Boundary-Layer Meteorology

, Volume 114, Issue 2, pp 287–313 | Cite as

Numerical modelling of the turbulent flow developing within and over a 3-d building array, part iii: a istributed drag force approach, its implementation and application

  • Fue-sang Lien
  • Eugene Yee


A modified k-ε model is used for the simulation of the mean wind speed and turbulence for a neutrally-stratified flow through and over a building array, where the array is treated as a porous medium with the drag on the unresolved buildings in the array represented by a distributed momentum sink. More specifically, this model is based on time averaging the spatially averaged Navier–Stokes equation, in which the effects of the obstacle-atmosphere interaction are included through the introduction of a distributed mean-momentum sink (representing drag on the unresolved buildings in the array). In addition, closure of the time-averaged, spatially averaged Navier–Stokes equation requires two additional prognostic equations, namely one for the time-averaged resolved-scale kinetic energy of turbulence,κ, and another for its dissipation rate, ε. The performance of the proposed model and some simplified versions derived from it is compared with the spatially averaged, time-mean velocity and various spatially averaged Reynolds stresses diagnosed from a high-resolution computational fluid dynamics (CFD) simulation of the flow within and over an aligned array of sharp-edged cubes with a plan area density of 0.25. Four different methods for diagnosis of the drag coefficient CDfor the aligned cube array, required for the volumetric drag force representation of the cubes, are investigated here. We found that the model predictions for mean wind speed and turbulence in the building array were not sensitive to the differing treatments of the source and sink terms in the κ and κ equations (e.g., inclusion of only the `zeroth-order' approximation for the source/sink terms compared with inclusion of a higher-order approximation for the source/sink terms in the κ and ε equations), implying that the higher-order approximations of these source/sink terms did not offer any predictive advantage. A possible explanation for this is the utilization of the Boussinesq linear stress–strain constitutive relation within the k–ε modelling framework, whose implicit omission of any anisotropic eddy-viscosity effects renders it incapable of predicting any strong anisotropy of the turbulence field that might exist in the building array.

Distributed drag force Drag coefficient 3-D building arrays Turbulence modelling Urban flows 


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  1. Brown, M. J., Lawson, R. E., DeCroix, D. S., and Lee, R. L.: 2001, Comparison of Centreline Velocity Measurements Obtained around 2D and 3D Building Arrays in a Wind Tunnel, Report LA-UR–01–4138, Los Alamos National Laboratory, 7 pp.Google Scholar
  2. Daly, B. J. and Harlow, F. H.: 1970, 'Transport Equations of Turbulence', Phys. Fluids 13, 2634–2649.Google Scholar
  3. Durbin, P. A. and Pettersson Reif, B. A.: 2001, Statistical Theory and Modelling for Turbulent Flows, John Wiley & Sons, New York, 285 pp.Google Scholar
  4. Getachew, D., Minkowycz, W. J., and Lage, J. L.: 2000, 'A Modi ed Form of the k–Model for Turbulent Flows of an Incompressible Fluid in Porous Media', Int. J. Heat Mass Transfer 43, 2909–2915.Google Scholar
  5. Launder, B. E. and Spalding, D. B.: 1974, 'The Numerical Computation of Turbulent Flows', Comp. Meth. Appl. Mech. Eng. 3, 269–289.Google Scholar
  6. Launder, B. E., Reece, G. J., and Rodi, W.: 1975, 'Progress in the Development of a Reynolds Stress Turbulence Closure', J. Fluid Mech. 68, 537–566.Google Scholar
  7. Leonard, B. P.: 1979, 'A Stable and Accurate Convective Modelling Procedure Based on Quadratic Upstream Interpolation', Comp. Meth. Appl. Mech. Eng. 19, 59–98.Google Scholar
  8. Lee, K. and Howell, J. R.: 1987, 'Forced Convective and Radiative Transfer within a Highly Porous Layer Exposed to a Turbulent External Flow Field', in Proc. ASME-JSME Thermal Engineering Joint Conference, Washington, DC, pp. 377–386.Google Scholar
  9. Lien, F.-S. and Leschziner, M. A.: 1994, 'A General Non-Orthogonal Finite-Volume Algo-rithm for Turbulent Flow at all Speeds Incorporating Second-Moment Closure, Part 1: Numerical Implementation', Comp. Meth. Appl. Mech. Eng. 114, 123–148.Google Scholar
  10. Lien, F.-S. and Yee, E.: 2004, 'Numerical Modelling of the Turbulent Flow Developing within and over a 3-D Building Array, Part I: A High-Resolution Reynolds-Averaged Navier– Stokes Approach', Boundary-Layer Meteorol., 112, 427–466.Google Scholar
  11. Lien, F.-S., Yee, E., and Wilson, J. D.: 2005, 'Numerical Modelling of the Turbulent Flow Developing within and over a 3-D Building Array, Part II: A Mathematical Foundation for a Distributed Drag Force Approach', Boundary-Layer Meteorol. 114, 243–283.Google Scholar
  12. Patankar, S. V.: 1980, Numerical Heat Transfer and Fluid Flow, Hemisphere Publishing Corporation, New York, 197 pp.Google Scholar
  13. Pinard, Jean-Paul J. D., and Wilson, J. D.: 2001, 'First-and Second-Order Closure Models for Wind in a Plant Canopy', J. Appl. Meteorol. 40, 1762–1768.Google Scholar
  14. Raupach, M. R. and Thom, A. S.: 1981, 'Turbulence in and above Plant Canopies', Annu. Rev. Fluid Mech. 13, 97–129.Google Scholar
  15. Raupach, M. R., Finnigan, J. J., and Brunet, Y.: 1996, 'Coherent Eddies and Turbulence in Vegetation Canopies: The Mixing Layer Analogy', Boundary-Layer Meteorol. 78, 351–382.Google Scholar
  16. Rhie, C. M. and Chow, W. L.: 1983, 'Numerical Study of Turbulent Flow Past an Airfoil with Trailing Edge Separation', AIAA J. 21, 1525–1532.Google Scholar
  17. Wilson, J. D.: 1985, 'Numerical Studies of Flow through a Windbreak', J. Wind Eng. Ind. Aerodyn. 21, 119–154.Google Scholar
  18. Wilson, J. D. and Yee, E.: 2003, 'Calculation of Winds Disturbed by an Array of Fences', Agric. For. Meteorol. 115, 31–50.Google Scholar

Copyright information

© Springer 2005

Authors and Affiliations

  • Fue-sang Lien
    • 1
  • Eugene Yee
    • 2
  1. 1.Department of Mechanical EngineeringUniversity of WaterlooWaterlooCanada
  2. 2.Defence R&D Canada – SuffieldAlbertaCanada

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