time,the unsteady flow numerical solution is obtained by the alone passage intime of the network [4], which must be stabilized only in space field foreach consecutive time moment, considering in each knot as fixed values thestabilized values of the function ψ at the foregoing time moment.



The necessary boundary conditions are imposed for the limit of the channeland for singular points such as the corners of the channel. (see fig. 1).8.1 On the channel walls the stream-line function is null Ψ = 0. We shallhave therefore the following conditions, if we consider the channel split in103 columns and 40 rows: with Ψ = 0 on I = 3 and 3 ≤ J ≤ 103, I = 43and 3 ≤ J ≤ 103, J = 3 and 3 ≤ I ≤ 43, J = 103 and 3 ≤ J ≤ 43.8.2 For the channel corners the stream-line function has the same valuesas these inside the channel [6] (the reflection condition).8.3 The perfect reflecting condition for the points on the limit of thechannel.8.4 On the plate

ψ(y) =

Y (L/2)∫

0

Up · dy =

y(L/2)∫

0

Up · dy = Up · y. (10)

8.5 At the initial time t = 0, for the sudden start of the plate, Up = 1.

Fig. 2 Unsteady flow with alternate vortices separationbehind the flat plate

9 Experimental results

The momentary visualization of the unsteady flow with alternate vorticesseparation behind the flat plate (fig. 3) are made for different Reynolds


number on the free surface of the viscous liquids in a special experimentalinstallation with a proper technics [6, 5].

Fig. 3 Unsteady flow with alternate vortices separationbehind the flat plate

10 Conclusions

After the obtaining of theoretical values for time period of vortices separationby numerical integration of unsteady flow for different Reynolds number,we intend to verify the experimental results concerning the dependence ofStrouhal number as function of Reynolds number, in comparision with thevery known constant value Sh ≈ 0, 21. According to whom, for a vortexflow-meter, for air, in the central area of the measuring domain appear avariation with 3% of the Strouhal number Sh = 0, 34 − 0, 37 [2, 3]. Thisvariation influence directly the precision of the flow-meter and representan internal accuracy limitation of the equipment due to the functioningprinciple.

References

[1] Cazacu, M.D., Cristescu,C., Olaru, V., Aristotel, I. Traductorul dedebit VORTEX cu desprindere de vartejuri. Ses.Jubiliar 45 ani aiICPE, Bucuresti, 30-31 oct.1995, Sectia Senzori si Traductoare, Vol.ST, 77–81.


[2] M. D. Cazacu, A. Ciocanea, I. Aristotel, Masurarea debitelor de lichidesi gaze pe principiul desprinderii alternate de vartejuri. Debitmetrulde tip VORTEX. Ses. Jubiliara 45 ani ai ICPE, Bucuresti, 30-31 oct.1995, Sectia Masurari electrice, Vol. MS, 41–44.

[3] M. D. Cazacu, A. Ciocanea, I. Aristotel, Influenta vascozitatii lamasurarea debitelor utilizand debitmetre de tip Vortex, Automatiza-tion Revue, Bucharest, nr.3 / 1995, 27–31.

[4] M. D. Cazacu, Mouvements unidimensionnels et nonpermanents desfluides compressibles dans le cas des petites variations de pression, avecdes applications au coup de belier (Doctoral Thesis).Bul. InstitutuluiPolitehnic, Bucuresti, 1958, Tom XX, fasc.3, 59–92.

[5] M. D. Cazacu, Flow visualization at the liquid free surface, The 5thInternational Symposium on Flow Visualisation, Prague, 21-25 Au-gust 1989, also Rev.de Mec.Appl., 34 (1989), no. 6, 617-628 and Jour-nal of Flow Visualization and Image Processing. Vol.1, No. 3, July-September 1993, 181–188.

[6] D. Dumitrescu, M. D. Cazacu, Theoretische und experimentelle Be-trachtungen uber die Stromung zaher Flussigkeiten um eine Platte beikleinen und mittleren Reynoldszahlen. ZAMM, 50(1970), 257–280.

Mircea Dimitrie Cazacu and Loredana NistorUniversity “Politehnica” of Bucharest,E-mail : [email protected]


Behaviour of the−42m piezoelectric crystal

containing a crack in antiplane state

Eduard - Marius CRACIUN

November 2, 2001

Abstract - We consider a piezoelectric crystal having as symmetry group−42m containing

a crack, in the third fracture mode and loaded by an initial applied electric field. We shall

prove that the incremental displacement of the crack tips are due to the initial applied

electric field. This observed phenomenon is being produced by the induced anisotropy of

the considered material.

Key words and phrases : piezoelectric crystal, crack in the third mode

Mathematics Subject Classification (2000) : 74M05

1 Introduction

In the paper we prove that in the case of a−42m piezoelectric crystal the

occurrence of the incremental displacement of the crack tips is due to theinitial applied electric field.In Section 2 we analyze the behaviour of the incremental displacement ofthe crack tips.In Section 3 is proved the main result of the paper those that the incremental

displacement of the tips vanish if the initial applied electric field of a−42m

piezoelectric crystal containing a crack vanishes.

2 Incremental displacement of the crack tips

We consider a prestressed and prepolarized piezoelectric crystal, having class−42m, containing a crack of length 2a situated in the symmetry plane x1x3

of the body.We assume that the crystal is initially loaded by a homogenous axial stressoσ. Also, we consider that, initial is applied a homogenous electric field

oE,

61

62 E.-M. Craciun

acting in the plane of the crack, but perpendicular to the direction of theaxial stress

oσ.

We suppose also that in the initial deformed equilibrium stateoB of the body,

at large distances from the crack, the body is loaded by an incremental shearstress τ > 0, acting in the direction of

oE.

In the above given conditions the crystal will be in antiplane incrementalstate, relative to the plane x1x2.

The involved boundary condition on the two faces of the crack are (see[1],[2]):

θ+23(x1) = θ23(x1, 0+) = θ−23(x1) = θ−23(x1, 0−) = −τ(x1) for− a < x1 < a,

(1)

∆+2 (x1) = ∆+

2 (x1, 0+) = ∆−2 (x1) = ∆−

2 (x1, 0−) = −w(x1) for− a < x1 < a.(2)

Here τ = τ (x1) is a given incremental shear stress and w = w(x1) is a givenincremental electrical surface density. In our final considerations we shallassume that τ = τ(x) = const. > 0 and w = w(x1) = 0, for −a < x1 < a.We shall reduce the boundary value problems to the Riemann-Hilbert prob-

lems, and solving them using Plemelj’s functions and Cauchy’s integral weget the following incremental displacements of the crack tips:

u3(±a, 0) = ∓τma (3)

where we have used following notations

m + in =b1 − b2

l1 − l2(4)

bα = Mα + iNα, lα = Pα + iQα (5)

with

m =∆M∆P + ∆N∆Q

∆P 2 + ∆R2, n =

∆N∆P −∆M∆Q

∆P 2 + ∆R2

∆M = M1 −M2, ∆N = N1 −N2,∆P = P1 − P2, ∆Q = Q1 −Q2. (6)

−42m piezoelectric crystal 63

3−42m piezoelectric crystal

The structure of material constants for a−42m piezoelectric crystal in Voigt’s

notation is described by the following relations (see [3] Appendix):

C11 C12 C13 0 0 C16

0 C11 C13 0 0 −C16

0 0 C33 0 0 00 0 0 C44 0 00 0 0 0 C44 00 0 0 0 0 C66

,

0 0 0 e14 0 00 0 0 0 e14 00 0 0 0 0 e36

,

ε11 0 00 ε11 00 0 ε33

.

The characteristic Eqs. in the case of our material becomes (see [1])

1ε11P(µ) = Ω2332µ

4 + 2Ω1332µ3 + [Ω1331 + Ω2332] µ2 + 2Ω1332µ + Ω1331 = 0.

(7)The involved instantaneous elasticities Ωklmn can be obtained taking into

account the special structure of the material constants for the−42m symme-

try group, as well as the special structures of the initial applied mechanicaland electrical fields. Long, but elementary calculus lead to the followingexpressions :

Ω2332 = C44 − χ11

E2, Ω1331 = C44+

σ −χ11

E2, Ω2331 = Ω1332 = −e14

oE .(8)

Here, for simplicity we have used the following notations

σ=

σ11 and

E=

E3 . (9)

Now, taking into account (8), the characteristic Eq. becomes :

1ε11P(µ) =

(C44 − χ11

E2

)µ4 − 2e14

E µ3 +

(2C44+

σ −2χ11

E2

)µ2−

2e14

E µ +

(C44+

σ −χ11

E2

)= 0. (10)

64 E.-M. Craciun

Taking into account the structure of material constants for the−42m sym-

metry group, we get:

Λ113 = Λ223 = χ11

E, Λ123 = Λ213 = e14, Λ122 = 0. (11)

cα =χ11

E +2e14µα + χ11

E µ2

α

ε11

(1 + µ2

α)(e14 + χ11µα

E

)− µα

(χ11

E +2e14µα + χ11

E µ2

α

) .

(12)

bα =1 + µ2

α

(1 + µ2α)

(e14 + χ11µα

E

)− µα

(χ11

E +2e14µα + χ11

E µ2

α

) . (13)

lα =[−e14

E +µα

(C44 − χ11

E2

)]bα +

[e14 + χ11µα

E

]cα. (14)

In the following we assume that initial is applied only an electric fieldoE,

the initial applied mechanic stress being zero, i.e.

oσ= 0. (15)

For simplicity, we shall use the following notation

χ11 = χ, e14 = e, ε11 = ε, c44 = c. (16)

Using this simplified alternative, we can see that the characteristic Eq.(7)

in the case of the−42m symmetry group takes the following form

(µ +1µ

)[(c− χ

E2)(µ +

1µ

)− 2eoE

]= 0. (17)

This Eq. has the following roots

µ1 = i, µ3 =−µ1= −i, µ2 =

coE

c− χ

E2

+ i

√√√√1− (c

oE

c− χ

E2

)2 , µ4 =−µ2 . (18)

−42m piezoelectric crystal 65

In the same time the parameters cα, bα and lα become

cα =χ

E

(1 + µ2

α

)+ 2eµα

εκα(19)

κα =(1 + µ2

α

) (l + χµα

E

)− µα

(χ

E +2lµα + χ

E µ2

α

)(20)

bα =1 + µ2

α

κα, α = 1, 2 (21)

lα =[−l

E +µα

(c− χ

E2

)]bα +

[l + χµα

E

]cα. (22)

We have also the following useful relation between cα and bα

cα =χ

E

(1 + µ2

α

)+ 2eµα

ε (1 + µ2α)

bα =χ

E

(1 + µ2

α

)+ 2eµα

εκα. (23)

Examinating Eqs.(18)-(22) we can conclude that are true the following exact

relations for any value of the initial applied electric fieldoE.

µ1 = i, c1 =i

ε, l1 = 2e, b1 = 0, l1 =

1ε

(−χ

E +ie

). (24)

The same Eq. leads to the validity of the following approximate relation,true if the parameter

E =e

c− χ

E2

(25)

has small absolute value :

µ2 = i + E , 1 + µ22 = 2iE , c2 =

i

ε, κ2 = 2e (1− iε) ,

b2 = iEe, l2 =

l

εi− cε

e− χ

E

ε. (26)

In this way, finally we get :

m + inb1 − b2

l1 − l2= − i

c. (27)

66 E.-M. Craciun

This important result shows that when the initial applied electric fieldoE

converges to zero, by m + in has the following behaviour : m + in → −i/c

whenE→ 0. Hence m → 0 when

E→ 0.

We can conclude that the nonvanishing incremental displacement u3 of thecrack tips is due to the initial applied electric field

oE. More exactly, to

the anisotropy induced by the field in the initial deformed and polarizedequilibrium configuration

oB of a piezoelectric crystal containing a crack and

having a material symmetry characterised by−42m crystal group.

4 Final remarks

We have considered a piezoelectric crystal prestressed and prepolarized, be-ing loaded by initial applied stress field and electric field. The initial de-formed equilibrium configuration is locally stable.The final result is to study of the incremental displacements of the cracktips.The above displacements vanish if the initial applied electric field is zero in

our piezoelectric crystal, with the symmetry class−42m. Here the observed

propriety is produced just by the induced anisotropy in the material.Acknowledgement. The author gratefully acknowledge the support pro-vided for this research by Romanian Academy under GAR no. 305/2001.

References

[1] E.M. Craciun, E. Soos, Incremental states in piezoelectric crystal Pro-ceedings of the 5-th GCM, Ed.Academiei Romane(2001), in press.

[2] E. Baesu, D. Fortune, E. Soos, Incremental behaviour of hyperelasticdielectrics piezoelectric crystal, ZAMP, (2000), (in press).

[3] Sirotin, I.I. and Skasholskaya M.P., Crystal Physics, Nauka,Moscow,(1975), (in Russian)

Eduard-Marius CraciunUniversitatea ”Ovidius” Constanta, B-dul Mamaia 124,8700, Constanta, RomaniaE-mail: [email protected]


An example of interaction between twogasdynamic objects: a piecewise constant solution

and a model of turbulence

Liviu Florin DINU

November 2, 2001

Abstract - The context of the considered interaction assumes a minimal nonlinearity

- in the form of a nonlinear subconscious. Consequently the interaction solution is es-

sentially constructed as an admissible solution. The present analysis has essentially two

objectives: (a) finding an explicit optimal form for the interaction solution, and (b) offer-

ing an exhaustively classifying characterization of this mentioned solution. Realising the

objective (a) is connected with: (a1) considering a singular limit of solution, (a2) consider-

ing a hierarchy of (natural) partitions of the singular limit, (a3) inserting some (natural)

gasdynamic factorizations at a certain level of the mentioned hierarchy and noticing a

compatibility (coherence) of these factorizations, (a4) identifying some inner connections

inside one of the mentioned partitions, (a5) predicting some exact details of the inter-

action solution, (a6) indicating some parasite singularities [= strictly depending on the

method] to be compensated [= pseudosingularities], (a7) re-weighting the singular limit of

solution. Realising the objective (b) is connected with finding some Lorentz arguments of

criticity. The interaction solution essentially appears to (exhaustively) include a subcritic

and respectively a supercritic contribution.

Key words and phrases : interaction of gasdynamic objects (turbulence, shock dis-

continuity), nonlinear subconscious, (compatible) gasdynamic factorizations, relativistic

ingredients (Lorentz arguments of criticity, subcritic / supercritic incidence).

Mathematics Subject Classification (2000) : 35L60, 35L65, 35L67, 35L99,76N99, 76Q05

A Fourier-Snell representation of the parallel linearized interaction betweena planar shock discontinuity and a planar compressible finite-core vortex theaxis of which is parallel to the shock has been considered first time by Ribner(1959) in a theoretical attempt consecutive to a pioneering and most sugges-

67

68 L.F. Dinu

tive experimental approach of Hollingworth and Richards (1956) concerningthe mentioned interaction. An ample and significant series of theoretical andexperimental developments has followed the two mentioned works [see Rib-ner (1985) for a thorough review]. A planar compressible finite-core vortexthe axis of which is parallel to the shock has the representation

[u(x∼, y∼), v(x∼, y

∼)] =

ε

2π

(1/r2∗)[− y∼, x∼] for r ≤ r∗

(1/r2)[− y∼, x∼] for r∗ ≤ r

, s ≡ p ≡ 0 (1)

where we shall use the Lagrangian reference frames x∼, y∼

(fixed on the undis-

turbed flow ahead of the shock) and x, y (fixed on the undisturbed flowbehind of the shock) in addition to the frame X, Y fixed on the shockdiscontinuity; we have

x∼ = X −MT, x = X −MT = x∼+(M −M) t∼; y∼

= y = Y ; t∼ = t = T,

where M and M are the Mach numbers respectively associated to the regionsbehind or ahead of the shock discontinuity [both are taken with respect tothe sound velocity behind].

The limit r∗ → 0 of (1) results in

[u(x∼, y∼), v(x∼, y

∼)] =

ε

2π·

[− y∼, x∼]

x∼2 + y

∼2, s ≡ p ≡ 0 (2)

∂v

∂x∼− ∂u

∂y∼

= εδ(x∼)δ(y∼). (3)

The present paper (see Dinu [1], [2]) has three main objectives:(i) to notice a gasdynamic factorization of the vorticity-shock interac-

tion; via

E(z2) ≡ (d01z2 +d02)2 +(d03z

2 +d04)2(z2− z2c) ≡ d203(z

2 +a2)(z2− b2)(z2− c2)

where

a =M

M> 1,

b2=M

M

[(2MM−1)−2M

√γ−1γ+1

MM

], c2=

M

M

[(2MM−1)+2M

√γ−1γ+1

MM

]

Interaction between two gasdynamic objects 69

[the coefficients d0j should be presented below] and to make use of thisfactorization to give an explicit, closed form to Ribner’s representation;

(ii) to identify a sequence of other five gasdynamic factorizations in theexplicit form of the vortex-shock interaction solution [since a vortex repre-sents a structured vorticity, the present factorizations appear to be inducedby that mentioned in (i) by structuring] and to take into account the realityof a factoring compatibility of these factorizations in order to select an ex-tensible (to the case of the oblique interactions) structure of the mentionedexplicit form; an optimal simplicity [see (5)-(10)] is seen to be induced inthe extensible structure by this factoring compatibility; we use the Lorentztransform

x =x + Mt√1−M2

=X√

1−M2, y = y, t =

t + Mx√1−M2

. (4)

in order to present the mentioned factorizations by

(ξ2 + η2 + ζi)2 − 4ξ2ζi =

=1

(x2 + y2)2[(zct− x

√z2c − ζi)2 − ζiy

2][(zct + x√

z2c − ζi)2 − ζiy2]

with

zc =M√

1−M2, ξ =

zcty

x2 + y2, η =

zcx√

t2 − x2 − y2

x2 + y2

Ep1 (ζi)[2(z2c − ζi)xξ + 2zc

√z2c − ζi tξ −

√z2c − ζi y(ξ2 + η2 + ζi)]

+ Ep2 (ζi)[−2zctξ + y(ξ2 + η2 + ζi)− 2

√z2c − ζi xξ]

= [√

z2c − ζiEp1 (ζi)− Ep

2 (ζi)][2√

z2c − ζi xξ + 2zctξ − y(ξ2 + η2 + ζi)]

= [√

z2c − ζiEp1 (ζi)− Ep

2 (ζi)]

[y/(x2 + y2)][(zct + x√

z2c − ζi)2 − ζiy2]

√z2c − ζi t[z2c(t

2 − x2)− ζi(x2 + y2)]± zcx[z2c(t2 − x2)− ζi(2t2 − x2 − y2)]

≡ (t√

z2c − ζi ∓ xzc)[(zct± x√

z2c − ζi)2 − ζiy2]

−2ξζiT v

1 (ζi) + (ξ2 + η2 + ζi)T v2 (ζi)

+√

z2c − ζi ·2ξζi[−yEv

1 (ζi)] + (ξ2 + η2 + ζi)[zctEv1 (ζi)− xEv

2 (ζi)]

= −[Ev2 (ζi)− Ev

1 (ζi)√

z2c − ζi][(ξ2 + η2 + ζi)(zct + x√

z2c − ζi)− 2ζiyξ]

=−[Ev2(ζi)−Ev

1(ζi)√

z2c−ζi][1/(x2+y2)](zct−x√

z2c−ζi)[(zct+x√

z2c−ζi)2−ζiy2]

70 L.F. Dinu

zc[z2c(t4 − t2x2 − t2y2 − x2y2)− ζi(t2y2 − x2y2 − y4 − t2x2)]

±√

z2c − ζi tx[z2c(t2 − y2)− (z2c − ζi)(x2 + y2)]

= [t(zct∓ x√

z2c − ζi)− z2cy2][(zct± x

√z2c − ζi)2 − ζiy

2]

where

Q1(z2) := d11z2 + d12, Q2(z2) := d01z

2 + d02, Q3(z2) := d03z2 + d04,

Ep1 (ζi) := MQ2(ζi) + zcQ3(ζi), Ep

2 (ζi) := zcQ2(ζi) + M(z2c − ζi)Q3(ζi)

Ev1 (ζi) := Q3(ζi), Ev

2 (ζi) := Q2(ζi)T v

1 (ζi) := −yEv2 (ζi), T v

2 (ζi) := −x(z2c − ζi)Ev1 (ζi) + tzcEv

2 (ζi),

with

d01 =2

γ + 1M

M(1− 2M2), d02 =

M

Md01 − 8

(γ + 1)2M

2

M2(1−M2),

d03 = − 2γ + 1

√1−M2, d04 =

M

Md03,

d11 =8

(γ + 1)2(1−M2), d12 = −M

Md11.

(iii) to use the mentioned extensible structure in order to indicate an ex-haustively classifying, deterministic and explicit characterization of Lighthill’sstatistic and implicit approach concerning the turbulence – planar shock in-teraction; cf. Fig.1. The incident turbulence, regarded as a perturbation,is modelled by a nonstatistical/noncorrelative superposition of some com-pressible finite core (or point core) planar vortices.

The linearized context implies the taking into consideration of a linearproblem with a nonlinear subconscious; the resultant perturbation is re-garded as a solution (“interaction solution”) of such a linearized problem.A nonlinear subconscious results when the nonlinearity is allowed only atthe zeroth order of a perturbation expansion: we linearize the perturbationof a piecewise constant admissible solution and prove that the requirementof admissibility is still active at the first order and essentially structures thelinearized description. The turbulence – planar shock interaction is associ-ated with a class of interaction elements. An interaction element models theinteraction between a planar shock and a single incident vortex correspond-ing to a certain inclination of the vortex axis with respect to the shock.The resulting “relativistically motivated” classification, which is essentiallyoblique, takes into account the importance of some subcritical, critical or


Figure 1: The simplest nonstatistical model of turbulence refraction (t > 0)

72 L.F. Dinu

supercritical inclinations of the incident vortices with respect to the shockin the mentioned interaction.

A final (extensive) version of the above mentioned analysis consists inreplacing the vorticity incident perturbation by a general gasdynamic inci-dent perturbation. In fact, it may be proven that the structure (i)–(iii) ofthe above mentioned interaction analysis persists in this final version.

The approach of the present paper (see Dinu [1], [2]) corresponds toa minimal nonlinearity [associated to the presence of a nonlinear subcon-scious]; still coupled with a “maximal” (exhaustive; explicit and oblique)classifying characterization of the turbulence – shock interaction.

This approach could be set in contrast with a lot of recent studies whichallow (analytically or numerically) a more complete considering of the non-linearity contribution yet in presence of the “minimal” case of a (strictly)parallel interaction; see for example Grove and Menikoff (1990), Han andYin (1993) or Inoue et all (1999).

The work of Han and Yin allows more nonlinearity yet in presence of aset of (approximating) restrictions [cf. its pag. 188]. These authors classifythe context of their work to be “complicated” [pag. 189]. Still, from sucha (“complicated”) context an analogue of the maximal (exhaustive; explicitand oblique) characterization included in this memoir does not emerge. Apossible cause for such an issue appears to be the absence of some struc-turing arguments (needed to replace a “complicated” context by a complexcontext).

More nonlinearity is (numerically) allowed in the parallel interactionsconsidered in the papers by Inoue et all or Grove and Menikoff.

A first aspect of the complex character concerns the modal (entropy /vor-ticity / sound) structure involved.

Remark. Even in presence of a suitable set of structuring argumentswe may need a “bit of chance” in order to get a succesful calculation. Forexample, the attempt to obtain an explicit/closed form for the parallel in-teraction solution may be fruitless if we are not aware of the presence of alot of “traps”:

(a) the emergent sound contribution cannot be computed directly; infact, if r∗ is the radius of the core of the planar incident vortex, this contri-bution can be put in an explicit form directly only in the limit r∗→0 [whichreplaces the incidence (1) by (2)] and only in the interior points of the soniccylinder [X > 0, x2 + y2 < t2]; incidentally it can be predicted (and verified)


in the exterior points of the sonic cylinder too; cf.

p = pr + ps, u = ur + us, v = vr + vs, (5)

[pr(x, y, t), ur(x, y, t), vr(x, y, t)] =

−K

4∑

i=1

kri (ζ)Q−(ζi)

[tk−(ζi) + xk−(ζi)]2 − ζiy2[k−(ζi)y,−k−(ζi)y, tk−(ζi) + xk−(ζi)]

(6)and

ps(x, y, t) = − K√t2 − x2 − y2

·H(t−√

x2 + y2) ·

4∑

i=1

ki(ζ)Q−(ζi)k−(ζi)y[tk−(ζi) + xk−(ζi)]

[tk−(ζi) + xk−(ζi)]2 − ζiy2+

4∑

i=1

ki(ζ)Q+(ζi)k+(ζi)y[tk+(ζi) + xk−(ζi)]

[tk+(ζi) + xk+(ζi)]2 − ζiy2

(7)

us(x, y, t) =K√

t2 − x2 − y2·H(t−

√x2 + y2) ·

4∑

i=1

ki(ζ)Q−(ζi)k−(ζi)y[tk−(ζi) + xk−(ζi)]

[tk−(ζi) + xk−(ζi)]2 − ζiy2+

vs(x, y, t) = − K√t2 − x2 − y2

·H(t−√

x2 + y2) ·4∑

i=1

ki(ζ)Q+(ζi)k+(ζi)y[tk+(ζi) + xk+(ζi)]


(8)

vs(x, y, t) = − K√t2 − x2 − y2

·H(t−√

x2 + y2) ·

4∑

i=1

ki(ζ)Q−(ζi)k(ζi)

(t + Mx)[tk−(ζi) + xk−(ζi)]−My2

[tk−(ζi) + xk−(ζi)]2 − ζiy2+

4∑

i=1

ki(ζ)Q+(ζi)k(ζi)

(t + Mx)[tk+(ζi) + xk+(ζi)]−My2


(9)

whereζ1 = −a2, ζ2 = b2, ζ3 = c2, ζ4 = −1,

74 L.F. Dinu

Q±(ζi) = Q1(ζi)[Q2(ζi)±Q3(ζi)√

z2c − ζi];

K =ε

2π· 1d2

03

, k±(ζi) =zc ±M

√z2c − ζi√

1−M2, k±(ζi) =

M zc ±√

z2c − ζi√1−M2

,

k(ζi)=

ζi√1−M2

√z2c−ζi

, ki(ζ)=

∏

j 6=i

(ζi−ζj)

−1

, kri (ζ)=

(2−i)(3−i)2

ki(ζ)√|ζi|;

kr2(ζ) = 0, kr

3(ζ) = 0; k−(ζ1) = 0; Q+(ζ1) = 0, Q−(ζ2) = 0, Q−(ζ3) = 0.

(b) the emergent vorticity contribution cannot be computed directlyeven in the limit r∗ → 0; its explicit form results by taking into account itsconnection with the emergent sound contribution:

uvorticity(x, y, t) = u−

(y∼, t∼ = T − X

M

)−M ∗p+

(y, t = T − X

M

)−

T∫

T−XM

∂p

∂x(x, y, θ)dθ − usound(x, y, t)

vvorticity(x, y, t) = v−

(y∼, t∼ = T − X

M

)+ (M −M)

∂ψ

∂y

(y, t = T − X

M

)−

T∫

T−XM

∂p

∂y(x, y, θ)dθ − vsound(x, y, t)

where we have to insert,

M ∗ =γ + 14M

[3− γ

γ + 1+

M

M

], T = t, T − X

M= − x

M, y

∼= y.

The entropy component (in the sense of Carrier) of the resultant solutionis:

s(x, y, t) ≡ γ2 − 14MM

(M −M)2p+

(− x

M, y

).

We end this paragraph by presenting the expression of the shock disconti-nuity perturbation ψ. Since lim

T→−∞ψ = 0, we obtain

ψ(y, t) =

t∫

−∞

[−γ + 1

4Mp+(y, θ) + u−(y, θ)

]dθ


where the subscripts + or − correspond respectively to the sides behind orahead of the shock discontinuity.

(c) finally, the explicit form of the nonsingular parallel interaction so-lution which corresponds to the incidence (2) results from a re-weighting are-set of the weight lost in the limit r∗ → 0; cf. Dinu and Dinu [3].

We have to strictly follow this recipe in order to reach an extensibleLorentz-type arguments structure of the mentioned parallel representation:

pr + ps ≡ p‖(x, y, t; ζ1, ζ2, ζ3, ζ4; zc; Q1, Q2, Q3)ur + us ≡ u‖(x, y, t; ζ1, ζ2, ζ3, ζ4; zc; Q1, Q2, Q3)vr + vs ≡ v‖(x, y, t; ζ1, ζ2, ζ3, ζ4; zc; Q1, Q2, Q3).

(10)

The structure of the limit r∗ → 0 of the parallel interaction solution reflects:

– the shape of the incident vortex [the emergent sound singularitiesare distributed along a (circular) sonic arc],

– the details of the modal [vorticity-shock] interaction some pseu-dosingularities [= compensated singularities: they are singularities for thecomponents (6)–(9) taken separately still they appear to be compensated inthe sums (5)] are present,

– the presence of a singularity in the incident contribution,– a memory of the various inner connections [cf. the compatibility of

the mentioned factorizations],– a gasdynamic specificity [in most cases the above mentioned factor-

izations become immaterial if the gasdynamic context is extended/lost; cf.Dinu and Dinu [4].

We could abstract these key phrases by saying that the details of theinteraction analysis in this paper (see Dinu [1], [2]) allow a structural char-acterization of the “prodigious memory” of the interaction solution.

A second aspect of the complex structure of the interaction solution con-sidered appears to be connected with the presence of a “relativistic” struc-ture. Modelling the incident turbulence by a superposition of compressibleplanar vortices appears to correspond to a first level of decomposition; next,in order to proceed, each incident vortex is decomposed (by a Fourier rep-resentation) into planar monochromatic waves – a second level of decom-position; finally, each incident planar monochromatic wave is Snell passedthrough the shock discontinuity. The composition of the mentioned levelsleads to a Fourier–Snell representation of the interaction solution. The main

76 L.F. Dinu

point of the present paper is that the result of the passage through the shockcan again be presented by two levels of recombination so that each incidentlevel of decomposition has a correspondent in the emergent solution. In fact,a “relativistic” character appears, at the first level of decomposition, to re-flect the importance of a critical (“relativistic”) inclination correspondingto

tgΘ = zc.

(cf. Dinu [1], [2]) where Θ is the the inclination of the vortex axis withrespect to the plane of the discontinuity. This remark is used in Dinu [1], [2]to particularly obtain the following oblique subcritic extension of the soundcontribution (10)

p(x, y, t) = 1 + [z∗c(Θ)− zc] ·p‖[x, y, t; a∗2, εbb

∗2, εcc∗2, υ∗2; z∗c(Θ);Q∗

1, Q∗2, Q

∗3] +

M [z∗c(Θ)− zc] ·u‖[x, y, t; a∗2, εbb

∗2, εcc∗2, υ∗2; z∗c(Θ);Q∗

1, Q∗2, Q

∗3]

u(x, y, t) =M

1 + M2

M2 zc[z∗c(Θ)− zc − 1

zctan 2Θ]

· cosΘ

√M2 + (M2 − M2) sin2 Θ

· u‖[x, y, t; a∗2, εbb∗2, εcc

∗2, υ∗2; z∗c(Θ); Q∗1, Q

∗2, Q

∗3]

+M

Mzc

[z∗c(Θ)− zc −

1zc

tan 2Θ]· cosΘ√

M2 + (M2 −M

2) sin2 Θ

· p‖[x, y, t; a∗2, εbb∗2, εcc

∗2, υ∗2; z∗c(Θ);Q∗1, Q

∗2, Q

∗3]

v(x, y, t) = v‖[x, y, t; a∗2, εbb∗2, εcc

∗2, υ∗2; z∗c(Θ);Q∗1, Q

∗2, Q

∗3]

w(x, y, t) = M

(2 +

M2

M2 · zc · [z∗c(Θ)− zc]

)· ( sign θ) sin Θ√

M2 + (M2 −M

2) sin2 Θ

· u‖[x, y, t; a∗2, εbb∗2, εcc

∗2, υ∗2; z∗c(Θ); Q∗1, Q

∗2, Q

∗3]

+(

1 +M2

M2 · zc · [z∗c(Θ)− zc]

)· ( sign θ) sin Θ√

M2 + (M2 −M

2) sin2 Θ

· p‖[x, y, t; a∗2, εbb∗2, εcc

∗2, υ∗2; z∗c(Θ);Q∗1, Q

∗2, Q

∗3]


where

z∗c(Θ) :=√

z2c − tan 2Θ,

a∗2 =a2+tan 2Θ, b∗2 = |b2−tan 2Θ|,

c∗2 = |c2−tan 2Θ|, υ∗2 =1+tan 2Θ

εb = sign (tan 2Θ− b2), εc = sign (tan 2Θ− c2)

Q∗1(z

∗2) := d11z∗2 + (d11tan2Θ + d12) ≡ Q1(z2)

Q∗2(z

∗2) := d01z∗2 + (d01tan2Θ + d02) ≡ Q2(z2)

Q∗3(z

∗2) := d03z∗2 + (d03tan2Θ + d04) ≡ Q3(z2).

and x, y, t have the extended subcritic Lorentz expressions [which reduce to(4) in the limit Θ → 0]

x =zc cosΘ√

M2+(M2−M

2) sin2 Θx+

M zc

Mt

+M zc

M· ( sign θ) sin Θ√

M2+(M2−M

2) sin2 Θz =

X√1−M2

;

y = y; z = z

t =M z∗c(Θ) cosΘ√

M2 + (M2 −M

2) sin2 Θx +

z2c

M z∗c(Θ)t

+z2c

M z∗c(Θ)· ( sign θ) sin Θ√

M2 + (M2 −M

2) sin2 Θz

We finally notice that the second level of decomposition /recombinationappears (see Dinu [1], [2]) to be essential for proving the exhaustive characterof the classification (iii).

References

[1] Dinu, L.F., Gasdynamic interactions with shocks: a linearized fundamen-tal solution, Revue Roumaine Math. Pures Appl., 46(2001), 267-287.

78 L.F. Dinu

[2] Dinu, L. F., Mathematical concepts in nonlinear gas dynamics, CRCPress, London [to appear].

[3] Dinu, L.F. and M.I. Dinu, Solutions with a nonlinear subconscions inthe linearized adiabatic gas dynamic [to appear].

[4] Dinu, L.F and M.I. Dinu, The parallel interaction between a planarvortex and a planar ionizing shock [to appear].

[5] Grove, J.W. and Menikoff, R., Anomalous reflection of a shock wave ata fluid interface, J. Fluid Mechanics, 219(1990), 313-336.

[6] Han, Z. and Yin, X., Shock dynamics, Kluwer, 1993

[7] Hollingworth, M.A. and Richards, E.J., On the sound generated by in-teraction of a vortex and a shock wave. British Aeronautical ResearchCouncil Report 18, No. 257(1956) FM 2371.

[8] Inoue. O., Hattori, Y., Onuma. S and Hyun, J.M., Shock-vortex in-teraction and sound generation. Proceedings of the 22nd InternationalSymposium on Shock Waves, 1999.

[9] Ribner, H.S., The sound generated by interaction of a single vortex witha shock wave. Institute of Aerophysics, University of Toronto, Report,61(1959).

[10] Ribner, H.S., Cylindrical sound wave generated by shock – vortex in-teraction, Journal, 23(1985) 1708–1715.

Liviu Florin DinuInstitute of Mathematics of the Romanian Academy,P.O. Box 1-764. RO-70700 Bucharest, RomaniaE-mail : [email protected]


An Analytical Model of the Glass Flow during thePressing Process

Alexandru DUMITRACHE

November 2, 2001

Abstract - Industrial glass is produced at temperatures above 600C, where glass be-

comes a highly viscous incompressible fluid, usually considered as Newtonian. An an-

alytical approximation to the viscous glass flow during the pressing phase is described,

including a general slip boundary condition. Explicit expression for flow velocity and pres-

sure gradient are obtained, by using a perturbation method based on the slowly varying

geometry of mould and plunger. Based on these results, the total force on the plunger for

given plunger velocity is calculated and an ordinary differential equation (ODE) for the

resulting plunger velocity can be derived, if the force is prescribed. The comparison of

present solution with numerically obtained FEM solutions showed a good agreement.

Key words and phrases : Reynolds’ equation, Asymptotic methods


1 Introduction

To obtain a glass form a two-stage process is often used. First, a blobof hot glass called gob falls into a configuration consisting of a mould andplunger. As soon as the entire glass drop has fallen into this mould, theplunger starts moving to press the glass. This process is called the pressingphase. At the end, the glass drop is reshaped into an preform of bottlecalled parison. After a short period of time, inserted for cooling purposes(the mould is kept at 500C) , the parison is blown to its final shape inanother mould. This process is called blowing phase. The contents of thispaper concerns with glass flow in the pressing phase in manufacturing glassjars or parisons. A typical feature of the shape of a parison is the fact thatwall thickness and radius vary very slowly (except for the bottom part). Thisslow variation will be utilized to obtain an explicit, analytical description ofvelocity and pressure field of the glass flow. The results will be comparedwith numerically obtained ”exact” solutions.

79

80 A. Dumitrache

2 Governing Equations and Dimensional Analysis

The motion of glass at temperatures above 600C can be described [1] bythe Navier-Stokes equations for incompressible Newtonian fluids, given by

ρ(∂v

∂t+ v .∇v

)= −∇.P, ∇.v = 0 (1a,b)

where v denotes the velocity field, and ρ the density. P is (minus) the fluidstress tensor

P = p I− τ , or Pij = p δij − τij (2)

where p is the pressure, I = (δij) is the unit tensor, defined by δij = 1 ifi = j, and = 0 otherwise, and τ is the deviatoric or viscous stress tensor.In Newtonian fluids, a linear relationship is assumed between τ and thedeformation rate of the fluid element, expressed in the rate-of-strain tensor.γ = ∇v + (∇v)T:

τ = η.γ or τij = η

(∂vi

∂xj+

∂vj

∂xi

). (3)

η depends on temperature, and may vary in space and time, but for auniform temperature as we have here, η remains constant. Together with∇.v = 0, ∇.P reduces to ∇p − η∇2v. As a result equations (1a,b) adopttheir common form

ρ(∂v

∂t+ v .∇v

)= −∇p + η∇2v, ∇.v = 0 (4a,b)

In view of the geometry of plunger and mould, we choose cylindrical coordi-nates (r, θ, z), while u, v, w will denote the z, r, θ component of the velocityv. We assume the problem to be axisymmetric, so that both w as well asany θ-derivatives vanish, and the problem reduces to a two-dimensional onein (r, z)-plane; see Figure 1.

To make a dimensional analysis we scale on a typical velocity V , typicallength D, density ρ and viscosity η:

z = Dz∗, r = Dr∗, u = V u∗, v = V v∗, p =ρV 2

Rep∗, t =

D

Vt∗

where Re = ρV D/η is Reynolds’ number. Substitute the above scaling intothe Navier-Stokes equations and further ignore the asterisks ∗.

According to [1], a typical velocity of the plunger, that can be used as acharacteristic velocity, is V = 10−1 m/s. A suitable choice for a characteris-tic length scale is the average wall thickness of the parison, say D ≈ 10−2m.

Analytical Model of the Glass Flow 81

According to [2], the dynamic viscosity of glass varies greatly with temper-ature, but for a temperature around 800C it is in the order of 104 kg/sm.The density of glass is 2500 kg/m3. Therefore, we obtain a Reynolds numberRe = 2.5×10−4.

Because the Reynolds number is small, we can ignore the inertia terms,and obtain the Stokes’ equations

∇p = ∇2v, ∇.v = 0 (5a,b)

Notice that since the Reynolds number is small, we use ρV 2/Re as thescaling pressure.

r

z

Rp

Rm

zpglass

plungermould

Figure 1: Sketch of configuration. Note that Rp = Rp(z−zp), Rm = Rm(z).

We can prove, via the energy equation, that temperature remains con-stant during the pressing process. If we start with a uniform temperaturefield, it will remain uniform everywhere, and it follows that the viscosityalso remains constant (excepting boundary layer).

To utilize the slowly varying geometry of plunger and mould, given byr = Rp(z) and r = Rm(z) respectively, it is convenient to introduce adimensionless small parameter ε and functions Rp and Rm (independent ofε), such that Rp(z) = Rp(εz) and Rm(z) = Rm(εz). In the sequel we willsolve equations (5 a,b) asymptotically for ε → 0 (valid beyond a certaindistance away from the plunger top).

Introduce the slow variable Z = εz. Then we assume that u, v, and p,written in the variable Z, have an asymptotic series expansion for small ε.

82 A. Dumitrache

To determine the power of ε to leading order of the series, we introduce thescaling:

u(r, z) = U(r, Z; ε), v(r, z) = εnV (r, Z; ε), p(r, z) = εmP (r, Z; ε).

We substitute the above scaling into equations (5a-b), and using the homo-geneity of the order of magnitude, we obtain n = 1, so v = εV and m = −1,so p = ε−1P . Introduce the expansion

U =∑

i=0

εiUi (r, z) , V =∑

i=0

εiVi (r, z) , P =∑

i=0

εiPi (r, z) .

For the rest, we only consider the leading order term, and finally we obtain

∂P0

∂Z=

∂2U0

∂r2+

1r

∂U0

∂r,

∂P0

∂r= 0,

∂U0

∂Z+

∂V0

∂r+

V0

r= 0. (6a,b,c)

2.1 Boundary conditions

To solve above equations, we need some boundary conditions. We will con-sider here a simple but adequate model, consisting of a linear relation be-tween shear stress and slip velocity. The coefficient of this relation (the slipfactor) may vary between zero and infinity, but depends on the problem andis to be obtained from experiments.

The slip boundary condition we adopt will be the slip velocity or tan-gential velocity difference vs = (v − vw).t proportional to tangential shearstress in (r, z)-plane,

vs = (v − vw).t = s(P.n).t , (7)

where slip factor s (a positive number) measures the amount of slip. Thereis no slip if s = 0, while there is no friction if s = ∞. The inverse, s−1, mightbe called the friction factor. In general, s = s(Z) may vary with position.In the examples below, it will be left constant.

To apply the above conditions to the moving plunger surface, we recallthat this is defined as:

r = Rp(z − zp(t)),

where z = zp(t) is the position of the top of the plunger at time t. Subscript“p” denotes the value at the plunger. Unless indicated otherwise, we willwrite in the following for short Rp = Rp(z − zp).

To determine the slip velocity, we note that the plunger is going downwith speed dzp

dt = vp. We make equation (7) dimensionless (ignore as before


the asterisks). Then we apply the slow variable Z = εz, and introduceZp(t) = εzp(t) such that

dZp

dt= εvp (8)

where by assumption (the now dimensionless) vp = O(1). Like before, Rp isto be read as Rp(Z − Zp). For small ε we obtain the boundary condition

U0 = sp∂U0

∂r+ vp at r = Rp. (9)

The other boundary conditions at the plunger is that the flow cannot pene-trate the wall of the plunger, which is v .n = 0, leading to the exact relation(in dimensionless form),

V = (U − vp)R′p at r = Rp, (10)

which can be used in exact form.The surface of the mould is given by

r = Rm(z),

(subscript “m” denotes the value at the mould). In a similar way as for theboundary conditions on the plunger we obtain the approximate boundarycondition

U0 = −sm∂U0

∂rat r = Rm, (11)

and the exact conditions

V = UR′m at r = Rm. (12)

In our slender-geometry approximation the exact shape of the free sur-face cannot be determined. Within the present approximation, we will dealwith the average level b of the free surface, as follows p = 0 at z = b(t),where b is a function of the time-dependent geometry, implicitly defined insuch a way that the (incompressible) glass volume is constant for all t.

3 The Results

Now we are ready to solve the equations (6a-c) with boundary conditions(9,10) and (11,12). At first, we consider equation (6a):

dP0

dZ=

∂2U0

∂r2+

1r

∂U0

∂r=

1r

∂

∂r

(r∂U0

∂r

)

84 A. Dumitrache

with solutionU0 =

14r2 dP0

dZ+ A(Z) ln(r) + B(Z) (13)

Using boundary conditions (9) and (11), we obtain A(Z) and B(Z).We want to know the value of the flux at some level z (or Z) through cross

section. The value of this flux depends on z, since the plunger goes downand causes the glass to move upward through a varying cross section. Sinceglass is an incompressible fluid, we have with Gauss’ divergence theorem

0 =∫

Ω∇.v dx =

∫

∂Ωv .ndS (14)

where v = Uez + εV er. From this follows that the flux is

2π

∫ Rm

Rp

rU(r, Z) dr = −πvpR2p . (15)

Using equation (13) and (15), we can determine the pressure gradient dP0dZ .

Finally we solve equation (6c) with boundary conditions (10) or (12).We can rewrite equation (6c) as

∂

∂r(V r) + r

∂U

∂Z= 0,

use (12), to obtain

V =A(Z)

r− 1

r

∫r∂U

∂Zdr =

1r

ddZ

∫ Rm

rrU(r, Z) dr. (16)

Finally, V is approximated (by V0).We discuss now the total force on the plunger and use the result to find

the velocity of the plunger.The total force on the plunger between top level z = zp(t) and glass level

z = b(t) is given by

f = 2π

∫ b

zp

(p− 2η

∂u

∂z

)R′

p + η

(∂v

∂z+

∂u

∂r

)Rp dz. (17)

Note that this glass level, as a function of time, is determined by using thecondition of constant glass volume.

We make equation (17) dimensionless by using the same scaling as before,and scale the force as f = 2πηV Df∗. This yields (we ignore again from hereon the asterisks ∗) :

f =∫ b

zp

(p− 2

∂u

∂z

)R′

p +(

∂v

∂z+

∂u

∂r

)Rp dz. (18)


As before, we will utilize the slowly varying duct radius to asymptoticallydetermine the total force. So we introduce in addition to the foregoingnotation

B = εb, f =1εF =

1εF0 + . . . .

and read Rp here as Rp(Z − Zp(t)). Substitute these into equation (18) toobtain:

F =∫ B

Zp

(P − 2ε2 ∂U

∂Z

)dRp

dZ+

(ε2 ∂V

∂Z+

∂U

∂r

)Rp dZ,

which is to leading order

F0 =∫ B

Zp

P0

dRp

dZ+

∂U0

∂r

Rp dZ. (19)

During the pressing phase, the plunger goes down as the glass moves up.So the velocity of the plunger (vp), the position of the top of the plunger(Zp), and the glass level (B) vary with time. Therefore we have to find asystem of 3 equations for vp, Zp, and B.

First, we observe that the (scaled) volume Ω of the total amount of glassis for all t equal to the constant.

Ω = π

∫ B

0R2

m dZ − π

∫ B

Zp

R2p dZ, where Rp = Rp(Z − Zp). (20)

(Ω/ε is the real dimensionless volume.) Furthermore,we found already ex-pression (19) for the total force on the plunger. After differentiating equa-tions (and (19 and 20) and using the defining relation (8) between Zp andvp, we obtain the system of differential equations, with the three unknownsZp, B, and vp. This system can be integrated numerically.

Examples

Two cases are considered: (i) with no-slip boundary conditions (sp = sm =0) (not represented here) and (ii) with mixed boundary conditions (sp = ∞and sm = 0). For case (i) there is a good agreement between the numericaland the analytic solution (the error is O(ε2) ∼ 1%.

For case (ii), (figure 2), in the axial velocity the numerical solution (∗)appears to differ rather a lot from the analytical solution (full line). This is,however, exactly consistent with the approximation of equation, where theerror is O(ε2) ∼ 4%.

86 A. Dumitrache

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8−0.005

0

0.005

0.01

0.015

0.02

0.025

0.03

r

axia

l com

pone

nt o

f vel

ocity

Mixed Boundary Conditions

t = 0

t = 0.7

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8−0.005

0

0.005

0.01

0.015

0.02

0.025

0.03Mixed Boundary Conditions

r

radi

al c

ompo

nent

of v

eloc

ity

t = 0

t = 0.7

t = 0

t = 0.7

a) Axial velocity b) Radial velocity.

Figure 2: Axial and radial velocity. Parabolic geometry, mixed slip.

4 Conclusions

We described a model for highly viscous incompressible glass flow during thepressing phase of the production. By using a perturbation method basedon the slowly varying geometry, we obtained explicit expressions for flowvelocity and pressure gradient.

Based on these results, we calculated the total force on the plunger forgiven plunger velocity, and an ODE for the resulting plunger velocity if theforce is prescribed.

Representative examples showed a very good agreement of the flow ve-locity between the present solution and numerically obtained “exact” FEMsolutions.

References

[1] de Snoo, S.L., Mattheij, R.M.M., and van de Vorst, G.A.L. Modelling ofGlass, in Particular Small Scale Surface Changes. Rana 96-11, EindhovenUniversity of Technology, 1996.

[2] van den Broek, W.A. Glass Morphology in Manufacturing Jars. MasterThesis, Eindhoven University of Technology, 1996.

Alexandru Dumitrache”Caius Iacob” Institute of Applied Mathematics, Calea 13 Septembrie no.13,76177-Bucharest, sect. 5, ROMANIAE-mail: [email protected]


The dynamic of mechanical forces on lungproperties

Cristian FALUP-PECURARIU, Dan MINEA,Oana FALUP-PECURARIU and Laura DRACEA

November 2, 2001

Abstract - The relationships between mechanical forces and structures of human body

have many implications. The lung is one of the most studied organ, subjected to varying

mechanical forces throughout life. This paper analyze the effects of this forces on fetal

lung growth, the first breathing, on signal transduction and gene expression in pulmonary

endothelial cells, in childhood and adult life. We highlighted the fact that mechanical

forces may be secondary to gradients in gravity motion or osmotic forces. The basic

science and clinical medical correlation with direct application are discussed in detail.

Key words and phrases : surface tension, fetal lung growth

Mathematics Subject Classification (2000) : 92C40

1 Introduction

This research area is broadly defined as cellular bioengineering, a new andrapidly growing field. Experimental work combined cell culture with quan-titative analysis to focus on two main areas: (1) biophysical mechanismswhich regulate fluid balance and transport in the lungs, and (2) mecha-nisms by which physical forces such as fluid shear stress and cyclic strainregulate the function of lung cells. The lung is a mechanically dynamic or-gan, and cells in the lung are subjected to shear stress due to fluid flow,tensile and compressive forces due to respiratory motion, and normal forcesdue to vascular or airway pressure. In the last ten years, it has becomeapparent that most cells sense their mechanical environment and respond tochanges. Although there are significant changes in airway mechanics duringmechanical ventilation and in some airway diseases, little is known abouthow such changes affect cellular functions in the lung [8, 10].

Mechanical forces have a powerful influence on cell behaviour, affectingproduction of potent signalling molecules such as nitric oxide in blood vessels

87

88 C. Falup, D.Minea, O. Falup, L. Dracea

and of substances vital for function such as surfactant in the lung. Theyalso influence the composition of the extracellular matrix [8, 4].

Mechanical forces may be secondary to gradients in gravity, motion,osmotic forces and interactions between cells and/or matrix components.With normal movement, following collision of the organism with an object,the ciliary beat pattern reverses and the ciliary beat frequency increases,providing an immediate correction in the directionality of movement.

2 The nature of physical forces in the lung.Quantitation of physical forces

Cells within a complex three-dimensional structure such as the lung arelikely to be subjected to a variety of different physical forces. One mightspeculate that most lung cells would be subjected to same degree of strain.Strain might be more prominent in cells of the alveolar wall and epitheliumduring breathing, while shear forces may have differential effects on variousportions of the vascular bed, with a gradient from the pulmonary arterial en-dothelium to the capillaries. Vascular endothelium might also be subjectedto strain and hydrostatic pressure. Changes in cellular shape(modulatedby matrix components) may result in changes in differentiation and/or pro-liferation. Conversely, physical forces have been shown to stimulate cellsto modulate the synthesis of matrix components [4] and to strengthen theintegrin cytoskeleton linkages of focal adhesion complexes (FAC) [2].

There have been several different approaches to quantifying the extentto which cells undergo mechanical deformation with changes in the extentof lung inflation. The interpretation of this studies is complicated becausethe degree of unfolding or stretching of the alveolar epithelium is dependenton the lung volume history prior to fixation [2] and because investigatorshave measured different end-points, such as the surface area of the alveolartissue-air interface [7], collagen fiber length, or epithelial basement mem-brane surface area [10].

3 Fetal lung growth. Physiology of the First Breath

During the last third of gestation, the lung grows proportionately with in-creasing body weight. Lung growth occurs as an increase in cell number,with DNA/g lung being constant during this period. Fetal lungs are keptinflated to a volume similar to postnatal functional residual capacity by theactive secretion of fluid (Kitterman, 1996). It has therefore been suggested

The dynamic of mechanical forces 89

that mechanical distension is the major factor influencing lung growth dur-ing fetal development [6].

During the first ”Breath of Life,” several physiologic responses occurvery rapidly. The infants chest is subjected to elevated pressures (≈ +100cm H2O) during delivery, but then is returned to zero, immediately follow-ing birth. This change in pressure is important for two reasons: (1) Asthe infant passes through the birth canal, the increased thoracic pressure,helps to clear the childs large airways of fluid. (2) Decreasing the chest wallpressure prepares the infant for beginning the process of initiating his firstbreath. Inspiratory pressures exceeding (≈ −70 cm H2O) must be devel-oped for the infant to initiate the first breath. Expelling the fetal fluid andcreating an increased negative thoracic pressure, both play an importantrole in providing a patent airway and establishing appropriate gas exchange

Surface tension can be described as a molecular cohesive force existingin the surface film of all liquids which tends to contract the surface to thesmallest possible area. A soap bubble on the end of a tube is an exampleof an air-liquid system in which the surface tension tends to shrink thesurface area. The bubble trans-mural pressure, when it is greater than 0,tends to expand the bubble. The surface tension at the inside and outsidesurfaces tends to prevent further expansion. Assuming that the thickness ofthe bubble is negligible, the relationship between trans-mural pressure (P),surface tension (T), and the radius (r) for a soap bubble at equilibrium isgiven by the following relationship, P = KT/r known as the Young-LaPlacerelationship. This expression states that with a constant surface tensionthe transmural pressure increases as the radius decreases. The substance,thought to contain dipalmitoyl lecithin as an essential component, floats onthe alveolar surface and helps to stabilize alveolar surface forces. Pulmonarysurfactant has a short half-life of 14 hours and therefore must be continuallysynthesized in the lungs.

Mechanical factors and fetal lung growth and differentiation:Factor effect Lung Surfactant Alveolar(example) growth maturation differentiationContinuous distension + Possible - (?) Type I cell favored;(tracheal ligation) type II cell inhibitedIntermittent distension + Possibly(fetal breathing Possible + (?) type II cell favoredmovements) (?)Underdistension(diaphragmatic hernia, - None(?) Type II cell favoredlung liquid drainage)


4 Mechanotransduction: how are mechanical signal trans-duced to cause changesin gene expression and cell metabolism

Cells adhere to neighboring cells and to the extracellular matrix (ECM) viatransmembrane receptors of the cadherin and integrin families, respectively.These receptors are bound in vivo to one other. Some proteins have beencalled cytoskeletal plaque proteins and, as a group, form the focal adhesioncomplex (FAC), which serves as a macromolecular scaffold which mechani-cally couples the cytoplasmic portion of integrins to the actin cytoskeleton.

• Flow-dependent Regulation of Gene Expression in Pulmonary VascularEndothelial Cells [1, 3, 5, 9]

Endothelial cells subjected to fluid shear stress undergo shape modifica-tion from a polygonal to ellipsoidal shape and become uniformly orientedin the direction of flow [3]. This change in shape is associated with theformation and redistribution of actin-containing microfilament bundles, orstress fibers. Focal contacts and the associated protein, vinculin, shift tothe proximal (relative to flow direction) cell margins. In this way, endothe-lial cells respond to shear stress and alter their morphology and functionsdynamically. When stimulated with shear stress, endothelial cells perceivethe stress and transmit the signal into the cell interior, resulting in the ac-tivation of a certain transcription factor (shear stress-related transcriptionfactor; SSTF). SSTF enters the cell nucleus and binds to a cis-element, shearstress responsive element (SSRE), in the promoter of the VCAM-1 gene, andsuppresses its transcriptional activity and decreases the mRNA levels. Thesynthesis and cell surface expression of VCAM-1 protein decrease as a result.Finally, adhesion of the lymphocytes to endothelial cells via the binding ofVLA-4 and VCAM-1 is suppressed.

• Shear stress-mediated regulation of endothelial gene expressionShear stress stimulates endothelial cells to release nitric oxide, a potent

vasodilating factor. Nishida et al (cit. by [1]) demonstrated that expressionof constitutive nitric oxide synthase (NOS) mRNA is markedly increased by24 hours exposure to shear stress (15 dynes/cm2) in bovine aortic endothe-lial cells. It is unclear at present whether inducible nitric oxide synthase isinfluenced by shear stress. Thrombomodulin (TM) is a membrane glyco-protein expressed in vascular endothelial cells which plays a central role inendothelial anti-thrombotic activity by inactivating thrombin. Expressionof its gene can be affected by shear stress.

• Shear-stress signal transductionThe fact that endothelial cell functions are regulated by shear stress

The dynamic of mechanical forces 91

strongly suggest the presence of a mechanism by which endothelial cells per-ceive shear stress, a mechanical force produced by blood flow, and transmitthe signal into the cell interior. Various second messengers, including Ca++,inositol triphosphate, diacylglycerol, cAMP, cGMP, phosphatidylinositol-3kinase, protein kinase C, tyrosine kinase and so on, have been reported tobe involved in the endothelial response to shear stress. That is mainly be-cause no sensor (receptor) for shear stress has ever been found. Althoughthe existence of a shear stress sensor remains unclear, there are some hy-potheses concerning the signal transduction pathway for shear stress: 1) theCa++ theory, 2) the membrane hyperpolarization theory, and 3) the integrintheory. In addition to indirect effects on transcriptional events, there maybe direct effects on nuclear structures from mechanically perturbed cells.The tensegrity model predicts that application of mechanical force to thecell will be transduced to the nucleus. The nuclear protein matrix is in-volved in chromosome organization and contains fixed sites for regulation ofDNA replication and transcription. Mechanical forces may therefore directlyafter gene expression and DNA synthesis, either by rearranging DNA regula-tory nuclear membrane- associated proteins by allowing previously restrictedDNA molecules to unfold (Roberts and DUrso, 1988) or by modulating nu-clear pore size and nucleo- cytosplasmic transport of mRNA (Feldherr andAkin, 1990) (cit. by [10]).

5 Conclusions

In recent years, there has been a growing recognition that physical forces areimportant in pathophysiologic processes in human disease states. Three ar-eas in which physical forces clearly are important are fetal lung development,surfactant metabolism, and mechanical ventilation. There is a renewed in-terest in the effects of mechanical forces (both pressure and volume) onthe lung functions in patients requiring mechanical ventilation. There areneeded more studies in this interdisciplinary field in order to answer to com-plex questions regarding therapeutic aspects.

References

[1] Ando J, Kamiya A., Flow-dependent regulation of gene expression invascular endothelial cells, Jap Heart J, 37(1996), 19-32.


[2] Choquet D, Felsenfeld DP, Sheetz MP., Extracellular matrix rigiditycauses strengthening of integrin-cytoskeleton linkages, Cell, 88(1997),39-48.

[3] Dewey CF, Bussolari SR, Gimbrone MA Jr, Davies PF., The dynamicresponse of vascular endothelial cells to fluid shear stress,J Biomech Eng,103(1981), 177-184.

[4] Leung DYM, Glagov S, Mathews MB., Cyclic stretching stimulates syn-thesis of matrix components by arterial smooth muscles cells in vitro,Science, 191(1976), 475-477.

[5] Masuda H, Shozawa T, Hosoda S, Kanda M, Kamiya A. Cytoplasmicmicrofilaments in endothelial cells of flow loaded accine carotid arteries,Heart and Vessels, 65-9, 1985

[6] Mercer RR, Laco JM, Crapo JD., Three-dimensional reconstructions ofalveoli in the rat lung for pressure-volume relationships,J Appl Physiol,62(1987), 1480- 1487.

[7] Oldmixon EH, Hoppin FC Jr., Alveolar septal folding and lung inflationhistory, J Appl Physiol, 71(1991), 6, 2369-2379.

[8] Waters C.M, Ridge K.,. Sunio G, Venetsanou K, Sznajder J.I, Mechanicalstretching of alveolar epithelial cells increases Na, K-ATPase activity, J.Appl. Physiol., 87(1999), 715-721.

[9] Wechezak AR, Viggers RF, Sauvage LR., Fibronectin and F-actin redis-triburion in cultured endothelial cells exposed to shear stress, Lab Invest,53(1985), 639-647.

[10] Wirtz HR, Dobbs L., The effects of mechanical forces on lung functions,Respiration Physiology 119(2000), 1-17.

Cristian Falup-Pecurariu, Dan MineaUniversity ”Transilvania” BrasovFaculty of Medicine, Department of Neurology

Oana Falup-Pecurariu, Laura DraceaUniversity ”Transilvania” BrasovFaculty of Medicine, Department of PedriaticsE-mail: [email protected]


Nonsteady shearing flow of a fluid of Maxwelliantype

Constantin FETECAU

November 2, 2001

Abstract - The flow of a fluid of Maxwellian type between two infinite parallel plates is

studied. Our solution, as it results from the comparative diagrams, seems to be in close

proximity of that of a second grade fluid for suitable choices of the material constants.

Key words and phrases : fluid of Maxwellian type, shearing flow, velocity field, stress

tensor

Mathematics Subject Classification (2000) : 76A05, 76D30, 76M99

1 Introduction

The rheological properties of materials are generally specified by their socalled constitutive equations. The concept of constitutive equation is inmany cases a severe simplification. In practice one uses equations thatmay be considered as a compromise between an adequate representation ofmaterial properties and mathematical simplicity. The simplest constitutiveequation for a fluid is a Newtonian one

S = 2µD (1)

in which S is the extra-stress tensor, D is the rate of deformation tensorand µ is the dynamic viscosity.

The mechanical behavior of many real fluids, especially those of lowmolecular weight (cf. Coleman et al. [3]), appears to be accurately describedby the equation (1) over a wide range of circumstances. However, in cases oftime dependent flows or to time dependencies at the boundaries, relaxationphenomena should be included. The simplest way to do this is to use amechanical model of Maxwell type (see Bohme [2]).

In this note we consider a mechanical model whose constitutive equation,as it was considered by Fetecau and Fetecau,

S − λ(S − LS − SLT ) = 2µD, (2)

93

94 C. Fetecau

is quite near of that of a Maxwell fluid. Here λ is a positive constant, Lis the velocity gradient and the superposed dot denotes the material timederivative. The exact solution for the flow of an incompressible fluid oftype (2) between two parallel infinite plates is established. This solution,for λ tending to zero, aspires to that corresponding to Navier-Stokes fluids.Further, as the final diagrams show (see Figs. 1 and 2), our velocity field isin close proximity of that obtained by Fetecau and Fetecau [5] for a secondgrade fluid.

2 Statement of the Problem

Consider a fluid of Maxwellian type, at rest, lying between two infiniteparallel plates at distance h apart. At time t = 0+ the lower plate begins tomove with the constant velocity V in a direction parallel to the upper onewhich is stationary. In a suitable cartesian co-ordinate system the velocityfield can be assumed to be of the form.

v = (v(y, t), 0, 0). (3)

For such a flow the tensors L and D have the components

L =

0 ∂yv(y, t) 0

0 0 0

0 0 0

and D =12

0 ∂yv(y, t) 0

∂yv(y, t) 0 0

0 0 0

while the dynamical equations reduce to the linear partial differential equa-tion of second order

−λ∂2t v(y, t) + ∂tv(y, t) = ν∂2

yv(y, t); 0 < y < h, t > 0. (4)

Here ν = µ/ρ is the kinematic viscosity and ρ is the constant density of thefluid.

Assuming that the fluid adheres to the walls, we have the boundaryconditions

v(0, t) = V, v(h, t) = 0; t > 0, (5)

as well as, the initial condition

v(y, 0) = 0; 0 ≤ y ≤ h (6)

Nonsteady shearing flow 95

Regarding the components of the stress tensor S, when the functionv(y, t) is determined, they can be easily obtained by mens of the equation(2) and the initial condition S(y, 0) = 0. Having in mind the components ofthe tensor L and D it is easily to show that S22 = S33 = S13 = S23 = 0.

3 Solution of the problem

Multiplying both sides of equation (4) by sin(nπy/h), integrating betweenthe limits y = 0 and y = h and using (5) and (6), we find that

−λd2vn

dt2+

dvn

dt+ νβ2

nvn = νV βn; vn(0) = 0, (7)

where vn = vn(t) is the finite Fourier sine transform of v(y, t) and βn = nπ/h.The general solution of the ordinary differential equation (71) is of the

form

vn(t)=c1exp

(1−

√1 + 4νλβ2

n

2λt

)+c2exp

(1 +

√1 + 4νλβ2

n

2λt

)+

V

βn, (8)

where the constant c2 has to be zero since vn(t) becomes unbounded fort →∞. Taking into account the initial condition (72) our solution acquiresthe simple form

vn(t) =V

βn

[1− exp

(1−

√1 + 4νλβ2

n

2λt

)]. (9)

Inverting this result by means of Fourier’s sine formula (see Sneddon [6])we get the velocity field

v(y, t) = V(1− y

h

)− 2V

h

∞∑

n=1

sin(βny)βn

exp

(1−

√1 + 4νλβ2

n

2λt

), (10)

that is likewise, as form, with the velocity field (see eq. (9.5) of Bandelliand Rajagopal [1])

v(y, t) = V(1− y

h

)− 2V

h

∞∑

n=1

sin(βny)βn

exp

(− νβ2

n

1 + αβ2n

t

), (11)

corresponding to the same flow of a second grade fluid.

96 C. Fetecau

Letting λ and α to tend to zero in (10) and (11) we attain to the samesolution

v(y, t) = V(1− y

h

)− 2V

h

∞∑

n=1

sin(βny)βn

exp(−νβ2

nt), (12)

corresponding to a Navier-Stokes fluid. For t →∞ all these solutions go tothe steady-state solution

v(y) = V(1− y

h

). (13)

The tangential component Sxy of the stress tensor S, corresponding tothe shearing flow of the fluid of Maxwellian type, will be obtained by meansof the linear differential equation

Sxy(y, t)− λ∂tSxy(y, t) = µ∂yv(y, t); 0 < y < h, t > 0, (14)

with the initial condition Sxy(y, 0) = 0; 0 ≤ y ≤ h. Its form is

Sxy =µV

h[exp(t/λ)− 1]+

4µV

hexp

(t

λ

) ∞∑

n=1

cos(βny)1 +

√1 + 4νλβ2

n

[1− exp

(−1−

√1 + 4νλβ2

n

2λt

)].

4 Conclusions

In this paper we have considered a mechanical model whose constitutiveequation (2) is quite near of that of a Maxwell fluid (actually, it correspondsto a Maxwell fluid having a negative relaxation time). The velocity field(10) corresponding to the shearing flow (3) of this fluid satisfies the linearpartial differential equation (4) and all initial and boundary conditions (5)and (6). Moreover, for λ → 0, as well as the solution (11) corresponding toa second grade fluid, it tends to the solution (12) for a Navier-Stokes fluid.For t → 0 all these solutions tends to the same steady state (13).

It is well known that in a Maxwell fluid the relaxation phenomena aretaken in consideration while in a second grade fluid, prevalently, are theretardation ones (cf. Dunn and Rajagopal [4]). In spite of this our velocityfield (10), as it results from the comparative diagrams, is in close proximityof that corresponding to a second grade fluid (11).

Nonsteady shearing flow 97

Figure 1: Variation of the velocity fields (10) and (11) for h = 30, V =10, λ = 0.05, α = 0.005, for values of t = 5, 10and12s.

Figure 2: Variation of the velocity fields (10) and (11) for h = 30, V =10, λ = 0.05, α = 0.005, for values of t = 10, 30 and 50s.

In Figs. 1 and 2, for comparison, the variations of these velocity fields,for fixed values of λ and α, are ploted for values of ν = 6, 21079, respec-tively, 1,138. In both cases the curves v1, v3 and v5 correspond to a fluid ofMaxwellian type and v2, v4 and v6 to a second grade one. From these dia-grams and many others, which are not included here, it results that for eachλ > 0 it corresponds one α > 0 so as to the velocity profiles correspondingto the two models to be close by.

98 C. Fetecau

References

[1] Bandelli, R. and Rajagopal, K. R., Start-up flows of second grade flu-ids in domains with one finite dimension,Int. J. Non-Linear Mechanics,30(1995), 6, 817-839.

[2] Bohme, G., Stromungsmechanik nichtnewtonscher Fluide. B. G. Teub-ner, Stuttgart-Leipzig-Wiesbaden 2000.

[3] Coleman, B. D., Markovitz, H. and Noll, W., Viscometric flows of non-Newtonian fluids. Springer-Verlag, Berlin-Heidelberg-New York, 1966.

[4] Dunn, J. E. and Rajagopal, K. R., Fluids of differential type: criticalreview and thermodynamic analysis, Int. J. Engng. Sci., 33(1995), 689-729.

[5] Fetecau, C. and Fetecau Corina, The Rayleigh-Stokes Problem for a fluidof Maxwellian type, Int. J. Non-Linear Mechanics, (in press).

[6] Sneddon, I. N., Functional Analysis, in Encyclopedia of Physics,Springer-Verlag, Berlin-Gottingen-Heidelberg, Volume II 1955.

Constantin FetecauTechnical University ”Gh. Asachi” IasiBlvd. Dimitrie Mangeron, nr. 67, 6600 - IasiROMANIAE-mail: [email protected]


An adaptive method for structured meshes inCFD problems

Florin FRUNZULICA and Lucian IORGA

November 2, 2001

Abstract - This paper proposes an iterative method for adaptive meshes obtained by

the deformation of the initial mesh, for computational fluid dynamics problems. The

deformation of the mesh is performed over an elastic, isotropic and continous domain.

The adaptive process is controlled with the principal strains and principal strain directions

of the mesh elements; one obtains the nodal forces on basis of the strain field, used for

the mesh deformation. The method is tested over a structured multi-zonal mesh used for

Euler 2D solver.

Key words and phrases : local error, dimensional modification, principal strains,

nodal displacements, finite element method, structured mesh, Euler solver.

Mathematics Subject Classification (2000) : 65N50, 76J20, 76M12

1 Introduction

During the last decade there were significant progress in the development ofstatic or dynamic adaptive methods used for the integration of diferentialequation systems, for practical use. These methods have the purpose ofachieving a greater spatial accuracy in order to obtain the right solution in aneffective way. The aim of greater spatial accuracy requires a greater numberon points in the mesh, which inevitably leads to an increase in computingcosts(due to an increase in hardware needs, computing time periods, etc).For an increased effeciency of the computing process it is usually prefferedto locally adapt the mesh based on the numerical solution of the problem,rather than a global discrete network.

The main ways to follow in order to obtain adaptive meshes are: 1) meshregenerating, 2) moving the inner nodes, 3) adding or extracting a numberof nodes, 4) a composed method, taking into account the previous 3 methodsalready presented. The author suggest the reader to consider references [1],

99

100 F. Frunzulica & L. Iorga

[2] for the case 1), [5], [10], [11] for the case 2), [3], [4], [9] for the case 3),[8] for the case 4).

The present method belongs to the 2) category, because the nodes aremoved towards the areas with large values of gradient of variable s (a solutionof the problem) . The deformation of the mesh is performed over an elastic,isotropic and continuous domain. The adaptive process is controlled withthe principal strains and principal strain directions of the mesh elements.

Due to the fact that our interest lies in the field of fluid flow modelling,the proposed method was tested, for beginning, on 2D structured meshesused for the study of internal or external flows using an Euler 2D solver.The proposed method can be easily extended to the 3D problem, the onlyproblem being the hardware resources needed.

2 The numerical integration of the Euler system

Compressible shock wave flow calculations can be done solving the Eulerequation system. This system, along with initial and boundary condi-tions, make up the exact mathematic model for perfect gas compressibleflows. When viscosity effects are negligible, various phenomenon, such asshock wave development and dynamics, vortex production, transport andthe transmission of acoustical perturbation are correctly represented by theEuler model.

For the numerical integration of the Euler system second order cell-centered finite volume discretization was used. Finite volumes are quadrilat-eral (in the 2D case), that make up a structured calculation mesh. Secondorder spatial accuracy results from liniar reconstruction of primitive vari-ables in every cell (finite volume). The gradient is determined using infor-mation from neighbouring cells or nodes, using the smallest square methodor by using the Gauss formula. Flow calculation through cell interference isdone upwind with a version of the well known ROE formula for the solvingof the local Riemann problem. For second order accuracy, primitive vari-able values on either side of the interference is calculated using gradientsin neighbouring cells. Where gradients are great, the flow limitator used toavoid short wave-length oscillations is of Van Leer type and is normally ap-plied to the interface. The integration formula is explicit, of Runge-Kuttatype with four intermediate steps. The time step is determined to everyiteration based on a numerical stability criteria.

In case of complicated domains, we can decide to divide the calculationdomain in simpler, structured domains. We have chosen, for reasons of

An adaptive method 101

implementation simplicity, to use a divization which on the frontier levelbetween sub-domains ensures a punctual continuity of nodes. In order toasses flows through the common frontier we used a buffer zone in the vicinityof the frontier, which includes a string of cells on both sides of the frontier. Inthe nodes on the common interface is assured primitive variable continuity.

3 An adaptive method

3.1 The element’s deformation function

The adaptive process is all about a controlled deformation of mesh cells,which leads to an enhancement or, on the contrary, a decrease of the numberof nodes, based upon a local error indicator. For this indicator the followingformula is proposed:

θ =|∇s| − |∇s|min

|∇s|max − |∇s|min

(1)

The physical variable s is chosen depending on the type of problem tostudy. In the case of fluid flow, it can be: density, pressure, or Mach number.As one can notice, the local error indicator can have values between 0 and1.

Increasing or decreasing density of nodes supposes to modify the dimen-sion of the element; by dimension of the element we mean the length inone-dimension, the area in two dimensions, and the volume in tree dimen-sions.We will consider φ to be the function that characterizes cell’s dimensionchange: ΩN = φ ·ΩO (N=new cell dimension, O=old cell dimension). Thisfunction can have one of the following values:

- φ > 1, when θ → 0, in the case of element size increase, and- φ < 1, when θ → 1, in the case of element size decrease (the mesh

becoming locally more dense).For the function φ we can propose the following form:

φ = φ1−θ 2

0 · φ θ 2

1 (2)

The value ρ = φ1/φ0 represents the ratio between the small size and thegreat size of the adapted mesh (0 < ρ < 1). When ρ approaches 0 thereis a significant increase in high gradient areas. If the control parameter ρis determined, parameter φ0 can be calculated using the length conserva-tion condition in the uni-dimensional case, area conservation and volumeconservation in the 2D case and 3D case, respectively:


∫

ΩdΩO =

∫

Ωφ · dΩN (3)

The function φ can be also written as a function of element specificdeformations ε1, ε2, ε3 assessed in the middle of the element, as seen in thefollowing line (for the 3D case):

φ = (1 + ε1) (1 + ε2) (1 + ε3) = f1 · f2 · f3 (4)

For an brick element, a thorough choice of the functions f1, f2, f3 wouldbe given by the izotropic distribution of the φ function on the main deforma-tion directions: f1 = f2 = f3 = φ1/3. In case of a preferential deformationof the element we can choose the functions as follows:

f1 = φk1 , f2 = φk2 , f3 = φk3 , unde k1 + k2 + k3 = 1 (5)

The direction given by the gradient of s, ~g = ∇s / |∇s| can be consideredas a preferential direction of deformation. But, during the process of defor-mation the direction given by ~g will be different than that of the maximumspecific deformation direction and, in order to take this into account, we willhave to modify the control functions (2) of the specific deformations:

f1 = φk1(1−θ2n2

1)0 φ

k1θ2n21

1 , f2 = φk2(1−θ2n2

2)0 φ

k2θ2n22

1 , f3 = φk3(1−θ2n2

3)0 φ

k3θ2n23

1

(6)where (n1, n2, n3) are the components of the direction ~g, referred to a coor-dinate system attached to the principal deformation directions.

Equations (6) are used for the calculation of the main deformation di-rections ε1, ε2, ε3 and implicitely for the calculation of the specific liniardeformation (εx, εy, εz) and angular (γxy, γyz, γzx) in the global mesh refer-ence system. For the 2D case, relations between main specific deformationsand specific angular and liniar deformations are as follows:

εx + εy = ε1 + ε2 , (7)εxεy − γ2

xy = ε1ε2 , (8)(εx − ε1) l1 + γxym1 = 0 or γxyl1 + (εy − ε1) m1 = 0 (9)

where ε1 is the maximum specific deformation and (l1,m1) is the main max-imum deformation direction.


An interesting alternative to this calculation process would be makingspecific angular deformations equal to zero. This choice seems to reduce therisk of strongly distorted elements.

In order not to alter the convergence of the 2D Euler solver we choseto limit an element’s area. In the calculations done we considered that theelement can be deformed until: (0.01...0.05)Ωe

O ≤ ΩeN ≤ (1.5...2)Ωe

O.

3.2 Determining the nodal displacements of the mesh

In order to determine the node displacements (u, v, w) knowing the element’sspecific deformations, we considered that the calculation domain to be mod-elled as being an elastic, homogeneous and izotrope material. In the 2D case,the model will be a flat plate, its thickness being δ, submitted to flat strainsand flat stress.

The idea of the adaptive algorithm is to determine the nodal forces thatlead to the state of the supposed specific deformation. For the calculation ofthe displacements using the finite element method, in the 2D and 3D case,the notion of concentrated force applied to the element nodes is somewhatimposed in order to be able to achieve a simple calculation model for theadaptation of the mesh.

We will consider the coordinates of the elementary nodes as well as thenodal displacements to be represented by the same set of interpolation func-tions (elements are iso-parametric):

~x =p∑

i=1

Ni (ξ, η, ζ) · ~xi , ~ue =p∑

i=1

Ni (ξ, η, ζ) · ~ui (10)

Between the specific deformations ε and the nodal displacement vectord for each element, stands the relation:

ε = [B] d (11)

where the matrix [B] contains the derivatives of the interpolation functions(∂Ni

∂x,∂Ni

∂y,∂Ni

∂z

)

i=1...p

.

It is known that the stiffness matrix for an element can be calculated asfollows:

[Ke] =∫

Ωe

[B]T [E] [B] dΩ (12)


where [E] is the elasticity matrix. If we multiply the matrix [Ke] at theright by the nodal displacement vector for each element de we will obtainthe following formula:

[Ke] de =∫

Ωe

[B]T [E] ε dΩ (13)

By adding the contribution of every element that have the node i incommon, we can obtain an estimation of the concentrated forces that ifapplied to node i inflict the local mesh deformation:

Fi =ei∑

e=1

∫

Ωe

[B]T [E] ε dΩ (14)

The next step is determining the mesh nodal displacements with thehelp of the finite element method. Without going into unnecessary detail,it would be worth reminding that the problem to be solved at this point is:

[K] d = F (15)

where [K] is the global stiffness matrix of the structure, d is the nodaldisplacement vector in the global coordinate system, and F is the nodalforce vector apllied to the structure. It is necessary to attach to the system(15) boundary conditions. These conditions can be: a) blocking type (thenodal displacements are blocked) and b) sliding type, in this case nodaldisplacement being allowed on the curve/surface that describes the domainfrontier.

The solving of system (15) with the associated boundary conditionscan be done with a procedure usually associated to the band matrix typeproblems. After solving this system we obtain the nodal displacements(ui, vi, wi)i=1...nn , with the help of which mesh node position can be up-dated:

~xNi ← ~xO

i + ~ui , i = 1...nn (16)

3.3 Adaptive algorithm

Imput data: original mesh, the values of s estimated in nodes, the charac-teristics of the elastic medium (E and ν). Steps: 1) estimating the valueof ∇s and selecting the principal strain directions; 2) one takes the valuesof the principal strains given by the functions (6) according with the val-ues of the error parameter θ ; 3) computing the linear and angular strains;


4) computing the nodal forces F , with the aid of (14 ); 5) determine thenodal displacements by using FEM solver; 6) establishing the position ofthe mass-centres for the elements of the new mesh; 7) updating the valuesof s in nodes and ∇s. The iterative process continues with step 1) until thenew position of the nodes will be obtained with the imposed precision.

4 Results and discussion

4.1 Adaptive mesh over a square domain

The first case study deals with an adaptive mesh built for a square, having35 x 35 equally spaced nodes. The variation of s is described by:

s(r) =

0 for |r| > ε

C e−ε2/(ε2−r2) for |r| ≤ ε(17)

where ε = 0.85 and r =√

x2 + y2. Boundary conditions: 1. freezing the 4corners; 2. considering the slip condition over all the edges. The adaptivemesh obtained in 3 iterations is shown in figure no. 2. One may notice anincreasement of the number of the nodes in the areas where ∇s has largevalues.

Fig.1: Function s(r) Fig.2: The adaptive mesh

4.2 Steady supersonic flow in 2D chanel

The second test was conducted on a 2D channel (fig.4), in which flow is su-personic (Mach = 1.6). Choosing ρ = 0.3 and maximum allowed reduction


order 0.35, the adapted mesh looks similar to the one in the figure no.3b.There is also a comparison between the isobar curves in the two situations(fig. no.4).

Fig.3: Mesh before (a) and after (b) the adaptive process

Fig.4: Isobar curves - (a) original mesh ; (b) adapted mesh

5 Conclusion

The adaptive algorythm is robust and relatively quick. Basically, for a 2Dmesh, the algorithm converges after a few iterations. Keeping the num-ber of nodes and mesh connectivities, the necessary hardware resources for


the adaptive process are relatively small. The Euler solver converges fairlyquickly on the adapted mesh, having as initial data the solution obtained onthe original mesh, and then extrapolated on the adapted mesh. The adaptiveprocedure is also suitable for multi-zoned meshes, in this case the adaptiveprocess being done in this case on sub-domains. The frontiers between sub-domains are either determined by the analist or allowed to deform, ensuringhowever the displacement continuity on the common frontiers.

The results obtained through 2D testing are encouraging, suggesting thatthe algorithm’s implementation on a structured 3D mesh and associationwith an 3D Euler solver would be rather effective.

References

[1] Anderson, D. A., Steinbrenner, J., Generating Adaptive Grids with aConventional Grid Scheme, AIAA Paper, 86-0427, New York 1986.

[2] Anderson, D. A., Equidistribution schemes, Poisson generations andadaptive grids, Appl. Math. Comput, 24(1987), 211-277.

[3] Berger, M., Oliger, J., Adaptive mesh refinement for hyperbolic partialdifferential equations, J. Comput. Phys., 53(1984), 484-512.

[4] Biswas, R., Strawn, R. C., Tetrahedral and hexahedral mesh adapta-tion for CFD problems, Applied Numerical Mathematics Journal, (toappear).

[5] Carcaillet, R., Optimization of three-dimensional computational gridsand generation of flow adaptive computational grids, AIAA Paper, 86-0156, Reno USA, 1986.

[6] Frunzulica, F., An adaptive method using grid deformation, NationalSimposium SISOM 2000.

[7] Frunzulica, F., An adaptive method using grid deformation, The sci-entific bulletin - University Politehnica of Bucharest, No.1/2001.

[8] Hwang, Wu S. J., Global and Local Remeshing Algoritms for Com-pressible Flows, J. Comput. Phys., 102(1992), 98-113.

[9] LeVeque, R. J., Wave propagation algorithmus for multi-dimensionalhyperbolic systems, J. Comput. Phys., 131(1997), 327-353.


[10] Liao, G., Anderson, D., A new approach to grid generation, Appl. Anal.,44(1992), 285-298.

[11] Liao, G., Pan, T. W., Su, J., A numerical grid generator basedon Moser’s deformation method, Numer. Methods Partial DifferentialEquations, 10(1994), 21-31.

Florin FrunzulicaDepartment of Aerospace Engineering, University ,,Politehnica” of Bucharest,Str. Gh. Polizu, No.1, 78126, Bucharest, Romania

E-mail: [email protected]

Lucian IorgaMechanical & Aerospace Engineering Fellow, Rutgers, The State Universityof New Jersey, U.S.A.

E-mail: l [email protected]


A Numerical Method for Richards’ Equation

Stelian ION

November 2, 2001

Abstract - The Richards’ equation represents the mathematical model of water flow into

porous media. It is a strong nonlinear parabolic equation in unsaturated flow domain and

linear elliptic in saturated flow domain. In the paper a numerical method is given to obtain

an accurate solution of one dimension Richards’ equations for layered soil. The method

consists in the finite volume scheme with imposed flux continuity for the space discretiza-

tion and the implicit backward differentiation formulae (BDF) for the time integration.

The numerical results are presented to illustrate the performance of the method.

Key words and phrases : partial differential equations, method of lines, differential

algebraic equation, discontinuous coefficients, flux continuity

Mathematics Subject Classification (2000) : 34A09, 35K15, 65H10, 65L80,65M20, 76S05

1 Introduction

Richards’ equation models water flow in a porous medium. This partialdifferential equation (PDE) combines the mass balance with Darcy law forvelocity and can cast in several forms, depending on whether pressure, watercontent or both are used as state variable. In the case of picewise homoge-neous soil it is advantageous to use the pressure as state variable because itis continuous function, while the water content is not so.

In this paper we restrict our consideration to the numerical approxi-mation of vertical unidimensional infiltration flow. The discrete numericalapproximation of the continuous model is presented in the third section.The space discretization is obtained by using the integral form of mass bal-ance on the proper net of the space domains. The net is defined so that aninterface between two different layers does not intersect its elements. Onthe interface the numerical flux is defined by a formula that reflects thecontinuity of the pressure head and the normal component of velocity.

109

110 S. Ion

2 Physical problem and Mathematical model

We are interested in modeling and finding quantitative evaluation of thewater infiltration into a heterogeneous soil. We suppose that the soil iscomposed by vertical homogeneous layers (see Fig.1).

6

layer 1

layer 2

5 h

z

interface

Fig.1 Layered soil

A widely used mathematical model of this physical problem is Richards’equation

∂tθ + div v = 0 (1)

where θ stands for volumetric water content and v denotes the Darcy ve-locity. To obtain a complete model of the water flow in many applicationswater content and velocity are given as functions of pressure head h

v = −K(h)∇(h + z)θ = θ(h) (2)

where K(h) represents the hydraulic conductivity. On the interface physicalconsiderations require the continuity of the pressure head and the normalcomponent of the velocity. So, we have

h(t, S) |−= h(t, S) |+v(t, S) · n |−= v(t, S) · n |+ (3)

Both functions K(h) and θ(h) reflect the hydraulic properties of the soiland share the common property to be constant for h > 0. Note that, the

A Numerical Method for Richards’ Equation 111

water content and hydraulic conductivity are discontinuous functions on theinterface.

3 Space Discretization. Time Integration

To obtain the space approximation of Richards’ equation we use the finite-volume method(FVM).

dz1/2 z1

dzi−1/2 zi

dzi+1/2 zi+1 zN

Vi

-¾

Interface

Fig. 2 Control volumes

In FVM techniques the control volumes Vi = (zi−1/2, z1+1/2) are intro-duced. The net zi−1/2 is defined so that any Vi is entirely included into asingle layer. Therefore, one interface is located at some zi−1/2. The centroidof Vi, zi, is used as location points for pressure head, hi(t) = h(t, z = zi).The space discretization is obtained from integral form of (1) over eachcontrol volume Vi (see Fig. 2)

∂t

∫ zi+1

2

zi− 1

2

θ(h)dt + vi+ 12(t)− vi− 1

2(t) = 0 (4)

Now, we consider as unknowns the values of the pressure head, hi(t)i=1.N ,that will be determined as the solution of a system of ordinary differentialequations (ODE) that approximate this integral form. To obtain this ODEthe integral is approximated by the middle point quadrature formula andthe velocities is defined by divided differences of pressure head.

∂t

∫ zi+1

2

zi− 1

2

θ(h)dt ≈ (zi+1/2 − zi−1/2)θ′(hi)∂thi

vi+ 12≈ −Ki+ 1

2

(hi+1 + zi+1)− (hi + zi)zi+1 − zi

(5)

The numerical hydraulic conductivity Ki+1/2 is differently evaluated, de-pending whether the point evaluation is an interface point or not.

Let z∗ = zi+1/2 be an interface point. We denote by h∗ the value of pres-sure head at this point and by K∗−(K∗

+) the left(right)limit of the hydraulic

112 S. Ion

conductivity value. We evaluate the velocity on the left (right) by takingconstant values of hydraulic conductivity, K∗−(K∗

+), and by linear interpo-lation of pressure head on each side of interface point, for example on theleft we have

v∗− = −K∗−

(h∗ + z∗)− (hi + zi)z∗ − zi

.

Next, we will impose the continuity conditions (3) and then a nonlinearequation it is obtained for h∗

H∗ =(

HiK∗−

z∗ − zi+ Hi+1

K∗+

zi+1 − z∗

)(K∗−

z∗ − zi+

K∗+

zi+1 − z∗

)−1

where H = h + z. Observing that the right term is between Hi+1 and Hi

we approximate the solution of the equation by the arithmetical mean ofthese values. So, we take for h∗ = (hi + hi+1)/2 and obtain for numericalhydraulic conductivity

1Ki+1/2

=1

K∗−

z∗ − zi

zi+1 − zi+

1K∗

+

zi+1 − z∗

zi+1 − zi(6)

If the point zi+1/2 is not interface point from the continuity of the hydraulicconductivity we obtain

Ki+1/2 = K(h∗) (7)

The system of differential algebraic equation is obtained from (5, 6, 7) andis given by

C(h)∂th + A(h)h + f(h) = 0

where A is a tridiagonal matrix, f is a vector and C is a diagonal matrixwith the property C(hi) = 0 , if hi > 0.

We note that the numerical scheme that approximates the space differen-tial operator ∂z(K(h)∂z(h + z)) is conservative and the numerical hydraulicconductivity giving by the formula (6) represent the physical conditions (3).In the case of linear elliptic equation with discontinuous coefficients it wasshown [6] that the conservativeness is a necessary condition for a numericalscheme to be convergence. Also for the linear case a formula like (6) wasobtained in the papers ([1], [3]).

For the time integration we use a backward differentiation formulae(BDF)

aCn+1 hn+1 − hn+10

4tn+1+ A(hn+1)hn+1 + f(hn+1) = 0

A Numerical Method for Richards’ Equation 113

where the constant a depends of the order of BDF and hn+10 is the predicted

value of the pressure head (it is known from the previous time steps). Formore details about BDF scheme see [2]. To know the values of pressure headhn+1 at the next time level tn+1 a nonlinear system of algebraic equation

G(x) = 0

must be solved. To solve this nonlinear equation the modified Picard methodis used [5]. This iterative method, that can be viewed as truncated Newtonmethod, is given by

1 x0 ←− hn

2 (a

∆tnCk + Ak)δxk = −G(xk)

3 xk+1 = xk + δxk

4 Numerical Results

We present the numerical results for two type of stratified soil. In the bothcases the soil consists in two different porous media, sand and clay, thatalternate in five equal width layers (0.20 m thickness). For each layer thehydraulic conductivity and water content function are giving by the vanGenuchten model [4]. The pressure head is giving at boundaries,hbot =−1.0m, hup = −0.5m and at initial time h(0, z) + z = −2.0m.

-0.8

-0.6

-0.4

-0.2

-1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2

Dep

th

pressure head

Time evolution of pressure head. Five layers of type b/g/b/g/b

time=4htime=12htime=24htime=48h

-0.8

-0.6

-0.4

-0.2

-2 -1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4

Dep

th

pressure head

Time evolution of pressure head. Five layers of type g/b/g/b/g

time=4htime=12htime=24htime=48h

Fig. 2a. Berino sand on the top Fig. 2b. Glendale clay on the top

Both figures represent the pressure head distribution at several momentsof time (4h, 12h, 24h and 48h). Here we wish to make a short comment aboutthe two pressure head profiles.

In the case of the sand on the top (Fig. 2a) followed by the clay (thesand is more permeable than clay) the pressure head has its maximum along

114 S. Ion

the first layer on the bottom, while in the case of the clay (Fig. 2b) followedby the sand the maximum is attained on the top. Physically this means thatthe water is accumulating in the first case and it is draining in the secondcase.

References

[1] K. Aziz and A. Settari, Petroleum Reservoir Simulation, Applied SciencePublishers, London, 1979.

[2] K.E. Brenan, S.L. Campbell and L.R. Petzold,Numerical Solution ofInitial Values Problems in Differential-Algebraic Equations, Classics inApplied Mathematics, SIAM, 1996.

[3] M. G. Edwards and C. F. Rogers, Finite volume discretization withimposed flux continuity for the general tensor pressure equation, Com-putational Geosciences 2 (1998), 259-290.

[4] van Genuchten, M. Th., A closed-form equation for predicting the hy-draulic conductivity of unsaturated soils, Soil Sci. Soc. Am. J., 44(1980), 892-898.

[5] F. Lehmann and Ph. Ackerer, Comparasion of Iterative Methods forImproved Solutions of the Fluid Flow Equation in Partially SaturatedPorous Media, Transport in Porous Media 31 (1998), 275-292.

[6] A. Samarski, V. Andreev, Methodes aux differences pour equations ellip-tiques, Editions Mir, Moscou, 1978.

Stelian IonInstitute of Applied Mathematics ,, Caius Iacob”Calea 13 Septembrie, No. 13, Sector 5Bucharest, P.O. Box 1-24, 70700E-mail : [email protected]


Boundary Integral Method for an OscillatoryStokes Flow Problem

Mirela KOHR

November 2, 2001

Abstract - In this paper we obtain a boundary integral method in order to determine

the oscillatory Stokes flow due to translational or rotational oscillations of a solid particle

in an unbounded viscous incompressible fluid. As an application of this boundary integral

method, we treate the case of high-frequency oscillations.

Key words and phrases : Boundary integral method, oscillatory Stokes flow.


1 Mathematical formulation of the problem

Let us consider an oscillatory flow due to the translational or rotationalvibrations of a solid particle with the velocity U exp(−iΩt), where Ω is thefrequency of oscillations, in an infinite quiescent Newtonian incompressiblefluid. If the Reynolds number is small, the velocity field v and the pressurefield p satisfy the following equations of the unsteady Stokes flow

ρ∂v∂t

= ρf −∇p + µ∇2v, ∇ · v = 0 in D, (1)

where ρ and µ are the density and the dynamic viscosity of the fluid, respec-tively, ρf is the body force and D is the domain of the flow, exterior to thesurface Σ of the solid particle. Supposing that the body force is constant, weintroduce the modified pressure P = p− ρ(f ·x). Also using the assumptionthat the flow is an oscillatory one, as well as the linearity of Eqs. (1), wededuce that the fields v and P can be written as follows

v(x, t) = u(x) exp(−iΩt), P (x, t) = q(x) exp(−iΩt). (2)

Next we introduce the non-dimensional quantities x′, u′ and q′ defined by

x = Lx′, u = V u′, q =µV

Lq′, (3)

115

116 Mirela KOHR

where L is a characteristic length of the particle and V is a characteristicvelocity. Then substituting Eqs. (2) and (3) into Eqs. (1), we obtain thenon-dimensional equations of oscillatory Stokes flow

λ2u′ = −∇′q′ + ∆′u′, ∇′ · u′ = 0 in D. (4)

Here λ2 = −iΩL2/ν is the frequency parameter and ν is the kinematicviscosity of the fluid.

Furtermore, we assume that the boundary Σ of the solid particle is aclosed surface of class C2. On this surface the field u′ must satisfy theboundary condition below

u′ = U′ on Σ, (5)

where U′ is the amplitude of oscillations of the particle. Moreover, since thefluid is in rest at infinity, we have to add the following conditions

|u′(x′)| → 0, |q′(x′)| → 0 as |x′| → ∞. (6)

Also for simplicity, we will further neglect primes from Eqs. (4)-(6).Now using the divergence theorem, we may prove the following unique-

ness result for the classical solution (u, q) of the oscillatory Stokes problem(4)-(6) (i.e., u ∈ C2(D) ∩ C0(D), q ∈ C1(D)):

Theorem 1. The system of Eqs. (4) has at most one solution that satisfiesthe boundary condition (5) and the decay conditions at infinity

|u(x)| = O(|x|−1), |∇u(x)| = O(|x|−2), |q(x)| = O(|x|−2) as |x| → ∞.(7)

2 The oscillatory free-space Green function

Let Gλ2= (Gλ2

ij )i,j=1,3 be the oscillatory free-space Green function, andlet Πλ2

= (Πλ2

1 , Πλ2

2 ,Πλ2

3 ) be the pressure vector of the oscillatory Stokesflow generated by an oscillatory point force placed at a point x0 of R3. Thesefunctions satisfy the equations

(λ2 −∆)Gλ2(x) +∇Πλ2

(x) = δ(x)I, ∇ ·Gλ2(x) = 0, (8)

where δ(x) is the three-dimensional delta function and x = x− x0.The components of the oscillatory Green function are given by [5], [6]

Gλ2

ij (x) =14π

δij

rA(λr) +

xixj

r3B(λr)

, i, j ∈ 1, 2, 3, (9)

Oscilatory Flow Problem 117

where r = |x| and λ ∈ C is that square root of λ2 which has positive realpart, and

A(k) = e−k(1 + k−1 + k−2)− k−2, B(k) = e−k(−1− 3k−1 − 3k−2) + 3k−2

(10)where k ∈ C \ 0. Also the components of the pressure vector Πλ2

aregiven by [6]

Πλ2

i (x) =14π

xi

r3, i ∈ 1, 2, 3. (11)

Furthermore, the stress tensor Sλ2associated to the oscillatory Green

function has the components [6]

Sλ2

ijk(x) = −Πλ2

j (x)δik +∂Gλ2

ij (x)∂xk

+∂Gλ2

kj (x)∂xi

. (12)

Using Eqs. (9)-(12), the components of the tensor Sλ2can be given ex-

plicitelly [6].Also note that the pressure tensor Λλ2

, associated to the stress tensorSλ2

, has the following components [6]

Λλ2

ik (x− y) ==14π

δik

r3(λ2r2 − 2) +

6xixk

r5

, i, k = 1, 2, 3. (13)

Next we consider a continuous vector function h = (h1, h2, h3) on thesurface Σ, and denote by Vs

λ2h and Vdλ2h the oscillatory single-layer and

double-layer potentials, given by

(Vsλ2h)j(x0) =

∫

ΣGλ2

ij (x)hi(x)dσ(x), x0 6∈ Σ, (14)

(Vdλ2h)j(x) =

∫

ΣSλ2

ijk(y − x)nk(y)hi(y)dσ(y), x0 6∈ Σ, (15)

where n is the outward unit normal vector to Σ. Additionally, let P sλ2h and

P dλ2h be the functions defined by

(P sλ2h)(x) =

∫

ΣΠλ2

i (x− y)hi(y)dσ(y), x0 6∈ Σ. (16)

(P dλ2h)(x) =

∫

ΣΛλ2

ik (y − x)nk(y)hi(y)dσ(y), x0 6∈ Σ. (17)

The functions (Vsλ2h, P s

λ2h) and (Vdλ2h, P d

λ2h) are smooth functions in eachof the domains R3 \D0 and D0 respectively, where D0 is the inner domain

118 Mirela KOHR

with the boundary Σ. Also these functions satisfy the equations of oscillatoryStokes flow, i.e.,

(λ2 −∆)(Vsλ2h)(x) +∇(P s

λ2h)(x) = 0, ∇ · (Vsλ2h)(x) = 0, x0 6∈ Σ, (18)

(λ2 −∆)(Vdλ2h)(x) +∇(P d

λ2h)(x) = 0, ∇ · (Vdλ2h)(x) = 0, x0 6∈ Σ. (19)

Let G be the steady Stokeslet (or the Oseen-Burgers tensor) of the Stokesflow, and let S be the corresponding stress tensor. These functions have thecomponents [1], [6]

Gij(x) =18π

δij

r+

xixj

r3

, Sijk(x) = − 3

4π

xixj xk

r5, i, j, k ∈ 1, 2, 3.

(20)We note the decomposition formulas

Gλ2

ij (x) = Gij(x) + Gλ2

ij (x), Sλ2

ijk(x) = Sijk(x) + Sλ2

ijk(x), (21)

where Gλ2

ij and Sλ2

ijk are continuous functions. Thus we have

Gλ2

ij (x) → Gij(x), Sλ2

ijk(x) → Sijk(x) as r → 0 (22)

and hence the oscillatory potentials have the same singular behaviour as inthe case λ = 0.

Let us introduce the notations

w+(x0) = limx→x0∈Σ

x∈R3\D0

w(x), w−(x0) = limx→x0∈Σx∈D0

w(x), (23)

for the limiting values of a function w on the two sides of Σ. Additionally,the formula below

(Vdλ2h)∗j (x0) =

∫ PV

ΣSλ2

ijk(y − x0)nk(y)hi(y)dσ(y), (24)

gives the principal value of the oscillatory double-layer potential on Σ. Alsothe symbol PV means that the integral in evaluated in the principal valuesense [6]. Similarly,

(Hλ2h)∗j (x0) = nk(x0)∫ PV

ΣSλ2

jik(x0 − y)hi(y)dσ(y) (25)

means the principal value of the surface force of the single-layer potentialon Σ.

Using the above arguments, we obtain the result below.


Theorem 2. Let Σ be a closed surface of class C2 and let h be a continuousvector function on Σ. Then the following properties hold on Σ,

(Vsλ2h)+ = (Vs

λ2h)− = Vsλ2h, (26)

(Vdλ2h)+ − (Vd

λ2h)∗ =12h = (Vd

λ2h)∗ − (Vdλ2h)−, (27)

(Hλ2h)+ − (Hλ2h)∗ = −12h = (Hλ2h)∗ − (Hd

λ2h)−. (28)

Moreover, at infinity the decay conditions occur,

(Vsλ2h)(x0) = O(|x0|−3), (P s

λ2h)(x) = O(|x0|−2) as |x0| → ∞. (29)

(Vdλ2,Γh)(x) = O(|x|−2), (P d

λ2h)(x) = O(|x|−2) as |x| → ∞. (30)

3 Boundary integral equation of the problem

In this section we give a boundary integral formulation in order to solvethe oscillatory Stokes problem (4)-(6). More exactly, we have:

Theorem 3. Let U be a given constant vector. Then the boundary integralequation

12h(x0) + (Vd

λ2h)∗(x0) + (Vsλ2h)(x0) = U, x0 ∈ Σ, (31)

has exactly one continuous solution h on Σ, and the boundary integral rep-resentations

u(x) = (Vdλ2h)(x) + (Vs

λ2h)(x), q(x) = (P dλ2h)(x) + (P s

λ2h)(x), x0 ∈ D(32)

yield the unique solution (u, q) of the oscillatory Stokes system (4), whichsatisfies the boundary condition (5) and the decay conditions (7).

Proof. Using the decomposition formula (21) we deduce that the single-layer and double-layer operators Vs

λ2 and V dλ2 that appear in Eq. (31) have

weakly singular kernels and hence, they are compact on the Banach spaceC0(Σ) of continuous functions on Σ. Therefore Eq. (31) is a Fredholmintegral equation of the second kind for which the Fredholm alternative canbe applied. Thus for a given constant vector U, Eq. (31) has a unique

120 Mirela KOHR

solution h in the space C0(Σ) if and only if the following homogeneousadjoint equation has only the trivial solution in the same space,

12h0(x0) + (H

λ2h0)∗(x0) + (Vs

λ2h0)(x0) = 0, x0 ∈ Σ. (33)

Let h0 ∈ C0(Σ) be a solution of Eq. (33). Consider the followingfunctions

u(0) = Vs

λ2h0, q(0) = P s

λ2h0, x0 6∈ Σ. (34)

These functions satisfy the equations

(λ2 −∆)(Vλ2h0)(x) +∇q(0)(x) = 0, ∇ · (Vs

λ2h0)(x) = 0 (35)

in each of the domains D and D0. Moreover, from Eqs. (28) and (33), wededuce that the surface force H

λ2h0, associated to the fields given by Eqs.

(34), has the following limiting value on the inner side of Σ,

(Hλ2h0)− = −Vs

λ2h0 on Σ. (36)

Straightforward computations and Eq. (36) lead to the identity

∫

D0

[λ2|u(0)|2 + 2e(0)jk e

(0)jk ]dv = −

∫

Σ|Vs

λ2h0|2dσ, (37)

where n is the outward unit normal vector Σ and 2e(0)jk = ∂u

(0)j /∂xk +

∂u(0)k /∂xj . This identity yields that u(0) = 0 on D0, and from Eq. (36) we

obtain the property(H

λ2h0)− = 0 on Σ. (38)

On the other hand, using the fact that the functions u(0) and q(0) de-termine an oscillatory Stokes flow in the unbounded domain D, with zerovelocity field on the surface Σ, and applying Theorem 1, we get u(0) =0, q(0) = 0 in D, and consequently

(Hλ2h0)+ = 0 on Σ. (39)

Finally, using Eqs. (28), (38) and (39) we deduce that h0 = 0 on Σ, andthus Eq. (31) has a unique continuous solution h on Σ. This function andEqs. (32) yield the solution of our problem.


4 Asymptotic solution for high-frequency oscillations

It is not difficult to see that in the case of high-frequency limit |λ|r →∞,the oscillatory Green function becomes [3], [6]

Gλ2

ji (x) = − 1λ2

∂Qi(x)∂xj

, (40)

whereQj(x) = Πλ2

j (x) =14π

xj

r3, j = 1, 2, 3. (41)

Furthermore, using the Eqs. (32) and (40), we obtain the velocity fieldu in the form

uj(x) = − 1λ2

∫

Σ

∂Qi(x)∂xj

hi(y)dσ(y) +∫

ΣQj(x)ni(y)hi(y)dσ(y). (42)

Therefore, in the case of high-frequency limit |λ|r →∞, u takes the form

u(x) = −∇φ(x), (43)

for x in the flow field. Note that φ is the velocity potential given by

φ(x) =1λ2

∫

ΣQi(x)hi(y)dσ(y) +

14π

∫

Σ

1rni(y)hi(y)dσ(y). (44)

Moreover, using the equation ∇ · u = 0 and Eq. (43), we deduce that thevelocity potential φ is a harmonic function.

The velocity field of the resulting irrotational flow vanishes at infinity,satisfies the impenetrability condition on the surface Σ, but it does notsatisfy the non-slip boundary condition on the same surface. At high fre-quences the flow is composed by an outer irrotational flow and a thin viscousboundary-layer of thickness |λ|−1 that resides on the particle surface. Theouter irrotational flow may be computed by solving the equations

φ(x0) = −2∫

ΣG(x,x0)Uk(x)nk(x)dσ(x)+2

∫ PV

Σ∇G(x,x0)·n(x)Φ(x)dσ(x),

(45)where x0 ∈ Σ, G(x,x0) = 1/(4πr) is the fundamental solution of the three-dimensional Laplace equation. Note that Eq. (45) has a unique solution φ[6]. Also for a point x0 away from the surface Σ, we have

φ(x0) = −∫

ΣG(x)Uk(x)nk(x)dσ(x) +

∫

Σ∇G(x) · n(x)φ(x)dσ(x). (46)

122 Mirela KOHR

Finally note that the equations of the thin boundary-layer can be foundin [3].

Remark. Another application of the boundary integral method, devel-oped in this paper, can be given in the case of small frequency oscillations.For more details see [2].

References

[1] M. Kohr, Modern Problems in Viscous Fluid Mechanics. Cluj UniversityPress, Cluj-Napoca, 2000 (in Romanian).

[2] M. Kohr, A boundary integral equation method for an oscillatory flowproblem, Mathematica (Cluj), to appear.

[3] M. Loewenberg, Axisymmetric unsteady Stokes flow past an oscillatingfinite-length cylinder. J. Fluid Mech., 265(1994), 265-288.

[4] C. Pozrikidis, A singularity method for unsteady linearized flow, Phys.Fluids A, 1(1989), 1508-1520.

[5] C. Pozrikidis, A study of linearized oscillatory flow past particles by theboundary-integral method, J. Fluid Mech., 202(1989), 17-41.

[6] C. Pozrikidis, Boundary Integral and Singularity Methods for LinearizedViscous Flow, Cambridge University Press, 1992.

Babes-Bolyai UniversityFaculty of Mathematics and Computer Sciences1 M. Kogalniceanu Str., 3400 Cluj-Napoca, RomaniaE-mail: [email protected]


Analytical Method for maximal drag airfoilsoptimization in cavity flows

Mircea LUPU, Adrian POSTELNICU, Ernest SCHEIBER

November 2, 2001

Abstract - In the paper there are solved direct and inverse boundary problems and

analytical solutions are obtained for optimization problems in the case of some nonlinear

integral operators. It is modeled the plane potential flow of an inviscid, incompressible

and unlimited fluid jet, which encounters a symmetrical, curvilinear obstacle (the deflector

of maximal drag), using the Jukowski-Roshko-Eppler cavity model. There are derived

integral singular equations, for direct and inverse problems and the movement in the

auxiliary canonical half-plane is obtained. Next, the optimization problem is solved in an

analytical manner. The design of the optimal airfoil is performed and finally, numerical

computations of the drag coefficient, which depends on the cavitation number, geometrical

and hydro-aerodynamical parameters are carried-out.

Key words and phrases : inviscid jets, cavity flow, singular integral equation, nonlinear

integral operator, optimization of airfoils.

Mathematics Subject Classification (2000) : 65M32, 65R20, 76G25

1 Introduction

As a general rule, the problems involving the optimization of the airfoils /hydrofoils, in the case of flows with free surfaces, lead to the study of non-linear integral- differential operators. This may be a reason for the scarcityof the purely analytical methods with respect to the analytical-numericalones, presented in the open literature. Important results are presented inrecent papers, [4], [5], [20].

We present in the paper an analytical method, which uses the half-planeas canonical domain, concerned with the optimization of the maximum dragdeflector in wake-flows.

The problems focusing on maximal drag are very important, in relationwith applications to the thrust reversal devices, control of the direction ofreactive vehicles, slowing by means of fluid jets, jet flaps systems from theairplanes wings, or turbine blades.

123

124 M. Lupu, A. Postelnicu, E. Scheiber

Concerning the monographical studies and results on the cavity flows, werefer to the recent works by Simona Popp [20], Maklakov [15] and Nicolescu[18].

The stationary potential plane flow of an inviscid fluid is considered inthe absence of mass forces (Hyp). Relating the velocity field w(z) = u(x, y)+iv(x, y) to the x0y frame in the physical flow domain Dz, z = x+iy, it followswithin the hypotesis (Hyp) formulated above, rot v = 0(v = grad ϕ(x, y)),div v = 0. The complex potential f(z) and the complex velocity w(z) aredefined through the analytic functions:

f(z) = ϕ(x, y) + iψ(x, y); w = u− iv =df

dz= V e−iθ. (1)

Here ϕ(x, y) is the velocity potential, ψ(x, y) is the stream function;V = (u2+v2)

12 and θ = arctan v

u , are the velocity magnitude and respectivelyits angle to the Ox axis. In the case of the direct problem the flow will bestudied using the hodographic method [8], [4], [20] and f(W ), z(W ), (W =V + iθ) will be obtained. In the case of a curvilinear domain Dz it isgenerally difficult to obtain directly f = f(z) and w = w(z) by solving theboundary problem, therefore it should be introduced a canonic auxiliarydomain Dζ , ζ = ξ + iη, |ζ| ≤ 1 (Levi Civita circle [13]) or the half-planeDζ , ζ = ξ + iη, η ≥ 0 [9], [10]. To the domain Dz, y ≥ 0 of the planesimmetrical jets it corresponds the domain Df , ϕ ∈ (−∞,∞), 0 ≤ ψ < ∞.We try to determine the analytic function f = f(ζ) which is the conformalmapping Df ↔ Dζ , with

fζ = 0; ϕξ = ψη; ϕη = −ψξ. (2)

To obtain the analyticity conditions for the velocity W (V, θ) we introducethe Jukovski function ω, by considering along the free lines V = V 0 :

ω = t + iθ, w = V 0e−ω, t = lnV 0

V, 0 ≤ V ≤ V 0, (3)

θψ = tϕ, θϕ = −tψ, ϕθ = −ψt, ϕt = ψθ, ωf = 0, fω = 0.(4)

In the case of free surface flow, the flow domain Dz is generally boundedby polygonal rigid walls (curvilinear) obstacles and stream lines. Along thesefree lines the velocity, pressure and density have constant values V 0, p0, ρ0 =

Maximal drag airfoils optimization 125

ρ, respectively. Applying Bernoulli’s law for the case of incompressible flowalong a stream line ψ = const. we obtain

12V 2 +

p

ρ=

12V 02 +

p0

ρ. (5)

Now we consider the following Theorems [9], [10]

Theorem 1. In the hypothesis (Hyp), if there is a conformal mapping f =f(ζ), fζ = 0 with Df ↔ Dζ then z = z(ζ) is analytic, with Dz ↔ Dζ .

Theorem 2. In the hypothesis (Hyp), if the function f is analytic in ζ andrealizes a conformal mapping between Df ↔ Dζ then ω = ω(ζ) is analyticand it is the conformal mapping between Dω ↔ Dζ .

At this stage, we shall find f = f(ζ) so that the boundaries of thedomains Dz, Df correspond to the boundary of Dζ , η = 0, ξ ∈ (−∞,∞), onwhich we have the streamlines ψ = const. As x′Ox is the axis of symmetry,we shall prove that for the found function f = f(ζ) the following conditionshold η = 0, ψ = const. and ∂ϕ

∂η |η=0 = 0 In this case, we have on η = 0

dx + idy =∂ϕ

∂ξ

1V

(cos θ + i sin θ)dξ. (6)

Separating the real and imaginary parts we find

x(ξ) =

ξ∫

ξ0

ϕξcos θ

Vdξ + x0, y(ξ) =

ξ∫

ξ0

ϕξsin θ

Vdξ + y0. (7)

2 Integral equations for direct and inverse problems

In the conditions stated in the previous section, we will consider the planeflow of an unlimited fluid jet, formed by a uniform upstream flow of velocity~V1 = V1~ı. The fluid encounters a symmetrical curvilinear obstacle BOB′

and the streamlines (BC) and (B′C ′) emanate from BandB′ respectively.They connect to the solid lines (CD) and (C ′D′) which are parallel to thesymmetry axis x′Ox (Fig. 1). On the free lines (BC) and (B′C ′) , themagnitude of the velocity V0 is known, hence 0 < V1 ≤ V 0. In this case wemay use the Jukovski- Roshko-Eppler model [18], [19], [20]: the finite areaof cavitation (CBOB′C) , followed by a Helmholtz wake, limited by the twoplates, where the fluid is motionless. Hence, the movement physical plane


is in the outer area; at infinity upstream in A0 and at infinity downstream(D,D′) the flow velocity is V 0i. The length of the airfoil (B′OB) is 2L, thedistance between the plates is (CC ′) = (DD′) = 2h and the depth of thecavitation domain is xC = xC′ = d (Fig. 1). We looking for the optimalshape of the airfoil (BOB′) which leads to maximum drag, when the cavitynumber Q = (V 0

V1)2 − 1 and the distance L are given.

We solved recently the same problem for the Helmholtz model withoutcavitation V1 = V 0, by otimising the shape of the impervious parachuteof maximal drag [11]. Also recently we solved the limited jet flow past anobstacle, finding the deflector (BOB′ of minimal drag [12].

In the case of direct problem without optimization the shape of the profileBOB′ is known and only the flow and the equations of the free lines are tobe obtained. If the obstacle is a triangle then many results are reported in[6], [16], [20], [22].

For an inverse problem the boundary (A0OBCDA0) is completely un-known, but V 0 is known on the free lines and the function V = V (θ) (orp = p(θ)) is given on the obstacle B′OB. The shape of the obstacle B′OBmust be found in order to maximize or minimize the drag. In order find theoptimal shape of maximal deflector we shall apply the Theorems 1 and 2.

Let us consider the biunivocal correspondance between the domains Dz

and Df with the halfplane Dζ , η ≥ 0, so that the boundary (A0OBCDAA0)be placed upon the η = 0 axis, ξ ∈ (−∞,∞) : A0(−∞), O(−1), B(a), C(1),D(∞) (Fig. 2). We will looking for the parameter a < 1.

We find f(ζ) = ϕ+ iψ analytic in Dζ , η ≥ 0, such that 4ψ = 0 and withthe boundary values ψ = 0 for ξ ∈ (−∞,∞), η = 0. The solution to thisDirichlet problem (Df ↔ Dζ) is [3], [10], [12]

f(ζ) = Aζ; A > 0;∂ϕ

∂ξ|η=0 = A;

∂ϕ

∂η|η=0 = 0. (8)

Once the boundary values θ = 0, ξ ∈ (−∞,−1) ∪ (1,∞); θ = θ(ξ), ξ ∈(−1, a); t = 0, ξ ∈ (a, 1) are known we determine the analytic function inDζ , η ≥ 0, ω(ζ) = t+ iθ. This is a mixed problem and we transform it into

a Dirichlet problem for the function S1(ζ) = R + iI =ω(ζ)√

(ζ − 1)(ζ − a), in

which case along the boundary we have: R = 0, ξ ∈ (−∞,−1) ∪ (a,∞) and

R = − θ(ξ)√(1− ξ)(ξ − a)

, ξ ∈ (−1, a). We obtain

ω(ζ) = ω1(ζ) = −√

(ζ − 1)(ζ − a)π

a∫

−1

θ(s)√(s− a)(s− 1)

· ds

s− ζ+ C1,


--

6

³

´

¨§

¨§

-

-

O

x

y

B

B′

A0

~V1

~V 0

~V 0

............ -

............ -

C D

C ′ D′

z

Fig. 1.Motion range in the physical plane Dz.

with the constant C1 = 0, if the velocity in O is such as: V (ξ = −1) = 0 andt(−1) = ∞. Applying the Sohotski - Plemelj formula [3], [4] to the integralpart, we obtain

t(ξ) = −√

(ξ − 1)(ξ − a)π

∫ a

−1

θ(s)√(s− 1)(s− a)

· ds

s− ξξ ∈ (−1, a). (9)

For the inverse problem, if we put t(ξ) = t(θ(ξ)), from V = V (θ), ξ ∈[−1, a] then (9) becomes a singular integral equation. The singularity of theintegral must be taken in the sense of Cauchy’s principal value [3], [4], [17].This problem was solved for several distributions [10], [12], [16].

Once θ = θ(ξ) obtained, for t = t(ξ) we find ω = ω(ζ) and using (7) wededuce z = z(ζ), the optimal design of the airfoil (OB).

If we suppose that, on the airfoil, t = t(ζ) is known, then we deduceanother formula for ω = ω(ζ).

Thus, using the analytical function S2 = R + iI =ω(ζ)

i√

ζ2 − 1whose

real part satisfies on the boundary the conditions: R = 0, ξ ∈ (−∞,−1) ∪(a,∞);R = − t(ξ)√

1− ξ2, ξ ∈ (−1, a) we deduce an analogous solution in D+

ζ :

ω(ζ) = −√

ζ2 − 1π

a∫

−1

t(s)√1− s2

ds

s− ζ(10)


-

6

A0 ξ D

η

a

BO

−1ψ = 0

θ = 0

ζ = ξ + iη

C

1t(ξ)

θ(ξ) t = 0

ψ = 0

Fig. 2.Correspondence of field Df , Dω with Dζ .

Applying the Sohotski-Plemelj formula for ξ ∈ (−1, a), η = 0+ we have

θ(ξ) = −√

1− ξ2

π

a∫

−1

t(s)√1− s2

ds

s− ξ, ξ ∈ (−1, a) (11)

If V = V (θ) or θ = θ(V ), θ = θ(t(ξ)) is given on the airfoil, then for theinverse problem we have a singular integral equation in the unknown functiont = t(ξ). Next the solution ω = ω(ζ) and z = z(ζ) may be deduced. Weremark that the relations (9) and (11) are inversion formulae for θ(ξ) ↔ t(ξ),and ξ ∈ (−1, a), respectively.

Using (6), we compute the curvature of the boundary K = K(ξ) =1

R(ξ)=|x′y”− y′x”|(x′2 + y′2)

32

=θ′(ξ)ϕ′(ξ)

V (ξ), where R(ξ) is the curvature radius. For

t = lnV 0

Vand V = K(ξ)

ϕ′

θ′, using (8) we deduce

ln θ′(ξ) =

√(ξ − 1)(ξ − a)

π

a∫

−1

θ(s)√(s− 1)(s− a)

ds

s− ξ+ln

AK(ξ)V 0

, ξ ∈ (−1, a).

For the direct problem, when the curvature K(ξ) of the profile is given theabove equation is a new form of the integro-differential equation obtained in[8], [16]. Denoting θ′(ξ) = Θ(ξ) and integrating by parts it results anothersingular integral equation (for the circle arc, K ≡ 1). This is in agreementwith the results contained in [4], [5], [15], [16].


3 The solution to the nonlinear optimization problem

The purpose of this section is to find the optimal deflector for maximal dragin the fluid cavity flow. This design corresponds to turbine blades or to theimpermeable parachute. We shall obtain the design of the correspondingprofile.

Using (6) and (8), with the arc element dS = ϕ′ξdξ

V (ξ)the length of the

profile OB is given by

S =∫ ξ

−1A

dξ

V (ξ), L = A

∫ a

−1

dξ

V (ξ). (12)

In this case the condition

A

∫ a

−1

dξ

V (ξ)=

A

V 0

∫ a

−1exp(t(ξ))dξ = L (13)

must be fulfilled. The above equality may be used to find A if L and V (ξ)are known. We will consider that the cavity number Q = (V 0

V1)2 − 1 is given

[1], [18], [20].Due to the fact that the actual problem is a plane one and x′Ox is

a symmetry axis, it is enough to deal only with the superior halfplaneA0OBCDAA0, where the profile is OB. In this case, the aerodynamicaldrag has only one component namely Px. From the Bernoulli formula (5),we compute the aerodynamic resultant for the whole airfoil BOB′ in thephysical plane z. Let us consider the circle C(O, R) which cuts the free linesin the points H and H ′. On the line Kz(OBHCRH ′B′O), we calculate theresultant force

2(Px + ıPy) =∮

Kz

(p− p0)dz.

Due to the symmetry and to the fact that on (BC) and (BC ′) we havep = p0 and on HC and H ′C ′ the resultant force vanishes, we get Py = 0and further

2Px =∫ h

0(p1 − p0)dy = (p1 − p0)h

or finally Px = P . Using (5) where V = V1 and the definition of Q, we get

P = ρV 21 Qh = ρV 2

1 QL(h

L). (14)

We pass now to the evaluation of the width h. Returning to (3),

w =df

dζ

dζ

dz= V 0e−ω(ζ)


and using (8), we get

dz =A

V 0eω(ζ)dζ (15)

Computing the velocity in the point D, where V → V1, θ = 0, V (D) = V1 =V 0 exp(−ω(D)); from (10),

ω(D) = limζ→∞

ω(ζ) =1π

∫ a

−1

t(s)√1− s2

ds

Now, returning to:

Q + 1 = exp(

2π

∫ a

1

t(s)√1− s2

ds

)

we get the condition:∫ a

−1

t(s)√1− s2

ds =π

2ln(Q + 1) + k. (16)

Using a quadrature for (15), we get for the point D

ih =∮

Kζ

A

V 0eω(ζ)dζ

due to the fact that ζ →∞ in D. We will evaluate this integral, by perform-ing the inversion:

ζ =1Z

, ζ →∞, Z → 0

ω(Z) = −√

1− Z2

π

∫ a

−1

t(s)√1− s2(sZ − 1)

ds; ω(0) =1π

∫ a

−1

t(s)√1− s2

ds.

In this case ih =∮

Cε

A

V 0eω(Z) dZ

Z2, where C(O, ε) is a circle of radius ε (small

enough) and the function appearing in the integral has just the singularityZ = 0, which is in fact a double pole. Applying the residues theorem,

(eω(Z))′Z=0 = eω(0)ω′(0)

we get, after some algebra:

h = −2∫ a

−1

st(s)ds√1− s2

· exp(

1π

∫ a

−1

t(s)√1− s2

ds

). (17)


Using (13), (14), (16) and (17), we obtain the resultant force on (OB)

P =12ρV 2

1 Q√

Q + 1( ∫ a

−1

(−s)t(s)√1− s2

ds

)( ∫ a

−1et(s)ds

)−1

(18)

which will be extremed, in the maximing sense. Let us consider the nonlinearfunctional

I[t] =(∫ a

−1

(−s)t(s)√1− s2

ds

)( ∫ a

−1et(s)ds

)−1

. (19)

The optimization problem becomes: find the velocity distribution t = t(ξ), ξ ∈(−1, a) on the OB airfoil, which maximizes the functional (19), along therestriction (16), with a given value of the cavitation number Q. Withinthe optimal solution, besides t = t(ξ), the parameter a will be also found.Using (19) and the restriction (16), the following equivalent functional isintroduced:

J [t, a] = I[t] exp

a∫−1

f(s)√1− s2

ds− k

g(a), g(a) =

a∫

−1

√1− s

1 + sds, (20)

where the factor g(a) = arcsin√

1− a2 +√

1− a2 was chosen for algebraicconvenience, to simplify the form of J . The assertion concerning the equiv-alence of functionals may be readily justified, by invoking the positivity ofI[t] and the fact that the exponential maximum value is 1, by analysing theintegrand and t(s). In this way, the design of the OB airfoil and finally,using (7), its (maximal) drag are obtained.

From (16) and V (ξ = −1) = 0, to ensure the convergence of the integrals,we put V (ξ) = V 0(ξ + 1)αj(ξ), with 0 < α < 1, j(−1) 6= 0.. Without losing

the generality, we choose α = 12 . From t(ξ) = ln

V 0

V (ξ), we have

t(ξ) = G(ξ)− ln

√1 + ξ

1− ξ, ξ ∈ (−1, a), (21)

where the term G(ξ) is generated by j(ξ). Introducing (21) in (20) we get

J [G, a] =U + h(a)∫ a

−1

√1− s

1 + seG(s)ds

exp

∫ a

−1

G(s)√1− s2

ds− f(a)− k

g(a)(22)


where

U =∫ a

−1

(−s)G(s)√1− s2

ds, f(a) =∫ a

−1ln

√1− s

1 + s

ds√1− s2

,

h(a) =∫ a

−1ln

√1− s

1 + s

sds√1− s2

=π

2+ arcsin a−

√1− a2 ln

√1− a

1 + a.

In order to find G(ξ), the functional J [G] is maximized to a functional H[G],whose maximum point may be readily computed and with the propertythat in this maximum point the two functionals have the same value (globalmaximum). Now we apply the Jensen’s inequality [7],[13],[15]:

If f(x) ≥ 0, g(x) are integrable functions on [a,b] then∫ b

af(x)eg(x)dx ≥

∫ b

af(x)dx exp

(∫ b

af(x)g(x)dx :

∫ b

af(x)dx

), (23)

where the equality case holds if and only if g(x) is a constant function.Applying the inequality (23) to the denominator of J [G, a], we get J [G, a] ≤H[U, a], where

H[U, a] =U + h(a)

g(a)

(exp

U + f(a) + k

g(a)

)−1

. (24)

For a given a∗∈(−1, 1), the global maximum of the positive H[U ] is obtainedby a simple differentiation with respect to U . We get

U = g(a)− h(a) =√

1− a2 +√

1− a2 ln

√1 + a

1− a. (25)

In this case, with (24) and (25), we obtain

J ≤ maxH[U, a] = M(a), (26)

where M(a) = exph(a)− f(a)− k

g(a)− 1. The equality in (23), (26) holds

when G(ξ) = G0 = const., i.e. with (25)

√1− a2 +

√1− a2 ln

√1 + a

1− a=

∫ a

−1

G0(−s)√1− s2

= G0

√1− a2.

Due to the small value of the global maximum, G(ξ) ≡ G0, we have

G(ξ) = 1 + ln

√1 + a

1− a


and therefore for (21),

t(ξ) = 1 + ln

√1 + a

1− a− ln

√1 + ξ

1− ξ, ξ ∈ (−1, a). (27)

Looking for the maximal value of M(a), we obtain as a result of the differ-entiation with respect to a, the following equation,whic is in fact (16),:

arcsin√

1− a2 ln(e

√1 + a

1− a) +

∫ π

arccos aln tan

θ

2dθ = k, a ∈ (−1, 1).

(28)By denoting its root, a = a∗, equation (28) becomes, with a∗ = cos ε:

ε ln(ecotε

2) +

∫ π

εln tan

θ

2dθ = k =

π

2ln(Q + 1). (29)

Denoting the root by a∗, the maximum has the expression

M(a∗) =1e

√1− a∗

1 + a∗=

1etan

ε

2

in agreement with that reported by Maklakov [15]; he used in his derivationas canonical domain the Levi-Civitta half-circle. Our numerical runs showthat equation (28), k = F (a) has a small root: a∗ ∈ (−1, 1), for k ∈ (0, 10).

This fact implies Q ∈ (0, exp20π− 1), in agreement with the theoretical

prediction: Q = (V 0

V1)2 − 1, V 0 > V1. Using (27) and (3), we get the

velocity distribution on the airfoil:

V =V 0

e

√1− a

1 + a

√1 + ξ

1− ξ, ξ ∈ (−1, a), a = a∗. (30)

Using (27) to compute θ(ξ) given by (11), we get

θ(ξ) =1π

ln

(e

√1 + a

1− a

)ln

√1−ξ1+ξ +

√1−a1+a√

1−ξ1+ξ −

√1−a1+a

+

+

√1− ξ2

π

∫ a

−1ln

√1 + s

1− s

ds√1− s2(s− ξ)

, ξ ∈ (−1, a),


where∫ a

−1

ds√1− s2(s− ξ)

is the first term multiplied by (−√

1− ξ2). For

the second integral of θ(ξ), we need the series

T (α) =2π

∞∑

n=0

α2n+1

(2n + 1)2=

1π

[Li 2(α)− Li 2(−α)] , 0 < α < 1, (31)

where Li 2(α) =∞∑

n=0

αn

n2is known as Euler’s dilogarithm series [15], [21].The

following relations may easily deduced:

dT

dα=

1πα

ln1 + α

1− α, T (α) =

1π

∫ α

0ln

1 + α

1− α

dα

α, T (±1) = ±π

4. (32)

Using successively the substitutions s = cos θ and t = tan ( θ2), along with

(31) and (32) , the integral I2 becomes

θ(ξ) =π

2+

1π

ln

√1−ξ1+ξ +

√1−a1+a√

1−ξ1+ξ −

√1−a1+a

−T (

√1 + ξ

1− ξ

√1− a

1 + a), ξ ∈ (−1, a). (33)

As we have previously remarked, the relations (27) and (33) may be inter-preted as solutions to equations (9) and (11) and they enable us to computeθ(ξ), when the velocity distribution t(ξ) is given and reciprocally (in inverseproblems). In the hodographic plane W (θ, V ) we find from (30) and (33)the following dependence on the airfoil

θ(V ) =π

2+

1π

ln1 + eV

V 0

1− eVV 0

− T

(eV

V 0

), 0 < V <

V 0

e. (34)

If in (34) the function θ(V ) = θ(t) is given on the airfoil, then (34) is thesolution of the inverse problem. For the Hemholtz model whitout cavitation(impermeable parachute), we have previously [14] obtained, in the same way,the velocity distribution and the value of the angle on the airfoil

t(ξ) = 1 + ln√

2ξ + 1

, V (ξ) =V 0

e

√1 + ξ

2, ξ ∈ (−1, 1), (35)

θ(ξ) =π

2+

1π

ln√

2 +√

ξ + 1√2−√ξ + 1

− T (

√1 + ξ

2), ξ ∈ (−1, 1). (36)

Hence, the hodographic formula will be the same.


4 Aerodynamical and geometrical parameters for the opti-mal deflector

The design of the optimal deflector may be obtained using the normalisedversions of the equations (6) and (7) with (25), (26)

X(ξ) =x(ξ)L

=

∫ ξ

−1

cos θ(s)ds

V (s)(a− s)∫ 1

−1

ds

V (s)(a− s)

, Y (ξ) =y(ξ)L

=

∫ ξ

−1

sin θ(s)ds

V (s)(a− s)∫ 1

−1

ds

V (s)(a− s)

,

(37)ξ ∈ (−1, a), a = a∗.

The optimal airfoil OB is given in (X,Y) coordinates, see Fig. 3. WithL and V0 given, using (13) and (27), or (30), we find the parameter A as:

A =V 0L

eg(a)

√1− a

1 + a. (38)

The velocity distribution V = V (S) , along the arc S may be also com-puted, using (12), (30) and (33)

S

L=

arcsin√

1− ξ2 +√

1− ξ2

arcsin√

1− a2 +√

1− a2, V =

V 0

e

tan ( ε2)

tan (α2 )

; (39)

a = cos(ε), ξ = cos(α),

with ξ ∈ (−1, a), S ∈ [0, L]. The cavitation height d is the abscisa of thepoint C and is computed using (7), normated with the length of the arc BC.

Further, with (18) and (19), by dividing toρV 2

1 L

2, we obtain the maximal

drag coefficient

C∗x(Q) =

Q√

Q + 12e

√1− a∗

1 + a∗, a∗ ∈ (−1, 1), (40)

which can be computed once Q is given and a∗ is obtained by solving (28).For the Helmholtz model without cavitation, i.e. the problem of the

impervious parachute in unlimited flow, when (BD), (B′D′) become free jetlines and V1 = V 0 the following value was obtained [14], [15]:

C∗x(0) =

8πe

≈ 0.93679,


which is the minimum in (40). This fact will be more closely highlighted inthe numerical results to be given below. In the case of the Helmholtz modelfor the normal flat plate (BOB′) (normal to the flow), without cavitation,it is known that [6], [8], [22]:

Cpx(0) =

2π

4 + π≈ 0.8798.

Further, using the J-R-E cavitational model applied to the normal flatplate, in [1], [6], [18], [20] it is shown that:

Cpx(Q) =

2π

4 + π(1 + Q).

For these cases of compressible flows with or without cavitation, wenotice here the results obtained by Simona Popp for the dihedral angle [20],flat plate [6], [19] or the Riabuchinski model [2], [20]. Our present resultsare very general for direct problems, moreover that in [20] there are reportedcomparisons for the Roshko, Wu, Tulin, Efros in the case of Ciaplighin gasusing the method of Imai-Lamla.

Galina Camenschi has solved in [2] the problem for compressible cavityflows, using the Efros-Kreisel-Gilbarg model (the method of the reversedjet); there are obtained various results by analytical and numerical meansfor the well-known approximations of the Ciaplighin gas, in the cavitationalcase.

Other direct problems were studied by Simona Popp [19], [20] for thecompressible case, using the J-R-E model, for the dihedral angle or for theflat plate in channel. We mention also [11], where the J-R-E model is usedfor the case of a cone in tunnel, an axially-symmetrical problem, by means ofthe (p,q) analytic functions. Other methods were introduced by P.P Kufarevand V.A. Shtanko [6], when (BOB′) are arcs.

If we take in equations (28) and (29) F (a) = π ln(Q+1), then we obtainfrom (40):

C∗x(Q) = 2

1 + Qe

sinhF (ε)

πtan

ε

2.

In equation (29) we develop in the integral around ε and taking the firstapproximation, we obtain

sinhF (ε)

πtan

ε

2≈ 4

π, Q ∈ (0, 2).

In this case,

C∗x(Q1) =

8πe

(1 + Q). (41)


Therefore, this result (which is in a fact a first approximation) shows alitle influence on the optimal parachute C0

x, for small values of Q. Takinginto account other therms in the series, beeter approximations for Cx may beobtained. Our values are comparable with those reported in the previouslystudies, or with the classical ones [1], [6], [15].

In Table 1 there are presented the results of our computations concerningC∗

x(Q), C∗x(Q1), Cp

x(Q), for Q ∈ [0, 10], by finding the corresponding root a∗

of (28). In Fig. 3, using the formulae (37), there are plotted the shapes of theoptimum airfoil for Q = 0.1, 0.5, 1, 1.5, 2, 4, with V 0 = 1. We remark thatfor small values of Q ≤ 2, these airfoils are close to the optimum parachuteairfoil [14].

Fig. 3. The plot of optimal airfoil OB.

By inspecting the expression of V from (30) and SL from (39), inthe neighbourhood of the point B (where the free line (BC) separates), we


Q a∗ C∗x C∗

x1 Cpx

0.06 -0.9990 0.9932 0.9930 0.93260.1 -0.9972 1.0358 1.0305 0.96780.5 -0.9501 1.4080 1.4052 1.31971 -0.8577 1.8801 1.8736 1.7596

1.5 -0.7588 2.3560 2.3420 2.19952 -0.6636 2.8340 2.8104 2.63944 -0.3537 4.7621 4.6840 4.39906 -0.1375 6.7063 6.5576 6.158610 0.1369 10.6305 10.3048 9.6778

Table 1.The geometrical and aerodynamical optimal parameters.

remark the jump due to limξa V (ξ) = V 0

e , limξa V (ξ) = V 0. Practically,this jump occurs because the shape of the airfoil around the point B appearsto be a infinitesimal spyral, reported in Tulin – type flows, [15], [20]. Thesame conclusions are obtained by using polar coordinates limξ→a θ(ξ) = −∞and the curvature jump K(ξ = a). This infinitesimal spyral at B and B′

don’t perturb the drag coefficient, hence they may be eliminated.A comparison of the Helmholtz model in the flat plate case and the

optimum cases of the parachute with or without cavity flow reveals that

Cpx =

2π

4 + π< C∗

x(0) =8πe

< Cpx(Q) =

8πe

(1 + Q)

For modern problems, we remark the results reported by Maklakov [15]for cavity liquid flows around airfoils and the studies by B. Nicolescu [18]dealing with flows with ”cavity bubbles” also around airfoils.

5 Final conclusions

We obtained in the present paper the correct analytical solution for theoptimum airfoil in the case of the cavity J-R-E model. Using intensivelynumerical techniques, we prove that for small values of the cavity numberQ in the range 0...2, the airfoil design is close to the case without cavitation[14]. But, when Q raises, the drag coefficient Cx also raises; in all thesesituations, the drag experienced by the curvilinear airfoil OB is greater asthe drag of the flat plate normal to the flow or when BOB′ is a dihedralangle.


The present results may be useful as a first approximation for the director inverse problems in the case of compressible or axially-symmterical cavityflows. Practically, our results are very useful for the blades of the shippropellers, wind turbines and lifting surfaces in aviation.

References

[1] Anton, I., The cavitation, vol. I,II, Romanian Academy PublishingHouse, Bucharest, 1984 (Romanian).

[2] Camenschi, Galina, On the cavity flows of the compressible fluid aboutof the flat plate, Studii Cerc. Math. Acad. Romana, 19(1967), no. 7, pp.957-962 (Romanian).

[3] Carabineanu, A., Bena, D., The method of conformal mapping forneighbouring domains, Ed. Acad. Romane, Bucuresti, 1993 (Roma-nian).

[4] Dragos, L., Fluid Mechanics, Romanian Academy Publishing House,Bucarest, 1999 (Romanian).

[5] A. M. Elizarov, N. B. Il’inskiy, A. V. Potashev, Inverse boundary valueproblems of aerohydrodynamics, Ed. Fiz-Mat, Moskva, 1994 (Russian).

[6] Gurevici, I. M., Theory of jets in ideal fluids, Academic Press, New-York, 1965.

[7] Hardy, G.H., Littlewood, J.E., Polya, G., Inequalities, Cambridge Univ.Press, 1959.

[8] Iacob, C., Introduction mathematique a la mecanique des fluides, Ed.Gauthier-Villars. Bucharest-Paris, 1959.

[9] Lupu, M., Obtaining and studying Beltrami type equations for axiallysymmetric MHD jets, in V. Barbu (editor), Differential equations andcontrol theory, Ed. Longman, London, 1991, pp. 171-182.

[10] Lupu, M., Scheiber, E., A study on some inverse boundary problems,St. Cerc. Math, 49 (1997), pp. 197-209.

[11] Lupu, M., Generalisation and study of the Jukovski - Roshko - Epplermodel in the case of axial-symmetrical jet flows, using (p−q) analyticalfunctions, Studii Cerc. Mecanica Aplicata, no. 3-4, 1991, pp. 167-183(Romanian).

[12] Lupu, M., Scheiber,E., Optimal design in jet aerodynamics in case of aninverse problem through analytical and numerical methods, An. Univ.Bucuresti, Math., 48 (1999), pp. 131-142.

[13] Lupu, M., Scheiber,E., Analytical methods for airfoils optimization inthe case of nonlinear problems in jet aerodynamics. Revue Roumainedes Math. Pures et Appliquees. no. 5-6, 2000.


[14] Lupu, M., Scheiber, E., Postelnicu, A., Optimal airfoil for symmetricalHelmholtz model in aerodynamics. Proceedings of the Satellite Confer-ence of 3d ECM, Symmetry and Antisymmetry, Transilvania Universityof Brasov, July 2000, pp. 185-198.

[15] Maklakov, D.V., Nonlinear problems in hydrodynamics, Ed. Yanus K.,Moskva, 1997 (Russian).

[16] Marcov, N., Carabineanu, A., The study of the fluid flows with freelines by nonlinear integral equations, St. Cerc. Math. (MathematicalReports), Romanian Academy, 1(51), no. 3, pp. 359-377.

[17] C. Meghea, I. Meghea, Differential and integral calculus Part 2, Tech-nical Publishing House, Bucharest, 2000 (Romanian).

[18] Nicolescu, B.N., Georgescu, A., Popa, N., Bolosteanu, B., Cavity flows -models and solutions, Applied and Industrial Mathematics Series, Uni-versity of Pitesti Press, 1999 (Romanian).

[19] Popp, Simona., Theoretical investigations and numerical evaluations ofwall effects in cavity flows, Symposium of Cavity Flows, Minnesota,USA, (1975), pp. 119-129, Trans ASME, Journal of Fluids Eng.

[20] Popp, Simona, Mathematical models in cavity flow theory, TechnicalPublishing House, Bucharest, 1985 (Romanian).

[21] A. P. Prudnikov, Y. A. Bricikov, O. I. Marishev, Integrals and series,Ed. Nauka, Moskva, 1981 (Rusian).

[22] Y. V. Sungurtsev, Plane flows of jets gas, Ed. M.G.U., Moskva (Ru-sian).

Mircea Lupu, Department of Mathematics, Transilvania University of Brasov,2200 Brasov, Romania, [email protected]

Adrian Postelnicu, Department of Thermo and Fluid Mechanics, Transilva-nia University of Brasov, 2200 Brasov, Romania, [email protected]

Ernest Scheiber, Department of Informatics, Transilvania University of Brasov,2200 Brasov, Romania, [email protected]


On a Boltzmann Model of Fermions

Dorin MARINESCU

November 2, 2001

Abstract - We consider a space homogeneuos model of fermions described by a nonlinear

Boltzmann type equation with unbounded collision terms. We prove the existence of a

global solution for the Cauchy problem, we validate the macroscopic conservation laws

and we obtain a H-theorem.

Key words and phrases : kinetic theory, Boltzmann equation, nonlinear equations

Mathematics Subject Classification (2000) : 47H10, 45G10, 47J35

1 Introduction

The statistical evolution of a rarefied gas of fermions can be described inthe frame of the kinetic theory by a modified Boltzmann equation [6]. Thelast years, it was an increased interest in studying the mathematical prop-erties of Boltzmann-type equations for systems of fermions. We mentionthe contributions of Hugenholtz [5], Landau and Ho [4] in the derivation ofBoltzmann-type equation for systems of fermions and the contributions ofGolse and Poupaud [3] in studying fluid limits for the Fermi-Dirac statistic.

In this paper we present the results of a study concerning qualitativeproperties of the solution for a Boltzmann-type equation describing the evo-lution of a space homogeneuos model of fermions.

To the end of this Section we introduce the kinetic equation for fermionsand we make some general assumptions for the kernel of the collision opera-tor. Section 2 is dedicated to the bounded case (for the collision kernel). Weinvestigate the global existence and uniqueness of the solution, the macro-scopic conservation laws and the H-Theorem. In Section 3 we extend the re-sults to the unbounded case by adapting Arkeryd’s argument to the classicalBoltzmann equation [1], [2], i.e. we prove the existence of a global solution(no uniqueness), the macroscopic conservation laws and the H-Theorem.

One considers the Cauchy problem for the following nonlinear kineticequation.

141

142 D. MARINESCU

∂f(t,−→ξ )

∂t= Q(f)(t,

−→ξ ), f(0, ·) = f0, 0 ≤ f0 ≤ 1. (1)

The unknown f : [0,∞) × R3 → [0, 1] (depending on the time – t and

velocity –−→ξ ) is the one particle distribution function. The collision operator

Q has the form

Q(f) := P (f)− S (f) , (2)

where the ”gain” operator P is defined by

P (g)(−→ξ ) :=

∫

R3×Sd−→ξ ∗d−→ω B(

−→ξ ,−→ξ ∗,−→ω )g′g′∗(1− g)(1− g∗), (3)

and the ”loss” operator S(g) := R(g) · g, with R defined as follows

R(g)(−→ξ ) :=

∫

R3×Sd−→ξ ∗d−→ω B(

−→ξ ,−→ξ ∗,−→ω )g∗(1− g′)(1− g′∗). (4)

In (3) and (4), S designates the unit sphere in R3 and g∗ = g(−→ξ ∗), g′ =

g(−→ξ ′), g′∗ = g(

−→ξ ′∗). The variables

−→ξ ,−→ξ ∗ are the pre-collision velocities and

−→ξ ′ =

−→ξ ′(

−→ξ ,−→ξ ∗,−→ω ),

−→ξ ′∗ =

−→ξ ′∗(

−→ξ ,−→ξ ∗,−→ω ) represent the post-collision

velocities and they satisfy the following properties.

−→ξ +

−→ξ ∗ =

−→ξ ′ +

−→ξ ′∗, (5)

−→ξ 2 +

−→ξ 2∗ =

−→ξ ′2 +

−→ξ ′2∗ . (6)

For each fixed −→ω ∈ S, the Jacobian of the map

(−→ξ ,−→ξ ∗) → (

−→ξ ′−→, ξ ′∗) (7)

equals to one and the kernel B : R3×R3×S→ R+ is a measurable functionsuch that

B(−→ξ ,−→ξ ∗,−→ω ) = B(

−→ξ ∗,

−→ξ ,−→ω ), (8)

B(−→ξ ,−→ξ ∗,−→ω ) = B(

−→ξ ′,

−→ξ ′∗,−→ω ). (9)

On a Boltzmann Model of Fermions 143

2 The Bounded Case

Consider Cb(R3) the space of real continuous and bounded functions on R3

with the norm ‖g‖∞ := sup−→ξ ∈R3

∣∣∣∣g(−→ξ )

∣∣∣∣.We suppose that the kernel B fulfills the following assumption.

Assumption 1:

∫

R3×Sd−→ξ ∗d−→ω B(·,−→ξ ∗,−→ω ) ∈ Cb(R3). (10)

An immediate consequence of (10) is that the operators P : Cb(R3) →Cb(R3) and R : Cb(R3) → Cb(R) given by (3), (4) are well defined Lipshitzoperators.

Let T > 0 and

C(0, T ;Cb(R3)) :=

f : [0, T ]× R3 → R3 [0, T ] 3 t → f(t, ·) ∈ Cb

(R3

)continous

(11)

be endowed with the norm

‖f‖ := supt∈[0,T ]

‖f(t)‖∞ . (12)

LetCb(R3) := g ∈ Cb(R3) | 0 ≤ g ≤ 1, (13)

and

BT := f ∈ C(0, T ;Cb

(R3

)) | f(t, ·) ∈ Cb

(R3

), (∀) t ∈ [0, T ]. (14)

The Cauchy problem in Cb(R3) associated to (1) can be formulated as fol-lows:

dtf(t) = P (f(t))−R(f(t)) · f(t), f(0) = f0. (15)

For each ν ≥ 0, f ∈ C(0, T ; Cb(R3)) and f0 ∈ Cb(R3) we consider theapplications

[0, T ] 3 t → Iν (f0, f) (t) ∈ Cb(R3), (16)

144 D. MARINESCU

Iν (f0, f) (t) := exp

−

t∫

0

[R(f(τ)) + ν]dτ

· f0+

+

t∫

0

exp

−

t∫

s

[R(f(τ)) + ν]dτ

· P (f(s))ds.

(17)

Note that, the problem (15) is equivalent to the following equation

Iν (f0, f) = f, (18)

inBT (f0) :=

f ∈ C(0, T ; Cb(R3)) |f(0) = f0, 0 ≤ f0 ≤ 1

, (19)

for f0 ∈ Cb(R3) and ν = 0.We have the following estimation.

Lemma 1. For all µ, ν ≥ 0 and T > 0 , if f0, g0 ∈ Cb(R3) and f ∈ BT (f0),g ∈ BT (f0), there exists a constant a C > 0 (independent of µ, ν, T , f0,g0, f and g) such that

‖Iν (f0, f)− Iµ(g0, g)‖ ≤ CT (|ν − µ|+ ‖f − g‖) + ‖f0 − g0‖∞ . (20)

Using (20) and Banach fixed point Theorem, first, we can prove theglobal existence and uniqueness of the solution of (18) when ν > 0.

Proposition 1. For each T, ν > 0 and f0 ∈ Cb

(R3

), (18) has a unique

solution in BT (f0) on the interval [0, T ].

Applying Proposition 1, by passing to limit, ν −→ 0, we obtain thefollowing existence result.

Theorem 1. For each T > 0 and f0 ∈ Cb

(R3

)the equation (15) has an

unique solution in BT (f0).

Obviously, if f is a solution of (17) in BT (f0), f also satisfies the equation

f(t) = f0 +∫ t

0Q(f)(s)ds, (21)

as well as


f(t) = f0 +∫ t

0Q(f)(s)ds, (22)

where f := f if 0 ≤ f ≤ 1 and f := 0 otherwise.Consider C(0, T ;L1(R3)) := g : [0, T ] → L1(R3)|0 ≤ g(t) ≤ 1 ∀t ∈

[0, T ], endowed with the norm

‖g‖1 := supt∈[0,T ]

‖g(t)‖L1 , (23)

and defineST (f0) := BT (f0) ∩ C(0, T ;L1(R3)). (24)

Using Theorem 1 we can also prove the following existence result.

Theorem 2. If f0 ∈ Cb

(R3

) ∩ L1(R3), for each T > 0, the equation (22)has an unique solution in ST (f0).

LetΦ(−→ξ ) := 1 +

−→ξ 2. (25)

The following conservation laws of the mass, momentum and energy arefulfilled.

Theorem 3. Let f be the solution of (21) and Φf0 ∈ L1(R3).Then,

∫

R3

f(t,−→ξ )d

−→ξ =

∫

R3

f0(−→ξ )d

−→ξ , (26)

∫

R3

−→ξ f(t,

−→ξ )d

−→ξ =

∫

R3

−→ξ f0(

−→ξ )d

−→ξ , (27)

∫

R3

−→ξ 2f(t,

−→ξ )d

−→ξ =

∫

R3

−→ξ 2f0(

−→ξ )d

−→ξ . (28)

In order to prove the H-Theorem we must show the following properties.

Lemma 2. If Φg ∈ L1(R3), then

h(g) := g ln(g) + (1− g) ln(1− g) (29)

belongs to L1(R3).

146 D. MARINESCU

Lemma 3. If Φf0 ∈ L1(R3), if f is the solution of Theorem 2 and if

q(f) :=∫

R3×Sd−→ξ ∗d−→ω B ln

(f ′f ′∗(1− f)(1− f∗)ff∗(1− f ′)(1− f ′∗)

)×

×[f ′f ′∗(1− f)(1− f∗)− ff∗(1− f ′)(1− f ′∗)],

(30)

then,h(f) ∈ C(0, T ;L1(R3)), (31)

q(f) ∈ L1([0, T ]× R3), (32)

and ∫

R3

d−→ξ h(f(t)) +

∫ t

0dτ

∫

R3

d−→ξ q(f(τ)) ≤

∫

R3

d−→ξ h(f0). (33)

If Φf0 ∈ L1(R3), then for f , the solution of Theorem 2 the functional

H(f)(t) :=∫

R3

h(f)(t)d−→ξ (34)

is well defined and from the inequality (33) one obtains immediately thefollowing H-Theorem.

Theorem 4 (H-Theorem) If f is the solution of Theorem 2 and Φf0 ∈L1(R3), then H(f)(t) is decreasing in time.

3 The Unbounded Case

We replace the hypothesis (10) for the collision kernel B by the followingmore general assumption.

Assumption 2:

B(−→ξ ,−→ξ ∗,−→ω ) ≤ C(1 +

−→ξ 2 +

−→ξ 2∗)λ,

(∀)(−→ξ ,−→ξ ∗,−→ω ) ∈ R3 × R3 × S, 0 ≤ λ < 1,

(35)

for C > 0.Let χn = χ−→ξ 2+

−→ξ 2∗≤n2 be the characteristic function of the ball of radius

n in R3 × R3 and let

Bn(−→ξ ,−→ξ ∗,−→ω ) = B(

−→ξ ,−→ξ ∗,−→ω )χn(

−→ξ ,−→ξ ∗). (36)


Let f0 ∈ L1(R3) ∩ Cb(R3), such that 0 ≤ f0 ≤ 1 and Φf0 ∈ L1(R3). Let(fn)∞n=1 be the sequence of solutions given by Theorem 2 corresponding toBn.

We observe that the sequence (fn)∞n=1 satisfies the hypotheses in Lemma5. Therefore, there exists a weakly convergent subsequence (fnj )∞j=1 inL1(R3) to some f, which satisfies (40). Denote also by (fn)∞n=1 this sub-sequence.

Under Assumption 2, we can prove:

Lemma 4. If

limn→∞

∫R3

φ(−→ξ )fnd

−→ξ =

∫R3

φ(−→ξ )fd

−→ξ , (37)

for each φ, such that (1 +−→ξ k)φ ∈ L1(R3), for all 0 ≤ k < 2, then

Q(fn) Q(f).

This Lemma is a consequence of Arkeryd’s Lemma [2] (Lemma 3.1, pp.13).

Lemma 5 (Arkeryd) If (fn)∞n=1 is a sequence of nonegative functions inL1(R3) satisfying the properties

‖Φfn‖L1 < C, (38)

∫R3

fn ln(fn)d−→ξ < C, (39)

where C is some positive constant, then there exists a subsequence (fnj )∞j=1

which converges weakly in L1(R3) to some function f and

limj→∞

∫R3

φ(−→ξ )fnjd

−→ξ =

∫R3

φ(−→ξ )fd

−→ξ , (40)

for each φ such that (1 +−→ξ k)φ ∈ L1(R3), for all 0 ≤ k < 2.

Using the results obtained in the bounded case, by Lemma 4 we canprove the following Theorem for the unbounded case.

148 D. MARINESCU

Theorem 5. If B satisfies (35) and Φf0 ∈ L1(R3), then there exists asolution f of equation (21), such that 0 ≤ f ≤ 1 and the following propertiesare satisfied

∫

R3

f(t,−→ξ )d

−→ξ =

∫

R3

f0(−→ξ )d

−→ξ , (41)

∫

R3

−→ξ f(t,

−→ξ )d

−→ξ =

∫

R3

−→ξ f0(

−→ξ )d

−→ξ , (42)

∫

R3

−→ξ 2f(t,

−→ξ )d

−→ξ =

∫

R3

−→ξ 2f0(

−→ξ )d

−→ξ . (43)

In addition, the H functional (34) decreases in time.

References

[1] L. Arkeryd. On the Boltzmann equation I: Existence. Arch. Rat. Mech.Anal., 45(1972) 1-16.

[2] L. Arkeryd. On the Boltzmann equation II: The full initial value problem.Arch. Rat. Mech. Anal., 45(1972) 17-34.

[3] F. Golse and F. Poupaud. Limite fluide des equations de Boltzmanndes semi-conducteurs pour une statistique de Fermi-Dirac. AsymptoticAnal., 6, No.2 (1992) 135-160.

[4] T. G. Ho and L. J. Landau. Fermi gas on a lattice in the van Hove limit.J. Stat. Phys., 87, No.3-4 (1997) 821-845.

[5] N. M. Hugenholtz. Derivation of the Boltzmann-equation for a Fermi-gas. Commun. Math. Phys., 111, No. 2 (1984) 393-420.

[6] L. E. Reichi. A Modern Course in Statistical Physics. E. Arnold, 1980.

Dorin MarinescuInstitute of Applied Mathematics Romanian Academy,Calea 13 Septembrie, No. 13Sector 5, Bucharest,P.O. Box 1-24, 70700,E-mail: [email protected]


A Two Dimensional Mathematical Model forSimulating Water and Chemical Transport in an

Unsaturated Soil

Anca Marina MARINOV

November 2, 2001

Abstract - Modeling of transport phenomena is a major concern in subsurface aquatic

environments. The main processes to be studied and analyzed are advection, convection,

diffusion, hydrodynamic dispersion, sorption, complexation, ion exchange and degrada-

tion. Groundwater flow and transport in porous media are linked directly to pore space

geometry, fluid properties and disolved constituents. We propose a 2D, deterministic

model to describe the transport and transformations of matter in the soil, crop and va-

dose environment. The model uses the Darcian equation for water flow and a solute

transport equation for the pollutant movement. The equations are solved by an ADI

method.

Key words and phrases : groundwater flow, hydrodynamic dispersion, transport of

solute, numerical solution

Mathematics Subject Classification (2000) : 34A09, 35K15, 65H10, 65L80,76S05

1 Introduction

The movement of contaminants in unsaturated soil is an important hydro-logic problem. Rainfall drives contamination into the soil through the vadosezone which extends from the ground surface to the water table, and thenpast the water table to the groundwater zone in which the chemicals maybe transposted laterally for distances of thousands of meters.

It is useful to be able to estimate the travel time for solutes from theground surface to the water table, the rate at which leachate leaves theunsaturated zone to become groundwater contamination, and the values ofthe pollutant’s concentration in the soil profile.

Estimation of the net groundwater recharge rate requires that one per-form a water balance at the ground surface. The water balance will be a

149

150 Anca Marina Marinov

function of the characteristics of the soil profile and the climate.The objective of this study was to solve the two-dimensional Darcian

partial differential equations to obtain the infiltration component of thehydrologic cycle and to solve the two-dimensional equation for the transportof the chemical contaminant in the unsaturated soil.

We propose a solute transport model to predict the contamination of thesoil and of the groundwater.

2 Infiltration and Soil Moisture Flow System

The assumed watershed section is shown in Fig.1. The x-axis is taken downthe slope (α) and the z-axis is perpendicular to the slope.

The left-hand end, (AD), and the bottom of the section, (AB), are as-sumed to terminate at impermeable layers. For the right-hand end of thesection we assume different situations (a-impermeable, b-permeable, in con-nection with a stream chanel).

Let us suppose: Darcy’s law is valid; The soil is homogeneous andisotropic; During the infiltration process the hysteresis is negligible.

Substituting Darcy’s velocities in the continuity equation, the rate ofchange of moisture in a volumic element can be expressed as

∂θ

∂t=

∂

∂x

(K(θ)

∂φ

∂x

)+

∂

∂z

(K(θ)

∂φ

∂z

)(1)

where θ (cm3/cm3) is the volumetric moisture content, φ (cm) is the totalpotential and K (cm/h) is the hydraulic conductivity. Equation (1) is theequation of moisture flow through porous materials. The total potential isgiven by:

φ(x, z) = h(x, z) + (L− x) sinα + z cosα (2)

where h(cm) is the pressure head and L(cm) is the length of the section.

Defining the moisture capacity of the soil C1(h) =∂θ

∂hequation (1)

becomes

C1(h)∂h

∂t=

∂

∂x

(K(h) ·

(∂h

∂x− sinα

))+

∂

∂z

(K(h) ·

(∂h

∂z+ cosα

)).

(3)It was assumed initially that the system was at a static equilibrium (the

total potential φ, at every point of the flow system was zero).Up to the time that the soil surface becomes saturated, the velocity

vzi,M+1/2 = −RI · cosα (4)

A Two Dimensional Mathematical Model 151

where RI (cm/h) is the rainfall rate.After the soil surface is saturated, and before surface runoff begins, it

is assumed that water will accumulate on the surface up to a maximumvertical depth (ponding depth) HM (cm). (HM ∼ 1.27 cm). If HB is thedepth of water on the surface at any time; before HM is satisfied, we have:

dHB

dt= RI −RE, 0 < HB ≤ HM. (5)

RE (cm/h) is the vertical infiltration rate.Equation (5) is solved for HB which is taken as the pressure head h on

the surface after surface saturation and before surface runoff begins.When the surface runoff starts, the vertical depth of water on the surface,

which is made up of HM plus the runoff depth, is used to define the upperboundary condition.

The other boundary conditions are as follows: vxN,j = 0 (BC−impermeable);vx1,j = 0 (AD − impermeable); vzi,1 = 0 (AB − impermeable). If BC isa boundary between the soil and a river, a pressure condition have to putthere (φ= constant).

3 The ADI Procedure

The ADI method of numerical analysis is an implicit scheme which breaksa system of differential equations into two subsystems. One subsystem isimplicit in one direction and explicit in the other, and the second subsysteminterchanges the order of the implicit-explicit directions. The two subsys-tems are solved alternately, the solution being complete for each time stepafter the second subsystem has been solved.

The integration of equation (3) has been done by an ADI scheme [2].The partial derivatives were replaced by:

∂h

∂t=

hn+1i,j − hn

i,j

∆t(6)

∂

∂x

[K

(∂h

∂x− sinα

)]n+1

=Kn

i,j + Kni+1,j

2∆x

[hn+1

i+1,j − hn+1i,j

∆x− sinα

]−

−Kni−1,j + Kn

i,j

2∆x

[hn+1

i,j − hn+1i−1,j

∆x− sinα

](7)

∂

∂z

[K

(∂h

∂z+ cosα

)]n

=Kn

i,j + Kni,j+1

2∆z

[hn

i,j+1 − hni,j

∆z+ cos α

]−


−Kni,j + Kn

i,j−1

2∆z

[hn

i,j − hn+1i,j−1

∆z+ cosα

](8)

∂

∂z

[K

(∂h

∂z+ cosα

)]n+2

=Kn+1

i,j + Kn+1i,j+1

2∆z

[hn+2

i,j+1 − hn+2i,j

∆z+ cos α

]−

−Kn+1i,j + Kn+1

i,j−1

2∆z

[hn+2

i,j − hn+2i,j−1

∆z+ cosα

](9)

where n represents a calculated value at time (n).Therefore, equation (3) becomes:

hn+1i−1,j

Kni,j + Kn

i−1,j

2∆x2+ hn+1

i,j

[−Kn

i−1,j + 2Kni,j + Kn

i+1,j

2∆x2− C1n

i,j

∆t

]+

+hn+1i+1,j

Kni,j + Kn

i+1,j

2∆x2= hn

i,j−1

(−Kni,j −Kn

i,j−1

2∆z2

)+

+hni,j

[Kn

i,j−1 + 2Kni,j + Kn

i,j+1

2∆x2− C1n

i,j

∆t

]+ hn

i,j+1

−Kni,j −Kn

i,j+1

2∆x2+

+Kn

i+1,j −Kni−1,j

2∆xsinα− Kn

i,j+1 −Kni,j−1

2∆zcosα (10)

at time-moment (n + 1), and

hn+2i,j−1

Kn+1i,j−1 −Kn+1

i,j

2∆z2+ hn+2

i,j

[−Kn+1

i,j−1 + 2Kn+1i,j + Kn+1

i,j+1

2∆z2+

C1n+1i,j

∆t

]+

+hn+2i,j+1

(−Kn+1

i,j −Kn+1i,j+1

2∆z2

)= hn+1

i−1,j

Kn+1i−1,j + Kn+1

i,j

2∆x2+

+hn+1i,j

[−Kn+1

i−1,j − 2Kn+1i,j −Kn+1

i+1,j

2∆x2+

C1n+1i,j

∆t

]+ hn+1

i+1,j

Kn+1i,j + Kn+1

i+1,j

2∆x2+

+Kn+1

i−1,j −Kn+1i+1,j

2∆xsinα +

Kn+1i,j+1 + Kn+1

i,j−1

2∆zcosα (11)

at time-moment (n + 2), respectively.


4 The Solute Transport Equation

The transport of solute in a mobile soil region is determined by chemicaldiffusion, convection and hydrodynamic dispersion.

The solute transport equation is defined in a macroscopic way, indicatingthat the state variables (solute concentration) and material characteristics(soil transport volume) are defined as averages over a Representative Ele-mentary Volume [REV], [1].

We have been assumed there are no immobile or stagnant soil waterregions. For the mobile soil region the connection-dispersion equation holds.

The continuity equation for the solute in the soil mobile region, for ansmall volume of soil is written as:

∂(θ · C)∂t

=∂

∂x

(θDx

∂C

∂x

)+

∂

∂z

(θDz

∂C

∂z

)− ∂(vxC)

∂x− ∂(vzC)

∂z(12)

where C (mg/cm3) is the solute concentration, in the mobile region, θ(cm3/cm3)is the mobile soil water content, vx, vz (cm/h) are the Darcian water veloc-ities.

vx · C (mg/cm2h) - is the convective solute flux density imposed bythe convective water flow, in x direction, Dx, Dz (cm2/h) are the apparentdiffusion coefficients in the mobile soil region.

Dx = De + Dm = Dex + λ|Vmx| (13)

where Vmx = Vx/θ is the average macroscopic pore water velocity.De (cm2/h) is an effective diffusion coeffcient, and λ(cm) is the soil solute

dispersivity.

De =Dif · a · eb·θ

θ(14)

Dif is the diffusion coefficient in pure water, a and b are the model’sparameters [3].

5 Numerical Solution of the Solute Transport Equation

In order to solve the equation (12) by finite difference technique, the sametime and space discretisation as for the water flow equation is adopted (TheADI method). For the (n + 1) − th time step, the equation (12) has been


approximated by:

θn+1i,j · Cn+1

i,j − θni,j · Cn

i,j

∆t=

1∆x

(θDx

∂C

∂x

)n+1

i+1/2,j

−(

θDx∂C

∂x

)n+1

i−1/2,j

−

− 1∆x

vn+1x (i + 1, j)

[en+1

1 (i + 1, j) · Cn+1(i, j) + en+12 (i + 1, j)Cn+1(i + 1, j)

] −− vn+1

x (i, j)[en+1

1 (i, j) · Cn+1(i− 1, j) + en+12 (i, j)Cn+1(i, j)

]+

+1

∆z

(θDz

∂C

∂z

)n

i,j+1/2

−(

θDz∂C

∂z

)n

i,j−1/2

−

− 1∆z

vnz (i, j + 1) [en

3 (i, j + 1) · Cn(i, j) + en4 (i, j + 1) · Cn(i, j + 1)] −

−vnz (i, j) · [en

3 (i, j)Cn(i, j − 1) + en4 (i, j)Cn(i, j)] . (15)

vx(i, j), vz(i, j) denote the entrance velocities in the i-th compartment,and

vx(i + 1, j), vz(i, j + 1) the exit velocities.If the flux in the i-th compartment is positive, then e1 = 1 and e2 = 0, if

this flux is negative e1 = 0, e2 = 1; if vz(i, j) > 0, e3(i, j) = 1, e4(i, j) = 1;if vz(i, j) < 0, e2(i, j) = 0, e4(i, j) = 1.

For the (n + 2)th-time step:

θn+2i,j · Cn+2

i,j − θn+1i,j · Cn+1

i,j

∆t=

1∆x

(θDx

∂C

∂x

)n+1

i+1/2,j

−(

θDx∂C

∂x

)n+1

i−1/2,j

−

− 1∆x

vn+1x (i + 1, j)

[en+1

1 (i + 1, j)Cn+1(i, j) + en+12 (i + 1, j)Cn+1(i + 1, j)

] −− vn+1

x (i, j)[en+1

1 (i, j) · Cn+1(i− 1, j) + en+12 (i, j)Cn+1(i, j)

]+

+1

∆z

(θDz

∂C

∂z

)n+2

i,j+1/2

−(

θDz∂C

∂z

)n+2

i,j−1/2

−

− 1∆z

vn+2z (i, j + 1)

[en+2

3 (i, j + 1)Cn+2(i, j) + en+24 (i, j + 1)Cn+2(i, j + 1)

] −−vn+2

z (i, j)[en+2

3 (i, j)Cn+2(i, j − 1) + en+24 (i, j)Cn+2(i, j)

]. (16)

To solve the equations (15) and (16) initial C0(i, j) conditions for con-centration a necessary.

The solute concentration at the soil surface (j = M +1/2) is set equal toCb(i). The upper boundary condition will be: the flux through the surface


of the soil is known. Thus:

∂

∂z(vz · C)i,j=M =

1∆z

vz(i,M + 1/2)[e3 · C(i,M) + e4 · C(i,M + 1/2)]−

−vz(i,M − 1/2)[e3 · C(i,M − 1) + e4 · C(i,M)] (17)

withC(i,M + 1/2) = Cb(i) (18)

and(vz)i,j=M+1/2 = RI · cosα; (19)

The lower impermeable boundary condition is:

(vz)i,j=1−1/2 = 0, (20)

Thus (∂

∂z(vz · C)

)

i,j=1−1/2

= 0, (21)

and∂

∂z

(θ ·Dz · ∂C

∂z

)

i,1

=(

θ ·Dz · ∂C

∂z

)

i,1+1/2

− 0 (22)

For the left wall (impermeable) (i = 1), (vx)1,j = 0. (23)

Thus∂

∂x(vx · C)1,j = 0, (24)

and∂

∂x

(θ ·Dx · ∂C

∂x

)

1,j

=(

θ ·Dx · ∂C

∂x

)

1+1/2,j

− 0. (25)

For the right wall (impermeable) (i = N), (vx)N,j = 0 (26)

∂

∂x(vx · C)N,j = 0, (27)

and∂

∂x

(θ ·Dx · ∂C

∂x

)

N,j

= 0−(

θ ·Dx · ∂C

∂x

)

N−1/2,j

(28)


6 Application

We have applied the model for a bidimensional sandy profile of soil (fig.1)with:

L = 2000cm, H = 100cm, N = 20, M = 20, dx = 100cm, dz = 5cm,dt = 0.1h, α = π/50, RI = 0.5cm/h, Ks = 2.08cm/h, θs = 0.364,

θ(h) = 0.098 +888.39

3339.83 + |h|1.674

K(θ) = 1.03 · 10−6e39.88θ

h0 = −((L − x sinα) + z cosα), Dex = 0.08cm2/h, Dez = 0.08cm2/h, λ =16.5cm, C0(i, j) = 0.01 mg/l, Cb(i, 1) = 200 mg/l for i = 5 : 15.


Some results obtained after 4 hours of continuous rain have been pre-sented in the figures 2-4.

References

[1] De Marsily, G., 1986. Quantitative hydrogeology. Groundwaterhydrology for engineers. Academic Press Inc.

[2] Marinov, A.M., 1992. Theoretical and experimental studies onunsaturated soil pollution. Mecanique Appliquee, Nr.2, Tome 37,p.213-220, Ed. Academiei Romane.


[3] Vanclooster, M., Viaene, P., Diels, J., Christiaens, K., 1994. Wave. Amathematical model for simulating water and agrochemicals in thesoil and vadose environment.

Anca Marina MarinovInstitute of Applied Mathematics ,, Caius Iacob”Calea 13 Septembrie, No. 13, Sector 5Bucharest, P.O. Box 1-24, 70700E-mail: [email protected]


A Numerical Analisys of Laminar TransportProcesses in Ducts with Cross-Sectional

Periodicity

Alexandru M. MOREGA

November 2, 2001

Abstract - This paper reports a numerical study on laminar, stationary flows in wavy-

walled tubes. Flows in the range Re = 10 − 160 are investigated, and their structure

and main features are presented through streamlines, axial velocity and pressure profiles.

Experimental evidence suggests that within this range of Reynolds number the flow is

axial symmetric, and almost periodic. However, periodicity is progressively lost for in-

creasingly Re flows. The numerical results are in good agreement with experimental data

- visualizations by the aluminum dust method and measured velocity profiles.

Key words and phrases : laminar flow, wavy-walled tube, finit element method

Mathematics Subject Classification (2000) : 65N30, 76D05

1 Introduction

Enhancement of heat and mass transfer under laminar, stationary and pul-sating flow conditions is a mater of great interest in many areas of medicaland biochemical engineering.

These flows are needed for the evaluation of heat and mass transfer inthe inertial and unsteady regions. Numerous experimental and numericalworks are devoted to the investigation of wavy-walled channels and tubesutilized in physiological and biomedical applications [9], flows in furrowedmembrane artificial lung [17], flows in arterial prostheses. Periodically con-stricted tubes are used also to investigate the flow of blood in vessels andits impact in occlusive vascular diseases, Chow and Soda [4]. The computedflow fields can also be relevant in the study of transport of Oxygen - Sobey[17] and stenoses-causing lipid particles. In particular, it is speculated thatthe trapping of these impurities in the separation region precedes the for-mation of a plaque of atheroma.

159

160 Alex. M. MOREGA

Belhouse et al. [1] pioneered the study of mass transfer under largeflow oscillations in a furrowed channel with application to high efficiencymembrane oxygenators. The numerical and experimental work of Patera[14] and Greiner [8] is devoted to the resonant heat transfer enhancementgrooved channels. Nishimura et al. [11, 12, 13] addressed pulsatile flows ingrooved channels and wavy-walled channels and tubes. Stone and Vanka[18] conducted a numerical study of developing flow and heat transfer in awavy passage. Analytic solutions [2, 3, 5, 4] and numerical [15] solutions forwavy-walled tubes are available, but for either a limited range of parameters,or for periodic flows.

The numerical studies in this area utilize either the vorticity-streamfunctionequations - which result after taking the curl of the Navier-Stokes equations- or the primitive variables formulation. More sophisticated approachesconsider coordinates transformation to solve for the irregular (wavy) walls.Either finite differences or finite element schemes are then used to integratethe models. In the attempt to reduce the computational effort, numerousnumerical models take advantage of the periodical nature of the geometry(tube) and assume periodicity (flow) boundary conditions. In this paper wewill evidence that this assumption may not be accurate for higher Reynoldsflows - flow periodicity sets in after a number of periods and in these casesit is more appropriate to consider the entire tube for the flow solution.

In this paper we report the numerical results of a FEM study on flowsin sinusoidal, wavy-walled tubes, for stationary, laminar regimes within theRe = 10− 160 flow range. The wavy-walled tube (Fig. 1) has the followingdimensions λ = 14mm, 2a = 3.5mm,Dmax = 10mm,Dmin = 3mm, whichcorrespond to the experimental setup described in [10].

Fig.1 The wavy-walled tube

2 The numerical model and its solution

The fluid is Newtonian and its flow is assumed incompressible. The inletand outlet sections are extended such that the inlet flow to the tube reachesthe Hagen-Poisseuille regime, and the exit section of the tube is not affectedby the outflow. The governing equation is the momentum (Navier-Stokes)

Laminar Transport Processes in Ducts 161

law

ρ

[∂u∂t

+ (u · ∇) · u]

= −∇p + µ∇2u,

and the mass continuity law

∇ · u = 0.

Here ρ is the mass density, µ is the dynamic viscosity, p is the pressure field,and u is the velocity field. Throughout the numerical study, we used thenondimensional form of the flow equations,

∂˜u

∂˜t

+(

˜u · ˜

∇)· ˜u= −∇ ˜

p +1

Re

˜∇

2˜u,

˜∇ · ˜

u= 0.

The space variables in ∇ are (r, z) = (r,z)Dmax

, u = uU0

, p = pρU2

0, t = t

Dmax/U0U0

is the inlet velocity (uniform profile), and Re = U0Dmaxµ/ρ is the Reynolds

number of the flow.

In general, the wavy-walled tube flow is a 3D problem. However, withinthe stationary laminar regime limits Re < 160), experimental evidence sug-gests that the flow is actually axial symmetric. Consequently, a simpler 2Daxial-symmetric model may be used in lieu of the full flow problem.

A zero, uniform pressure boundary condition was set for the tube exit.The inlet velocity profile is axial and uniform (uz = 1, ur = 0). No-slipconditions are specified for the tube wall (uz = ur = 0), and the axialboundary has symmetry conditions (∂uz

∂r = ur = 0).

For the solution phase we used the finite element (FEM) package FIDAP[6]. The flow problem was solved by Galerkin method in the primitive vari-ables (ur, uz, p) formulation, as implemented by FIDAP. The models werecreated by GAMBIT [7], starting from a number of fixed points and by usingNURBS constructals. Figure 3 shows details of the medium mesh (43307quadrilateral elements) that was used throughout the numerical study.

162 Alex. M. MOREGA

Figure 2. FIDAP FEM structured mesh (medium)

Accuracy tests were conducted for structured meshes of different sizes (22392,43307 and 63767 quadrilateral elements). In all cases we used the successivesubstitution method to solve the set of nonlinear flow equations, with anattenuation factor of 0.5 for all degrees of freedom. The pressure model wasdiscontinuous, with a 10−6 penalty factor. With this strategy, 6 − 11 iter-ations were needed to reach the stationary steady state (10−2 tolerance forthe velocity components and residual). Figure 3 shows the normalized axialvelocity in the 10th enlargement, for Re = 146 (experimental data from [10]).

Figure 3. Accuracy tests


3 Results and discussion

The first flow reported here is at Re = 10. Figure 4 shows the flow patterns(in the 9th hollow) through the streamlines for this case and – for compari-son – for Re = 52, 158 (documented later). Flow separation occurs near thepoint of greatest rate of tube divergence [15], at constrictions.

Re =10 Re =52 Re =158

Figure 4. Flow structure -19 streamlines

The separation region grows fast with Re, but then slower as the flowfills the hollow. Also, with increasing Re, the streamlines in the main, axialflow, are straighter and straighter, aligned with the axis.

3th cross-section 5th cross-section 9th cross-section

Figure 5. Flow structure Re=52

Figure 5 (19 streamlines) depicts the flow at Re = 52. Flow reversal inboth Stokes and viscous flows can occur when point sources of momentum,or other singularities (e.g. sharp corners or small enough included angle),are placed near a plane boundary. In the wavy-walled tube flow such sin-gularities are neither on the boundary nor within the flow. However, Stokesseparation that is evidenced in the wavy-walled tube flow exhibits similari-ties with both theoretical mechanisms: a certain minimum wall curvature isnecessary, and a source of momentum is present in the form of the drivingflux [15].

Figure 6 shows the streamlines structure for the flow at Re = 79. Al-though periodicity is preserved throughout the largest part of the tube, itsets in later, after four hollows. Stokes recirculation fills the hollow, and itscenter is shifted toward the downstream converging section.

164 Alex. M. MOREGA



The last case presented here is the flow for Re = 158 (Fig. 7). Stokesrecirculation is similar to that of Re = 79, however its center is even moreshifted towards the outlet constriction of the hollow. The detachment andreattachment points (rings) that delimit the core and secondary flows areclearly distinguishable.



Next, we investigated flow periodicity, as evidenced by the pressure dropcurve, axial velocity axial profile and axial velocity profiles in the largestand smallest cross sections. The axial velocity profiles at the enlargementsfor Re = 52, 79, 158 are presented in Fig. 8.

Re =52 Re =79 Re =158

Figure 8. Axial velocity at Dmax.

The velocity profile at constriction in the periodic flow section is nearly


parabolic for low Re numbers and it flattens at higher Re numbers. Ap-parently, periodicity sets in after 3, 5 and respectively 7 hollows. The samebehavior is distinguishable from the axial velocity profiles at the constric-tions (Fig. 9).

Re =52 Re =79 Re =158

Figure 9. Axial velocity at Dmin.

Figure 10 shows best that with increasing Re periodicity is lost in favorto a different type of flow. If for Re = 52 periodicity is attained within threehollows and for Re = 79 after 5, for Re = 158 this pattern is attained onlyafter 7 hollows, whereas the amplitude of the oscillations is decreasing.

Re =52 Re =79 Re =158

Figure 10. Axial velocity along the axis

Whiles axial velocity along the tube axis is strictly periodic, pressure is asuperposition of a linear decaying profile and a periodic profile. The highestpressure drop occurs in the constrictions and, as expected, the numericalmodel evidences the pressure recovery in the divergent sections (Fig. 11).Flow periodicity is also apparent from this graph.

166 Alex. M. MOREGA

Re =52 Re =79 Re =158

Figure 11. Pressure axial profile along the axis

4 Conclusions

The study we conducted on stationary wavy-walled tube flows leads to thefollowing conclusions:

Within the range Re = 10 − 160 the flow is periodic, but periodicity isprogressively lost at higher Re, when eventually a different type offlow sets in.

The velocity profile at constrictions in the periodic flow section is nearlyparabolic for low Re numbers, and it flattens at higher Re numbers.

The separation region grows fast with Re, but then slower as it fills thehollow.

Increasing Re number reduces the pressure drop.

The highest pressure drop occurs in the constrictions, and it is followed byrecovery in the divergent section of each period.

Beyond Re = 158 the flow becomes unsteady. Initially, windows of steadi-ness are interrupted by intervals of unsteadiness. A further increase of Renumber progressively makes the unsteady intervals prevail over the steadyones. This regime is currently under investigation.

References

[1] Belhouse, B.J., Belhouse, F.H., Curl, C.M., MacMillan, T.I., Gun-ning, A.J., Spratt, E.H., Macmurray, S.B. and Nelems, J.M., A high


efficiency membrane oxygenator and pulsatile pumping system,Trans.ASME, Artif. Internal Organs, 19(1973), p. 72, .

[2] Bellinfante, D.C., Proc. Camb. Phil. Soc., 58 (1962), p. 405, .

[3] Burns, J.C. and Parkes, T.,J. Fluid Mech.,29 (1967), p. 731,.

[4] Chow, J.C.F. and Soda, K., Phys. Fluids, 15(1972), p. 1700,.

[5] Dodson, A.G., Townsend, P. and Walters, K., Rheol. Acta, 10(1971),p. 508,.

[6] FIDAP v.8.6, Fluent Inc. USA, 2001.

[7] GAMBIT v.1.3, Fluent Inc. USA.

[8] Greiner, M., An experimental investigation of resonant heat trans-fer enhancement in grooved channels,Int. J. Heat Mass Transfer,34(1991), p. 1383,.

[9] Lee, J.S. and Fung, Y.C., ASME, J. Appl. Mech., 37(1970), p.9, .

[10] Nishimura, T., Bian, Y.N., Matsumoto, Y. and Kunitsugu, K.,Fluidflow and mass transfer characteristics in a sinusoidal wavy-walled tubeat moderate Reynolds numbers for steady flow, Heat Mass Trans-fer(submitting), 2001.

[11] Nishimura, T. and Kojima, N., Mass transfer enhancement in a sym-metric sinusoidal wavy-walled channel for pulsatile flow,Int. J. HeatMass Transfer, 38(1995), p. 1719, .

[12] Nishimura, T. and Matsune, S., Mass transfer enhancement in a sinu-soidal wavy channel for pulsatille flow,Heat Mass Transfer, 32(1996),p. 65,.

[13] Nishimura, T., Oka, N., Yoshinaka, Y. and Kunitsugu, K, Influenceof imposed oscillatory frequency on mass transfer enhancement ofgrooved channels for pulsatile flow, J. Heat Mass Transfer, 43(2000),p. 2365,.

[14] Patera, A.T. and Mikic, B.B., Exploiting hydrodynamic instabili-ties, resonant heat transfer enhancement,Int. J. Heat Mass Transfer,29(1991), p. 1127,.

168 Alex. M. MOREGA

[15] Ralph, M.E., Steady flow structures and pressure drops in wavy-walledtubes, J. Fluids Eng., p. 255, 1987.

[16] Savvides, G.N. and Gerrard, J.H., J. Fluid Mech., 138(1984), p. 129,.

[17] Sobey, I.J., On flow through furrowed channels. Part 1. Calculatedflow patterns,J. Fluid Mech., 96(1980), p. 1,.

[18] Stone, K. and Vanka, S.P., J. Fluids Eng., 121(1999), p. 713, .

Alexandru MoregaDepartment of Electrical Engineering,POLITEHNICA University of Bucharest313 Splaiul Independentei, 6-Bucharest, 77206 ROMANIAE-mail: [email protected], [email protected], [email protected]


Domain Decomposition Approach for 3D FlowComputation in Hydraulic Francis Turbine

Sebastian MUNTEAN, Romeo F. SUSAN-RESIGA, IoanANTON and Victor ANCUSA

November 2, 2001

Abstract - The paper presents a methodology for computing the three-dimensional flow

in a Francis turbine. The turbine domain is decomposed in two subdomains correspond-

ing to the interblade channels for the distributor and runner, respectively. An iterative

method is developed for coupling the velocity and pressure fields on the distributor-runner

interface, such that a continuous hydrodynamic field is obtain for the whole turbine do-

main. In order to compute a runner steady relative flow, the mixing interface approach

is embedded within the iterative domain decomposition technique. Numerical results are

presented for the GAMM Francis turbine and the excellent agreement with experimental

data validates our methodology.

Key words and phrases : 3D inviscid incompressible flow, Francis turbine, domain

decomposition


1 Introduction

Developments in computer software and hardware made possible the com-putation of three-dimensional flows in turbomachines, [5]. However, com-puting the real flow (viscous and turbulent) through a hydraulic turbine stillrequires large computer memory and CPU time. As a result, a simplifiedsimulation technique must be employed to obtained useful results for turbinedesign and/or analysis, using currently available computing resources.

The turbomachines flow is essentially unsteady due to the rotor-statorinteraction. On the other hand, rigorously speaking, the geometrical period-icity of the stator/rotor blade rows cannot be used since there are differencesin flow from one interblade channel to another. However, with carefully cho-sen and experimentally validated assumptions, one can devise a methodologyfor computing the turbine flow, such that very good and engineering usefulresults are obtained. This paper presents such a methodology, and validates

169

170 S. Muntean, R.F. Susan-Resiga, I. Anton, V. Ancusa

the numerical results with experimental data for the GAMM Francis turbinemodel, [4].

Figure 1: GAMM Francis turbine [5] and flow survey axes.

Figure 1 shows the turbine meridian view with the actual main dimen-sions and the four survey axes used to investigate the velocity and pressurefields. The first axis, AA’ corresponds to the distributor inlet (spiral caseoutlet) where the velocity field is supposed to be known. This is the only in-formation required for our methodology in order to compute the flow throughthe whole turbine. The distributor (stay vanes and guide vanes) domain,Figure 2 is conventionally bounded by the revolution surfaces generated bythe axis AA’ and BB’, and the upper/lower rings respectively. The BB’ axisconventionally marks the distributor outlet and runner inlet, although thischoice was motivated here by the availability of experimental data. Theconical surface generated by rotating the BB’ segment around the turbineaxis plays an important role in our methodology for coupling the distributorabsolute flow with the runner relative flow. The next survey axis CC’ islocated right after the runner blades. Only velocity data are available onthis axis. The last survey axis DD’ is conventionally considered as being thedraft tube inlet. The runner computing domain, Figure 2, starts with theconical surface generated by BB’ and ends with a disc of radius DD’.

Section 2 presents the assumptions we made on the flow, as well asthe boundary conditions for the distributor/runner solution domain. Theiterative domain decomposition scheme, which employs the mixing interfaceapproach is outlined in section 3. Extensive comparisons of numerical resultsand experimental data are presented in section 4, and the conclusions aresummarized in the last section.

3D Flow Computation 171

Figure 2: Three-dimensional view of the distributor (left side) and the run-ner (right side) domains.

2 Absolute and relative flow equations. Boundary condi-tions.

The governing equations for the incompressible and inviscid flow are:

∇ · v = 0 ,d(ρv)

dt= ∇p + ρg (1)

The left hand side in the momentum equation has the following expressionin the inertial frame of reference:

∂

∂t(ρv) +∇ · (ρvv) (2)

When the equations of motion are solved in a rotating frame of reference,the relative velocity is introduced, w = v − ω × r and the left-hand side ofthe momentum equation can be written as:

∂

∂t(ρw) +∇ · (ρww) + 2ω ×w + ω × ω × r + ρ

∂ω

∂t× r (3)

The method developed in this paper is designed to solve only for steady flows(absolute or relative), and therefore the partial derivatives with respect totime vanish in (2) and (3).

The absolute flow equations are the natural choice for the distributor,but for the runner one may choose to solve either the absolute or the rela-tive flow. Since our goal is to develop an iterative technique to couple the


distributor and runner hydrodynamic fields, it is convenient to use absolutevelocity conditions at the runner inlet section. The FLUENT code [7] isable to compute the boundary conditions for the relative velocity, when therelative flow solver is employed. However, the conditions on the blade arehomogeneous for the relative flow, w · n = 0, but non-homogeneous for ab-solute flow, v · n = (ω × r) · n. The FLUENT code chooses the suitablecondition on the blade simply by specifying that this boundary be of walltype.

At the runner outlet (draft tube inlet) pressure conditions should bespecified. However, both numerical and engineering considerations lead tothe conclusion that this is not the best choice. Numerical experiments haveshown that enforcing a measured pressure distribution at the runner outletproduces spurious recirculation after the flow leaves the blades, in disagree-ment with the measured velocity field. On the other hand, from engineeringviewpoint, neither velocity nor pressure distributions are usually known inindustrial practice. As a result, one should employ a condition that is avelocity pressure relationship. Such condition can be devised by assumingthat in this section there is no radial flow, i.e. vr = 0 or negligible. Theradial projection of the momentum equation becomes,

∂p

∂r=

ρvu2

r. (4)

This is the so-called radial equilibrium outlet condition. However, this condi-tion defines the pressure up to an additive constant. Therefore, a referencepressure value should be specified at r = 0.

In order to compute the 3D flow through the Francis turbine, computa-tional domains have been defined separately for the distributor and for therunner, Figure 2. For each domain, the flow is computed separately and aniterative technique, to be presented in the next section, is used to matchthe pressure and velocity fields at the distributor-runner interface BB’. Theboundary conditions for computing the flow within a solution domain areas follows:

• velocity field is prescribed on the inflow section for both distributorand runner domains;

• pressure distribution is imposed on the distributor outlet section;

• radial equilibrium condition (4) is prescribed on the runner outlet sec-tion; periodic conditions are imposed on the periodic boundary;


• wall conditions (i.e. normal velocity) are imposed in the stay, guide,and runner blades, as well as on the distributor upper/lower rings andrunner band and crown, respectively.

The computational domains are discretized with 3D (tetrahedral) cellsand a segregated solver is used within the FLUENT code, [7]. Figure 2shows the spatial position of the survey axes on which numerical results arepresented in order to perform the comparison with available experimentaldata.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7s [−].

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

cr, c

u, c

z [−

].

cr (Muntean, coupled)/(Muntean, segregated)cr (Sottas&Ryhming, experimental)cu (Muntean, coupled)/(Muntean, segregated)cu (Sottas&Ryhming, experimental)cz (Muntean, coupled)/(Muntean, segregated)cz (Sottas&Ryhming, experimental)

cr=−0.132

cu=0.219

cz=0.0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7s [−].

0.9

0.95

1

1.05

1.1cp

[−].

(Sottas&Ryhming, experimental)(Muntean, numerical coupled)(Muntean, numerical segregated)

Figure 3: Velocity components (left side) and pressure coefficient (right side)on the distributor inlet axis AA’. Velocity components are imposed andpressure coefficient is computed.

On the distributor inlet section, the velocity field is specified as shown inFigure 3. Constant velocity profiles are considered for numerical simulation,although viscous effects and spiral case actual design produce a variationnear the upper/lower rings. The velocity coefficients values, see Figure 3,were chosen such that cr insures the global discharge measured value, cz = 0insures a plane flow, and cu corresponds to the ideal flow angle of attackon the stay vanes. No circumferential variation of the velocity is assumed.These conditions are met by an ideal spiral case. However, the simplificationwe made do not significantly affect the hydrodynamic field, as shown byvarious authors [2]. Our previous results support this assertion, as shownin [6]. Note that only this velocity distribution needs to be known for themethodology we present in this paper to compute the entire turbine flow.


3 Domain Decomposition Iterative Method

Since the flow computation is performed separately for the distributor andrunner domains, a coupling technique is required in order to obtain a contin-uous velocity and pressure fields across the conical surface generated by BB’.In this paper, an iterative coupling method is employed, with the algorithmoutlined below.

First, the distributor flow is computed, using the inlet velocity fromFigure 3, and an arbitrary (e.g. constant, or experimental when available)outlet pressure distribution. Next, iterations are performed as follows:

• compute the runner flow, using inlet velocity distribution obtained atthe distributor outlet, and the radial equilibrium outlet condition foroutlet pressure;

• compute the distributor flow, keeping the inlet velocity constant andusing the outlet pressure distribution obtained at the runner inlet. Thestopping criterion is that the pressure distribution on the distributor-runner interface is practically unchanged from one iteration to another.

Stability and convergence is achieved by using under-relaxation for bothvelocity and pressure, i.e,

pout disti = (relax)pin runner

i−1 + (1− relax)pout disti−1 (5)

vin runneri = (relax)vout dist

i−1 + (1− relax)vin runneri−1 (6)

respectively. The relaxation parameter value was relax = 0.3, but furtherinvestigations can be carried out in order to speed up the convergence. Nu-merical investigations have shown that 5...7 iterations lead to an engineeringacceptable convergence, provided that a good pressure distribution is usedfor the first distributor computation.

The chief difficulty in using the above algorithm is related to transferringdata from distributor to runner and vice versa. When looking at the Figure2, one can easily observe that the distributor outlet surface do not match therunner inlet surface, although both lie on the same cone segment. Moreover,the distributor has 24 blades, while the runner has only 13. The problem iseven more complicated by the use of an unstructured mesh. The solutionis to perform a circumferential averaging, which is equivalent to the fullmixing of the wakes (or any other circumferential non-uniformities). As aresult, this technique is known as the ”mixing interface method”. In orderto implement the above algorithm, we have developed a FORTRAN code,


using the IMSL library. The procedure preserves the flow rate and is calledat each iteration step to perform the ”mixing”, discharge correction, andrelaxation.

4 Numerical Results

Extensive comparisons of numerical results with available experimental data,[1], were performed for the GAMM Francis turbine model in order to validatethe above methodology. The experimental results correspond to the opti-mum operating point, i.e. guide vane opening 25, discharge Q = 0.372m3/sand specific hydraulic energy Eref = 58.42 J/kg.

In our previous work, single cascade flows and cascades in tandem [6]were investigated in order to assess the importance of the viscous effects aswell as the suitable boundary conditions. On the other hand, we have per-formed separate computations for the GAMM Francis turbine distributorand runner, using inlet/outlet conditions according to the actual measure-ments. However, in practice only the distributor inlet conditions are known,and it is this methodology that allows the entire turbine flow computationwithout information on intermediate survey axes.

This section presents our numerical results obtained with the mixinginterface approach compared with previous individual distributor and run-ner computations, as well as with the experimental results. Note that forall following figures, ”segregated” means separate computations for eitherdistributor or runner, while ”coupled” denotes results obtained with themixing interface approach. In order to avoid confusion we mention that thesame appellation is used by the FLUENT code to denote two different ap-proaches to solve the flow equations, and there is no connection at all withour approach.

Figure 3 (right side) presents the pressure coefficient on the distribu-tor inlet axis AA’. The pressure distribution is practically constant, butthe pressure level is slightly lower for the coupled approach. Since we em-ploy an inviscid flow model, one should expect that the computed pressureshould be lower than the measured value because the hydraulic losses areneglected. The dashed line was obtained by imposing the measured pressuredistribution on the distributor outlet surface, according to the points fromFigure 4. We mention once again that these data are not currently availablein engineering practice. Moreover, when computing separately the runnerflow, the pressure obtained at the runner inlet is significantly different fromwhat we have imposed at the distributor outlet. Thus, in order to obtain acontinuous pressure field an iterative coupling technique is required.


0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7s [−].

−0.4

−0.2

0.0

0.2

0.4

0.6

cz, c

u, c

m [

−].

cz (Sottas&Ryhming, experimental)cu (Sottas&Ryhming, experimental)cm(Sottas&Ryhming, experimental) (Muntean, numerical coupled) (Muntean, segregated)

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7s [−].

0.40

0.45

0.50

0.55

0.60

0.65

0.70

cp [

−].

(Sottas&Ryhming, experimental)(Muntean, numerical coupled)(Muntean, numerical segregated)

Figure 4: Velocity components (left side) and pressure coefficient (right side)on the runner inlet axis BB’. Comparison between the experimental data,the computed results and imposed conditions.

Figure 4 (left side) shows the velocity profile on the mixing interface BB’,corresponding to the distributor outlet and runner inlet. Note here that thedashed lines are the imposed velocity profiles (in agreement with the ex-periment) for separate runner calculation. The actual computed results arepresented with solid lines, and an excellent agreement with experimentaldata is observed. This is important, since correct runner results heavily relyon accurate inflow conditions. Figure 4 (right side) presents the pressuredistribution on the mixing interface. The numerical results are taken fromthe runner inlet, since the pressure is a prescribed boundary condition atdistributor outlet. One can easily observe that a separate runner compu-tation, even with a relatively good inlet velocity field, leads to 20% errorin pressure level. The coupled computation, solid line, not only eliminatesthe pressure discontinuity at the mixing interface, but also produces muchbetter results in agreement with experimental data.

The velocity field on the draft tube inlet is presented in Figure 5 (leftside). All velocity components are better predicted by the coupled compu-tation, especially near the turbine axis. Once again, the separate runnercomputation predicts a large swirl, in disagreement with experiment. Thecoupled computation correctly predicts the tangential velocity, and producesa significant improvement in axial and meridional velocity components nearthe turbine axis. Since the draft tube flow, and ultimately the pressurerecovery coefficient, depends on the amount of swirl after the runner, onecan see that the coupled computation significantly improves the absoluteflow direction predictions. Note that these good results are also due to the


0.0 0.2 0.4 0.6 0.8 1.0 1.2s [−].

−0.4

−0.2

0.0

0.2

0.4

0.6

cz, c

u, c

m [

−].

cz (Sottas&Ryhming, experimental)cu (Sottas&Ryhming, experimental)cm(Sottas&Ryhming, experimental) (Muntean, numerical coupled) (Muntean, numerical segregated)

0.0 0.2 0.4 0.6 0.8 1.0 1.2s [−].

−0.15

−0.10

−0.05

0.00

0.05

cp [

−].

(Sottas&Ryhming, experimental)(Muntean, numerical coupled) (Muntean, numerical segregated)

Figure 5: Velocity components (left side) and pressure coefficient (right side)on the draft tube inlet axis DD’. Comparison between the experimentaldata and the computed results.

radial equilibrium outlet condition employed on the runner domain outlet.The pressure distribution on the DD’ axis is shown in Figure 5 (right side).The coupled results are in excellent agreement with experiment, while theseparate computation predicts a much larger pressure drop near the turbineaxis, despite using the same boundary condition.

5 Conclusions

The paper presents a domain decomposition method, developed by the au-thors for computing the flow through the entire Francis turbine. Due tothe current limitations in computer hardware the 3D flow computation isperformed in two domains, corresponding to the distributor and runner in-terblade channels. The main advantage of computing the flow through theentire turbine is that only the velocity distribution at the distributor inlet(usually taken as a constant velocity profile corresponding to the spiral caseoutlet) needs to be known. The radial equilibrium condition employed atthe runner outlet/draft tube inlet section is found to be the best choice inorder to avoid spurious back flow after the runner.

Extensive comparison of numerical results with experimental data is per-formed, in order to validate and assess the accuracy of the numerical method.As a first conclusion, it seems that the inviscid flow model is suitable forcomputing the 3D hydraulic turbine flow at and around the best efficiencyoperating point. Comparison with velocity and pressure profiles on the threesurvey axes show an excellent agreement of the coupled computations with


experiments.In conclusion, the method presented in this paper is a reliable and effi-

cient tool for computing 3D turbomachinery flows. It can be easily employedfor design and optimization investigations, once a 3D inviscid flow solver (e.g.the FLUENT commercial CFD code) is available.Ackowledgements This work was partially supported from the follow-ing grants: GAR No. 81/2001, ANSTI-C No. 4161/2000 and GAR No.120/1999.

References

[1] F.Avellan, P.Dupont, M.Farhat, B.Gindroz, P.Henry, M.Hussain, Ex-perimental flow study of the GAMM turbine model. In G.Sotas andI.L.Ryhming (eds.) 3D-computation of incompressible internal flows,Vieweg Verlag, Braunschweig, NNFM 39(1993), 33-59.

[2] E.Goede, Stacking Technique for Multistage 3D Flow Computation inHydraulic Turbomachinery. In G.Sotas and I.L.Ryhming (eds.) 3D-computation of incompressible internal flows, Vieweg Verlag, Braun-schweig, NNFM 39(1993), 93-100.

[3] S.Muntean, R.Susan-Resiga, I.Anton, V.Ancusa, Mixing InterfaceMethod Applied to Three-Dimensional Flow Through Francis Turbine.InI.Anton, V.Ancusa and R.Resiga (eds.) Proceedings of the Workshopon Numerical Methods for Fluids Mechanics and Magnetic Liquids,Timisoara, (2001), 25-44.

[4] E.Parkinson, Test Case 8: Francis Turbine, Turbomachinery WorkshopERCOFTAC II, 1995.

[5] G.Sotas, I.L.Ryhming (eds.) 3D-computation of incompressible internalflows, Proceedings of the GAMM workshop, Notes Numerical Fluid Me-chanics (NNFM), Vieweg Verlag, Braunschweig, 1993.

[6] R.Susan-Resiga, S.Muntean, I.Anton, Analiza numerica a curgerii ˆinretele radiale dispuse ˆin tandem, Grant ANSTI-C Nr. 4161, 2000.

[7] ***, FLUENT 5. User’s Guide, Fluent Incorporated, 1998.

Sebastian Muntean Romanian Academy-Timisoara Branch, Center forAdvanced Research in Engineering Sciences, Bv. Mihai Viteazul 24, 1900-Timisoara, ROMANIA. E-mail : [email protected]

Romeo Susan-Resiga, Ioan Anton, Victor Ancusa“Politehnica” University of Timisoara, Bv. Mihai Viteazul 1, 1900-Timisoara,ROMANIA. E-mail : [email protected],[email protected],[email protected]


Approximate orthogonalization of linearlyindependent functions with applications to

Galerkin-like discretization techniques

Elena PELICAN and Constantin POPA

November 2, 2001

Abstract - Galerkin-like discretization techniques are widely used for both differential

and integral equation. They essentially use an initial set of linearly independent function

and ”project” the problem onto the linear space spanned by them. As a consequence, we

get a square, nonsingular, but ill-conditioned linear system. This bad aspect is usually

related to the ”non-orthogonality” of initial set of functions. In the present paper we

describe two algorithms, based on some approximate orthogonalization methods proposed

by Z. Kovarik, which eliminate these difficulties. We analyse their effect on the condition

number of the matrix of the linear system obtained after the Galerkin discretization of

the variational formulation of our problem. We describe at the end of the paper numerical

experiments on the one dimensional steady state transfer equation.

Key words and phrases : approximate orthogonalization, Galerkin discretization,

variational equations.

Mathematics Subject Classification (2000) : 65F10, 65F20

1 Classical Kovarik’s algorithms

In the paper [3], Z. Kovarik proposed the following approximate orthogo-nalization algorithm: let Φ0 = φ0

1, ..., φ0n be a linearly independent system

in a (real) Hilbert space (of functions) (H, 〈·, ·〉, || · ||H). For k = 0, 1, 2, . . .we consider the Gram matrix Gk associated to Φk, (Gk)ij = 〈φk

j , φki 〉 and

construct the next system Φk+1 by

Φk+1 = Φk · Γk (1)

where

Γk =

I + a1Hk + . . . + aqk

(Hk)qk

2(I + Gk)−1 (2)

179

180 E. Pelican, C. Popa

with Hk = I − Gk, aj = 2−2j (2j)!(j!)2

, qk ≥ 0 positive integers and the multi-plication in (1) defined by

φk+1i =

n∑

j=1

(Γk)jiφkj , i = 1, . . . , n. (3)

With respect to (1)-(3) the following convergence result is proved in [3].

Theorem 1. If||G0||2 < 1 (4)

and Φ∞ = φ∞1 , . . . , φ∞n ⊂ H is the system defined by Φ∞ = Φ0 · (G0)−12

then, for all i = 1, . . . , n

limk→∞

||φki − φ∞i ||H = 0. (5)

Moreover, the elements of Φ∞ are mutually orthogonal.

Remark 1. The norm || · ||2 is the spectral norm of the matrix G0 (see e.g.[2]). The inequality (4) can be obtained by an appropriate scalling of theelements of the matrix G0 (see e.g. [6]).

Remark 2. The matrices Hk and Γk, k ≥ 0 in (2) are symmetric and pos-itive definite (see [6] and [7]).

In the particular case H = IRm, the following ”matricial” versions ofKovarik’s algorithms (1)-(3) can be constructed: let A = A0 be an n ×mreal matrix (n ≤ m) with linearly independent rows. For k = 0, 1, 2, . . . wedefine the next matrix Ak+1 by

Ak+1 = ΓkAk, Γk =

I + a1Hk + . . . + aqk

(Hk)qk ,2(I + Ak(Ak)t)−1 (6)

where Hk = I − Ak(Ak)t and aj , qk are as in (2). As a direct consequenceof Theorem 1 we obtain the following result.

Corollary 1. If A∞ is the n × m matrix defined by A∞ = (AAt)−12 A,

then ||Ak − A∞||2 → 0, k → ∞. Moreover, the matrix A∞ has mutuallyorthogonal rows.

Remark 3. Extensions of the above algorithms (6) to arbitrary rectangularmatrices can be found in the papers [6] and [7].

Approximate orthogonalization 181

2 Kovarik’s algorithms applied to Galerkindiscretizations

Let a : H ×H → IR be a bilinear functional and f ∈ H. We consider thevariational problem: find u∗ ∈ H such that

a(u∗, v) = 〈f, v〉, ∀ v ∈ H. (7)

If Φ0 = φ01, . . . , φ

0n ⊂ H is as in Section 1, we construct in a classical way

(see e.g. [4]) the linear system associated to (7)

A0x = b0, (8)

where the n× n matrix A0 and the vector b0 ∈ IRn are defined by

(A0)ij = a(φ0j , φ

0i ), b0

i = 〈f, φ0i 〉, i, j = 1, . . . , n. (9)

As it is well known (see [4]), with some additional conditions on the bilinearform a, the approximate solution of the variational problem (7) is obtainedby solving the linear system (8). But, usually the matrix A0 is ill-conditionedand this bad aspect determines a lot of computational troubles for bothdirect and iterative solvers. In order to avoid these difficulties, we shalldescribe in what follows two methods for improving the condition numberof A0.Method 1. We (formally) apply Kovarik’s algorithms (1)-(3) to the system(”discretization basis functions”) Φk, k ≥ 0. In each case we define thecorresponding matrix Ak and the vector bk ∈ IRn by

(Ak)ij = a(φkj , φ

ki ), bk

i = 〈f, φki 〉, i, j = 1, . . . , n. (10)

Method 2. We apply the ”matricial” versions (6) of Kovarik’s algorithmsto the matrix A0 = A0 and b0 = b0 ∈ IRn and obtain the matrices Ak andvectors bk ∈ IRn, k ≥ 0.

Remark 4. The behaviour of the above Method 2 is completely charac-terized by Corollary 1 (for m = n and A replaced by A0). Concerning theMethod 1, the following results hold.

Theorem 2. In the above hypothesis, the sequences (Ak)k≥0, (bk)k≥0 can berecursively generated as follows:

Gk = Γk−1Gk−1Γk−1, Ak+1 = ΓkAkΓk, bk+1 = Γkbk, k = 1, 2, . . . (11)

with Γk defined as in (2).


Proof. Using (9), (10), (3), the bilinearity of a and the fact that Γ0 issymmetric, we succesively obtain

(A1)ij = a(φ1j , φ

1i ) = a(

∑n

p=1(Γ0)pjφ

0p,

∑n

q=1(Γ0)qiφ

0q) =

n∑

p,q=1

(Γ0)qi(Γ0pj)a(φ0

p, φ0q) =

n∑

p,q=1

(Γ0)qi(Γ0)pj(A0)qp = (Γ0)tiA0(Γ0)j ,

(12)

where by (Γ0)r ∈ IRn we denoted the r-th row of Γ0 (considered as columnvector). In the same way, by taking into account that (G1)ij = 〈φ1

j , φ1i 〉 and

replacing a(·, ·) by 〈·, ·〉 we obtain G1 = Γ0G0Γ0. Then, by an inductionargument the first two equalities in (11) hold. The third one is directlyobtained from (10), (1) and (3).

Theorem 3. If the bilinear form a is continuous on H thenlimk→∞ ||Ak − A∞(a)||2 = 0, where A∞(a) is the n × n matrix defined by(A∞(a))ij = a(φ∞j , φ∞)i, i, j = 1, . . . , n.

Proof. From (5) and the continuity of a we get for all i, j = 1, . . . , nlimk→∞(Ak)ij = limk→∞ a(φk

i , φki ) = a(φ∞j , φ∞i ) = (A∞(a))ij .

Remark 5. From (6) and (11) it results that the above two methods arealmost of the same order of complexity (from the view point of the compu-tational effort per iteration). The advantage of the Method 1 is that itis directly connected to the initial (continuous) problem (7) and the basisfunctions Φ0. This aspect allows us to think at some similar applications forGalerkin discretizations of first kind integral equations (see e.g. [1]).

3 Numerical experiments

We considered in our numerical experiments the one dimensional steadystate transfer problem (with internal heat source f)

−u′′(x) + α u′(x) = f(x), x ∈ (0, 1), u(0) = u(1) = 0. (13)

The variational formulation of (13) is the following (see [4]): find u∗ ∈H = H1

0 (0, 1) such that (7) holds with a given by a(u, v) =∫ 10 u′(x)v′(x) +

αu′(x)v(x)dx. For a given n ≥ 2, let Φ0 = φ01, . . . , φ

0n−1 be the set of

picewise linear finite element basis functions (see [4]). According to (9)we obtain the linear system of the form (8) associated to the variationalformulation (7) of (13). On H = H1

0 (0, 1) we considered the scalar product

Approximate orthogonalization 183

(see [4]) 〈u, v〉 =∫ 10 u′(x)v′(x)dx. For improving the condition number of

the matrix A = A0 we used the following algorithms A1 and A2, whichcorrespond to the methods 1 and 2 from Section 2, respectively. In bothalgorithms we used only the first Kovarik algorithm from (2) and (6), withqk = 2, ∀ k ≥ 0.Algorithm A1. Set A0 = A, G0 = 1

2(A0 + At0) and for k = 0, 1, 2, . . . do

Hk = I −Gk, Γk = I +12Hk +

38(Hk)2, Gk+1 = ΓkGkΓk, Ak+1 = ΓkAkΓk.

Algorithm A2. Set A0 = A and for k = 0, 1, 2, . . . do

Hk = I −Ak(Ak)t, Γk = I +12Hk +

38(Hk)2, Ak+1 = ΓkAk.

We made numerical experiments for three different values of the convectionconstant α. For each value of α we constructed the mesh size h = 1/nsuch that the Peclet constant, Pe = αh rests in the interval (0, 2) (whichensures the stability of the computed solution (see [4]). In the second rowof each of the Tables from 1 to 6, starting with column 3, we indicated, inboldface the numbers of Kovarik’s algorithm sweeps inside the algorithmsA1 and A2. All the other numbers in columns from 3 to 9 representthe corresponding values of the spectral condition number of Ak, given byk2(Ak) = ||Ak||2||A−1

k ||2. All the numerical experiments have been madewith the numerical linear algebra software OCTAVE.

Table 1. Algorithm A1 with α = 100n Pe 0 4 7 10 12 14 15

200 0.5 998.68 41.578 23.58 11.71 3.587 1.562 1.5614100 1 250.12 23.7 1.34 5.1245 2.4987 1.158 1.15466 1.5 110.37 16.8 5.5 1.01 1.01 1.01 1.01


300 0.5 1508.6 130.74 19.851 14.567 6.5136 5.2143 5.2143150 1 377.47 70.5 18.9 12.234 4.265 3.265 3.114100 1.5 168.11 25.47 8.008 6.25 3.564 2.456 2.456


400 0.5 2514 338.952 12.674 5.1 4.775 3.5621 3.5621200 1 962 287.62 26.81 6.9876 6.1423 4.52 2.16133 1.5 84.375 15.74 5.012 1.09 1.0567 1.057 1.057



200 0.5 998.68 323.19 49.040 7.4744 2.2417 1.0405 1.0001100 1 250.12 80.957 12.304 2.0063 1.0197 1.0 1.066 1.5 110.37 35.737 5.4699 1.1485 1.0 1.0 1.0


300 0.5 1508.0 488.0 74.045 11.256 3.2781 1.2041 1.0108150 1 377.47 122.16 18.548 2.9062 1.1349 1.0 1.0100 1.5 168.11 54.452 8.2870 1.4661 1.0006 1.0 1.0


400 0.5 2514 1178 52 5 4 1.4 1.05200 1 962 349.876 23.73 6.9 4.965 1.9345 1.0612133 1.5 84.375 29.631 4.5546 1.0729 1.0 1.0 1.0

References

[1] Engl, H.W. Regularization methods for the stable solution of inverseproblems, Surv. Math. Ind., 3(1993), 71-143.

[2] Golub, G.H., van Loan, C.F. Matrix Computation, The John’s HopkinsUniv. Press, Baltimore, 1983.

[3] Kovarik, Z. Some iterative methods for improving orthogonality, SIAMJ. Numer. Anal., 7(3)(1970), 386-389.

[4] Oden, J.T., Reddy, J.N. An Introduction to the Mathematical Theoryof Finite Elements, John Wiley & Sons, Inc., 1976.

[5] Popa, C. Least-squares solution of overdetermined inconsistent linearsystem using Kaczmarz’s relaxation, Intern. J. Comp. Math., 55(1995),79-89.

[6] Popa, C. Extension of an approximate orthogonalization algorithm toarbitrary rectangular matrices, Linear Alg. Appl., 331(2001), 181-192.

[7] Popa, C. A method for improving orthogonality of rows and columnsof matrices, Intern. J. Comp. Math., 77(2001), 469-480.

Elena Pelican, Constantin Popa”Ovidius ”University, Blvd. Mamaia, No 124, 8700- Constanta, RomaniaE-mail: [email protected], [email protected]


Appearance of pores through black lipidmembranes due to collective thermic movement of

lipid molecules

Dumitru POPESCU, Stelian ION and Maria Luiza FLONTA

November 2, 2001

Abstract - In this paper we pointed out a new mechanism of stochastic pore appearance

through black lipid membranes, based on normal movement of lipid molecules on lipid

bilayer. For this aim we used a naturally condition consisting in: the change of total

deformation energy of lipid bilayer to be less then thermic energy of all molecules belonging

to deformed region on bilayer surface. From this condition we obtained the following data:

the radius of collective movement region, the wave length spectrum of bilayer deformation,

both compatible with pore formation, as the pores radius. These results was obtained

as function on bilayer thickness, cross section area of lipid molecules, temperature and

parameters characterizing inter-molecular forces.

Key words and phrases : lipid bilayer, stochastic pores, thickness fluctuation

Mathematics Subject Classification (2000) : 65H05, 92C10

1 Introduction

The mechanism of pore formation through lipidic bilayers is not well known.One of the proposed mechanism of pores appearance is based on structuraldefects which may form in the ordered structure of lipid bilayers [2]. Thereare two ways for defect generation. One way consists in lateral movement inopposed directions of two or three neighboring molecules giving birth to afree lipid region. Another way is that of cluster formation. It is known thatthe lipid matrix of natural membranes is composed of many type of lipids.Due to lateral movement and selective association between lipids it is possi-ble the clusters formation [18]. The region between neighboring clusters is adefect. On the other hand the lipid movement normal to the bilayer surfaceresults in fluctuation of its thickness [10, 11, 15]. These fluctuations are thecollective motion of phospholipid molecules as a consequence of intramolec-ulare forces. Colective motion normal to bilayer surface is another way for

185

186 D. Popescu, S. Ion and M. L. Flonta

pores formation [17]. A model based on selfoscillation of lipid bilayer wasproposed [17]. In all model of pores formation were tacitly neglected thebilayer elastic properties. Theory of elastic properties of liquid crystals iswell developed [9, 20]. But elastic theory was used for membrane thicknessfluctuations [10, 11] and for calculation of deformation energies of bilayermembrane and its effect on gramicidine channel properties [6].

2 Free Energy Variation

In this paper we will consider the plan bilayers in which the hydrophobicchains are normal to lipid bilayer. In this case the lipid bilayer is as a liquidcrystal of type smectic A. The elasticity theory well-developed for liquidcrystals by de Gennes [9], may be applicable to plan lipid bilayers. In theinitial state the bilayer surfaces are plane and his free energy is minimum.After a perturbation the surface of a monolayer move with the distanceu(x,y) from initial position. According to Huang theory [14] the free energychange per surface unit is:

4F = hB(u

h

)2+ hK

(∂2u

∂x2+

∂2u

∂y2

)2

+ γ

[(∂u

∂x

)2

+(

∂u

∂y

)2]

(1)

In smectic liquid crystals there are three ways of free energy change dueto the change in thickness and the change in surface area. Thickness changeproduces a compression and a modification of the axis of the molecule. Thelast modification is known as splay distortion. In the formula of free energychange due to a perturbation of u(x, y) amplitude the first term representsan elastic energy for compression of layer and is characterized by compres-sion elastic constant B; the second term represent the elastic energy forsplay distortion and is characterized by elastic coefficient K; the third termis the free energy of superficial tension change characterized by coefficient ofsurface tension γ. In our case the deformation of a halfbilayer has a cylindri-cal symmetry, so the total free energy change of a deformation of a surfaceof radius R at the level of the plane surface is:

F = 2π

∫ 1

0

[B

u2

h+ hK

(∂u

r∂r+

∂2u

∂r2

)+ γ

(∂u

∂r

)2]

rdr (2)

We consider only a local deformation of a membrane surface having a formof wave of length λ, with amplitude equal to half thickness of bilayer h:

u(r) = h cos2πr

λ(3)

Pores through black lipid membranes 187

We are interested to find the parameters λ and R that satisfy :

2πa

πR2

∫ R

0rF (r, λ)dr = (3N −Ng − 4)

kT

2(4)

Where,N is number of atoms from a molecule and Ng is the number ofthe internal bonds of the molecule. For simplicity we suppose that 3N−Ng−4 = 6 . After some cumbersome calculus the last equation can be writtenas:

a0(y)x4 + b0(y)x2 + c0(y) = 0 (5)

Where the functions a0, b0, c0 and the variables x, y are given by:

x = h2π

λ, y = R

2π

λ

a0(y) =K

Bh2

(1 +

sin 2y

y− 3

cos 2y − 12y2

+4y2

∫ 1

0

sin2 ty

tdt

)

b0(y) =γ

Bh

(1− sin 2y

y− cos 2y − 1

2y2

)

c0(y) = 1 +sin 2y

y+

cos 2y − 12y2

− (3N −Ng − 4))kT

Bha(6)

Unfortunately the equation (4) is too complicate to obtain an explicitformula for the unknowns λ and R but it can be obtained a parametricalrepresentation. In order to solve it we consider x as unknown and y asparameter so, our equation is reduced to an algebraic equation for x thatcan be solved very easy. Since the functions a0 and b0 are always positivesthe equation has the solutions only if the function c0 is negative. If thiscondition is satisfied there is only one positive solution. The parametricform of it is given by:

λ(y) = 2πh

√−b0(y) +

√b20(y)− 4a0(y)c0(y)2a0(y)

−1

R(y) =yλ(y)2π

(7)

3 The Analyse of Thickness Fluctuations which can Gener-ate Transbilayer Pores

We selected a reference bilayer with its unsolved hydrophobic core thicknessof 28.5A, coresponding to hydrophobic channes of 11 methilene groups.


The cross section area of polar group a is equal to 19.3A, and tempera-ture is 300K. The other parameters from eqution (4) characterizing theintermolecular forces have the following values: the splay coefficient K =0.933 ∗ 10−11N, the compression coefficient B = 5.75 ∗ 107N/m2 and thecoefficient of superficial tension, γ , is equal to 15 ∗ 10−4N/m. Solving theequation (4) for the reference bilayer we have found both the radius, R, ofbilayer surface involved in collective thermal motion and wave length, λ, ofthe bilayer deformation for which the transbilayer pores can appear.

Fig.1 The dependence of deformation wavelength on the radius of perturbed surface.

The dependence between these solutions R = R(λ) was represented infig. 1 and the significant values of them are written in the table I. Also ,besides R and λ the table contains the number of the molecules implied inthe collective motion and the deformation energies.

The values was calculated when the deformation energy equals to thethermal energy of all molecules involved in collective thermal motion. Thereference bilayer is characterized by 2h=28.5 A, ao=19.3 A2, T=300 K,


K = 0.983 ∗ 10−11N, B = 5.36 ∗ 107N/m2, γ = 15 ∗ 10−4N/m

Table IParameters λm − λ2:1 − λM Rm −R2:1 −RM Em − E2:1 − EM/kT N

2h=28.5A 53.65-59.00-193.13 12.76-15.06-29.39 79.52 -110.81-421.83 27-137

2h=25 A 47.30-50.81-156.26 1.02 25.24 0.51 311.02 .17- 104

ao=38.6A2 72.62-125.1-166.27 23.2-34.12-58.32 131.3-284.23-830.53 52-277

T=320 K 52.38-56.72-215.82 9.91-10.74-28.10 47.94-56.37-385.64 16-128

The lipid molecules have to move simultaneously perpendicular to sur-face bilayer in order to cause the transbilayer pores appearance, belong tothe region of radius R. Their number N is equal to πR2/ao. For the ref-erence bilayer, studied here, the number of molecules is N ∈ (0.17, 104)(Tab.I). But the radius of perturbed surface can’t be smaller than the ra-dius of the cross section area of polar group, so the smallest radius of theperturbed region has to be equal to Rgp =

√a0/π = 2.479A. In the case

of lipid bilayers for which the minimum radius is smaller than Rgp we canaffirm that even a single lipid molecule can move up to the middle of thebilayer. It is possible that this molecule to pass in the other monolayer ofthe bilayer, that is equivalent to the flip-flop diffusion, a known process inthe membrane biophysics. On the other hand, looking at the fig. 1, wecan observe that the functions R = R(λ) and λ = λ(R) are not one-to-onefor all values of the corresponding variable. In the table I, the solution ofeq.(4) and the other parameters were written in special way. Beside theirextreme values which define the real interval, there is an internal valueswhich divides each range in two sub-intervals. On the first sub-interval thedependence in not unique. For example, for each value of λ belonging tothe range (λm, λ2:1) the radius R takes two values. This is very interestingfrom the point of view of the pores appearance, because for each of thesevalues it is possible to appear two different pores. In order to see the ef-fect of hydrophobic core thickness on the transbilayer pore appearance, wesolved the equation (4) for the2h = 25A. The change of cross section ofpolar group has an important effect on both radius of perturbed surface andwave length of deformation. In order to see this effect the equation (4) wassolved for a0 = 38.6A2, the other parameter remaining unchanged. Compar-ing the graph drown for reference bilayer (a0 = 19.3A2) with graph for thephospholipid bilayer (a0 = 38.6A2) one can see three essential modifications:• The size of perturbed surface increases. In the last case R ∈ (15.88, 55.81)A(Tab.I). Much more lipid molecules have to involved in collective thermalmotion to generate transbilayer pores: N ∈ (52, 277)


• Because Rm = 15.18A the flip-flop diffusion isn’t take place;• The uniqueness interval of R and λ for the corresponding function in-creases. The temperature increase influences only the wave length of defor-mation.

4 CONCLUSIONS

This work may be continued with the analyse of effect of bilayer elasticproperties and surface tension on the transbilayer pore formation. Also thepore radius may be calculated.

References

[1] Alvarez, O., and R. Latore, Voltage-dependent capacitance in lipidbilayers made from monolayers, Biophys. J, 21(1978),1-17.

[2] Abidor, I. G., V. B. Arakelian, L.V. Chernomordik, Yu. A. Chiz-madzhev, V. F. Pastushenko and M.R. Tarasevich, Electric breakdownof bilayer lipid membranes. I. The main experimental facts and theirqualitative discussion, Bioelectrochem. and Bioenerg., 6(1979), 37-52.

[3] Buldt, G., H. U. Gally, A. Seeling, J. Seeling, and G. Zaccai, Neutrondiffraction studies on selectivity deuterated phospholipid bilayers, Na-ture (London), 27(1978), 182-184.

[4] Chizmadzhev, Yu. A., V. B. Arakelian, V. F. Pastushenko, Electricbreakdown of bilayer lipid membranes. III. Analysis of possible mech-anism of defect origination, Bioelectrochem. and Bioenerg., 6(1979),63-70.

[5] Dilger, J. P., The thickness of monoolein lipid bilayers as determinedfrom reflectance measurements, Biochim. Biophys. Acta, 645(1981),357-363.

[6] Elliot, J. R., D. Needham, J. P. Dilger, and D. A. Haydon, The ef-fects of bilayer thickness and tension on gramicidin single-channel life-time,Biochim. Biophys. Acta, 735(1983), 95-103.

[7] Engelman, H., H. P. Duwe, and E. Sackman, Bilayer bending elasticitymeasurement by Fourier analysis of thermally excited surface undula-tions of flaccid vesicles, J. Physique Lett.,46(1985), L-395-L-400.


[8] Fettiplace, R., D. M. Andrews, and D.A. Haydon, The thickness, com-position and structure of some lipid bilayers and natural membranes,J.Membr. Biol., 5(1971), 277-296.

[9] de Gennes, P. G., The Physics of Liquid Crystals, Clarendon Press,Oxford, 1974.

[10] Hladky, S. B., and D. W. R. Gruen, Thickness fluctuations in blacklipid membranes, Biophys. J., 38(1982),251-258.

[11] Hladky, S. B., and D. A.Haydon, Ion movement in gramicidin chan-nels, Current Topics in Membrane Transport, 21(1984), 327-372.

[12] Helfrich, P., and E. Jakobsson, Calculation of deformation energiesand conformations in lipid membranes containing gramicidin channels,Biophys. J., 57(1990), 1075-1084.

[13] Helfrich, W., Elastic properties of lipid bilayers: theory and possibleexperiments, Z. Naturforsch, 28C(1973), 693-703.

[14] Huang, H.W., Deformation free energy of bilayer membrane and itseffect on gramicidin channel lifetime, Biophys. J., 50(1986), 1061-1070.

[15] Miller, I. R., Energetics of fluctuations in lipid bilayer thick-ness,Biophys. J., 45(1984), 643-644.

[16] Neher, E.,and H. Eibl, The influence of phospholipid polar groups ongramicidin channels,Biochim. Biophys. Acta, 464(1977), 37-44.

[17] Popescu, D. C. Rucareanu and Gh.Victor, A model for appearance ofthe statistical pores in membranes due to the selfoscillations, Bioelec-trochem. and Bioenerg., 25(1991), 91-105.

[18] Popescu, D., Gh. Victor, Association probabilities between the singlechain amphiphiles into a binary mixtures, Biochim. Biophys. Acta,1030(1990), 238-250.

[19] Schneider, M. B., J. T. Jenkins, and W. W. Webb, Thermal fluc-tuations of large quasi-spherical bimolecular phospholipid vesicles, J.Physique, 45(1984), 1457-1472.

[20] Stephen, M. J., and J. P. Straley, Physics of liquid crystals, Rev. Mod.Phys., 46(1974), 617-704.


[21] White, S. H., Formation of solvent-free black lipid bilayer membranesfrom glyceryl monooleate dispersed in squalene, Biophys. J., 23(1978),337-347.

Dumitru Popescu, Maria Luiza FlontaCentre of Neurobiology and Molecular Biophysics,Faculty of Biology, University of Bucharest,Spl.Independentei, nr.91-95, Bucharest, Romania.

Stelian IonInstitute of Applied Mathematics ,, Caius Iacob”Calea 13 Septembrie, No. 13, Sector 5Bucharest, P.O. Box 1-24, 70700E-mail : [email protected]


Optimality and Non-Optimality Criteria forSingular Control

Mihai POPESCU

November 2, 2001

Abstract - The non-negativity, respectively, negativity conditions of the second variation

of the performance index, give the sufficient optimality and non-optimality criteria for the

singular control.

Key words and phrases : singular control, second variation, performance index

Mathematics Subject Classification (2000) : 34H05

1 Introduction

Maximum Principle gives a necessary condition for the optimality of the ad-missible trajectories. Among the trajectories satisfying the Maximum Prin-ciple, there are ones for which the associated control is not determined bythis principle; these are the singular extremals. The singular case is properto differential systems, linear in the control variable, and corresponds to thevanishing of the Huu. This fact renders improper the use of the well-knownRiccati matrix equation, whose solution of which allows the optimality anal-ysis for the non-singular case. The practical problems involving singular con-trol appear in the spatial dynamics. Thus, the trajectories for rocket, havinglimited thrust, exhibit a singularity with respect to the fuel consumption.Lawden (1963) showed that the singular arc of intermediate thrust satisfiesthe Pontryagin’s principle and hence may minimize the fuel consumptionin the orbital transfer with final time not precised. By using the gener-alized Legendre-Clebsch conditions Kelley (1967) proved non-optimality ofthe intermediate thrust arc (Lawden’s spiral). In 1971, Bell showed the non-optimality of Lawden’s spiral by the construction of a state variation class.At last, Gift (1993) [6] also achieved the necessary second order optimalityconditions and proved the non-optimality of the intermediate thrust arcsof the Lawden’s spiral. The objective of the present study is to developan optimality and non-optimality analysis model for singular control. This

193

194 M. Popescu

consists in the construction of the symmetrical matrix P(t) which satisfiesthe sufficient optimality and non-optimality criteria.

2 The optimal control problem

Letx = f(x,u, t) (1)

be a system submitted to the constraints

x(t0) = x0, ψ [x (tf) , tf ] = 0, (2)

where

x = (x1, x2, . . . , xn)T ∈ X, u = (u1, u2, . . . , um)T ∈ U (3)

stand for the state vector and control vector, while f and ψ are columnvectors of dimension n and s, respectively. The initial time t0 and the initialstate x0 are specified, while the final time tf may be or not specified.

Consider the performance index of the form

J = F [x (tf) , tf ] +

tf∫

t0

f0(x,u, t)dt, (4)

where F and f0 are the scalar functions at least twice continuously differ-entiable in each of their arguments.

The optimal control problem is stated as follows:Given equation (1), and the conditions (2), find the control u ∈ U such

that the performance index (4) be minimized. The use of differential con-straints (1) and final condition (2) requires the transformation of the per-formance index in the extended form:

J =[F + νTψ

]t=tf

+

tf∫

t0

[H (x,u, λ, t)− λTx

]dt, (5)

where H (x,u, λ, t) = f0(x,u, t)+λTf(x,u, t), λ denotes a n order vector ofLagrange, time dependent multipliers, while ν is a constant, s order vector.For the purpose of evaluating the second order variation, the control vectorfamily u(t, ε) is defined, and the associated state vector is x(t, ε).

Optimality and Non-Optimality Criteria 195

By Taylor’s series development, one obtains:

J(ε) = J(0) + εJ1 + ε2J2 + . . . . (6)

The optimality conditions required to the optimal control vector u(t, 0) tominimize the functional J become:

J1 = 0, J2 ≥ 0. (7)

The first eq. (7) leads to the transversality conditions. For tf specified, itresults:

J2 =

tf∫

t0

[12δxTHxxδx + δuTHuxδx +

12δuTHuuδu

]dt

+12δxT

(Fxx + νTψxx

)δx

∣∣∣∣tf

.

(8)

3 Singular control with final constraints

For the singular control (Hu = 0, Huu = 0, t ∈ [t0, tf ]) the second variation,Eq. (8), becomes:

J2 =

tf∫

t0

[12δxTHxxδx + δuTHuxδx

]dt +

12δxT

(Fxx + νTψxx

)δx

∣∣∣∣tf

. (9)

and the following relations hold

δx = fxδx + fuδu, δx (t0) = 0, (10)

ψxδx|tf = 0. (11)

In order to satisfy the final constraints (Eq. (2)) and the variation (Eq.(11)), the admissible control is defined by u0(t) + δu(t) ∈ U1 ∩ U2, where

U1 = u(t); ui ≤ ui(t) ≤ ui, t ∈ [t0, tf ] , i = 1, . . . , m , (12)

U2 = u(t); ψ [x (tf) , tf ] = 0, x = f(x,u, t), x (t0) = x0 . (13)

Taking into account the constraints (10), the second variation (9) isexpressed as

J2 =

tf∫

t0

[12δxTHxxδx + δuTHuxδx

+δλT (fxδx + fuδu− δx)]dt

+12δxT

(Fxx + νTψxx

)δx

∣∣∣∣tf

.

(14)

196 M. Popescu

Finally, integrating the last term in the integrand by parts, and choosingδλ = 1

2P(t)δx, the integrand of the expression (14) is one quadratic form inδx and δu, P(t) is a continuously differentiable, simetric matrix function oftime.

Furthermore, it is easy to verify that

J2 =

tf∫

t0

[12δxT

(P + Hxx + fT

x P + Pfx)

δx

+δuT(Hxx + fT

u P)δx

]dt

+12δxT

(Fxx + νTψxx −P

)δx

∣∣∣∣tf

.

(15)

Likewise, the relation (11) of the final conditions variation is satisfied;consequently:

∑si=1 ψi

xδxi (tf) +∑n

i=s+1 ψixδxi (tf) = 0 or

M1δxs (tf) + M2δxn−s (tf) = 0, (16)

where M1 =(ψi

xj

), i, j = 1, 2, . . . , s, M2 =

(ψk

xl

), l, k = s + 1, s + 2, . . . , n

and δxs = (δx1, δx2, . . . , δxs)T, δxn−s = (δxs+1, δxs+2, . . . , δxn)T. If M1 is

non-singular, from (16) we get

δxs (tf) = −M−11 M2δxn−s (tf) , (17)

which implies

δx (tf) =

δxs (tf)· · · · · · · · ·δxn−s (tf)

=

−M−1

1 M2

· · · · · · · · ·I

· δxn−s (tf) . (18)

4 Optimality conditions for singular control

This result provides

J2 =

tf∫

t0

[δxδu

]T

·[P + Hxx + fT

x P + Pfx(Hux + fT

u P)T

Hux + fTu P 0

]

·[

δxδu

]dt

+12

(δxn−s)T [

MT(Fxx + νTψxx −P

)M

](δxn−s)

∣∣∣∣tf

.

(19)

Optimality and Non-Optimality Criteria 197

The non-negativity of the second variation (19) holds if

P + Hxx + fTx P + Pfx ≥ 0 for all t ∈ [t0, tf ] , (20)

Hux + fTu P = 0 for all t ∈ [t0, tf ] , (21)


)M

∣∣tf≥ 0. (22)

The inequalities (20), (21), (22) represent the sufficient optimality conditionsof the partial singular control.

5 Non-optimality conditions for singular control

For δx 6= 0 sufficiently small the prevailing term in the second variationexpression is

I =

tf∫

t0

δuT(Hux + fT

u P)δxdt. (23)

We consider the control variation defined by

δu =

k for t ∈ [t1, t1 + ∆t] ⊂ [t0, tf ]0 for t ∈ [t0, tf ] \ [t1, t1 + ∆t] ,

(24)

where k is an arbitrary constant and ∆t > 0 with ∆t = ε → 0, so that wemay write

I(ε) =

t1+ε∫

t1

kT(Hux + fT

u P)δxεdt =

t1+ε∫

t1

G(t, ε)dt, (25)

where G(t, ε) = kT(Hux + fT

u P)δxε. Taking into account that I(ε) is an

integral with parameter and δx(t1)0, we have:I ′(0) = 0, I ′′(0) = kT

(Hux+ fT

u P)δx

∣∣∣t=t1

= kT(Hux + fT

u P)fuk

∣∣t=t1

.

Developing in Taylor series, we obtain

I =12

[kT

(Huxfu + fT

u Pfu)kε2

]∣∣∣∣t=t1

. (26)

Since any point of the interval [t0, tf ] may be chosen as t1, the second vari-ation sign is the same with the sign of I, given by (26).

Consequently, the negativity of the second variation is ensured by satis-fying the inequalities

P + Hxx + fTx P + Pfx ≤ 0 for all t ∈ [t0, tf ] , (27)

198 M. Popescu

Huxfu + fTu Pfu < 0 for all t ∈ [t0, tf ] , (28)


)M

∣∣tf≤ 0, (29)

which represent sufficient non-optimality conditions for the singular control.

References

[1] Archenti, A. R., and Vinh, N. X., Intermediate Thrust Arcs and theirOptimality in a Central, Time-Invariant Force Field, J. Opt. Th. Ap-plic., 11(1973), 293–304.

[2] Bell, D. J., The Non-Optimality of Lawden’s Spiral, Astronautica Acta,16 (1971), 317–324.

[3] Bell, D. H., and Jacobson, D. H., Singular Optimal Control Problems,Academic Press, New York, 1975 (Chap. 2, pp. 37–56; Chap. 3, pp. 92–94).

[4] Gabasov, R., and Kirillova, F. M., High-Order Necessary Conditionsfor Optimality, SIAM Journal and Control, 10 (1972), 127–168.

[5] Lawden, D. F., Calculation of Singular Extremal Rocket Trajectories,Journal of Guidance, Control and Dynamics, 5 (1992), 6, 1 361–1 365.

[6] Gift, J. G., Second Order Optimality Priciple for Singular Control Prob-lems, Journal of Optimization Theory and Applications, 76 (1993), 3,477–484.

[7] Popescu, M. E., Optimal Control in Pursuit Problem, Journal of Guid-ance, Control and Dynamics, 15 (1992), 6, 661–665.

[8] Popescu, M. E., Applications of Canonical Transformation in Optimiz-ing Orbital Transfers, Journal of Guidance, Control and Dynamics, 20(1997), 4, 774–779.

[9] Popescu, M. E., Optimal Transfer from Equilateral Libration Points,Acta Astronautica, 15 (1987), 4, 209–212.

[10] Popescu, M. E., Functional Analysis Methods in the Study of the Opti-mal Transfer, Journal of Guidance, Control and Dynamics, 13 (1990),2, 374–376.

[11] Popescu, M. E., and Pelletier, F., Optimal Control for a LagrangianFunctional, Nonlinear Analysis,(to appear).

Mihai PopescuInstitute of Applied Mathematics of the Romanian Academy,Bucharest, ROMANIAE-mail: [email protected]


The study of the water stability in canals withrectangular cross section by means of frequency

analysis

Lucica ROSU, Liliana SERBAN, Dan PASCALE,Cornel CIUREA, Carmen MAFTEI

November 2, 2001

Abstract - The movement of water inside irrigation canals with automatic controllers

occurs in unsteady flow state, due to variations of the consumed flow and to the operation

of the gates. The basic equations of the unsteady water motion are the Saint-Venant

equations, non-linear equations of the hyperbolic type. The mathematical model of the

automatic system was obtained by analytical integration. The study of the operation sys-

tem, by means of frequency analysis, supposes the study of the solutions of the equations,

which define the unsteady motion. The paper outlines the influence of some parameters

of the canal on the automatic negative feedback system stability, in the case of BIVAL

control

Key words and phrases : unsteady water motion; analytical integration; water sta-

bility in irrigation canals.


1 Introduction

The basic equation of the unsteady water motion in open canals, are theSaint-Venant equations, which form a system of non-linear equations, withvariable coefficients, belonging to the differential equations, of the hyper-bolic type. The mathematical model of the automatic system (composed ofthe assembly canal-regulator) was obtained by analytical integration. Theanalysis of the operation system suppose the study of the solutions of theequations which define the unsteady motion, ζ(t), for the level disturbanceand q(t).

199

200 L. Rosu, L. Serban, D. Pascale, C. Ciurea, C. Maftei

2 The mathematical model

The basic equations of unsteady water motion (the Saint-Venant equations)are: the dynamic equation (1) and the continuity equation (2):

∂Z

∂s+

1g· ∂V

∂t+

1g· V · ∂V

∂s+

Q2

K2= 0 (1)

∂A

∂t+

∂Q

∂s= 0 (2)

In these two equations, the following hydraulic elements are functions ofspace(s) and of time (t): Z = Z(s, t) - the water level measured to a da-tum; Q=Q(s, t) - the water discharge; V =V (s, t) average velocity of water;J=J(s, t) - hydraulic slope of the stream (J=Q2/K2), A=A(s, t) -the cross-sectional area of flow; K=A · C · √R - the conveyance factor of the canalsection; C=Ry/n-Chezy’s resistance factor , where, n -roughness factor andg- the acceleration due to gravity. The initial conditions are known for:Q=Q0(s) and Z=Z0(s) when t=0. The upstream and downstream bound-ary conditions are the canal check-gate structures, for which the opening isdetermined by the selected control method.

Starting with the observation of the periodicity of the consumption flowrate and level variation charts, for the linearizing of the Saint-Venant equa-tions the main hypotheses of the low-amplitude oscillation theory and theproperty of Fourier transform are applied. The initial reference state is con-sidered the state of permanent motion. Non-linear equations with variablecoefficients, (1) and (2), become linear equations of second order with con-stant coefficients, which make possible to obtain the general solutions forthe periodical motion case.

All reference values, characteristic for the permanent state, are put downnamely Q0, h0, A0, V0,K0, as well as the values called disturbances, namelyq, ζ, a, v, k. At a given moment, t, the characteristic quantities will be: Q =Q0 + q; h = h0 + ζ ; A = A0 + a; V = V0 + v; K = K0 + k.

The linearized equations of the unsteady motion and their solutions,q(t)and ζ(t), are achieved for two special situations: equation (3) and (4)for the axis s orientated in wave motion’s direction and equation (5) and (6)for the axis s orientated in the opposite direction of the wave motion:

∂2ζ

∂t2− (

c2 − V 20

) · ∂2ζ

∂s2+ 2 · V0 · ∂2ζ

∂s · ∂t+ α

∂ζ

∂s+ β · ∂ζ

∂t= 0 (3)

B0 · ∂ζ

∂t+

∂q

∂s= 0 (4)

Water stability in chanel 201

∂2ζ

∂t2− (

c2 − V 20

) · ∂2ζ

∂s2− 2 · V0 · ∂2ζ

∂s · ∂t− α

∂ζ

∂s+ β · ∂ζ

∂t= 0 (5)

B0 · ∂ζ

∂t+

∂q

∂s= 0 (6)

In these equations, c is the progressive waves velocity, B0 is the canalwidth on water plane, while α and β are coefficients which depend on thegeometrical and hydraulic elements of the canal.

The solutions of wave motion, for automatic controller BIVAL type, ζfrom the linearized equations (3), (5) and q from (4), (6), are achieved im-posing the initial conditions and boundary or the conjugation conditions,specific to this controller type. The BIVAL control concept (Figure 1) de-termines the canal check gate opening, a, based on the variation of twolevels in the downstream canal pool: at the upstream end, Zu, and at thedownstream end, Zd.

The water-checked level, Zi, is given by the relationship:

Zi = k · Zu + (1− k) · Zd (7)

Where: k- the weight coefficient (0 < k < 1).

Fig.1

Assuming that q(ti)d is a variation of water discharge in the downstream

section of the pool, produced at a certain moment ti, then the correctionvalue which must be given to the gate opening, ∆a(ti), is:

∆a(ti) =12· a0

∆h0· ζ(ti)

u +q(ti)u

Q0· a0 (8)


Where: ζu(ti)− the water-level variation and qu(ti) - the water dischargevariation at the upstream end of the canal pool; a0 -initial opening of thegate; ∆h0-the initial head differential: ∆h0 = ZA − Zo,u ; ZA = ct - thewater level into the upstream reservoir of the pool, Zo,u -the water- level atthe upstream end of the canal pool; Q0-the initial water discharge througha check gate.

In the case of BIVAL control concept, the level variation at the up-stream and downstream end of the pool, ζu and ζd, are determined from theexpression of the general solutions of the motion equations:

ζd(s, t) = a1ep1s cos (ωt + q1s + ϕ1) + a2e

p2s cos (ωt + q2s + ϕ2) (9)ζu(s, t) = a1e

−p2s cos (ωt− q2s + ϕ1) + a2e−p1s cos (ωt− q1s + ϕ2)(10)

In the above equations: a1 , a2 , ϕ1 , ϕ2 are integration constants; ω is theoscillation frequency (according to the period of the discharge and levelfunctions); the terms p1 , p2 , q1 , q2 , are amounts depending on the geometricand hydraulic characteristics of the canal.

3 The study of the water stability in canals

Automatic negative feedback control hydraulic systems rise an importanttheoretical and practical problem, that of operational stability. The studyfor wave motion stability was based on the method of linear system analysisin frequency range. The operational analysis of such a system implies thestudy of the solutions the equations describing motion, ζ(t) and q(t). Theautomatic hydraulic system is stable at low input value (flow rate q(t))perturbations, if the output value (water level ζ(t)) perturbations dampenin time. The same system becomes unstable at low perturbations if theζ(t)perturbation increases indefinitely in time.

The pulsation of the sine input signal, qo(t), having a unitary amplitudeand leading to maximum values of the output signal amplitude, ω(t), rep-resents the resonant pulsation of the system. This pulsation highlights thesystem’s unsteady state. The system state at different pulsation ω (π/24div 15π) of the sinusoidal input signal with unit amplitude, qo, was exam-ined by analyzing the system response that means water level variation atthe pool end, ζuand ζd. The analysis revealed either the way in which theparameter k (weight coefficient) and S (bottom slope of canal) affect the sys-tem stability or the resonant pulsation values at which the system loses itsstability. In the harmonic analyses of the function qo(t), it was considered a

Water stability in chanel 203

number n=12 of harmonics. In these conditions, the time unit interval tookvalues from ∆t=2 hours, for the pulsation ω=π/24 (the period of sine signalT=48hours), to ∆t=20seconds, for the pulsation ω=15π (T=8 minutes).

The system behavior, at high and low frequency, is represented in thegraph in the Figure 2.

Fig.2

It was considered a single canal pool of rectangular cross section (sup-plied from a constant-level upstream reservoir) with the length L=1000m,the bottom canal width, b=1m, the water depth h=1.75m, the side slopem=0, the bottom slope of canal S=0.005%, the roughness n=0.02, the down-stream customer discharge Qd=1m3/s and H=0, 137. The Froude numberhas the expression H=Vo/c , where, Vo-average velocity of water at initialmoment; c- the celerity for small gravitational wave.

Fig.3

The parameter u, a measure of the flow resistance, is defined by therelationship u=2 · g· So · L/V 2

o , where So- the initial energy gradient and


g- the acceleration due to gravity. In this graph is presented the water levelvariation ζ, as a function of the pulsation ω and the weight coefficient k. Itmay be noticed that for the value k=0, 5 the resonance phenomenon occursat the pulsation ω=7π.

The change of upstream gate opening, ∆a, in the case of rectangularcross section of the canal (h=1.75m) is represented in the Figure 3. Theinput values were: Qd=1m3/s; q=1m3/s; S=0.019; b=1m; m=0; n=0.02;h=1.75m; H=0.32; u=37, 48 (for k=0.5); u=18, 64 (for k=0) and u=18.64(for k=1). The upstream gate opening curve, ∆a, reveals the unstablebehavior of the system at high pulsation (ω=6div 15π).The same conclusionregarding the unstable operation of the upstream command control systemcan be found out in the references [1].

4 Conclusions

The use of frequency analysis of the automatic feedback controllers opera-tion assumes certain facility: hold out the possibility to obtain very usefulinformation concerning the influence of different geometrical and control pa-rameters on the wave-motion stability and to obtain a global image of thesephysical phenomenon progress.

It was emphasized the unstable operation of the BIVAL command controlsystem at a high pulsation (ω=6πdiv 15π), for the weight coefficient k=0.The same conclusion, regarding the unstable operation of the upstream con-trol system, can be found out by analysis of the graph of the water levelvariation |ζ|max at the end pool, in case of a short pool, for k=0.5.

Although the analytical model brings in certain approximations due tolinearization, it proves to be a practical useful method to emphasize theconditions governing the occurrence of the instability phenomenon in theoperation of automatic hydraulic systems.

References

[1] Hancu, S., and others, 1982, Hydraulics of automated irrigation systems,CERES, Bucharest.

[2] Rosu L., 1999, Design and assessment of automated operation irrigationcanals, Ed. Ovidius University Press, Constanta.

Rosu Lucica, Serban Liliana, Dan Pascale, Cornel Ciurea, Carmen MafteiOvidius University, Blvd. Mamaia 124, 8700-Constantza, ROMANIAE-mail: [email protected]


Translation flows of non-local memory-dependentmicropolar fluids

Valeriu Al. SAVA

November 2, 2001

Abstract - Abstract- The solution of the translation flow problem for a non-local memory-

dependent micropolar fluid is obtained.

Key words and phrases : translation flow, micropolar fluids


1 Basic theory

The theory of non-local, memory-dependent micropolar fluids introduced byEringen [1] deals with a class of fluids which possess internal structures withsome internal characteristic length. When the internal characteristic lengthbecomes comparable to the external characteristic length, nonlocal effectsbecome prominent.

Balance laws of non-local micropolar fluids are:[1], [3]Mass:

D%/Dt + %vk,k = 0, in Ω; (1)

Microinertia:

Djkl/Dt + (εkrmjrl + εlrmjrk)ϑm = 0, in,Ω; (2)

Momentum:

tkl,k + %(fl −Dvl/Dt) + %fl = 0, inΩ; (3)

Moment of momentum:

mkl,k + εlmntmn + %(ll −DIl/Dt) + %(ll − εlmnxlfm)ϑk = 0, inΩ; (4)

Energy:

−%Dε/Dt+ tklakl +mklbkl + qk,k +%h−%fkvk−%(lk− εklmxlfm)ϑk +%h = 0(5)

205

206 V. Al. Sava

whereakl = vl,k + εlkmϑm, bkl = ϑk,l (6)

are the deformation-rate tensor and % =mass density, jkl =microinertia ten-sor, vk =velocity vector, ϑk =giration vector, tkl =stress tensor, mkl =couplestress tensor, fk =body force density, lk =body couple density, fk =nonlocalbody force residual , lk =nonlocal body couple residual, Ik = jklϑl :spindensity, ε =internal energy, h =energy source density, h =nonlocal energyresidual, qk =heat vector.

The body residual are subject to

∫

Ω

(%f , %l, %h)dω = 0. (7)

We employ the usual summation convention over the repeated indicesand an index followed by a comma to denote the partial derivative; εklm

indicate the usual alternating tensor.In [1] Eringen gave a set of constitutive equations for non-local, memory-

dependent micropolar fluids. For the present work we produce here only theresults of the linear constitutive theory of micro-isotropic (i.e. jkl = jδkl)and deformation-rate dependent fluids :

tkl = −πδkl + Tkl, Tkl = 2

t∫

−∞

∫

Ω

A1klmn(s = t)amn(τ)dτdω,

mkl = Mkl = 2

t∫

−∞

∫

Ω

A2klmn(s = t)bmn(τ)dτdω,

(8)

where

Aβklmn = [σβ

1 + 3σβ2 + σβ

3 j′]δklδmn + σβ4 j′δklδmn + (σβ

5 + 3σβ6 j′)δkmδnl

+(σβ7 + 3σβ

8 j′)δknδml + (σβ9 + σβ

10)j′δkmδnl + (σβ

11 + σβ12)j

′δklδml,

σβs = σβ

s (%′, t = τ, t = s, κ), s = 1, 2, · · · , 12; β = 1, 2.(9)

are the material functions. Here and subsequent the accent on top indicatesthe history of the motion. Note that %, j, and κ depend only on time t. Thisis in accordance with the fact that fluids possess no preferred configuration.

Translation flows of micropolar fluids 207

2 Translation motion

We consider the steady translation motion generated in a micro-isotropicand deformation-rate dependent fluid. As is customary, the basic flow isgiven by

vx = f(z), vy = g(z), vz = 0;ϑx = h(z), ϑy = l(z), ϑz = 0.

(10)

Equations of continuity (1) and microinertia (2) are satisfied, showing thatthe fluid is incompressible, % and j are constants. The thermodynamicpressure π is replaced by an unknown pressure p. Body and surface forceand couple residuals and body force and couples can be ignored.

Equation of motion (3) and (4) reduce to

∂Tzx/∂z = ∂p/∂x, ∂Tzy/∂z = ∂p/∂y, (11)

∂Mzx/∂z + Tyz − Tzy = 0, ∂Mzy/∂z + Tzx − Txz = 0, (12)

where p(x, y) is the pressure. Considering that nonlocal effects are importantonly in z−direction, constitutive equations (8) reduce to

Txz =t∫

−∞

d∫−d

α1(t− τ, |z − z|)azx(z) + α2(t− τ, |z − z|)axz(z) dzdτ,

Tzx =t∫

−∞

d∫−d

α2(t− τ, |z − z|)azx(z) + α1(t− τ, |z − z|)axz(z) dzdτ,

Tzy =t∫

−∞

d∫−d

α2(t− τ, |z − z|)azy(z) + α1(t− τ, |z − z|)ayz(z) dzdτ,

Tyz =t∫

−∞

d∫−d

α1(t− τ, |z − z|)azy(z) + α2(t− τ, |z − z|)ayz(z) dzdτ,

(13)

Mxz =t∫

−∞

d∫−d

γ2(t− τ, |z − z|)bxzdzdτ

Mzx =t∫

−∞

d∫−d

γ1(t− τ, |z − z|)bxzdzdτ

Myz =t∫

−∞

d∫−d

γ2(t− τ, |z − z|)byzdzdτ

Mzy =t∫

−∞

d∫−d

γ1(t− τ, |z − z|)byzdzdτ

(14)

208 V. Al. Sava

where

axz(z) = l(z), azx(z) = f ′(z)− l(z), bxz(z) = h′(z),ayz(z) = −h(z), azy(z) = g′(z) + h(z), byz(z) = l′(z)

(15)

α1 = 2[σ17 + (3σ1

8 + σ111 + σ1

12)j], α2 = 2[σ15 + (3σ1

6 + σ19 + σ1

10)j],γ1 = 2[σ2

7 + (3σ28 + σ2

11 + σ212)j], γ2 = 2[σ2

5 + (3σ26 + σ2

9 + σ210)j],

(16)

Equations (11) integrate to give

Tzx = z∂p/∂x, Tzy = z∂p/∂y, (17)

where some constants are set equal zero, since at the plane z = 0,Tzx = Tzy = 0. Further progress requires some knowledge regarding thematerial functions αa and γa. Following Eringen [2, 3] we assume

αa, γa(t−τ, |z − z|) =

α0a,γ0

a(t−τ)r

(1− |z−z|

r

), when |z − z| ≤ r

0, when |z − z| > r,(18)

where a = 1, 2 and r is an internal characteristic length (e.g. the radius ofgiration of the molecules) and α0

a, γ0a are functions of (t − τ). The internal

characteristic length r may be different for the stress and couple stress.We note that

∂

∂z

[1r

(1− |z − z|

r

)]=

1r2

sgn(z − z). (19)

The partial derivatives of (11) and (12) with respect to z give

∂2

∂z2Tzx = 0,

∂2

∂z2Tzy = 0,

∂3

∂z3Mzx + ∂2

∂z2 Tyz = 0,∂3

∂z3Mzy − ∂2

∂z2 Txz = 0.

(20)

Substituting (13) to (15) and (19) into (20) one obtains

α02L[f ′] + (α0

1 − α02)L[l] = 0, α0

2L[g′]− (α01 − α0

2)L[h] = 0 (21)

γ01L[h′′] + α0

1L[g′] + (α01 − α0

2)L[h] = 0

γ01L[l′′]− α0

1L[f ′] + (α01 − α0

2)L[l] = 0, (22)

Translation flows of micropolar fluids 209

where α0a, γ

0a (a=1,2) and the linear difference operator L[.] are defined by

α0

a, γ0a

=

∞∫

0

α0

a(τ), γ0a(τ)

dτ, (23)

L[u] = u(z + r)− 2u(z) + u(z − r). (24)

From (21) we solve for

L[f ′] =(1− α0

1

α02

)L[l], L[g′] = −

(1− α0

1

α02

)L[h] (25)

With these, we eliminate f and g from (22) to obtain

L[h′′]− a2L[h] = 0, L[l′′]− a2L[l = 0 (26)

wherea2 =

[(α0

2

)2 − (α0

1

)2] /

γ01 α0

2 . (27)

The differential-difference equations (26) have the general solution

h, l (z) = A1, A2 sinh(az) + B1, B2z + C1, C2. (28)

We assume strict adherence boundary conditions :

f(z) = 0, g(z) = 0, h(z) = 0, l(z) = 0 at z = ±d. (29)

Using the last two of these conditions in (28) one obtains

h, l (z) = A1, A2[sinh(az)− z

dsinh(ad)

], (30)

where A1, A2 are arbitrary constants,Carrying (30) into (25) and performing the integration under conditions

(29)1,2, we have

f, g (z) =A2,−A1

a

(1− α0

1

α02

)[cosh(az)− cosh(ad)]+

H,G2

(z2 − d2

),

(31)where H and G are arbitrary constants.

Determination of A1, A2,H and G requires that we substitute (30) and(31) into (11) and (12). If we do so we obtain the final solution of theproblem in the form :

f, g(z) =A [cosh(az)− cosh(ad)] + B[z2 − d2]

∂p/∂x, ∂p/∂y, (32)

210 V. Al. Sava

where A =α0

1 − α02

aα02(α

01 + α0

2)d

sinh(ad), B =

2α0

1 + α02

and

h, l(z) =d

(α01 + α0

2) sinh(ad)

(sinh(az)− z

dsinh(ad)

)∂p

∂x,−∂p

∂y

. (33)

We remark that this solution can reduces to solution for Navier-Stokesfluids.

References

[1] A.C.Eringen, Memory-dependent orientable non-local micropolar fluids,Int.J.Engng.Sci.,29 (1991), 12, 1515-1529.

[2] Eringen A.C., Continuum Physics, vol4.PartIII,p.249,New-York: Aca-demic Press 1976.

[3] A.C.Eringen, Linear theory of non-local elasticity and dispersion of planewawes, Int.J.Engng.Sci.,10(1972),4,425-435.

Valeriu Al. Sava,Universitatea tehnica ”Gh.Asachi”, Bd. Copou 11,6600-Iasi, ROMANIA,E-mail: [email protected]


Nonreflecting Far-Field Conditions for UnsteadyAerodynamics and Aeroacoustics

Romeo F. SUSAN-RESIGA and Hafiz M. ATASSI

November 2, 2001

Abstract - The exterior unsteady aerodynamics / aeroacoustic problem for a thin airfoil

in uniform mean flow with a transverse vortical velocity disturbance (gust) is formulated

for a finite computational domain. Exact radiation conditions are imposed on the outer

boundary using the non-local Dirichlet-to-Neumann map. The boundary value problem

for the unsteady velocity potential is solved using the Finite Element Method (FEM).

The main difficulty addressed and solved in this paper is the FEM implementation of the

radiation condition, which is originally formulated for the unsteady pressure and it has

to be implemented for the unsteady velocity potential. Our approach uses only first order

normal derivatives, thus allowing the use of standard finite elements.

Key words and phrases : unsteady aerodynamics, aeroacoustics, nonreflecting condi-

tions, finite element


1 Introduction

A common method used for the numerical solution of wave problems in anunbounded domain [5] is based on truncating the domain via an artificialboundary B, thus forming a finite computational domain Ω bounded by B.A so-called nonreflecting boundary condition (NRBC) is imposed on B toensure that, according to the causality principle, only outgoing waves arepermitted and no spurious wave reflection occurs from B.

For linear aeroacoustic problems the boundary B can be chosen such thatoutside the computational domain the mean flow is uniform. In this case theconvective wave equation can be reduced to the Helmholtz equation and awhole variety of NRBCs can be employed. The collection of NRBCs thathave been proposed can be divided into two sets: nonlocal and local NRBCs.Nonlocal NRBCs, the main example of which is the Dirichlet-to-Neumannmap [8], are exact, while local NRBCs are usually approximate.

211

212 R. F. Susan-Resiga and H. M. Atassi

Whatever the choice of NRBC, it is applied to the unsteady pressure.However, when the Goldstein’s splitting of the unsteady velocity field [6] isemployed one ends up solving a boundary value problem for the acousticvelocity potential. This paper addresses the issue of implementing NRBCsin the context of using the finite element method for solving the acousticpotential boundary value problem. The main issues to be dealt with are theimplementation of the convective derivative in the relationship potential-pressure while using unstructured mesh discretization, and moreover theimplementation of second order partial derivatives coming from the pressurenormal derivative. Note that when using standard finite element formulation,only first order derivatives are naturally computed on the boundary. Weexemplify our approach for the exact non-local Dirichlet-to-Neumann map,but any NRBC which establish a relationship between the pressure normalderivative and the pressure values on B can be used.

2 Thin Airfoil in a Transverse Gust

The governing equations for linear unsteady aerodynamics and aeroacousticsare the linearized Euler equations. The flow variables, velocity V , pressure p,density ρ, and entropy s, are considered to be the sum of their mean valuesU , p0, ρ0, and s0 and their perturbations u′, p′, ρ′, and s′. For a uniformmean flow field U = i1U0, and no entropy perturbations, the linearizedEuler equations reduce to:

D0ρ′

Dt+ ρ0∇ · u = 0 , ρ0

D0u

Dt= −∇p , (1)

where D0/Dt ≡ ∂/∂t + U0∂/∂x1 is the material derivative associated withthe mean flow.

Consider a thin airfoil at a zero angle of attack in the preceding meanflow. According to the velocity splitting theorem, [1] and [6], p. 220, thevelocity perturbation can be considered as the sum

u(x, t) = uv(x, t) + ua(x, t) , (2)

of a vortical component uv and an irrotational component ua. Here weassume no incident acoustic waves. The vortical component is solenoidal,∇ · uv = 0, and it will be purely convected by the mean flow, D0uv/Dt =0. Because the governing equations have been linearized we can consider,without loss of generality, a harmonic upstream vortical disturbance (gust)of the form

uv = i2a2 exp[i(k1x1 − ωt)] , (3)

Nonreflecting Far-Field Conditions 213

where i2 is the unit vector perpendicular to the airfoil, a2 ¿ U0 is the gustamplitude, ω is the gust angular frequency, and k1 = ω/U0 is the wave num-ber. The interaction between the gust and the airfoil will produce a potentialfield ua governed by the equations (1). If we introduce the unsteady velocitypotential φ, defined by ua = ∇φ, the unsteady pressure perturbation p′ willbe

p′ = −ρ0D0φ

Dt. (4)

Since the pressure and density perturbations are related through the meanflow speed of sound c2

0 = p′/ρ′, we obtain the convective wave equationsatisfied by both p′ and φ,

(D2

0

Dt2− c2

0∇2

)p′

φ

= 0 . (5)

We nondimensionalize length with respect to the airfoil half-chord length,L, time with respect to L/U0, the incoming gust with respect to a2, the un-steady potential with a2L, the unsteady pressure with ρ0a2U0 and introduceM = U0/c0 the Mach number. In addition, from now on k1 will denote thereduced frequency k1L = ωL/U0. We then factor out the time dependencep′(x, t) = p′(x) exp(−ik1t) and φ(x, t) = φ(x) exp(−ik1t). As a result, equa-tion (4) becomes

p′ = − ∂φ

∂x1+ ik1φ (6)

With the Prandtl-Glauert transformation,

X1 = x1

X2 = x2

√1−M2 and

PΦ

=

p′

φ

exp(iKMx1) (7)

the convective wave equation (5) becomes the Helmholtz equation,

(∇2 + K2)

PΦ

= 0 , (8)

where K = k1M/(1−M2) and the Laplace operator is considered with re-spect to the X1, X2 variables. At this point, one can choose which variable tosolve for. In [11] we have solved the boundary-value problem for P . The mainadvantage in this case is that one can directly employ the Finite ElementMethod (FEM) [7] thanks to the simple boundary conditions. However, aspecial approximation for the leading edge singularity had to be developed,and the solution can be determined only up to a multiplicative constant.


This constant is computed after obtaining a so-called unit-strength singu-larity solution. The main drawback of this approach is that it cannot begeneralized for loaded airfoils, since no boundary conditions can be specifiedin this case on the airfoil surface.

Scott and Atassi [10] have solved the boundary-value problem for Φusing finite differences on a structured grid. However, we are using the FEMon unstructured triangulation, thus achieving the maximum flexibility indealing with complex geometries and local mesh refinement.


X

X

n

nouter boundary2

1

B

A

Ω

CLE TEW

Figure 1: Computational domain and boundary segments for a thin, un-loaded airfoil.

Because the solution Φ is antisymmetric with respect to X2, we havechosen as computational domain Ω a half circle in the upper-half-plane,Figure 1. On the airfoil surface, the total unsteady velocity perturbationvanish, therefore,

∂Φ∂n

=1√

1−M2exp

[iK

MX1

]on [LE − TE] . (9)

Note that the nonhomogeneous Neumann condition (9) for Φ implies a ho-mogeneous Neumann condition for pressure, ∂P/∂n = 0, which has beenused in [11]. On the upstream boundary segment, A − LE the solution iscontinuous, and in our case we will set, without loss of generality,

Φ = 0 on [A− LE) . (10)

Across the wake the pressure P is continuous, but Φ will have a jump. Forour setup we have P will vanish in the wake W, and from (6) and (7) we get

P = iK

MΦ− ∂Φ

∂X1= 0 on (TE − C] . (11)


Integrating (11) we obtain the expression for Φ in the wake,

Φ(X1) = ΦTE exp[iK

M

(X1 −XTE

1

)]on (TE − C] , (12)

where XTE1 = 1 and ΦTE is the (unknown) value of Φ at the trailing edge

TE. Note that in order to satisfy both the homogeneous Helmholtz equationand (12) in the wake W, the solution must behave like

Φ(X1, X2) = ΦTE exp[iK

M

(X1 −XTE

1

)− K

M

√1−M2X2

], (13)

in the wake W neighborhood, displaying an exponentially decaying behaviorin a direction normal to the wake. Both (10) and (12) imply the homogeneousDirichlet condition P = 0 on [A− LE)

⋃(TE − C], used in [11].

The boundary-value problem for Φ is originally formulated for an infinitedomain. According to the causality principle no waves are reflected frominfinity. The mathematical expression for this far-field radiation condition isthe Sommerfeld condition,

∂P

∂R− iKP = o(

√R) as R →∞ , (14)

where R =√

X21 + X2

2 is the radius. For practical computations, the compu-tational domain is truncated, and the radiation condition (14) is replaced bya nonreflecting condition on the outer boundary B. For example, if B is cho-sen to be a circle of radius RB, one can solve analytically the exterior Dirich-let problem for P , then differentiate the solution with respect to R and takeR → RB. As a result, we have constructed a mapping P |B → (∂P/∂n)|Bcalled Dirichlet-to-Neumann (DtN) map. The expression for the DtN map∂P/∂n = MP on a circle is [8]:

MP (RB, θ) =K

π

∞∑

n=0

′H

(1)n

′(KRB)

H(1)n (KRB)

2π∫

0

P (RB, θ) cos n(θ − θ′) dθ′ , (15)

where H(1)n are the Hankel functions of the first kind, and the prime on the

sum indicates that the first term is halved.Note that the radiation condition is written for the pressure P and not

for Φ. As a result, we should write P and ∂P/∂n in terms of Φ on the circleB. We start by writing the derivative ∂Φ/∂X1 using polar coordinates (R, θ),

∂Φ∂X1

=∂Φ∂R

cos θ − 1R

∂Φ∂θ

sin θ (16)


to obtain the pressure on B as

P (R, θ) = −∂Φ∂R

cos θ +1R

∂Φ∂θ

sin θ + iK

MΦ . (17)

The normal derivative of the pressure is

∂P

∂n≡ ∂P

∂R= −∂2Φ

∂R2cos θ +

sin θ

R

∂2Φ∂θ∂r

− sin θ

R2

∂Φ∂θ

+ iK

M

∂Φ∂R

. (18)

When employing the FEM for solving the boundary-value problem, theabove equation has the main drawback of including second order derivativesin the radial (normal) direction, while the variational formulation introducesonly first order normal derivatives. As a result, by using the Helmholtz equa-tion in polar coordinates,

∂2Φ∂R2

+1R

∂Φ∂R

+1

R2

∂2Φ∂θ2

+ K2φ = 0 (19)

we can replace the second order radial derivative in (18) to obtain:

∂P

∂R=

sin θ

R

∂

∂θ

(∂Φ∂R

− ΦR

)+

(cos θ

R+ i

K

M

)∂Φ∂R

+cos θ

R2

∂2Φ∂θ2

+ cos θ K2Φ

(20)Equations (17) and (20) are the key ingredients for a successful imple-

mentation of the radiation condition (DtN map in this case) when using thepotential Φ. Their main advantage is that only first order radial (normal)derivatives need to be evaluated, while the tangential derivatives are eas-ily computed along a circle. This feature is particularly important when anunstructured triangulation is used inside the computational domain.

4 Finite Element implementation

The computational domain from Figure 1 is triangulated, then we employthe clasical discrete approximation for the solution,

Φh(X) =∑

i

Ni(X)Φi , (21)

where Ni is the shape function for node i (we are using here the piecewiselinear “hat” functions) and Φi are the nodal values, with the sum performedover all nodes. The standard Galerkin Finite Element equations are,

∑

j

Φj

∫

Ω

(−∇Nj · ∇Ni + K2NjNi

)dΩ +

∫

∂Ω

Ni∂Φ∂n

dΓ = 0 . (22)


Of course, due to the localized support of the shape functions, the abovesum is performed over the nodes j adjacent to the node i.

For the airfoil nodes, where the Neumann condition (9) is prescribed, thesecond term in (22) is moved in the right-hand-side and it can be computedanalytically1. For the nodes on the outer boundary B, one has to computefirst [4],

Mij = Mji =∫

BNiMNjdΓ =

KRBπ

∞∑

n=0

′H

(1)n

′(KRB)

H(1)n (KRB)

θi+4θ∫

θi−4θ

θj+4θ∫

θj−4θ

Ni(θ)Nj(θ′) cos n(θ − θ′)dθdθ′ =

KRBπ

∞∑

n=0

′H

(1)n

′(KRB)

H(1)n (KRB)

4(1− cosn4θ)2

n4(4θ)2cosn(θi − θj) . (23)

For n = 0 we have

limν→0

4(1− cos ν4θ)2

ν4(4θ)2= (4θ)2 .

In (23) the indices i and j obviously correspond to nodes on the outerboundary B. It can be seen that the non-local DtN map couples all thenodes on B, thus producing a dense block in the otherwise sparse systemmatrix. Since the Finite Element method employs the local flux rather thanthe normal derivative, in order to apply the DtN map on the outer boundaryB we introduce the local flux

Fi = RB4θ

(∂Φ∂R

)

i

= −∑

j

Φj

∫

Ω

(−∇Nj · ∇Ni + K2NjNi

)dΩ (24)

computed according to (22) for the node i on B, where RB4θ is the boundarysegment length computed by assuming equally spaced nodes. If the nodes

1If xl < xc < xr, then we have

xrZxl

Nc(x) exp(iax)dx =

xcZxl

x− xl

xc − xlexp(iax)dx +

xrZxc

xr − x

xr − xcexp(iax)dx

= − [iaxc − iaxl − 1] exp(iaxc) + exp(iaxl)

a2(xc − xl)+

[−iaxc + iaxr + 1] exp(iaxc)− exp(iaxr)

a2(xr − xc)


on B are ordered counterclockwise, then

(RB4θ)Pi = −Fi cos θi + 0.5 (Φi+1 − Φi−1) sin θi

+(iKRB4θ

M

)Φi (25)

(RB4θ)2(

∂P

∂n

)

i

= 0.5 [(Fi+1 −4θΦi+1)− (Fi−1 −4θΦi−1)] sin θi +

Fi

(4θ cos θi + iKRB4θ

M

)+

[Φi+1 + ((K RB4θ)2 − 2)Φi + Φi−1

]cos θi (26)

With (23), (25), and (26) we can now write the nonreflecting condition onB as

(RB4θ)2(

∂P

∂n

)

i

−∑

j

Mij [(RB4θ)Pj ] = 0 , (27)

where the sum is computed for all nodes j on B.

5 Numerical results

To illustrate the accuracy of the boundary conditions numerical implemen-tation we present an example for an incoming gust with reduced frequencyk1 = 5.0 and a mean uniform flow Mach number M = 0.7. An unstructuredtriangulation with 11669 nodes is employed for the computational domainfrom Figure 1. The system of linear equations arising from the FEM dis-cretization is solved with a BiConjugate Gradient iterative method and anIncomplete LU factorization preconditioner, using the Portable, ExtensibleToolkit for Scientific Computations [3].

The physical quantity of interest is the normalized unsteady pressureamplitude p′/(ρ0U0a2) both on the airfoil and the far-field acoustic radiation.I order to assess the accuracy of our numerical results, we use as a referencethe solution of the gust problems obtained from the Possio singular integralequation [9]. For the far-field (i.e. on the outer boundary B) we use theGreen’s theorem to express P in terms of its value along the airfoil [2].

Figure 2 shows the pressure on the airfoil. An excellent agreement ofthe FEM results with the quasi-analytical integral equation solution is ob-tained. The small wiggles in the FEM solution come from the numericaldifferentiation of the potential Φ required to compute the pressure.

Figure 3 presents the far-field directivity for the radiated sound, i.e.the polar plot of the quantity

√r|p′|/(ρ0U0a2) computed here on the outer

boundary B. The comparison shows an impressive accuracy of our FEM


-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

-1 -0.5 0 0.5 1x1

Possio (real)FEM (real)

???????????????????????????

???????????????????????????

???????????????????

????????????

?????????????????????????

???????????????????????????????????????????

?Possio (imag)FEM (imag)

ccccccccccccccccccc

cccccccc

ccccccccccccccccc

ccccccccccccccc

ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc

c

Figure 2: Dimensionless pressure, p′/(ρ0U0a2), on the thin airfoil subject toa transverse gust with reduced frequency k1 = 5.0 at Mach number M = 0.7.

0

0.05

0.1

0.15

0.2

0.25

-0.1 -0.05 0 0.05 0.1 0.15 0.2

integralFEM

r rr rr rr r rr r rr rr r rr rr r r r rr rr rrrr

rrrrrrrrrrrrr

rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr

rrrrrrrrrrrrrrrrrrrrrrrrrrrr

rrrrrrr

rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr

r

Figure 3: Acoustic radiation from a thin airfoil subject to a transverse gustwith reduced frequency k1 = 5.0 at Mach number M = 0.7. Polar plot ofthe directivity,

√r|p′|/(ρ0U0a2).


method, due to the accurate non-reflecting condition employed, and its care-ful numerical implementation.

6 Conclusions

The paper is focused on formulating and implementing exact boundarynonreflecting conditions for exterior aeroacoustic problems. Although theboundary value problem is formulated for the acoustic velocity potential,the radiation conditions are imposed on the unsteady pressure. On the otherhand, the exact nonreflecting condition employed here is written as a non-local Dirichlet-to-Neumann map for a circle, thus coupling all the presurevalues on the outer boundary.

The main features of our FEM implementation of the nonreflecting con-ditions can be summarized as follows:

• When computing the pressure normal derivative, the second order ra-dial derivative of the velocity potential is replaced by a second or-der tangential derivative. As a result, only first order radial (normal)derivative of the velocity potential need to be computed allowing theuse of standard Finite Element techniques.

• Instead of the radial derivative of the velocity potential we are us-ing the local normal flux, which is the natural quantity computed onthe boundary in the FEM formulation. The tangential derivatives forboth potential and potential flux are easily computed on the circleusing second order schemes. The scaling factors introduced when nu-merically computing the unsteady pressure and its normal derivativekeep the matrix entries of order one, thus enhancing the accuracy ofthe solution.

The above nonreflecting conditions can be applied for any loaded airfoilaeroacoustic problem, provided that the outer boundary is chosen to be acircle after the Prandtl-Glauert coordinate transformation is employed, andthe circle radius is large enough to have a practically uniform mean flowoutside the computational domain.

To assess the accuracy of the boundary conditions and their Finite Ele-ment implementation we solve the model problem of a thin unloaded airfoilin a transverse gust. Our numerical solution is compared with a solutionobtained by solving the Possio integral equation. The agreement is excellentfor both near-field and far-field unsteady pressure.


References

[1] H. M. Atassi, Unsteady Aerodynamics of Vortical flows: early and recentdevelopments. In Aerodynamics and Aeroacoustics, pages 121–172. ed.Fung, K.-Y., World Scientific, 1994.

[2] H. M. Atassi, M. Dusey, and C. M. Davis, Acoustic radiation from athin airfoil in nonuniform subsonic flow. AIAA Journal, 31(1):12–19,1993.

[3] S. Balay, W. D. Gropp, Lois C. McInnes, and B. F. Smith. PETSCcusers manual. Technical Report ANL-95/11 - Revision 2.1.0, ArgonneNational Laboratory, 2000.

[4] O. G. Ernst, A finite-element capacitance matrix method for exteriorhelmholtz problems. Numerische Mathematik, 75:175–204, 1996.

[5] D. Givoli, Numerical Methods for Problems in Infinite Domains.Springer-Verlag, New York, 1998.

[6] M.E. Goldstein, Aeroacoustics. McGraw-Hill International Book Com-pany, New York, 1976.

[7] F. Ihlenburg, Finite Element Analysis of Acoustic Scattering. Springer-Verlag, New York, 1998.

[8] J. B. Keller and D. Givoli, Exact non-reflecting boundary conditions.Journal of Computational Physics, 82:172–192, 1989.

[9] C. Possio, L’azione aerodinamica sul profilo oscillante in un fluido com-pressible a velocita iposonora. L’Aerotehnica, 18(4), 1938.

[10] J. R. Scott and H. M. Atassi, A finite-difference frequency domainnumerical scheme for the solution of the gust response problem. Journalof Computational Physics, 119:75–93, 1995.

[11] R. F. Susan-Resiga and H. M. Atassi, Domain-Decomposition Methodfor Time-Harmonic Aeroacoustic Problems. AIAA Journal, 39(5):802–809, 2001.

Romeo F. Susan-ResigaUniversitatea “Politehnica” Timisoara, Bd. Mihai Viteazu 1,1900-Timisoara, ROMANIAE-mail: [email protected]


Hafiz M. AtassiUniversity of Notre Dame, Notre Dame, IN 46556, U.S.A.E-mail: [email protected]


On the Uniqueness of the Solution of the Initialand Boundary Value Problem for Third Grade

Fluids

Victor TIGOIU

November 2, 2001

Abstract - This paper shows that the solution of the initial and boundary value problem

for a third grade fluid (different from the model employed by Fosdick and Rajagopal, see

Tigoiu [9]) is unique. This result completes the oldest one from [9] and produce an a priori

estimate which is usefulness in the proof of the asymptotic stability of the rest state also.

Key words and phrases : third grade fluids, uniqueness, asymptotic stability

Mathematics Subject Classification (2000) : 76A05, 76A10

1 Introduction

In the last years some new results concerning with polynomial fluids ofsecond and third grade are made into evidence. On a part the existence,uniqueness and dependance on the initial data are discussed in papers likeDunn and Fosdick [2], Cioranescu and Guirault [1], Galdi and Sequeira [4],Passerini and Patria [7], for second grade fluids and Fosdick and Rajagopal[3], Passerini and Patria [7], Tigoiu [9], [10] for third grade fluids. On theother part the problem of the asymptotic stability of the rest state has beenalso discussed (especially in connection with the question of Joseph [5] on the”nonexistence” of polynomial fluids of grade greater then 1 ) in [3], Patria[8] and Tigoiu [10], [11].

It was proved in [10], [11] that there is a strong connection (for instance)between some stability results and some a priori estimate obtained in theproof for uniqueness (for weakly perturbed flows). We shall develop in thefirst part of our paper this ideas. In the second part we shall employ withthe proof of uniqueness for the solution of the initial and boundary valueproblem for the general class of third grade fluids introduced in [9] in thenonlinear case.

223

224 V. Tigoiu

2 Uniqueness for the solution of the initial and boundaryvalue problem. Linear case.

Let an incompressible third grade fluid given by (see [9])

T = −pI + µA1 + α1(A2 −A21) + β1A3 + β2(A1A2 + A2A1)+

+β3(trA21)A1,

(1)

with β1 < 0, µ ≥ 0, β1 + 2(β2 + β3) ≥ 0. In (1) A1, A2, A3 are thefirst three Rivlin - Ericksen tensor fields µ, α1, β1, β2, β3 are constantconstitutive modules which obey the above thermodynamic restrictions (see[9]) and p is the hydrostatic pressure field.

In this section we consider a weakly perturbed (from the rest state) flow.Consequently, the response of the fluid (1) will be insensitive to the secondorder (nonlinear) terms. The flow and continuity equations will be giventhen by

∂v∂t

− µdivA1 − α1∂

∂tdivA1 − β1

∂2

∂t2divA1 + gradp = ρb,

divv = 0.

(2)

The attached initial and boundary value problem is

v(0,x) = vo(x),∂v∂t

(0,x) = ao(x),

v(t,x)|∂Ω = 0.

(3)

where Ω is supposed to be a rigid and fixed domain ( Ωt = Ω for anyt ∈ [0,∞), see also for such problems [3], [5]).

Let be (vi, pi), i = 1, 2 two solutions of the problem (2) - (3) corre-sponding to the initial data (vo

i ,aoi ), i = 1, 2, respectively. Like is habitual

in this kind of problems, we denote by

v ≡ v1 − v2; vo ≡ vo1 − vo

2; ao ≡ ao1 − ao

2; p ≡ p1 − p2; b ≡ b1 − b2. (4)

Due to the linearity of equations (2), the problem verified by (v, p) is similarbut with null initial data. We adapt, in a nontrivial manner, for our problem,a known technique (see Lions [6]). We suppose for this that v ∈ (W 1,2

o (Ω))3

with condition (2)2 and with∂v∂t

∈ (L2(Ω))3,∂A1

∂t∈ (L2(Ω))6,

Uniqueness 225

∂2A1

∂t2∈ (L2(Ω))6, p ∈ W 1,2(Ω) and b ∈ (L2(Ω))3, a conservative field.

After some long but straightforward calculations (multiplying (2)1 with v

and with∂v∂t

and integrating over Ω) we arrive at

ρ

2d

dt‖ v ‖2 +

µ

2‖ A1 ‖2 +

α1

4d

dt‖ A1 ‖2 +

β1

4d2

dt2‖ A1 ‖2 −

− β1

2‖ ∂A1

∂t‖2= 0,

β1

4d

dt‖ ∂A1

∂t‖2 +

µ

4d

dt‖ A1 ‖2 +

α1 + 2ρc2o

2‖ ∂A1

∂t‖2≥ 0,

(5)

where we have employed Friedrichs and Korn’s inequalities and we denotedco the corresponding domain dependent constant (see also [10]). Relations(5) lead us to the a priori inequality

d

dt

β1

4d

dt‖ A1 ‖2 +

α1 + µ

4‖ A1 ‖2 +

ρ

2‖ v ‖2 +

β1

4‖ ∂A1

∂t‖2

+

+µ

2‖ A1 ‖2 +

α1 − β1 + 2ρc2o

2‖ ∂A1

∂t‖2≥ 0,

(6)

for all t ∈ (0, To). We remark that the quantity in accolades is not, strictlyspeaking, an energy. For this reason we shall call our method ”a quasiener-getic method”. So, we denote

E(t) ≡ β1

4d

dt‖ A1 ‖2 +

α1 + µ

4‖ A1 ‖2 +

ρ

2‖ v ‖2 +

+β1

4‖ ∂A1

∂t‖2 − ξβ1

4‖ A1 ‖2,

(7)

for all ξ ∈ R and we evaluate the quantity E(t) + ξE(t)

E(t) + ξE(t) ≥ −ξ2β1 + ξ(α1 + µ + 2ρc2o)− 2µ

4‖ A1 ‖2 +

+ξβ1 − 2(α1 − β1 + 2ρc2

o)4

‖ ∂A1

∂t‖2 .

(8)

We simply remark now that, if ξ ∈ (−∞,−λ), λ > 0 and

−λ ≡ min2(α1 − β1 + 2ρc2

o)β1

,α1 + µ + 2ρc2

o +√

(α1 + µ + 2ρc2o)2 − 8µβ1

2β1

,

226 V. Tigoiu

then it follows E(t) + ξE(t) ≥ 0. Employing the constitutive inequalityβ1 < 0 and Friedrichs’s inequality we obtain

d

dt‖ A1 ‖2 +

α1 + µ + 2ρc2o

β1‖ A1 ‖2≤ 4

β1F (0)e−ξt, (9)

for all t ∈ [0, To). Denoting now ξo ≡ α1 + µ + 2ρc2o

β1and putting into

evidence that ξo < 0 and ξo − ξ > 0, we obtain from (9), for eacht ∈ [0, T0), the a priori estimate for the L2 norm of the field A1

‖ A1(t) ‖2≤ 4

β1

1− e−(ξo−ξ)t

ξo − ξF (0)+ ‖ A1(0) ‖2 e−(ξo−ξ)t

e−ξt. (10)

We denote now M1(t) ≡ 1− e−(ξo−ξ)t

ξo − ξ; P ≡ 1 +

α1 + µ− ξβ1

β1;

M(t) ≡ 2M1(t); N(t) ≡ PM1(t) + e−(ξo−ξ)t, and we easily remark that0 < M1(t) < 1/(ξo − ξ); P > 0. Consequently we can state the followinglemma

Lemma 1. Each solution of the problem (2) - (3) verifies the a priori esti-mate

‖ A1(t) ‖2≤

N ‖ A1(0) ‖2 + M ‖ ∂A1

∂t(0) ‖2

e−ξt,

‖ ∂A1

∂t(t) ‖2≤

N ′ ‖ A1(0) ‖2 + M ′ ‖ ∂A1

∂t(0) ‖2

e−ξ1t,

(11)

for all t ∈ [0, To) and where N’, M’ depend on N, M, on constitutivemodules µ, α1, β1 and on To also.

From Lemma 1 we obtain the main result of this part of the paper

Theorem 1 (uniqueness) Let it be (v1, p1) and (v2, p2) two solutionsof the problem (2) − (3) corresponding to the same initial data. Thenv1(t,x) = v2(t,x) a.e. in Ω, for all t ∈ [0, To).

Moreover, independent on the statement of the theorem it results also that∂v1

∂t(t,x) =

∂v2

∂t(t,x) a.e. in Ω and for all t ∈ [0, To).

Remark 1. The a priori estimate (11) has been used in [11] in order toprove the asymptotic stability of the rest state.

Uniqueness 227

3 Uniqueness for the solution of the initial and boundaryvalue problem. Nonlinear case.

Here we shall employ a known stability criterion (see [9], Chap. III.B) andconsequently (without a significant lost of generality) we shall restrict to theclass of third grade fluids with 3β1 + 2β2 = 0, α1 > 0. The constitutivelaw is then given by

T = −pI + µA1 + α1(A2 −A21) + β1A3 − 3

2β1(A1A2 + A2A1)+

+12β1(trA2

1)A1,

(12)

with µ ≥ 0, α1 > 0, β1 < 0. The flow equations are

ρv = divT + ρb (13)

where T is given in (12) and b is the body forces field (supposed to beconservative). The initial value problem is given in (3)1 and for boundaryconditions we have

v(x, t) |∂Ω= 0, ∇v(x, t) |∂Ω= L0(t,x). (14)

Like in the previous section we shall consider a pair of solutions (vi, pi),i = 1,2 and we shall employ the notations (4). We remark that the initialand boundary conditions are given by

v(x, t) = 0,∂v∂t

(x, t) = 0, v(x, t) |∂Ω= 0, ∇v(x, t) |∂Ω= 0. (15)

We introduce (12) (written for v1 and v2 ) into the flow equations,

we multiply by v and∂v∂t

, respectively, we integrate over Ω and aftersubtracting the resulting relations we arrive to

ρ

∫

Ω

∂v∂t

· vdx + ρ

∫

Ω

∇v2[v] · vdx +12

∫

Ω

(T1 − T2) ·A1(v)dx = 0,

ρ

∫

Ω

∂v∂t

· ∂v∂t

dx + ρ

∫

Ω

∇v[v] · ∂v∂t

dx + ρ

∫

Ω

∇v[v2] · ∂v∂t

dx+

+ρ

∫

Ω

∇v2[v] · ∂v∂t

dx +12

∫

Ω

(T1 − T2) ·A1(∂v∂t

)dx = 0,

(16)

228 V. Tigoiu

where A1(v) = ∇v + (∇v)T , and A1(∂v∂t

) = ∇∂v∂t

+ (∇∂v∂t

)T =∂A1

∂t.

Here T1 and T2 stand for the values of the effective stress from (12),evaluated for v1 and v2 respectively. In order that relations (16) makesense we suppose

v ∈ L∞(0, T ; W 1,40 (Ω) ∩W 3,4(Ω)),

∂v∂t

∈ L∞(0, T ; W 1,40 (Ω) ∩W 2,4(Ω),

∂2v∂t2

∈ L2(0, T ;W 1,20 (Ω)),

∂A1

∂t∈ L∞(0, T ; W 1,4

0 (Ω) ∩W 2,4(Ω),

∂2A1

∂t2∈ L2(0, T ;W 1,2

0 (Ω))

(17)

A very long and carefully made calculus lead us to obtain from (16)

−β1

4d

dt‖ ∇A1 ‖2

L2(Ω)≤d

dt

β1

4d

dt‖ A1 ‖2

L2(Ω) +2µ + α1

4‖ A1 ‖2

L2(Ω) +

+ρ

2‖ v ‖2

L2(Ω) −β1

4‖ ∇A1 ‖2

L2(Ω) −β1

4‖ ∇A1[v] ‖2

L2(Ω) −

−β1

4‖ ∇A1[v2] ‖2

L2(Ω) +β1

4‖ ∂A1

∂t‖2

L2(Ω)

+

+2µ + α1

4B2(T,v,v2) ‖ A1 ‖2

L2(Ω) +ρ

2B1(T,v,v2) ‖ v ‖2

L2(Ω) −

−β1

4B3(T,v,v2) ‖ ∇A1 ‖2

L2(Ω) −β1

4B5(T,v,v2) ‖ ∇A1[v] ‖2

L2(Ω) −

−β1

4B6(T,v,v2) ‖ ∇A1[v2] ‖2

L2(Ω) −β1

4B4(T,v,v2) ‖ ∂A1

∂t‖2

L2(Ω)

(18)

where Bi(T,v,v2) are combinations of ci(T,v2), cj(T,v)andc20(Ω) which are

given bellow

c1(T,v2) = supt∈(0,T )

essmaxx∈Ω

| v2(t,x) |, c2(T,v2) = supt∈(0,T )

essmaxx∈Ω

| ∇L(2) |,

c3(T,v2) = supt∈(0,T )

ess

maxx∈Ω

| ∂L(2)

∂t|

, c4(T,v2) = supt∈(0,T )

essmaxx∈Ω

| L(2) |,

Uniqueness 229

c5(T,v2) = supt∈(0,T )

ess

maxx∈Ω

| ∂v2

∂t|

, c6(T,v2) = supt∈(0,T )

ess

maxx∈Ω

| ∂

∂t∇A(2)

1 |

,

c7(T,v) = supt∈(0,T )

ess

maxx∈Ω

| ∂v∂t

|

, c8(T,v) = supt∈(0,T )

ess

maxx∈Ω

| ∂

∂t∇A1 |

.

Here c20(Ω) is again the constant which results from the application of

Friedrichs and Korn inequalities. In order to obtain (18) the imbeddingtheorems of Sobolev and Kondrachew have also been employed.

If we chose now c′(T,v,v2) = maxBi(T,v,v2); i = 1, 6 then theinequality (18) is transformed into

−β1

4d

dt‖ ∇A1 ‖2

L2(Ω)≤d

dtf(t) + c′(T,v,v2)f(t)+

+β1

4d

dt

(‖ ∂A1

∂t(t) ‖2

L2(Ω) exp(−B4(T,v,v2)t))

exp(B4(T,v,v2)t),

(19)

where we denoted

f(t) =β1

4d

dt‖ A1 ‖2

L2(Ω) +2µ + α1

4‖ A1 ‖2

L2(Ω) +

+ρ

2‖ v ‖2

L2(Ω) −β1

4‖ ∇A1 ‖2

L2(Ω) −β1

4‖ ∇A1[v] ‖2

L2(Ω) −

− β1

4‖ ∇A1[v2] ‖2

L2(Ω) +β1

4‖ ∂A1

∂t‖2

L2(Ω)

We are now ready to give the following lemma

Lemma 2. If the estimate (19) is true then, there is T1 > 0 such that

d

dtf(t) + c′(T,v,v2)f(t) ≥ 0. (20)

for all t ∈ (0, T1).

It is to remark, for the proof, that from (17) the map t 7−→‖ ∇A1(t) ‖2L2(Ω)

is from C1((0,T)). As ‖ ∇A1(t) ‖2L2(Ω)≥ 0 for all t > 0 it results that there

is T0 > 0 such thatd

dt‖ ∇A1 ‖2

L2(Ω) ≥ 0 , for all t ∈ (0, T0). Similarly,

there is T ′0 > 0 such thatd

dt

(‖ ∂A1

∂t(t) ‖2

L2(Ω) exp(−B4(T,v,v2)t))≥ 0,

230 V. Tigoiu

for all t ∈ (0, T ′0). We denote now T1 ≡ minT0, T′0 and then the estimate

(20) is true for all t ∈ (0, T1).We remark now that from the initial data (3)1 and from (20) we obtain

0 ≤ β1

4d

dt‖ A1 ‖2

L2(Ω) +2µ + α1

4‖ A1 ‖2

L2(Ω) +

+ρ + 2c′(T,v,v2)

2‖ v ‖2

L2(Ω) −β1 − 4c′(T,v,v2)

4‖ ∇A1(t) ‖2

L2(Ω)

(21)

Now we can state the following principal lemma

Lemma 3. For any c > 0 there is T2 > 0, such that for any t ∈ (0, T2)the inequality

d

dt

[‖ A1(t) ‖2

L2(Ω) −

−c ‖ ∇A1(t) ‖2L2(Ω)

(1− exp(−M2

ct)

)]exp(−M1t

≤ 0

(22)

holds. In (22) we have

M1(T,v,v2) =2µ + α1 + 2(ρ + 2c′(T,v,v2))c2

0(Ω)−β1

> 0,

M2(T,v,v2) =β1 − 4c′(T,v,v2)

β1> 0.

(23)

For the proof, it results from (21), with Friedrichs and Korns’s inequalitiesthat

d

dt‖ A1(t) ‖2

L2(Ω) − M1(T,v,v2) ‖ A1(t) ‖2L2(Ω) −

− M2(T,v,v2) ‖ ∇A1(t) ‖2L2(Ω) ≤ 0,

(24)

where we have employed the notations (23). Then for any c > 0 it resultsthat

d

dt

(‖ A1(t) ‖2

L2(Ω) −c ‖ ∇A1(t) ‖2L2(Ω)

)exp(−M1t)

+

+cd

dt

[‖ ∇A1(t) ‖2

L2(Ω) exp(−(M1 +

M2

c)t

)]exp

(M2

ct

)≤ 0.

(25)

Uniqueness 231

We see now that the map t 7−→ ‖ ∇A1(t) ‖2L2(Ω) exp

(−(M1 +

M2

c)t

)

is C1((), T )). Then a similar reasoning to those performed for (20) gives:there is 0 < T2 < T1 such that for any t ∈ (0, T2) we shall haved

dt

[‖ ∇A1(t) ‖2

L2(Ω) exp(−(M1 +

M2

c)t

)]≥ 0 and consequently we ob-

tain (22). Employing now the boundary conditions (16) we arrive to

‖ A1(t) ‖2L2(Ω) −c ‖ ∇A1(t) ‖2

L2(Ω)

[1− exp(−M2

ct)

]≤ 0, (26)

for any t ∈ (0, T2) and c > 0. On the other hand, for any fixed t ∈ (0, T2)the map

c 7−→‖ A1(t) ‖2L2(Ω) −c ‖ ∇A1(t) ‖2

L2(Ω)

(1− exp(−M2

ct)

)≡ F (c) (27)

is a continuous function of c ∈ R+. Let us now consider a sequence cn > 0,cn −→ 0. As F (cn) ≤ 0 for any n, it results that lim

n→∞F (cn) ≤ 0 and it

follows immediately that ‖ A1(t) ‖2L2(Ω)≤ 0 for any t ∈ (0, T2) (then by

continuity we prolong the result on the whole interval). So we arrived to thefollowing main result

Theorem 2 (uniqueness) Let be (vi, pi), with i = 1,2 two solutions ofthe problem (12), (13), (3)1,2 and (14). Then, for T > 0, we have

v1(x, t) = v2(x, t), (28)

a.e. in Ω × [0, T ). Moreover, the equality is established also between thefirsts Rivlin - Ericksen tensors

A(1)1 (x, t) = A(2)

1 (x, t), (29)

a.e. in Ω× [0, T ).

References

[1] Cioranescu D. and Girault V., Weak and classical solutions of a familyof second grade fluids, Int. J. Non -Linear Mechanics, 32, 2, (1997),317-335.

232 V. Tigoiu

[2] Dunn J.E. and Fosdick R. L., Thermodynamics, stability and bounded-ness of fluids of complexity two and fluids of second grade, Arch. Rat.Mech. Anal., 56(1974), 191 - 252.

[3] Fosdick R. L. and Rajagopal K. R., Thermodynamics and stability offluids of third grade, Proc. Roy. Soc. London, A, 339(1980), 351 - 377.

[4] Galdi G. P. and Sequeira A., Further existence results for classical so-lutions of the equations of second grade fluids, Arch. Rat. Mech. Anal.,128 (1994), 297.

[5] Joseph D. D., Instability of the rest state of fluids of arbitrary gradegreater then one, Arch. Rat. Mech. Anal., 75(1981), 251 - 256.

[6] Lions J.L., Quelques Methodes de Resolution des Problemes aux Limitesnon Lineaires, Dunod, Gauthier-Villard,Paris, 1969.

[7] Passerini A. and Patria M. C., Existence, uniqueness and stability ofsteady flows of second and third grade fluids in an unbounded ”pipe-like” domain. Int. J. Non-Lin. Mech., 35(2000), 1081- 1103.

[8] Patria M. C., Stability questions for a third grade fluid in exteriordomains, Int. J. Non - Linear Mechanics, 24, 5(1989), 451 -457.

[9] Tigoiu V., Wave propagation and thermodynamics for third grade flu-ids, St. Cerc. Mat., 39, 4(1987), 279 - 347.

[10] Tigoiu, V., Weakly perturbed flows in third grade fluids , ZAMM, 80,6(2000), 423- 428.

[11] Tigoiu V., Does a Polynomial Fluid of Third Grade Exist ? (A Hinton the Validity of Using Coleman and Noll’s Approximation Theorem),Proceedings of the 5-th Int. Sem. ”Geometry, Continua and Microstruc-tures”, (2001), 207 - 220.

Victor TigoiuFacultatea de Matematica, Universitatea din BucurestiStr. Academiei 14, 70109 Bucuresti, RomaniaE-mail: [email protected]

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