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**PENETRABLE ROUGHNESS FLOWS in NATURE and in ENGINEERING**

Yevgeny A. Gayev Institute of Fluid Mechanics of UNAS Kyiv, May , 2004 NATO Advanced Study Institute

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**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

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**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

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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…

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**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.

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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… !

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**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…

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**2.3. What is the 'Spraying System'?**

Fountains, sprays in every day life 1 - Hannover. 2 - Osnabrűck. 3 - Kiev

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**Fountains, sprays in every day life**

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

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**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.

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**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

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

Conventional "bottle" nozzle ZaNPP: cooling water temperatures in January and June 1999

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**Typical distributions of wind and air temperature within the SCS**

Log-portion Distorted portion

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

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**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

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**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

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**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.

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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

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**General mathematical model**

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

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**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:

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**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

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**(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

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**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

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**(B) Flow enters a duct with infinite EPRs 2d Navier – Stokes equations**

Dimensionless variables ~ !

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**(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

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**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

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**(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 …

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**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.

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

The carried medium to be predicted together with the carrying one

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**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

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**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?

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**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

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**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

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**Conclusions from the 'constant viscosity' models**

Dimensionless criteria for the initial EPR region Universal coordinates for the external BL for the main EPR region

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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…

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**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 *

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**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

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**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

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**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

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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

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**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…

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**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.

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**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.

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**Spectra within an extended easily penetrable roughness array**

Measurements in Surrey university WT [Gayev,Savory] (2) Behind 20 rows (1) Behind 5 rows

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**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

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**Spectrum measurements within the EPR**

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

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**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.

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**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. ! ! ! ???

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**Spectrum measurements over the EPR**

z=230 mm z=180 mm Spectrum curves rise up till the elevation z ~1,5h – 3h

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**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…

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**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…

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**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 …

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**…and the peaks continue rising for next elevations.**

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

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**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?

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**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.

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**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.

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**For the turbulent kinetic energy**

For the dissipation rate

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**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…

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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.

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**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…

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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

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**Thank you very for your patient attention!**

NATO ASI Flows in Obstructed geometries + =

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