1

PENETRABLE ROUGHNESS FLOWS

in NATURE and in ENGINEERING

Yevgeny A. GayevInstitute of Fluid Mechanics of UNAS

2

Introduction- Examples of canopy (?) flows

- Who was the first in the area- Concept of Easily Penetrable Roughness (EPR)

Experimental data: In forests; in wind tunnels; in vegetated river flows; in spraying systems (SQS)

Theoretical considerations- General mathematical model and its particular cases; 1d-simplifications- EPR made up of immobile elements (model of a 'forest' ; EPR in a duct)- EPR made up of mobile particles (model of a 'droplet layer' )- Heat and mass transfer in the EPRs- Models of a polidisperse and multi- speed droplet layers

Turbulence in the penetrable layers - Wind tunnel measurements of mean characteristics - Theoretical models of the turbulence in EPRs

- Spectral appearance of the turbulence in EPRs

Results and discussion, prospective problems

Concluding remarks

C o n t e n t s

3

1. Variety of areas where 'tall roughnesses' may be met

Forests and agro- eco- cenosis

River flows invegetated beds

Urban settlements Heat exchangers Spraying coolers

Storming ocean

After P.Mestayer

After R.Bortkovsky

4

1.1. A historical overview

Ludvig Prandtl, Klaus Oswatitsch

"Fűrer durch die Strömungslehre"

L. Prandtl seemed to be the first in the area… but

…the real achievements, however, should be attributed

to meteorologists and (later) river hydraulics experts…

100 yearsof the BL theory

5

1.2. Important articles in the fieldFor natural forests:Wright I.L., Lemon E. Photosynthesis under field conditions. Agronomy Journal, 1966, 58, 3. Meroney R.N. Characterictics of wind and turbulence in and above model forests. J. Applied Meteorology, 1968, 7, 5.Konstantinow A.R. e.a. Application experience of gradient masts for determining evaporation and heat exchange in forest. - Proc. GGO, 1969, iss. 81.Plate E.J. Aerodynamic Characteristics of Atmospheric Boundary Layers. - U.S. Atomic Energy Commission, 1971. Menzhulin G.W. On the theory of a stationary meteorological regime of a vegetation canopy. - Proc. GGO, 1973, 297. Shaw R.H. Secondary wind speed maxima inside plant canopy. J. Applied Meteorology, 1977, 16. Dubov A.S., Bickova L.P. e.a. Turbulence in a Vegetation Canopy. - Leningrad: Hydrometeoizdat, 1978. Raupach M.R., Thom A.S. Turbulence in and above plant canopies. Ann. Review Fluid Mech., 13, 1981.Brutsaert W. Evaporation into the Atmosphere, 1982. Finnigan J. Turbulence in Plant Canopies. Ann. Review Fluid Mech., 2000, v. 32.

For river hydraulics:Kouwen N., e.a. Flow retardance in vegetated channels. J. of the Irrigation and Drainage Div., Proc. ASCE, 95(IR2), 1969.Knight D.W., Macdonald J.A. Hydraulic resistance of artificial strip roughness. Proc. ASCI, J. Hydraulics Div., HY6, 1979. Nuding A. Fliesswiederstandsverhalten in Gerinnen mit Ufergebuesch. - Technische Hochschule Darmstadt, Institut fuer Wasserbau, Nr. 35, 1991. Nepf H.M. Drag, turbulence, and diffusion in flow through emergent vegetation. Water Resources Research, 1999,35, N 2, pp. 479 - 489.

For urban ecology:Rotach M. W. Turbulence Within and Above an Urban Canopy. Zuericher Geographische Schriften, H. 45, 1991.

Davidson M.J., Belcher S.E., Hunt J.C.R. Atmospheric flow through groups of buildings and dispersion from localized sources. - In: Wind Climate in Cities. NATO ASI, Karlsruhe, 1993.

In oceanology:Bortkovsky R.S. Air-see exchange of heat and moisture during storms. D.Reidel, Dortrecht.Wu J. Spray in the atmospheric surface layer: laboratory study. J.Geophysical Research, 1973, 78, N 3.

In engineering fluid mechanics:Nickitin I.K. Complex turbulent flows and processes of heat and mass exchange.- Kiev, 1980. Ghosh S., Hunt J.C.R. e.a. Dynamics of turbulent air-flow in droplet driven sprays. Applied Scientific Resarch, 1993, 51.

Gayev Ye.A. Aerothermal theory of an Easily Penetrable Roughness. Particular application to the atmospheric flow in and over longscale Spray Cooling System. - Il Nuovo Cimento, C20, 1997.

6

2.1. Experimental data:measurements in forests and in agricultural crops

[Rauner-1958; Inoue-1963; Lemon&Wright-1965; Allen-1968; Dubov&Marunich-1971] [Thom&Raupach-1970; Oliver-1971; Cionco-1972; Shaw-1974]

!

Log-like profiles over the forest

Distorted shapes of U(z) within the forestData for turbulence will be provided later…

7

Log-like profiles outside the vegetated area

2.2. Experimental data:measurements in river flows

Two variants of problem formulation:

(A) Vertical-plane problem (B) Horizontal-plane problem

[Kouwen-1970;]

Data for turbulence will be provided later…

Distorted shapes of U(z) within the vegetated area

8

2.3. What is the 'Spraying System'?Fountains, sprays in every day life

1 - Hannover. 2 - Osnabrűck. 3 - Kiev

9

Fountains, sprays in every day life

1 – Guildford (UK)2 – Karlsruhe (De).

10

Few words about Spraying Cooling Systems (SCS)

Panoramic view of the Zaporizhzhya NPP's spraying cooling system (SCS)Specification: 1 – NPP's reactors 61000 MWt; 2 – spraying channel № 1, dimensions 4000100 m;

3 – spraying channel № 2; 4 – array of fountains h=6 m; 5 – additional cooling towers.

11

2.3. Experimental data:in-situ measurements in industrial spraying coolers

Remote electricalanemometers and psychrometers

at 10 levels of the 15m mast

12

Conventional "bottle" nozzle

ZaNPP: cooling water temperatures in January and June 1999

Plan view of the Zaporizhzhya's Nuclear Power Plant Spraying Cooling System

13

Typical distributions of wind and air temperature within the SCS

Log-portion

Distorted portion

14

2.5.Data generalization: similar to "universal" profiles within forests

15

3.1. The terms in use:Canopy

Forest canopy, etc.Too narrow…

Layerwith distributed force

[J. Hunt]Too mathematically…

Penetrable roughness[W. Brutsaert]

High roughness[Cermak e.a.-1971]

Conclusion 1: there are many similar features for (at least mean quantities of)

flows within differing obstruction layers.

A uniform theory may be possible.

Penetrable obstruction

Not correct…

Easily Penetrable Roughness, EPRAn adjective allowing some mathematical operations

like additivity of forces

Porous medium In filtration theories…

Roughness sublayer [Mestayer]

16

3.2. Fluid Mechanics' point of view: from 'small' to 'tall' and penetrable roughnesses

Height of the roughness

Is neglected

Motion and exchange processes within the roughness are of most interest.

Besides, motion of the roughness elementsmay be practically important, too.

Almost all Fluid Mechanics case problems may be generalized in order to learn properties of the (Easily) Penetrable Roughnesses

h<<H

h ~ (0,1 – 0,3)Hh ~ (0,3 – 0,9)H ?

Sand roughness

17

2. 2. What happens within the PR?

Kind permission for using this photo given by Prof. J.E.Cermak (Colorado University) is gratefully acknowledged

Bulk results of the

intensive vorticity:

♪ a mean force to each local portion of the fluid

♪ intensive mixing to be accounted via exchange

coefficients μT etc.

18

3.1. A main conclusion from the experiments:source terms to be included into equations that govern the process

EPR ,0

EPR ,);(||)()(

0

121

**

z

zdrzrsuUuUсFnzf

kF

n(x,z) or s(x,z) account for density of the resistant elements, i.e. elements of the EPR; they thus represent an architectonics of the penetrable roughness

U(x,z) and u(x,z) account for motion of the carrying media (air or water) and the carried media (elements of the EPR)

19

~~~~~~~~~~~~~~~~~~~

~~~~~~~~~~~~~~~~~~~~~

General mathematical model

~~~~~~~~~~~~~~~~~~~

.

1

Pr

1

0

02

0

01

0

011

0121

0

121

1

(r)E}S{e(r)βT}α{t(r)}z

tv(r)

x

t{u(r)m(r)c

(r)dr,e)n(r)S(Eβz

j

Sc}

z

EV

x

E{Uρ

(r)dr,t)n(r)S(Tαz

j}

z

TV

x

T{Ucρ

(r),Su(r)}{Uρc}z

uv(r)

x

um(r){u(r)

,z

V

x

U

S(r)dr,n(r)u)(Uρcz

τ)

z

UV

x

U(Uρ

E

T

kx

kx

20

Model verification by a sequence of sub-models:

4.1. EPR made up of immobile elements.

)0,()0,( hxUhxU

0

z

V

x

U

],[ ,0

],0[ ,1

1 hz

hzknU

zz

UV

x

UU

k

0 0 VUz UzUx )( 0

UUz Boundary conditions:

)0,()0,( hxhx

Conjugation conditions:

Boundary Layer Approach;Is it always valid?

Is valid?

Is valid?

Is valid?

21

1 – Initial Region:

4.1. Numerical results:general structure of the unrestricted EPR flow

6 –Main Region,profiles of a final shape:

.1 0 ),( UxUfxd

UdU

xAxU 1)(0if k=1

)exp()( 21

0 xnScxU x if k=2

0)(

Ufzd

d

)(

)( ,

)(

)(

Ach

zAch

Ash

zAshUU hh

if k=1

Boundary layer over the EPR

Stagnation Zone 7 is possible if A>A critical ~2,5

))(1)(ln( 22 AshAshAl

22

4.2. Pressure driving flows in ducts(fully developed and time dependent flows)

(A) Infinite EPRs in an endless plain duct (B) Flow enters a duct with infinite EPRs

(D) Flow enters a duct with a finite EPRs(C) Infinite porous insert in a plain duct

(E) Pipe lines (heat echangers) of various cross sections with filters

23

(A)Endless duct with an infinite Easily Penetrable RoughnessNavier – Stokes equations become 1d

)1,1,( ,0

]1,1[],0[ ,

β''Re

1

hhz

hhzkAU

U zz

0 :0 Uz

0 :1 Uz

Analytical solution for linear EPR, k=1

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0 0.02 0.04 0.06 0.08 0.1 0.12

U(z) for linear force, Re=1, h=.25 (Beta=-1!)

A=10100

1000

),(Re;2

1hAf 1)(2 2

1

0 dzzUq

Resistance coefficient via flow&EPR parameters

because

0

20

40

60

80

100

120

140

0 10 20 30 40 50 60 70 80 90 100

Beta*Re vs A for linear force; h=.3, Re=var

Re=5 10 20

Numerical solution for quadratic EPR, k=2

24

(B) Flow enters a duct with infinite EPRs2d Navier – Stokes equations

)1,( ,0

]1,1[],0[ ,

Re

12

2

2

22

hhz

hhzAV

z

V

x

V

z

p

z

V

x

VU

0

z

V

x

U

20

2000 ρ

ττ ,

ρ , , , ,

UU

pp

U

VV

U

UU

H

zz

H

xx Dimensionless

variables

)1,( ,0

]1,1[],0[ ,

Re

12

2

2

22

hhz

hhzAU

z

U

x

U

x

p

z

VU

x

U ~

~

!

25

(B) Some results for flow entering a duct

Mean velocity is gradually transformed from an uniform to a final shape (1d) profile

Pressure distributions in the duct

Sear stress distributions in the duct

26

Length Lx of the initial region

♪ Different curve behavior for small Re

♪ For large Rean approach is observed

to the limit case

already found from Boundary Layer Approx

[Schlichting]

Conclusion: Boundary Layer Approach is valid for large Re

27

(D) Flow enters a duct with a finite EPRs (penetrable backward facing steps)

Vortical motion behind "penetrable steps" h=0,3, l=1 in a duct flow Re=100 depending on A=100 (above) or A=10 (below)

*2

Re

1)( fUpgradUU

0Udiv

♪ there is no vorticity for easily penetrable EPR (small A);

♪ the vorticity is only appearing for A~10;

♪ there is an intensive vorticity for A~100;

♪ another calculation method is required if one needs precise knowledge within the PR with large A.

More details: Gayev, Shikhaliev …

28

(F) Pulsating flow in a duct with EPRsbiological applications are possible

)cos(0 tpp

Solution has been obtained in an analytical form using complex numbers.

There is an animation graphical program…

(a) Smooth walls in the duct (Richardson' phenomenon)

(b) EPRs near walls in the duct (opposite currents are larger)

Conclusion. Three regimes depending on frequency may be observed:

♪ at slow pulsations, ω<5, the flow resembles the Puaseule flow at each time moment;

♪ at frequent pulsations, ω>50, a phase shift occur, and the opposite currents become larger.

U U p a tAU z h

z ht zz

20

0 1( cos )

, [ , ]

, [ , ] .

29

5.1. EPR made up of mobile elements (droplet layer model)

kuUAzz

UV

x

UU x

p)(

τ

0

z

V

x

U

kuUBz

u

x

uu )(

The carried medium to be predicted

together with the carrying one

30

6.1.A. Heat transfer in droplet layer 6.1.B. Mass transfer in droplet layer

)(Pr

1TtA

z

j

z

TV

x

TU T

T

)( TtBz

t

x

tu T

)(Sc

1 EeA

z

j

z

EV

x

EU E

E

)( EeBz

e

x

eu E

10 btbe

6.2. Mutual action of the heat and mass transfer

)()( EeBTtBz

t

x

tu ET

Profiles of dry and wet air temperature and droplet temperature

Humidity profiles formed by droplet

layer

31

~~~ ~~~)],()[( ,2])()[(2

0

τruUr

zd

uddrruUrA

zd

d

7.1. Model of a polidisperse droplet layereach r-sort of droplets is a separate medium

zz

UV

x

UU

,)()(0

drrAuU k

7.1.1. Investigation of the one-dimensional model(model flow in a duct)

1 ,0 ,0 0 -τ hUhzUz

How to find parameters and of an 'equivalent' monodisperse droplet layer? *А *a

0

* )( drrAA 1

0

** ))(

)((

dr

r

rAA

32

7.1. Some results for a polidisperse droplet layer

Air velocity profiles and velocity profiles of two droplet media ("heavy" and "light") in two cross-sections of the droplet layer

33

8. Models of multi- speed droplet layers

2 droplet media:

♪ rising up

♪ falling down

4 droplet media:

♪ starting with u0=+1, initially rising up and then falling down;

♪ starting with u0= -1, initially rising up and then falling down.

Conclusion: various structures of the 'obstruction medium' may be represented in the EPR concept

34

for the initial EPR region

Universal coordinates

for the external BL

for the main EPR region

}1

,;,{ VA

UzxA

h

h

UUUU

zdz

,)(

12

},{hU

U

h

z

Conclusions from the 'constant viscosity' modelsDimensionless criteria

H

hh

T

2

knhA

)(

)( ,

)(

)(

Ach

zAch

Ash

zAshUU hh

,2

1

ah

ma

khB

Sc,,,Pr,,2

02

1T0T

02

TE

2

02

1TT

02

TT

Dcm

ShbB

SnhA

acm

ShB

SnhA

35

9. Turbulence in the penetrable layersa number of experiments in various obstruction layers were carried out…

[Meroney;Savory; etc][Raupach,Finnigan e.a.]

In forests…

[Kouwen; Sherenkov,Bennovitsky]

In water flumes…In models of urban settlements…

[Gayev e.a.]

In spraying coolers…

[Meroney;Raupach;Gayev,Savory; etc.]

In wind tunnels…

36

9. Some more data for turbulence

Geometries of canopy elements studied

Down- and Up-canopies in the wind tunnel of Surrey University [Gayev,Savory](working section dimensions: 1,5 m width, 2 m height; length ~5 mnumber of obstructions up to 500)

*

Flow Canopy elements(stems not to scale)

y =

120

mm

x = 60 mm

xx

xx

Typical measurementarray locations

Canopy element layout and measurement locations

37

9.1. Mean velocity profiles over the EPR

y / T-1 0 1

z / h

0

1

2

0.3

Contours of normalised mean velocity (U/Uoo) at X=10 rows (D-Canopy)

♪ U-profiles are very distinct at the EPR beginning

♪ U-profiles become united far away into the EPR

38

9.2. Turbulence intensity profiles over the EPR

y / T-1 0 1

z / h

0

1

20.1 0.1

0.2 0.2

0.3 0.3

0.4 0.4

0.5

0.4

0.40.3

0.3

0.2

0.5

0.2

0.5

0.2

Contours of normalised mean velocity (U/Uoo) at X=10 rows (D-Canopy)

♪ U'-profiles are very distinct at the EPR beginning

♪ U'-profiles become more united far into the EPR

39

40

Bulk properties of the flow

x (cm)

0 50 100 150 200 250 300 350

U (

m/s

)

0

1

2

3

4

5Maxima (xw/h=0.5)Minima (xw/h=0.5)Maxima (xw/h=1.0)Minima (xw/h=1.0)Maxima (xw/h=1.5)Minima (xw/h=1.5)Maxima (xw/h=2.0)Minima (xw/h=2.0)Theory

Variation of longitudinal velocity along the EPR's (i.e. D-canopy) and comparison with theory

D-canopy, 8m/s Stems only, 8m/s

h = 12 cm

h = 7 cm

x (cm)

0 50 100 150 200 250 300 350

(

cm)

0

10

20

30

40

50

D-canopy, Uoo = 6m/s

D-canopy, Uoo = 4m/s

D-canopy, Uoo = 2m/s

U-canopy, 90 deg, 8m/sU-canopy, 45 deg, 8m/s

U-canopy, 0 deg, 8m/s

h = 120 mm

h = 70 mm

D-canopy, 8m/s

Stems-onlycanopy, 8m/s

Height of external boundary layerabove different canopies

41

~~~~

9.3. Theoretical modeling of the EPR turbulence(a) Algebraic closures: sufficient for most calculations

, ),(,

],0[ ,

2

0

hzhzllz

Ul

hz

h

T

T,1 z

UT

Within the EPR

Outside the EPR

Comparison with Allen's measurements Calculation for a vegetated channel flow[Gayev,Wenka,Rodi]. More details: Bennovitsky,Gayev

)1ln()( *

hh z

hzUUzU

)sinh(

)sinh()(

Ah

zAUzU h

42

Well, EPR flows are similar in terms of mean (overaged) properties.

Are they so similar, i.e. have common features in term of the turbulence?

Bulge (i.e. sec. max.) problem in forest flows…

43

9.4. 'Fine structure' of the turbulence

It might have been expected that (1) vortices are proportional to the 'grid size' within the EPR, and (2) dissipate to small scales over the EPR…

It is suggested to examine this expectation by a spectrum measurements.

44

Spectra over a smooth surface

Features are known:1. Energy containing vortices 1<f<200 Hz;2. Inertial sub-layer E~f^(-5/3) for 20<f<500 Hz;3. Dissipation E~f^(-4) for 700<f<3 000 Hz;4. Vortices calm down with the height z.

Spectra in some points within D-wake

1. Energy of the vortices is much larger;

2. Peaks on spectrum curves are present for some points in the wake.

3. Again, vortices calm down with the height.

Measurements in Surrey university WT [Gayev,Savory]

45

Spectra within an extended easily penetrable roughness arrayMeasurements in Surrey university WT [Gayev,Savory]

(1) Behind 5 rows

(2) Behind 20 rows

46

Spectral appearance of the EPR turbulence

Spectrum curves almost over the surface with the 'tall trees' h=70 mm,on the elevation z=2 mm taken as a reference level

47

Spectrum measurements within the EPR

Spectrum curves almost coincide for all the elevations 0 < z <1h:here z=40 mm

48

Spectrum curves almost coincide for all the elevations 0 < z <1h

z=60 mmz=100 mm

Conclusion: turbulence is rather homogeneous within the EPRalthough the EPR is significantly inhomogeneous.

49

Spectrum curves began rise up over the EPR, z > h =70 mm …contrary to the case of a smooth surface.

! ! ! ???

z=140 mm z=160 mm

Spectrum measurements over the EPR fetch

50

Spectrum curves rise up till the elevation z ~1,5h – 3h

z=180 mm z=230 mm

Spectrum measurements over the EPR

51

Then, spectrum curves fall down like it were over the smooth surface…

z=200 mm z=260 mm

52

For more high elevations, spectrum curves continue to fall down…

z=220 mm z=280 mm

53

For more highest elevations, a spike appears on spectrum curves …

z=240 mm z=320 mm

54

z=300 mm z=480 mm

…and the peaks continue rising for next elevations.

55

z=480 mm z=540 mm

Vortices dissipate for only elevations z=6h (5 rows) and z=8h (20 rows) .

What the reason of such behavior of the vortices?

56

Finnigan’s e.a. image of coherent vortices over the PR [Ann. Rev. Fluid Mech., 2000, v.32]

Vorticity within and over the EPRs is an interesting subject

for further investigations.

57

)1,( ,0

]1,1[],0[ ,

Re

122

2

2

2

22

hhz

hhzVUAU

z

U

x

U

x

p

z

VU

x

U

Source/sink terms as a mean for capturing PR features

)1,( ,0

]1,1[],0[ ,

Re

122

2

2

2

22

hhz

hhzVUAV

z

V

x

V

z

p

z

V

x

VU

In momentum equations:

… there is only one possibility given by dimensional analysis, that is the force term.

58

)()( T UfPxxx

kU

jjii

)()( T UfPx

k

xx

kU kk

jkjii

For the turbulent kinetic energy

For the dissipation rate

59

How should these terms look like?

]/[ 32 smskUUf k ~)(

Dimensional analysis for the turbulent kinetic energy – two variants

,~)( 3sUUf k

Dimensional analysis for the dissipation rate – four variants

,~)( 22 kUsUf

,~)( 42UsUf

sUUf ~)(

,~)( 2/32UksUf

]/[ 32 sm

… it might be worth to examine all of these terms

to express peculiarity of various possible PR structures…

60

Main conclusionsA variety of problems that look out from the first glance as different ones may be treated, in fact, as problems of a penetrable roughness (i.e. canopy, porous layer etc) flows.

The EPR model given here allows investigation of PRs differing in nature and in structure, e.g. constructed from immovable, movable, multi speed and, may be, waving, flexible etc. elements.

An approach for distinguishing the “initial” and “main” regions of the canopy (=EPR) flow was suggested along with some analytical estimation formulas.

Approaches based on continuous media models capture important averaged flow features but more sophisticated models are required for predicting EPR flow turbulence characteristics.

The latter may be done by including source/sink terms into corresponding equations.

PR flows should be, because of their novelty, investigated further and deeper.

It should be thought over how the 'useful features' of the PR flows discovered by meteorologists and hydrologists may be utilized in engineering science and technology.

61

Prospective problems

EPR models for "waving" and flexible elements EPR models for shallow water flowswith the water surface affected by the EPR

Models for SQS constructed with 're-freshen' gaps

And much moreto give job for many meteorologists, hydraulics,

wind engineers, engineering people and those of other expertise area…

62

AcknowledgmentDeutsche Forschungsgemeinschaft: Institute of Hydromechanics, University of Karlsruhe, Germany (Prof. G. Ernst, Prof. W. Rodi)

Royal Society: University of Surrey, United Kingdom (Prof. E. Savory, Prof. N. Toy)

Colorado University, USA (Prof. R. Meroney)

University of California, Davis, USA (Prof. R.Shaw)

My native Institute of Hydromechanics Ukr. Nat. Academy of Sci.

NATO Scientific Affairs Divisionfor such a unique possibility to carry out this Institute

63

Thank you very for your patient attention!

+ =NATO ASI

Flows in Obstructed geometries