Wind velocity profiles and shear stresses on a lake downwind from a canopy: Interpretation of three experiments in a wind tunnel

Transcription

1 ST. ANTHONY FALLS LABORATORY Engineering, Environmental and Geophysical Fluid Dynamics Project Report No. 493 Wind velocity profiles and shear stresses on a lake downwind from a canopy: Interpretation of three experiments in a wind tunnel by Dane A. Jaster, Angel L.S. Pere, Fernando Porte-Agel and Hein G. Stefan Prepared for National Center for Earth Surface Dynamics May 2007 Minneapolis, Minnesota 1

2 The niversity of Minnesota is committed to the policy that all persons shall have equal access to its programs, facilities, and employment without regard to race, religion, color, sex, national origin, handicap, age or veteran status. 2

3 Abstract We are interested in the shear stresses exerted by wind on a lake surface, especially if a lake has a small surface area. We have therefore begun to study the development of the atmospheric boundary layer over a small lake surrounded by a vegetation canopy of trees or cattails. Wind tunnel experiments have been performed to simulate the transition from a canopy to a flat solid surface. These experiments and the data collected are described in SAFL Report 492. In the first experiment we used several layers of chicken wire with a total height of 5cm, a porosity of 98% and a length of 2.4m (8 ft) in flow direction to represent the vegetation canopy, and the floor of the wind tunnel consisting of plywood was used to represent the lake. The chicken wire represents a porous step that ends at x=0. This experimental setup was considered to be a crude representation of a canopy of trees or other vegetation that ends at the shore of a lake. In a second experiment we used an array of pipe cleaners inserted in a styrofoam board to represent the canopy. The porosity of that canopy was 78%. In a third experiment we used a solid step with a smooth surface which could be a simplified representation of a high bank or buildings on the upwind side of a lake. Wind velocity profiles were measured downstream from the end of the canopy or step at distances up to x=7m. sing the velocity profile at x=0, the absolute roughness of the two canopies was determined to be 1.3 cm and 0.5 cm, respectively, and the displacement height was determined to be 2.3cm and 6.68cm. The roughness of the wind tunnel floor downstream from the canopy was determined to be m =0.03mm. Three distinct layers were identified in the measured velocity profiles downstream from the canopy: the surface layer in response to the shear on the wind tunnel floor, an outer layer far above the canopy, and a mixing/blending layer in between. With sufficient distance downwind from the canopy the mixing layer should disappear, and the well-known logarithmic velocity profile should form. The shear stress on the surface downwind from the canopy was unaffected by wind sheltering after x/h=100, and the effect was less than 10% after x/h=60. A separated flow region formed downstream of each of the three canopies. The distance to reattachment was about 8 times the displacement height in the canopy. After a distance x/h=25 an internal boundary layer could be identified. It was characteried by rising shear stresses with distance from the canopy. Between x/d=8 and x/h=25 a turbulent shear layer touches down on the surface. Shear stresses in this range are highly variable, depending on canopy roughness and porosity. The velocities profiles downwind from the canopy are shaped by two attributes: the canopy roughness ( 0 ) and the canopy porosity: the velocity profile at the end of the canopy is given by the canopy roughness, while the velocity profiles downwind from the canopy are shaped by both roughness and the porosity of the canopy. Wind velocity profiles took a much longer distance than x/h=100 to overcome the canopy effect. This leads to the conclusion that surface shear stresses make a much faster transition than velocity profiles downstream from the canopy. In other words, momentum transfer is faster than mass transfer downstream from the canopy. 3

4 Table of contents 1. Introduction 5 2. Wind tunnel experiment Velocity data Aerodynamic characteriation of the wind velocity profiles Aerodynamic characteriation of the experimental canopies Velocity deficit Shear layers Apparent shear velocity and roughness downwind from the canopy Flow regions Region (1) Separated flow region Turbulent shear layer between flow over the canopy and separated flow Turbulent shear layer: Region (2) Internal boundary layer: Region (3) Shear stress on flat surface downwind from canopy Application of wind tunnel results to a lake Summary and conclusions References...52 Appendix I...58 Appendix II...62 Appendix III.63. 4

5 1. Introduction The surface mixed layer (SML) of a lake is the most dynamic one within the limnological system. Forced convection induced by wind stress on the water surface, and free thermal convection induced by heat loss from the water to the atmosphere are the main processes that generate SML turbulence and control the transport of momentum, heat, and mass (gas) across the air/water interface. Surface layer turbulent mixing, and its effects on stratification, is crucial for many ecological, geochemical, and physical processes that occur in lakes. Despite of numerous studies, wind stress especially on small lake surfaces is still not well understood (Wüest and Lorke 2003). Direct observations of wind over or near a lake are not only uncommon but also questionable because of uncertainty about the spatial inhomogeneity of the instantaneous and time averaged wind field. More common are measurements at local weather stations, typically several kilometers from a lake. The lack of observation necessitates that numerical lake water quality models calibrate wind speed by a wind sheltering coefficient (Hondo and Stefan, 1993). The calibrated wind speed is unlikely to represent wind speed at a particular point in space and time; rather the calibration provides an areally and temporally averaged wind speed that minimies the mean residual between the model results and data. While practical for ero or one-dimensional lake models, this approach neglects meaningful and significant details of non-uniform wind speed distributions over lakes that are likely to be important for physically realistic threedimensional lake models (e.g. Edinger 2001, Wang and Hutter 1998; Wang, Hutter, and Bäuerle 2001; Wang 2003). Momentum transfer at a lake surface The transport of momentum across the air-water interface generally occurs from the atmosphere to the water. As the wind impinges on the water surface, the 5

6 water surface generates a drag on the wind, slowing the wind nearest the water surface. The wind shear stress can be parameteried as 2 2 τ = [ / L ] C D ρ a F (1.2) In the previous equation, C D is an empirical drag coefficient, ρ a is the density of air, and is the time averaged streamwise wind velocity at some specified height above the water surface. The drag coefficient is a function of wave roughness, which is a rather complicated function of both wind speed and the state of wave development or wave age (Wüest and Lorke 2003). The wave age A W is simply the ratio of wave phase speed to either the wind velocity or to roughly 30 times the air shear velocity *. By this association of C D with A W and with, a relationship between C D and 10 (wind speed measured at 10 m height) can be established. In small and medium lakes, wind speed is generally low, i.e. < 5 m/s, representing the greatest variability in drag coefficients. Because of sheltering and limited fetch, wave fields on lakes are often immature. In addition to spatial wind speed variability, these interactions are the primary complications in determining momentum transfer from the atmosphere into the lake surface boundary layer. In this paper, we present results from a wind tunnel experiment designed to provide some information on the transition from a vegetation canopy to a water surface (Figure 1.1). The purpose of the experiment was to measure and analye the velocity profiles and surface shear stresses downstream from the canopy. 6

7 Figure 1.1. Schematic view of a tree canopy around a lake 2. Wind tunnel experiments Three experiments were conducted to study the development of the atmospheric boundary layer in the transition from a vegetation canopy of trees or cattails to a lake surface. The boundary layer wind tunnel of the Saint Anthony Falls Laboratory at the niversity of Minnesota was used; its test section is 1.7 m wide, 1.8 m high, and 16 m long (Figure 2.1). More details on the wind tunnel characteristics are given by Farell and Iyengar (1999). In this study, the tunnel was operated in closed circuit and the average wind velocity was on the order of 10 to 15 m/s. In the first experiment the canopy was simulated by several layers of chicken wire placed over the wind tunnel floor. The porosity of the wire mesh determined from its weight and volume was 98.0 %. This model canopy covered the total 7

8 width of the wind tunnel and extended over a length of 1.2m (4 ft) in the flow direction. The canopy had a height of approximately 5 cm (Figure 2.2). The chicken wire represents a porous step that ends at x=0. This experimental set-up was considered to be a crude representation of a canopy of tress or other vegetation that ends at the shore of a lake. The floor of the wind tunnel consisted of painted plywood representing a smooth surface. The experiment was run at an air temperature of 28 o C. The Reynolds number based on model canopy height of 0.05 m was therefore approximately Re = (15)(0.05)(67000) = Based on wind tunnel height of H = 1.8 m it would have been Re = A Pitot tube connected to a precision differential manometer.was used to measure the time-averaged wind velocity at different positions in the boundary layer. The Pitot tube had 3mm outer diameter and was mounted on a traversing mechanism that allowed vertical positioning above the wind tunnel floor. Measurements were made along the centerline of the wind tunnel test section, i.e. 0.85m from the lateral walls. At each position of the Pitot tube an average velocity over seconds was recorded, and stored in the data base used in the analysis. In particular, vertical profiles of wind velocity were collected over the chicken wire mesh near the transition and also at seven different positions downwind of the transition (x=0.2 m, 0.4 m, 0.6 m, 1.1 m, 1.8 m, 2.76 m and 5.4 m). For the second experiment a model canopy was created from pipe cleaners. Pipe cleaners were inserted into a foam board of 2.5cm thickness (Figures 2.3 8

9 and 2.4), and the board was placed on the wind tunnel floor. It covered the total width of the wind tunnel and extended over a length of 2m in flow direction. Foam board without pipe cleaners was installed on the wind tunnel floor downstream from the canopy and represented an aerodynamically smooth surface. The canopy made of pipe cleaners had a height of approximately 7.5 cm and a porosity of about 78%. The experiment was run at an air temperature of 28 C. Wind velocity profiles were measured at the end of the canopy (x=0) and at nine positions downwind of the canopy (x = 0.2 m, 0.4 m, 0.6 m, 1.1 m, 2.0 m, 3.25 m, 4.07 m, 5.94 m and 6.68 m). The wind tunnel setting was 10 m/s. Reynolds number based on model canopy height of m was therefore approximately Re = (10)(0.075)(67000) = In a third experiment we used a solid step which could be a representation of a high bank or buildings on the upwind side of the lake. The step was created from styrofoam boards (Figure 2.5). It covered the total width of the wind tunnel and extended over a length of 2m in flow direction. Foam board was also installed on the wind tunnel floor downstream from the canopy and represented an aerodynamically smooth surface. The wind tunnel velocity setting was 10 m/s. The solid step had a height of approximately 5.0 cm and 0.0% porosity. The experiment was run at an air temperature of 27 C. The Reynolds number based on the model canopy height of m was approximately Re = Based on the wind tunnel height of H = 1.8 m it would have been Re = Wind velocity profiles were measured at the end of the canopy (x=0) and at 9

10 nine positions downwind of the canopy (x = 0.2 m, 0.4 m, 0.6 m, 1.1 m, 2.0 m, 2.76 m, 4.07 m, 5.94 m and 6.68 m). Canopy characteristics for all three experiments are summaried in Table 2.1. From wind velocity profiles measured at the end of the canopy or step (x=0) the absolute roughnesses of the canopies were determined to be 1.3 cm for the chicken wire, 0.5 cm for the pipe cleaners, and 0.001cm for the styrofoam step, respectively. The displacement heights were determined to be 2.3 cm and 6.6 cm and 5.08cm for the three canopy configurations. The roughness of the wind tunnel floor downstream from the canopy was determined to be 0.001cm. The measured independent variables for the three experiments are summaried in Table 2.1. In addition to canopy porosity and height they include the wind tunnel velocity setting and the reference velocity for each experiment. The displacement height in the canopy, the roughness of the canopy and the roughness of the wind tunnel floor are derived from the measured velocity profiles. Table 2.1 Measured independent variables for the three experiments Canopy geometry Canopy porosity 98% 78% 0% Height of canopy h (m) Displacement height in canopy d (m)

11 Relative displacement height d / h Velocities and shear stresses Wind tunnel velocity setting (m/s) Reference wind velocity r (m/s)* Canopy shear velocity * c (m/s) Relative canopy shear vel. * c / r Shear stress on canopy τ c (N/m 2 ) Roughnesses Absolute roughness of canopy 0c (m) Relative roughness of canopy 0c / h Absolute roughness of wind tunnel floor w/o canopy 0s (m)** Relative roughness of wind tunnel floor w/o canopy 0s / h Roughness ratio: canopy to wind tunnel floor 0c / 0s * r is the velocity measured at the end of the canopy (x=0) at a height =6h above the wind tunnel floor. (h=height of the canopy) 11

12 Figure 2.1: Test section of the St. Anthony Falls Laboratory wind tunnel (looking upwind) Figure 2.2: Canopy represented by chicken wire mesh, 98% porosity, 5cm height 12

13 Figure 2.3 Wind tunnel test section with canopy made of pipe cleaners Figure 2.4: Close-up view of canopy made of pipe cleaners 13

14 Figure 2.5: Test section with the foam boards looking downstream. 3. Velocity data In Figure 3.1 are the plots of the measured velocity profiles downstream from the canopy. The data are plotted against distance from the wind tunnel floor using linear coordinates. The data have been normalied to canopy height h for vertical distance and reference velocity r (measured at x=0 and =6h). The canopy height is a very logical choice of a reference length. The reference velocity is a more arbitrary choice, which was made after consideration of several downwind and upwind locations. It was considered the best possible choice as a free stream velocity. 14

15 3.1 Aerodynamic characteriation of wind velocity profiles without a canopy. Before the data in Figure 3.1 are analyed a velocity profile obtained in the wind tunnel without a canopy is shown in Figure 3.2 for reference. This wind velocity profile without a canopy is a fully developed turbulent velocity profile as shown by the linearity of the semi-logarithmic plot in Figure 3.3. The data in this plot can be fitted to the Prandtl-von Karman equation * 1 = ln κ 0 (3.1) In this equation, () is the wind velocity, * is the shear velocity, κ is the von Karman constant (=0.4), is the height, and 0 is the aerodynamic roughness height. Equation (3.1) can be rewritten as k k * ( ln ) ln( ) * = 0 (3.2) Equation 3.2 resembles a straight line equation given by the function y = ax + b (3.3) where the dependent variable y =, and the independent variable x = ln (). As is well-known one can find * from the slope of the straight line drawn through the data on a semi-logarithmic plot, and 0 by extrapolating the straight line to the height = 0 where =0. The straight line and the extrapolation are shown in Figure 3.3. From this plot the absolute roughness of the wind tunnel floor 0 was determined to be roughly m = 0.03mm. This value of 0 is consistent with previous data collected in the same wind tunnel. It should be recalled that the wind tunnel floor is made of plywood and, in our experiments, represents the water surface of the lake which will have a different roughness. 15

16 3.2 Aerodynamic characteriation of the experimental canopies The aerodynamic characteriation of the canopy is essential to our efforts. The top of the canopy is not a solid wall. The wind therefore penetrates some distance into the canopy. A description of this process was given by Finnegan (2000), and some information can also be found in Stull (1988) and Garratt (1992). The turbulent velocity profile above a canopy can be fitted to the equation * 1 d = ln κ 0 (3.4) Equation (3.4) contains the canopy roughness ( 0 = 0c ), a displacement height (d) and a shear velocity (*=* c ) as parameters. All three can be estimated by using the classical semi-log plot of a velocity profile measured across the canopy. We have used the velocity profiles measured at the edge (x=0) of the canopy (Figure 3.4) to determine all three parameters. The displacement distance (d) was determined by an iterative process. A plot of velocity () vs. (-d) on a semi-log graph was made for different values of d. If the selected (d) was too small, the plot curved upwards, and if (d) was too large, the plot curved downwards. Iterations were completed until the data plot showed no curvature (see Fig. 9.8 in Boundary Layer Meteorology by Stull, 1988). The displacement height (d) was found to be m for the chicken wire canopy, and m for the pipe cleaner canopy. For the former this displacement height is about 46% of the canopy height h=0.05m, and for the latter it is about 88% of the canopy height h= 0.075m. An estimate d = 2/3h is often used for real vegetation canopies consisting of trees or other plants. 16

17 For the canopy with 0% porosity the displacement height was d=h. Once (d) had been determined, the canopy roughness ( 0i ) could be obtained by extrapolation of the linear fitted plot of vs. ln(-d) to ero velocity = 0 (Fig. 3.5). The canopy roughness was determined to be approximately m for the chicken wire and 0.005m for the pipe cleaner and m for the solid step. By comparison the roughness of uncut grass in the real world is about 0.01m (see Fig 9.6 in Boundary Layer Meteorology by Stull, 1988). The shear velocity (*) at x=0 was computed using Equation 3.5. It is proportional to the slope of the line in Figure 3.5. The shear velocity was found to be 1.73 m/s for the chicken wire canopy, 1.00 m/s for the pipe cleaner canopy and 0.48 m/s for the solid step. d = * ln κ 0 (3.5) Results from the analysis of the canopy data (x=0) for the three experiments are summaried in Table 2.1. The shear stress on the canopy τ c was calculated using the shear velocity c * from the plot in Figure 3.5. τ c = ρ c * 2 was calculated with ρ=1.28kg/m 3 and is reported in Table 2.1. The roughness of the canopies differs vastly. The chicken wire canopy is 1300 times rougher than the wind tunnel floor; the pipe cleaner canopy is 500 times as rough. The solid step has about the same roughness as the wind tunnel floor. The relative penetration of the wind into the porous canopy also differs widely. d/h=0.46 indicates a penetration to more than half the canopy height; d/h =0.88 indicates 17

18 penetration to 12% of the canopy height; and d/h=1 designates no penetration (solid step). 3.3 Velocity deficit All wind velocity profiles in Figure 3.1 show a deficit relative to the wind velocity profile without a canopy (Figure 3.2). The roughest canopy has a much larger velocity deficit than the smoothest canopy. A direct comparison of velocity profiles at the end of the canopy (x=0), and further downwind (x/h=55 and x/h=90) is shown in Figure 3.4. Even at a distance as large as x/h=90, the effect of the canopies is still evident in the velocity profiles. There is still a measurable velocity deficit downwind from the two very rough and porous canopies. Downwind from the step (0% canopy porosity) the velocity profile at x/h=90 has has metamorphosed almost completely into on typical of the wind tunnel without a canopy (Figure 3.4). In Figure 3.4 the velocity profile for the non-porous canopy (step) is distinctly different from those for the two porous canopies. 18

19 /h Distance from Canopy 0.0m 0.2m 0.4m 0.6m 1.1m 1.8m 2.76m 5.4m / r Canopy porosity = 98% /h Distance from Canopy 0.0m 0.2m 0.4m 0.6m 1.1m 2.0m 3.25m 4.07m 5.94m 6.68m / r Canopy porosity = 78% 19

20 /h Distance from Canopy 0.0m 0.2m 0.4m 0.6m 1.1m 2.0m 2.76m 4.07m 5.94m 6.68m ` / r Canopy porosity = 0% Figure 3.1 Normalied velocity profiles downstream from the canopy /h / r Figure 3.2 Normalied velocity profile in wind tunnel with no canopy. h=5cm was used, and r was measured at =6h 20

21 10 1 /h / r Figure 3.3 Semi-log plot of the normalied velocity profile in the wind tunnel with no canopy. A height h=5cm and a velocity r measured at a height of =6h were used for reference 21

22 Porosity 98% 78% 0% No canopy /h / r x/h = 0 3 Porosity 98% 2 78% 0% /h / r x/h = 0 22

23 Porosity 98% 78% 0% No canopy /h / r x/h = Porosity 98% 78% 0% No canopy /h / r x/h = 90 Figure 3.4 Velocity profiles at three distances from the canopy 23

24 0.10 -d (m) Velocity (m/s) Figure 3.5 Determination of canopy roughness ( 0c ) for the 98% porosity canopy 3.4 Shear layers To analye the measured velocity profiles downstream from the canopy semilogarithmic plots of the data were made (Figure 3.6). All of these plots have a linear portion at their lower end. These portions are representative of a turbulent shear layer as is illustrated in Figure 3.7 for distances of x/h=55 and x/h=90. The linear portion of each semi-log velocity profile can be fitted to the Prandtl-von Karman equation 24

25 * 1 = ln κ 0 (3.1) In this equation, () is the wind velocity, * is the shear velocity, κ is the von Karman constant (=0.4), is the height, and 0 is the aerodynamic roughness height. The above equation can be rewritten as k k * ( ln ) ln( ) * = 0 (3.2) Equation 4.2 resembles a straight line equation given by the function y = ax + b (3.3) where the dependent variable y =, and the independent variable x = ln (). As is well-known one can find * from the slope of the straight line drawn through the data on a semi-logarithmic velocity profile plot, and 0 by extrapolating the straight line to the height = 0 where =0. The straight lines and their extrapolations for the velocity profiles for all three canopies are shown in Figure 3.6. Absolute roughness varies from 0 =0.001mm to 0 =0.1mm, and the average of 0 =0.03mm is on the same order as 0 for the wind tunnel floor without a canopy (Figures 3.2 and 3.3). Each slope * and intercept 0 determined from a semi-log velocity profile give an apparent shear velocity and an apparent roughness that satisfy equation (3.1). It is not implied that the assumptions that underly equation (3.1) are satisfied. For further analysis the wind velocity profiles were divided into three layers: (1) a surface layer downwind from the canopy and adjacent to the wind tunnel floor, (2) a blending (mixing) layer and (3) an outer layer. The surface layer is affected first by the characteristics of the canopy, and by the roughness of the surface downwind from the canopy. The outer layer is far above the canopy. The blending layer is between the outer layer and the surface layer. In the blending layer momentum is transferred from the outer layer to the surface layer. This 25

26 vertical subdivision of the wind field was based on the Cartesian and the semi-log velocity profile plots in Figures 3.1 and (m) Distance from Canopy 0.2m 0.4m 0.6m 1.1m 1.8m 2.76m 5.4m Velocity (m/s) Canopy porosity = 98% Distance from Canopy 0.4m (m) m 1.1m 2.0m 3.25m 4.07m 5.94m Velocity (m/s) 6.68m 26

27 Canopy porosity = 78% 1 (m) Distance from Canopy 0.4m 0.6m 1.1m 2.0m 2.76m 4.07m 5.94m 6.68m Velocity (m/s) Canopy porosity = 0% Figure 3.6 Semi-log plot of normalied velocity profiles downstream from canopy 27

28 10 1 /h Porosity % 78% 0% No canopy / r Distance x/h=55 28

29 10 /h Porosity 98% 78% 0% No canopy / r Distance x/h>90 Figure 3.7 Semi-log plots of velocity profiles at distances x/h=55 and x/h>90 downwind from the canopy, and a velocity profile with no canopy The apparent roughness 0 is plotted as a function of distance (x) downstream from the canopy in Figure 3.8. Apparent shear velocity is plotted vs. distance in Figure

30 Porosity 98% 78% 0% Z 0 /h x/h Figure 3.8 Apparent absolute roughness 0 vs. distance from canopy 30

31 */* r Porsity 98% 78% 0% x/h */* r Porosity 98% 78% 0% x/h Figure 3.9 Normalied apparent shear velocity vs. distance. Values in the transition from 0<x/h<20 have been omitted in the lower plot. 31

32 4. Apparent shear velocity and roughness downwind from the canopy 4.1 Flow regions The velocity profiles immediately downwind from the canopy are affected first by the shear stress between the high speed flow over the canopy and the stagnant air behind the canopy. Further downwind from the canopy the wind tunnel surface begins to exert a shear stress on the air flowing above it, and in response a surface (boundary) layer is expected to develop. There seems to be a general pattern in the distribution of slopes and intercepts in Figure 3.6, suggesting that the apparent * and 0 are correlated with each other in a way that depends on distance. Going from the end of the canopy in downstream direction, three regions of semi-log velocity profiles can be distinguished: Region (1) begins at the edge of the canopy (x=0), the slopes are very flat representing a very high * and 0. Over a short distance immediately downstream from the canopy, the slopes steepen to an approximately constant value representing a constant *. In Region (2) there are about 3 or 4 semi-log velocity profiles with constant slope (constant *) but decreasing intercept (decreasing 0 ) with distance. A minimum 0 is reached. Region (3) is beyond region (2). The slopes flatten (higher *) and the intercept Z 0 eventually assumes a constant value (constant 0 slightly higher than the minimum 0 ). 32

33 Further analysis of the three regions is necessary to obtain the surface shear stress distribution downstream from the canopy. One needs to ask whether there is a separated flow downwind from the canopy, and where the internal boundary layer really begins. 4.2 Region (1) Region (1) has two parts stacked on top of each other: On the bottom is a separated flow region, and on top is a turbulent shear layer between the high velocity flow over the canopy, and slower air in the separated flow region Separated flow region For all three canopies, the velocity profiles at x=0 (Figure 3.1) show flow separation. The height at which separation occurs is different for each canopy, and an apparent function of canopy porosity. The height at which the Cartesian plot of the velocity profile shows flow separation (=0) is listed in Table 4.1, and can be seen in Figure 3.4 for x/h=0. This height is a rough guess because the Pitot tubes do not work well at low velocity. The height of flow separation should be close, if not identical to the displacement height d in Table 2.1, - and it is. The separated flow region extends over a distance that can only be guessed from the velocity profiles in Figure 3.1. Estimates of the distance to re-attachment are given in Table 4.1. They are normalied to the displacement height d, not the canopy height h. The 98% porosity canopy has only a very short separated flow region. It ends before x/d=8. The shear velocity * for the velocity profile at 33

34 x=0 does not capture this very small separated region because the velocities closest to the wind tunnel floor at x=0 are too low to be measured with any accuracy. The separated flow region is shown as E in Figure 4.1, which is a modified version of a schematic flow behind dunes by Walker and Nickling (2002). Figure 4.1 Conceptual model of the transitional flow regions downwind from a transverse porous canopy and a flat terrain. A = outer flow, C = blending layer, E = separated flow, F = turbulent shear layer, H = turbulent shear one/ blending layer, I = internal boundary layer (modified from Walker and Nickling, 2002) Turbulent shear layer between flow over the canopy and separated flow Immediately after the end of the canopy, i.e. near x/h=0, one would expect a high shear stress (shear velocity) between the wind blowing over the canopy, and the 34

35 stagnant air behind the canopy. Indeed, slopes of the semi-logarithmic velocity plots (Figure 3.6) give high apparent shear velocities * at x=0 and x=4h (Table 4.1). A turbulent shear layer between the air flow over the top of the canopy and the stagnant air behind the canopy starts at the end of the canopy x=0 and grows vertically both downwards and upwards with distance from the canopy. This turbulent shear layer is on top of the separated flow below. Where the separated flow region ends, the turbulent shear layer reaches the wind tunnel floor. In this turbulent shear layer close to the canopy, the flat slopes of the semi-log velocity profiles (Fig. 3.5) give not only high shear velocities, but also high apparent roughnesses 0. Both a carry-overs from the turbulence in the porous canopy rather than a measure of floor roughness of the wind tunnel. At distances as short as x/h=4 and x/h=8 from the canopy, the wind tunnel floor has not had time to act on the air above it because of the presence of a separated flow cell. The linear portions of the semi-logarithmic velocity profiles can only represent a turbulent shear or blending layer between the fast air stream above the canopy and the stagnant air behind the canopy. It may even be questioned whether a semi-log velocity profile should be applied to the immediate transition region downwind from a canopy. 4.3 Region (2): Turbulent shear layer: Region (2) is characteried by a more or less constant shear velocity *. It is the continuation of the turbulent shear layer discussed above (Zone F in Figure 35

36 4.1). The constant * (constant shear stress) in region (2) is paired with an apparent roughness 0 that diminishes strongly with distance from the canopy. The downstream extent (x/h) of Region (2) and the minimum 0 are given in Table 4.1. Region (2) extends over a distance from roughly x/h=8 up to roughly x/h=25. Although the linear fit to the semi-log plots in Region (2) is pretty good (Figure 3.5), it can be noted for all three canopies that the last two or three points at the lower end of the semi-log plot seem to give a flatter slope (larger *) than the overall fitted line. The slope of the flatter line is very similar to the slopes (*) that are obtained in Region (3). This could mean that the shear velocities (shear stresses) on the flat surface of Region (2) are actually higher than we have reported. Data with higher spatial resolution need to be collected to test this observation. 4.4 Region (3): Internal boundary layer Region (3) extends beyond x/h=25 and is characteried by increasing, albeit slowly, shear velocity * (surface shear) and absolute roughness 0. In Figure 4.1 region (3) is identified as the internal boundary layer I, which may actually start in region (2). As the internal boundary layer grows thicker from the bottom up, the turbulent shear one H, which is the continuation of F, is reduced in height. 36

37 Table 4.1 Characteristics of flow regions downwind of canopy Canopy porosity 98% 78% 0% Height (m) where flow separation occurs at x=0 Height /h where flow separation occurs at x=0 Distance x (m) where re-attachm. to floor has occur. Distance x / d where re-attachm to floor has occur. Const. min. shear velocity *(m/s) in turb. shear layer Norm. const. min. shear vel. */ r in turb. shear layer Beginning x/h of turb.shear layer (constant *) Ending x/h of turb.shear layer (constant *) Minimum absolute roughness of wind tunnel floor 0s (m) Minimum absolute roughness of wind tunnel floor 0s / h Constant absolute roughness of wind tunnel floor 0s (m) Constant absolute roughness of wind tunnel floor 0s / h Region (1) Separated flow one , < < x < 0.6 < 0.4 < 4 6< x/d< 9 < 8 Region (2) Turbulent shear layer Region (3) Internal boundary layer

38 5. Shear stress on flat surface downwind from canopy Shear stresses ( τ ) on the wind tunnel floor were calculated from shear velocities for distances x/h>20 and plotted in Figure 5.1. The normalied plot of (τ/τ r ) uses the shear stress τ c at the furthest downstream measurement point as a reference (Table 4.2). The normalied shear stress as a function of distance is of particular interest because it characteries the reduction in shear force applied to the surface downstream from the canopy. This reduction is usually referred to as the wind sheltering effect. For lake water quality models it is often determined by model calibration Porosity 98% 78% 0% τ/τr x/h 38

39 τ/τr Porosity 98% 78% 0% x/h Figure 5.1 Normalied shear stress on the wind tunnel floor vs. distance from the canopy. Values in the transition 0 < x/h < 20 have been omitted in the lower plot. As can be seen in Figure 5.1, a reduction in surface shear stress due to wind sheltering by a canopy occurs no further than a distance of about 100 canopy heights. For practical purposes x/h < 60 may be a good limit. Normalied shear stress starts at ero in the separated flow one, which starts at the foot of the canopy and can extend up to 8 canopy heights downstream from the canopy (Region 1 in Fig. 4.1). Immediately downwind from the separated flow one the turbulent shear layer touches down, and the shear stress can be higher than the downwind reference shear stress. According to the experimental results this may occur only over a short distance 4 < x/h < 10 in Region (2). The largest reduction 39

40 in shear stress appears to occur over distance of 8 x/h 25. The lowest relative shear stresses are in the range 0.3 < τ/τ r < For trees of10m height wind sheltering would extend over a distance of less than 1000m. In very rough terms, if the wind fetch of a lake is smaller than 100 canopy heights wind sheltering will be significant; if the wind fetch is larger than 1000 canopy heights, wind sheltering can probably be ignored. By comparison, Figure 3.4 shows that the velocity profiles at x/h = 90 continue to have a significant velocity deficit compared to the wind tunnel velocity profile without a canopy. This leads to the conclusion that surface shear stresses make a much faster transition than velocity profiles downstream from the canopy. In other words, momentum transfer is faster than mass transfer downstream from the canopy. Shear stresses τ on a surface are related to wind speed by a drag coefficient. If we use equation (1.2) with shear stresses τ = τ r and wind velocities = r from Table 5.1, we obtain the C D - values in the last row of Table 5.1. These drag coefficients are identical to C 10 values for water surfaces (Wuest and Lorke, 2003) (C 10 means that wind velocities measured at an elevation of 10m are to be used in equation (1.2)). The equality of the drag coefficients indicates that wind tunnel floor is a good representation of a water surface. 40

41 Table 5.1 Experimental shear stress parameters Canopy Porosity 98% 78% 0% Canopy height h (m) Reference wind velocity r (m/s) Reference shear velocity * r (m/s) Reference shear stress τ r (N/m 2 ) Normalied reference shear velocity * r / r Drag coefficient C D in equation (1.2) r is the reference wind velocity measured at x=0 and =6h * r is the shear velocity at the furthest downwind velocity profile.. τ r is the shear stress on the wind tunnel floor at the furthest downwind velocity profile. This will also be used as the reference shear stress. The surface shear stresses downwind from a canopy can be summaried as follows: 1) There is a separated flow region immediately downwind from the canopy. Its maximum length is 8 times the displacement height in the canopy. A crude estimate of displacement height is 0.66 times the canopy height. Displacement 41

42 height is actually dependent on canopy porosity. In the separated flow region the surface shear stress will be very small or even negative. For the modeling of momentum transfer to a lake surface we recommend to ignore it. 2) Wind speed over a lake (water) surface is about 1.08 times the wind speed measured at 10m elevation above a cut grass surface, as is typical for airport weather stations. With this wind speed a farfield (no wind sheltering effect) shear stress on the water can be calculated from equation (1.2). The drag coefficient C D is from to It varies more or less linearly from at a wind speed of 5m/s to at a wind speed of 25m/s (Wuest and Lorke, 2003). 3) sing the surface shear from 2) above as a reference (τ r ), the reduction in surface shear stress due to wind sheltering and as a function of distance from the canopy can be estimated using Figure 5.1. This figure shows wind tunnel results for three canopies of different roughness and porosity. Canopy height is included in the normalied distances. 4) Wind sheltering appears to affect surface shear stresses on a lake surface significantly only for a distance less than 100 times the canopy height. Roughness of the canopy has a large effect on wind sheltering. Higher roughness lengthens the wind sheltering region. According to Figure 5.1 the distance is less than 50 times the height of canopies of low roughness; it is less than 80 times the height of very rough canopies. 42

43 5) There is some uncertainty about the shear stress between the wind reattachment point at and the beginning of the internal boundary layer (Regions 2 in Figure 4.1). This is roughly the distance 5 < x/h < 25. ntil additional information has been collected we recommend extrapolation of the experimental curves in Figure 5.1 down to ero shear stress at a distance x/d=8. 6. Application of wind tunnel results to a lake Water quality (computer) models can simulate vertical mixing and heat transfer in lakes. Mixing processes are driven by heat and momentum transfer at the lake surface, which in turn are linked to wind speed, air temperature, solar radiation and relative humidity. The approximate power input from the wind to a lake surface (2-D) is the integral τ*dx over the entire wind fetch (F). That power input is needed in 1-D lake stratification models (e.g. Ford and Stefan, 1980). In 3-D lake models (e.g. Edinger, 2001) the shear stress distribution on the water surface of a lake has to be specified as a boundary condition. The reduction in power input due to wind sheltering depends on the wind fetch F. The above integral is for 2-D flow along the wind direction. It has to be applied to different wind fetches according to the shape of a lake surface in order to obtain the wind power input over an entire lake surface. In many model applications the weather data come from a land-based station some distance away from a lake. Wind stresses on the lake surface are calculated from measured wind velocities using empirical drag coefficients that may account for wind fetch and the state of the water surface in terms of wave 43

44 characteristics (Wuest and Lorke, 2003). Wind sheltering by the terrain, vegetation and buildings surrounding a lake is accounted for by a wind sheltering coefficient that is often determined by model calibration. The results of the experimental study described herein may provide a means to estimate wind sheltering based on canopy type and height surrounding a lake. The three canopies used in the wind tunnel experiments simulate to some degree of realism a vegetation cover (trees, reeds, crops) or buildings on the land surrounding a small lake. The wind velocities measured downstream from the canopy and the shear stress distributions derived from them include the wind sheltering effect of an upwind canopy or buildings on the transition to a flat and smooth surface downwind. We assume that the solid surface downwind from the canopy in the experiment represents a lake surface. Although the roughnesses of the two surfaces (solid and water) do not scale correctly, we believe that this deficiency does not affect the results seriously. Our experimental results indicate that the canopy characteristics (height, roughness and porosity) exert a more important control over the wind field and shear stresses downstream from the canopy, than the roughness of the flat surface downwind from the canopy. Characteriation of the canopy is therefore most important to estimate the wind sheltering effect. The experimental canopies were characteried in Table 2.1, and the experimental results are characteried in Table 4.1 and a number of figures. To apply the observed wind sheltering effect of the experimental canopies to a real lake setting, we need to normalie the velocity and shear stress data as shown in Section 4. We selected the experimental canopy height (h) as the logical reference length for distances and elevations. As a reference velocity we selected the wind speed at the end of the canopy at x=0 and =6h. As a reference shear stress we select the shear stress at the furthest down-wind point x/h ~100 from the canopy. 44

45 For a lake application we can consider the canopy characteristics outlined in Table 6.1 (The format is the same as Table 2.1). The canopy is assumed to be higher than 1.5m. Lower vegetation will have a wind sheltering effect only by the change in roughness going from land to water; there will be no flow separation due to a step in the transition from land to water. In other words, when there is no separated flow region the flow transition is a pure roughness transition. The wind tunnel results have shown that a separated flow region is on the order of 8 times the displacement height. For a canopy height of 1.5m that represents about 12m or less. Compared to the wind fetch even on a small lake, 12m is negligeable, and a minimum canopy height of 1.5m therefore seems justified. We are also assuming that the relative displacement height d/h is only a function of canopy porosity. This allows us to calculate a displacement height (d) for a given canopy geometry as shown in Table 6.1. The length of a separated flow region x is then estimated to be about 8 times the displacement height. The reference wind velocity at the lake has to be calculated from the wind speed measured at a weather station, typically at an airport. The calculations have to account for differences in elevation (10m at the airport vs. 6h m at the canopy) and differences in roughness (e.g. 0 =0.006 m for cut grass at an airport vs. 0c =0.05 to 0.5 for a tree canopy). The canopy roughness length can be obtained from the literature (e.g. Fig. 9.6 in Stull, 1988). It is probably not a good idea to assume that the relative canopy roughness 0c /h depends only on canopy 45

46 porosity, and that its values can be extracted from Table 2.1. This assumption leads to a canopy roughness that is too large (see Table 6.1). The absolute roughness of the surface downwind from the canopy can be estimated starting with Charnock s relationship. This roughness is probably not too crucial for the surface shear stress in the transition region (0 < x/h < 100). Further downwind it will become important, and its magnitude will depend on wind fetch and wave age (Wuest and Lorke, 2003). The water surface roughness effect will be included in the value of the drag coefficient C D which is used to estimate the shear stress τ on a water surface in the absence of wind sheltering, (equation 2.1) as was mentioned in the Introduction. That coefficient is on the order of C D =0.001 to for wind speeds from 3 to 15 m/s. It was shown in Section 5 (Table 5.1) that C D is the same in the wind tunnel experiments. Wind velocities are usually measured and recorded 10 m above ground level at a weather station often at an airport. What remains to be discussed is the relationship between the wind velocity measured at an airport, the approach wind velocity over the canopy of a lake, and the reference wind velocity in the wind tunnel. How can the wind velocity 10 measured at 10m elevation at an airport with surface roughness 10 be translated into a canopy wind velocity c at height =6h with a canopy roughness 0c? The question is answered in the Appendix I. 46

47 Actually the question does not have to be answered if only the shear stress distribution (including wind sheltering) on a lake is to be determined. The experimentally determined reduction of shear stress by wind sheltering is given in Figure 5.1. In that figure shear stresses are reported relative to a far downwind reference ( τ r ) which can be estimated from equation (1.2) with the wind speed at 10m height from an airport and a drag coefficient C 10 from to 0,0015 (Wuest and Lorke, 2003). Nevertheless, we will show in the Appendix how wind measurements at 10m elevation above a surface of cut grass, as is typical for airport weather stations, can be translated into wind velocities at 10m elevation above a lake surface, or at 6 canopy heights above a vegetation (tree) canopy. Table 6.1 Independent parameter values for a lake site Canopy geometry Canopy porosity 98% 78% 0% Relative displacement height d / h Height of canopy h (m) Displacement height in canopy d (m) Length of separated flow region 8d (m) * Roughness Absolute roughness at airport weather station (cut grass) 0 (m) Fig. 9.6 Stull (1988)

48 Absolute roughness of canopy 0c (m) from Fig. 9.6 Stull (1988) Relative roughness of canopy 0c / h Absolute roughness of lake surface w/o canopy 0s (m)** Fig. 9.6 Stull (1988) Roughness ratio: canopy to lake surface c / 0s Velocity Wind velocity (m/s) at 10m elev. at airport Reference wind velocity c (m/s)* over the canopy Reference wind velocity L (m/s)* over the lake Relative canopy shear velocity * c / c 15** Canopy shear velocity * c (m/s) Shear stress Shear stress on canopy τ c (N/m 2 )

49 * r is the wind velocity measured at the end of the canopy (x=0) at a height =6h. (h=height of the canopy) A kinematic feature is the shape of the velocity profile over the canopy in the experiment compared to the prototype. It depends on the roughness of the canopy. Rough canopies have velocity profiles with smaller vertical gradients than smooth canopies (Figure 3.4). When the two profiles are overlayed, it is apparent that the rough boundary velocity profile has a larger velocity deficit to fill downstream from the canopy. 7. Summary and conclusions Three wind tunnel experiments have been performed to simulate the wind transition from a canopy to a flat solid surface. The experiments are motivated by the desire to learn how the shear stresses exerted by wind on a lake surface are reduced by wind sheltering from a canopy of trees, by buildlings or by bluffs, In the first experiment we used several layers of chicken wire with a total height of 5cm, a porosity of 98% and a length of 8m in flow direction to represent the vegetation canopy, and the floor of the wind tunnel consisting of plywood was used to represent the lake. The chicken wire represents a porous step that ends at x=0. This experimental set-up was considered to be a crude representation of a canopy of trees or other vegetation that ends at the shore of a lake. In a second experiment we used an array of pipe cleaners inserted in a 49

50 styrofoam board to represent the canopy. The porosity of that canopy was 78%. In a third experiment we used a solid step with a smooth surface which could be a simplified representation of a high bank or buildings on the upwind side of a lake. Wind velocity profiles were measured downwind from the end of the canopy or step at distances up to x=7m. sing the velocity profile at x=0, the absolute roughness of the two canopies was determined to be 1.3 cm and 0.5 cm, respectively, and the displacement height was determined to be 2.3cm and 6.68cm.The roughness of the wind tunnel floor downstream from the canopy was determined to be m =0.03mm. Three distinct layers were identified in the measured velocity profiles downstream from the canopy: the surface layer in response to the shear on the wind tunnel floor, an outer layer far above the canopy, and a mixing/blending layer in between. With sufficient distance downwind from the canopy the mixing layer should disappear, and the well-known logarithmic velocity profile should form. The shear stress on the surface downwind from the canopy was unaffected by wind sheltering after x/h=100, and the effect was less than 10% after x/h=60. A separated flow region formed downstream of each of the three canopies. The distance to reattachment was about 8 times the displacement height in the canopy. After a distance x/h=25 an internal boundary layer could be identified. It was characteried by rising shear stresses with distance from the canopy. Between x/d=8 and x/h=25 a turbulent shear layer touches down on the surface. Shear stresses in this range are highly variable, depending on canopy roughness 50