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TEST OF TURBULENCE MODELS FOR WIND FLOW OVER TERRAIN WITH SEPARATION AND RECIRCULATION H. G. KIM∗ and V. C. PATEL† Iowa Institute of Hydraulic Research, The University of Iowa, 300 South Riverside Drive, Iowa City, IA 52242-1585, U.S.A.

(Received in final form 4 August 1999)

Abstract. Several two-equation turbulence models using isotropic eddy viscosity and wall functions are assessed by solution of the neutral atmospheric boundary layer over a flat surface and wind flow over two- and three-dimensional models and real terrain. Calculations are presented for wind flow over the Sirhowy Valley in Wales, an embankment along the Rhine in Germany and the Askervein Hill in Scotland. Comparisons of predictions with previous work, and laboratory and field data, show that the RNG-based k– model gives the best agreement with respect to the flow profiles and length of the separated flow region. The results of this model are analyzed with a non-linear stress-strain relation to gauge the potential effect of turbulence anisotropy. Keywords: Flow separation, k– turbulence model, k–ω turbulence model, Renormalization group (RNG) model, Preferential dissipation modification (PDM) model, Wall functions.

1. Introduction In predominantly mountainous countries such as Korea and Japan, where large areas of land are covered by continuous mountains and valleys, flow separation behind mountains and recirculation in valleys is an important flow feature. With residential and industrial areas located on the flat bottom of valleys, emitted pollutants are difficult to diffuse into the outer atmosphere under certain meteorological conditions. The pollutants become trapped in recirculation zones within valleys. Reliable prediction of pollutant dispersion is clearly impossible without detailed and accurate information about the flow field. Prediction of the flow field in mountainous terrain is the subject of the present paper. A numerical model to predict pollutant dispersion over complex terrain was proposed by Kim and Lee (1998). From comparisons with wind-tunnel experiments using model hills, it was found that a low-Re-number model, which resolved the flow in the sublayer, was needed to make an accurate prediction of the separation region. In a real atmosphere, however, the Reynolds number is very high and due to roughness there is no viscous sublayer. ∗ Visiting Postdoctoral Research Associate, on leave from Advanced Fluids Engineering Research Center, Pohang University of Science and Technology, Pohang, Korea. † Author for correspondence: E-mail: [email protected]

Boundary-Layer Meteorology 94: 5–21, 2000. © 2000 Kluwer Academic Publishers. Printed in the Netherlands.

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For high Reynolds number flow on rough surfaces, it is impossible to avoid using the wall-functions approach (Patel, 1998) in which empirical correlations are used to set the near-wall boundary conditions. In the present investigation, therefore, the numerical method of Kim and Lee (1998) is used with wall functions, and several different but somewhat related two-equation turbulence models, to assess their relative performance in predicting recirculating atmospheric wind flow. This assessment is based on comparisons with available laboratory and field data.

2. Numerical Method and Turbulence Models In this so-called microscale (horizontal scales less than 20 km) resolution of atmospheric flow we solve the Reynolds-averaged equations of continuity and momentum for steady, incompressible flow, namely,   ∂Uj ∂Ui ∂(Ui Uj ) 1 ∂p ∂ ν = 0, =− + − ui uj − 2ij k $j Uk . (1) ∂xj ∂xj ρ ∂xi ∂xj ∂xj Here, Ui and ui are the mean and fluctuation velocities in xi directions, respectively, ρ and ν are the air density and kinematic viscosity, respectively, p is pressure, and $i is the angular velocity of the earth’s rotation. The energy equation is not solved for a neutrally stratified atmosphere. In Equation (1), the Reynolds stress −ui uj is related to the mean flow by the Boussinesq approximation of linear and isotropic eddy viscosity 2 −ui uj = 2νt Sij − kδij , 3

(2)

  ∂Uj i where νt = Cµ k 2 / is the eddy viscosity and Sij = ∂U /2 is the mean + ∂xj ∂xi strain-rate tensor. The eddy viscosity in Equation (2) is obtained from one of five two-equation turbulence models, namely, the k– model (Jones and Launder, 1972), the modified k– model (Duynkerke, 1988), the preferential dissipation modification (PDM) k–  model (Leschziner and Rodi, 1981), the RNG-based k– model (Yakhot and Orszag, 1986), and the k–ω model (Wilcox, 1988). The basic k– model, which is representative of all five models, comprises the transport equations for turbulence kinetic energy (k ≡ ui ui /2)) and its dissipation rate ():   ∂(Ui k) νt ∂k ∂ − ui uj Sij − , = ∂xi ∂xi σk ∂xi (3) ∂(Ui ) ∂ = ∂xi ∂xi



νt ∂ σ ∂xi



 − (C1 ui uj Sij + C2 ). k

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TABLE I Turbulence constants in two-equation models. Models

Cµ

C1

C2

σk

σ

Standard constants Modified constants RNG k–

0.09 0.033 0.085

1.44 1.46 1.42 − C1R

1.92 1.83 1.68

1.0 1.0 0.7179

1.3 2.38 0.7179

Note: C1R = η(1 − η/4.38)/(1 + 0.015η3 ).

Detering and Etling (1985) and Duynkerke (1988) modified the k– model to simulate the neutrally and stably-stratified atmospheric boundary layer (ABL) by adjusting the model constants to include the Coriolis effect on turbulence, and to correct the over-estimation of the eddy viscosity in the upper atmosphere where the wind shear is weaker. Castro and Apsley (1997) adopted the PDM model of Leschziner and Rodi (1981) to account for streamline curvature effects but they obtained only minor improvements. This is the k– model, except the production term in the -equation is replaced as follows: −C1 ui uj Sij → −2.24ui uj Sij − 0.8νt (2Sns )2 ,

(4)

where Sns is the mean-strain rate tensor in streamline coordinates. The RNG k– model of Yakhot and Orszag (1986) is based on renormalization group theory, in which p the k and  equations are modeled by double-expansion in a parameter η = 2Sij Sij k/, which is the ratio of the turbulence time scale to the mean-strain rate scale, and the model constants are theoretically determined. Equations (3) apply to the RNG model but the turbulence coefficients are different. The coefficients in these k– based models are listed in Table I. As already mentioned, all of the above models are used with wall functions, in which U=

u∗ z ln , κ z0

k=

u2∗

, 1/2

Cµ

=

u3∗ , κz

(5)

where u∗ is the friction velocity, z0 is the roughness height, z is distance from the surface, and κ is the Kármán constant (= 0.41). The first grid point is located in the logarithmic region, 30 < z+ , where z+ = u∗ z/ν (Rodi, 1993). The fifth model considered is the k–ω model, which uses a transport equation for ω instead of , where  = Cµ ωk. Patel and Yoon (1995) successfully simulated separated flow over rough surfaces with this model but had to employ a very fine near-wall grid to resolve the flow. Here it is used with wall functions similar to those of Equation (5). In all of the simulations presented here the numerical method of Kim and Lee (1998) is used with appropriate terrain geometry, roughness and boundary conditions. Wind-tunnel or field measurements at a reference site were used to prescribe

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the upstream conditions. The homogeneous Neumann condition was aplied for all flow variables at the outer boundaries. With the exception of the Askervein Hill, all of the calculations relate to nearly two-dimensional topography. Therefore, for these cases, only two-dimensional flow was simulated using orthogonal curvilinear meshs. In the calculation for the Sirhowy Valley, for example, 110 × 50 grid points were used and the grid dependency of the results was assessed with solutions on a 165 × 75 grid. The maximum difference in mean velocities between the two calculations was less than 5%. For the three-dimensional simulation of the Askervein Hill, a mesh of 100 × 70 × 30 was used, with an orthogonal 100 × 70 grid on the ground and the 30 points in the vertical direction distributed non uniformly, with concentration near the surface. The numerical method employs a finite-volume discretization in boundary-fitted coordinates and the SIMPLEC algorithm to couple pressure and velocity on a nonstaggered grid. Details of the numerical method, such as transformation of the equations from general to computational coordinates and discretization formulae, are given in Kim (1997). Solutions were obtained with all of the above described turbulence models but only representative results are presently here to illustrate the similarities and differences. Also, several attempts were made to obtain solutions with a so-called non-linear two-equation model but they were not successful. It appears that others have experienced similar numerical difficulties with these models, particularly for three-dimensional flows. The few solutions that have been published were obtained with near-wall damping functions, which, as already noted, are not appropriate for high-Reynolds-number flows on rough surfaces. However, it is of interest to explore the anisotropy of the normal stresses that is predicted by non-linear eddyviscosity models. To this end, we have analyzed the three-dimensional flow over the Askervein Hill predicted with the linear RNG model to calculate the anisotropy implied by the nonlinear model of Craft et al. (1997):

−ui uj

∗

  k 2 1 = 2νt Sij − kδij + CI νt Sik Skj − Skl Skl δij 3  3 k +CII νt (ik Skj + j k Ski )    k 1 +CIII νt ik j k − lk lk δij  3  2  k 2 +CIV Cµ νt 2 Ski lj + Skj li − Skm lm δij Skl  3  2  k 1 +CV Cµ νt 2 Sik Sj l − Smk Sml δij Skl  3 2 k +CVI Cµ νt 2 Sij Skl Skl 

TEST FOR TURBULENCE MODELS

+CVII Cµ νt

k2 Sij kl kl , 2

9 (6)

where ij = (∂Ui /∂xj −∂Uj /∂xi )/2 is the mean vorticity tensor and CI ∼ CVII are model constants. As noted above, efforts to obtain solutions with this replacing the linear model, Equation (2), with wall functions, Equation (5), were not successful. 3. Atmospheric Boundary Layer over Flat Terrain For a preliminary assessment of the various turbulence models, we consider the neutrally stable atmospheric boundary layer on a flat surface without the Coriolis effect. This is a straightforward calculation under the assumption of one-dimensional (1-D) flow. Figure 1 shows results of calculations with the k– model with the standard and modified constants (Table I) for an atmospheric boundary layer with thickness δ = 500 m and free-stream velocity U∞ = 10 m s−1 . It is found that the velocity and the turbulence kinetic energy profiles predicted with the two models are in close agreement but there is significant difference in the profile of the eddy viscosity. In particular, the standard model is in excellent agreement with the empirical formulae suggested by Brost and Wyngaard (1978). The following empirical function is known to be a good approximation to the eddy viscosity in a neutral or a stably-stratified atmosphere (Duynkerke, 1988): νt z z 1.5 −1 φm , (7) = 1− κu∗ δ δ δ where φm is a function of stability, and equal to unity for a neutral atmosphere. It is of interest to note that the present results with the modified constants are very similar to the prediction of Jung (1994; Figures 3 and 10) who used the same modified constants. In particular, the model with modified constants predicts an eddy viscosity that does not approach zero at the upper edge of the boundary layer. Figure 2 shows another comparison of the two turbulence models in a neutral boundary layer. The calculations with the models are compared with field measurements at the reference site of Askervein Hill, which was evaluated as fully developed boundary-layer data (Mickle et al., 1988). Again it is found that the standard model gives better agreement with the field data than the modified-constant model. The modified constants referred to above were adjusted to secure agreement with measurements in the atmospheric boundary layer over a rough flat terrain. However, as pointed out by Apsley and Castro (1997), the mean-velocity and shear-stress profiles in nearly equilibrium 1-D flows are insensitive to the value of Cµ (this appears in the transport equations in the combination Cµ1/2 k). They recommended the standard constants, which are optimized by more extensive measurements.

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Figure 1. Simulation of 1-D atmospheric boundary layer in neutral condition. : k– model with standard constants; #: k– model with modified constants; ——: empirical formula (Brost and Wyngaard, 1978).

Figure 2. Vertical profiles of mean horizontal velocity, turbulence kinetic energy and Reynolds stress at the reference site of Askervein Hill. Symbols: field measurements by four instruments (Mickle et al., 1988). ——: standard constants (1D-computation); – – – –: modified constants (1D-computation).

4. Wind Flow over a Triangular Ridge Next, we consider wind flow over a two-dimensional triangular ridge (Figure 5) within a boundary layer. Experimental data for this arrangement were obtained by Arya and Shipman (1981). Table II shows the reattachment location predicted with different turbulence models along with experimental observations and some previous calculations. The

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TABLE II Measured and predicted reattachment point behind a triangular ridge. Classification

Source

Reattachment point

Experiment k– (modified) Modified constants

Arya and Shipman (1981) Mouzakis and Bergeles (1991) Jung (1994)

13.0H 10.0H 12.0H

Present computations

9.8H 11.5H 9.8H 13.5H 16.5H

  k– model     k–ω model   PDM model    RNG model    Modified constants

Figure 3. Vertical profiles of horizontal velocity at various locations behind the ridge. Symbols: experiments (Arya and Shipman, 1981); ——: RNG k– model (present); – – –: modified constants (present); – · – · –: PDM k– model (present).

distance to reattachment is measured from the centre of the ridge, (x, z) = (0, H ) being the vertex of the ridge. Figures 3 and 4 show vertical profiles of horizontal velocity and Reynolds shear stress at three locations behind the ridge. Calculations with three turbulence models are compared with experimental data. The RNG model predicts the separation size as 13.5H × 1.8H , which is closest to the wind-tunnel measurement of 13H × 2.5H . An empirical correlation by Hosker (1984) gives the height of the separation zone as follows:   zs L = 1.0 + 1.5 exp −1.3 , H H

(8)

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H. G. KIM AND V. C. PATEL

Figure 4. Vertical profiles of Reynolds stress at various locations behind the ridge. Symbols: experiments (Arya and Shipman, 1981); ——: RNG k– model (present); – – –: modified constants (present); – · – · –: PDM k– model (present).

where zs is the height of the separation cavity, and L and H are the height and width of the obstacle, respectively. According to this formula, the height of the separation cavity would be 1.45H , close to the value predicted by the RNG model but about 40% lower than that observed in the experiments. It is interesting to note from Table II that Jung (1994) reported a reattachment length of 12H while the present model with the same modified constants predicts 16.5H . It is conjectured that this discrepancy is due to larger numerical diffusion in the hybrid scheme employed by Jung (1994). The reattachment length of 10H predicted by Mouzakis and Bergeles (1991) is close to the present result with the k– model (9.8H ), although these authors employed a modified value of Cµ , and wall functions including pressure gradients. Figure 3 shows reasonable agreement between the measurements and present results except in the separation region (x/H = 8). Measurements in this region are suspect because the instruments used could not resolve the direction of the flow. In addition to this limitation of the measurement technique, Arya and Shipman (1981) reported that the experimental uncertainty was about 15%. The RNG model predicts well the Reynolds stress profiles except at x/H = 8. The measured profiles extend to larger vertical distances as would be expected from the larger height of the measured separation cavity. In general, the present predictions with the RNG model are in better agreement with the experiments than those of Mouzakis and Bergeles (1991). Another example of flow over a triangular ridge is shown in Figure 5. In this case, the effect of a porous fence some distance upstream of a triangular ridge is investigated. This configuration is of interest in preventing wind-swept erosion of piles of loose material, such as sand and coal. Wind-tunnel experiments were

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Figure 5. Orthogonal grid around a triangular ridge with an upstream fence. Geometric Parameters: Ridge Height, H (= 15 cm); Ridge Base Length, B(= 19.25 cm); Fence Height, h(= 20 cm); Fence Distance, G(= 32 cm).

performed by Lee and Park (1997), with and without the fence. The numerical computations were carried out with a fence of 50% porosity using the RNG model and the orthogonal grid of Figure 5, and compared with the flow without the fence. The computed streamlines for the two cases are shown in Figure 6. The experimentally determined reattachment points at 11H and 8H , without and with the fence respectively, are in excellent agreement with the predictions shown in the figure. Placement of a porous fence shortens the reattachment length, because the fence decelerates the wind and increases the turbulence intensity.

5. Wind Flow over Real Terrain 5.1. W IND

FLOW IN A PERIODIC VALLEY

The Sirhowy Valley in South Wales has an extensive topography composed of nearly two-dimensional periodic ridges and valleys. Figure 7 shows the valley shape at the field site where measurements were made by Mason and King (1984) and Mason (1987). For the numerical simulation, periodic conditions were imposed on the upstream and downstream faces of the computational domain extending a distance of about 1600 m. A roughness height of 1 cm (Mason, 1987) was imposed. The numerical simulations were carried out both for westerly and easterly winds. The predicted separation and reattachment points are summarized in Table III. The field data are compared with the numerical results in Figure 7 with respect to velocity measurements at a height of 8 m above the valley surface for easterly wind, and the locations of the separation and reattachment points. This figure and Table III show that the RNG model offers satisfactory prediction of the size of the recirculation region.

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TEST FOR TURBULENCE MODELS

TABLE III Separation and reattachment points in Sirhowy Valley. Wind directions

Classification

Westerly

Separation point (m) Reattachment point (m) Length (m)

Easterly

5.2. W IND

Separation point (m) Reattachment point (m) Length (m)

Field data

k– model

RNG model

Modified constants

245

293

270

156

1023

596

1055

1197

778

303

785

1041

845

870

800

675

1254

1205

1215

1250

409

335

415

575

FLOW OVER AN EMBANKMENT

Another example of flow over real terrain is shown in Figure 8, which shows a comparison between the field measurements of Hauf and Neumann-Hauf (1982) and present computations with the RNG model in wind flow over a 2-D embankment constructed along the Rhine River in Germany. The embankment has a height of 2.9 m and base-width of 17.7 m; the ground was covered with grass and the corresponding roughness height was 1.5 cm. In the figure, simulation results for perpendicular (90◦ ) and inclined (45◦ ) oncoming winds to the length direction of the embankment are depicted together. When the oncoming wind has an inclination angle of 45◦ , the speed-up at the top of embankment and the size of the separation cavity on the lee side are reduced due to a decrease in the effective slope of the embankment. Although quantitative measurements were not reported, the RNG model appears to predict the sizes of recirculation region observed in the field experiments. 5.3. W IND

FLOW OVER ASKERVEIN HILL

The field experiment conducted at Askervein Hill in Scotland (57◦ 110 N 7◦ 220 N) under an international collaboration project offers a comprehensive data set for verification of models of flow and turbulence over hilly terrain (Walmsley and Taylor, 1996). As the final example of the present study, the three-dimensional wind field over the Askervein Hill (the left-most hill that resembles an ellipsoid in Figure 9) and surrounding area was simulated using the RNG-based k– turbulence model.

TEST FOR TURBULENCE MODELS

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possible to correlate the observed changes in the normal stresses with the various terms in Equation (6). While it is certain that the isotropic model is much too simple, lack of data at this level of detail preclude any judgement about the realism of the non-linear model considered here. The results of the two models show some trends observed in the wind-tunnel measurements of Finnigan et al. (1990) for the flow over a two-dimensional ridge. It appears that the failure to obtain solutions with this non-linear model is most likely related to the inability of the wall functions to a priori prescribe changes of the type shown in Figure 12. This conclusion is underscored by the fact that most previous solutions with such non-linear models have been obtained with either twodimensional flows, where the role of the wall functions is greatly diminished, or in three-dimensional flows with near-wall modelling, which avoid wall functions.

6. Conclusions Several two-equation turbulence models were applied with wall functions to atmospheric flows in a variety of situations, ranging from a neutral boundary layer on a flat surface to the flow over obstacles and in valleys. The models were assessed by comparison with available wind-tunnel and field data. It is found that the best overall performance is obtained with the RNG-based k– model. This model is a good candidate for prediction of pollutant transport under neutral conditions, and for extension to stable and unstable boundary layers. Finally, the brief investigation into one of the emerging non-linear eddy-viscosity models suggests that a careful study is needed to investigate the role of wall functions in these models, with emphasis on the prediction of three-dimensional flow over rough surfaces.

Acknowledgements This research was supported by Iowa Institute of Hydraulic Research (IIHR) at The University of Iowa and Advanced Fluids Engineering Research Center (AFERC) at Pohang University of Science and Technology, Korea.

References Apsley, D. D. and Castro, I. P.: 1997, ‘A Limited-Length-Scale k– Model for the Neutral and StablyStratified Atmospheric Boundary Layer’, Boundary-Layer Meteorol. 83, 75–98. Arya, S. and Shipman, M.: 1981, ‘An Experimental Investigation of Flow and Diffusion in the Disturbed Boundary Layer over a Ridge – I. Mean Flow and Turbulence Structure’, Atmos. Environ. 15, 1173–1184. Brost, R. and Wyngaard, J. C.: 1978, ‘A Model Study of the Stably Stratified Planetary Boundary Layer’, J. Atmos. Sci. 36, 1821–1822.

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Castro, I. P. and Apsley, D. D.: 1997, ‘Flow and Dispersion over Topography: A Comparison Between Numerical and Laboratory Data for Two-Dimensional Flows’, Atmos. Environ. 31, 839–850. Craft, T. J., Launder, B. E., and Suga, K.: 1997, ‘Prediction of Turbulent Transitional Phenomena with a Nonlinear Eddy-Viscosity Model’, Int. J. Heat Fluid Flow 18, 15–28. Deterling, H. W. and Etling, D.: 1985, ‘Application of the E– Turbulence Model to the Atmospheric Boundary Layer’, Boundary-Layer Meteorol. 33, 113–133. Duynkerke, P. G.: 1988, ‘Application of the E– Turbulence Closure Model to the Neutral and Stable Atmospheric Boundary Layer’, J. Atmos. Sci. 45, 865–880. Finnigan, J. J., Raupach, M. R., Bradley, E. F., and Aldis, G. K.: 1990, ‘A Wind Tunnel Study of Turbulent Flow over a Two-Dimensional Ridge’, Boundary-Layer Meteorol. 50, 277–317. Hauf, T. and Neumann-Hauf, G.: 1982, ‘The Turbulent Wind Flow over an Escarpment’, BoundaryLayer Meteorol. 24, 357–369. Hosker, R. P.: 1984, ‘Flow and Diffusion near Obstacles’, in D. Randerson (ed.), Atmospheric Science and Power Production, pp. 241–325. Jones, W. P. and Launder, B. E.: 1972, ‘The Prediction of Laminarization with a Two-Equation Model of Turbulence’, Int. J. Heat Mass Transfer 15, 301–314. Jung, S. J.: 1994, ‘Application of the E– Turbulence Numerical Model to a Flow and Dispersion Around Triangular Ridge (I)’, J. Korea Air Pollution Res. Assoc. 10, 116–123. Kim, H. G.: 1997, ‘Numerical Investigation of the Pollutant Dispersion over Complex Terrain’, Ph.D. Thesis, Dept. of Mech. Eng. Pohang Univ. of Sci. and Tech. Kim, H. G., Lee, C. M., Lim, H. C., and Kyong, N. H.: 1997, ‘An Experimental and Numerical Study on the Flow over Two-Dimensional Hills’, J. Wind Eng. Ind. Aerodyn. 66, 17–33. Kim, H. G. and Lee, C. M.: 1998, ‘Pollutant Dispersion over Two-Dimensional Hilly Terrain’, KSME Int’l J. 12, 96–111. Lee, S. J. and Park, C. W.: 1997, ‘Surface Pressure Variations on a Triangular Prism by Porous Fences in a Simulated Atmospheric Boundary Layer’, J. Wind Eng. Ind. Aerodyn. 73, 45–58. Leschziner, M. A. and Rodi, W.: 1981, ‘Calculation of Annular and Twin Parallel Jets Using Various Discretisation Schemes and Turbulence-Model Variations’, J. Fluids Eng. 103, 352–360. Mason, P. J. and King, J. C.: 1984, ‘Atmospheric Flow over a Succession of Nearly Two-Dimensional Ridges and Valleys’, Quart. J. Roy. Meteorol. Soc. 110, 821–845. Mason, P. J.: 1987, ‘Diurnal Variations in Flow over a Succession of Ridges and Valleys’, Quart. J. Roy. Meteorol. Soc. 113, 1117–1140. Mickle, R. E., Cook, N. J., Hoff, A. M., Jensen, N. O., Salmon, J. R., Taylor, P. A., Tetzlaff, G., and Teunissen, H. W.: 1988, ‘The Askervein Hill Project: Vertical Profiles of Wind and Turbulence’, Boundary-Layer Meteorol. 43, 143–169. Mouzakis, F. N. and Bergeles, G. C.: 1991, ‘Numerical Prediction of Turbulent Flow over a TwoDimensional Ridge’, Int’l J. Numer. Meth. Fluids 12, 297–296. Patel, V. C.: 1998, ‘Perspective: Flow at High Reynolds Number and over Rough Surfaces – Achilles Heel of CFD’, J. Fluids Eng. 120, 434–444. Patel, V. C. and Yoon, J. Y.: 1995, ‘Application of Turbulence Models to Separated Flow over Rough Surfaces’, J. Fluids Eng. 117, 234–241. Raithby, G. D., Stubley, G. D., and Taylor, P. A.: 1987, ‘The Askervein Hill Project: A Finite Control Volume Prediction of Three-Dimensional Flows over the Hill’, Boundary-Layer Meteorol. 39, 247–267. Rodi, W.: 1993, ‘Turbulence Models and their Application in Hydraulics’, A State-of-the-Art Review, 3rd edn., IAHR Monograph, 104 pp. Salmon, J. R., Bowen, A. J., Hoff, A. M., Johnson, R., Mickle, R. E., and Taylor, P. A.: 1988, ‘The Askervein Hill Experiment: Mean Wind Variations at Fixed Heights above the Ground’, Boundary-Layer Meteorol. 43, 247–271.

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Teunissen, H. W., Shokr, M. E., Bowen, A. J., Wood, C. J., and Green, D. W. R.: 1987, ‘Askervein Hill Project: Wind-Tunnel Simulations at Three Length Scales’, Boundary-Layer Meteorol. 40, 1–29. Walmsley, J. L. and Taylor, P. A.: 1996, ‘Boundary-Layer Flow over Topography: Impacts of the Askervein Study’, Boundary-Layer Meteorol. 78, 291–320. Wilcox, D. C.: 1988, ‘Reassessment of the Scale-Determining Equation for Advanced Turbulence Models’, AIAA J. 26, 1299–1309. Yakhot, V. and Orszag, S. A.: 1986, ‘Renormalization Group Analysis of Turbulence. I. Basic Theory’, J. Scientific Computations 1, 3–51.