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The aerodynamics of shelter belts
Agricultural Meteorology- ElsevierPublishing Company, Amsterdam- Printed in The Netherlands
T H E A E R O D Y N A M I C S OF S H E L T E R BELTS ERICH J. PLATE Radiological Physics Division, Argonne National Laboratory, Argonne, IlL (U.S.A.) (Received December 1, 1969) (Resubmitted May 25, 1970)
ABSTRACT PLATE, E. J., 1971. The aerodynamics of shelter belts. Agr. MeteoroL, 8: 203-222.
The effectiveness of a shelter is determined not only by its total drag but also by the distribution of the drag generated momentum defect in the sheltered area. Many aerodynamic factors affect both the drag and the momentum defect distribution, such as the boundary layer profiles of the approaching flow and the shape and the porosity of the shelter. How the interaction of these factors shapes the velocity distributions in the sheltered region is discussed qualitatively in this paper, on the basis of a flow model which consists of different regions. In each of them a different combination of aerodynamic factors is acting. Emphasis is on the region directly downwind from the shelter. It is shown that separation from the top of the shelter belt gives rise to a separation streamline which divides the low velocity flow below from the high velocity flow aloft. The blending of the flow across this streamline, which determines the recovery of the wind profile and the reduction in sheltering efficiency, is caused by the gradient in velocity across the streamline, while its location is determined by the drag on the shelter and the pressure distribution behind it. It is also shown that more turbulence is produced in a high density shelter than in a porous one. Finally, some conclusions are drawn with regard to research needs for improving our understanding of shelter belt aerodynamics. INTRODUCTION Shelter belts belong in the class of earliest devices that man used to improve climatic conditions to serve his agricultural needs. The necessity for them may have been as readily apparent to early farmers in Denmark who saw their crops suffering from exposure to the relentless winds blowing off the North and Baltic Seas, as it became to the Oklahoma farmer in the thirties whose topsoil was lifted off the ground and carried away in duststorms. But only in the last thirty or so years have shelter belts been studied in a systematic manner, with the goal of finding the optimum shelter belt which with the minimum loss in land and perhaps Agr. MeteoroL, 8 (1971) 203-222
204
E.J. PLATE
water, at lowest cost in planting and maintaining, yields the optimum protection, expressed as maximum yield, of the adjacent crops. It is probably fair to say that this general problem has at present not been solved. Too many variables enter into its solution, and too many different disciplines--agronomy, meteorology, forestry, to name a few--must contribute to it. There is no question that a shelter belt arrangement is a system whose optimization requires input information from many sources, and if such an analysis should ever become economically interesting contributions from all of them will have to be considered. One of the most important inputs will be the aerodynamics of the wind flow about shelter belts, and it is with this part of the problem that the present paper is concerned. In principle, the aerodynamic action of a shelter belt is easy to understand. The shelter belt exerts by its drag a force on the wind field which is compensated by a loss of momentum of the air, according to Newton's second law. In an incompressible fluid a reduction in momentum implies a reduction in velocity, and thus the drag is converted into the wind speed reduction desired for sheltering. Obviously, the greater the drag the greater is the reduction in wind speed. However, in wind shelters one is not interested in total wind speed reduction but in an optimum reduction in a thin air layer near the ground in which the crops that need protection are found. In this connection " o p t i m u m " can mean a number of things. It can mean reduction of wind speed to a minimum value at a given point, or often a reduction to a largest safe wind speed over the longest possible lee distance, or a reduction in evaporation, or an enhancement of CO 2 supply. Each of these requirements may have its own most efficient shelter in a given situation and the shelter which exerts the largest drag may not be among them. The most frequent design requirement is that the wind be reduced below the dangerous level over a maximum distance, Discovering the shelter which does this most efficiently has been the objective of much of the research on shelter belts. It is an enormously complex analytical problem since it requires a solution of the full turbulent Navier Stokes equation for a complete treatment, and it is not surprising that most shelter belt research has been done experimentally and by trial and error, by evaluating the wind reduction of existing belts. Most often quoted are the results of N.~GELI (1941) who obtained an optimum solution of sorts by showing that a medium dense screen reduced the velocity by at least 20 over a larger distance than either a very dense screen or a screen with very high porosity. Similar results were found in a wind tunnel, by BLErqK and TRIENES(1956). As far as shape is concerned, an optimum that is generally accepted has not been found, and most field research was conducted on existing shelter belts for a post facto assessment of their effect. Reviews of field results are given by VAN DER L1NDE (1962) and GEIGER (1965). The only theoretical treatment of the shelter belt problem known to the writer was given by KAISER (1959) who assumed that the sheltering results from diffusion of the momentum defect downwind from the shelter as if it were a passive Agr. MeteoroL, 8 (1971)203-222
AERODYNAMICS OF SHELTER BELTS
205
scalar. This model is physically unrealistic and somewhat oversimplified, yet it does point to the decisive role which the drag plays in the shelter problem, and it also leads to a prediction of velocity profiles which are described by the error function over a part of the air layer in the sheltered region. It would indeed be too much to expect an analytical solution which covers all details of the shelter belt flow. However, many aspects of the flow field are very similar to well known aerodynamic situations. It is the purpose of this paper to point out these aerodynamic features. The results of this discussion suffice to yield criteria for modeling shelter belts in wind tunnels, and it is also hoped that they may help those engaged in actively planning shelter belts or shelter belt research to become aware of those characteristics of a potential shelter material which might be beneficial or detrimental in meeting the desired shelter requirements. THE FLOW FIELD AT A SHELTER BELT
The complexity of the flow around awind shelter is evident from the schematic situation shown in Fig.1. A boundary layer flow is approaching a wedge-shaped obstruction which has been placed on a flat plate. Not less than seven flow zones of different aerodynamic behavior can be distinguished. In Zone 1, the flow field is mostly determined by the conditions in the undisturbed boundary layer far upstream from the wedge. In Zone 2, the flow field is displaced and distorted due (~ (~
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(~
STANDING EDDY ZONE
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POTENTIAL OUTER FLOW
ill
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)
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Fig.1. The flow zones of a boundary layer disturbed by a shelter belt. (After PLATE and [,IN, 1965.)
Agr. Meteorol., 8 (1971) 203-222
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E . J . PLATE
to the presence of the wedge, with the lower boundary of Zone 2 given by the separation induced shear layer which starts at the edge of the wedge and forms the transition to the highly retarded flow in Zone 3. When the wedge is solid, back flow may occur, leading to a separation bubble with a reattachment point, at a distance L downstream from the wedge. Downstream of the reattachment point the flow is again in the direction of the mean wind, and in the layer 5 it gradually increases in velocity until at some large distance the "inner layer" 5 has blended with the outer flow. A new and thicker boundary layer is thus formed which adjusts to the local boundary conditions at the ground until the effect of the obstruction can only be inferred by comparing the boundary layer thickness with that which would have existed if the wedge had not been there. The flow in the region downstream from reattachment is that of the adjustment of an initial velocity profile to the local boundary conditions, and can be determined in principle from an initial profile downwind ofreattachment by methods of boundary layer calculations, as has been done by PLATE (1967) for distances larger than 35h, where h is the height of the obstruction. At present, however, such analyses depend on empirical observations and assumptions, and further research is needed to eliminate the empirical constants. Here we shall be concerned mostly with Zones 2 and 6, which are most important in relation to the sheltering problem. THE FLOW IN THE SHELTERED REGION BEHIND A SOLID BELT
A detailed study of the flow field directly downwind of the obstruction has been made, for a solid obstruction, by CHANG (1966) who used experimental data obtained in a wind tunnel at Colorado State University. The upper parts of some of his mean velocity profiles are shown in Fig.2A. The velocity profiles are poorly defined in the separation bubble below the parts of the profiles shown because large pressure gradients exist as indicated in Fig.2C which distort pitotstatic tube measurements, and because the low mean velocities, at high turbulence levels, cast serious doubts on measurements obtained with hot wire anemometers in this region. Similar measurements behind obstacles consisting of a fence have been given by GOOD and JOUBERT (1968) and for an obstacle with a quarter circle cross section which faces the flow with the round surface by MUELLERet al. (1963). Measurements behind a sharp-edged plate placed perpendicular to an air stream were reported by ARIE and ROUSE (1956). All these measurements have in common a very rapid change of velocity across a curve which can be identified as the location of all velocities equal to one half the velocity uo~ outside of the boundary layer. This curve is identical to the separation streamline near the upper edge of the
Fig.2. Experimental data on the flow field downwind of a shelter (wind tunnel results of CHAN~, 1966). A. Mean velocities. B. Turbulence characteristics. C. Pressure. Agr. Meteorol., 8 (1971) 203-222
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shelter. It shall be taken as the intrinsic x-axis, with the z-coordinate measured from it. The flow field has all the characteristics of the flow that results if an upper airstream with initially uniform velocity u~ is joined along the separation streamline with a lower stream of zero velocity, or, in the case of a porous shelter with velocity u b through the shelter, with a lower stream of velocity u b. The solution for this flow situation is well known (see SCHLICHTING, 1968) as long as it can be assumed that the lower velocity stream extends to minus infinity, and that the streamline curvature does not affect the velocity distribution, a The solution is based on a shearing stress given by the eddy viscosity assumption: Ou
z, = p8 Oz
(1)
where the eddy viscosity e is a function of x, but not of z, and is defined by:
= ~
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(3)
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and the variable ~ is equal to ~ = a(z/x) where ~r is an empirical constant. For Ub = 0, the experimental data of CHANG (1966) show that eq.3 is indeed in excellent agreement with experimental data. Some of his results are reproduced in Fig.3. For positive values of ~ the agreement is nearly perfect, partly due, of course, to the experimental determination of a. For negative values of ~, the lower boundary starts distorting the results, and some deviations appear. The distortion is, however, within the experimental error upwind of reattachment. By using eq.2 and eq.3 in eq.l, a bell-shaped Gaussian distribution of the shear stress is obtained, which is in agreement with laboratory observations. This is seen in Fig.2B, where all the turbulence data are reproduced which correspond to the situation of Fig.2A. A detailed comparison of the shear stress distribution with the Gaussian distribution is given in Fig.4 which leaves no doubt that the model is essentially correct. The model also predicts a constant value of a, i.e., the velocity profiles should collapse onto a single curve if the ordinates measured i Recently, the effect of curvature on half jet velocity profiles has been considered by UCHIDA and SUZUKI(1968), but their results are not directly applicable to the present problem.
Agr. Meteorol., 8 (1971) 203-222
209
AERODYNAMICS OF SHELTER BELTS
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f r o m the u = ~ u ® curve, are divided b y x/a. The experimental values o f x/a have been calculated f r o m the m e a n velocity d a t a a n d are shown in Fig. 5. A linear relation between this q u a n t i t y a n d x is indeed found, with a value oftr = 14.5, which is in r e a s o n a b l e a g r e e m e n t with the e x p e r i m e n t a l value o f 13.8 r e p o r t e d b y SCHLICHTING (1968). W h e t h e r this value r e m a i n s c o n s t a n t for p o r o u s screens o r d e p e n d s on 2 is n o t known. The intercept o f the straight line with the x-axis yields the virtual origin x 0 o f the intrinsic x-axis to a b o u t Xo = - 6h.
Agr. Meteorol., 8 (1971) 203-222
210
E. J. PLATE 1.5
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x/(rh versus x/h, f r o m m e a n velocity data.
Another aspect of flow with a porous shelter which is at present not fully understood is how the wind profile develops near the ground downwind of the shelter. It is evident that the analytical model of the two parallel currents loses its usefulness as soon as the presence of the lower boundary makes itself felt on the blending region between the currents. Also, it follows from the governing equations that the shear stress gradient shapes the mean velocity profile. A strong initial gradient across the blending region leads to rapid distortion of the mean velocity profiles, while a weak shear stress gradient requires a longer distance to modify the mean profiles by the same amount. Consequently, large values of Ub/Uo~ are approaching 1 by the same amount over longer distances than small values, that is,
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Fig.6. Sheltering at different porosities, according to N.~GEL! (1941).
Agr. Meteorol., 8 (1971) 203-222
AERODYNAMICSOF SHELTERBELTS
211
blending between inner and outer flow proceeds at the slower rate. But large values of Ub/Uoo mean "little" sheltering, and large velocity gradients near the ground. It can easily be seen that there exists an "optimum porosity" at which Ub is a minimum over the longest distance. As already mentioned, such an optimum has indeed been found in natural environments, as is illustrated in Fig.6 which is due to N~,GELI (1941). The sheltering was determined by measuring the actual wind velocity at a height of 1.4 m, and dividing it by the velocity that would have existed in the absence of a shelter. A medium dense shelter belt showed the best results. Systematic wind tunnel tests by JENSEN (1954) and by BLENK and TRIENES (1956) showed a maximum sheltering to be associated with porosities (defined as percentage of open area in the total area of the screen) of 35-50 %. THE DISPLACEMENTOF THE SEPARATIONSTREAMLINE The error function profile is seen to be a realistic representation of the mean velocity distribution in the neighborhood of the separation streamline. But the development was assumed to take place along the curve given by u/u~ = 0.5 which was found experimentally. A prediction of the flow field between the shelter and the reattachment point is possible only if the location of this curve can be determined from no more information than a knowledge of the velocity distribution in the undisturbed boundary layer, and of the characteristics of the shelter. Such a prediction is at present not possible. We may, however, gain some understanding of how this location depends on the drag on the shelter, and on the base pressure behind it, i.e., how it depends on the pressure distribution about the shelter. The effect of the shelter on the separation streamline may be considered in an analysis which neglects the effect of the lower boundary except by letting the ground form a streamline. The flow shown in Fig.7 represents a simplified model of this situation. For generality, a porous screen is considered. The screen introduces a momentum sink which leads to a deceleration of the flow upstream from uo~ to Ub, where ub is the velocity through the shelter, and to an upward deflection U~o,P~o
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Agr. MeteoroL, 8 (1971) 203-222
212
E . J . PLATE
of the streamlines as a result of mass conservation. A measure of the streamline displacement, or of the vertical extent of the sheltered area, can be obtained by applying the conservation of mass and of momentum principles to the free body diagram indicated by A B C D in Fig.7. The change in momentum must be balanced by the external forces in the x-direction. I f the effect of blending is neglected, which is likely to be permissible near the screen, then the sum ~,co of momentum fluxes and forces on CD:
Zco= (put + pOh* + (pu~ + po0(Z - h')
(5)
where the subscript 0o refers to conditions in the undisturbed flow far from the lower boundary, the subscript b refers to conditions slightly behind the screen, and p is the pressure. For definitions of the length parameters in eq.5, see Fig.7. For A B one gets: (6)
Zan = + (pu~ + poo)l
The momentum flux across B C is + pu~ • Qnc where Qnc is the volume flux across BC, which evidently is the difference in volume flux into A B and out of CD, so that: Y'nc = - p u ~ ( u ~ - Ub)h*
(7)
The additional external force acting on the fluid is the drag on the screen, so that Newton's second law leads to: (8)
D = - Zco + ZAs + Z•c
The solution 1 of this equation for h*/h is, with eq.5, 6, 7: h"
(9)
Co
where Co is the drag coefficient, and Q the base pressure coefficient, defined by: h
2
(Pfront - -
pb)dZ
O ~o Co = i~2 pu2£h . . . . . pu 22/~ . . . . .
; Q - p~l/2-pu~Pb
(10)
When u b is equal to zero, the height h* of the sheltered region is proportional to Coh/Q. This result is in good agreement with experimental data, both for a flat plate in a free stream, like that of ARm and ROUSE (1956) and for a plate in a boundary layer, as shown by PLATE (1964). He found that Co = 1.65Q and h" = t A similar derivation for a solid plate in a stream of fluid has been given by REICHARDT(1945) (as quoted by Birkhoff, 1950) who applied these ideas to predict the width of a cavitation bubble. Agr. Meteorol., 8 (1971) 203-222
A E R O D Y N A M I C S O F S H E L T E R BELTS
213
1.67h, the latter result based on data of NAGAnHUSHANAIAH (1961). It is readily apparent that eq.9 is an approximation only since both the approach velocity profile and the diffusion of the interface between flow in the shelter and outside of the shelter are neglected. More accurate results by GOOD and JOUBERT (1968) have shown that CD = 1.82Q; however, these writers have not given parallel results on h*. When ub ~ u~, the pressure at both front and back of the screen approaches p~ and thus both CD and Q approach zero at the same rate, so that in the limit ub = u~o; h* = h, as it must. For small values ofub/u~, it is likely that h*/h decreases below the value at uJu~ = 0 because the ratio CD/Q remains roughly constant. Consequently, a porous screen results in a smaller height of the sheltered volume as compared to a solid screen. THE DRAG ON THE SHELTER
The drag on the shelter is governed not only by the shape and porosity of the shelter but also by the aerodynamic characteristics of the approaching boundary layer flow. This has been made clear in the researches of PLATE (1964) and GOOD and JOUBERT (1968). The latter have shown that if the height of the (solid) shelter is less than 0.5 times the thickness of the boundary layer, as is usual for real shelter belts, then the approaching boundary layer flow sets the scales for modeling all the features of the problem. Thus, if the velocity profile that would exist at the location of the shelter in the absence of the shelter is logarithmic and described by a profile of the form: u _ 1 In z t/,
K
(11)
Z0
where x is von Karman's constant (~0.4), then the shear velocity u, sets the velocity scale for the drag and the roughness height z o of the surface configuration sets the length scale. Consequently one may expect to find that a drag coefficient C. of the shelter belt based on u.: C, -
D 112 pu2, h
(12)
is a universal function of h/z o. The experiments of GOOD and JOUBERT (1968) were made with a smooth surface only. In that case, there is no characteristic length of the rough surface, and the only length which can be used to represent the surface characteristics must be based on the kinematic viscosity v and the shear velocity u.. Consequently, it is found that:
c
Agr. Meteorol., 8 (1971) 203-222
214
E.J. PLATE
for a smooth surface (GOOD and JOUBERT, 1968) while for a rough surface one may put:
C,=f2
zo-
(14)
The functions f l and J2 are, of course, dependent also on shelter geometries. As yet, only eq. 13 has been tested experimentally and only for sharp-edged shelters, with the results shown in Fig.8. ,/8 I
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t
Fig.8. Drag coefficients of fences in thick boundary layers. In Fig.8 the data are fitted by a straight line, whose equation has been given by GOOD and JOUBERT (1968) to:
u.h
C. = 277 l o g - - - - 268
(15)
Analogously, one expects to find, for a fully rough surface: C. = A l o g - -
h
+ B
(16)
Zo
The results in eq.15, 16, have an important consequence for modeling a solid shelter in a wind tunnel. It is seen that identical drag coefficients are obtained if, for a rough approach flow, the ratio h/zo is kept constant--provided that the geometries of the shelter are identical in model and prototype and that h/~ < O.15 in the model--which implies: hmodel hprototype
--
ZO model
(17)
Zo prototype
Agr. MeteoroL, 8 (1971) 203-222
AERODYNAMICS OF SHELTER BELTS
215
Consequently, in selecting a profile of the mean velocity distribution upstream of the shelter one determines the scale factor for the experiments. It should be noted that BLENK and TRIENES (1956) did not attempt to scale N~,GELI'S (1941) flOWS in this manner; they needed an arbitrary conversion factor of 0.5 for reducing the downwind distance to make their experimental data conform to the results of Niigeli. This is a clear indication that modeling was not properly accomplished. How the drag coefficient of a porous shelter is related to the approach velocity profile can at present only be surmised. It should be defined more suitably by using Ub as reference velocity: Cp
-
D 1/2 pu 2 h
(18)
in which Cp, D and Ub are unknowns whose interrelations are not known. The drag coefficient Cp depends on the porosity and also on the Reynolds number of the elements forming the shelter. For screens consisting of wires, some systematic experiments have been made to determine this dependency by stretching screens across the whole cross section of a wind tunnel and measuring the pressure drop, at a given approach velocity. Typical are the investigations of SCHUBAUER et al. (1950). In a natural shelter, it is usually not possible to specify the porosity and the sizes of the shelter elements, and all estimates of the drag must rely either on direct measurement of the flow near similar installations or on guesses. Such measurements must naturally be made at a time during the growing season which corresponds to that during which shelter is required by the crops, or else differences in foliage or shelter growth might be sufficient to render valueless any conclusions based on the experiment. THE BASE PRESSURE
The main difficulty in predicting drag forces on the shelter stems from our inability to determine the base pressure, i.e., the pressure at the back of the shelter. It is readily apparent from eq.10 that the larger the difference between pressure on the front and the base pressure, the larger the drag. The pressure on the front of the shelter is generally determined, for a solid shelter, by the flow pressure of the approaching flow on the face of the shelter, independent of the shape of the shelter. This is strictly true for a plate suspended in the free stream, with velocity u~, and pressure p~, of a wind tunnel. The pressure on the front in that case can be calculated from inviscid flow theory (ARIE and ROUSE, 1956) and the maximum pressure at the center of the plate is found to be exactly equal to the stagnation pressure Pst P~ + 1/2 pu 2 calculated from Bernoulli's equation. For a shelter immersed in a boundary layer, the approach flow is modified by the presence of the ground. This modification begins, according to surface pressure measurements of G o o d and JOUBERT (1968), along a smooth floor at a distance o f x = 15 (~/h)°'Th =
Agr. Meteorol., 8 (1971) 203-222
216
E.J. PLATE
upstream of the shelter where 6 is the boundary layer thickness; upon approaching the shelter the flow is retarded more strongly near the surface than at higher elevations and eventually separation occurs at a short distance in front of the shelter. The separation "bubble" does not extend over the full height of the shelter, and a stagnation points exists at about 0.5-0.7h from the ground at the face of the shelter. Below the stagnation streamline the flow is downward; a standing eddy forms in the corner between ground and shelter. Above this eddy the flow is accelerating towards the upper edge of the shelter. This flow field changes the pressure distribution from what would be expected had the ground not exerted any friction; in particular, the pressure maximum appears at the stagnation point and not at ground level. Clearly such a flow field does not exist in front of a porous shelter, because the porosity does not permit a pressure build-up in front that leads to separation. It is likely that at the rear of the screen the pressure is about constant, as for the solid fence case, and that the velocity distribution across the screens is fairly uniform. The latter is evident in some of the velocity profiles reported by BLENK and TRIENES (1956). A similar observation has also been made at the edge of a forest. A low level jet is often observed which extends over the first few tree rows and then settles down to a roughly uniform velocity over a substantial part of the tree height (REIFSNYDER,1955; MERONEY, 1968). This uniform velocity makes it likely that the pressure across the front of the shelters is also fairly uniform, as long as the shelter is of approximately constant porosity. In that case the pressure drop across the shelter is proportional to the square of the velocity through the shelter, with a factor of proportionality depending on the porosity only. The pressures in front and in the rear of the shelter are not independent because the air flowing over the top of the shelter provides a "leak" connecting the two pressures. However, the mechanism by which the pressure at the back side is governed is not yet understood. The prediction of the "base" pressure is a problem of considerable importance in aerodynamics and much work has been done on it, particularly for applications of supersonic aerodynamics (i.e., BROWN and STEWARTSON, 1969, for a recent survey). For low speed aerodynamics the problem appears to have aspects which complicate its solution even further, such as the turbulence in the blending region which is likely to be an important factor in the transmission of pressure gradients, or the smaller pressure differences between base pressure and free stream pressure, which leads to a curved separation streamline. The pressure field that arises is illustrated for the data of CHANG (1966) in Fig.2C, in which isobars have been indicated. It is noteworthy that substantial pressure gradients exist both in the longitudinal direction and in the vertical. These pressure gradients indicate that in the sheltered region it is not justified to simplify the Navier Stokes equations by using the boundary layer assumptions. Measurements along the ground have shown that the negative pressure downstream of the shelter is approximately constant over a distance of about Agr. MeteoroL, 8 (1971) 203-222
217
AERODYNAMICSOF SHELTERBELTS
3 times the height h and then starts rising very rapidly until, in the neighborhood of reattachment, it reaches a maximum which is slightly above the pressure in the free stream. Further downstream it slowly decreases and reaches asymptotically the free stream pressure, as is required by the boundary layer character of the redeveloping flow. This is illustrated in Fig.9. The figure was obtained by plotting non-dimensional pressure differences (Pwall -- Poo)/PU~ VS. x/h, and it is seen that the distributions are independent of velocity, and scale uniquely for a particular shelter.
-03
oC,:=.,~ " d ~ ) ~1""
-03
o
5
I0
ra
5
I0
9.1
5 5 5 5 5 5
I0 5 5 5 20 20
18.2 4.5 9.1 18.3 6.1 9. I
l O • • •
-0.4
o
v -0.5
0
I0
I
2O
50
40
50
5
60
20
70
4,5 --
_
12.2
80
90
X h
Fig.9. Pressure distributions along the ground. (After PLATE and LIN, 1965.)
The difference between ground and free stream pressure gives rise to a vertical force on the flow in the blending region which tends to push the separation streamline towards the ground. This is called the "Coanda" effect. The curvature of the streamlines, as expressed by the radius of curvature R, can be approximately inferred by applying the momentum conservation law to a small element out of the blending region, as indicated in Fig.10. Let the element have a velocity u which, by the definition of a streamline, is directed normal to the radius of curvature R, and let the pressure across the element change by an amount (Op/az)dz. The centrifugal force on the element then must be balanced by the resultant of pressure forces and shear forces on the front and rear faces of the element, or neglecting the latter, the momentum balance equation yields:
Agr. Meteorol., 8 (1971) 203-222
218
E.J. PLATE
0p
p + ~-~z +~....I..- .
7
/
J
/
L~Z
If~ t
~
~
-.,...~.
~
Strmm ne
R
Fig.10. Illustrating the Coanda effect.
1
1
dp
R
pu 2
69z
(19)
or in n o n - d i m e n s i o n a l form: h R
2 h x
Pwa. - P~ 1]2 pu 2
2
u~o
uz
~3p' 0rl
where p ' -
P - p~ Pwall -- Po~
-g(r/)
(20)
The pressure gradient function g(q) in eq.20 is p r o b a b l y d e p e n d i n g on the geo m e t r y only, a n d m a y be assumed to obey a similarity law d e p e n d i n g on the same similarity variable q as the mean velocity distribution. F o r the curve u = 0.5 u~, the function 2(uZ~/u z) (Op'/Or/) is a constant = ~ a n d the radius o f curvature becomes: h R
_ ~ Pw, n - Po~ 1/2 pu~
(21)
with a m a x i m u m value of: h R
h - ctQ--x
(22)
While this e q u a t i o n represents only an a p p r o x i m a t i o n , it shows that the larger the pressure difference Pw,. - P+ the smaller is the radius o f curvature. A solid fence, with a large base pressure coefficient yields a m o r e r a p i d turning o f the u = 0.5 u~ curve t o w a r d s the g r o u n d than a p o r o u s screen. The C o a n d a effect is at least in p a r t responsible for the r a p i d loss o f sheltering efficiency o f a
Agr. Meteorol., 8 (1971) 203-222
AERODYNAMICSOF SHELTERBELTS
219
solid screen. A porous screen, with its higher velocity in the blending region and its lower base pressure, exhibits significantly smaller streamline curvature. THE TURBULENCEIN THE SHELTEREDREGION More important than the Coanda effect on the wind field behind the shelter is the turbulence which is found in the air flow. The turbulence, whose vertical component is w' and whose longitudinal component is u', gives rise to turbulent shear stresses z = - p u ' w ' which not only adjust the mean velocity distribution but which are also responsible in producing turbulence. The strong mean velocity gradients generated by separation from the shelter edge interact with the shear stresses existing in the undisturbed flow to produce more turbulence for small values of Ub than for large values. To see this, consider the energy balance of the turbulent motion, integrated over a volume consisting of a slice of (infinitesimal) length Ax, of infinite vertical extent and of unit width: Ax
i-
dq 2 ~dz
= Ax
0
i
•
7
Off
~-z dz - D
(23)
0
where small terms have been neglected. The overbars denote time averages. In eq.23 q2 is the kinetic energy of the turbulent motion at a point:
q2 =
~(U,2 ..~ /),2 ..~ W,2)
(24)
and D is the dissipation of kinetic energy into heat in the volume. The term on the left is the rate of change of turbulent energy in the volume which for steady flows can be expressed by: dq 2 d - ~ - dz = ~ 0
-fiq2 dz
(25)
0
This term is determined mostly by the first term on the right in eq.23 which represents the turbulence productions, which in the stages of growth of the separation layer exceeds considerably the dissipation. By inserting eq.1, 2, into the production term of eq.23, and by neglecting the dissipation term as compared with the production, it follows readily that in the blending region the turbulence increases at a rate proportional to (u 2 - u2)(u~o - Ub) and independent of x, until the presence of the ground modifies the turbulence structure. Consequently, a more solid fence has associated with it a larger turbulence level, i.e., the total amount of turbulence downstream of a porous screen is lower than that behind a solid screen. This will certainly affect the turbulence transport processes in the sheltered region, thus explaining why higher evaporation rates are observed behind a solid screen than behind a porous screen, as was observed by BLENKand TRIENES Agr. Meteorol., 8 (1971) 203-222
220
E.J. PLATE
(1956). However, perhaps a more important factor in determining exchange processes behind a solid screen is the return flow in the separation bubble which has associated with it strong vertical currents. These currents do not exist behind porous screens, so that for either of these reasons if wind protection is the most important task of the shelter belt, a porous screen has great advantages over a solid one. SUMMARY AND CONCLUSIONS
The discussion of this paper has shown that the information required for an aerodynamically most efficient shelter belt design is not yet available. But it has become clear what information is needed, and how it can be obtained. First, one requires a knowledge of the velocity profile at the location of the shelter before the shelter is to be planted or constructed. This information is readily available, as a good estimation of the u. and z o values of the undisturbed boundary layer can be had by either taking some local measurements or by consulting the existing literature on the velocity distribution near natural surfaces, such as GElGER (1965, p.275). But for going beyond this, one must make a number of assumptions which are backed by experimental evidence only for a solid shelter belt. A drag coefficient and a base pressure coefficient must be determined to predict the location of the separation streamline, and a velocity Ub must be assumed. Then it is possible to calculate the height h* from eq.9 to a first approximation, and to calculate the blending velocity profile by using eq.3, perhaps with tr the solid screen value. To fill in the gaps in our knowledge, it is evidently necessary to re-analyze existing wind tunnel data and field data, and to perform parallel experiments in wind tunnels and in the field. The materials of which natural shelters are made preclude the use of wind tunnels for the direct design of an optimum shelter. Since leaf area, growth rate, density and shape of shelter belts vary too much to be predictable in advance, the greatest use of a wind tunnel may be found in one of the two following areas. The wind tunnel can be useful in evaluating the efficiency and arrangement of wind breaks made of non-random materials, which may consist of regular arrangements of slats, reed, bamboo or other material of similar kind. These can be studied effectively in a wind tunnel because their geometry is reproducible, and their drag characteristics can be determined in the same manner as was described for screens. It appears feasible to investigate in this manner the optimum design of a wind break with respect to cost and efficiency. Some experiments by Japanese writers (SATO et al., 1952) along these lines have already been performed. It is, however, necessary during these experiments to give due regard to the approach velocity profile, its roughness height Zo and its shear velocity u. as well as to the other aerodynamic factors of the surroundings which might affect the sheltering. Agr. MeteoroL, 8 (1971) 203-222
AERODYNAMICSOF SHELTERBELTS
221
The second application is to the evaluation of the effect of terrain modifications in the neighborhood of the shelter on existing shelter belts. Field experiments yield the necessary information on the approach wind distribution and on the sheltering from which a model of the undisturbed situation can be made. As was pointed out earlier, geometric similarity might suffice for this purpose in addition to scaling of the z0-values by the scale of the shelter model. I f the experiments yield significantly different sheltering coefficients, improvements can be made by trial and error, by blocking or widening the interstices between the shelter elements. Design of new shelters and classification of existing shelters may be aided by some fundamental research which would improve our understanding of the aerodynamic effects of shelter belts. From the discussion in previous sections it is clear that the effect of shelter shape and porosity on the degree of sheltering is incompletely understood, as is the effect of ground roughness on the drag of the shelter. To this one may add systematic investigations of the effect of finite width shelters and of the effect of orientations not perpendicular to the wind. A study presently under way in the Fluid Dynamics and Diffusion Laboratory of Colorado State University is concerned with the modification of wind field by multiple shelters and with their optimum spacing. ACKNOWLEDGEMENTS The experiments on which much of this paper is based were performed by graduate students of the author while he was at Colorado State University. This work was performed under the auspices of the U.S. Atomic Energy Commission. REFERENCES ARE, M. and RousE, H., 1956. Experiments on two-dimensional flow over a normal wall. J. Fluid Mechanics, 1: 129-141. BLENK, H. and TRIne.s, H., 1956. Str6mungstechnische Beitr~ige zum Windschutz. In: Grundlagen der Landtechnik, 8. V.D.I. Verlag, Diisseldorf, 65 pp. BROWN, S. N. and STEWARTSON,K., 1969. Laminar Separation. In" N. SEARS(Editor), Annual Reviews of Fluid Mechanics, 1. Annual Reviews, Inc., Palo Alta, Calif., pp.45-72. CHANG, S. C., 1966. Velocity Distributions in the Separated Flow behind a Wedge-shaped Model Hill. Thesis, Colorado State Univ., Denver, Colo., 101 pp., unpublished. GEIGER,R., 1961. Das Klima der bodennahen Luftschicht. Vieweg, Braunschweig, 4th ed., 646 pp. GOOD, M. C. and JOtmERT,P. C., 1968. The form drag of two-dimensional bluff-plates immersed in turbulent boundary layers. J. Fluid Mechanics, 31: 547-582. HAUrZKY, J., 1968. Gas diffusion near buildings. In: D. H. SLADE(Editor), Meteorology and Atomic Energy, 1968. U.S. Atomic Energy Commission, Division of Technical Information Extension, Oak Ridge, Tenn., pp.221-255. JENSEN, M., 1954. Shelter Effect. Danish Technical Press, Copenhagen, 264 pp. KAISER,H., 1959. Die Str6mung an Windschutzstreifen. Ber. Deut. Wetterdienstes, 7(53): 36 pp. MEROSrY, R. N., 1968. Characteristics of wind and turbulence in and above model forests. J. Appl. Meteorol., 7(5): 780--788. Agr. Meteorol., 8 (1971) 203-222
222
z . J . PLATE
MUELLER, T. J., KORST, H. H. and CHOW, W. L., 1963. On the separation, reattachment, and redevelopment of incompressible turbulent shear flow. J. Basic Eng., 63-AHGT-5, 86: 221-226. NAGABHUSHANAIAH,H. S., 1961. Separation Flow Downstream o f a Plate Set Normal to a Plane Boundary. Thesis, Colorado State University, Denver, Colo., 150 pp., (unpublished). N.~GELI, W., 1941. Untersuchungen tiber die Windverh~iltnisse im Bereich yon Windschutzstreifen. Mitt. Schweiz. Anst. Forst. Versuchswesen, 23: 221-276. PLATE, E. J., 1964. The drag on a smooth flat plate with a fence immersed in its turbulent boundary layer. A S M E Paper 64-FE-17:12 pp. PLATE, E. J., 1967. Diffusion from a ground level line source into the disturbed boundary layer far downstream from a fence. Intern. J. Heat Mass Transfer, 10: 181-194. PLATE, E. J. and LIN, C. Y., 1965. The Velocity Field Downstream from a Two-dimensional Model Hill. (Final report, Part I, U.S. Army Material Agency, Contract DA-AMC-36-039-63-G7, CER65EJP-CYL14, 75 pp). REICHARDT, H., 1945. Die Gesetzmiissigheiten yon Kavitationsblasen an umstrdmten Rotationskdrpern. Max Planck Institut for Str6mungsforschung, GOttingen, UM6628, 34 pp. REIFSNVDER, W. E . 1955. Wind profiles in a small isolated forest stand. Forest Sci., I: 289-297. SATO, K., TAMACHI,M., TERANDA,K., WATANABE,Y., KATOH, T., SAKANONE,Y., and IWASAKI, M., 1952. Studies on Wind Breaks. Nippon Gakujutsu-Shiukokai, Tokyo, 201 pp. ScnLiCrmN~, H., 1968. Boundary Layer Theory. McGraw-Hill, New York, N.Y., 747 pp. SCHUBAUER,G. B., SPANGENBER~,W. G. and KLEBANOFr,P., 1950. Aerodynamics of damping screens. Natl. Advisory Comm. Aeron., Tech. Note, 2001:16 pp. TOWNSEND, A. A., 1956. The Structure o f Turbulent Shear Flow. Cambridge University Press, London, 317 pp. UCmOA, S. and SOZUKI, T., 1968. On a similar solution for a turbulent half-jet along a curved streamline. J. Fluid Mechanics, 33: 379-398. VAN DER LINDE, J., 1962. Trees outside the forest. In: Forest Influences. F.A.O., Rome, pp. 141-208.
Agr. Meteorol., 8 (1971) 203-222
T H E A E R O D Y N A M I C S OF S H E L T E R BELTS ERICH J. PLATE Radiological Physics Division, Argonne National Laboratory, Argonne, IlL (U.S.A.) (Received December 1, 1969) (Resubmitted May 25, 1970)
ABSTRACT PLATE, E. J., 1971. The aerodynamics of shelter belts. Agr. MeteoroL, 8: 203-222.
The effectiveness of a shelter is determined not only by its total drag but also by the distribution of the drag generated momentum defect in the sheltered area. Many aerodynamic factors affect both the drag and the momentum defect distribution, such as the boundary layer profiles of the approaching flow and the shape and the porosity of the shelter. How the interaction of these factors shapes the velocity distributions in the sheltered region is discussed qualitatively in this paper, on the basis of a flow model which consists of different regions. In each of them a different combination of aerodynamic factors is acting. Emphasis is on the region directly downwind from the shelter. It is shown that separation from the top of the shelter belt gives rise to a separation streamline which divides the low velocity flow below from the high velocity flow aloft. The blending of the flow across this streamline, which determines the recovery of the wind profile and the reduction in sheltering efficiency, is caused by the gradient in velocity across the streamline, while its location is determined by the drag on the shelter and the pressure distribution behind it. It is also shown that more turbulence is produced in a high density shelter than in a porous one. Finally, some conclusions are drawn with regard to research needs for improving our understanding of shelter belt aerodynamics. INTRODUCTION Shelter belts belong in the class of earliest devices that man used to improve climatic conditions to serve his agricultural needs. The necessity for them may have been as readily apparent to early farmers in Denmark who saw their crops suffering from exposure to the relentless winds blowing off the North and Baltic Seas, as it became to the Oklahoma farmer in the thirties whose topsoil was lifted off the ground and carried away in duststorms. But only in the last thirty or so years have shelter belts been studied in a systematic manner, with the goal of finding the optimum shelter belt which with the minimum loss in land and perhaps Agr. MeteoroL, 8 (1971) 203-222
204
E.J. PLATE
water, at lowest cost in planting and maintaining, yields the optimum protection, expressed as maximum yield, of the adjacent crops. It is probably fair to say that this general problem has at present not been solved. Too many variables enter into its solution, and too many different disciplines--agronomy, meteorology, forestry, to name a few--must contribute to it. There is no question that a shelter belt arrangement is a system whose optimization requires input information from many sources, and if such an analysis should ever become economically interesting contributions from all of them will have to be considered. One of the most important inputs will be the aerodynamics of the wind flow about shelter belts, and it is with this part of the problem that the present paper is concerned. In principle, the aerodynamic action of a shelter belt is easy to understand. The shelter belt exerts by its drag a force on the wind field which is compensated by a loss of momentum of the air, according to Newton's second law. In an incompressible fluid a reduction in momentum implies a reduction in velocity, and thus the drag is converted into the wind speed reduction desired for sheltering. Obviously, the greater the drag the greater is the reduction in wind speed. However, in wind shelters one is not interested in total wind speed reduction but in an optimum reduction in a thin air layer near the ground in which the crops that need protection are found. In this connection " o p t i m u m " can mean a number of things. It can mean reduction of wind speed to a minimum value at a given point, or often a reduction to a largest safe wind speed over the longest possible lee distance, or a reduction in evaporation, or an enhancement of CO 2 supply. Each of these requirements may have its own most efficient shelter in a given situation and the shelter which exerts the largest drag may not be among them. The most frequent design requirement is that the wind be reduced below the dangerous level over a maximum distance, Discovering the shelter which does this most efficiently has been the objective of much of the research on shelter belts. It is an enormously complex analytical problem since it requires a solution of the full turbulent Navier Stokes equation for a complete treatment, and it is not surprising that most shelter belt research has been done experimentally and by trial and error, by evaluating the wind reduction of existing belts. Most often quoted are the results of N.~GELI (1941) who obtained an optimum solution of sorts by showing that a medium dense screen reduced the velocity by at least 20 over a larger distance than either a very dense screen or a screen with very high porosity. Similar results were found in a wind tunnel, by BLErqK and TRIENES(1956). As far as shape is concerned, an optimum that is generally accepted has not been found, and most field research was conducted on existing shelter belts for a post facto assessment of their effect. Reviews of field results are given by VAN DER L1NDE (1962) and GEIGER (1965). The only theoretical treatment of the shelter belt problem known to the writer was given by KAISER (1959) who assumed that the sheltering results from diffusion of the momentum defect downwind from the shelter as if it were a passive Agr. MeteoroL, 8 (1971)203-222
AERODYNAMICS OF SHELTER BELTS
205
scalar. This model is physically unrealistic and somewhat oversimplified, yet it does point to the decisive role which the drag plays in the shelter problem, and it also leads to a prediction of velocity profiles which are described by the error function over a part of the air layer in the sheltered region. It would indeed be too much to expect an analytical solution which covers all details of the shelter belt flow. However, many aspects of the flow field are very similar to well known aerodynamic situations. It is the purpose of this paper to point out these aerodynamic features. The results of this discussion suffice to yield criteria for modeling shelter belts in wind tunnels, and it is also hoped that they may help those engaged in actively planning shelter belts or shelter belt research to become aware of those characteristics of a potential shelter material which might be beneficial or detrimental in meeting the desired shelter requirements. THE FLOW FIELD AT A SHELTER BELT
The complexity of the flow around awind shelter is evident from the schematic situation shown in Fig.1. A boundary layer flow is approaching a wedge-shaped obstruction which has been placed on a flat plate. Not less than seven flow zones of different aerodynamic behavior can be distinguished. In Zone 1, the flow field is mostly determined by the conditions in the undisturbed boundary layer far upstream from the wedge. In Zone 2, the flow field is displaced and distorted due (~ (~
UNDISTURBEDBOUNDARYLAYER (OUTER LAYER) REGIONOF HILL INFLUENCE (MIDDLE LAYER )
(~
REGIONOF REESTABLISHING BOUNDARY LAYER (INNER LAYER )
(~
BLENDING REGION BETWEEN MIDDLE AND OUTER LAYER
~)
BLENDING REGION BETWEEN INNER AND MIDDLE LAYER
(~
STANDING EDDY ZONE
(~)
POTENTIAL OUTER FLOW
ill
I
o,.,
-m~x)
)
"
Fig.1. The flow zones of a boundary layer disturbed by a shelter belt. (After PLATE and [,IN, 1965.)
Agr. Meteorol., 8 (1971) 203-222
206
E . J . PLATE
to the presence of the wedge, with the lower boundary of Zone 2 given by the separation induced shear layer which starts at the edge of the wedge and forms the transition to the highly retarded flow in Zone 3. When the wedge is solid, back flow may occur, leading to a separation bubble with a reattachment point, at a distance L downstream from the wedge. Downstream of the reattachment point the flow is again in the direction of the mean wind, and in the layer 5 it gradually increases in velocity until at some large distance the "inner layer" 5 has blended with the outer flow. A new and thicker boundary layer is thus formed which adjusts to the local boundary conditions at the ground until the effect of the obstruction can only be inferred by comparing the boundary layer thickness with that which would have existed if the wedge had not been there. The flow in the region downstream from reattachment is that of the adjustment of an initial velocity profile to the local boundary conditions, and can be determined in principle from an initial profile downwind ofreattachment by methods of boundary layer calculations, as has been done by PLATE (1967) for distances larger than 35h, where h is the height of the obstruction. At present, however, such analyses depend on empirical observations and assumptions, and further research is needed to eliminate the empirical constants. Here we shall be concerned mostly with Zones 2 and 6, which are most important in relation to the sheltering problem. THE FLOW IN THE SHELTERED REGION BEHIND A SOLID BELT
A detailed study of the flow field directly downwind of the obstruction has been made, for a solid obstruction, by CHANG (1966) who used experimental data obtained in a wind tunnel at Colorado State University. The upper parts of some of his mean velocity profiles are shown in Fig.2A. The velocity profiles are poorly defined in the separation bubble below the parts of the profiles shown because large pressure gradients exist as indicated in Fig.2C which distort pitotstatic tube measurements, and because the low mean velocities, at high turbulence levels, cast serious doubts on measurements obtained with hot wire anemometers in this region. Similar measurements behind obstacles consisting of a fence have been given by GOOD and JOUBERT (1968) and for an obstacle with a quarter circle cross section which faces the flow with the round surface by MUELLERet al. (1963). Measurements behind a sharp-edged plate placed perpendicular to an air stream were reported by ARIE and ROUSE (1956). All these measurements have in common a very rapid change of velocity across a curve which can be identified as the location of all velocities equal to one half the velocity uo~ outside of the boundary layer. This curve is identical to the separation streamline near the upper edge of the
Fig.2. Experimental data on the flow field downwind of a shelter (wind tunnel results of CHAN~, 1966). A. Mean velocities. B. Turbulence characteristics. C. Pressure. Agr. Meteorol., 8 (1971) 203-222
urn-13.7 msec"°
0 5 10 15 msec"l i
i
i
i
J
40 35 3O 25
eo 20
N
15
.--"! "~
I
/-°I
~
I0 5
5
A
,o
,~
2o
2~
3o
3~
4o
x (cm)
,~
5o
5~
6o
~
_ _
m
o u"/.~o 0
um = 13.7 msec"
( dlmefl$1onllsI )
005
*
7'o
• w"/u~o
U,W'-~;/U¢CO
40 ,) 5,5 30 25
u_.= I
E
"20
5
B
I0
15
20
25
30
35
40 45 xIcm)
50
55
60
:5O
_2o "i5
C 0
/
....
~ i
/f
'
. 5
~
I./ I0
---.
~
,i' 15
~ '~'''~'.
f 20
.4 25
/I 30
t'l , f I 35 40
X (cm)
\\ ~
NUMBERSDENOTE
./
I / 4.5
l 50
I 5.5
I 60
65
70
208
E.J. PLATE
shelter. It shall be taken as the intrinsic x-axis, with the z-coordinate measured from it. The flow field has all the characteristics of the flow that results if an upper airstream with initially uniform velocity u~ is joined along the separation streamline with a lower stream of zero velocity, or, in the case of a porous shelter with velocity u b through the shelter, with a lower stream of velocity u b. The solution for this flow situation is well known (see SCHLICHTING, 1968) as long as it can be assumed that the lower velocity stream extends to minus infinity, and that the streamline curvature does not affect the velocity distribution, a The solution is based on a shearing stress given by the eddy viscosity assumption: Ou
z, = p8 Oz
(1)
where the eddy viscosity e is a function of x, but not of z, and is defined by:
= ~
1
x (u~ + Ub)
(2)
The resulting velocity distribution is given by the error function: U
m
u~o+Ub(l_2erfO 2
(3)
where: erf~ = ~ f
e'm2dm
and
2-
Uo~--Ub Uoo + Ub
(4)
0
and the variable ~ is equal to ~ = a(z/x) where ~r is an empirical constant. For Ub = 0, the experimental data of CHANG (1966) show that eq.3 is indeed in excellent agreement with experimental data. Some of his results are reproduced in Fig.3. For positive values of ~ the agreement is nearly perfect, partly due, of course, to the experimental determination of a. For negative values of ~, the lower boundary starts distorting the results, and some deviations appear. The distortion is, however, within the experimental error upwind of reattachment. By using eq.2 and eq.3 in eq.l, a bell-shaped Gaussian distribution of the shear stress is obtained, which is in agreement with laboratory observations. This is seen in Fig.2B, where all the turbulence data are reproduced which correspond to the situation of Fig.2A. A detailed comparison of the shear stress distribution with the Gaussian distribution is given in Fig.4 which leaves no doubt that the model is essentially correct. The model also predicts a constant value of a, i.e., the velocity profiles should collapse onto a single curve if the ordinates measured i Recently, the effect of curvature on half jet velocity profiles has been considered by UCHIDA and SUZUKI(1968), but their results are not directly applicable to the present problem.
Agr. Meteorol., 8 (1971) 203-222
209
AERODYNAMICS OF SHELTER BELTS
"~'m--~ ( l + e r f ( ) ~ 1 1
U¢)
0.4
•A
SYMBOL
0.3 0.2
0.1 - 1.6
- 1.2
-0.8
-0.4
x/h 3
"
4
•
5
•
6
I
0.4
0
o
0.8
J
1.2
1.6
¢ Fig.3. Mean velocities fitted by error function.
1.0
& o
x/h = 8 x / h = 10
a
x/h
O.8
OA 0.2 -I.I
-0.9
-0.7
-0.5
-0.3
-0.1
0.1
0.3
0.5
0.7
0.9
I.I
o'~
Fig.4. Shear stress distribution fitted by Gaussian curve.
f r o m the u = ~ u ® curve, are divided b y x/a. The experimental values o f x/a have been calculated f r o m the m e a n velocity d a t a a n d are shown in Fig. 5. A linear relation between this q u a n t i t y a n d x is indeed found, with a value oftr = 14.5, which is in r e a s o n a b l e a g r e e m e n t with the e x p e r i m e n t a l value o f 13.8 r e p o r t e d b y SCHLICHTING (1968). W h e t h e r this value r e m a i n s c o n s t a n t for p o r o u s screens o r d e p e n d s on 2 is n o t known. The intercept o f the straight line with the x-axis yields the virtual origin x 0 o f the intrinsic x-axis to a b o u t Xo = - 6h.
Agr. Meteorol., 8 (1971) 203-222
210
E. J. PLATE 1.5
1.0
• u(o=9.1 msec-*
xo/h=6.2
x um:13.7 msec-~
o'=14.6
~ e Z
x
/
x
e
0.5
I 2
0
I 4
I 6
J 8
J I0
I 12
x/h
Fig.5.
x/(rh versus x/h, f r o m m e a n velocity data.
Another aspect of flow with a porous shelter which is at present not fully understood is how the wind profile develops near the ground downwind of the shelter. It is evident that the analytical model of the two parallel currents loses its usefulness as soon as the presence of the lower boundary makes itself felt on the blending region between the currents. Also, it follows from the governing equations that the shear stress gradient shapes the mean velocity profile. A strong initial gradient across the blending region leads to rapid distortion of the mean velocity profiles, while a weak shear stress gradient requires a longer distance to modify the mean profiles by the same amount. Consequently, large values of Ub/Uo~ are approaching 1 by the same amount over longer distances than small values, that is,
IOO °
w 80
•" r
WIND
i .-.,,I
_z '-
s
•
o'~°
..~s"
~eo oO "°Do
..;r ,....... ,,,/
/
•/ ' / 1
,"-"
,
u.
WINOWARO ;
.
I
LEEWARD
. . . . . . . . DECIDUOUSWOODSIN WINTER ,oo , .
.
.
.
.
MEDIUM-DENSE SHELTERBELT
bl
o
5'
o
s
,o
,s
~o
25
3o
MULTIPLESOF SHELTERBELT HEIGHT
Fig.6. Sheltering at different porosities, according to N.~GEL! (1941).
Agr. Meteorol., 8 (1971) 203-222
AERODYNAMICSOF SHELTERBELTS
211
blending between inner and outer flow proceeds at the slower rate. But large values of Ub/Uoo mean "little" sheltering, and large velocity gradients near the ground. It can easily be seen that there exists an "optimum porosity" at which Ub is a minimum over the longest distance. As already mentioned, such an optimum has indeed been found in natural environments, as is illustrated in Fig.6 which is due to N~,GELI (1941). The sheltering was determined by measuring the actual wind velocity at a height of 1.4 m, and dividing it by the velocity that would have existed in the absence of a shelter. A medium dense shelter belt showed the best results. Systematic wind tunnel tests by JENSEN (1954) and by BLENK and TRIENES (1956) showed a maximum sheltering to be associated with porosities (defined as percentage of open area in the total area of the screen) of 35-50 %. THE DISPLACEMENTOF THE SEPARATIONSTREAMLINE The error function profile is seen to be a realistic representation of the mean velocity distribution in the neighborhood of the separation streamline. But the development was assumed to take place along the curve given by u/u~ = 0.5 which was found experimentally. A prediction of the flow field between the shelter and the reattachment point is possible only if the location of this curve can be determined from no more information than a knowledge of the velocity distribution in the undisturbed boundary layer, and of the characteristics of the shelter. Such a prediction is at present not possible. We may, however, gain some understanding of how this location depends on the drag on the shelter, and on the base pressure behind it, i.e., how it depends on the pressure distribution about the shelter. The effect of the shelter on the separation streamline may be considered in an analysis which neglects the effect of the lower boundary except by letting the ground form a streamline. The flow shown in Fig.7 represents a simplified model of this situation. For generality, a porous screen is considered. The screen introduces a momentum sink which leads to a deceleration of the flow upstream from uo~ to Ub, where ub is the velocity through the shelter, and to an upward deflection U~o,P~o
....... U~Owp ~
U~
1-"-
. / "
/ °
I " . ~-_:.__ , ~
..
blending region
).L..I---%.\\Nz;'//-,\\\'
~ X
Fig.7. Momentum balance with boundary layer type velocity profile neglected.
Agr. MeteoroL, 8 (1971) 203-222
212
E . J . PLATE
of the streamlines as a result of mass conservation. A measure of the streamline displacement, or of the vertical extent of the sheltered area, can be obtained by applying the conservation of mass and of momentum principles to the free body diagram indicated by A B C D in Fig.7. The change in momentum must be balanced by the external forces in the x-direction. I f the effect of blending is neglected, which is likely to be permissible near the screen, then the sum ~,co of momentum fluxes and forces on CD:
Zco= (put + pOh* + (pu~ + po0(Z - h')
(5)
where the subscript 0o refers to conditions in the undisturbed flow far from the lower boundary, the subscript b refers to conditions slightly behind the screen, and p is the pressure. For definitions of the length parameters in eq.5, see Fig.7. For A B one gets: (6)
Zan = + (pu~ + poo)l
The momentum flux across B C is + pu~ • Qnc where Qnc is the volume flux across BC, which evidently is the difference in volume flux into A B and out of CD, so that: Y'nc = - p u ~ ( u ~ - Ub)h*
(7)
The additional external force acting on the fluid is the drag on the screen, so that Newton's second law leads to: (8)
D = - Zco + ZAs + Z•c
The solution 1 of this equation for h*/h is, with eq.5, 6, 7: h"
(9)
Co
where Co is the drag coefficient, and Q the base pressure coefficient, defined by: h
2
(Pfront - -
pb)dZ
O ~o Co = i~2 pu2£h . . . . . pu 22/~ . . . . .
; Q - p~l/2-pu~Pb
(10)
When u b is equal to zero, the height h* of the sheltered region is proportional to Coh/Q. This result is in good agreement with experimental data, both for a flat plate in a free stream, like that of ARm and ROUSE (1956) and for a plate in a boundary layer, as shown by PLATE (1964). He found that Co = 1.65Q and h" = t A similar derivation for a solid plate in a stream of fluid has been given by REICHARDT(1945) (as quoted by Birkhoff, 1950) who applied these ideas to predict the width of a cavitation bubble. Agr. Meteorol., 8 (1971) 203-222
A E R O D Y N A M I C S O F S H E L T E R BELTS
213
1.67h, the latter result based on data of NAGAnHUSHANAIAH (1961). It is readily apparent that eq.9 is an approximation only since both the approach velocity profile and the diffusion of the interface between flow in the shelter and outside of the shelter are neglected. More accurate results by GOOD and JOUBERT (1968) have shown that CD = 1.82Q; however, these writers have not given parallel results on h*. When ub ~ u~, the pressure at both front and back of the screen approaches p~ and thus both CD and Q approach zero at the same rate, so that in the limit ub = u~o; h* = h, as it must. For small values ofub/u~, it is likely that h*/h decreases below the value at uJu~ = 0 because the ratio CD/Q remains roughly constant. Consequently, a porous screen results in a smaller height of the sheltered volume as compared to a solid screen. THE DRAG ON THE SHELTER
The drag on the shelter is governed not only by the shape and porosity of the shelter but also by the aerodynamic characteristics of the approaching boundary layer flow. This has been made clear in the researches of PLATE (1964) and GOOD and JOUBERT (1968). The latter have shown that if the height of the (solid) shelter is less than 0.5 times the thickness of the boundary layer, as is usual for real shelter belts, then the approaching boundary layer flow sets the scales for modeling all the features of the problem. Thus, if the velocity profile that would exist at the location of the shelter in the absence of the shelter is logarithmic and described by a profile of the form: u _ 1 In z t/,
K
(11)
Z0
where x is von Karman's constant (~0.4), then the shear velocity u, sets the velocity scale for the drag and the roughness height z o of the surface configuration sets the length scale. Consequently one may expect to find that a drag coefficient C. of the shelter belt based on u.: C, -
D 112 pu2, h
(12)
is a universal function of h/z o. The experiments of GOOD and JOUBERT (1968) were made with a smooth surface only. In that case, there is no characteristic length of the rough surface, and the only length which can be used to represent the surface characteristics must be based on the kinematic viscosity v and the shear velocity u.. Consequently, it is found that:
c
Agr. Meteorol., 8 (1971) 203-222
214
E.J. PLATE
for a smooth surface (GOOD and JOUBERT, 1968) while for a rough surface one may put:
C,=f2
zo-
(14)
The functions f l and J2 are, of course, dependent also on shelter geometries. As yet, only eq. 13 has been tested experimentally and only for sharp-edged shelters, with the results shown in Fig.8. ,/8 I
[
[
I
I I I [I
I
I
/e
/
//~,/// • 2r J t
70C 600 cH
~,.~J"BDATA POINTSFROMPLATE
50O
,,96,,-
tl,,. j ~
4OO fl
3ooiio2
" ~ DATAOFGOODAND JOuBERT(1968)
J
I
I 3
I
I 5
I
I I 11 7 103 u~._hh
t
Fig.8. Drag coefficients of fences in thick boundary layers. In Fig.8 the data are fitted by a straight line, whose equation has been given by GOOD and JOUBERT (1968) to:
u.h
C. = 277 l o g - - - - 268
(15)
Analogously, one expects to find, for a fully rough surface: C. = A l o g - -
h
+ B
(16)
Zo
The results in eq.15, 16, have an important consequence for modeling a solid shelter in a wind tunnel. It is seen that identical drag coefficients are obtained if, for a rough approach flow, the ratio h/zo is kept constant--provided that the geometries of the shelter are identical in model and prototype and that h/~ < O.15 in the model--which implies: hmodel hprototype
--
ZO model
(17)
Zo prototype
Agr. MeteoroL, 8 (1971) 203-222
AERODYNAMICS OF SHELTER BELTS
215
Consequently, in selecting a profile of the mean velocity distribution upstream of the shelter one determines the scale factor for the experiments. It should be noted that BLENK and TRIENES (1956) did not attempt to scale N~,GELI'S (1941) flOWS in this manner; they needed an arbitrary conversion factor of 0.5 for reducing the downwind distance to make their experimental data conform to the results of Niigeli. This is a clear indication that modeling was not properly accomplished. How the drag coefficient of a porous shelter is related to the approach velocity profile can at present only be surmised. It should be defined more suitably by using Ub as reference velocity: Cp
-
D 1/2 pu 2 h
(18)
in which Cp, D and Ub are unknowns whose interrelations are not known. The drag coefficient Cp depends on the porosity and also on the Reynolds number of the elements forming the shelter. For screens consisting of wires, some systematic experiments have been made to determine this dependency by stretching screens across the whole cross section of a wind tunnel and measuring the pressure drop, at a given approach velocity. Typical are the investigations of SCHUBAUER et al. (1950). In a natural shelter, it is usually not possible to specify the porosity and the sizes of the shelter elements, and all estimates of the drag must rely either on direct measurement of the flow near similar installations or on guesses. Such measurements must naturally be made at a time during the growing season which corresponds to that during which shelter is required by the crops, or else differences in foliage or shelter growth might be sufficient to render valueless any conclusions based on the experiment. THE BASE PRESSURE
The main difficulty in predicting drag forces on the shelter stems from our inability to determine the base pressure, i.e., the pressure at the back of the shelter. It is readily apparent from eq.10 that the larger the difference between pressure on the front and the base pressure, the larger the drag. The pressure on the front of the shelter is generally determined, for a solid shelter, by the flow pressure of the approaching flow on the face of the shelter, independent of the shape of the shelter. This is strictly true for a plate suspended in the free stream, with velocity u~, and pressure p~, of a wind tunnel. The pressure on the front in that case can be calculated from inviscid flow theory (ARIE and ROUSE, 1956) and the maximum pressure at the center of the plate is found to be exactly equal to the stagnation pressure Pst P~ + 1/2 pu 2 calculated from Bernoulli's equation. For a shelter immersed in a boundary layer, the approach flow is modified by the presence of the ground. This modification begins, according to surface pressure measurements of G o o d and JOUBERT (1968), along a smooth floor at a distance o f x = 15 (~/h)°'Th =
Agr. Meteorol., 8 (1971) 203-222
216
E.J. PLATE
upstream of the shelter where 6 is the boundary layer thickness; upon approaching the shelter the flow is retarded more strongly near the surface than at higher elevations and eventually separation occurs at a short distance in front of the shelter. The separation "bubble" does not extend over the full height of the shelter, and a stagnation points exists at about 0.5-0.7h from the ground at the face of the shelter. Below the stagnation streamline the flow is downward; a standing eddy forms in the corner between ground and shelter. Above this eddy the flow is accelerating towards the upper edge of the shelter. This flow field changes the pressure distribution from what would be expected had the ground not exerted any friction; in particular, the pressure maximum appears at the stagnation point and not at ground level. Clearly such a flow field does not exist in front of a porous shelter, because the porosity does not permit a pressure build-up in front that leads to separation. It is likely that at the rear of the screen the pressure is about constant, as for the solid fence case, and that the velocity distribution across the screens is fairly uniform. The latter is evident in some of the velocity profiles reported by BLENK and TRIENES (1956). A similar observation has also been made at the edge of a forest. A low level jet is often observed which extends over the first few tree rows and then settles down to a roughly uniform velocity over a substantial part of the tree height (REIFSNYDER,1955; MERONEY, 1968). This uniform velocity makes it likely that the pressure across the front of the shelters is also fairly uniform, as long as the shelter is of approximately constant porosity. In that case the pressure drop across the shelter is proportional to the square of the velocity through the shelter, with a factor of proportionality depending on the porosity only. The pressures in front and in the rear of the shelter are not independent because the air flowing over the top of the shelter provides a "leak" connecting the two pressures. However, the mechanism by which the pressure at the back side is governed is not yet understood. The prediction of the "base" pressure is a problem of considerable importance in aerodynamics and much work has been done on it, particularly for applications of supersonic aerodynamics (i.e., BROWN and STEWARTSON, 1969, for a recent survey). For low speed aerodynamics the problem appears to have aspects which complicate its solution even further, such as the turbulence in the blending region which is likely to be an important factor in the transmission of pressure gradients, or the smaller pressure differences between base pressure and free stream pressure, which leads to a curved separation streamline. The pressure field that arises is illustrated for the data of CHANG (1966) in Fig.2C, in which isobars have been indicated. It is noteworthy that substantial pressure gradients exist both in the longitudinal direction and in the vertical. These pressure gradients indicate that in the sheltered region it is not justified to simplify the Navier Stokes equations by using the boundary layer assumptions. Measurements along the ground have shown that the negative pressure downstream of the shelter is approximately constant over a distance of about Agr. MeteoroL, 8 (1971) 203-222
217
AERODYNAMICSOF SHELTERBELTS
3 times the height h and then starts rising very rapidly until, in the neighborhood of reattachment, it reaches a maximum which is slightly above the pressure in the free stream. Further downstream it slowly decreases and reaches asymptotically the free stream pressure, as is required by the boundary layer character of the redeveloping flow. This is illustrated in Fig.9. The figure was obtained by plotting non-dimensional pressure differences (Pwall -- Poo)/PU~ VS. x/h, and it is seen that the distributions are independent of velocity, and scale uniquely for a particular shelter.
-03
oC,:=.,~ " d ~ ) ~1""
-03
o
5
I0
ra
5
I0
9.1
5 5 5 5 5 5
I0 5 5 5 20 20
18.2 4.5 9.1 18.3 6.1 9. I
l O • • •
-0.4
o
v -0.5
0
I0
I
2O
50
40
50
5
60
20
70
4,5 --
_
12.2
80
90
X h
Fig.9. Pressure distributions along the ground. (After PLATE and LIN, 1965.)
The difference between ground and free stream pressure gives rise to a vertical force on the flow in the blending region which tends to push the separation streamline towards the ground. This is called the "Coanda" effect. The curvature of the streamlines, as expressed by the radius of curvature R, can be approximately inferred by applying the momentum conservation law to a small element out of the blending region, as indicated in Fig.10. Let the element have a velocity u which, by the definition of a streamline, is directed normal to the radius of curvature R, and let the pressure across the element change by an amount (Op/az)dz. The centrifugal force on the element then must be balanced by the resultant of pressure forces and shear forces on the front and rear faces of the element, or neglecting the latter, the momentum balance equation yields:
Agr. Meteorol., 8 (1971) 203-222
218
E.J. PLATE
0p
p + ~-~z +~....I..- .
7
/
J
/
L~Z
If~ t
~
~
-.,...~.
~
Strmm ne
R
Fig.10. Illustrating the Coanda effect.
1
1
dp
R
pu 2
69z
(19)
or in n o n - d i m e n s i o n a l form: h R
2 h x
Pwa. - P~ 1]2 pu 2
2
u~o
uz
~3p' 0rl
where p ' -
P - p~ Pwall -- Po~
-g(r/)
(20)
The pressure gradient function g(q) in eq.20 is p r o b a b l y d e p e n d i n g on the geo m e t r y only, a n d m a y be assumed to obey a similarity law d e p e n d i n g on the same similarity variable q as the mean velocity distribution. F o r the curve u = 0.5 u~, the function 2(uZ~/u z) (Op'/Or/) is a constant = ~ a n d the radius o f curvature becomes: h R
_ ~ Pw, n - Po~ 1/2 pu~
(21)
with a m a x i m u m value of: h R
h - ctQ--x
(22)
While this e q u a t i o n represents only an a p p r o x i m a t i o n , it shows that the larger the pressure difference Pw,. - P+ the smaller is the radius o f curvature. A solid fence, with a large base pressure coefficient yields a m o r e r a p i d turning o f the u = 0.5 u~ curve t o w a r d s the g r o u n d than a p o r o u s screen. The C o a n d a effect is at least in p a r t responsible for the r a p i d loss o f sheltering efficiency o f a
Agr. Meteorol., 8 (1971) 203-222
AERODYNAMICSOF SHELTERBELTS
219
solid screen. A porous screen, with its higher velocity in the blending region and its lower base pressure, exhibits significantly smaller streamline curvature. THE TURBULENCEIN THE SHELTEREDREGION More important than the Coanda effect on the wind field behind the shelter is the turbulence which is found in the air flow. The turbulence, whose vertical component is w' and whose longitudinal component is u', gives rise to turbulent shear stresses z = - p u ' w ' which not only adjust the mean velocity distribution but which are also responsible in producing turbulence. The strong mean velocity gradients generated by separation from the shelter edge interact with the shear stresses existing in the undisturbed flow to produce more turbulence for small values of Ub than for large values. To see this, consider the energy balance of the turbulent motion, integrated over a volume consisting of a slice of (infinitesimal) length Ax, of infinite vertical extent and of unit width: Ax
i-
dq 2 ~dz
= Ax
0
i
•
7
Off
~-z dz - D
(23)
0
where small terms have been neglected. The overbars denote time averages. In eq.23 q2 is the kinetic energy of the turbulent motion at a point:
q2 =
~(U,2 ..~ /),2 ..~ W,2)
(24)
and D is the dissipation of kinetic energy into heat in the volume. The term on the left is the rate of change of turbulent energy in the volume which for steady flows can be expressed by: dq 2 d - ~ - dz = ~ 0
-fiq2 dz
(25)
0
This term is determined mostly by the first term on the right in eq.23 which represents the turbulence productions, which in the stages of growth of the separation layer exceeds considerably the dissipation. By inserting eq.1, 2, into the production term of eq.23, and by neglecting the dissipation term as compared with the production, it follows readily that in the blending region the turbulence increases at a rate proportional to (u 2 - u2)(u~o - Ub) and independent of x, until the presence of the ground modifies the turbulence structure. Consequently, a more solid fence has associated with it a larger turbulence level, i.e., the total amount of turbulence downstream of a porous screen is lower than that behind a solid screen. This will certainly affect the turbulence transport processes in the sheltered region, thus explaining why higher evaporation rates are observed behind a solid screen than behind a porous screen, as was observed by BLENKand TRIENES Agr. Meteorol., 8 (1971) 203-222
220
E.J. PLATE
(1956). However, perhaps a more important factor in determining exchange processes behind a solid screen is the return flow in the separation bubble which has associated with it strong vertical currents. These currents do not exist behind porous screens, so that for either of these reasons if wind protection is the most important task of the shelter belt, a porous screen has great advantages over a solid one. SUMMARY AND CONCLUSIONS
The discussion of this paper has shown that the information required for an aerodynamically most efficient shelter belt design is not yet available. But it has become clear what information is needed, and how it can be obtained. First, one requires a knowledge of the velocity profile at the location of the shelter before the shelter is to be planted or constructed. This information is readily available, as a good estimation of the u. and z o values of the undisturbed boundary layer can be had by either taking some local measurements or by consulting the existing literature on the velocity distribution near natural surfaces, such as GElGER (1965, p.275). But for going beyond this, one must make a number of assumptions which are backed by experimental evidence only for a solid shelter belt. A drag coefficient and a base pressure coefficient must be determined to predict the location of the separation streamline, and a velocity Ub must be assumed. Then it is possible to calculate the height h* from eq.9 to a first approximation, and to calculate the blending velocity profile by using eq.3, perhaps with tr the solid screen value. To fill in the gaps in our knowledge, it is evidently necessary to re-analyze existing wind tunnel data and field data, and to perform parallel experiments in wind tunnels and in the field. The materials of which natural shelters are made preclude the use of wind tunnels for the direct design of an optimum shelter. Since leaf area, growth rate, density and shape of shelter belts vary too much to be predictable in advance, the greatest use of a wind tunnel may be found in one of the two following areas. The wind tunnel can be useful in evaluating the efficiency and arrangement of wind breaks made of non-random materials, which may consist of regular arrangements of slats, reed, bamboo or other material of similar kind. These can be studied effectively in a wind tunnel because their geometry is reproducible, and their drag characteristics can be determined in the same manner as was described for screens. It appears feasible to investigate in this manner the optimum design of a wind break with respect to cost and efficiency. Some experiments by Japanese writers (SATO et al., 1952) along these lines have already been performed. It is, however, necessary during these experiments to give due regard to the approach velocity profile, its roughness height Zo and its shear velocity u. as well as to the other aerodynamic factors of the surroundings which might affect the sheltering. Agr. MeteoroL, 8 (1971) 203-222
AERODYNAMICSOF SHELTERBELTS
221
The second application is to the evaluation of the effect of terrain modifications in the neighborhood of the shelter on existing shelter belts. Field experiments yield the necessary information on the approach wind distribution and on the sheltering from which a model of the undisturbed situation can be made. As was pointed out earlier, geometric similarity might suffice for this purpose in addition to scaling of the z0-values by the scale of the shelter model. I f the experiments yield significantly different sheltering coefficients, improvements can be made by trial and error, by blocking or widening the interstices between the shelter elements. Design of new shelters and classification of existing shelters may be aided by some fundamental research which would improve our understanding of the aerodynamic effects of shelter belts. From the discussion in previous sections it is clear that the effect of shelter shape and porosity on the degree of sheltering is incompletely understood, as is the effect of ground roughness on the drag of the shelter. To this one may add systematic investigations of the effect of finite width shelters and of the effect of orientations not perpendicular to the wind. A study presently under way in the Fluid Dynamics and Diffusion Laboratory of Colorado State University is concerned with the modification of wind field by multiple shelters and with their optimum spacing. ACKNOWLEDGEMENTS The experiments on which much of this paper is based were performed by graduate students of the author while he was at Colorado State University. This work was performed under the auspices of the U.S. Atomic Energy Commission. REFERENCES ARE, M. and RousE, H., 1956. Experiments on two-dimensional flow over a normal wall. J. Fluid Mechanics, 1: 129-141. BLENK, H. and TRIne.s, H., 1956. Str6mungstechnische Beitr~ige zum Windschutz. In: Grundlagen der Landtechnik, 8. V.D.I. Verlag, Diisseldorf, 65 pp. BROWN, S. N. and STEWARTSON,K., 1969. Laminar Separation. In" N. SEARS(Editor), Annual Reviews of Fluid Mechanics, 1. Annual Reviews, Inc., Palo Alta, Calif., pp.45-72. CHANG, S. C., 1966. Velocity Distributions in the Separated Flow behind a Wedge-shaped Model Hill. Thesis, Colorado State Univ., Denver, Colo., 101 pp., unpublished. GEIGER,R., 1961. Das Klima der bodennahen Luftschicht. Vieweg, Braunschweig, 4th ed., 646 pp. GOOD, M. C. and JOtmERT,P. C., 1968. The form drag of two-dimensional bluff-plates immersed in turbulent boundary layers. J. Fluid Mechanics, 31: 547-582. HAUrZKY, J., 1968. Gas diffusion near buildings. In: D. H. SLADE(Editor), Meteorology and Atomic Energy, 1968. U.S. Atomic Energy Commission, Division of Technical Information Extension, Oak Ridge, Tenn., pp.221-255. JENSEN, M., 1954. Shelter Effect. Danish Technical Press, Copenhagen, 264 pp. KAISER,H., 1959. Die Str6mung an Windschutzstreifen. Ber. Deut. Wetterdienstes, 7(53): 36 pp. MEROSrY, R. N., 1968. Characteristics of wind and turbulence in and above model forests. J. Appl. Meteorol., 7(5): 780--788. Agr. Meteorol., 8 (1971) 203-222
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z . J . PLATE
MUELLER, T. J., KORST, H. H. and CHOW, W. L., 1963. On the separation, reattachment, and redevelopment of incompressible turbulent shear flow. J. Basic Eng., 63-AHGT-5, 86: 221-226. NAGABHUSHANAIAH,H. S., 1961. Separation Flow Downstream o f a Plate Set Normal to a Plane Boundary. Thesis, Colorado State University, Denver, Colo., 150 pp., (unpublished). N.~GELI, W., 1941. Untersuchungen tiber die Windverh~iltnisse im Bereich yon Windschutzstreifen. Mitt. Schweiz. Anst. Forst. Versuchswesen, 23: 221-276. PLATE, E. J., 1964. The drag on a smooth flat plate with a fence immersed in its turbulent boundary layer. A S M E Paper 64-FE-17:12 pp. PLATE, E. J., 1967. Diffusion from a ground level line source into the disturbed boundary layer far downstream from a fence. Intern. J. Heat Mass Transfer, 10: 181-194. PLATE, E. J. and LIN, C. Y., 1965. The Velocity Field Downstream from a Two-dimensional Model Hill. (Final report, Part I, U.S. Army Material Agency, Contract DA-AMC-36-039-63-G7, CER65EJP-CYL14, 75 pp). REICHARDT, H., 1945. Die Gesetzmiissigheiten yon Kavitationsblasen an umstrdmten Rotationskdrpern. Max Planck Institut for Str6mungsforschung, GOttingen, UM6628, 34 pp. REIFSNVDER, W. E . 1955. Wind profiles in a small isolated forest stand. Forest Sci., I: 289-297. SATO, K., TAMACHI,M., TERANDA,K., WATANABE,Y., KATOH, T., SAKANONE,Y., and IWASAKI, M., 1952. Studies on Wind Breaks. Nippon Gakujutsu-Shiukokai, Tokyo, 201 pp. ScnLiCrmN~, H., 1968. Boundary Layer Theory. McGraw-Hill, New York, N.Y., 747 pp. SCHUBAUER,G. B., SPANGENBER~,W. G. and KLEBANOFr,P., 1950. Aerodynamics of damping screens. Natl. Advisory Comm. Aeron., Tech. Note, 2001:16 pp. TOWNSEND, A. A., 1956. The Structure o f Turbulent Shear Flow. Cambridge University Press, London, 317 pp. UCmOA, S. and SOZUKI, T., 1968. On a similar solution for a turbulent half-jet along a curved streamline. J. Fluid Mechanics, 33: 379-398. VAN DER LINDE, J., 1962. Trees outside the forest. In: Forest Influences. F.A.O., Rome, pp. 141-208.
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