A WIND TUNNEL STUDY OF THE VELOCITY FIELD ABOVE A MODEL PLANT CANOPY

Transcription

1 CSI RO AUST RALIA CSIRO LAN D and WATER A WIND TUNNEL STUDY OF THE VELOCITY FIELD ABOVE A MODEL PLANT CANOPY Julie M. Styles Technical Report No (November 997)

2 A Wind Tunnel Study of the Velocity Field Above a Model Plant Canopy by Julie M. Styles Supervisors: Dr Michael Raupach Division of Land and Water CSIRO Dr Frank Houwing Department of Physics Australian National University A thesis submitted in partial fulfilment of the requirements for the Degree of Bachelor of Science (Honours) The Australian National University Canberra ACT 000 Australia 9 November, 997

3 This thesis is my original work and has not been submitted, in whole or in part, for a degree at this or any other university. Nor does it contain, to the best of my knowledge and belief, any material published or written by any other person, except as acknowledged in the text. This thesis may be freely copied and distributed for private use and study. No part of this thesis may be included in a publication without the prior written permission of the author. Any reference must be fully acknowledged. Signed 7 November, 997 Acknowledgements Several people contributed to the success of this project. The project relied on the use of the CSIRO Pye Laboratory wind tunnel and Laser Doppler Velocimeter. I would like to thank all at the Pye Laboratory who allowed me to take part in the larger experiment currently being undertaken there, giving me valuable time in the wind tunnel and assisting with the experimental phase, analysis and write-up of this thesis. Thanks especially to those who gave feedback on my seminar presentation and thesis and to: Margi Böhm and Dale Hughes for introducing me to the wind tunnel and its mechanisms; Margi, John Finnigan and Mike Raupach for sharing their knowledge of boundary layer meteorology; Mike for finding time for me during a particularly busy period; Greg Heath for design assistance and the use of his excellent slides; Frank Houwing for support as departmental supervisor.

4 Abstract A wind tunnel experiment investigates semi-empirical equations for predicting the friction velocity u, roughness length z 0 and zero-plane displacement d of a plant canopy or rough surface based on the geometry and density of the plants or roughness elements. The velocity statistics are measured with a Laser Doppler Velocimeter (LDV) which is known to overcome many of the shortcomings of instruments used in previous wind tunnel studies. Results are obtained for a model plant canopy which is denser than many of the wind tunnel models upon which the equations are based. The equations are found to be consistent with these new data. The present experiment establishes the use of the LDV as a measurement technique for wind tunnel canopy flows and provides a significant validation of the semi-empirical equations. iii

5 Contents Acknowledgements...ii Abstract...iii List of Figures... v List of Tables... vii List of Symbols...viii. Introduction.... Theory The Atmospheric Boundary Layer Momentum Budget Gradient-Diffusion Theory Mean Velocity Profiles Zero-Plane Displacement....6 Drag Characteristics Method Wind Tunnel Laser Doppler Velocimeter LDV System Settings Frequency shift selection Bandpass Filters Coincidence Time Experimental Scheme Analysis Velocity Statistics Coordinate Transformation Data Rotation Results and Discussion Friction Velocity Velocity Statistics Zero-Plane Displacement Mean Velocity Profile Drag Characteristics Conclusions Appendices Appendix : Reynolds Averaging Appendix : LDV System A. Overview of the LDV System A. ColorLink Multicolor Receiver... 6 A.3 Signal Processor... 6 References iv

6 List of Figures Figure : Classification of atmospheric boundary layer regions...4 Figure : Wind tunnel schematic...7 Figure 3: Direction of fringes and measured velocity component relative to the laser beams... 8 Figure 4: Grid points defining location of measurements relative to roughness elements... Figure 5: Measurement locations (L) at each Position, showing grid points (gp) and peg/bulb numbering (P)...3 Figure 6: Laser Doppler Velocimeter probe orientation... 6 Figure 7: Height above ground z versus Reynolds stress uw ' ' at (a) Position and (b) Position Figure 8: Height above ground z versus normalised Reynolds stress uw ' ' u at each Position Figure 9: Height above ground z versus normalised standard deviation of streamwise velocity σ u u at each Position Figure 0: Height above ground z versus normalised standard deviation of vertical velocity σ w u at each Position Figure : Height above ground z versus mean streamwise velocity u at each Position Figure : Variation of uw ' ' with height and location within the canopy at (a) Position ; (b) Position ; (c) Position 3; and (d) Position Figure 3: Variation of uv ' ' with height and location within the canopy at (a) Position ; (b) Position ; (c) Position 3; and (d) Position Figure 4: Variation of vw ' ' with height and location within the canopy at (a) Position ; (b) Position ; (c) Position 3; and (d) Position Figure 5: Variation of u with height and location within the canopy at (a) Position ; (b) Position ; (c) Position 3; and (d) Position Figure 6: Variation of w with height and location within the canopy at (a) Position ; (b) Position ; (c) Position 3; and (d) Position v

7 Figure 7: Variation of v with height and location within the canopy at (a) Position ; (b) Position ; (c) Position 3; and (d) Position Figure 8: Comparison of u obtained from different probe angles Figure 9: Comparison of σ u obtained from different probe angles Figure 0: Height above ground z versus drag coefficient C d at each Position Figure : Mean velocity profile and region of application of the logarithmic law... 5 Figure : Momentum diffusivity K m at each Position Figure 3: Power law fit to mean velocity data Figure 4: Comparison of dependence of predicted roughness parameters on roughness density λ with experimental data for (a) u u h ; (b) dh; and (c) z0 h Figure A: LDV system components... 6 Figure A: Signal Processor components Plate : CSIRO Pye Laboratory wind tunnel. Plate : Three-dimensional TSI Laser Doppler Velocimeter. vi

8 List of Tables Table : Friction velocity u for each Position and probe configuration Table : Zero-plane displacement at each Position Table 3: Physical and aerodynamic parameters at each Position Table 4: Comparison of predicted aerodynamic roughness parameters with experiment Table A: LDV system components and functions vii

9 List of Symbols A effective shelter area b breadth of roughness elements b, b blue laser velocity vector c mean level of momentum absorption c d empirical constant, =7.5 c w empirical constant, = C d drag coefficient C R drag coefficient of an isolated roughness element C S drag coefficient of substrate surface d zero-plane displacement D mean inter-element distance f d Doppler frequency f m measured frequency f s frequency shift g acceleration due to gravity g, g green laser velocity vector h canopy height K M momentum diffusivity coefficient K* momentum diffusivity coefficient in the inertial sublayer p static pressure Ro Rossby number t time u streamwise velocity u h mean streamwise velocity at the top of the canopy u i streamwise, transverse or vertical velocity u friction velocity U free-stream velocity v transverse velocity V effective shelter volume, velocity w vertical velocity x streamwise space coordinate x i streamwise, transverse or vertical space coordinate y transverse space coordinate z vertical space coordinate z 0 roughness length depth of the roughness sublayer z w α power law exponent γ uh u δ boundary layer depth Kronecker delta tensor δ ij ε ijk η j alternating unit tensor j th component of unit vector parallel to the earth s axis of rotation κ von Karman constant, =0.40 viii

10 λ ν ρ σ τ τ 0 τ R τ S Ψ Ω roughness density kinematic viscosity of air air density standard deviation shear stress, transit time ground shear stress shear stress on substrate surface shear stress on roughness elements mean velocity profile influence function angular velocity of earth ix

11 . Introduction As global climate change predictions play an increasingly important role in shaping our environmental policies and practices, one of the current priorities in meteorological research is to gain a better understanding of the interaction between vegetation and the atmosphere. It is hoped that it will be possible to parametrise the influence of vegetation canopies in terms of appropriate characteristic quantities which depend on the amount of roughness presented by the canopy. This thesis is concerned with the mean velocity profile of flow over a plant canopy or rough surface, which is found to depend on certain roughness parameters characteristic of the canopy. The friction velocity u represents the effect of the shear stress at the ground; the roughness length z 0 represents the drag induced by the presence of the canopy; and the zero-plane displacement d represents the displacement of the flow above the canopy. These parameters help in understanding and quantifying the momentum and scalar transfer processes occurring between the canopy and the atmosphere. Specific applications include the parametrisation of surface drag for use in atmospheric models, pollutant dispersal modelling and calculation of wind loads on buildings or trees (Arya, 988). Based on a number of field and wind tunnel experiments, using both physical and empirical constraints, a set of equations has been derived to relate these roughness parameters to the geometry and density of the plants or roughness elements (Raupach, 99; Raupach, 994). These equations are potentially useful for predicting the influence of a plant canopy on the mean flow above with the only velocity measurement required being u h, the mean velocity at the top of the canopy. The prediction for d requires no velocity measurements at all. Other theories and relationships have been put forward to predict these roughness parameters, but these either fail to correctly describe the observations; rely on knowledge of the drag coefficients of the roughness elements; or require measurements of velocity statistics in some region above or within the canopy (Seginer, 974; De Bruin and Moore, 985). The main aim of this project is to investigate these roughness parameters in detail for flow over a model plant canopy which is relatively dense compared to the wind tunnel models used in the derivation of the equations under question, and assess the validity of the equations in this situation. In the course of this investigation it will be necessary to look at the statistical

12 properties of the velocity field and interpret them in terms of the equations of motion, in particular the equation for conservation of momentum. The present validation of the Raupach (994) equations is also significant because of the high accuracy of velocity measurements provided by a state-of-the-art Laser Doppler Velocimeter (LDV, described in Section 3). Previous experiments of this type have relied on such instruments as X-wires which utilise a pair of hot wires in a cross configuration to measure two velocity components, or triple wires which add a third wire to this configuration. X-wires have been shown to be inaccurate at the high turbulence intensities that are found within and near the top of a canopy and have an acceptance angle of only ± 45 o. Triple wires increase the acceptance angle to ± 90 o but still cannot resolve flow reversals and are also limited to two velocity components (Legg et al., 984). The LDV overcomes these problems. It can measure all three velocity components, resolves flow reversals, is much less hindered by high turbulence intensities and is non-invasive, requiring no direct contact with the flow. The thesis is divided into five main sections. In the section following this introduction, the basic tools for analysis of boundary layer flow, including the momentum equation and various theories used in order to overcome the closure problem of this equation, are presented. This includes a brief derivation of the equations under question for predicting the roughness parameters of a canopy and an explanation of the theories behind these equations. Section 3 describes the instruments and analyses used to obtain the velocity statistics of flow over a model plant canopy. The velocity statistics and quantities derived therefrom are presented with discussion of relevance and errors in Section 4. Comparison of measured roughness parameters with predicted values are also made in the this section. Section 5 provides a conclusion including a summary of the findings and their significance.

13 . Theory. The Atmospheric Boundary Layer The atmospheric boundary layer can be defined as the region of the atmosphere that responds to surface influences in time scales of less than a day. The boundary layer can be divided into various regions where different length scales and different driving mechanisms are important, as shown in Figure. The choice of appropriate length scales is important in obtaining a phenomenological description of the turbulent processes through dimensional analysis. In the typical daytime outer layer, turbulence is generated by both thermal convection and by the shear layer between the top of the boundary layer and the free stream (Stull, 988). On windy, cloud covered days the atmosphere is neutrally buoyant and turbulence is driven by wind shear arising from the no-slip condition at the surface (Garratt, 99). A wind tunnel environment closely approximates neutral atmospheric conditions. Coriolis effects due to the earth s rotation control the velocity structure in the outer layer. The governing length scale in this region is δ, the depth of the boundary layer. The surface layer defines the region where surface effects dominate and Coriolis effects become less important. Within the surface layer, the roughness sublayer extends from the ground up to -5 canopy heights, where the detailed structure of the roughness elements influence the turbulence and flow properties. Significant spatial variations in the properties of the flow and departures from behaviour predicted from surface layer theories are observed in the roughness sublayer. Turbulence here is generated by both wake production in the low pressure region behind roughness elements and by shear production (Raupach et al., 986). The flexibility of plant parts may also play an important role in the production of turbulence in the roughness sublayer (Finnigan, 985). Destruction of turbulence through viscous dissipation is especially important close to the ground. The important length scales in the roughness sublayer include the roughness length z 0 representing the drag induced by the roughness elements, as well as lengths defining the geometry and spacing of the roughness elements. Above the roughness sublayer, the inertial sublayer can be thought of the region of overlap of the roughness sublayer and outer layer. In this region shear production and viscous dissipation of turbulent kinetic energy are negligible (Finnigan and Brunet, 995). The 3

14 dominant turbulent process here is the conversion of large-scale eddies to small-scale eddies via the energy cascade process. The only governing length scale in the inertial sublayer is the height above the ground displaced by the presence of the canopy, z - d. z =δ Outer (Ekman) Layer z<<δ Inner (Surface) Layer Inertial Sublayer z >>z 0 Roughness Sublayer Figure : Classification of atmospheric boundary layer regions. z is the height above the ground; δ is the depth of the boundary layer; and z 0 is the roughness length.. Momentum Budget The main equation governing turbulent boundary layer flow is the Navier-Stokes equation, a form of Newton s second law which states that the time rate of change of momentum of a fluid particle is equal to the sum of the forces acting on it. Instantaneous values at particular points in space are difficult to measure and visualise when considering three-dimensional turbulent flows, so the Navier-Stokes equations are often averaged over both space and time. A spatial average requires that the canopy be homogenous over the averaged region, i.e. tree spacings, tree heights and foliage density should be approximately constant. This enables canopy effects as a whole to be well represented by the spatial average, despite considerable spatial variations in the flow within the canopy. Temporal averages require that the time scales of interest be short compared with scales at which average properties of the flow change appreciably. The present wind tunnel flow was designed to be stationary and horizontally homogeneous far downstream of the entrance of the tunnel and of the leading edge of the roughness elements. Wind tunnel coordinates are defined as follows: subscript denotes the mean streamwise direction along the tunnel; subscript denotes the horizontal cross-stream direction with the positive direction assigned arbitrarily; subscript 3 denotes the vertical 4

15 direction with positive upwards. The symbol x i with i =,,3 denotes position and is interchangeable with x, y and z, while u i denotes velocity and is interchangeable with u, v and w. Using the summation convention, the instantaneous Navier-Stokes equation for velocity component u i is: Dui Dt p ui = gδi 3 Ω εijkηjuk + ν () ρ x x i j where D/Dt = / t + u i / x i is the substantive derivative for the instantaneous advective velocity u i. The first term on the right-hand side of Equation () represents the pressure gradient force; the second term represents the force due to gravity; the third term represents the Coriolis force due to the earth s rotation; and the last term represents viscous forces. The instantaneous velocity u i is then decomposed into a time averaged component u i and a deviation u i ' from this average: ui = ui + ui'. These components are substituted into Equation () and the equation is then averaged over time to give: ui t δ ε η ν u ( u u i i' j') p ui + u j = g i 3 Ω ijk j uk + () x x ρ x x j j i j where the Boussinesq approximation has been made. This approximation, standard for boundary layer applications, states firstly that density changes resulting from pressure changes are negligible. Secondly, it is assumed that density changes resulting from temperature changes are important only in their influence on buoyancy, that is in combination with the acceleration due to gravity (Garratt, 99). Another assumption implicit in the time averaged equation is that of ergodicity: that the temporal average is the same as an ensemble average, for which the averaging operator and differentiating operators commute (Monin and Yaglom, 97; Kaimal and Finnigan, 994). This is a good assumption if the averaging period is long compared with the time scale at which the turbulent motions are correlated (Kaimal and Finnigan, 994). The time-averaged velocity is then decomposed into the spatial average u i and a deviation u i " from this average: ui = ui + ui". After substitution into Equation () the equation is averaged over space. The time and spatially averaged Navier-Stokes equation is then (Raupach and Shaw, 98): 5

16 u i t u u u u u i i" j" i' j' + u j + + x x x j j j p p" = δ 3 ε η + ν ui + ν ui " g i Ω ijk j uk (3) ρ x ρ x x x i i The product rule of differentiation and the continuity equation for incompressible flow give the relation: j j u j u x i j ui uj = (4) x j Therefore the time and spatially averaged Navier-Stokes equation becomes: u t u u ui" uj" ui' uj' x x x i i j j j j p p" = δ 3 ε η + ν ui + ν ui " g i Ω ijk j uk (5) ρ x ρ x x x i i The process of averaging over some combination of time and space is known as Reynolds averaging and the implications and definitions of this process are discussed in Appendix. The Rossby number is defined as the ratio of the magnitudes of the nonlinear advective terms to the Coriolis term, Ro = j V V V, where V and L are typical Ω V Ω L velocity and length scales, respectively, and Ω is the angular velocity of the earth s rotation, ~0-4 rad s -. The Rossby number can be used to estimate the relative importance of these terms (Panofsky and Dutton, 984). In the wind tunnel the velocity scales defining the turbulent motion range from -0 ms -, and length scales range from ~ m. The wind tunnel Rossby number ranges from , hence Coriolis effects are everywhere negligible compared to the advective terms. The roof of the wind tunnel is carefully adjusted to eliminate pressure gradients along the tunnel. The flow is designed to be stationary and horizontally homogeneous so that time derivatives and x and y derivatives disappear. In the present context the time and space averaged vertical velocity should be zero within measurement error since there are no vertical outlets for the flow. Further, the second and sixth terms on the right hand side of Equation (5) represent the form and viscous drag, respectively, on the plant parts within the canopy j 6

17 (Raupach and Shaw, 98). These terms can be incorporated into an average drag force D ρ (Kaimal and Finnigan, 994). In the surface layer with the above approximations, the time and space averaged Navier-Stokes equation for the u velocity component reduces to: z ν u D + + = 0 (6) z ρ ( u" w" u' w' ) I II III IV Term I in Equation (6) is called the dispersive flux of momentum and is non-zero if there are spatial correlations between the time averaged quantities, i.e. if departures of w from the space average correlate at different points with departures of u from the space average. Term II is the well known single-point Reynolds stress or diffusive momentum flux. The Reynolds stress represents the vertical flux of streamwise momentum due to mixing induced by velocity fluctuations. Term III is the viscous stress term representing the exchange of momentum between fluid particles due to viscous forces. Term IV is non-zero only for heights within the canopy. The combined stresses in Equation (6) (terms I, II and III) define the shear stress on a surface parallel to the ground, above the canopy. Previous experiments have shown that Term I is insignificant compared to Term II (Raupach et al., 986; Raupach et al., 980). Viscous effects are also found to be insignificant except within ~ mm of solid surfaces (Garratt, 99). Equation (6) therefore predicts that, above the canopy, the Reynolds stress is constant with height and therefore equal to the stress at the ground, τ 0. That is, in the surface layer above the canopy, Equation (6) reduces to: ρ u' w' = constant = τ0 (7) Within the plant canopy, the drag term in Equation (6) accounts for the momentum sink provided by the plant parts. This causes the Reynolds stress to decrease within the canopy. The drag force can be parametrised in terms of a drag coefficient. As will be shown in Section.5, Equation (6) allows the drag coefficient to be calculated, from which the mean level of momentum absorption can be determined. The present experiment investigates the above Navier-Stokes equations and accounts for experimental variabilities such as the extent of the spatial variation in first and second order velocity moments. Hereafter, the angle brackets denoting spatial averaging will generally be omitted, and it should be clear from the context whether the quantity in question has been spatially 7

18 averaged or not. The distinction is only important in regions very close to or within the canopy where departures from the spatial mean are observed..3 Gradient-Diffusion Theory In the inertial sublayer of the atmospheric boundary layer (Figure ) the flow has been successfully treated using gradient-diffusion theory, otherwise known as K-theory. This theory assumes that the flux of a property is proportional to the gradient of the property, in analogy with the process of molecular diffusion. The proportionality constant is known as the diffusivity coefficient. The momentum flux τ can then be expressed in terms of the velocity gradient in the vertical (z) direction: τ ρ u = K M (8) z where K M is the momentum diffusivity coefficient and ρ is the air density. This procedure greatly simplifies the equations governing the mean properties of the flow. It has, however, been found to be misleading in the region close to the top of the canopy, and completely inapplicable within the canopy where counter-gradient fluxes have been observed (see for example Raupach and Legg, 984; Denmead and Bradley, 985). K-theory relies on the scale of the turbulent transport mechanisms being much smaller than the scale over which the gradient of the property changes appreciably (Corrsin, 974). This condition is rarely satisfied within the canopy where the turbulent transport of momentum is dominated by intermittent energetic gusts having vertical length scales as large as the height of the canopy (Finnigan, 979a, b; Raupach et al., 986)..4 Mean Velocity Profiles The most easily measured and well-known characteristic of boundary layer flows is the mean velocity profile. The Navier Stokes equation for u (Equation (7)) did not reveal an analytic solution for the mean velocity because the unknown quantity uw ' ' was found to dominate. The Navier-Stokes equation can be used to formulate dynamical equations for uw ' ' and other second order velocity moments, but these will involve unknown triple correlations. Similarly, the equation for the triple correlations involves fourth-order correlations, and so on. This situation is known as the turbulence closure problem. In order to solve the Navier-Stokes equation it is necessary to turn to further hypotheses. An expression 8

19 for u can be found through dimensional analysis if appropriate length and velocity scales can be chosen. Turbulent flow over a boundary is characterised by at least two different length scales (Tennekes and Lumley, 97). In the wall or surface layer close to the roughness elements at the ground, the no-slip condition gives rise to a viscosity-dominated length scale ν u where ν is the kinematic viscosity and the friction velocity u is the characteristic velocity for the turbulence fluctuations. Over a rough surface it is expected that length scales characterising the geometry of the roughness elements also come into play in the wall region. At large Reynolds numbers, however, the boundary layer thickness δ is much larger than ν u and it is expected that in the outer region, close to the top of the boundary layer, δ is the governing length scale. The two regions can then be treated separately through dimensional analysis and the descriptions thus found must be reconciled by a process of asymptotic matching whereby the velocity gradient is forced to be continuous. The assumption is that if δu ν is large enough, then there will exist a region where z >> ν u and z << δ simultaneously, so that the velocity gradients obtained through dimensional analysis for the two regions must be equal in the overlap region, known as the inertial sublayer. This leads to the well known logarithmic law in the inertial sublayer for flow over rough surfaces: u u z d = ln (9) κ z 0 where uz ( ) is the mean streamwise velocity at height above the ground z; κ 0.40 is the von Karman constant; d is the zero-plane displacement; z 0 is the roughness length; and u is the friction velocity representing the effect of wind shear stress on the ground and defined by u = τ 0 ρ where τ 0 is the surface shear stress and ρ is the air density. A full derivation of this important result via the asymptotic matching process can be found in Tennekes and Lumley (97, Chapter 5) or Tennekes (98). The zero-plane displacement accounts for the fact that momentum is being absorbed throughout the canopy rather than just at the surface, and is usually h, where h is the canopy height (Kaimal and Finnigan, 994; Arya, 988). Experimental values for d over coniferous forest range from 0.6h to 0.9h (Jarvis et al., 976, Table V). The roughness length z 0 is a measure of the aerodynamic drag of the canopy, or the effectiveness of the canopy as a momentum absorber. It has been estimated as 30h for wind tunnel models and sand surfaces, 0.5h for various crops and grass surfaces (Arya, 9

20 988), and from 0.0h to 0.0h for coniferous forest (Jarvis et al., 976, Table V). A review of many surfaces gives z0 h in the range 00. < z0 h < 0. (Garratt, 99). Several different arguments have been presented that lead to Equation (9) (see Monin and Yaglom, 97, Section 5). All rely on the initial assumption that viscosity can be neglected at high Reynolds numbers, so that the only important length scale in the inertial sublayer is the displaced height z d (Monin and Yaglom, 97; Tennekes, 98). Application of the logarithmic law is limited to the inertial sublayer. Below this, the roughness sublayer defines the region that is affected by the detail of the roughness elements at the ground. In this layer, spatial variations and departures from the logarithmic law are observed. The height of the roughness sublayer, z w, has been found to extend as high as 5h above a rough tree canopy (Garratt, 980), while over crop canopies z w has been found to be only slightly above h (Raupach et al. 99). In wind tunnel studies, the range of z w found is from h to 5h (Raupach et al. 99). The upper limit of the logarithmic region is in the outer layer where the flow becomes insensitive to the surface. The log law has, however, found to be valid for some distance into the outer layer because the departure from the log law occurs very slowly. In a wind tunnel environment, the depth of the logarithmic region may be very small or even non-existent due to the relatively high ratio of roughness element height to boundary layer depth. In such cases the mean wind profile may be better represented by an empirical power law of the form: uz () z d uz ( ) = z d r r α (0) for some reference height z r and exponent α. A power law has been found to be a good average representation of the velocity profile over the whole of the boundary layer. It has often been used in wind tunnel experiments in preference to the logarithmic law which requires detailed measurements in the relatively short region corresponding to the atmospheric logarithmic layer (Plate, 97). It has been found that the exponent in the power law increases with increasing surface roughness, ranging from 0.6 over flat land to 0.8 over woodland forest and over tall buildings (Plate, 97; Eng. Sci. Data Item 706, 97). The power law approach, however, doesn t allow for comparison of the characteristic roughness parameters z 0 and d defining the logarithmic proportionality with those published for various roughness densities in both field and wind tunnel experiments. 0

21 .5 Zero-Plane Displacement Thom (97) first put forward the proposal that the zero-plane displacement level could be identified with the mean level of momentum absorption in the canopy. Based on his wind tunnel study of a model crop consisting of rigid, tall, circular cylinders, he measured the drag force per unit column of crop with a moment balance on which several elements within the canopy were mounted. The theoretical relation for the integrated drag per unit horizontal area of crop, neglecting any mutual sheltering of crop elements, is given by: h ' f = ρ a() z u () z Cd () z dz 0 () where ρ is the air density, a(z) is the projected area of vegetation elements per unit horizontal area per unit height at the level z, u ( z) is the mean wind speed at height z and C d (z) is the drag coefficient of the elements which has been well characterised for circular cylinders. The integral is taken to a height h', slightly above the top of the canopy, to account for the finite length of the cylinders. Using the known wind speed dependence of the drag coefficient of an isolated long circular cylinder, Thom was able to compare his moment-balance measurements of the drag force to values calculated from Equation (). Any discrepancy would be attributed to the sheltering effect of neighbouring elements which could be incorporated into the shelter factor for momentum, p = f τ 0 = f f where f M is the measured drag force, equal to the shear d M stress τ 0 at the surface. Thom found that for his model crop, the measured force was not significantly different from the calculated force, that is p d. Field experiments have yielded p d = 3 to 4, while the range found for wind tunnel experiments is p d = to 3 (Raupach and Thom, 98; Brunet et al., 994). Thom (97) calculated the mean level, c, at which the drag force acts from the relation: h ' ρ az () u() z Cd () zzdz 0 c = f () From the mean velocity profile above the canopy, Thom estimated the zero-plane displacement d by the standard technique of finding the best parameters to fit the logarithmic mean wind law, Equation (9), to the data, using the known friction velocity u and von Karman constant κ = Thom found no significant difference between his values for d and c, and therefore suggested that the status of d is raised from that of an empirical constant of integration to that of the level of mean momentum sink (Thom, 97, p.4).

22 In practice, when the drag coefficient of the roughness elements has not been previously characterised, an assumption may be made about its relation to wind speed or it may be calculated directly from Reynolds stress measurements. Knowledge of the shelter factor p d is not required in the use of Equation () since it appears in both the numerator and denominator. Thom suggested, as a first approximation, that C d may be assumed constant, in which case the finite difference form of Equation () becomes: d = i i a u z ( δz) i i i i a u i i ( δz) where the subscript i refers to the level z i. Alternatively, if the Reynolds stress can be measured within the canopy, the Navier- Stokes equation (Equation (6)) can be used to find the drag coefficient. Assuming that form drag is the main drag mechanism within the canopy, and that the dispersive momentum flux can be neglected, Equation (6) becomes (Brunet et al., 994): i (3) duw ' ' dz = C ()() z a z u () z d (4) where the form drag ff ( z) making up the drag term D ( z) in Equation (6) has been parametrised in terms of the drag coefficient C d via the relation f () z = ρc ()() z a z u () z. The values for C d thus found can be substituted into the finite F d difference forms of Equations () and () to find d. Jackson (980) showed that the interpretation of d as the mean level of momentum absorption is consistent with the derivation of the logarithmic mean wind law which implicitly assumes that the flow in the inertial sublayer depends on the surface details through the elevation of the level at which the surface stress appears to act..6 Drag Characteristics Turbulent flow is considered to be dynamically fully rough when hu ν > 55, where the dimensionless parameter hu ν is a form of Reynolds number (Raupach, 99). A typical wind tunnel value of this Reynolds number is ~0 4. In this regime the drag characteristics z 0, d and u have been found to depend only on the geometric properties of the surface. The derivation of a relation between the roughness element geometry and the characteristic quantities d, z 0 and u will be presented below, following Raupach (99).

23 The total stress τ 0 is divided into a component τ S acting on the substrate surface and a component τ R acting on the roughness elements. The effect of a sheltering element is to produce a reduction in ground shear stress behind the element and a reduction in the drag force on other elements. The wake of an isolated bluff roughness element is parametrised in terms of an effective shelter area A in which the surface stress equals zero and an effective shelter volume V in which the drag force equals zero. The effective regions A and V are chosen to account for the integrated stress and force deficits created by the presence of the roughness element. It is hypothesised that for an isolated roughness element of height h and breadth b, the shelter area A and volume V scale as: A b uh u V for h>>b; and b huh u Hence: A bhuh u V for h<<b. bh uh u where c c ( b h) p = ' and ( ) A= cbhuh u (5) V = c bh u u h c = c ' b h p ; p = when bh 0 and p = 0 when bh. The second hypothesis made is that if the roughness elements are distributed either uniformly or randomly, then their combined effective shelter area and volume can be calculated by the random superposition of individual shelter areas and volumes. and τ s Explicitly, the idea of the effective shelter area is that τ s = 0 inside the shelter area A, = τ 0 outside A, where τ s0 is the unsheltered ground shear stress. Hence if one s roughness element with effective shelter area A is placed in the flow with total area S, the S A A ground shear stress becomes τs0 = τs0. The principle of random superposition S S then says that additional roughness elements will reduce the ground shear stress by the same factor A S, so that if n roughness elements each with effective shelter area A are randomly positioned over area S then the average ground shear stress is given by: n A τ ( ) S n = τs0 (6) S 3

24 The unsheltered surface stress is parametrised as τ S0 s h = ρcu, where C s is the unobstructed drag coefficient for the substrate surface and uh is the mean streamwise velocity at the top of the canopy. Defining roughness density λ = nbh S and taking the limit as S while λ is held constant then, using Equation (5), Equation (6) becomes: uh τs( λ) = ρcsuh c λ exp u (7) In a similar manner the average drag force for n elements is found to be: n nφ V τ R ( n) = (8) S Sh where Φ represents the average drag force on an isolated element. For n elements the undisturbed drag force per unit area is nφ/s. In terms of the roughness density λ and again allowing S this becomes: u h τr( λ) = λρcruh exp c λ u (9) where C R is the drag coefficient for an isolated roughness element. Equation (9) makes sense in predicting that the stressτ R acting on the roughness elements increases linearly with λ for small λ but is attenuated at larger roughness densities as mutual sheltering occurs. Making the approximation c c = c', defining γ = uh u and using the relation τ0 = τs + τr = ρu, Equations (7) and (9) give: u h γ = = S + u ( C λc ) R c' λγ exp In the limit of low λ the exponential term is negligible. At high λ, the hypotheses made are untenable and it is then assumed that γ approaches a constant value, γ ( λmax ) (0) 03.. A simplified version of Equation (0) (Raupach, 994) assumes that the low λ limit applies for all λ < λ max, so that: u h γ = = S + u u h γ = = S + u ( C λc ) R ( C λ C ) max R λ < λ λ λ max max () 4

25 To relate z 0 to u h u it is necessary to account for the fact that the logarithmic mean wind law, Equation (9), does not hold within the roughness sublayer which extends from the ground to a height above the canopy, z w. For h < z < z w, it has been suggested that a profile influence function u z d Ψ may be added to the right hand side of Equation (9) to κ z d w account for the departure of the momentum diffusivity within the roughness sublayer from the surface layer form. The relation between γ = uh u and z 0 then becomes: z h d 0 ( h ) ( ) h = exp Ψ exp h κγ () where Ψ h is the value of Ψ at z = h. A form for Ψ can be found if the momentum diffusivity in the roughness sublayer is assumed to be constant at K = κ u ( z d) gradient-diffusion theory leads to: uz () u u = = z K κ m ( ) ( z d) u uz () = + z d z c * κ w w m w. Then where the constant c* can be found by the requirement that the mean velocity function be continuous at z = z w, that is, by equating Equations (9) and (3) at z = z w. Substituting for c*, Equation (3) becomes: u z zw d zw uz () = ln zw d + (4) κ z0 zw d Comparison of Equation (4) with the profile influence function added to Equation (9) reveals: (3) Ψ z d zw d z zw = ln + (5) z d z d z d w Analogy of the flow around z = h with a free shear layer leads to the suggestion that the momentum diffusivity length scale ( z d) scales with the vertical length scale of the w shear layer ( h d), that is, ( z d) = c ( h d). From Equation (5), at height h Ψ then becomes: w w ( c ) = + Ψ ln h w c w where c w is a constant, found to be, giving Ψ h w (6) 5

26 Finally, a relation between d and u h u is found using the hypothesis that d is the mean level of momentum absorption by the roughness elements, or the centroid of the drag force profile, as proposed by Thom (97). If this is assumed to be governed by the spread of the strong shear layer behind an element which is limited by the inter-element distance D, then at high λ values the proportionality dh λ holds. At low λ the limit is dh 0. An empirical fit to the data gives (Raupach, 994): ( cd λ ) d = exp h c λ d (7) where the constant c d = 75.. Equations (), (), (6) and (7) then specify the characteristic parameters uh u, z 0 and d in terms of the roughness element geometry λ. They have been confirmed using numerous sets of data from both wind tunnel experiments and field experiments with roughness densities ranging from λ ~ 00. to λ ~. The present study provides an independent, reliable verification of these equations for a roughness density not previously studied. 6

27 3. Method 3. Wind Tunnel The CSIRO Pye Laboratory wind tunnel used in this experiment is an open-circuit blower type designed to simulate turbulent flow in the neutral atmospheric surface layer (Plate ). The working section is 0.6 m long,.78 m wide and 0.65 m high. The flow was developed upstream of the measuring section by an array of inexpensive pegs 8 mm in height. The measuring section contained an array of light bulbs 9 mm in height and 5 mm in breadth, the heat from which will be utilised in other experiments as a scalar source. The peg density was chosen to produce a roughness density close to that of the bulbs in order to maintain close to equilibrium conditions throughout the measuring section. The tunnel features a moveable roof to maintain a longitudinal static pressure gradient across the working section of less than % of the dynamic pressure ρu, where U is the velocity in the free stream above the top of the boundary layer. A schematic of the tunnel is shown in Figure. A full description of the tunnel and its design considerations can be found in Wooding s (968a and 968b) reports. z (m) Wind pegs Position Position Position 3 bulbs Position x (m) 0 Figure : Wind tunnel schematic. Approximate location of measuring positions is shown. Pegs and bulbs are not to scale in either dimension. 3. Laser Doppler Velocimeter Velocity components were measured with a TSI Laser Doppler Velocimeter (LDV) system (Plate ). An overview and description of the system components is given in Appendix. Micron-sized seeding particles (theatrical fog) were fed into the tunnel to act as fluid 7

28 tracers. The LDV is designed to measure the Doppler frequency in each of three directions by three pairs of laser beams focused at the measuring volume. Each pair is at a different frequency so that three independent interference fringes are formed in the measuring volume. The fringes formed by the intersecting beams are parallel to the bisector of the beams, and the measured velocity component is perpendicular to the fringes, as shown in Figure 3. The length scales of the measuring volume depend on the angle of the lasers. For the measurements presented here the shortest length scale of the measuring volume was nominally about 50 µm. For each pair of lasers, one of the beams is frequency shifted so that the interference fringes move at the frequency of the shift. As a seeding particle traverses the measuring volume, the interference fringes are scattered back to the detector at a frequency f m equal to the frequency shift f s plus or minus the Doppler frequency f d of the particle, f m= fs ± fd, with the sign depending on the direction of the particle relative to the fringe movement. A particle travelling with the fringes will give a frequency signal less than the frequency shift, while a particle travelling against the fringes will give a higher frequency signal. The Doppler frequency is given by the component of particle velocity perpendicular to the fringes divided by the fringe separation, f d = V d. To ensure that the velocity components of a single particle are being measured, the system is set up so that each pair of lasers is required to log a signal within a time period set in the order of 5-0 µs. Each measuring station is sampled for a minimum period of 0 seconds which was found to correspond to about 0 5 particle measurements on average. f Lasers shifted unshifted Measured velocity component Fringes Figure 3: Direction of fringes and measured velocity component relative to the laser beams. 8

29 3.3 LDV System Settings The LDV input parameters and output signals are controlled with the software program Flow Information Display (FIND). The most important of the input settings are discussed below Frequency shift selection When one of the lasers in a pair is shifted slightly in frequency, the interference fringes formed at the measuring volume move at a frequency equal to the frequency shift, in the direction from the higher to the lower frequency beam. This allows flow reversals to be resolved and increases sensitivity to slow particles. The actual frequency shift is set at 40 MHz, but for the purposes of signal processing, this shift is altered in a frequency mixer to a value selected by the user. The optimum frequency shift depends on the expected velocity range of the particles in the direction of the laser measuring coordinate. A different frequency shift is selected for each coordinate. A frequency shift too low will not distinguish between positive and negative velocities, while a frequency shift too high will reduce the capacity to filter out high frequency noise. The fringe shift direction can also be selected and should be set opposite to the direction of mean (positive) flow. The minimum frequency shift necessary to cover the entire velocity range is then equal to twice the Doppler frequency corresponding to the expected maximum negative velocity. From the relation f m= fs fd, with fs = fd for a particle travelling with maximum negative velocity in the direction of the fringes, the measured frequency is equal to the Doppler frequency, and hence the measurement is equivalent to one taken with no frequency shifting. This ensures that the minimum number of fringes crossed by a particle traversing the centre of the measurement volume is the same as would be crossed with no frequency shifting. While the maximum negative velocity in each of the wind tunnel coordinates can generally be estimated, it is necessary to transform these velocities to laser coordinates, noting that each of the wind tunnel coordinates is likely to contribute to the maximum negative velocity for a particular laser coordinate. The transformation is discussed in Section Once the transformation is known, it can be determined whether positive or negative velocities in the wind tunnel coordinates contribute to a negative velocity in the laser 9

30 coordinate. The contributions from the expected maxima or minima from each wind tunnel coordinate are then added up according to the transformation to determine the maximum negative velocity in the laser coordinate. That is, for a particular laser coordinate, the appropriate positive or negative maximum velocity in each of the wind tunnel coordinates is input to the transformation matrix to determine the maximum negative velocity in the laser coordinate. The optimal shift frequency found for each of the laser coordinates was found to be around 5 MHz, and this value was used in the present experiment. This value was calculated based on estimated maximum and minimum velocities of 0 and -5 ms -, respectively, in the streamwise direction and 7 and -7 ms -, respectively, in both the vertical and transverse directions, for a free stream velocity above the boundary layer of 3 ms Bandpass Filters The bandpass filter settings determine the high- and low-pass filter combination used in the signal processing. The high- and low-pass filters are available in the ratio :0. The high-pass filter is required to eliminate the DC pedestal generated by the transit time of the particle, while the low-pass filter cuts out high frequency noise. It is necessary to ensure that the values chosen for these filters do not impinge on the expected frequency range of the Doppler signals. Since the diameter of the measuring volume is in the order of 50 µm and the maximum expected velocity of a particle traversing the volume in any direction is ~0 ms -, the pedestal frequency is about 0.4 MHz, and this defines the minimum setting for the highpass filter. In the present experiment the low- and high-pass filters were always set at and 0 MHz. With the appropriately chosen frequency shift, the minimum net signal frequency was always well above MHz, while the range of velocities encountered was small enough that the maximum net signal frequency was well below 0 MHz. The importance of choosing the correct frequency shift and filter settings was demonstrated when examination of the velocity time series with incorrect settings clearly revealed an artificial cut-off at negative velocities. The data obtained with the above settings was carefully examined to ensure that no biasing was occurring Coincidence Time The coincidence time determines the maximum time allowed before each of the lasers must lodge a signal in order to be confident in assigning the signal from each laser to a single particle. If one or more signals are missing, then the signal is discarded. The coincidence 0