1. PENETRABLE ROUGHNESS FLOWS in NATURE and in ENGINEERING
Yevgeny A. Gayev
Institute of Fluid Mechanics of UNAS
Kyiv, May , 2004
NATO Advanced Study Institute

2. C o n t e n t s 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

3. 1. Variety of areas where 'tall roughnesses' may be met
River flows in
vegetated beds
Forests and agro- eco- cenosis
Storming ocean
After R.Bortkovsky
Urban settlements
Heat exchangers
Spraying coolers
After P.Mestayer

4. 1.1. A historical overview
L. Prandtl seemed to be the first in the area… but
Ludvig Prandtl, Klaus Oswatitsch
"Fűrer durch die Strömungslehre"
100 years
of the BL theory
…the real achievements, however, should be attributed
to meteorologists and (later) river hydraulics experts…

5. 1.2. Important articles in the field
For 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
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 forest
Data for turbulence
will be provided later… !

7. 2.2. Experimental data: measurements in river flows
Two variants of problem formulation:
(A) Vertical-plane problem
(B) Horizontal-plane problem
Log-like profiles outside the vegetated area
[Kouwen-1970;]
Distorted shapes of U(z) within the vegetated area
Data for turbulence
will be provided later…

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
2 – Karlsruhe (De).
1 – Guildford (UK)

10. Few words about Spraying Cooling Systems (SCS)
Panoramic view of the Zaporizhzhya NPP's spraying cooling system (SCS)
Specification: 1 – NPP's reactors 61000 MWt; 2 – spraying channel № 1, dimensions 4000100 m;
3 – spraying channel № 2; 4 – array of fountains h=6 m; 5 – additional cooling towers.

11. anemometers and psychrometers
2.3. Experimental data: in-situ measurements in industrial spraying coolers
Remote electrical
anemometers and psychrometers
at 10 levels of the 15m mast

12. Plan view of the Zaporizhzhya's Nuclear Power Plant Spraying Cooling System
Conventional "bottle" nozzle
ZaNPP: cooling water temperatures in January and June 1999

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. Conclusion 1: there are many similar features for (at least mean quantities of)
flows within differing obstruction layers.
A uniform theory may be possible.
3.1. The terms in use:
Layer
with distributed force
[J. Hunt]
Too mathematically…
Canopy
Forest canopy, etc.
Too narrow…
High roughness
[Cermak e.a.-1971]
Roughness sublayer [Mestayer]
Porous medium
In filtration theories…
Penetrable obstruction
Not correct…
Penetrable roughness
[W. Brutsaert]
Easily Penetrable Roughness, EPR
An adjective allowing some mathematical operations
like additivity of forces

16. 3.2. Fluid Mechanics' point of view: from 'small' to 'tall' and penetrable roughnesses
h<<H
h ~ (0,1 – 0,3)H
h ~ (0,3 – 0,9)H ?
Sand roughness
Motion and exchange processes within the roughness are of most interest.
Besides, motion of the roughness elements
may be practically important, too.
Height of the roughness
Is neglected
Almost all Fluid Mechanics case problems may be generalized in order to learn properties of the (Easily) Penetrable Roughnesses

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
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)
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

19. General mathematical model
~~~~~~~~~~~~~~~~~~~~~
~~~~~~~~~~~~~~~~~~~
~~~~~~~~~~~~~~~~~~~

20. Boundary Layer Approach;
Model verification by a sequence of sub-models: EPR made up of immobile elements.
Boundary Layer Approach;
Is it always valid?
Is valid?
Boundary conditions:
Conjugation conditions:

21. 4.1. Numerical results: general structure of the unrestricted EPR flow
Boundary layer over the EPR
1 – Initial Region:
6 –Main Region,
profiles of a final shape:
if k=1
if k=1
Stagnation Zone 7 is possible
if A>A critical ~2,5
if k=2

22. (E) Pipe lines (heat echangers) of various cross sections with filters
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. Navier – Stokes equations become 1d
Endless duct with an infinite Easily Penetrable Roughness
Navier – Stokes equations become 1d
Analytical solution for linear EPR, k=1
Numerical solution for quadratic EPR, k=2
Resistance coefficient via flow&EPR parameters
because

24. (B) Flow enters a duct with infinite EPRs 2d Navier – Stokes equations
Dimensionless variables ~ !

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 Re
an 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)
♪ 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. Conclusion. Three regimes depending on frequency may be observed:
(F) Pulsating flow in a duct with EPRs biological applications are possible
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.

29. 5.1. EPR made up of mobile elements (droplet layer model)
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
6.2. Mutual action of the heat and mass transfer
 Profiles of dry and wet air temperature and droplet temperature
Humidity profiles  formed by droplet layer

31. 7.1. Model of a polidisperse droplet layer each r-sort of droplets is a separate medium
Investigation of the one-dimensional model
(model flow in a duct)
~~~
~~~
How to find parameters and of an 'equivalent' monodisperse droplet layer?

32. 7.1. Some results for a polidisperse droplet layer
…\0\"Drops_of_2sizes.BMP"
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. Conclusions from the 'constant viscosity' models
Dimensionless criteria
for the initial EPR region
Universal coordinates
for the external BL
for the main EPR region

35. 9. Turbulence in the penetrable layers a number of experiments in various obstruction layers were carried out…
[Kouwen; Sherenkov,Bennovitsky]
In water flumes…
[Raupach,Finnigan e.a.]
In forests…
In models of urban settlements…
[Meroney;Savory; etc]
[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 m
number of obstructions up to 500)
Canopy element layout and measurement locations *

37. 9.1. Mean velocity profiles over the EPR
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
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

40. Bulk properties of the flow
Height of external boundary layer
above different canopies
Variation of longitudinal velocity along the EPR's
(i.e. D-canopy) and comparison with theory

41. 9.3. Theoretical modeling of the EPR turbulence (a) Algebraic closures: sufficient for most calculations
~~~~
Within the EPR
Outside the EPR
Comparison with Allen's measurements
Calculation for a vegetated channel flow
[Gayev,Wenka,Rodi]. More details: Bennovitsky,Gayev

42. Well, EPR flows are similar in terms of mean (overaged) properties
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 Spectra in some points within D-wake
Measurements in Surrey university WT [Gayev,Savory]
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.
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.

45. Spectra within an extended easily penetrable roughness array
Measurements in Surrey university WT [Gayev,Savory]
(2) Behind 20 rows
(1) Behind 5 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=100 mm
z=60 mm
Spectrum curves almost coincide for all the elevations 0 < z <1h
Conclusion: turbulence is rather homogeneous within the EPR
although the EPR is significantly inhomogeneous.

49. Spectrum measurements over the EPR fetch
z=160 mm
z=140 mm
Spectrum curves began rise up over the EPR, z > h =70 mm …
contrary to the case of a smooth surface.
! ! ! ???

50. Spectrum measurements over the EPR
z=230 mm
z=180 mm
Spectrum curves rise up till the elevation z ~1,5h – 3h

51. Then, spectrum curves fall down like it were over the smooth surface…
z=260 mm
z=200 mm
Then, spectrum curves fall down like it were over the smooth surface…

52. For more high elevations, spectrum curves continue to fall down…
z=280 mm
z=220 mm
For more high elevations, spectrum curves continue to fall down…

53. For more highest elevations, a spike appears on spectrum curves …
z=320 mm
z=240 mm
For more highest elevations, a spike appears on spectrum curves …

54. …and the peaks continue rising for next elevations.
z=480 mm
z=300 mm
…and the peaks continue rising for next elevations.

55. What the reason of such behavior of the vortices?
z=540 mm
z=480 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
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. Source/sink terms as a mean for capturing PR features
In momentum equations:
… there is only one possibility given by dimensional analysis, that is the force term.

58. For the turbulent kinetic energy
For the dissipation rate

59. How should these terms look like?
Dimensional analysis
for the turbulent kinetic energy – two variants
Dimensional analysis
for the dissipation rate – four variants
… it might be worth to examine all of these terms
to express peculiarity of various possible PR structures…

60. Main conclusions
A 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 flows
with the water surface affected by the EPR
Models for SQS constructed with 're-freshen' gaps
And much more
to give job for many meteorologists, hydraulics, wind engineers, engineering people and those of other expertise area…

62. Acknowledgment
Deutsche 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 Division
for such a unique possibility to carry out this Institute

63. Thank you very for your patient attention!
NATO ASI
Flows in Obstructed geometries + =