large-eddy simulation of turbulent flowsmech.vub.ac.be/thermodynamics/phd/bamdad.pdf ·...

Dept. of Fluid MechanicsVrije Universiteit BrusselPleinlaan 2, B-1050Belgium

Large-Eddy Simulation of TurbulentFlows

Application to Low Mach Number and

Particle-Laden Flows

Bamdad Lessani

Ph.D. Thesis

March 2003Promoter: Prof. Chris Lacor

Proefschrift ingediend tot het behalen van de graad vandoctor in de Toegepaste Wetenschappen

Acknowledgment

I would like to thank several people who helped me to complete thisthesis. I would especially like to thank my advisor, Professor Chris Lacor,for his enthusiasm about this research, continuous support, endless patienceand unfailing encouragement that he has given to me.

I would also like to thank my reading committee members, ProfessorsCharles Hirsch, Gregoire Winckelmans, Pierre Comte, Martine Baelmans,Patrick Kool, Johan Deconinck and Jean Vereecken for their helpful com-ments on a draft of this dissertation.

This work has benefited greatly from discussion with many graduate stu-dents, among them Sergey Smirnov, Jan Ramboer, Tim Broeckhoven andMark Brouns. Especial thanks to Sergey Smirnov who also provided me withthe incompressible carrier flow solver for my particle-laden flow simulations.

This work would not be possible without constant help of Alain Wery,our system administrator, who was always willing to kindly solve my net-work related problems. Many thanks to Jenny D’Haes for her administrativeassistance.

Financial support for this work was partially provided by the FlemishScience Foundation (FWO) under the grants G.0170.98. and G.0130.02.This support is gratefully acknowledged.

Abstract

Large-Eddy Simulation (LES) of low Mach number compressible turbu-lent flows has been investigated.

An efficient algorithm has been developed and successfully applied to twotest cases such as the channel flow and cavity flow. An implicit time accurateapproach combined with multigrid, residual smoothing, local time steppingand preconditioning was used. In general, due to the restriction imposed onthe time step by the physics of the flow on one hand, and the stiffness of theequations at low Mach number on the other hand, the advantage of a fullyimplicit approach over an explicit one for LES is not obvious. It was shownthat the present approach, for the test cases considered, allows an efficiencygain of a factor 4 to 7 compared to the use of a purely explicit approach.

Different multigrid strategies like V-cycle, W-cycle and their sawtoothcounterparts have been tested and finally, V-sawtooth cycle was demon-strated to be the most robust. A residual smoothing which also takes intoaccount the clustering of the mesh is used as an additional method for theconvergence acceleration. The physical time derivative is discretized with amulti-step method. Both the trapezoidal method and the second order back-ward Euler method have been tested, and the advantage of the trapezoidalmethod over the backward Euler has been shown. A dual time steppingmethod in which a pseudo-time derivative is added to the equation, is usedto solve the implicit equation. At each implicit time step, the steady statesolution in pseudo-time gives the values of independent variables at the newtime step.

Preconditioning is applied to the pseudo-time derivative to remove thestiffness of the equations. The preconditioning parameter should also takeinto account the unsteady effects. As a result of unsteadiness, a global pre-conditioning parameter has to be used in the entire flowfield.

Large-eddy simulation of particle-laden flows is another part of this thesis.A one-way coupling method, which neglects the effect of particles on thecarrier flow, is used. An incompressible finite volume solver is used for thecarrier flow computations.

Particles are supposed to be spherical and the particle motion is governedonly by the drag force. To calculate the fluid velocity at particle locationsdifferent interpolation methods such as the first, second and third order in-terpolations were tested.

Lycopodium, glass and copper particles with different diameters havebeen considered. The results of the channel flow simulation are in goodagreement with the DNS and other LES calculations found in the literature.

Contents

1 LES of Low Mach Number flows 31.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Literature Survey . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 Filtered Navier-Stokes Equations . . . . . . . . . . . . . . . . 71.4 Subgrid-Scale Modeling . . . . . . . . . . . . . . . . . . . . . . 9

1.4.1 Dynamic Smagorinsky Model . . . . . . . . . . . . . . 11

2 Convergence Acceleration Techniques 152.1 Explicit Runge-Kutta Schemes . . . . . . . . . . . . . . . . . . 15

2.1.1 Some of the Characteristics of the Runge-Kutta Schemes 162.1.2 Dual Time Stepping With Explicit Runge-Kutta . . . . 18

2.2 Analysis of Temporal Discretization . . . . . . . . . . . . . . . 202.3 Multigrid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.3.1 Grid Transfer Operators . . . . . . . . . . . . . . . . . 292.4 Residual Smoothing . . . . . . . . . . . . . . . . . . . . . . . . 332.5 Preconditioning . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.5.1 Non-Conservative Preconditioned Equations . . . . . . 362.5.2 Conservative Preconditioned Equations . . . . . . . . . 372.5.3 Dual Time Stepping . . . . . . . . . . . . . . . . . . . 38

2.6 Time Correction . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3 Comparative Study 493.1 Influence of Preconditioning on Multigrid . . . . . . . . . . . . 493.2 Influence of the Number of Inner Iterations . . . . . . . . . . . 503.3 Influence of the Time Correction Technique . . . . . . . . . . 533.4 Relation Between the Time Integration and the Physical Time

Step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543.5 The Time Step and Preconditioning Parameters . . . . . . . . 55

4 Final Results 594.1 Channel Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.1.1 Test Case Description and Turbulence Statistics . . . . 60

1

2 CONTENTS

4.1.2 One Dimensional Energy Spectra . . . . . . . . . . . . 654.2 Cavity Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.2.1 Test Case Description and Turbulence Statistics . . . . 664.2.2 The Mean Velocity Field . . . . . . . . . . . . . . . . . 684.2.3 The Three-Dimensional Picture . . . . . . . . . . . . . 72

5 Particle Laden Flows 775.1 Introduction and Literature Survey . . . . . . . . . . . . . . . 775.2 Incompressible Navier-Stokes Solver . . . . . . . . . . . . . . . 805.3 Finite Volume Compact Scheme . . . . . . . . . . . . . . . . . 82

5.3.1 Convective Fluxes . . . . . . . . . . . . . . . . . . . . . 835.3.2 Viscous Fluxes . . . . . . . . . . . . . . . . . . . . . . 855.3.3 Application to Channel Flow . . . . . . . . . . . . . . . 86

5.4 Calculation of the Particle Trajectories . . . . . . . . . . . . . 905.4.1 Interpolation . . . . . . . . . . . . . . . . . . . . . . . 925.4.2 Numerical Details . . . . . . . . . . . . . . . . . . . . . 95

5.5 Particle-Laden Channel Flow . . . . . . . . . . . . . . . . . . 985.5.1 Lycopodium Particles . . . . . . . . . . . . . . . . . . . 995.5.2 Glass Particles . . . . . . . . . . . . . . . . . . . . . . 1065.5.3 Copper Particles . . . . . . . . . . . . . . . . . . . . . 106

6 Conclusion and Future Work 1136.1 Compressible Low Mach Number Flows . . . . . . . . . . . . . 1136.2 Particle-Laden Flows . . . . . . . . . . . . . . . . . . . . . . . 115

Chapter 1

LES of Low Mach Numberflows

1.1 Introduction

Turbulent flows have an infinite variety ranging from the flow of blood in ourbody to the atmospheric flows. Everyday life gives us an intuitive knowledgeof turbulence in fluids; during air travel, one often hears the word turbulencegenerally associated with the fastening of seat-belts. The flow passing anobstacle or an airfoil creates turbulence in the boundary layer and developsa turbulent wake which will generally increase the drag exerted by the flowon the obstacle. The majority of atmospheric or oceanic currents cannot bepredicted accurately and fall into the category of turbulent flows, even inthe large planetary scales. Galaxies look strikingly like the eddies which areobserved in turbulent flows such as the mixing layer, and are, in a way ofspeaking, the eddies of a turbulent universe. Numerous other examples ofturbulent flows arise in aeronautics, hydraulics, nuclear and chemical engi-neering, oceanography, meteoroly, astrophysics and internal geophysics. Aclear understanding of this physical phenomena is one of the most essentialand important problems of applied science.

In the past, due to the complexity of the governing equations of the flow,analytical solutions of these equations were restricted to very simple casesand, as a result an experimental approach was the only method to tackle thisproblem. The experimental approach is mainly based on similitude so thatmeasurements made on one system, which is generally in the laboratory, canbe used to describe the behavior of other systems.

A large part of the fluid mechanics community are still involved in ex-perimental analysis, but by the rapid growth of computer power and thereduction of their prices, experiments are becoming more and more expen-

3

4 CHAPTER 1. LES OF LOW MACH NUMBER FLOWS

sive compared to the numerical approaches. Furthermore, in many situationslike hypersonic flows or combustion problems with high temperature, mea-surements are very difficult or even impossible.

In the numerical approach the governing partial differential equations ofthe flow, which after some simplifications are called Navier-Stokes equations,are discretized in space and a time marching method is used to deal with thetime derivative term.

Three main procedures are now in use for the computation of turbulentflows, Direct Numerical Simulation (DNS), Large-Eddy Simulation (LES),and an statistical approach based on the Reynolds Averaged Navier-Stokes(RANS) equations.

The most accurate approach is DNS and the Navier-Stokes equations aresolved without averaging or approximation other than numerical discretiza-tions whose errors can be estimated and controlled. This method does notneed any modeling and all the motions contained in the flow are resolved.The results of a DNS contain very detailed information about the flow ata large number of grid points. These results can be used to produce sta-tistical information. DNS is used as a research tool for understanding themechanisms of turbulence production, energy transfer, dissipation, noise pro-duction, drag reduction, and many other physical aspects of turbulence. In adirect numerical simulation all of the kinetic energy dissipation, which occurson the smallest scales, is captured. The size of the grid must not be largerthan a viscously determined scale, called the Kolmogorov scale, η. If L isthe characteristic length of the physical domain, the number of points in onedirection should be of the order,

N ∼ L

η

with the kinematic viscosity ν(m2/s) and the dissipation per unit massε(m2/s3) the Kolmogorov length scale η = (ν3/ε)1/4. The dissipation perunit mass ε can be approximated as ε ∼ u3/L where u is a characteristic ve-locity of the flow. Finally the number of points for a resolved DNS in threedimensions can be estimated as,

N ∼ (L

η)3 ∼ (

uL

ν)9/4 = Re9/4

For the high Reynolds numbers the number of points enormously increasesand obviously DNS is restricted to relatively low Reynolds numbers.

A method to overcome this problem is called Large-Eddy Simulation(LES) in which the low-frequency modes in space (large eddies) are directlysimulated and the energy exchange with the high-frequency modes (small

1.2. LITERATURE SURVEY 5

eddies), which are not explicitly simulated, is modeled using a subgrid scalemodel. The justification for such a treatment is that the larger eddies containmost of the energy and are clearly test case dependent whereas the smallereddies are more universal and easier to model.

To separate the large from the small scales, LES is based on the definitionof a filtering operation. The Navier-Stokes equations are filtered and theeffect of small scales appears through a subgrid-scale (SGS) stress term.To close the system, an expression for the SGS stresses must be obtained.Similarly to DNS, LES provides a three-dimensional, time dependent solutionof the Navier-Stokes equations. Thus, it still requires fairly fine meshes.Among the objectives of LES are to study more complex configurations closerto those of engineering interest and at Reynolds numbers beyond the reachof DNS.

If one is only interested in knowing the averaged quantities ReynoldsAveraged Navier-Stokes (RANS) approach is the best choice. In RANS ap-proaches to turbulence, all of the unsteadiness is averaged out. If the flowis steady, time averaging is used as a statistical averaging. While the smallscales are somewhat universal, the large ones are affected very strongly bythe boundary conditions. The complexity of turbulence makes it impossiblefor a single RANS model to represent all turbulent flows. Some adjustmentof the constants is often required.

Another approach known under different names such as Unsteady ReynoldsAveraged Navier-Stokes (URANS), Very Large-Eddy Simulation (VLES),Semi-Deterministic Simulation (SDS), and sometimes Coherent StructureCapturing (CSC) calculates the average field and certain low-frequency modesin time, and models the remaining random part, (for a review see Sagaut[96]).

1.2 Literature Survey

Large-eddy simulation has become one of the major tools to investigate thephysics of the turbulence. However, the majority of computations found inthe literature are either incompressible ([32], [33], [50], [52], [13], [59], amongothers) or compressible with moderate or high Mach number ([80]), [108],[26], [67], [24], among others) and fewer articles can be found for LES of lowMach number compressible flows. We note that, if the Mach number in theentire flowfield is low, compressibility can be neglected and incompressibleequations can be used. Many problems, however, contain some regions withvery low Mach numbers while other regions are compressible so that thecompressible equations must be used. If the size of low-speed regions is smallthey will have little effect on the convergence, but if the size of these regions


is large they completely dominate the convergence process. Low-speed flowswith strong heat addition are another type of problems that are compressiblebecause of density variations induced by heat addition. The most commonexamples are combustion problems and flows with heat transfer. For all ofthese low-speed problems a compressible code must be used and the problemof stiffness should be carefully addressed.

The LES of low Mach number with compressible codes has not beenadequately discussed in the literature. To the author’s knowledge, the onlyLES of very low Mach number compressible channel flow is that of Wanget al. [116]. They studied a turbulent channel flow with significant heattransfer using a compressible code. In their article the numerical procedureis briefly described and they put emphasis on the physical results. Arad etal. [3] used an implicit time marching method combined with multigrid forLES of compressible flows with a minimum Mach number of 0.15 withoutusing preconditioning. In the large-eddy simulation of Pierce [87], appliedto reacting flows, a variable-density approach has been used to deal withthe problem of compressibility at low Mach number. Chidambaram et al.[16] studied the isotropic decaying turbulence at low Mach number with acompressible preconditioned solver.

For a review of the implicit methods that preserve the possibility of un-steady applications at all speeds see Mary et al. [75], Shuen et al. [100],Buelow et al. [10] and Dailey et al. [22]. Dailey et al. [22] are among thefew investigators who have reported the application of multigrid to precondi-tioned solvers, especially unsteady solvers. They demonstrated the effective-ness of this method for unsteady lid driven laminar cavity flow and pulsatilechannel flow.

In this thesis we deal with the large-eddy simulation of very low Machnumber flow with a compressible code combined with preconditioning andmultigrid, with a focus on the numerical difficulties and the methods toovercome them. The LES code is an extension of the compressible ReynoldsAveraged Navier-Stokes (RANS) solver EURANUS, Lacor et al. [60], [61],Alavilli et al. [1], Kang et al. [51], Rizzi et al. [92] and Zhu et al. [120].Combined with preconditioning, Euranus uses a unified computing methodfor all Mach numbers, ranging from low subsonic [37] to hypersonic regimes,[1], [120].

The preconditioning technique was applied successfully to different steadyand unsteady RANS flows, Hirsch et al. [40]. EURANUS contains an efficientmultigrid solver, Zhu et al. [120], which has also been used for LES Lessaniet al. [70], based on the Full Approximation Storage (FAS) scheme, Brandt[9].

The LES formulation and the filtering procedure are presented in chapter

1.3. FILTERED NAVIER-STOKES EQUATIONS 7

1. The Smagorinsky model is used for subgrid-scale modeling. Both theclassical model and the dynamic model, Germano et al. [32], are availablein the code. The detailed formulation of the time discretization such as theRunge-Kutta methods, preconditioning, multigrid, time extrapolation andtime correction are explained in chapter 2. Dual time-stepping is used toupdate the equations in time and the physical time derivative is discretizedeither with a backward differencing method or with a trapezoidal scheme.The advantage of trapezoidal scheme over backward differencing, the rela-tionships between preconditioning and multigrid, and other related issues arediscussed in chapter 3. In chapter 4 the method is applied to channel andcavity flows. Chapter 5 is about particle-laden turbulent flows. And finally,in chapter 6 a brief summary of the results and conclusions is given.

1.3 Filtered Navier-Stokes Equations

In LES one computes the motion of large-scale structures, while modelingthe nonlinear interactions with the small-scales. The governing equations forthe large eddies are obtained after filtering. The filtering operation can bewritten in terms of a convolution integral:

f(x) =∫D

G(x − x′)f(x′)dx′ (1.1)

Where f is a turbulent field, G is some spatial filter and D is the flow domain.We note that the minimum width of the spatial filter could be equal to thegrid spacing. For compressible flow the Favre averaging is used rather thanthe standard one. A Favre-filtered variable is defined as, φ = ρφ/ρ. Thefiltered Navier-Stokes equations are:

∂ρ

∂t+

∂ρui∂xi

= 0 (1.2)

∂ρui∂t

+∂ρuiuj

∂xj= − ∂p

∂xi+

∂σij∂xj

(1.3)

∂ρE

∂t+

∂(ρEuj + puj)

∂xj=

∂σijui∂xj

− ∂qj∂xj

(1.4)

The momentum equation can be further worked out as:

∂ρui∂t

+∂ρuiuj

∂xj= − ∂p

∂xi+

∂σij∂xj

+∂(σij − σij)

∂xj︸︷︷︸A

− ∂τij∂xj︸︷︷︸B


Two unclosed terms appear, term A is due to the nonlinearity of the viscousstresses, and term B is the divergence of the SGS stresses

τij = ρ(uiuj − uiuj) (1.5)

σij and σij are defined as:

σij = µ(∂ui∂xj

+∂uj∂xi

− 2

3

∂uk∂xk

δij)

σij = µ(∂ui∂xj

+∂uj∂xi

− 2

3

∂uk∂xk

δij)

Term A is always neglected, i.e., σij = σij, while term B is modeled

τij − 1

3τkkδij = −2µt(Sij − 1

3Smmδij) (1.6)

Where Sij =12(ui,j + uj,i) is the magnitude of large-scale stress-rate tensor,

the eddy viscosity is,µt = ρC∆2|S| (1.7)

with ∆ = (∆x∆y∆z)1/3 the filter width and |S| = (2SijSij)

1/2. The traceof the SGS stresses, τii, can be either modeled, Moin et al. [80], or simplyneglected, Erlebacher et al. [29]. In the present case it is neglected. Thefiltered energy equation (1.4),

∂ρE

∂t+

∂(ρEuj + puj)

∂xj=

∂σijui∂xj

− ∂qj∂xj

− ∂(ρEuj − ρEuj)

∂xj︸︷︷︸C

− ∂(puj − puj)

∂xj︸︷︷︸D

+∂(σijui − σijui)

∂xj︸︷︷︸E

− ∂(qj − qj)

∂xj︸︷︷︸F

(1.8)

Four unclosed terms C, D, E and F appear, here:

qj = −k∂T

∂xjand qj = −k

∂T

∂xj

Term F is neglected i.e. qj = qj. Term E is the subgrid-scale contributionto the viscous dissipation. Vreman et al. [114] showed that at moderate orhigh Mach number modeling term E improves the results. In the present

1.4. SUBGRID-SCALE MODELING 9

calculations the Mach number is always very low and term E is neglected.The subgrid-scale heat-flux Qi is modeled by using an eddy diffusivity SGSmodel.

Qi = ρcp(T ui − T ui) = −µtcpPrt

∂T

∂xi(1.9)

Prt = 0.5 or is calculated dynamically using the dynamic procedure of Ger-mano et al. [32]. Using the equation of state, term D can be closed

puj − puj = ρR(T uj − T uj) =γ − 1

γQj

The last remaining term is the convective term C, with E the total energyfor unit mass E = ε + uiui/2:

ρ(ujE − ujE) = ρ(ujε − uj ε) + ρ(˜ujuiui2

− ujuiui2

) =

ρ(ujε − uj ε) + ρ(˜ujuiui2

− ujuiui2

)︸︷︷︸G0

+ρ(ujuiui

2− uj

uiui2

) =

Qj

γ− uj

τii2

(1.10)

It can be argued that τii/2 is small compared to the thermodynamic pressureand it can be neglected. Term G is neglected. For the present calculationsthis approximation did not cause any problem, but we do not know howaccurate this estimation is in a general case. Equation (1.10) may also bewritten in the following way,

ρ(ujE − ujE) =Qj

γ+ Dj

with Dj = ρ( ˜ujuiui

2− uj

uiui

2) a turbulent transport term. Neglecting τii and

G in equation (1.10) is equivalent to neglecting the turbulent transport termDj.

And finally, the filtered energy equation (1.8),

∂ρE

∂t+

∂(ρEuj + puj)

∂xj=

∂σijui∂xj

− ∂qj∂xj

− ∂Qj

∂xj(1.11)

1.4 Subgrid-Scale Modeling

In the context of large-eddy simulation τij defined in (1.5) is called thesubgrid-scale stress. The name “stress” comes from the way in which it


is treated rather than its physical nature. It is in fact the momentum fluxcaused by the action of the unresolved scales.

The most commonly used subgrid-scale (SGS) model is the one proposedby Smagorinsky [101]. It is based on the notion that the principal effects ofthe SGS stress are increased transport and dissipation. As these phenomenaare due to the viscosity in laminar flows, it seems reasonable to have an eddyviscosity model which is modeled according to equation (1.6).

A variety of SGS models have been used by different researchers, such astwo-point closures, scale-similar models, and one equation models, to namea few among others. Kraichnan [58], by using a two-point closure model forisotropic turbulence, computed the energy transfer from the resolved to theunresolved scales. He then defined an eddy-viscosity in wave space, whichhampers its extension to finite-difference schemes and to complex geometries.To overcome this shortcoming, based on the theory of isotropic turbulence,Metais and Lesieur [84] derived the structure function model which is verysimilar to the Smagorinsky model, see also Ducros et al. [25]. Based on theassumption that the most active subgrid scales are those closer to the cut-off, and that the scales with which they interact most are those right abovethe cutoff, Bardina et al. [5] developed the scale-similar models. Anotherapproach is the one-equation models, in which in a similar way to RANS, atransport equation is solved for the subgrid-scale energy to obtain the veloc-ity scale. One-equation models have been used by Schumann [98], Horiuti[43] and Carati et al. [12], among others. However, as stated by Piomelli[88], based on the results obtained using one-equation models, the expenseinvolved in solving an additional equation does not seem to be justified byimprovements in the accuracy. They are more methods for the subgrid-scalemodeling in the literature. The interested reader can refer to the book ofSagaut [96] for a complete review.

In the present thesis the Smagorinsky model is used and will be explainedfurther in detail. The Smagorinsky model can be derived in a number of waysincluding with a sort of mixing-length assumption in which the eddy viscosityis assumed to be proportional to the SGS characteristic length scale (alsocalled filter width) ∆, and to a characteristic turbulent velocity v∆ = ∆|S|.Where |S| = (2SijSij)

1/2 is a typical velocity gradient at ∆, determined withthe aid of the large scale (filtered-field) deformation tensor Sij. In the contextof finite volume no explicit filtering is used, and the filter is implicit in thenumerical grid which does not support scales smaller than the grid size. Thefilter width used in the Smagorinsky model is taken as the third root of thecontrol volume. The eddy viscosity model can finally be written as,

νt = C[∆(1− e−y+25 )]2|S| (1.12)

The coefficient C is a constant which should be combined with a near wall


damping (1 − e−y+25 ) to vanish near the solid boundaries. The near wall

damping is a function of the wall coordinate y+ = uτyw

ν. uτ = (τw/ρ)1/2 is

the friction velocity, τw is the wall shear stress, ρ is the density of the fluidand yw is the distance from closest the wall.

In the present thesis the filter width ∆ = (∆x∆y∆z)2/3, with ∆x, ∆yand ∆z the dimensions of computational cell. However, generally speaking,the width of the filter ∆ need not have anything to do with the grid size hother than the obvious condition that ∆ > h.

Another way is to be calculated using a dynamic procedure which is thesubject of the next section. Clearly, the Smagorinsky eddy viscosity is asimple algebraic model, but since the small scales tend to be more homoge-neous and isotropic than the large ones, it is hoped that even simple modelscan describe their physics accurately. Moreover, since the SGS stresses onlyaccount for a fraction of the total stresses, modeling errors should not af-fect the overall accuracy of the results as much as in the Reynolds AveragedNavier-Stokes (RANS) modeling approach.

For the temperature, one introduces an eddy diffusivity αt such that,

T ui − T ui = −αt∂T

∂xi

The eddy diffusivity is related to the eddy viscosity via the turbulent Prandtlnumber,

Prt =νtαt

(1.13)

turbulent Prandtl number can be set to a constant .5 ≤ Prt ≤ 1 or calculateddynamically, Moin et al. [80].

1.4.1 Dynamic Smagorinsky Model

Originally, this model is due to Germano et al. [32]. Later, Moin et al. [80]generalized the model for the large-eddy simulation of compressible flows.Lilly [71] proposed using a least squares technique to minimize the differencebetween the closure assumption and the resolved stresses. The formulationsof this section are based on the article of Moin et al. [80] combined with theleast square approach of Lilly [71].

The key element of the dynamic model concept is the utilization of thespectral data contained in the resolved field. This information is obtainedby introducing a test filter with a larger filter width than the resolved gridfilter, which generates a second field with scales larger than the resolved field.Assume that for a quantity φ, the spatially test-filtered quantity is denotedby a caret as φ. The width of the test filter is denoted by ∆. The test-filtered stresses Tij are defined by direct analogy to subgrid-scale stresses τij,


equation (1.5). Before giving the formulation of Tij, the standard averagingis re-introduced in equation (1.5),

τij = ρuiuj − ρui ρuj/ρ (1.14)

The test-filtered stresses Tij are defined in a similar way as τij,

Tij = ρuiuj − ρui ρujρ (1.15)

The Leonard stresses1 Lij are given by,

Lij = ρuiuj − 1

ρρuiρuj (1.16)

the Leonard stresses are completely computable from the resolved variables,and are related to equations (1.15) and (1.14) by the following algebraicrelation,

Lij = Tij − τij (1.17)

By applying the test filter to τij defined in equation (1.6), the test-filteredvalues τij may be written as,

τij − 1

3τkkδij = −2Cαij (1.18)

with αij given as,

αij = ∆2ρ ˜|S|(Sij − 1

3Smmδij)

In order to continue modeling, we need to make the approximation; Cαij =Cαij which is equivalent to considering that C is constant over an intervalat least equal to the test filter cutoff length,

τij − 1

3τkkδij = −2Cαij (1.19)

The same model of (1.6) is also used for the test field stresses,

Tij − 1

3Tkkδij = −2Cβij (1.20)

1Originally, these stresses were introduced by Leonard [64] in the context of an incom-pressible flow. If the subgrid scale velocity u′

i = ui − ui is defined, the SGS stresses canbe decomposed into three parts,

τij = uiuj − uiuj = (ui + u′i)(uj + u′

j) − uiuj = Lij + Cij + Rij

where Lij = uiuj − uiuj are the Leonard stresses, Cij = uiu′j + u′

j ui are the cross termsand Rij = u′

ju′i are the SGS Reynolds stresses.


with,

βij = ρ∆2 ˆ|S|(ˆSij − 1

3ˆSmmδij)

Now by inserting Tij from (1.20) and τij from (1.19) into (1.17) we have,

Lij − 1

3Lkkδij = −2C(βij − αij) (1.21)

with Lkk = Tkk − τkk.To solve for C, this expression is contracted with Mij = βij − αij, after

the appropriate spatial averaging,

C = −1

2

< (Lij − 13Lkkδij)Mij >

< M2ij >

(1.22)

< · · · > denotes space averaging.According to the formulation of Mij one can easily check that Mkk = 0,

as a result, in the numerator of (1.22) the term 13LkkδijMij =

13LkkMii = 0,

and equation (1.22) could be further simplified as,

C = −1

2

< LijMij >

< M2ij >

(1.23)

For the energy equation, the subgrid scale turbulent Prandtl number Prtin equation (1.9) is to be determined. If the Favre averaging in (1.9) isreplaced by the standard averaging,

Qi = cp(ρTui − ρT ρuiρ

) (1.24)

in the same way, Φi the heat flux at the test filter scale is defined as,

Φi = cp(ρTui − ρT ρuiρ ) (1.25)

by subtracting (1.25) from (1.24) the value of Θi = Φi − Qi is directly com-putable,

Θi = cp(¯ρuiT − 1ˆρρuiρT ) (1.26)

By applying the test filter to Qi defined in (1.9), the test-filtered value maybe written as,

Qi = −Ccp

Prt

∆2ρ|S| ∂T

∂xi(1.27)


the same model is used for the test filter scale,

Φi = −Ccp

Prtˆρ∆2| S| ∂ T

∂xi(1.28)

Now by taking Θi from (1.26), Φi from (1.28), and Qi from (1.27), insertingthem into Θi = Φi − Qi, and contracting the equation by Ni, we will finallyhave,

Prt = −C< N2

i >

< KjNj >(1.29)

with,

Ni = ˆρ∆2| S| ∂ T∂xi

−

∆2ρ|S| ∂T

∂xi(1.30)

and,

Ki = ¯ρuiT − 1ˆρρuiρT (1.31)

For the channel flow calculation the test filter is only applied along thetwo homogeneous directions. To give an example, the test filter of a variableφi,j,k is defined as, (assuming i and k indices correspond to the homogeneousdirections):

φi,j,k =φi,j,k

4+ (φi+1,j,k + φi−1,j,k + φi,j,k−1 + φi,j,k+1)/8

+(φi+1,j,k+1 + φi+1,j,k−1 + φi−1,j,k+1 + φi−1,j,k−1)/16(1.32)

Chapter 2

Convergence AccelerationTechniques

2.1 Explicit Runge-Kutta Schemes

We are concerned with the numerical solution of:

dU

dt= F (U) (2.1)

An s-stage Explicit Runge-Kutta (ERK) scheme from time level (n) to (n+1)is accomplished as,

Un+1 = Un +∆ts∑j=1

bjF (U j) (2.2)

In which,

U i = Un +∆ti−1∑j=1

aijF (U j)

As an example a three-stage scheme can be written,

U1 = Un

U2 = Un +∆t(a21F1)

U3 = Un +∆t(a31F1 + a32F

2)Un+1 = Un +∆t(b1F

1 + b2F2 + b3F

3)

(2.3)

For the three-stage method there are 3(3+1)2

= 6 coefficients available, in

general for a basic s-stage Runge-Kutta scheme there are s(s+1)2

coefficients

available, i.e., the degree of freedom (DOF) is s(s+1)2

.

15

16 CHAPTER 2. CONVERGENCE ACCELERATION TECHNIQUES

2.1.1 Some of the Characteristics of the Runge-KuttaSchemes

Order Conditions

Order conditions are nonlinear equations in the coefficients that must be sat-isfied to make the time discretization error of the method of order O(∆tn).The Mathematica [91] package can be used to find these nonlinear equa-tions, the result is expressed in terms of unknown coefficients aij, bj andci =

∑sj=1 aij.

The number of order conditions for each order, up to order 10, can bewritten as,

nq = 1, 1, 2, 4, 9, 20, 48, 115, 286, 719 (2.4)

It means that for a first order RK method we need one equation to be sat-isfied, for second order method 1+1=2 equations, for third order 1+1+2=4equations, for fourth order 1+1+2+4=8 equations, and so on. For a firstorder s-stage RK method the only equation is

τ(1)1 =

s∑i=1

bi − 1 = 0 (2.5)

for a second order s-stage these two equations are needed

τ(1)1 =

∑si=1 bi − 1 = 0

τ(2)1 =

∑si=1 bici − 1

2= 0

(2.6)

and for a third order s-stage Runge-Kutta method the following four equa-tions have to be satisfied,

τ(1)1 =

∑si=1 bi − 1 = 0

τ(2)1 =

∑si=1 bici − 1

2= 0

τ(3)1 = 1

2

∑si=1 bic

2i − 1

3!= 0

τ(3)2 =

∑si=1,j=1 biaijcj − 1

3!= 0

(2.7)

τ(q)k is a shorthand notation denoting equation number k of the equationsthat impose qth-order accuracy in time.

Error

The error in a qth-order explicit Runge-Kutta scheme may be quantified in ageneral way by taking the L2 and L∞ principal error norm,

A(q+1) = ||τ (q+1)||2 =

√√√√nq+1∑j=1

(τ(q+1)j )2 (2.8)

2.1. EXPLICIT RUNGE-KUTTA SCHEMES 17

A(q+1)∞ = ||τ (q+1)||∞ = MAX|τ (q+1)

j | (2.9)

For example for a 4th-order method, according to eq. (2.4), there are nineequations related to 5th-order accuracy which are not satisfied and the errorwill be:

A(5) = ||τ (5)||2 =

√√√√n5=9∑j=1

(τ(5)j )2

Linear Stability

The linear stability function for an s-stage explicit Runge-Kutta method isgiven by:

P (z) = 1 + α1z + α2z2 + α3z

3 + · · ·+ αszs (2.10)

αi coefficients depend on the values of bj and aij. For a value of z such that|P (z)| < 1 the method is stable.

To provide an example, suppose the three-stage Runge-Kutta schemegiven in (2.3) is used to discretize the following model problem,

dy

dt= λy y(0) = y0 (2.11)

here λ is a complex constant. The amplification yn+1

yn is equal to,

yn+1

yn= p(z) = 1 + α1z + α2z

2 + α3z3

with α1 = b1+b2+b3, α2 = b2a21+b3a31+b3a32, α3 = b3a32a21, and z = λ∆t.

For more examples about model problems, see Moin [81].

Dispersion and Dissipation Error

The dispersion and dissipation errors of the Runge-Kutta method are cal-culated by comparing the phase and amplitude of the complex polynomial,P (z), with the exponential function, ez, along the imaginary axis.

Suppose z = iy, phase, φ, and amplitude, α, errors will be

φ = arg(eiy)− arg(P (iy)) = iy − arg(P (iy)) (2.12)

α =|P (iy)||eiy| = |P (iy)| (2.13)


Efficiency

The relative efficiency of two Runge-Kutta schemes depends on their stability-limited time steps and their number of steps,

η(stab) =∆t1/s1

∆t2/s2

(2.14)

Where s is the number of stages and ∆t is the time step. Scheme 1 is more

efficient for η(stab) > 1.

2.1.2 Dual Time Stepping With Explicit Runge-Kutta

For convenience and feasibility of analytical treatment, let us consider againthe model problem,

dy

dt= λy y(0) = y0 (2.15)

From the analytical solution to this problem we have,

y(t +∆t)

y(t)= eλ∆t

which means that, in order to have a bounded solution the real part of λshould be negative.

For a linear convection-diffusion equation ut + aux = αuxx, the complexconstant λ is called the Fourier footprint, equation (2.31), which depends onthe type of the spatial discretization, and the values of a and α.

If an explicit Runge-Kutta is used to solve (2.15), at stage k we have,

yk+1 = y0 + αk (∆tλ)︸︷︷︸zo

yk (2.16)

to be stable, zo = ∆tλ should lie inside the stability region of the Runge-Kutta scheme, or according to the linear stability condition of a Runge-Kuttamethod,

|P (∆tλ)| < 1

with P (z) given by equation (2.10).Now suppose a backward Euler method is used to discretize the time

derivative dydy. A pseudo-time derivative dy

dτis added to the equation to solve

the implicit system,dy

dτ+

yn+1 − yn

∆t= λyn+1 (2.17)

2.1. EXPLICIT RUNGE-KUTTA SCHEMES 19

An explicit Runge-Kutta can be used to march to a steady-state in pseudo-time. At stage k we have,

yk+1 = y0 + αk∆τ(λyn+1 − yn+1

∆t) + αk

yn

∆t∆τ (2.18)

The superscript k counts pseudo-time steps and the index n counts implicit(physical) time steps. The values at time level n + 1 are still unknown. Thesteady-state solution of equation (2.18) in pseudo-time τ gives the value ofyn+1. By replacing the unknown values at time level n + 1 by the knownvalues at pseudo-time level k,

yk+1 = y0 + αk∆τ(λyk − yk

∆t) + αk

yn

∆t∆τ (2.19)

after further rearranging,

yk+1 = y0 + αk (∆τλ − ∆τ

∆t)︸︷︷︸

ze

yk + αkyn

∆t∆τ (2.20)

by comparison of (2.16) and (2.20) one can see,

ze = zo − ∆τ

∆t

which means that, when an explicit Runge-Kutta is used to solve an implicitsystem in pseudo-time, the Fourier footprint is shifted to the left along thereal axis in a complex plane. For small values of the physical time step∆t, which is the case in a large-eddy simulation, this shift may cause ze togo beyond the stability limit of the Runge-Kutta method, i.e., P (ze) > 1,and make the solution unstable. We note that in (2.20), αk

yn

∆t∆τ is constant

during the inner-iterations (iterations in pseudo-time) and does not have anyinfluence on the stability.

A method to overcome this problem is to treat the yk

∆tterm in equation

(2.19) implicitly within the Runge-Kutta cycle,

yk+1 = y0 + αk∆τ(λyk − yk+1

∆t) + αk

yn

∆t∆τ (2.21)

After further rearranging,

yk+1 = y0 + αk (∆τλ

1 + αk∆τ∆t

)︸︷︷︸zi

yk +αk

∆τ∆t

(yn − y0)

1 + αk∆τ∆t

(2.22)


Now, for any value of ∆t or ∆τ we have,

∆τλ

1 + αk∆τ∆t

< ∆τλ → zi < zo

Thus, there is no shift to the left and a small value of ∆t would not causeany problem.

We also note thatαk

∆τ∆t

(yn−y0)1+αk

∆τ∆t

is constant during the inner-iterations and

does not affect the stability.

In conclusion, according to what we have learned from the model problem,we know that in the case of the Navier-Stokes equations, the same procedureshould be followed in order to stay in the Runge-Kutta stability region whenthe physical time step ∆t is small.

Special care should be taken to facilitate the inversion of matrices. Thisproblem will be explained in section (2.5.3).

2.2 Analysis of Temporal Discretization

In an implicit approach to the Navier-Stokes equations, the physical timederivative is usually discretized with a multi-step method, like the trapezoidalmethod or the second order backward differencing method. The aim of thissection is to observe the behavior of different time discretization schemestested on a simple one dimensional linear equation and to choose the onewhich fits the best for LES calculations.

The one dimensional convection equation, ut+cux = 0 is considered. Thespatial derivative is approximated using a seven point, sixth order centraldifference scheme:

ux = 1.5ui+1 − ui−1

2∆x− 0.6

ui+2 − ui−2

4∆x+ 0.1

ui+3 − ui−3

6∆x(2.23)

Suppose u(x, t) is represented by a single Fourier mode, u(x, t) = v(t)eIkx

and insert this value into equation (2.23),

v(t)

v(t)+ Ic

1.5sin(k∆x)− 0.3sin(2k∆x) + 0.26

sin(3k∆x)

∆x= 0

which can be written as,

v(t)

v(t)+ Ick∗ = 0 (2.24)

2.2. ANALYSIS OF TEMPORAL DISCRETIZATION 21

Table 2.1: Coefficients of equation (2.26)

β1 β0 β−1 γ1 γ2

Second order backward 1.5 -2 0.5 1 0Trapezoidal method 1 -1 0 0.5 0.5

k∗ is called the effective wave number. The maximum value of (k∗∆x)maxfor this scheme is (k∗∆x)max = 1.585 1. The space derivative is discretizedwith a high-order central method to minimize the error coming from thespatial discretization. For the temporal discretization different methods havebeen used, such as the fourth order (for a linear equation) explicit Runge-Kutta method, second order backward Euler, the trapezoidal scheme, andtwo implicit second order Runge-Kutta methods. The second order backwardEuler and the trapezoidal schemes can be written in a general form,

β1un+1i + β0u

ni + β−1u

n−1i

∆t+ γ1(cux)

n+1 + γ2(cux)n = 0 (2.26)

with the values of βi and γi given in table (2.1). The fourth order explicitRunge-Kutta scheme is given by equation (2.2) with s = 4 and,

aij =

0 0 0 0

1/4 0 0 00 1/3 0 00 0 1/2 0

bj =[0 0 0 1

](2.27)

Two implicit Runge-Kutta schemes, have also been tested. Their coefficientsare set according to Marx [74], and they are called DIRK2.1 and DIRK2.2.They can be written as,

U1 = Un +∆tβ11F (U1)Un+1 = Un +∆t(β20F (Un) + β21F (U1) + β22F (Un+1))

(2.28)

1It is also interesting to mention that the spatial discretization error changes the speedof propagation of the kth Fourier mode. By multiplying v(t)eIkx to both sides of equation(2.24),

v(t)eIkx +k∗∆x

k∆xv(t)(Ick)eIkx = 0

ut + ck∗∆x

k∆xux = 0 (2.25)

The speed of propagation of a single Fourier mode is ck∗∆xk∆x instead of c. For this spe-

cial case (i.e., convection equation discretized with a pure central scheme) this error isequivalent to the dispersion error which cause some wiggles to appear on the left (or rightdepending on the ratio of k∗/k) of the propagating wave.


BACKWARD DIFFERENCING

0. 40. 80. 120. -1.0

1.0

3.0

5.0 Backward DifferencingExact Solution

TRAPEZOIDAL RULE

0. 40. 80. 120. -1.0

1.0

3.0

5.0 Trapezoidal ruleExact Solution

Figure 2.1: Left: 2nd order backward differencing, right: trapezoidal scheme.

with (β11 = −0.2060, β20 = 0.2698, β21 = −0.9698 and β22 = −0.3) forDIRK2.1 and (β11 = −0.2651, β20 = 0.0545, β21 = −0.7545 and β22 =−0.3) for DIRK2.2. These two implicit Runge-Kutta scheme, as well as thebackward Euler and trapezoidal schemes are second order accurate in timeand A-stable, i.e., stable in the left part of the complex plane including theaxis.

Figures (2.1) and (2.2) show the effect of the time discretization on thepropagation of a Gaussian wave. CFL = 1 for all cases and c = ∆t = ∆x = 1.The results are compared with the exact solution which is the initial solutionpropagating with a convective velocity equal to one. From figure (2.1) onecan see that the second order backward differencing leads to a larger decreaseof amplitude compared to the trapezoidal scheme. This is in accordance withthe well known fact that the backward differencing is more dissipative thanthe trapezoidal scheme which has no dissipation.

Figure (2.2) compares two second order implicit DIRK2.1, DIRK2.2 anda fourth order explicit Runge-Kutta. Obviously, for this test case the bestresult is obtained with the fourth order explicit method. However, for thecompressible Navier-Stokes equations due to the restrictions imposed on thetime step through the CFL condition, the explicit method becomes unefficientand an implicit method combined with dual time stepping has to be used.

In general, if the spatial derivative is approximated with a central schemeon a uniform mesh of spacing ∆x,

∂u

∂x

∣∣∣∣∣n

i

=1

∆x

N∑l=−N

aluni+l (2.29)

Assume uni is represented by a single Fourier mode,

uni = vneIk∆xi (2.30)


DIRK2.1

0. 40. 80. 120. -1.0

1.0

3.0

5.0 DIRK2.1Exact Solution

DIRK2.2

0. 40. 80. 120. -1.0

1.0

3.0

5.0 DIRK2.2Exact Solution

4th order RK.

0. 40. 80. 120. -1.0

1.0

3.0

5.0 4th order RK.Exact Solution

Figure 2.2: Left: implicit DIRK2.1, middle: implicit DIRK2.2, right: explicitfourth order Runge-Kutta

By inserting this Fourier mode into equation (2.29),

∂u

∂x

∣∣∣∣∣n

i

= vn1

∆x

N∑l=−N

aleIk∆x(i+l)

the Fourier footprint of the residual can be defined as,

=

∑Nl=−N ale

Ik∆x(i+l)

eIk∆xi=

N∑l=−N

aleIk∆xl (2.31)

In general the Fourier footprint is a complex number = x+iy and is directlyrelated to the type of the spatial discretization. For example, if a convectiveterm is discretized with a purely central method, the Fourier footprint wouldbe purely imaginary with no real part. In contrast, a pure diffusion term hasonly negative real part without any imaginary value. The Fourier footprintof an upwind method, which has some hidden diffusion term contains bothimaginary and real parts.

To calculate the dissipation and dispersion errors of the time discretiza-tion, the model problem dy

dt= λy is again considered. If the model problem

is discretized with a trapezoidal method the amplification factor P (λ∆t) =yn+1/yn is,

P (λ∆t) =yn+1

yn=

1 + λ∆t/2

1− λ∆t/2(2.32)

For the implicit Runge-Kutta method given by equation (2.28) the amplifi-cation factor is equal to,

P (λ∆t) =1 + β20λ∆t + β21λ∆t

1−β11λ∆t

1− β22λ∆t(2.33)

For the fourth-order explicit Runge-Kutta we have,

P (λ∆t) = 1 + b4(λ∆t) + a43b4(λ∆t)2 + a43a32b4(λ∆t)3 + a43a32a21b4(λ∆t)4

(2.34)


And finally, the amplification factor of the second order backward Euler is,

P (λ∆t) =−β0 ±

√β2

0 − 4β−1(β1 − γ1λ∆t)

2(β1 − γ1λ∆t)(2.35)

Having more than one root is the key characteristic of multi-step methods.The second root is spurious. By putting the numerical values of βs fromtable (2.1) into the above equation for the first root2 we will have,

P (λ∆t) =2 +

√1 + 2(λ∆t)

3− 2(λ∆t)(2.36)

To separate the dissipation coming from the spatial discretization from thedissipation of the time stepping method, λ is supposed to be a pure imaginarycomplex number3. Now the dispersion and dissipation errors of the timediscretization scheme is calculated by comparing the phase and amplitudeof the complex polynomials P (λ∆t) calculated in equations (2.32), (2.33),

(2.34), and (2.35) with the analytical amplification factor y(t+∆t)y(t)

= eλ∆t,

obtained from the analytical solution of the model problem y′ = λy. Thedissipation and phase (dispersion) error of different schemes as a function ofthe imaginary variable λ∆t are shown in figure (2.3).

For a linear convection equation, the maximum value of λ∆t depends onthe type of the spatial discretization, and the CFL number. For example, forthe central scheme given in equation (2.23), the amplification factor of thesemi-analytical4 equation is equal to,

v(t +∆t)

v(t)= e−Ick

∗∆t

from the above equation one can see that the maximum value of the argumentof the exponential function depends on the type of the discretization and the

2To calculate the phase and amplitude of the square of a complex number, first its realand imaginary parts should be separated with the following formula,

z =√a + ib =

√r + a

2+ i

√r − a

2b ≥ 0

with r =√a2 + b2.

3This is in line with the analytical solution of the convection equation ut + cux = 0. Ifu(x, t) is represented by a single Fourier mode u(x, t) = v(t)eIkx, the resulting equationfor v(t) will be,

v(t) = −Ickv(t)

in the model problem, v(t) is represented by y, and −Ick by λ.4It is called semi-analytical because the spatial derivative is discretized but the time

derivative is calculated analytically.


Dissipation of different schemes for central differencing.

Imaginary Axis

Dissipation

.00 1.00 2.00 3.00 .50

.75

1.00

1.25

1.50 DIRK2.1DIRK2.2TRAPEZOIDALExplicit RK2nd Order Euler

Dispersion of different schemes for central differencing.

Imaginary Axis

Dispersion

.00 1.00 2.00 3.00 -2.00

-1.20

-.40

.40

1.20

2.00 DIRK2.1DIRK2.2TRAPExplicit RK2nd Order Euler

Figure 2.3: Dissipation and phase error of DIRK2.1, DIRK2.2, trapezoidalscheme and four-stage fourth order Runge-Kutta scheme

CFL number,

ck∗∆t =c∆t

∆xk∗∆x = CFLk∗∆x

for the spatial discretization of equation (2.23) the maximum value of k∗∆xwas 1.585, and the CFL number was set equal to 1. Thus, the imaginary vari-able λ∆t varies between 0 and 1.585 and the maximum amount of dissipationand dispersion can be obtained from figure (2.3) which is at λ∆t = 1.585.Another useful insight can be deduced from this analysis about the relation-ship between the CFL number and the accuracy of the spatial differencing.We know that the more accurate schemes have higher peak values for theireffective wave number k∗∆x. As a result, with the same CFL number, ahigher order method will be more dissipated and disperesed by the errors oftemporal discretization than a lower order method. That is why in generalfor the higher order spatial discretization a smaller CFL number is used.

According to figure (2.3) the explicit four-stage, fourth-order Runge-Kutta scheme (dot-dashed line) is dispersive for big λ∆t, which is normalfor an explicit scheme, but it has a very small dispersion error for λ∆t up to 1.5. The trapezoidal scheme (dotted line) is not dissipative but it has alarger dispersion error than DIRK2.1 (bold line) and DIRK2.2 (dashed line).The dissipation of DIRK2.1 is very small which makes it a good candidate forLES calculations. According to the plot, the worst time discretization is thesecond order backward Euler (dot-dot-dashed line) with the biggest disper-sion and dissipation errors. The disadvantage of the DIRK2.1 and DIRK2.2schemes is that they are two-stage implicit methods which makes their effi-ciency in terms of CPU time questionable. When a two-stage implicit schemeis used, one needs to obtain two steady-state solutions in pseudo-time to go


from the physical time level n to the time level n + 1. As a result, eventhough DIRK2.1 and DIRK2.2 perform nicely in a linear problem, their ap-plicability for LES, where the computational time is an important parameter,is unclear.

In the following sections more comparisons will be made between thesecond order implicit multistage schemes (backward Euler and trapezoidalscheme) and the explicit Runge-Kutta scheme to assess the efficiency of theseimplicit approaches in the context of large-eddy simulation.

2.3 Multigrid

Multigrid can be used to improve the rate of convergence to a steady state.For unsteady calculations, a pseudo-time derivative is added to the set ofequations and multigrid is used in pseudo-time during the inner iterations.The concept of acceleration by the introduction of multiple grids was firstproposed by Fedorenko [30].

The idea of using multigrid emerges from the fact that most iterativemethods eliminate the high frequency errors very fast, but fail to eliminatethe low frequency errors at the same rate. In addition, the notion of highand low frequency error is related to the coarseness of the grid, in that anylow frequency error on a certain grid will become a high frequency error onanother sufficiently coarse grid. Therefore, the idea is to solve the equationson a sequence of grids ranging from fine to coarse, so that the completespectrum of errors can be eliminated with a rate proper to the eliminationof the high frequency errors.

First theories of multigrid methods were applied only to elliptic equations,later, extensions to hyperbolic systems were developed. There is by now afairly well-developed theory of multigrid methods for elliptic equations basedon the concept that the updating scheme (i.e., Runge-Kutta or any othertime marching technique) acts as a smoothing operator on each grid [9],[36]. The smoothing is also possible for hyperbolic systems. It is possible toaccelerate the evolution of a hyperbolic system to a steady state by using largetime steps on coarse grids so that disturbances will be more rapidly expelledthrough the outer boundary. A number of effective multigrid solvers ([47],[46], [2], [83], among others) have been constructed for the Euler equations ofgas dynamics, which are hyperbolic. Transonic and subsonic flows have beencomputed with these solvers. Some multigrid methods ([104], [73], [112],[1], [41], [11], [119], among others) have also been devised for the numericalsolution of the compressible Navier-Stokes equations.

It is also interesting to mention that, when multigrid is used for solvingnon-linear systems of equations, the coarse grid equations are solved for an

2.3. MULTIGRID 27

approximation of the complete finer grid solution. In case of a linear system,however, the coarse grid equations are solved for the “correction” on the finergrid. Therefore, Brandt [9] calls his algorithm for solving non-linear systemsof equations the Full Approximation Storage (FAS) scheme.

On a structured mesh, a sequence of independently generated coarsermeshes can be generated by eliminating alternate points in each coordinatedirection. In this section we will explain the V-sawtooth multigrid. For fur-ther readings, the interested reader can refer to Hackbush [36], or McCormick[95].

In order to give a precise description of the multigrid scheme, subscriptk may be used to indicate the grid. The finest grid level is on grid k − 1,and the next coarser levels are k and k + 1. For example, with k = 1 thesecorrespond to levels 0,1 and 2, respectively. Suppose the general equation onthe finest grid level k − 1 is represented as,

∂U0

∂t= Res0(U0) (2.37)

Res0(U0) is the residual of the finest grid. To explain the method, a three-level multigrid is assumed but the extension to more grid levels is straight-forward. Before going to the next coarser level k, the solution vector on gridk must be initialized as,

U(0)k = T k

k−1Uk−1 (2.38)

where Uk−1 is the current value on grid k − 1 (here is the finest grid andwas called U0), and T k

k−1 is called a transfer (also called restriction) operatorwhich brings the information of the fine grid to the coarse grid. Next it isnecessary to transfer a residual forcing function such that the solution on gridk is driven by the residuals calculated on grid k − 1. The forcing function isdefined as,

Fk = Qkk−1Resk−1(Uk−1)− Resk(U

(0)k ) (2.39)

where Qkk−1 is a transfer/restriction operator for the residuals. Then Resk(Uk)

is replaced by Resk(Uk) + Fk in the time stepping scheme which can be aRunge-Kutta method. A (q+1)-stage scheme on grid level k is reformulatedas,

U(1)k = U

(0)k + α1∆tk[Resk(U

(0)k ) + Fk]

· · · = · · ·U

(q+1)k = U

(0)k + αq+1∆tk[Resk(U

(q)k ) + Fk]

The result of Uk = U(q+1)k then provides the initial data for grid k + 1. We

note that a number of smoothing sweeps (i.e., Runge-Kutta iterations) can


Level K+1

S

S

S

S

S

S

S

S

R

RP

PR

RP

RP

P

Level K−1

Level K

Figure 2.4: Structure of multigrid V-sawtooth (left) and W-sawtooth (right)cycles; letter designation are defined as S: solve equations, R: restrict solutionand residuals, P: prolongate corrections.

be applied before going to the next coarser grid. To go to the next coarserlevel k + 1, the solution vector on grid k + 1 is again initialized as,

U(0)k+1 = T k+1

k Uk

and the forcing function Fk+1 is,

Fk+1 = Qk+1k (Resk(Uk) + Fk)− Resk+1(U

(0)k+1)

a (q+1)-stage scheme gives,

U(1)k+1 = U

(0)k+1 + α1∆tk+1[Resk+1(U

(0)k+1) + Fk+1]

· · · = · · ·U

(q+1)k+1 = U

(0)k+1 + αq+1∆tk+1[Resk+1(U

(q)k+1) + Fk+1]

Finally, the resulting solution on the coarsest level Uk+1 = U(q+1)k+1 has to

be transferred back to grid k with the aid of a prolongation (also calledinterpolation) operator T k

k+1,

Unewk = Uk + T k

k+1(Uk+1 − T k+1k Uk)

the value of Unewk can either be smoothed again with a Runge-Kutta method

or directly prolongated to the next finer grid k−1. If it is directly prolongatedwithout any smoothing, the multigrid cycle is called sawtooth cycle. In caseof a sawtooth cycle the final value on grid level k − 1 is,

Unewk−1 = Uk−1 + T k−1

k (Unewk − T k

k−1Uk−1)

If on grid level k instead of going to the next finer level k−1, the solution andresidual are again restricted to the coarser level k+1, the cycle is called a W-cycle. In figure (2.4), V-sawtooth and W-sawtooth cycles are schematicallyshown. The empty circles denote no smoothing.

2.3. MULTIGRID 29

Coarse Grid

b a b

1 1 2 2

C

aFine Grid

Figure 2.5: One dimensional restriction operator.

2.3.1 Grid Transfer Operators

The intergrid transfer operators employed in the present multigrid methodsuch as the restriction operator of residuals Qcoarse

fine , the restriction operatorof variables T coarse

fine and the prolongation operator of variables T finecoarse are

discussed in this section. First and second order prolongations, as well aslinear and quadratic restrictions are discussed. The order of the prolongationoperator can be defined as the highest degree plus one of polynomials that areinterpolated exactly. With this definition the piecewise constant prolongationand the linear prolongation, are first and second order, respectively.

The residual-restriction operator is frequently defined as the adjoint of thecorrection-prolongation operator, meaning that one operator is the transposeof the other, see Swanson et al. [105] for discussion of the adjoint property.According to this definition, the linear restriction is the adjoint of the firstorder prolongation and the quadratic restriction is the adjoint of the secondorder interpolation. Hackbush [36] proved that to have a mesh-independentrate of multigrid convergence, the sum of the order of the prolongation oper-ator mp, and the order of the restriction operator mr, must exceed the orderof the differential equation 2m, being considered.

mp + mr > 2m

For example in Navier-Stokes equations 2m = 2, and a choice could be thecombination of a linear restriction mr = 1 with a second order prolongationmp = 2 to satisfy the above condition.

Linear Restriction

The linear restriction of the fine grid residuals ResF to a coarser mesh ac-cording to figure (2.5) may be written as,

ResC = QcfResF =

∑ni=1 ∆

aiResai

∆C

(2.40)

with Resa1, Resa2 the residuals and ∆a1, ∆

a2 the length of the cells (a,1),(a,2).

∆C is the length of the coarse mesh. In two and three dimensions ∆ repre-sents the surface and the volume of the cell, respectively.


a a

4

a a

b b

b

b

bb

b

b

c c

c c

C1 2

3 4

1 2

3 4

5 6

7 8

1 2

3

Figure 2.6: Two dimensional restriction operator.

In a similar way the restriction of the fine grid variables UF is defined as,

UC = T cfUF =

∑ni=1 ∆

aiU

ai∑n

i=1 ∆ai

(2.41)

For a one dimensional problem according to figure (2.5) n = 2 in equations(2.40) and (2.41), for a two dimensional n = 4, see figure (2.6), and for athree dimensional problem n = 8 (figure not shown). The cells with an aindex are the surrounding cells of point C. In one dimension there are twocells (n=2), in two dimension four cells (n=4), and in three dimension eightcells (n=8).

Quadratic Restriction

By increasing the number of cells involved in the restriction operator, a moreaccurate operator can be obtained. In 1D according to the figure (2.5) fourcells are involved and the quadratic restriction of residual may be written as,

ResC = QcfResF =

1

∆C

(3

4

2∑i=1

∆aiResai +

1

4

2∑i=1

∆biResbi) (2.42)

The restriction of the fine grid variables UF is defined as,

UC = T cfUF =

38

∑2i=1 ∆

aiU

ai + 1

8

∑2i=1 ∆

biU

bi

38

∑2i=1 ∆

ai +

18

∑2i=1 ∆

bi

(2.43)

In two dimensions according to figure (2.6) 16 neighboring cells are involvedand the restriction of the residual is,

ResC =1

∆C

(9

16

4∑i=1

∆aiResai +

3

16

8∑i=1

∆biResbi +

1

16

4∑i=1

∆ciResci) (2.44)

2.3. MULTIGRID 31

whereas for the restriction of the variables,

UC = T cfUF =

964

∑4i=1 ∆

aiU

ai + 3

64

∑8i=1 ∆

biU

bi +

164

∑4i=1 ∆

ciU

ci

964

∑4i=1 ∆

ai +

364

∑8i=1 ∆

bi +

164

∑4i=1 ∆

ci

(2.45)

In three dimensions (figure not shown) 64 cells are involved and the residualrestriction operator is,

ResC =1

∆C

(27

64

8∑i=1

∆aiResai +

9

64

24∑i=1

∆biResbi+

3

64

24∑i=1

∆ciResci+

1

64

8∑i=1

∆diResdi )

(2.46)and the restricted variable on the coarse mesh is,

UC =27512

∑8i=1 ∆

aiU

ai + 9

512

∑24i=1 ∆

biU

bi +

3512

∑24i=1 ∆

ciU

ci +

1512

∑8i=1 ∆

diU

di

27512

∑8i=1 ∆

ai +

9512

∑24i=1 ∆

bi +

3512

∑24i=1 ∆

ci +

1512

∑8i=1 ∆

di

(2.47)the point C is at the cell center of a coarser cube. The cubes with index aare the surrounding fine cubes of point C, cubes b have a common face withcubes a, cubes c have a common edge with cubes a and cubes d have only acommon corner with cubes a.

First Order Prolongation

The simplest prolongation is the piecewise constant prolongation which isa first order operator. For a 1D problem, as illustrated in figure (2.7), thecorrection of a fine cell δUf is simply the correction of the coarse cell δUa,

δUf = T FC δUC = δUa (2.48)

Note that cell i − 1 gets the same correction as cell i with this approach. Intwo and three dimensions the fine grid value is the value of the coarse grid inwhich the cell center of the fine element is located, for example figure (2.8)shows how the piecewise constant prolongation operates in two dimensions.The cell center of the fine element Uf is located in a coarse grid with a centerat Ua

1 , and the value of fine element simply equals,

δUf = T FC δUC = δUa

1 (2.49)

Second Order Prolongation

To improve the accuracy, a linear prolongation can be used which is second-order accurate according to the definition. In one dimension, as illustrated


i−1

U Ua b

Ufd

d d

i

Figure 2.7: One dimensional prolongation operator.

U U

U

d

U

U

a

b c

b1

1

2

1

f

d

d

d

d

Figure 2.8: Two dimensional prolongation operator.

2.4. RESIDUAL SMOOTHING 33

d

U U

U U

U

U

U

U

U

a

b

b

b

c

c

c

d

1

1

2

3

1

2

3

1

f

d d

d d

d

d d

d

Figure 2.9: Three dimensional prolongation operator.

in figure (2.7) the linear operator is,

δUf = T FC δUC =

3

4δUa +

1

4δU b (2.50)

In two dimensions, as shown in figure (2.8), the cell center of the fine elementUf is located in a bigger volume with four corners Ua

1 , U b1 , U b

2 , and U c1 . These

four corners are the cell centers of a coarser element. If a uniform mesh isassumed the bilinear prolongation in two dimensions may be written as,

δUf =9

16δUa

1 +3

16(δU b

1 + δU b2) +

1

16δU c

1 (2.51)

In three dimension the cell centers of the coarser grid make a cube in whichthe cell center of the fine element is located, see figure (2.9). A trilinearprolongation in a uniform mesh gives,

δUf =27

64δUa

1 +9

64(δU b

1 +δU b2 +δU b

3)+3

64(δU c

1 +δU c2 +δU c

3)+1

64δUd

1 (2.52)

In order to account for the mesh non-uniformity a volume-weighted averagecan be used. Based upon the experience from a large number of numeri-cal simulations, Zhu [122] and Eliasson [28], a constant linear prolongationworks well for both uniform and non-uniform meshes. The weighted linearprolongation does not give significant improvement in convergence, but it israther complicated and costly.

2.4 Residual Smoothing

The local stability range of the basic time-stepping scheme can be extendedby applying a procedure called implicit residual smoothing. This technique


was first introduced by Lerat [65] and later devised by Jameson [45] forRunge-Kutta schemes. In a similar way to the other convergence accelerationtechniques, for unsteady calculations a pseudo-time derivative is added to theset of equations and residual smoothing is used in pseudo-time during theinner iterations.

The smoothed residual can be calculated by using an elliptic operator.If Res is the original residual and Res is the smoothed residual, in onedimension the residual smoothing may be written as, 5

Resi − Resi = ε∆2iResi

with ∆2i defined as,

∆2iResi = Resi+1 − 2Resi + Resi−1

The same formula is used in three dimensions, and to avoid expensive calcu-lation, a factored form should be used as following:

(1− εi∆2i )(1− εj∆

2j)(1− εk∆

2k)Resi,j,k = Resi,j,k (2.53)

In one dimension by considering a linear convection equation

∂w

∂t+ a

∂w

∂x= 0

discretized with a central scheme, and using a Fourier analysis, see Swansonet al. [105], one can show that a sufficient condition for stability is,

ε ≥ 1

4((

CF Ls

CF L)2 − 1)

where CF Ls and CF L are the CF L numbers of the smoothed and un-smoothed time marching scheme, respectively.

Then, by considering the diffusion equation

∂w

∂t= ν

∂2w

∂x2

and using the same procedure employed for the convection equation, thesmoothing coefficient may be determined as,

ε ≥ 1

4(Ds

D− 1)

5To give a physical insight, one can look at the above equation as a 1D diffusionequation, discretized in time with a first order backward Euler with Resi the value of thefunction at the new time step, and ε = α∆t/∆x2. Obviously, after one time step, the highfrequency components of Resi are quickly damped due to the diffusion.

2.5. PRECONDITIONING 35

with Ds and D the diffusion number of the smoothed and unsmoothedscheme, D is defined as D = ν∆t

∆x2 . For a three dimensional calculationdifferent definitions of ε are available in the literature. For the test casesconsidered in this thesis, two types of residual smoothing, which take intoaccount the effect of the mesh aspect ratio, are used. The first one is due toSwanson et al. [104] and the smoothing factor is defined as,

εi =1

4

(CF Ls

CF L

1

1 + 0.0625(λj/λi + λk/λi)

)2

− 1

(2.54)

λi, λj and λk are the spectral radii scaled with the cell face area, in the threedirections of i, j and k,

λi = @U · @Si + c|@Si|and similarly for λj and λk. The three vectors @Si, @Sj and @Sk are the vectorsnormal to the cell face with a length equal to its surface. On a Cartesian mesh@Si, @Sj and @Sk have non-zero components only in the x, y and z coordinates,respectively. But on a general mesh (non-Cartesian) each of them may have

three components along x, y and z coordinates. @U is the velocity vector atthe cell face and c is the speed of sound. The other formulation used is dueto Radespiel et al. [90] and the smoothing coefficient is defined as,

εi =1

4

CF Ls

CF L

1 + max(√

λj/λi,√

λk/λi)

1 + max(λj/λi, λk/λi)

2

− 1

(2.55)

Even though theoretically an explicit Runge-Kutta scheme can be made un-conditionally stable with the implicit residual smoothing, there is a practicallimit on the time step when solving the hyperbolic problem. A good practicalvalue for CFLs

CFLin equations (2.54) and (2.55) is CFLs

CFL= 2.

Note that apart from the central type residual smoothing, also upwindtype, Blazek et al. [7], and forward-backward implementations, Zhu et al.[121] have been formulated.

2.5 Preconditioning

Preconditioning removes the stiffness of the compressible Navier-Stokes equa-tions at low Mach number and, as a result, it allows the use of a unifiedmethod that is accurate and efficient for both compressible and incompress-ible flows. Using preconditioning has two advantages. First, it acceleratesthe convergence to a steady state. Second, by modifying the artificial dissi-pation it improves the accuracy of the steady state solution. However, for


LES there is no artificial dissipation in the momentum and energy equationsand preconditioning would not have any effect on them in this respect. Inaddition, in the context of dual time stepping the real steady state in pseudo-time is never reached and the subiteration procedure stops as soon as theerror of the residual drop is at the same order of magnitude as the othererrors of calculation.

The first contribution to the development of preconditioning methodswas the artificial compressibility method introduced by Chorin [18]. Turkel[110], [111], generalized the Chorin’s formulation by allowing artificial timederivatives in all the equations and not just the continuity equation. TheChoi and Merkle [17] preconditioning system is closely related to Turkel’ssystem. The major difference is that the Choi-Merkle system was derivedusing temperature as a dependant variable in the energy equation. Van Leeret al. [62], developed an alternate preconditioning formulation that was basedupon a characteristic re-scaling of the equations. Their analysis providedan interesting physical and geometrical interpretation of the acoustic andparticle wave processes. However, the Van Leer system at low mach numbersis closely related to the Turkel and Choi-Merkle systems, for further reading,see Darmofal et al. [23].

2.5.1 Non-Conservative Preconditioned Equations

The preconditioning method used in the present calculations is very similar tothe one introduced by Choi and Merkle [17] with temperature as a dependantvariable in the energy equation. Furthermore, the extension of Turkel [109]to Chorin’s [18] artificial compressibility method, with a second parameter inthe momentum equations, is used to keep the generality of the formulation.This method has been applied by Hakimi [37] to different test cases, includingReynolds averaged turbulent flows with different turbulence models as wellas non-Newtonian fluids. The non-conservative preconditioned Navier-Stokesequations for steady-state flows take the following form,

1β2 p,t + (ρuj),j = 0αui

β2p,t + ρui,t + ρui,juj = −p,i + τij,j

ρcp(T,t + T,iui)− (p,t + uip,i) = −qi,i + τijui,j

(2.56)

The time derivative of density, ρ,t, in the continuity equation has been re-placed by 1

β2 p,t andαui

β2p,t has been added to the momentum equations. The

energy equation keeps its original form and is represented in terms of tem-perature as dependant variable.


2.5.2 Conservative Preconditioned Equations

Starting with equations (2.56), the continuity equation is multiplied by uiand is added to the momentum equations,

(1 + α)uiβ2

p,t + ui(ρuj),j + ρui,t + ρui,juj = −p,i + τij,j (2.57)

further rearranging gives the momentum equations in conservative form,

(1 + α)uiβ2

p,t + ρui,t + (ρujui),j = −p,i + τij,j (2.58)

For the energy equation the momentum equations are multiplied by ui andadded to the energy equations,

αu2i

β2p,t + ρuiui,t︸︷︷︸

ρ(u2

i2

),t

+ ρui,jujui︸︷︷︸ρuj(

u2i2

),j

+uip,i − uiτij,j + ρcpT,t + ρcpuiT,i

−(p,t + uip,i) + qi,i − τijui,j = 0 (2.59)

(αu2

i

β2−1)p,t+ρ(

u2i

2),t+ρuj(

u2i

2),j+ρcpT,t+ρcpuiT,i = −qi,i+(τijui),j (2.60)

We know that, H = cpT +u2

i

2:

(αu2

i

β2− 1)p,t + ρH,t + ρuj(

u2i

2),j + ρcpuiT,i = −qi,i + (τijui),j (2.61)

(αu2

i

β2− 1)p,t + ρH,t + [(ρ

u2i

2uj),j − u2

i

2(ρuj),j] + [(ρcpT uj),j

−T (ρcpuj),j] = −qi,i + (τijui),j (2.62)

(αu2

i

β2−1)p,t+ρH,t−(cpT +

u2i

2)︸︷︷︸

H

− 1β2 p,t︷︸︸︷

(ρuj),j +((ρcpT +ρu2

i

2)︸︷︷︸

ρE+p

uj),j = −qi,i+(τijui),j

(2.63)

(αu2

i

β2− 1)p,t + ρH,t +

H

β2p,t + ((ρE + p)uj),j = −qi,i + (τijui),j (2.64)

and finally the energy equation in conservative form can be written as,

(αu2

i + H − β2

β2)p,t + ρH,t + ((ρE + p)uj),j = −qi,i + (τijui),j (2.65)


The resulting equations of the preconditioned system with conservative fluxestake the form,

1β2 p,t + (ρuj),j = 0(1+α)ui

β2 p,t + ρui,t + (ρujui),j = −p,i + τij,j

(αu2

i +H−β2

β2 )p,t + ρH,t + ((ρE + p)uj),j = −qi,i + (τijui),j

(2.66)

By applying the finite volume method to these equations:∫ 1β2 p,tdV +

∫ρujnjdS = 0∫

( (1+α)ui

β2 p,t + ρui,t)dV +∫(ρujui + pδij − τij)njdS = 0∫

(αu2

i +H−β2

β2 p,t + ρH,t)dV +∫((ρE + p)uj + qj − τijui)njdS = 0

(2.67)

or in matrix form:

Γ−1 ∂Q

∂t= Res (2.68)

with,

Γ−1 =

1β2 0 0 0 0(1+α)uβ2 ρ 0 0 0

(1+α)vβ2 0 ρ 0 0

(1+α)wβ2 0 0 ρ 0

α(u2+v2+w2)+H−β2

β2 0 0 0 ρ

(2.69)

Res = −

1Ω

∫ρujnjdS

1Ω

∫(ρuju1 + pδ1j − τ1j)njdS

1Ω


1Ω


1Ω

∫((ρE + p)uj + qj − τijui)njdS

(2.70)

Ω is the volume of the cell and Q = [p, u, v, w, H].

2.5.3 Dual Time Stepping

The formulation presented in the previous sections is not time accurate andis only useful for steady state calculations. By introducing a pseudo-time, τ ,the same procedure can be followed for a time accurate calculation and forunsteady solvers. The general equations can be written as following:

1β2 p,τ + ρ,t + (ρuj),j = 0αui

β2p,τ + ρui,τ + ρui,t + ρui,juj = −p,i + τij,j

ρcpT,τ − p,τ + ρcp(T,t + T,iui)− (p,t + uip,i) = −qi,i + τijui,j

(2.71)


or with the conservative fluxes:1β2 p,τ + ρ,t + (ρuj),j = 0(1+α)ui

β2 p,τ + ρui,τ + (ρui),t + (ρujui),j = −p,i + τij,j

(αu2

i +H−β2

β2 )p,τ + ρH,τ + (ρE),t + ((ρE + p)uj),j = −qi,i + (τijui),j

(2.72)

By applying the finite volume method to these equations:∫ 1β2 p,τdV +

∫ρ,tdV +

∫ρujnjdS = 0∫

( (1+α)ui

β2 p,τ + ρui,τ )dV +∫(ρui),tdV +

∫(ρujui + pδij − τij)njdS = 0∫

(αu2

i +H−β2

β2 p,τ + ρH,τ )dV +∫(ρE),tdV +

∫((ρE + p)uj + qj − τijui)njdS = 0

(2.73)In matrix form,

Γ−1 ∂Q

∂τ+

∂U

∂t= Res (2.74)

with U = [ρ, ρu, ρv, ρw, ρE], and Γ and Res given in equations (2.69) and(2.70), respectively. β and α in equation (2.69) are the preconditioning pa-rameters. α is always taken as a constant around -1, and β can be definedlocally as, Venkateswaran et al. [113],

β = Min(K1Max((uiui)1/2,

ν

δmin,

l

π∆t), c) (2.75)

With δmin the smallest cell length, l the biggest characteristic dimension ofthe domain, ∆t the physical time step and c the speed of sound. In theanalytical studies of Venkateswaran et al. [113], K1 = 1.

The physical time derivative is discretized with a multi-step scheme:

∂U

∂t−Res =

β1Un+1 + β0U

n + β−1Un−1

∆t−γ1Res(Un+1)−γ2Res(Un) (2.76)

The pseudo-time derivative can also be written in terms of U the conservativevariables,

P−1 ∂U

∂τ+

∂U

∂t= Res (2.77)

with P−1 = Γ−1 ∂Q∂U

. The steady state solution of equation (2.77) in pseudo-time τ gives the value of Un+1.

Now a Runge-Kutta method can be used to reach the steady state inpseudo-time. As explained in section (2.1.2), for stability reasons the β1

∆tU

term should be treated implicitly within the Runge-Kutta cycle. For exampleat the stage i,

U i − U0

∆τ+ αiP (

β1Ui + β0U

n + β−1Un−1

∆t) =

Pαi(γ1Res(U i−1) + γ2Res(Un)) i = 1, q (2.78)


By adding and subtracting β1U0,

U i − U0

∆τ+ αiP (

β1Ui − β1U

0 + β1U0 + β0U

n + β−1Un−1

∆t) =

Pαi(γ1Res(U i−1) + γ2Res(Un)) (2.79)

(I + αi∆τβ1

∆tP )(U i − U0) = Pαi∆τ(−β1U

0 + β0Un + β−1U

n−1

∆t+γ1Res(U i−1) + γ2Res(Un)) (2.80)

U i − U0 = (I + αi∆τβ1

∆tP )−1Pαi∆τ(−β1U

0 + β0Un + β−1U

n−1

∆t+γ1Res(U i−1) + γ2Res(Un)) (2.81)

the difference on the left hand side, U i − U0, is replaced 6 by a difference of(Qi − Q0), multiplied by a Jacobian, ∂U

∂Q,

∂U

∂Q(Qi − Q0) = (I + αi∆τ

β1

∆tP )−1 ∂U

∂QΓαi∆τ ×

6This is a second order approximation, however, when the steady-state is reached inpseudo-time the approximation will be exact. To check this, suppose a general case whereU = [u1, u2, · · · , un] and Q = [q1, q2, · · · , qn] are two vectors with n elements. The elementsof U , denoted by ul are a function of the elements of Q denoted by ql, i.e.,

ul = ul(q1, q2, · · · , qn) l = 1, n

The Taylor series expansion of ul at time level i around time level 0 is,

u(i)l = u

(0)l +

n∑j=1

(q(i)j − q

(0)j )(

∂ul

∂qj)(0) + O(q(i)

j − q(0)j )2 l = 1, n

where the superscript i counts the time level in pseudo-time. At the steady state, τ → ∞,the leading truncation error approaches to zero,

limτ→∞

n∑j=1

O(q(i)j − q

(0)j )2 = 0

and the following equality is correct,

u(i)l − u

(0)l =

n∑j=1

(q(i)j − q

(0)j )(

∂ul

∂qj)(0) l = 1, n

in a matrix form we have,

U i − U0 =∂U

∂Q(Qi −Q0)


(−β1U0 + β0U

n + β−1Un−1

∆t+ γ1Res(U i−1) + γ2Res(Un)) (2.82)

Qi − Q0 =∂Q

∂U(I + αi∆τ

β1

∆tP )−1 ∂U

∂QΓαi∆τ ×

(−β1Ui−1 + β0U

n + β−1Un−1

∆t+ γ1Res(U i−1) + γ2Res(Un)) (2.83)

U0 on the right side of equation (2.82) was replaced by U i−1 in equation(2.83) that is a more recent value.

The matrix ∂Q∂U

(I + αi∆τ β1

∆tP )−1 ∂U

∂Q= (I + αi∆τ β1

∆tΓ∂U∂Q

)−1 should be cal-

culated analytically. We note that if we started from equation (2.74) insteadof equation (2.77), we would have to invert the matrix (I + αi∆τ β1

∆tΓ∂U∂Q

)−1

instead of ∂Q∂U

(I + αi∆τ β1

∆tP )−1 ∂U

∂Q.

For the compressible case, if from the beginning the proper set of variablesis not used, this matrix may be a full 5 × 5 matrix and computationallyexpensive to invert. For instance to evaluate (I + αi∆τ β1

∆tP )−1 ∂U

∂Q, first one

needs to calculate P = ∂U∂Q

Γ, with,

∂U

∂Q=

1RT

ρu1

cpTρu2

cpTρu3

cpT−ρcpT

u1

RT

ρu21

cpT+ ρ ρu1u2

cpTρu1u3

cpT−ρu1

cpT

u1

RTρu1u2

cpT

ρu22

cpT+ ρ ρu2u3

cpT−ρu2

cpT

u3

RTρu1u3

cpTρu3u2

cpT

ρu23

cpT+ ρ −ρu3

cpTHRT

− 1 ρu1HcpT

ρu2HcpT

ρu3HcpT

−ρHcpT

+ ρ

(2.84)

and,

Γ =

β2 0 0 0 0−(1+α)u1

ρ1ρ

0 0 0−(1+α)u2

ρ0 1

ρ0 0

−(1+α)u3

ρ0 0 1

ρ0

β2−α4V 2−Hρ

0 0 0 1ρ

(2.85)

@V 2 = u21 + u2

2 + u23. Obviously, the matrix P = ∂U

∂QΓ, is a full 5× 5 matrix. It


is the same problem with Γ∂U∂Q

=

β2

RTβ2ρu1

cpTβ2ρu2

cpTβ2ρu3

cpT−β2ρcpT

−αu1

ρRT

−αu21

cpT+ 1 −αu1u2

cpT−αu1u3

cpTαu1

cpT

−αu2

ρRT−αu1u2

cpT

−αu22

cpT+ 1 −αu2u3

cpTαu2

cpT

−αu3

ρRT−αu1u3

cpT−αu3u2

cpT

−αu23

cpT+ 1 αu3

cpTβ2−α4V 2

ρRT− 1

ρ(β

2−α4V 2

cpT)u1 (β

2−α4V 2

cpT)u2 (β

2−α4V 2

cpT)u3 −β2−α4V 2

cpT+ 1

(2.86)

However, for the incompressible flow, the Jacobian ∂U∂Q

will have a very simple

form of: ∂U∂Q

= diag(0, ρ, ρ, ρ, ρ) and the product of Γ∂U∂Q

will also have a simple

form of: Γ∂U∂Q

= diag(0, 1, 1, 1, 1). As a result (I + Γαq∆τ β1

∆t∂U∂Q

)−1 will be adiagonal matrix equal to:

(I + αi∆τβ1

∆tΓ

∂U

∂Q)−1 =

1 0 0 0 00 1

1+αiβ1∆τ

∆t

0 0 0

0 0 1

1+αiβ1∆τ

∆t

0 0

0 0 0 1

1+αiβ1∆τ

∆t

0

0 0 0 0 1

1+αiβ1∆τ

∆t

(2.87)

For the test cases considered in this thesis the Mach number is very low andno difference was observed between the results of compressible and incom-pressible Jacobians.

For the general compressible case, one should start from a proper setof variables using a non-conservative formulation to have a preconditionerwhich can be easily inverted.

The non-conservative formulation in equation (2.71) can be written in amatrix form as,

Γ−1w

∂Q

∂τ+ M

∂W

∂t= A

∂W

∂x+ B

∂W

∂y+ C

∂W

∂z(2.88)

with,

Γ−1w =

1β2 0 0 0 0αu1

β2 ρ 0 0 0αu2

β2 0 ρ 0 0αu3

β2 0 0 ρ 0

−1 0 0 0 ρcp

(2.89)


and,

M =

1 0 0 0 00 ρ 0 0 00 0 ρ 0 00 0 0 ρ 0

−T R 0 0 0 ρ(cp − R)

(2.90)

Both sides of equation (2.88) are multiplied by M−1 and the pseudo-timederivative of Q is written in terms of W , i.e., ∂Q

∂τ= ∂Q

∂W∂W∂τ

,

M−1Γ−1w

∂Q

∂W

∂W

∂τ+ M−1M

∂W

∂t= M−1A

∂W

∂x+ M−1B

∂W

∂y+ M−1C

∂W

∂z(2.91)

P−1w

∂W

∂τ+

∂W

∂t= Aw

∂W

∂x+ Bw

∂W

∂y+ Cw

∂W

∂z(2.92)

with P−1w = M−1Γ−1

w∂Q∂W

, Aw = M−1A, Bw = M−1B and Cw = M−1C.

The Jacobian ∂Q∂W

equals,

∂Q

∂W=

RT 0 0 0 Rρ0 1 0 0 00 0 1 0 00 0 0 1 0−T R 0 0 0 1

(2.93)

and the matrix P−1w is equal to,

P−1w =

RTβ2 0 0 0 Rρ

β2

RTαu1

ρβ2 1 0 0 Rαu1

β2

RTαu2

ρβ2 0 1 0 Rαu2

β2

RTαu3

ρβ2 0 0 1 Rαu3

β2

(RT )2−RTβ2

ρ(cp−R)β2 0 0 0 R2Tβ2(cp−R)

+ 1

(2.94)

A Runge-Kutta scheme is used in pseudo-time τ and first order implicitEuler7 is used in physical time t.

W i − W 0

∆τ+αiPw

W n+1 − W n

∆t= αiPw(Aw

∂W

∂x+Bw

∂W

∂y+Cw

∂W

∂z)n+1 (2.95)

The superscript i counts pseudo-time steps and the index n counts physicaltime steps. The steady state solution of equation (2.95) in pseudo-time τ

7The use of any other multi-step method for the discretization of the physical timederivative is straightforward, but a first order implicit Euler scheme is chosen to simplifythe algebraic manipulations.


gives the value of W n+1. The values at time level n+1 are unknown and arereplaced by the known values at pseudo-time level i − 1,

W i − W 0

∆τ+αiPw

W i−1 − W n

∆t= αiPw(Aw

∂W

∂x+Bw

∂W

∂y+Cw

∂W

∂z)i−1 (2.96)

For stability reasons explained previously, the αiPwW i−1

∆tterm on the left

hand side is treated implicitly in the Runge-Kutta cycle,

W i − W 0

∆τ+ αiPw

W i − W n

∆t= αiPw(Aw

∂W

∂x+ Bw

∂W

∂y+ Cw

∂W

∂z)i−1 (2.97)

by adding −αi∆τ∆t

PwW 0 to both sides, and bringing −αiPwWn

∆tfrom the left

side to the right,

W i − W 0 + αi∆τ

∆tPw(W

i − W 0) = Pw∆ταi(−W 0 − W n

∆t+

(Aw∂W

∂x+ Bw

∂W

∂y+ Cw

∂W

∂z)i−1) (2.98)

(W i − W 0)(I + αi∆τ

∆tPw) = Pw∆ταi(−W 0 − W n

∆t+

(Aw∂W

∂x+ Bw

∂W

∂y+ Cw

∂W

∂z)i−1) (2.99)

W i − W 0 = (I + αi∆τ

∆tPw)

−1Pw∆ταi(−W 0 − W n

∆t+

(Aw∂W

∂x+ Bw

∂W

∂y+ Cw

∂W

∂z)i−1) (2.100)

Now suppose U = [ρ, ρu, ρv, ρw, ρE], is the conservative variables. In orderto write the equations in terms of the conservative variables U , the transfor-mation ∆W = ∂W

∂U∆U is used,

∂W

∂U(U i − U0) = (I + αi

∆τ

∆tPw)

−1Pw∆ταi(−∂W

∂U

U0 − Un

∆t+

(Aw∂W

∂U

∂U

∂x+ Bw

∂W

∂U

∂U

∂y+ Cw

∂W

∂U

∂U

∂z)i−1) (2.101)

multiplying both sides by ∂U∂W

,

U i − U0 =∂U

∂W(I + αi

∆τ

∆tPw)

−1Pw∂W

∂U∆ταi(−U0 − Un

∆t+

(∂U

∂WAw

∂W

∂U

∂U

∂x+

∂U

∂WBw

∂W

∂U

∂U

∂y+

∂U

∂WCw

∂W

∂U

∂U

∂z)i−1) (2.102)


by using the simple algebraic equality, ∂W∂U

∂U∂W

= 1,

U i − U0 =∂U

∂W(I + αi

∆τ

∆tPw)

−1 ∂W

∂U

∂U

∂WPw

∂W

∂U∆ταi(−U0 − Un

∆t+

(Ac∂U

∂x+ Bc

∂U

∂y+ Cc

∂U

∂z)i−1)(2.103)

Ac, Bc and Cc are respectively equal to Ac =∂U∂W

Aw∂W∂U

, Bc =∂U∂W

Bw∂W∂U

, andCc =

∂U∂W

Cw∂W∂U

.

U i − U0 =∂U

∂W(I + αi

∆τ

∆tPw)

−1 ∂W

∂UPc∆ταi(−U0 − Un

∆t+

(Ac∂U

∂x+ Bc

∂U

∂y+ Cc

∂U

∂z)i−1) (2.104)

Where Pc =∂U∂W

Pw∂W∂U

is the preconditioner in U variables. Now the conser-vative fluxes can be retrieved,

U i − U0 =∂U

∂W(I + αi

∆τ

∆tPw)

−1 ∂W

∂UPc∆ταi(−U i−1 − Un

∆t+

(∂F

∂x+

∂G

∂y+

∂E

∂z)i−1) (2.105)

with ∂F∂x

= Ac∂U∂x

, ∂G∂y

= Bc∂U∂y, and ∂E

∂z= Cc

∂U∂z. U0 on the right hand side of

equation (2.104) is replaced by U i−1 in equation (2.105) that is a more recentvalue.

To work with another set of variables Q instead of U, one simply needsto use the transformation U i − U0 = ∂U

∂Q(Qi − Q0) on the left hand side of

equation (2.105),

Qi − Q0 =∂Q

∂U

∂U

∂W(I + αi

∆τ

∆tPw)

−1 ∂W

∂UPc∆ταi(−U i−1 − Un

∆t+

(∂F

∂x+

∂G

∂y+

∂E

∂z)i−1) (2.106)

The advantage of this formulation is that for any set of variables the matrixwhich should be inverted, i.e., (I + αi

∆τ∆t

Pw)−1, remains the same. And if

from the beginning Pw is chosen properly, then (I + αi∆τ∆t

Pw)−1 can always

be easily calculated.The matrix (2.94) has a good structure and is very easy to find its inverse

Pw. The general matrix A with a structure similar to Pw is,

A =

a11 0 0 0 a15

a21 a22 0 0 a25

a31 0 a33 0 a35

a41 0 0 a44 a45

a51 0 0 0 a55

(2.107)


and its inverse A−1,

A−1 =

b11 0 0 0 b12

−a21b11+a25b21a22

1/a22 0 0 −a21b12+a25b22a22−a31b11+a35b21

a330 1/a33 0 −a31b12+a35b22

a33−a41b11+a45b21a44

0 0 1/a44 −a41b12+a45b22a44

b21 0 0 0 b22

(2.108)

with bij defined as,[b11 b12

b21 b22

]=

1

a11a55 − a15a51

[a55 −a15

−a51 a11

](2.109)

The structure of (I +αi∆τ∆t

Pw) is also similar to A and the same method canbe used to calculate its inverse.

2.6 Time Correction

The equations in pseudo time are always solved approximately and only atentative solution is obtained. Hirt and Harlow [42] introduced a “GeneralCorrective Procedure” that reduces the number of subiterations and main-tains the required accuracy. The method has also been tested by Marx [74]for two-dimensional incompressible unsteady calculations, and showed to beeffective in reducing the number of inner iterations. Weber et al. [117] ap-plied it to their LES calculations and concluded that the use of this procedureimproves the accuracy of the results when the given degree of convergence is“almost” enough to obtain a good solution.

In this section the general corrective procedure of Hirt and Harlow isbriefly explained. Consider the following equation,

∂U

∂t= Res(U) (2.110)

suppose the time derivative is discretized implicitly with a trapezoidal scheme,

Un+1 − Un

∆t=

1

2(Res(Un+1) + Res(Un))

to solve the implicit system a pseudo-time τ derivative is introduced,

∂U

∂τ+

Un+1 − Un

∆t=

1

2(Res(Un+1) + Res(Un))

the steady state solution of the above equation in pseudo-time τ gives thevalue of Un+1. When the steady state is reached, the time derivative ∂U

∂τis

2.6. TIME CORRECTION 47

“theoretically” equal to zero. However, in practice some convergence criterionstops the subiteration procedure and leaves a small error ε coming from thenot fully converged iterative procedure. After the last inner iteration thesystem that is really solved is,

Un+1 − Un

∆t=

1

2(Res(U

n+1) + Res(Un)) + ε (2.111)

with Un+1

an approximate solution at time level n+1 and ε the error relatedto the drop of the residual during the inner iterations. This error can beeasily suppressed by using a corrective procedure which put ε equal to zeroin equation (2.111) and solve for an intermediate solution Un+1,

Un+1 =1

2∆t(Res(U

n+1) + Res(Un)) + Un

the implicit system at the next time level is,

∂U

∂τ+

Un+2 − Un+1

∆t=

1

2(Res(Un+2) + Res(U

n+1))

which gives the value of Un+2

after the computational steady state is reached,

Un+2 − Un+1

∆t=

1

2(Res(U

n+2) + Res(U

n+1)) + ε (2.112)

and again in the above equation ε is set to zero and Un+2 is calculated,

Un+2 =1

2∆t(Res(U

n+2) + Res(U

n+1)) + Un+1

This correction procedure reduces the error accumulation of time integrationand might allow the use of a coarser convergence criteria. Nevertheless, itshould be noted that this procedure does not completely eliminate the errors.In equation (2.112) the residuals are still calculated from U and not from U .It is also possible to calculate the residuals in function of U . This variant hasalso been tested by Marx [74], but he concluded that the best results wereobtained when using the procedure of Hirt and Harlow [42].

Chapter 3

Comparative Study

The efficiency of an implicit dual time-stepping method in comparison witha purely explicit approach depends on the number of inner iterations Ni andthe ratio of the physical time step to the time step of a purely explicit schemeR = ∆t

∆τ. The gain achieved will be, gain= R

Ni. As a result, the main goal is to

choose the physical time step ∆t as large as possible and make the numberof inner iterations Ni as small as possible.

To be able to decrease the number of inner iterations, one has to usedifferent convergence acceleration techniques and set the corresponding pa-rameters properly. For example, to have a fast convergence in pseudo-timethe optimum values of α and β parameters in preconditioning should beused. Different types of multigrid cycle should be tested. The maximumvalue of the physical time step which also maintain the required temporalaccuracy should be set. The necessary number of inner-iterations that isdirectly proportinal to the resiual drop should be found. The relationshipbetween multigrid and preconditioning should be studied and so on.

In the coming sections some aspects of this problem will be studied. Theonly aim is to create some feeling about the different parameters which aresubsequently used for the final calculations on finer meshes.

3.1 Influence of Preconditioning on Multigrid

The success of multigrid depends on the ability of the smoother to removethe high frequency errors which cannot be supported on coarser meshes.In fact efficient smoothing can be obtained if one can insure that all errorcomponents containing high frequency modes are efficiently damped on finestgrid since multigrid will smooth the lower frequencies.

Darmofal et al. [23] analysed the smoothing properties of a four stageRunge-Kutta method for preconditioned Euler and Navier-Stokes equations

49

50 CHAPTER 3. COMPARATIVE STUDY

via the Fourier footprint of the discrete spatial operator. They showed pre-conditioning effectively removes the modes along the real axis1 and bringsthem to the regions with higher damping properties. As a result, for lowMach number flows preconditioning is necessary to remove the stiffness ofthe equations and to increase the high frequency damping ability of thesmoother.

The use of Fourier footprint to analyse the smoothing abilities of pre-conditioned Euler and Navier-Stokes equations can also be found in otherreferences, [63], [72], [37], and [34].

In order to show this dependence a test was carried out on a channel atReτ = 180 and M=0.06, with a mesh of 33×33×33 points. The streamwise,normal and spanwise dimensions are 4πδ×2δ×2πδ, with δ the channel half-width. Figure (3.1) shows part of the convergence graphs of the continuityand first momentum equations for four different combinations of multigridand preconditioning. The number of inner iterations is fixed to 50.

According to the figure, there is almost no difference between singlegrid and multigrid without preconditioning (dotted and dot-dashed lines).When preconditioning is activated multigrid becomes quite efficient (solidand dashed lines). When both preconditioning and multigrid are switchedoff, the residual drop of the first momentum equation is around one order ofmagnitude, and for the continuity equation less than two. However, whenpreconditioning and multigrid are combined together, the residual drops aremore than four and three orders of magnitude for the first momentum andcontinuity equations, respectively.

Due to the addition of artificial dissipation, the multigrid works moreefficiently for the continuity equation than for the first momentum equationin which the artificial dissipation is switched off2. For these calculationsα = −1 and β is globally defined as β2 = 3.V 2

cent, where Vcent is the velocityat the centerline of the channel. A three-level V-sawtooth multigrid is used.

3.2 Influence of the Number of Inner Itera-

tions

The goal is to find the minimum number of inner iterations or the minimumdrop of residual in pseudo time during inner iterations. The error relatedto the drop of the residual in inner the iterations should neither be bigger

1According to Darmofal, these modes with a poor propagation property, are a directresult of the low Mach number stiffness and correspond to the low speed convection modespresent in the system.

2Adding artificial dissipation to momentum equations would immediately lead to thelaminarization of the flow at this Reynolds number.

3.2. INFLUENCE OF THE NUMBER OF INNER ITERATIONS 51

33*33*33 meshDeltat=.01, Trap., Inner.it=50

iterations

Res1

0. 53. 107. 160. -4.00

-2.50

-1.00

.50 MG, PrecoNo MG, PrecoNo MG, No PrecoMG, No Preco

33*33*33 meshDeltat=.01, Trap., Inner.it=50

iterations

Res2

0. 53. 107. 160. -5.0

-3.0

-1.0

1.0 MG, PrecoNo MG, PrecoNo MG, No PrecoMG, No Preco

Figure 3.1: The effect of preconditioning on multigrid for the channel flowcalculation with a fixed number of inner iterations, left: continuity equation,right: first momentum equation

nor smaller than the errors of spatial and temporal discretizations. A biggerresidual drop error may lead to wrong results and a smaller residual droperror compared to other errors, unnecessarily increases the computationaltime.

Finding the proper amount of residual drop for unsteady calculations witha theoretical approach is very difficult, if not impossible. It strongly dependson the other errors coming from boundary conditions, spatial and temporaldiscretization, turbulence modeling etc...

In order to have a rough estimation about the minimum number of inneriterations the same channel flow has been tested at Reτ = 180. Precon-ditioning, multigrid and residual smoothing are all switched off. The timederivative is discretized using a second order backward differencing schemewith the physical time step ∆t = 0.0025 δ

uτ. Figure (3.2) shows the effect of

the number of inner iterations on the results. Four different cases with dif-ferent number of inner iterations of 12, 18, 25 and 120 have been considered.The initial solution is a fully developed and statistically steady simulationobtained with an explicit method. This solution is advanced in time for 1.5 δ

uτ

seconds with an implicit method using the above mentioned number of inneriterations. In figure (3.2) the profiles of the normal fluctuations

√v′v′ and

mean velocity are shown. The results obtained with 25 (dashed line) and120 (bold line) inner iterations follow closely each other, but the deviationstarts when the number of inner iterations is decreased to 18 (dotted line).With 12 inner iterations (dot-dashed line) the results are completely wrong.In figures (3.3) and (3.4) part of the residual history are plotted. With 25inner iterations the residual drop of the continuity equation is around one


Effect of the number of inner iterationsvv turbulence intensity versus y

y/delta

v_rms/u*

-1.00 -.67 -.33 .00 .00

.27

.53

.80 Inner-it = 120Inner-it = 25Inner-it = 18Inner-it = 12

Effect of number of inner iterationsmean velocity profiles versus y

log(y+)

<u+mean>

.80 1.37 1.93 2.50 8.0

12.0

16.0

20.0 Inner_it = 120Inner_it = 25Inner_it = 18Inner_it = 12

Figure 3.2: Left: y components of turbulence intensity√

v′v′, right: meanvelocity profile.

Inner iteration = 1202nd order Backward Differencing, DT=0.0025.

iterations

Residual

1300. 1400. 1500. 1600. -6.0

-2.0

2.0

6.0 res1res2res3res4res5

Inner iterations = 252nd order Backward Differencing, DT=0.0025.

iterations

Residual

0. 40. 80. 120. -2.50

-1.50

-.50

.50 res1res2res3res4res5

Figure 3.3: Drop of the residuals for 120 (left) and 25 (right) inner iterations.

3.3. INFLUENCE OF THE TIME CORRECTION TECHNIQUE 53

Inner Iteration = 18Backward Differencing, DT=0.0025

iterations

Residual

10720. 10760. 10800. 10840. -2.00

-1.20

-.40

.40 res1res2res3res4res5

Inner iterations = 12Backward Differencing, DT=0.0025

iterations

Residual

7150. 7170. 7190. 7210. -1.20

-.40

.40

1.20 res1res2res3res4res5

Figure 3.4: Drop of the residuals for 18 (left) and 12 (right) inner iterations.

Effect of the time correction.mean velocity profiles versus y

log(y+)

<u+mean>

.40 1.10 1.80 2.50 4.0

9.3

14.7

20.0 With correction.Without correction.exp

Effect of the time correctionvv turbulence intensity versus y

y/delta

v_rms/u*

-1.00 -.67 -.33 .00 .00

.27

.53

.80 With correctionWithout correctionExplicit

Figure 3.5: Effect of time correction on mean velocity (left) and√

v′v′ com-ponent of turbulence intensity (right).

order of magnitude and one can conclude that for the present calculation, tohave a proper result, the drop of the continuity equation should be minimumone order of magnitude. However, this value is very marginal and for othercalculations, to be on the safe side, a bigger residual drop is always used.

3.3 Influence of the Time Correction Tech-

nique

The time correction technique explained in section (2.6) is tested on the samechannel flow. From figure (3.2) one can see that with 18 inner iterations theerror related to an insufficient drop of residual deteriorates the results. This


error can partly be eliminated by using the time correction technique. Figure(3.5) shows the influence of the time correction on the profile of mean velocityand the normal velocity fluctuations

√v′v′ after the solution is advanced in

time for 1.5 δuτ

seconds from a fully developed solution of an explicit scheme.

One can see from figure (3.5) that when the time correction method isused the velocity and turbulence intensity profiles of the implicit approach(bold line) follow more smoothly the profile of the explicit approach (dottedline) compared to the case where no time correction is used (dashed line).

However, it should be mentioned that by further decreasing the numberof inner iterations the time correction technique will not be effective any-more. The reason is, that the procedure does not completely eliminate theerror related to an insufficient residual drop, but somehow prevents the erroraccumulation.

The time correction method has also been applied to the case with 12inner iterations (results not shown). With 12 inner iterations, the residualdrop error was too big to be eliminated by the time correction technique andthe results, even with the time correction, were not satisfactory. On the otherhand, when the number of inner iterations is big enough and the residual droperror is small, there is no need to use the time correction procedure.

In summary, this procedure is only effective at some marginal point wherethe number of inner iterations is “almost” sufficient to get good results andby using the time correction those small errors may be eliminated.

3.4 Relation Between the Time Integration

and the Physical Time Step

It is well known that the backward differencing scheme, unlike the trapezoidalscheme, has a dissipation error proportional to the time step, see Hirsch [39],Moin [81]. In spite of this, the backward differencing is much more popularthan the trapezoidal scheme in the CFD community.

In order to see the influence of time discretization on the LES results, achannel flow at Reτ = 180 with a mesh of 33× 33× 65 points in the x, y andz directions is considered. The streamwise, normal and spanwise dimensionsare 4πδ × 2δ × 2πδ, with δ the channel half-width. The initial solution is afully developed and statistically steady simulation obtained with an explicitmethod.

This solution is advanced in time for 1.5δ/uτ seconds with an implicitmethod. The time derivative is discretized either with a backward differenc-ing (β1 = 1.5, β0 = −2, β−1 = 0.5, γ1 = 1. and γ2 = 0.0) or a trapezoidal

3.5. THE TIME STEP AND PRECONDITIONING PARAMETERS 55

Backward Differencing.mean velocity profiles versus y

log(y+)

<u+mean>

.50 1.17 1.83 2.50 3.0

9.7

16.3

23.0 ExplicitDT=.01DT=.005DT=.0025

Backward Differencing.vv turbulence intensity versus y

y/delta

v_rms/u*

-1.00 -.67 -.33 .00 .00

.27

.53

.80 ExplicitDT=.01DT=.005DT=.0025

Figure 3.6: Effect of the time step with the backward-differencing, left: meanvelocity, right:

√v′v′ turbulence intensity.

scheme (β1 = 1., β0 = −1, β−1 = 0, γ1 = 0.5 and γ2 = 0.5),

∂U

∂t−Res =

β1Un+1 + β0U

n + β−1Un−1

∆t− γ1Res(Un+1)− γ2Res(Un) (3.1)

Figures (3.6) and (3.7) show the mean velocity profile and the turbulenceintensities calculated with the backward differencing scheme and the trape-zoidal scheme, respectively.

The trapezoidal scheme is much more flexible than the backward differ-encing for larger time steps. By increasing the time step, the results startdeviating from the reference solution (obtained with an explicit time accu-rate Runge-Kutta method) at ∆t = 0.005δ/uτ (dotted line) for the backwarddifferencing and at ∆t = 0.02δ/uτ (dot-dot-dashed line) for the trapezoidalscheme. This behavior is in accordance with the flat plate boundary layerresults of Weber et al. [117].

3.5 The Time Step and Preconditioning Pa-

rameters

In preconditioning, the most important parameter to be chosen is the βparameter. In general, the optimum value of this parameter depends onthe local speed of the flow, the viscosity, and for unsteady calculations onthe magnitude of the physical time step. For steady calculations a localpreconditioning parameter3 has always been more successful than a global

3A parameter which depends on the local speed of the flow rather than a global quantitylike the free-stream velocity.


Trapezoidal Rule.mean velocity profiles versus y

log(y+)

<u+mean>

.50 1.17 1.83 2.50 3.0

9.7

16.3

23.0 ExplicitDT=.0025DT=.005DT=.01DT=.02

Effect of Time Step on Trapezoidal Method.vv turbulence intensity versus y

y/delta

v_rms/u*

-1.00 -.67 -.33 .00 .00

.27

.53

.80 ExplicitDT=.0025DT=.005DT=.01DT=.02

Figure 3.7: Effect of the time step with the trapezoidal scheme, left: meanvelocity, right:

√v′v′ turbulence intensity.

one. Examples in the literature are numerous, see for example Van Leer etal. [62], and Turkel [110], [111]. In contrast, for time accurate calculationsthe number of articles is rather limited.

As explained in section (2.5.3), based on a perturbation analysis of the lowMach number limit for the Euler and Navier-Stokes equations, Venkateswaranet al. [113] proposed equation (2.75) for the calculation of the precondition-ing parameter β.

The preconditioning parameters α is always taken as a constant around-1, and β is calculated as,

β = Min(K1Max((uiui)1/2,

ν

δmin,

l

π∆t), c) (3.2)

In order to assess the applicability of this formula for unsteady viscous flows,a series of tests has been carried out. A two dimensional lid driven cavityflow at Re = Uh

ν= 10000 is considered. Artifical dissipation is added to all

equations and the trapezoidal method is used to discretize the time deriva-tive. A three-level V sawtooth multigrid is also used. Both the dimension hof the square cavity, and the velocity U of the lid are considered to be unity.A 65× 65 mesh is used with a cosine distribution of the mesh points,

x, y =1− cosθ

20 < θ < π

with δmin/h = 6.045×10−4 being the closest point to the wall. Three differenttime steps, ∆t1 = 0.002 h

U, ∆t2 = 0.01 h

U, and ∆t3 = 0.1 h

Uare considered. In

equation (2.75) the first argument of the Max operator (uiui)1/2 is called

3.5. THE TIME STEP AND PRECONDITIONING PARAMETERS 57

the inviscid velocity scale. The biggest value of the inviscid velocity scale iswhen the velocity is equal to the lid velocity U = 1,

max(uiui)1/2 = 1(m/s)

The second argument νδmin

is called the diffusion velocity scale and for thistest case its maximum value is at the first point away from the wall,

max(ν

δmin) =

10−4

6.045× 10−4= 0.165(m/s)

The third argument lπ∆t

is called the unsteady velocity scale. For ∆t1 =0.002 h

Uit is equal to,

h

π0.002 hU

=1

π0.00211

= 159.23(m/s)

As a result, the unsteady velocity scale is the dominant term in equation(2.75) and β will be constant in the whole field.

Figure (3.8) shows the residual history of the continuity, momentum andenergy equations for ∆t1 = 0.002 h

Uand for different β paramaters ranging

from 1 to 500. It can be seen from the figure that the best convergence is forβ between 500 (dot dashed line) and 100 (dotted line). This is in accordancewith what equation (2.75) suggests,

β = Min(K1Max(1, 0.165, 159.23), 300) = Min(K1159.23, 300)

depending on the values of K1, which is normally around 2, the β parameteris around 300.

Figure (3.9) shows the same residuals for ∆t2 = 0.01 hU. The optimum β

is between 50 (dotted line) and 100 (dot dashed line). The unsteady velocityscale is,

h

π0.01 hU

=1

π0.0111

= 31.84(m/s)

with an adjusting coefficient K1 3 the value of β calculated from equation(2.75) will be in the same range of β obtained from this numerical experience.

And finally, in figure (3.10) the residuals are shown for a higher value ofthe physical time step, ∆t3 = 0.1 h

U, with an unsteady velocity scale equal to,

h

π0.1 hU

=1

π0.111

= 3.184(m/s)

Based on the convergence histories of figure (3.10), the optimum β is between1 (dashed line) and 10 (dotted line), which is in agreement with the abovevelocity scale.


Effect of BETPAR, 65*65 clustered mesh, DT=0.0022D cavity flow, unsteady calculation, with multigrid.

iterations

Res1

0. 333. 667. 1000. -8.0

-4.7

-1.3

2.0 BETPAR = 1BETPAR = 10BETPAR = 100BETPAR = 500


iterations

Res2

0. 333. 667. 1000. -8.0

-4.7

-1.3



iterations

Res4

0. 333. 667. 1000. -6.0

-3.3

-.7

2.0 BETPAR=1BETPAR=10BETPAR=100BETPAR=500

Figure 3.8: Convergence history for different β’s, ∆t = 0.002LU, left: conti-

nuity, middle: first momentum, right: energy equations.


iterations

Res1

0. 333. 667. 1000. -8.0

-5.0

-2.0



iterations

Res2

0. 333. 667. 1000. -9.0

-5.7

-2.3



iterations

Res4

0. 333. 667. 1000. -6.0

-3.7

-1.3

1.0 BETPAR=1BETPAR=10BETPAR=50BETPAR=100

Figure 3.9: Convergence history for different β’s, ∆t = 0.01LU, left: continu-

ity, middle: first momentum, right: energy equations.


iterations

Res1

0. 300. 600. 900. -9.0

-5.3

-1.7

2.0 BETPAR = 0.1BETPAR = 1BETPAR = 10BETPAR = 100


iterations

Res2

0. 300. 600. 900. -10.0

-6.0

-2.0

2.0 BETPAR = 0.1BETPAR = 1BETPAR = 10BETPAR = 100


iterations

Res4

0. 300. 600. 900. -9.0

-5.3

-1.7

2.0 BETPAR=.1BETPAR=1BETPAR=10BETPAR=100

Figure 3.10: Convergence history for different β’s, ∆t = 0.1LU, left: continu-

ity, middle: first momentum, right: energy equations.

Chapter 4

Final Results

Channel and cavity flows are considered to test the ability of the methodon finer meshes. Before going into the details of the specific test case, somepoints concerning the numerical method may be highlighted,

• The calculation of the viscous fluxes needs much more computationaleffort compared to the one of the convective fluxes. To have a fastercomputation the viscous fluxes are only calculated on the finest gridlevel and restricted to the coarser levels via the forcing function of themultigrid, see also Vatsa et al. [112], Lacor et al. [61].

This has another advantage; the subgrid-scale eddy viscosity which isa part of the viscous fluxes routine is evaluated only on the finest level.This is in agreement with the basic assumption of LES which needs arather fine mesh to directly simulate the big vortices and model thesmaller and more isotropic ones.

• Different multigrid strategies like V-cycle and W-cycle as well as theirsawtooth counterparts have been tested and finally, the V-sawtoothcycle showed to be the most robust for the present calculations.

• First order prolongation and quadratic restriction are used as multigridtransfer operators.

• A trapezoidal method is used to discretize the physical time derivative.

• A five-stage explicit Runge-Kutta scheme, with three evaluations ofdissipation, Jameson [48], is used as the smoother for the multigrid.For the equation,

dU

dt= Resc(U) + Resd(U)

59

60 CHAPTER 4. FINAL RESULTS

with Resc and Resd the convective and diffusive residuals, the solutionat an intermediate stage k is updated as,

Uk+1 = U0 + αk(Resc(Uk) + βkResd(U

k) + (1− βk)Resd(Uk−1))

α and β are equal to, α = 14, 1

6, 3

8, 1

2, 1, β = 1, 0, 0.56, 0, 0.44. This

scheme has a good damping property and a large stability region, whichallows to use a bigger time step.

• For the spatial discretization a central method with Jameson-Schmidt-Turkel (JST) type artificial dissipation, Jameson et al. [49], is used.The JST type dissipation adds a first order dissipation near discontinuities1,which are always absent in the present LES calculations, and a thirdorder background dissipation. If the third order artificial dissipation isadded to the momentum equations, it will lead to the immediate lam-inarization of the flow. To avoid this, the artificial dissipation is onlyadded to the continuity equation but not to the momentum and energyequations. According to our experience this also suffices to guaranteea good overall convergence in the inner iterations.

4.1 Channel Flow

4.1.1 Test Case Description and Turbulence Statistics

The geometry of the channel is shown in figure (4.2). A mesh of 65×65×65points is used in the x, y and z directions, the streamwise, normal andspanwise dimensions are 4πδ × 2δ × 4

3πδ, with δ the channel half-width.

Uniform meshes with spacing ∆x+ = ∆xuτ

ν 35 and ∆z+ = ∆zuτ

ν 12 are

used in the streamwise and spanwise directions. A non uniform mesh withhyperbolic tangent distribution is used in the wall-normal direction. The firstmesh point away from the wall is at y|+1 = ∆yuτ

ν 0.49 and the maximum

spacing (at the centerline of the channel) is 13.8 wall units. The followingtransformation gives the location of grid points in the y direction,

yi =1

atanh[ξiarctanh(a)] (4.1)

with,ξi = −1 + 2(j − 1)/(Ny − 1) (j = 1, 2, · · · , Ny)

Here a is the adjustable parameter of the transformation (0 < a < 1); a largevalue of a distributes more points near the walls. In the present computa-tions, a = 0.98346, Ny = 65.

1Like shock waves.

4.1. CHANNEL FLOW 61

10−1

100

101

102

103

10−12

10−11

10−10

10−9

10−8

10−7

10−6

10−5

10−4

y+

Cde

lta2

Cdelta2

10−1

100

101

102

103

10−5

10−4

10−3

10−2

10−1

100

y+

Prt

Prt

Figure 4.1: Variation of C∆2 defined in Eq. (1.7) (left figure), and turbulentPrandtl number defined in Eq. (1.29) (right figure) with distance from thewall.

The Reynolds number based on the friction velocity is Reτ = 180 andthe Mach number at the centerline is M = 0.06. A dynamic procedure,Germano et al. [32], is used to calculate the Smagorinsky coefficient C andthe turbulent Prandtl number Prt. In figure (4.1) the product C∆2, (with∆2 = ∆x∆y∆z)2/3) and the turbulent Prandtl number Prt are plotted as afunction of the wall coordinate y+.

The preconditioning parameters α = −1 and β = 0.14 Lx

π∆t. For this

test case, ∆t = 0.01δ/uτ and Lx/δ = 4π, and in equation (2.75) the thirdargument of the Max operator equals, Lx

π∆t= 400uτ . The smallest cell length

is the y coordinate of the first mesh point away from the wall and the secondargument of the Max operator in equation (2.75) equals, ν

0.49ν/uτ= 1.15uτ .

The first argument of the Max operator, (uiui)1/2 reaches its maximum value

at the centerline of the channel which is equal to, (uiui)1/2 = 18uτ . As

a result, everywhere in the channel, Max((uiui)1/2, ν

δmin, lπ∆t

) = Lx

π∆tand

when β is calculated according to equation (2.75), it should be constantin the whole field. The β parameter in terms of the velocity is equal toβ = 3.16Ucenter, with Ucenter the centerline velocity.

The physical constraint requires ∆t to be less than the time scale of thesmallest resolved scale of motion, i.e., CF Lc =

∆tUc

∆x≤ 1, with Uc a convective

velocity. For the channel flow the maximum Uc is at the center of the channel.In order to check if the assumed time step satisfies this condition or not letus calculate the CF Lc at the channel center. In the present calculation atthe center of the channel Uc/uτ = 18.58 and ∆x = Lx/(Nx − 1) = 4πδ/64.The CF Lc may be written as,

CF Lc =∆tUc

∆x=

(0.01δ/uτ )(18.58uτ )

4πδ/64= 0.946 < 1 (4.2)


Lx

xy

z

Lz

Ly

Figure 4.2: Channel flow test case.

Therefore, the physical condition is satisfied.

For this test case the effect of residual smoothing was not clear. Afterusing residual smoothing the convergence of the continuity equation wasimproved but the convergence of the momentum and energy equations werebetter without residual smoothing. The reason can be the absence of artificialviscosity in those equations. To be on the safe side, the residual smoothingwas switched off for this test case.

The mean velocity profile and turbulence intensities are shown in figures(4.3) to (4.6). The results are compared with the DNS data of Kim et al. [52],and the LES of Morinishi [82]. The results of Morinishi are also obtained witha second order spatial discretization but on a coarser mesh of 32× 65× 32.In the same figures the results of a coarse mesh simulation, i.e., 33× 33× 65points in x, y and z directions, are also plotted for comparison.

In figure (4.7), resolvable < u′v′ >, viscous < ν ∂u∂y

>, modeled (SGS)

< νt∂u∂y

> and total (resolvable+viscous+modeled) shear stresses are shown.As expected, the total shear stress is a straight line with a slope equal toone.

Fluid statistics were averaged for 11.5δ/uτ . The physical time step isaround 400 times bigger than the time step of a purely explicit method,the residual drop of the continuity equation is fixed to -2.5 and this requiresalmost 100 inner iterations. Increasing the number of inner iterations and theabsolute value of the residual drop showed to be unnecessary as the resultsremained the same for higher number of inner iterations. The gain achievedcompared with a purely explicit method is roughly equal to 400

100= 4.


mean velocity profiles versus y

log(y+)

<u+mean>

-1.00 -.50 .00 .50 1.00 1.50 2.00 2.50 .0

5.0

10.0

15.0

20.0

25.0 Present 65x65x65Morinishi 33x65x33DNS KimPresent 33x33x65

Figure 4.3: Mean streamwise velocity of channel flow compared with DNS ofKim [52] and second order LES of Morinishi [82].

uu turbulence intensities vs. y

y/delta

u_rm

s/u*

-1.00 -.80 -.60 -.40 -.20 .00 .00

.50

1.00

1.50

2.00

2.50

3.00

3.50 Present 65x65x65DNS KimMorinishi 33x65x33Present 33x33x65

Figure 4.4:√

u′u′ turbulence intensities for channel flow with fine 653 andcaorse 33×33×65 meshes compared with DNS of Kim [52] and second orderLES of Morinishi [82].


vv turbulence intensities vs. y

y/delta

v_rms/u*

-1.00 -.80 -.60 -.40 -.20 .00 .00

.20

.40

.60

.80


Figure 4.5:√

v′v′ turbulence intensities for channel flow.

ww turbulence intensities vs. y

y/delta

w_r

ms/

u*

-1.00 -.80 -.60 -.40 -.20 .00 .00

.40

.80


Figure 4.6:√

w′w′ turbulence intensities for channel flow.


Resolved modeled and total turbulence shear stress

y/delta

-1.00 -.80 -.60 -.40 -.20 .00 .00

.33

.67

1.00 1 Viscous Stress2 Modeled3 Resolvable(1+2+3) Total

Figure 4.7: Resolvable < u′v′ >, viscous < ν ∂u∂y

>, modeled < νt∂u∂y

> and

total (resolvable+viscous+modeled) shear stress.

4.1.2 One Dimensional Energy Spectra

The energy spectra at two different locations of y+ = 6.09 and y+ = 144.54compared with DNS of Kim et al. [52] are shown in figures (4.8) and (4.9).The mesh used for DNS is not the same as the mesh of the present calcula-tion and in DNS the energy spectra are calculated at slightly different walldistances of y+ = 5.39 and y+ = 149.53.

We will briefly explain how the streamwise (Euu(kx), Evv(kx), Eww(kx))and spanwise (Euu(kz), Evv(kz), Eww(kz)) energy spectra are calculated. Asan example, it will be shown how streamwise energy spectra are calculated.

Suppose an xz plane parallel to the wall and at a distance of y = yc fromthe wall. If φ(x, y, z) denotes the velocities φ = (u, v, w), the first step is tocalculate, at an arbitrary spanwise location zc, the energy spectrum of theperiodic function φ(x, yc, zc) varying in the streamwise x direction.

If the number of cells (not points, because finite volume is used) in thestreamwise x direction is Nx, and Lx is the period of φ(x, yc, zc), the complexspectrum of φ(x, yc, zc) denoted by φ(kx) is calculated as,

φ(kx) =1

Nx

Nx−1∑j=0

φ(xj, yc, zc)e−ikxxj kx =

2π

Lx

k − Nx

2≤ k ≤ Nx

2− 1

(4.3)with φ(xo, yc, zc) and φ(xNx−1, yc, zc) the values of function at the first andlast cells, respectively, and xj =

Lx

Nxj.

Once φ(kx) has been calculated, the energy spectrum Eφφ(kx) can be


10−1

100

101

102

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

kx

Str

eam

wis

e 1D

ene

rgy

spec

tra

at y

+ =

6.0

9Euu−LESEvv−LESEww−LESEuu−DNSEvv−DNSEww−DNS

100

101

102

10−6

10−5

10−4

10−3

10−2

10−1

100

kz

Spa

nwis

e 1D

ene

rgy

spec

tra

at y

+ =

6.0

9

Euu−LESEvv−LESEww−LESEuu−DNSEvv−DNSEww−DNS

Figure 4.8: One dimensional energy spectra, left: streamwise, right: span-wise, at y+ = 6.09 compared with DNS of Kim et al. [52] at y+ = 5.39.

written as,

Eφφ(kx) = φ(kx)φ∗(kx) (4.4)

with φ∗(kx) the complex conjugate of φ(kx). The same procedure is appliedto other streamwise lines of the xz plane. And finally, Eφφ(kx) are averagedover the spanwise direction.

For Eφφ(kz) the function is supposed to be periodic in the z direction andafter the energy spetra are calculated, they are averaged in the streamwisedirection.

4.2 Cavity Flow

4.2.1 Test Case Description and Turbulence Statistics

A lid driven cavity flow is considered as another example to test the abilityof the present approach. The motion of a Newtonian fluid within a lid-driven rectangular three-dimensional cavity is maintained by the continuousdiffusion of kinetic energy from the moving wall. This energy is initially con-fined to a very thin viscous layer of fluid next to the moving boundary. Theblocking action of the bounding walls and a great variety of hydrodynamicphenomena unevenly redistribute this energy over most of the cavity volume.

The geometry is shown in figure (4.10). The flow is driven by the topwall, z = h, in the x direction moving with a velocity U = 1. In the vicinityof the lid the maximum Mach number of the flow is equal to M = 0.003,but according to the velocity distributions, figure (4.11), the average Mach

4.2. CAVITY FLOW 67

10−1

100

101

102

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

kx

Str

eam

wis

e 1D

ene

rgy

spec

tra

at y

+ =

144

.54


100

101

102

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

kz

Spa

nwis

e 1D

ene

rgy

spec

tra

at y

+ =

144

.54



number of the entire flow field is much smaller and roughly equal to M =0.001. The Reynolds number is Re = Uh

ν= 10000 where h is the dimension

of the cube. A 333 mesh is used with a cosine distribution of the mesh points.Starting from a stagnant flow, the flow is fully developed after 32h/U , afterwhich statistics were collected during a period of 32h/U .

The mean velocity profiles along the vertical and horizontal symmetrylines are shown in figure (4.11). The statistics of the fluctuating field arecompared in figures (4.11) to (4.17). The agreement of the present simulationwith the experiment of Prasad et al. [89], and the DNS data of Leriche et al.[66] is satisfactory.

A constant coefficient Smagorinsky model with C = 0.0085, see equation(1.12), combined with near wall damping is used for subgrid scale mod-eling. The preconditioning parameters α = −1 and β = 0.1 L

π∆t, with

∆t = 0.01h/U . The β parameter in terms of the velocity is β = 3.16U .By using residual smoothing a better convergence was obtained for all equa-tions. The smoothing parameter is set according to Radespiel et al. [90], andthe CFL number is increased by a factor of two, CFLsmooth

CFL= 2.

To have an estimation for the value of CF Lc =∆tUc

∆x, which necessarily

should be smaller than one to have a time accurate simulation, we calculatethe CF Lc number of a computational cell in the vicinity of the moving lid.We suppose that Uc is roughly equal to the velocity of the lid Uc = U . Thelength of a computational cell away from the side walls and in the center ofthe top wall is around ∆x 0.05h. The CF Lc equals,

CF Lc =∆tUc

∆x=

(0.01h/U)(U)

0.05h= 0.25 < 1 (4.5)


h

x

y

z

a b

c

d

U = 1

Figure 4.10: Cavity flow test case.

The physical time step ∆t is around 1000 times bigger than the maxi-mum time step allowed by the stability condition of an explicit scheme in acompressible flow. To have a well resolved wall-layer near the moving lid, oneneeds a more clustered mesh near the wall compared to that of the channelflow. This explains why the explicit time step for the cavity flow is muchsmaller than that of the channel flow, and using an implicit approach is evenmore justified for the cavity flow calculation.

In almost 130 inner iterations the residual of the continuity equationdrops four orders of magnitude. As a rough estimation, and without takinginto account the extra work of the multigrid, the implicit approach is around1000/130 7.5 times faster than a purely explicit approach.

4.2.2 The Mean Velocity Field

The dominant feature of the mean flow is the large-scale recirculation whichappears to span the cavity. Figure (4.18) shows the typical time-averagedflow structure on a central vertical plane (the symmetry plane). The flowconsists of one primary eddy and three secondary eddies, see Kosef et al.([56] and [57]) for experimental visualizations.

The eddies on the lower right and left corners of the cavity are calledthe “downstream secondary eddy” and the “upstream secondary eddy”, re-spectively. The eddy on the upper left corner is called the “upper secondaryeddy”.

The contact of the rotating fluid mass with the side walls (xz planes aty = 0 and y = h) leads to an axial (along y) flow towards the mid-plane(xz plane at y = h/2). This can be explained in the following way; thefluid elements which rotate at a large distance from the side walls are inequilibrium under the influence of the centrifugal force which is balanced

4.2. CAVITY FLOW 69

Cavity flow, Re=10000

x/h or z/h

(U or W)/Uo

.00 .40 .80 1.20 -.40

.40

1.20

2.00 Simulation UExp USimulation WExp W

Figure 4.11: Mean velocity profile along two symmetry lines (ab and cd),lines are used for the simulation data and markers for experimental data.


x/h

10*u’/Uo

.00 .40 .80 1.20 .000

.100

.200

.300 LESExp PrasadDNS Leriche

Figure 4.12: Statistics of the fluctuating velocity 10√u′u′U

along the x axis,line ab, y = z = h/2, lines are used for the simulation data and markers forexperimental data.



x/h

10*w’/Uo

.00 .40 .80 1.20 .00

.40

.80

1.20 LESExp PrasadDNS Leriche

Figure 4.13: Statistics of the fluctuating velocity 10√w′w′U

along the x axis,line ab, y = z = h/2.


x/h

100*u’w’/Uo^2

.00 .40 .80 1.20 -.040

.040

.120


Figure 4.14: Statistics of the fluctuating velocity 100u′w′U2 along the x axis,

line ab, y = z = h/2

4.2. CAVITY FLOW 71


z/h

10*u’/Uo

.00 .40 .80 1.20 .00

.20

.40


Figure 4.15: Statistics of the fluctuating velocity 10√u′u′U

along the z axis,line cd, (x = y = h/2), lines are used for the simulation data and markersfor experimental data.


z/h

10*w’/Uo

.00 .40 .80 1.20 .00

.20

.40


Figure 4.16: Statistics of the fluctuating velocity 10√w′w′U

along the z axis,line cd, (x = y = h/2).



z/h

100*u’w’/Uo^2

.00 .40 .80 1.20 -.100

-.060

-.020


Figure 4.17: Statistics of the fluctuating velocity 100u′w′U2 along the z axis,

line cd, (x = y = h/2).

by a radial pressure gradient. The peripheral velocity of the elements nearthe side walls is reduced, thus decreasing the centrifugal force, whereas thepressure gradient in y direction remain the same. This set of circumstancescauses the fluid elements near the side walls to flow radially inwards (to thecenter of the side walls), and for reasons of continuity that motion must becompensated by an axial flow (along y) towards the mid-plane2. Accordingto Leriche et al. [66], the maximum axial velocity due to this mechanismis less than 1% of the lid velocity. For a complete discussion about axiallysymmetrical boundary layers, see Schlichting [97].

As a result of this axial (y) flow, there is not a single main eddie but twolarge eddies, one on each side of the symmetry plane at y = h/2, which havesimilar structure. The flow near the symmetry plane is a transition zonebetween the two systems.

4.2.3 The Three-Dimensional Picture

A fourth major secondary structure, which is not seen in xz planes, called“corner vortex” has a significant influence on the flow and is the most ev-ident manifestation of three-dimensionality in the flow. The corner vortexoriginates from the adjustment of the shear and pressure forces acting on therecirculating fluid. An example of a corner vortex is shown in figure (4.21,

2The same phenomenon is observed in a teacup! After stirring, the radial inward flownear the bottom brings the tea leaves to the center.

4.2. CAVITY FLOW 73

left). The sense of rotation of these vortices is consistent with the directionof the axial (y) flow towards the symmetry plane.

Koseff et al. [56] used a dye streak to visualize the vortical motion of theflow in the downstream secondary eddy (DSE) region. They showed that theDSE is divided into two parts and each part moves in two opposite directions(along the y axis) towards the side walls (xz planes at y = 0 and y = h).And then, due to the presence of the walls they change their directions andthey continue moving in the opposite x direction. This explanation is inaccordance with the present simulation. In figure (4.20) one can see the twocorner vortices, but in figure (4.20, left), which is in a plane further fromthe DSE, there are no vortices on the lower part of the cavity. However, infigure (4.20, left) one can see two other vortices on the upper part of thecavity. The presence of these two upper vortices was not reported on theexperimental work of Koseff et al. ([56], [57]).

Figures (4.20, right) and (4.21, right) show the instantaneous vectorlinesprojected on two different cross sections. In both pictures the Taylor-Gortler-like (TGL) vortices are clearly visible. The longitudinal Taylor-Gortler-like(TGL) vortices are formed because the surface of separation between theprimary eddy and the DSE is effectively a concave “wall”. The curved surfaceleads to the formation of TGL structure. In the experimental work of Koseffet al. [57] using the rheoscopic liquid technique, in which crystalline particlesare placed in the flowfield and the flow is illuminated with a sheet of light,the TGL vortices have been visualized. The corner vortex, the DSE and theTGL vortices are all integrally inter-related. The length of the TGL vorticesis smaller than the total length of the cavity. As can be seen in figures (4.19),no vortical motion can be observed in a plane far away from the DSE region.

Koseff [55] showed that the size and distribution of the TGL vortices isuneven and varies with time, and this behavior is a function of Reynoldsnumber. At lower Reynolds numbers the TGL vortices are more uniform insize and evenly distributed. In the present calculation Re=10000 and thevortices meander in the y direction. Thus, in figures (4.20, left) and (4.21,left) due to the time averaging, the time averaged vectorlines do not showthe TGL vortices.


Y

X

Z

Figure 4.18: Time averaged vectorlines projected on the symmetry plane.

Y

Z

X

Y

Z

X

Figure 4.19: Streamlines at plane x = 0.17/h, left: time averaged, right:instantaneous.

4.2. CAVITY FLOW 75

Y

Z

X

Y

Z

X

Figure 4.20: Streamlines at plane x = 0.38/h, left: time averaged, right:instantaneous.

Y

Z

X

Y

Z

X

Figure 4.21: Streamlines at plane x = 0.8/h, left: time averaged vectorlinesshowing the presence of corner vortices. Right: instantaneous vectorlinesshowing the presence of Taylor-Gortler like vortices.

Chapter 5

Particle Laden Flows

5.1 Introduction and Literature Survey

Particle-laden turbulent flows are important in numerous industrial pro-cesses, such as coal combustion, dust deposition and removal in clean rooms,droplets deposition in gas-liquid flows, etc... Accurate prediction of particle-laden turbulence is important in order to gain a better understanding ofparticle transport by turbulence as well as ultimately improve the design ofthe engineering devices in which two-phase flows occur.

Traditional methods are usually based on the Reynolds-Averaged Navier-Stokes (RANS) equations in which the entire spectrum of velocity fluctua-tions is represented indirectly using various parametrizations, see for exampleChang et al. [14] as well as Berlemont et al. [6]. A primary shortcoming ofRANS methods for the prediction of particle-laden turbulent flows is relatedto deficiencies associated with the model used to predict properties of theEulerian turbulence field. This problem can be solved by using a more so-phisticated approach like DNS but the drawback is that it is only applicableto low or moderate Reynolds numbers. In between there is the large-eddysimulation (LES) which is less sensitive to modeling errors than in RANScalculation and less restricted to low Reynolds number than DNS, see for areview Yeung [118], Crowe et al. [21], Michaelides [78] and Peng et al. [86].

Concerning the effect of the particles on the carrier flow, two differentapproaches called one-way and two-way coupling can be used. In a one-waycoupling model the presence of particles has negligible effect on the carrierflow but in a two-way coupling model the effects of particles is taken intoaccount in the carrier flow.

Elghobashi [27] has proposed the map shown in figure (5.1) for the effectof particles on carrier-phase turbulence. For volume fractions1 less than 10−6

1The volume fraction is defined as the ratio of the total volume of the particle to the

77

78 CHAPTER 5. PARTICLE LADEN FLOWS

10−6

One−way coupling

Two−waycoupling

10−30

turbulence

10−4

10−2

100

102

couplingFour−way

Tp

Te

fraction

Particles volume

Negligible

effect on turbulence

Particlesenhance

turbulence

Particlesdecay

Figure 5.1: Proposed map for particle-turbulence modulation

the presence of the particles would have no effects on turbulence. For volumefractions between 10−6 and 10−3, the particles can augment the turbulence,if the ratio of particle response time2 to the turnover time of a large eddy isgreater than unity, or attenuate the turbulence, if the ratio is less than unity.Chen et al. [15] came to the same conclusion by conducting an experiment ina vertical wind tunnel. They denoted turbulence modulation, which generallydecreases turbulence fluctuations, when the dispersed-phase motion accom-modates to the continuous-phase motion, and turbulence generation, whichgenerally increases turbulence fluctuations, when the continuous-phase veloc-ity field is disturbed by particles wakes. Their studies show that turbulencegeneration (modulation) dominates turbulence modification, tending to in-crease (decrease) turbulence levels, when dispersed-phase elements have large(small) relaxation times compared to characteristic turbulence timescales.

For volume fractions greater than 10−3, particle-particle collisions becomeimportant, and the turbulence of the carrier phase can be affected by theoscillatory motion due to particle collisions. Elghobashi [27] identifies thiseffect as four-way coupling.

If two-way coupling effects are significant, then the subgrid-scale turbu-lence model might need modification. This would be especially importantif the particulate phase couples strongly with the small-scale turbulence.Whether this is the situation or not may depend upon the matching of the

total volume of the carrier flow.2The particle response time (also called relaxation time) is defined as, τp = ρpd2

18νρf, where

ρp and ρf are the density of the particle and the flow, respectively. d is the diameter ofthe particle and ν is the kinematic viscosity of the flow.

5.1. INTRODUCTION AND LITERATURE SURVEY 79

time-scale between the particles and the subgrid-scale turbulence.

Armenio et al. [4] investigated the effect of the turbulent fluctuations onthe particle motions by computing particle statistics obtained by the filteredDNS of turbulent channel flow at Reτ = 175. They used filtering to eliminatethe effect of the subgrid scale fluctuations on the particles. Their minimumfilter ratio was equal to two and the maximum was equal to four. Theirresults show that the subgrid fluctuations have a limited effect on particlestatistics. The difference between DNS data and filtered results is moreevident for particles initially located in the buffer layer, and decreases forparticles released in the core of the channel.

They explain this behaviour as follows. In the core of the channel andaway from the near-wall region, the particle motion is, to a large extent,governed by the large-scale, energy-carrying structures, and the effect ofsmall-scale fluctuating velocity field is negligible. In the buffer layer and nearthe wall, on the other hand, the scale of the largest structures decreases, andthe particle motion becomes more sensitive to fluctuations. However, theeffect of filtering was only significant at the largest filter ratio.

Wang et al. [116] investigated the effect of subgrid scales on particlestatistics in a channel flow at Reτ = 640 by solving a transport equationfor the subgrid-scale (SGS) kinetic energy, which allowed them to model thefluctuating SGS velocities as a random process with Gaussian distributionand amplitude obtained from the SGS kinetic energy; these velocities werethen added to the filtered ones. According to their results, the addition ofSGS velocities has negligible effect on particle statistics.

Boivin et al. [8] conducted several a priori tests of subgrid-scale turbu-lence models utilizing results from direct numerical simulation (DNS) of aforced homogeneous isotropic turbulent flow using a two-way coupling and in-cluding the back effect of the particles. They stated that the dynamic subgridmodel yields more accurate predictions than the conventional Smagorinskymodel. This conclusion was further confirmed by a posteriori tests. Theyconsider heavy particles with diameters very small compared to the gridspacing, but with relaxation times larger than the time scales of the subgridmotions and comparable to the time scales of the large eddies. They alsopointed out that subgrid-scale modeling of the fluid turbulence in gas-solidflows may benefit from the fact that a large part of the total dissipationin the flow arises from fluid-particle interactions, especially at larger massloadings. Consequently, modeling errors could have a smaller impact on LESpredictions than in a single-phase flows, especially for large particle responsetimes and at high mass loading.

Miller et al. [79] calculated a transitional droplet laden mixing layerwith direct numerical simulation (DNS) and made a subgrid-scale analysis.


They defined a “subgrid Stokes number”: Stsg =τpτsg

, where τp is the particle

relaxation time and τsg is the subgrid time scale based on the subgrid velocityvariance and the filter width.3 They showed that subgrid effects on thedroplet trajectories can only be neglected safely for a subgrid Stokes numbersubstantially greater than unity.

The near wall behavior of particles has been studied by many researchers.To mention a few among others, Haarlem et al. [35] performed a DNS of achannel bounded by a no-slip and free-slip surface and they investigated theeffect of wall boundary condition. The DNS of particle flow in the wall regionof a horizontal channel was carried out by Pedinotti et al. [85].

In this chapter the numerical method to solve the incompressible Navier-Stokes is explained first. Then, the finite volume method is briefly explainedas well as the use of compact schemes. An overview of the particle motionsimulation including the calculation of particle trajectories, particle velocitystatistics, different types of interpolation, initialization, time advancementand walk search algorithm is given in section (5.4). Later, the results of theparticle-laden channel flow with different types of particles are represented.The effect of interpolation on the results is also investigated.

5.2 Incompressible Navier-Stokes Solver

The method used to solve the incompressible Navier-Stokes equations is asubset of the pressure correction method originally applied by Harlow andWelch [38] for the computation of free surface incompressible flows. It iscalled fractional step or projection method, developed independently byChorin [19] and Temam ([106], [107]).

The fractional step method is slightly different from the original pres-sure correction method in a way that the pressure term is removed in theintermediate step. In this section we explain the fractional step method.

The governing equations for an incompressible flow are,

∂ui∂xi

= 0 (5.1)

∂ui∂t

+∂uiuj∂xj

= −1

ρ

∂p

∂xi+ ν

∂2ui∂xj∂xj

(5.2)

where xi’s are the Cartesian coordinates, and ui’s are the correspondingvelocity components. The integration method is an explicit four stage Runge-

3τsg = ∆f/u′′u′′1/2, where ∆f is the filter width, and u′′ is the fluctuating velocity

with respect to the Favre density weighted filtered velocity u, i.e., u′′ = u− u.

5.2. INCOMPRESSIBLE NAVIER-STOKES SOLVER 81

Kutta scheme, equation (2.2), with s = 4 and,

aij =

0 0 0 0

1/4 0 0 00 1/3 0 00 0 1/2 0

bj =[0 0 0 1

]

The fractional step method, or time-split method, is used to solve the systemof equations (5.2). To explain the fractional step method we assume, forthe sake of clarity, that the time derivative is discretized with an explicitforward Euler method. The extension to a multi-stage Runge-Kutta methodis straightforward as the same procedure is followed at every stage.

The discretized form of equation (5.2) may be written as,

un+1i − uni∆t

+∂uni unj

∂xj= −1

ρ

∂p

∂xi+ ν

∂2uni∂xj∂xj

(5.3)

In an incompressible flow the pressure cannot be updated from either themomentum, nor the continuity equation. However, with the following methoda pressure field is calculated which satisfies the continuity equation.

First, by dropping the gradient of pressure from equation (5.3) an inter-mediate velocity u∗

i is obtained,

u∗i − uni∆t

+∂uni unj

∂xj= ν

∂2uni∂xj∂xj

(5.4)

The velocity at the next time level un+1i is equal to u∗

i plus a correctionvelocity ucori ,

un+1i = u∗

i + ucori (5.5)

The velocity field at time level n + 1 must be divergence free, i.e.,

∂un+1i

∂xi= 0 → ∂u∗

i

∂xi= −∂ucori

∂xi(5.6)

By inserting the value of un+1i from (5.5) into equation (5.3),

(u∗i + ucori )− uni

∆t+

∂uni unj∂xj

= −1

ρ

∂p

∂xi+ ν

∂2uni∂xj∂xj

(5.7)

after subtracting equation (5.4) from equation (5.7),

ucori∆t

= −1

ρ

∂p

∂xi(5.8)


By applying the divergence operator to both sides of equation (5.8),

1

∆t

∂ucori∂xi

= −1

ρ

∂2p

∂xi∂xi(5.9)

from (5.6) the value of∂ucor

i

∂xiis known and is inserted into (5.9),

1

∆t

∂u∗i

∂xi=

1

ρ

∂2p

∂xi∂xi(5.10)

Once the pressure field, which satisfies the continuity equation, is calculatedfrom equation (5.10), the velocity at time level n + 1 can be obtained fromequation (5.5) and (5.8),

un+1i = u∗

i −∆t

ρ

∂p

∂xi(5.11)

5.3 Finite Volume Compact Scheme

The space derivatives are discretized with a finite volume compact schemeusing a cell-averaged approach. First, in order to highlight the advantage ofa cell-averaged approach over a point-wise approach for high order unsteadycalculations let us consider the following equation,

∂U

∂t+∇ · F = ∇ · Q (5.12)

with U the vector of independent variables, F the vector of convective fluxes,and Q the vector of viscous fluxes. The finite volume method can be appliedto a volume of control V ,∫ ∂U

∂tdV +

∫∇ · F dV =

∫∇ · QdV (5.13)

Let U denote the cell-averaged value of U ,

U =1

V

∫UdV (5.14)

With a cell-averaged approach, the volume integral of the time derivative inequation (5.13) can be exactly calculated and this term therefore does notcontribute to spatial errors,

∂U

∂tV +

∫∇ · F dV =

∫∇ · QdV (5.15)

5.3. FINITE VOLUME COMPACT SCHEME 83

However, if a point-wise approach was used, the time derivative would adda spatial second-order error to the equations.4

After this short introduction about the necessity of a cell-averaged ap-proach for unsteady calculations, we proceed with the methods of discretizingthe space derivatives with compact schemes. Using compact schemes in a fi-nite volume context is part of the research conducted at the department.The interested reader can refer to Smirnov et al. [102], [103] and Lacor et al.[94] for further readings.

5.3.1 Convective Fluxes

Before going to three dimensions, a one dimensional case is considered todemonstrate the main aspects of the present approach. Suppose the cell-averaged values of the independent variable u are known, the first step is tocalculate the values of u at the interfaces. In 1D the interfaces are points, butin 2D and 3D these are lines and surfaces, respectively, which require specialattention. We will address this problem later in this section. A compactinterpolation formula may be written as,

βui−3/2 + αui−1/2 + ui+1/2 + αui+3/2 + βui+5/2 = (5.16)

aui+1 + ui

2+ b

ui+2 + ui−1

2+ c

ui+3 + ui−2

2

The coefficients a, b, c and α, β can be derived by developing all the variablesin a Taylor series about point i+1/2, and requiring all the coefficients of theresulting expansion to vanish up to some definite term. It is worthwhile tomention that the coefficients obtained for the cell-averaged approach wouldbe different from the coefficients of a point-wise approach for the same orderof accuracy. The reason is the way the Taylor expansions are calculated forthe cell-averaged values. For example the Taylor expansion of ui+1 aroundpoint i + 1

2would be,

ui+1 =1

xi+ 32− xi+ 1

2

∫ xi+3

2

xi+1

2

u(x)dx =1

xi+ 32− xi+ 1

2

×

∫ xi+3

2

xi+1

2

(ui+ 1

2+ (x − xi+ 1

2)∂u

∂x|i+ 1

2+

1

2(x − xi+ 1

2)2

∂2u

∂x2|i+ 1

2+ · · ·

)dx

4It is easy to check that,U = Ucenter + O(h2)

with U the cell averaged value, Ucenter the value at the center of the cell, and h the gridspacing.


The approximation (5.16) is fourth order accurate if the following relationsare satisfied,

a + b + c = 1 + 2α + 2β

a + 7b + 19c = 6(α + 4β)

In the present study the following fourth order interpolation is used,

1

4ui− 1

2+ ui+ 1

2+

1

4ui+ 3

2=

3

4(ui+1 + ui) (5.17)

The extension to two and three dimensions is straightforward. In 3D, for aCartesian mesh, the formulation may be written as,

1

4[u]i− 1

2,j,k + [u]i+ 1

2,j,k +

1

4[u]i+ 3

2,j,k =

3

4(ui+1,j,k + ui,j,k) (5.18)

[φ] denotes the interface averaged values of a primitive variables φi+ 12,j,k.

Once the values of the primitive variables at the interfaces have been calcu-lated, the next step is to calculate the fluxes at the interfaces.

∫∇ · F dV =

∫F · dS =

nf∑i=1

F i∆Si

with F i the interface-averaged values of fluxes, ∆Si the surface area of eachface and nf the number of faces of the control volume. In 3D and 2D theinterfaces are surfaces and lines, respectively.

To calculate the fluxes at the interface, the interface-averaged value of theproduct of primitive variables [uv] is needed. Special care should be takenin calculating the interface-averaged of the products to keep the high orderof accuracy of the method. The interface-averaged of the product of u andv is,

[uv] =1

yA − yB

∫ B

Auvdy (5.19)

to calculate [uv] the following approximation may be used,

[uv] = [u][v] + O(∆y)2 (5.20)

with ∆y = yA − yB. The truncation error of equation (5.20) can be easilycalculated by developing u and v in a Taylor series around point M , figure(5.2). Kobayashi [54] proposed to recover 4th order of accuracy by approxi-mating the leading truncation error of order 2 by finite-difference formula ofsecond order accuracy,

[uv] = [u][v] +∂u

∂y|M ∂v

∂y|M∆y2

12+ O(∆y)4 (5.21)


D

i,j i+1,j

A

B

M

C

Figure 5.2: Two dimensional Cartesian mesh

The derivatives ∂u∂y|M and ∂v

∂y|M are approximated by a classical second order

finite-difference method,

∂u

∂y|M =

uA − uB∆y

+ O(∆y)2 (5.22)

This method can also be generalized for the use on arbitrary meshes, Smirnovet al. [102].

5.3.2 Viscous Fluxes

In a finite volume context the viscous fluxes are evaluated as follows,

∫∇ · QdV =

∫Q · dS =

nf∑i=1

Qi∆Si (5.23)

with Qi the interface-averaged values of viscous fluxes, ∆Si the surface areaof each face and nf the number of faces of the control volume. The calculationof viscous fluxes can be done in two steps,

• consider the rectangle ABCD in figure (5.2). The interface-averagedvalues of primitive variables [u], [v] at AB, BC, CD and DA are known.By applying the Gauss theorem the cell-averaged values of the velocity

gradients ∂u∂xi

and ∂v∂xi

are calculated.

• in a similar way to equation (5.18), the interface-averaged values of

gradients, ([ ∂u∂xi

], [ ∂v∂xi

]), are evaluated from their cell-averaged ( ∂u∂xi

, ∂v∂xi

)counterparts.

Once the interface-averaged values of gradients are known, the calculation of∑nf

i=1 Qi∆Si in equation (5.23) is straightforward.


5.3.3 Application to Channel Flow

The compact scheme is applied to the channel flow test case at Reτ = 180.However, to save some computational time, the compact scheme is only usedfor the convective fluxes in the streamwise and spanwise directions. Theviscous and convective fluxes in the normal directions are calculated witha standard second order method. The viscous fluxes in the spanwise andstreamwise directions are obtained with a classical (not compact) fourth ordermethod. The overall accuracy of the method is then O(∆x)4 + O(∆z)4 +O(∆y)2.

Convective Fluxes

The convective flux in the normal direction (xz planes) is calculated witha standard second order difference. The values at the interface of two cells(i, j, k) and (i, j +1, k) are obtained with a linear interpolation φ = (u, v, w),

φ|i,j+ 12,k =

∆y1φi,j+1,k +∆y2φi,j,k∆y1 +∆y2

(5.24)

with ∆y1, ∆y2 the distances between the cell centers of (i, j, k), (i, j + 1, k)and the center of the interface (i, j + 1

2, k).

In the streamwise direction (yz planes) the interface-averaged values ofthe primitive variables are calculated according to equation (5.18). Then,the interface-averaged of the product of the primitive variables is, ψ, φ =(u, v, w),

[φψ] = [φ][ψ] +∂φ

∂z|M ∂ψ

∂z|M∆z2

12+ O(∆y)2

with M a point in the middle of the interface. The derivatives ∂φ∂z|M and

∂ψ∂z|M are approximated by a classical second order finite-difference method.

In the spanwise direction (xy planes) the same procedure is applied,

1

4[φ]i,j,k− 1

2+ [φ]i,j,k+ 1

2+

1

4[φ]i,j,k+ 3

2=

3

4(φi,j,k+1 + φi,j,k) (5.25)

the products may be written as,

[φψ] = [φ][ψ] +∂φ

∂x|M ∂ψ

∂x|M∆x2

12+ O(∆y)2

with a second order approximation for ∂φ∂x|M and ∂ψ

∂x|M .


Viscous Fluxes

In the normal direction (xz planes) the following difference formulation isused to calculate the gradients of φ = (u, v, w),

∂φ

∂y|i,j+ 1

2,k =

φi,j+1,k − φi,j,k∆y

(5.26)

with ∆y the distance between the cell centers.

In the streamwise and spanwise directions a fourth order difference methodis used. In the streamwise direction (x direction) it may be written,

[∂φ

∂x]i+ 1

2,j,k =

5

4

φi+1,j,k − φi,j,k∆x

− 1

4

φi+2,j,k − φi−1,j,k

3∆x

and similarly for the spanwise direction (z direction),

[∂φ

∂z]i,j,k+ 1

2=

5

4

φi,j,k+1 − φi,j,k∆z

− 1

4

φi,j,k+2 − φi,j,k−1

3∆z

Poisson Equation

As explained in section (5.2) a Poisson equation for the pressure (equation(5.9)) should be solved to satisfy the continuity equation. In the presentcalculation a collocated grid is used, which means that the same controlvolume is used for the momentum and continuity equations. By integratingequation (5.9) over a volume V and using the Gauss theorem,∫

ucori nidS =∫ ∂p

∂xinidS (5.27)

where ni is the normal unit vector of the faces. To evaluate the right handside, the interface-averaged values of the pressure gradient [ ∂p

∂xi] are needed.

The way to calculate these gradients, is as explained in section (5.3.2).Concerning the boundary conditions for the pressure, periodic boundary

conditions are imposed for the pressure in the homogeneous directions witha driving source term in the streamwise momentum equation. The gradientof pressure normal to the wall is set to zero.

Channel Flow Calculation

Before doing a simulation with particles, it was necessary to test the abilityof the code in solving the carrier flow.

Therefore, large-eddy simulations were performed at a Reynolds number,based on friction velocity and channel half-width, of Reτ = uτ δ

ν= 180. The


mean velocity profiles versus y

log(y+)

<u+mean>

-1.00 -.50 .00 .50 1.00 1.50 2.00 2.50 .0

5.0

10.0

15.0

20.0

25.0 meanu DNS-Kim

Figure 5.3: Mean streamwise velocity of channel flow compared with DNS ofKim [52].

uu,vv,ww turbulence intensities vs. y

y/delta

u,v,w_rms/u*

-1.00 -.80 -.60 -.40 -.20 .00 .00

1.00

2.00

3.00 uuvvww uu-DNS vv-DNS ww-DNS

Figure 5.4: Turbulence intensities for channel flow compared with DNS ofKim [52].


10−1

100

101

102

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

101

kx

Str

eam

wis

e 1D

ene

rgy

spec

tra

at y

+ =

6.0

9


100

101

102

10−6

10−5

10−4

10−3

10−2

10−1

100

kz

Spa

nwis

e 1D

ene

rgy

spec

tra

at y

+ =

6.0

9



10−1

100

101

102

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

kx

Str

eam

wis

e 1D

ene

rgy

spec

tra

at y

+ =

144

.54


100

101

102

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

kz

Spa

nwis

e 1D

ene

rgy

spec

tra

at y

+ =

144

.54




flow was resolved using 65×65×65 grid points in the x, y and z directions.The streamwise, normal and spanwise dimensions are 4πδ × 2δ × 4

3πδ, with

δ the channel half-width. The grid spacing in wall coordinates in the x and zdirections is ∆x+ = ∆xuτ

ν 35 and ∆z+ = ∆zuτ

ν 12. A non-uniform mesh

with hyperbolic tangent distribution is used in the wall-normal direction.The first mesh point away from the wall is at y+ = ∆yuτ

ν 0.49 and the

maximum spacing (at the centerline of the channel) is 13.8 wall units. ASmagorinsky model with damping near the wall is used for the subgrid scalemodeling. The Smagorinsky constant is equal to C = 0.007, (see equation(1.7)). Fluid statistics were averaged for 6δ/uτ . The mean velocity profileand turbulence intensities are shown in figures (5.3), and (5.4). The resultsare compared with the DNS data of Kim et al. [52]. The energy spectra attwo different locations of y+ = 6.09 and y+ = 144.54 compared with DNS ofKim et al. [52] are also shown in figures (5.5) and (5.6).

5.4 Calculation of the Particle Trajectories

The particle tracking routines can be added to any LES solver without majormodification of the original routines. The particle tracking routines shouldbe called at each time step immediately after the carrier flow velocities havebeen updated. The time step used for the particles must be equal to the timestep of the carrier flow. The three components of the particles velocities aswell as their x,y and z coordinates are updated at every iteration.

The particle equation of motion used in the simulations describes themotion of particles with densities substantially larger than that of the sur-rounding fluid and diameters small compared to the Kolmogorov scale. Thetotal (drag+body) force Fi of a single particle in a uniform flow field can begenerally expressed by,

Fi = CDApρf2|v − u|(ui − vi) + gδi1(ρpVp) i = 1, 2, 3 (5.28)

Where Ap is the exposed frontal area of the particle to the direction of theincoming flow, and Vp is the volume of the particle. For a spherical particle

Ap = π d2

4and Vp = 4

3π(d

2)3, with d the particle diameter. vi is the velocity

of the particle and ui is the velocity of the fluid at the particle position.The body force gδi1(ρpVp) acts along the positive streamwise direction corre-sponding to flow in a vertical channel flow. The fluid and particle densitiesare denoted ρf and ρp, respectively. The particle acceleration dvi

dt= Fi

ρpVpcan

finally be written as,

dvidt

=ρfρp

3

4

CD

d|v − u|(ui − vi) + gδi1 i = 1, 2, 3 (5.29)

5.4. CALCULATION OF THE PARTICLE TRAJECTORIES 91

Equation (5.29) may also be written in the following form,

dvidt

=1

τp(ui − vi) + gδi1 (5.30)

Where 1τp

=ρf

ρp

34CD

d|v − u| is the particle response or relaxation time. If the

drag coefficient is defined with the Stokes’s drag law CD = 24Rep

, which is valid

for small particle Reynolds number Rep = |v − u|d/ν, the Stokes relaxationtime τp may be written as,

τp =ρpd

2

18νρf(5.31)

Previous computations of particle-laden turbulent channel flow have shownthat the particle Reynolds number Rep does not necessarily remain small(e.g., see Rouson et al. [93]). Therefore, an empirical relation for CD fromClift et al. [20] valid for particle Reynolds number up to about 40 wasemployed,

CD =24

Rep(1 + 0.15Re0.687

p ) (5.32)

The particle Reynolds number of the present calculations never exceeds thismaximum value. It should also be noted that (5.29) is appropriate for de-scribing the motion of smooth rigid spheres and neglects the influence ofvirtual mass, buoyancy, and the Basset history force on particle motion. Forparticles with material densities large compared to the fluid these forces arenegligible compared to the drag. The effect of lift force, while relevant toproblems of particle deposition, is less significant to this work and thereforethe effect of shear-induced lift in the equation of motion has been neglected.Finally, the volume fraction of particles is assumed small enough such thatparticle-particle interactions are negligible.

Once the velocities of the particles at the new time level are calculatedwith equation (5.29), their positions will be obtained by solving the followingequation:

dxi,pdt

= vi i = 1, 2, 3 (5.33)

where xi,p is the particle position. Equations (5.29) and (5.33) are integratedin time using a second-order Adams-Bashforth method. In general, for theequation,

dΦ

dt= Res(Φ)

the Adams-Bashforth method is,

Φn+1 = Φn +∆t(3

2Res(Φn)− 1

2Res(Φn−1))


Sometimes, it is more convenient to work with the nondimensionalized equa-tions. Let the superscript ‘+’ denote variables nondimensionalized by innerscales, i.e., u+ = u/uτ , x+ = xuτ/ν, t+ = tu2

τ/ν and g+ = gν/u3τ , the

nondimensional forms of equations (5.29) and (5.33) may be written as,

dv+i

dt+=

ρfρp

3

4

CD

d+|v+ − u+|(u+

i − v+i ) + g+δi1 i = 1, 2, 3

dx+i,p

dt+= v+

i i = 1, 2, 3

with the Reynolds number defined as,

Rep = |v+ − u+|d+

If one wants to take the effect of particles on the carrier flow also intoaccount, a two-way coupling method may be used.

According to equation (5.29), the force exerted on the particle of massmp is equal to,

fi =dvidt

mp

the same force is exerted on the fluid from the particle and the accelerationof the fluid Si from this force is equal to,

Si = − fiρV

with V the volume of the computational cell. If the cell contains Nparc

particles the total force exerted on the fluid element is,

Fi = ΣNparc

s=1 (fi)s

For the two-way coupling calculation the source term, Fi

ρV, should be added

to the momentum equations to take into account the effects of the particleson the carrier flow. The momentum equations are then written as,

∂ui∂t

+∂uiuj∂xj

= −1

ρ

∂p

∂xi+ ν

∂2ui∂xj∂xj

− FiρV

(5.34)

5.4.1 Interpolation

To calculate the drag force exerted on the particles, the fluid velocity at theparticle locations is needed.

Three types of interpolations are explained, the first one is the simplestone and is the first order interpolation, then the second and third orderinterpolations are explained.


First Order Interpolation

For the first order interpolation if the particle is located in cell i, and thecell center velocity of the carrier flow, or the cell-averaged velocity of thecarrier flow in cell i is equal to U, then the velocity at particle location isalso supposed to be U.

Second Order Interpolation

In a second order approximation, first, the values of the carrier flow velocityare supposed to be stored at the center of the cells, then a quadratic Lagrangeinterpolation5 is used to calculate the velocity of the fluid at the particlelocations.

For a quadratic interpolation, in one dimension three cells are involved,in two dimensions nine cells and in three dimensions 27 cells. The Lagrangeinterpolation for a 1D problem is written as,

f(x) = F1(x − x2)(x − x3)

(x1 − x2)(x1 − x3)+F2

(x − x1)(x − x3)

(x2 − x1)(x2 − x3)+F3

(x − x1)(x − x2)

(x3 − x1)(x3 − x2)(5.35)

Where F1, F2, and F3 are the values of the function at the points x1, x2 and x3,respectively. In this case x1, x2, x3 are the coordinates of the cell centers, andF1, F2, F3 are the cell center values. For three dimensional problems theabove mentioned formula is used consecutively in three directions.

Third Order Interpolation

In the present code, a cell averaged approach is used for the higher (morethan two) order spatial approximation. In a cell averaged approach, unlikethe pointwise approach, the values in some specific points (the center of thecell for example) are not specified and the Lagrange formulation cannot bedirectly used.

To deal with this problem, suppose a 1D mesh with x1, x2, x3, x4 the meshcoordinates and U1 the cell-averaged value of the function in the interval[x1, x2] and U2 the cell-averaged value of the function in the interval [x2, x3]and U3 the cell-averaged value of the function in the interval [x3, x4].

Suppose the variation of the function U(x) is approximated with a poly-nomial L(x) which is not known yet, i.e., U(x) L(x). All we know is thatthree cells are involved and L(x) should be a second order polynomial.

5We note that, if a quadratic interpolation is used, then the accuracy of the interpo-lation is third order. But for the current calculations where a cell-averaged approach isused for the carrier flow, the assumption that the flow velocities are stored at the centerof the cells brings a second order error into simulations.


Eventhough the coefficients of the second order polynomial L(x) are un-known, we know its integral in the interval [x1, x2] is equal to the cell-averagedvalue of U multiplied by the length of the cell. By defining a functionφ(x) =

∫ xx1

L(x)dx we have,

∫ x2

x1

L(x)dx = (x2 − x1)U1 = φ(x2)

similarly integrating from x1 to x3 leads,∫ x3

x1

L(x)dx =∫ x2

x1

L(x)dx+∫ x3

x2

L(x)dx = (x2−x1)U1+(x3−x2)U2 = φ(x3)

and in the same way after integrating from x1 to x4,∫ x4

x1

L(x)dx = (x2 − x1)U1 + (x3 − x2)U2 + (x4 − x3)U3 = φ(x4)

Or in general one can write,

∫ x

x1

L(x)dx = φ(x) → d

dx

∫ x

x1

L(x)dx = φ′(x) → L(x) = φ′(x)

Now to calculate L(x) one first has to calculate φ(x) and then the derivative ofφ(x) will be equal to L(x). With four points at (x1, φ(x1) = 0), (x2, φ(x2)), (x1, φ(x3))and (x1, φ(x4)), the function φ(x) can be estimated with a third order La-grange interpolation,

φ(x) = 0× (x − x2)(x − x3)(x − x4)

(x1 − x2)(x1 − x3)(x1 − x4)+ (5.36)

φ(x2)(x − x1)(x − x3)(x − x4)

(x2 − x1)(x2 − x3)(x2 − x4)+

φ(x3)(x − x1)(x − x2)(x − x4)

(x3 − x1)(x3 − x2)(x3 − x4)+

φ(x4)(x − x1)(x − x2)(x − x3)

(x4 − x1)(x4 − x2)(x4 − x3)

Finally, L(x) is calculated by taking the derivative of φ(x), and it is a secondorder polynomial.

L(x) = φ′(x) (5.37)


5.4.2 Numerical Details

Particle Velocity Statistics

To be able to draw the profiles of the particle velocities, the cell-averagedvalues of the particles are needed. The instantaneous cell-averaged velocityvector of the particles is,

Vp =

∑Nparc

s=1 vsNparc

(5.38)

whereVp is the cell averaged velocity vector of the particles, vs is the velocityvector of a single particle inside the cell, and Nparc is the total number ofparticles in the cell.

Once Vp is obtained the time or/and space averaging can be performedand, in the same way that the turbulence statistics of the carrier flow are cal-culated, the particle fluctuations can be calculated. The particle fluctuatingvelocity vector, v′

p, can be defined as,

v′p =< Vp > −Vp (5.39)

Where < · · · > denotes the time or/and space averaging. For example,if we denote the three components of the cell averaged velocity vector ofthe particles as, Vp = (V 1p, V 2p, V 3p) and the fluctuating part as v′

p =(v′

1p, v′2p, v′

3p) then < v′1pv

′2p > is calculated as,

< v′1pv

′2p >=< V 1pV 2p > − < V 1p >< V 2p > (5.40)

Memory Consideration

If the total number of particles is Npar and the number of cells is Ncell, theextra memory needed for the particle tracking routines will be 15Npar +13Ncell.

The extra 15 arrays with the length of Npar are as follows; three arraysfor the three components of velocity, three arrays for the coordinates of theparticles, three arrays for the indices of the cells where the particles arelocated, and six arrays to store the particles acceleration and velocity attime level n-1 used in the Adams-Bashforth method.

The extra 13 arrays with the length Ncell are three arrays for the particlescell averaged velocities, one array for the number of particles in cells and ninearrays for the time averaged values of the three components of velocity andtheir products.


p2

Yymin

ymin + rp

yp1

y

Figure 5.7: Schematic representation of an elastic collision.

Boundary Conditions

If a particle crosses the boundary, its velocity and position should be modifiedaccording to the type of the boundary condition. There are two types ofboundary conditions in the channel flow, the wall boundary condition andthe periodic boundary condition.

• Wall boundary condition: if (xp1, yp1, zp1) is the coordinates of the par-ticle, (up1, vp1, wp1) the particle velocity, (ymin,ymax) the y coordinatesof the wall and rp the radius of the particle. The particle is assumedto hit the wall as soon as its center is one radius from the wall, i.e.,(yp1 < ymin + rp or yp1 > ymax − rp).

The wall boundary condition for the velocity, assuming that the wallis a xz plane is,

up2 = up1vp2 = −vp1wp2 = wp1

(5.41)

Where state 2 is when the boundary condition has been applied. Fromthe above condition it is obvious that a fully elastic collision is assumed.

For the coordinates of the particle,

xp2 = xp1yp2 = 2(ymax − rp)− yp1yp2 = 2(ymin + rp)− yp1zp2 = zp1

(5.42)

Figure (5.7) is a schematic representation of an elastic collision fromthe lower wall.

• Periodic boundary condition: in the channel flow the periodicity isapplied in the streamwise and spanwise directions. In the periodic


boundary condition the velocities of the particles are not changed andonly the position is modified. For example suppose a particle has leftthe domain in the positive streamwise direction, i.e. xp1 > xmax, thenthe periodicity is applied as follows,

xp2 = xp1 − (xmax − xmin)yp2 = yp1zp2 = zp1

(5.43)

If the particle leaves the domain in the negative streamwise direction,i.e. xp1 < xmin then the boundary condition will be,

xp2 = xp1 + (xmax − xmin)yp2 = yp1zp2 = zp1

(5.44)

A similar rule is valid in the z direction.

Walk Search Algorithm

To calculate the drag force, one needs to find the fluid velocity at the particlelocation, and it is necessary to know the indices of the cell in which theparticle is located. A very simple algorithm called walk search 6 can beused. In 1D suppose xp is the coordinate of the particle and xi is the meshcoordinates and xp > xi, the walk search algorithm is,

• Step 1: if(xp > xi.And.xp < xi+1 the particle is in the cell, otherwisego to the Step 2.

• Step 2: i = i + 1 Go to Step 1.

If xp < xi the algorithm is similar but now one sweeps through the meshin the opposite direction, i.e., the i index is decreased.

In 3D one has to apply sequentially the same algorithm in three directions.In the worst case the number of operations is (Ni+Nj+Nk), with Ni, Nj andNk the number of cells in i, j and k directions. Once the indices of a particleare obtained, they are saved and at the next time step, searching starts fromthe previous location of the particle to find its new location, which requiresonly a few operations.

6It is interesting to mention that walk search algorithm is also used in the contextof unstructured grid generation, Mavriplis [76], as a way to find the cell where a newlygenerated point is located.


For a curvilinear mesh, first an arbitrary cell is chosen, then the dot prod-ucts of the outward normal unit vectors of the cell faces with the vectorsconnecting the particle position to the cell face centers are calculated. De-pending on the values of these dot products, the next appropriate neighboringcell is chosen. The process continues until all dot products are negative, orin other words when the point is located inside the cell.

For the time being, we are dealing with a Cartesian mesh, and the walksearch algorithm can be used in its simplified form.

5.5 Particle-Laden Channel Flow

Once a time-averaged steady state solution has been obtained for the Eu-lerian velocity field, i.e. the carrier flow, the particles are assigned randomlocations throughout the channel. This can be done in two ways. First,one may consider the whole volume of the channel and distribute the parti-cles randomly. With this approach some near wall cells, which have a verysmall thickness, may not receive any particle. A second way, to have a moreuniform distribution and make sure that the cells receive about the samenumber of particles, is to go through the cells one by one and randomly puta particle inside each cell. With this approach, the initial volume fractionof the particles will be higher close to the walls where the mesh cells areclustered. The initial particle velocity was assumed to be the same as thefluid velocity at the particle locations.

Generally, at one time step, the carrier flow calculation is more timeconsuming7 than the calculation of particle trajectories. Therefore, anotherway to obtain a closer initial solution to the fully developed results is tocalculate the particle trajectories in a “frozen” turbulent field, and use theoutput of this calculation as an initial solution for a realistic simulation.

Starting with this initial solution, the flow and particles are advanced intime simultaneously until a time-averaged steady state is reached for the par-ticles. The development time, i.e., the time required for particles to becomeindependent of their initial conditions, depends on the particle response time.In the calculation of Wang et al. [116] the development times of the parti-cles range from 0.5δ/uτ for the Lycopodium particles which are the lightestparticles, to 6δ/uτ for the copper which are the heaviest.

In the present calculations, the development time is set to 6δ/uτ for allparticles. After an equilibrium condition had been reached, particle statisticswere accumulated for another 6δ/uτ .

7At least for an incompressible calculation where at each stage the Poisson equationshould be calculated.

5.5. PARTICLE-LADEN CHANNEL FLOW 99

Table 5.1: Parameters of Lycopodium particles

τpδ/uτ

τ+p

rpδ

r+p

ρp

ρf

4.73× 10−2 8.53 1.4× 10−3 0.2 603.4

5.5.1 Lycopodium Particles

The particle response time τp =ρpd2

18νρf, the radius of the particle rp, and the

particle to fluid density ratio ρp

ρfare shown in table (5.1). The values are

expressed in terms of both channel variables (δ and uτ ) and in wall units.These data are exactly the same as the data of Wang et al. [116] with whichthe present results are compared.

In the present calculation the channel half width δ = 0.01m, and thecarrier fluid is air with a density of ρf = 1.16Kg/m3

The Effect of the Order of Interpolation

Fukagata et al. [31] studied the effect of the order of interpolation on therms particle velocity fluctuations of glass and copper particles. They exam-ined three different types of interpolation such as the Nearest Grid Point 8

(first order), linear interpolation (second order) and a sixth order Lagrangeinterpolation.

Wang et al. [115] found, that linear interpolation causes large errorsin rms fluid velocity fluctuations at the particle locations. Fukagata et al.[31] showed, however, that these errors do not have a harmful effect on thecalculation of particle velocity fluctuations.

The results presented by Fukagata et al. [31] for the mean velocity andturbulence intensities of glass and copper particles show that for heavy par-ticles, the interpolation errors have negligible effect on the time averagedstatistics. They argue that this might be due to the large Stokes relaxationtime of glass and copper particles, which implies that the particles are lesssensitive to the small fluctuations in fluid velocity which may be smoothedout by a low order interpolation scheme.

In order to check the argument of Fukagata et al. [31], the effect of inter-polation on Lycopodium particles, which are lighter than glass and copperparticles, is examined in the present study. The density of Lycopodium is700kg/m3 compared to 2500 and 8500 for glass and copper, respectively.

8The Nearest Grid Point method is the one which we explained previously, see section(5.4.1), and called first order interpolation.


A channel flow at Reτ = 180 with a rather coarse mesh of 33 × 65 × 33points in the x, y and z directions is considered. Uniform meshes with spac-ing ∆x+ = ∆xuτ

ν 70.6 and ∆z+ = ∆zuτ

ν 23.6 are used in the streamwise

and spanwise directions. A non uniform mesh with hyperbolic tangent dis-tribution (see equation (4.1)) is used in the wall-normal direction. The firstmesh point away from the wall is at y+ = ∆yuτ

ν 0.49 and the maximum

spacing (at the centerline of the channel) is 13.7 wall units. The develop-ment time and averaging time for particle statistics are both equal to 6δ/uτseconds.

Figures (5.8) and (5.9) show the mean velocity profiles and the turbu-lence intensities of particles for different types of interpolation. From figure(5.9) one can easily see that the particle turbulence intensities have a highermagnitude when a higher order of interpolation is used. This may be ex-plained in the way, that by increasing the degree of interpolation, the smallscale fluctuations are captured more precisely, which results in a better rep-resentation of the particle fluctuations. Also shown in the figures, are thefluid turbulence intensities. One can see from the figures that the streamwisevalues of the particle turbulence intensities are higher than that of the fluid,but the spanwise and normal values are smaller. The same behavior wasobserved in the simulation of Wang et al. [116]. We will discuss more aboutthe particle velocity fluctuations in the coming sections.

The shape of the mean velocity profile of the particles, figure (5.8), alsochanges slightly. From the same figure one can also notice that by increas-ing the degree of interpolation the magnitude of particle velocities slightlydecreases.

Even though the effect of interpolation is not very significant, and forheavier particles like glass particles it may even be negligible, to be on the safeside, a third order interpolation is always used for the rest of the calculations.

Fine Mesh Calculation

For the final simulation of Lycopodium a finer mesh with more points inthe spanwise direction is used. The new grid parameters in wall units are,∆x+ = 70.6, ∆z+ = 11.8, ∆y+

min = 0.49 and ∆y+max = 13.7, and the number

of points in the x, y, and z directions is 33× 65× 65.The results are shown in figures (5.10) to (5.13). The mean velocity

profile of particles (figure (5.10)) shows that the particles, apart from a smallregion around the buffer layer y+ 30, closely track the fluid flow. Thesame behavior was observed in the LES of Wang et al. [116]. In the directnumerical simulations of Rouson et al. [93] (results not shown), the meanvelocity profile of particles, slightly lags that of the fluid in the near wallregion (y+ < 10). Rouson et al. [93] attribute the lag in the Lycopodium


Lycopodium, one way couplingmean velocity profiles versus y

log(y+)

<u+mean>

1.00 1.50 2.00 2.50 12.0

14.7

17.3

20.0 Part Vel 1st orderPart Vel 2nd orderPart Vel 3rd orderDNS-Kim carrier flow

Figure 5.8: Effect of the order of interpolation on the mean streamwise ve-locity of the particles.

uu,vv,ww turbulence intensities of particles vs. y

y/delta

u,v,w_rms/u*

-1.00 -.80 -.60 -.40 .00

1.17

2.33

3.50 <u’u’> first order<u’u’> second order<u’u’> third order<u’u’> carrier flow<v’v’> first order<v’v’> second order<v’v’> third order<v’v’> carrier flow<w’w’> first order<w’w’> second order<w’w’> third order<w’w’> carrier flow

Figure 5.9: Effect of the order of interpolation on the rms velocity fluctua-tions of the particles.


profile to preferential concentration of Lycopodium particles in the low-speedstreaks. The discrepancy between LES and DNS results may be related tothe fact that the near-wall structures are less well resolved in the LES and,consequently, preferential concentration of particles in low-speed regions isless significant than in the DNS, resulting in an over-prediction of the meanvelocity of the Lycopodium particles.

Comparisons of rms particle velocity fluctuations from the present calcu-lations with both the LES results of Wang et al. [116] and the DNS data ofRouson et al. [93] are shown in figures (5.11), (5.12) and (5.13).

Apart from some small differences in the streamwise particle velocityfluctuations, the general agreement between the present LES and the LES ofWang et al. [116] is quite satisfactory. In figure (5.11) the predicted particlestreamwise turbulence intensities (bold line) are somewhat higher than theone of Wang et al. (crosses). The reason can be the different type of thespatial discretization and/or the subgrid-scale model that they use in theircarrier flow calculation. Wang et al. used a finer mesh with 65 × 65 × 65grid points, and for the subgrid scale modeling, they used the Lagrangiandynamic approach of Meneveau et al. [77].

As mentioned before, one can see from the figures that the streamwisevalues of the particle turbulence intensities are higher than that of the fluid,but the spanwise and normal values are smaller.

It is difficult to speculate on the causes of the difference between LES pre-dictions and DNS results. Wang et al. [116] investigated the errors causedby neglecting the particle transport by subgrid-scale (SGS) velocities. It iswell known that in LES the smallest scales of motion are not resolved. Thus,only the large-scale velocity field is directly available in an LES computationfor determining particle motion. Wang et al. [116] investigated the effect ofSGS velocity field on particle transport by adding SGS fluctuations to thefluid velocity used in the particle equation of motion (5.29). The magnitudeof the SGS fluctuations u′′

i was determined by solving a transport equation9

for SGS kinetic energy, q2 = u′′2i /2 as proposed by Schuman [99]. The com-

ponent fluctuations u′′i were then scaled by a random number sampled from

a Gaussian distribution and added to ui at the particle locations. The com-

9The transport equation can be written as,

∂q2

∂t+ uj

∂q2

∂xj= 2νt|S|2 +

∂

∂xj(13l∆√q2/2

∂q2

∂xj) + ν

∂2

∂xj∂xj−√

12cεq3

l∆

where

cε = π(2

3k0)3/2, l∆ = min(∆, cεy)

with the Kolmogorov constant k0 = 1.6.


Table 5.2: Parameters of glass particles

τpδ/uτ

τ+p

rpδ

r+p

ρp

ρf

54× 10−2 97.2 2.5× 10−3 0.45 2155

plete velocity, i.e., ui + u′′i , was subsequently used in (5.29) to determine the

particle velocity.

Their findings showed that for the 28 µm Lycopodium, there is a negli-gible effect of the SGS fluid velocity on particle fluctuations and the changein the particle velocities due to the addition of the SGS fluid velocity is lessthan 1%. This conclusion is also valid for larger or heavier particles, as theinertia makes the particle response less sensitive to the small-scale, fluctuat-ing, velocity field. For particles with material densities large compared to thecarrier flow the response of particles to the frequency spectrum of turbulencecan be shown to be proportional to 1/(τpw) (w is the frequency). Thus, forincreasing values of the relaxation time τp and/or frequency the filtering ofhigh frequency motions by particle inertia is significant.

A possible explanation for the discrepancies between the LES and DNSresults may be suggested by considering the coherent structures near thewall. From the experiments of Kline et al. [53], it has been well knownthat streaks of respectively low and high longitudinal velocity exist closeto the wall, between y+ = 5 and y+ = 40 ∼ 50. Several numerical andexperimental studies have shown that the fluid elements located above thelow-speed streaks are ejected away from the wall. Conversely, the high speedfluid elements correspond to the sweep events, i.e., pumping of the high speedflow of the outer region to the near wall region.

There are still a lot of controversies about the interpretation of the low-and high-speed streaks, and how they are related to the sweep and ejectionevents. But what is obvious, is the important role of these structures on thetransport of parcels of fluid as well as the particles. In LES the inadequacyof the computational grid resolution to resolve the streaks at their properscale, may be a reason for the difference between the LES and DNS results.Furthermore, this problem could be more serious for the particles than forthe fluid elements, due to their higher inertia and density compared to thefluid which helps them to keep better their inertia when they are transported.


Lycopodium,33*65*65 meshmean velocity profiles versus y

log(y+)

<u+mean>

-1.00 -.50 .00 .50 1.00 1.50 2.00 2.50 .0

5.0

10.0

15.0

20.0

25.0 <U> particlesLES CarrierDNS-Kim Carrier

Figure 5.10: Mean streamwise velocity of Lycopodium particles in turbulentchannel flow.

Lycopodiumuuparticle turbulence intensities vs. y

log(y+)

u_rms/u*

-.20 .20 .60 1.00 1.40 1.80 2.20 2.60 .00

.80

1.60

2.40

3.20

4.00 <up’up’> part<up’up’> DNS part<up’up’> LES Wang part<u’u’> LES carrier

Figure 5.11: Root-mean-square velocity fluctuations (u′u′)1/2 of Lycopodiumparticles in turbulent channel flow compared with DNS of Rouson et al. [93]and LES of Wang et al. [116].


Lycopodiumvv particle turbulence intensities vs. y

log(y+)

v_rms/u*

-.20 .20 .60 1.00 1.40 1.80 2.20 2.60 .00

.25

.50

.75

1.00 <vp’vp’> part<vp’vp’> DNS part<vp’vp’> LES Wang part<v’v’> LES carrier

Figure 5.12: Root-mean-square velocity fluctuations (v′v′)1/2 of Lycopodiumparticles in turbulent channel flow compared with DNS of Rouson et al. [93]and LES of Wang et al. [116].

Lycopodiumww particle turbulence intensities vs. y

log(y+)

w_rms/u*

-.20 .60 1.40 2.20 .00

.40

.80

1.20 <wp’wp’> part<wp’wp’> DNS part<wp’wp’> LES Wang part<w’w’> LES carrier

Figure 5.13: Root-mean-square velocity fluctuations (w′w′)1/2 of Lycopodiumparticles in turbulent channel flow compared with DNS of Rouson et al. [93]and LES of Wang et al. [116].


5.5.2 Glass Particles

The results for glass particles, which have a bigger diameter compared tothe Lycopodium, are presented in figures (5.14) to (5.17). The parameters ofglass particles are shown in table (5.2). They also have a higher relaxationtime, Stokes number, and density than the Lycopodium particles.

In figure (5.14) the particle mean velocity profile is compared with theDNS of Rouson et al. [93], and LES of Wang et al. [116]. LES predictions ofthe mean velocity are in good agreement with the DNS and the other LEScalculations. Unlike the Lycopodium particles, that closely track the fluid,the velocity of glass particles near the wall is higher than that of the fluid.This is a result of the transport of high velocity particles from the outerregion to the near-wall region. The glass particles have a higher inertia thanLycopodium particles and they can memorize better their velocities.

Particle turbulence intensities are shown in figures (5.15), (5.16), and(5.17). We note that the streamwise rms velocities are again smaller thanthe DNS values while the wall normal and spanwise rms velocities in the LESare slightly greater than the corresponding values in the DNS.

When compared with the Lycopodium, it is clear from the figures thatnear the wall, the streamwise fluctuation levels increase for a higher value ofStokes number while the wall normal and spanwise fluctuations are reduced.

This anisotropy can be measured by defining the following parameter,

βi =< f2

i >

< f1f1 > + < f2f2 > + < f3f3 >(no sum)

where fi is either the fluid or particle velocity fluctuations. If the fluctuationsare isotropic, βi’s are equal to 1/3 and the deviation from this value is ameasure of the anisotropy in fluctuations.

Shown in figures (5.21), (5.22), and (5.23) is a comparison of the anisotropyfor Lycopodium, glass, and copper particles with the carrier flow. It is evidentthat the anisotropy of the particle fluctuations is larger than the fluid, andthis anisotropy increases with the particle Stokes number. Copper particlestransport is discussed in the next section.

5.5.3 Copper Particles

The parameters of copper particles, which have a bigger diameter and densitythan the glass particles, are given in table (5.3). The results are shown infigures (5.18) to (5.20). For the particle mean velocity, there is a discrepancybetween the present calculation and the LES of Wang et al. [116]. Thenumber of grid points and the grid spacing in wall coordinates are the sameas the one of Wang et al. [116]. However, Wang et al. [116] used a different


Glass particle, 33*65*33mean velocity profiles versus y

log(y+)

<u+mean_particle>

.00 .50 1.00 1.50 2.00 2.50 .0

5.0

10.0

15.0

20.0

25.0 <U> particlesDNS particlesLES particles (Wang)LES Carrier Flow

Figure 5.14: Mean streamwise velocity of glass particles in turbulent channelflow compared with DNS of Rouson et al. [93] and LES of Wang et al. [116].

Glass, 33*65*33uu particle turbulence intensities vs. y

log(y+)

u_rms/u*

.00 .50 1.00 1.50 2.00 2.50 .00

.80

1.60

2.40

3.20

4.00 <up’up’> particles<up’up’> DNS<up’up’> LES Wang particles<u’u’> LES carrier

Figure 5.15: Root-mean-square velocity fluctuations (u′u′)1/2 of glass parti-cles in turbulent channel flow compared with DNS of Rouson et al. [93] andLES of Wang et al. [116].

Table 5.3: Parameters of copper particles

τpδ/uτ

τ+p

rpδ

r+p

ρp

ρf

4.50 811.40 3.94× 10−3 0.71 7241.4


Glass, 33*65*33vv particle turbulence intensities vs. y

log(y+)

v_rms/u*

.00 .50 1.00 1.50 2.00 2.50 .00

.25

.50

.75

1.00 <vp’vp’> part.<vp’vp’> DNS<vp’vp’>LES Wang part.<v’v’> LES carrier

Figure 5.16: Root-mean-square velocity fluctuations (v′v′)1/2 of glass parti-cles in turbulent channel flow compared with DNS of Rouson et al. [93] andLES of Wang et al. [116].

Glass, 33*65*33ww particle turbulence intensities vs. y

log(y+)

w_rms/u*

.00 .50 1.00 1.50 2.00 2.50 .00

.25

.50

.75

1.00 <wp’wp’> particles<wp’wp’> DNS<wp’wp’> LES Wang particles<w’w’> LES carrier

Figure 5.17: Root-mean-square velocity fluctuations (w′w′)1/2 of glass parti-cles in turbulent channel flow compared with DNS of Rouson et al. [93] andLES of Wang et al. [116].


Copper Particlemean velocity profiles versus y

log(y+)

<u+mean_particle>

.00 .50 1.00 1.50 2.00 2.50 .0

5.0

10.0

15.0

20.0

25.0 <U> particlesDNS particlesLES particles (Wang)LES Carrier Flow

Figure 5.18: Mean streamwise velocity of copper particles in turbulent chan-nel flow compared with DNS of Rouson et al. [93] and LES of Wang et al.[116].

type of subgrid-scale modeling and spatial discretization. Note also that allcalculations exhibit a slight plateau near the wall. The plateau may resultfrom the transport of high velocity particles from the outer region of thechannel to the near-wall region and this feature appears to be somewhatmore pronounced in the DNS.

Comparisons of rms particle velocity fluctuations from the present calcu-lation to both DNS and LES results are shown in figures (5.19) and (5.20).As may be observed there is a general good agreement between the presentpredictions and the DNS and LES.


Copper Particleuu particle turbulence intensities vs. y

log(y+)

u_rms/u*

.00 .50 1.00 1.50 2.00 2.50 .00

1.00

2.00

3.00

4.00

5.00 <up’up’> particles<up’up’> DNS<up’up’> LES Wang particles<u’u’> LES carrier

Figure 5.19: Root-mean-square velocity fluctuations (u′u′)1/2 of copper par-ticles in turbulent channel flow compared with DNS of Rouson et al. [93]and LES of Wang et al. [116].

Copper Particlevv particle turbulence intensities vs. y

log(y+)

v_rms/u*

.00 .50 1.00 1.50 2.00 2.50 .00

.20

.40

.60

.80

1.00 <vp’vp’> particles<vp’vp’>LES Wang part.<v’v’>LES carrier

Figure 5.20: Root-mean-square velocity fluctuations (v′v′)1/2 of copper par-ticles in turbulent channel flow compared with LES of Wang et al. [116].


Components of anisotropy tensor for fluid and Lycopodium particles

y/Delta

Beta

-1.00 -.80 -.60 -.40 -.20 .00 .00

.20

.40

.60

.80

1.00 beta1 Fluidbeta2 Fluidbeta3 Fluidbeta1 Lycobeta2 Lycobeta3 Lyco

Figure 5.21: Anisotropy comparison of the fluid and the Lycopodium parti-cles.

Components of anisotropy tensor for fluid and Glass particles

y/Delta

Beta

-1.00 -.80 -.60 -.40 -.20 .00 .00

.20

.40

.60

.80

1.00 beta1 Fluidbeta2 Fluidbeta3 Fluidbeta1 Glassbeta2 Glassbeta3 Glass

Figure 5.22: Anisotropy comparison of the fluid and the glass particles.


Components of anisotropy tensor for fluid and copper particles

y/Delta

Beta

-1.00 -.80 -.60 -.40 -.20 .00 .00

.20

.40

.60

.80

1.00 beta1 Fluidbeta2 Fluidbeta3 Fluidbeta1 Copperbeta2 Copperbeta3 Copper

Figure 5.23: Anisotropy comparison of the fluid and the copper particles.

Chapter 6

Conclusion and Future Work

6.1 Compressible Low Mach Number Flows

The numerical problems of large-eddy simulation of compressible low Machnumber flows have been studied, Lessani et al.[68], [69]. The main numericalaspects may be itemized as follows:

• For stability reasons, as explained in section (2.1.2), a semi-implicitRunge-Kutta method should be used during the inner iterations inpseudo-time. A way to simplify the matrix inversion, in the case of theNavier-Stokes equations, has been explained in section (2.5.3).

• The advantage of the trapezoidal method over the commonly usedsecond-order backward differencing for the time derivative discretiza-tion has been shown. The trapezoidal method allows the use of a biggertime-step compared to the second-order backward differencing method,section (3.4).

• The necessity of using preconditioning along with multigrid in order tohave an effective damping ability has been demonstrated, section (3.1).In the absence of preconditioning, as a result of the high frequencyerrors which cannot be damped properly on fine mesh, the multigridbecomes quite ineffective.

• In order to prevent the laminarization of the flow on one hand, andto have a good damping property1 on the other hand, the third or-der Jameson type artificial dissipation is only added to the continuityequation and not to the momentum and energy equations. Numeri-cal experience has shown that this suffices to guarantee a good overallconvergence in the inner iterations.

1Specially for the continuity equation where there is no physical dissipation.

113

114 CHAPTER 6. CONCLUSION AND FUTURE WORK

• To have a faster calculation, the viscous fluxes are only calculated onthe finest grid. This has another advantage, that is in agreement withthe basic assumption of LES which needs rather fine mesh.

• V-sawtooth multigrid was the most robust for the present calculations.The analytical reason is not clear.

• The effect of the residual smoothing was not clear. For the channelflow it did not bring any improvement, but for the cavity flow, helpedto reduce the number of inner iterations.

• Time correction technique was not a big help to reduce the numberof inner iterations. However, for some particular number of inner it-erations, using the time correction technique leads to better results,section (2.6).

• A fully implicit approach is used and compared with a purely explicitone. It was shown that the implicit approach, depending on the testcase considered, is 4 to 7 times faster than a purely explicit method.

One may argue against comparing the efficiency of a fully implicit ap-proach with a purely explicit one, and ask for a comparison between differentimplicit approaches, or at least semi-implicit approaches.

A common semi-implicit approach, which is especially used for incom-pressible calculations, is to discretize the viscous part implicitly2, and thenon-linear convective part explicitly. An approximate factorization methodlike Alternating Direction Implicit (ADI) is generally used to deal with theimplicit part. The error of the factorization is of the same order as the errorof the time discretization scheme and is second order in time. This method isquite useful in the framework of incompressible flows, but it does not bringany improvement for the compressible low Mach number flows where thetime step is mainly restricted by the convective CFL condition based on thebiggest eigenvalue of the system.

In a fully implicit approach where both convective and viscous terms arecalculated implicitly, one first has to linearize the non-linear convective terms,and then an implicit solver can be used to march in time. The linearizationbrings another second order error into equations.

If the equations are discretized with a finite difference method and thetime accuracy is not the main issue, the linearization and factorization errorsapproach to zero in the steady state. But if time accuracy is important,one cannot easily neglect these errors, even though they are always second

2It is almost always done with a trapezoidal method, which is also called Crank-Nicolsonwhen applied to the diffusion term.

6.2. PARTICLE-LADEN FLOWS 115

order in time. To solve this problem, for the unsteady calculations one hasto always add a pseudo-time to reach the steady state in pseudo-time andminimize the above mentioned errors.

Unfortunately, the use of an ADI solver in the context of a finite volumemethod is not straightforward. In a finite volume method the fluxes aredirectly calculated and the method is supposed to be applicable to any kindof complex mesh including structured curvilinear or even unstructured grid.However, by incorporating some of the features of the finite difference methodinto the finite volume approach such as the coordinate transformation whichin the finite volume context should be applied to the fluxes, the use of theADI in combination with the finite volume method could be again possible.

Recently, Hsu and Jameson [44] used a hybrid scheme to calculate com-plex unsteady compressible flows with a finite volume method. Their ap-proach is a two step method which combines ADI and dual time stepping.At the first step, the linearized implicit equations are solved with an ADImethod which provides a formal second order accuracy in time. At the secondstep, a pseudo-time is introduced and with a number of inner iterations theerrors due to both linearization and factorization are reduced. They testeda two dimensional pitching airfoil and showed that this hybrid approach is40 percent cheaper than a pure dual time stepping scheme.

The approach of Hsu and Jameson seems to be an interesting approachand it is worthwhile to be further investigated in the future in an LES context.

The second interesting method, which has not been studied in this thesis,is the use of an implicit approach in pseudo-time and to march in time with anADI method. The implicit treatment brings in the complexity of coordinatetransformation which is necessary for an ADI method, but the overall speedup may compensate for the extra computational cost.

6.2 Particle-Laden Flows

Large-eddy simulation of particle-laden channel flow was carried out at Reτ =180. An incompressible finite volume solver, based on a cell-averaged ap-proach, was used for the carrier flow calculations. The methodology of calcu-lating the space derivatives with a finite volume compact scheme was brieflydescribed.

The velocity and location of the particles were calculated using a secondorder Adams-Bashforth formula. The drag force of the particles is calculatedassuming the particles are smooth rigid spheres and the influence of virtualmass, buoyancy, shear-induced lift, and Basset history force were neglectedon particles motion. The empirical relation used for drag coefficient CD isvalid for particle Reynolds number up to 40 and during the calculation, the

116 CHAPTER 6. CONCLUSION AND FUTURE WORK

particle Reynolds number never exceeded this value.Different interpolation methods such as first, second and third order in-

terpolations were tested. In the final calculations, a third order Lagrangeinterpolation was used.

The Lycopodium, glass, and copper particles were considered. A one-waycoupling method, which neglects the effect of particles on the carrier flow, isused. The agreement between the present calcuations and the existing DNSand LES data in the literature is satisfactory. The anisotropy of particlefluctuations were compared. It was demonstrated that the anisotropy of theparticle fluctuations increases with the Stokes number.

Two-way coupling of carrier flow and particles will be considered in thefuture. For the incompressible flow, after the addition of particle sourceterms in the momentum equations, special care should be taken in solvingthe Poisson equation to keep the stability and efficiency of the code.

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