Boundary-Layer Meteorology

, Volume 114, Issue 2, pp 245–285 | Cite as

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

  • Fue-sang Lien
  • Eugene Yee
  • John D. Wilson


In this paper, we lay the foundations of a systematic mathematical formulation for the governing equations for flow through an urban canopy (e.g., coarse-scaled building array) where the effects of the unresolved obstacles on the flow are represented through a distributed mean-momentum sink. This, in turn, implies additional corresponding terms in the transport equations for the turbulence quantities. More specifically, a modified k--ε model is derived for the simulation of the mean wind speed and turbulence for a neutrally stratified flow through and over a building array, where groups of buildings in the array are aggregated and treated as a porous medium. This model is based on time averaging the spatially averaged Navier--Stokes equations, in which the effects of the obstacle--atmosphere interaction are included through the introduction of a volumetric momentum sink (representing drag on the unresolved buildings in the array).The k--ε turbulence closure model requires two additional prognostic equations, namely one for the time-averaged resolved-scale kinetic energy of turbulence,κ, and another for the dissipation rate, ε, of κ . The transport equation for κ is derived directly from the transport equation for the spatially averaged velocity, and explicitly includes additional sources and sinks that arise from time averaging the product of the spatially averaged velocity fluctuations and the distributed drag force fluctuations. We show how these additional source/sink terms in the transport equation for κ can be obtained in a self-consistent manner from a parameterization of the sink term in the spatially averaged momentum equation. Towards this objective, the time-averaged product of the spatially averaged velocity fluctuations and the distributed drag force fluctuations can be approximated systematically using a Taylor series expansion. A high-order approximation is derived to represent this source/sink term in the transport equation for κ . The dissipation rate (ε) equation is simply obtained as a dimensionally consistent analogue of the κ equation. The relationship between the proposed mathematical formulation of the equations for turbulent flow within an urban canopy (where the latter is treated as a porous medium) and an earlier heuristic two-band spectral decomposition for parameterizing turbulence in a plant canopy is explored in detail.

Canopy flows Disturbed winds Drag coefficient Turbulence closure Urban winds Wind models 


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  1. Ayotte, K. W., Finnigan, J. J., and Raupach, M. R.: 1999, 'A Second-Order Closure forNeutrally Strati ed Vegetative Canopy Flows', Boundary-Layer Meteorol. 90, 189–216.Google Scholar
  2. Belcher, S. E., Jerram, N., and Hunt, J. C. R.: 2003, 'Adjustment of a Turbulent Boundary Layer to a Canopy of Roughness Elements', J. Fluid Mech. 488, 369–398.Google Scholar
  3. Brown, M. J., Lawson, R. E., DeCroix, D. S., and Lee, R. L.: 2001, Comparison of Centerline Velocity Measurements Obtained around 2D and 3D Building Arrays in a Wind Tunnel, Report LA-UR-014138, Los Alamos National Laboratory, 7 pp. Cowan, I. R.: 1968, 'Mass, Heat, and Momentum Exchange between Stands of Plants and their Atmospheric Environment', Quart. J. Roy. Meteorol. Soc. 94, 318–332.Google Scholar
  4. Daly, B. J., and Harlow, F. H.: 1970, 'Transport Equations of Turbulence', Phys. Fluids 13, 2634–2649.Google Scholar
  5. DeCroix, D. S., Smith, W. S., Streit, G. E., and Brown, M. J.: 2000, 'Large-Eddy and Gaussian Simulations of Downwind Dispersion from Large Building HVAC Exhaust', in 11th Joint Conference on the Applications of Air Pollution Meteorology with the A&WMA, American Meteorological Society, Boston, MA, pp. 53–58.Google Scholar
  6. Finnigan, J. J.: 1985, 'Turbulent Transport in Flexible Plant Canopies', in B. A. Hutchison, and B. B. Hicks (eds. ), The Forest–Atmosphere Interaction, D. Reidel Publishing Company, Boston, pp. 443–480.Google Scholar
  7. 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
  8. Ghosal, S. and Moin, P.: 1995, 'The Basic Equations for the Large-Eddy Simulation of Turbulent Flows in Complex Geometry', J. Comput. Phys. 118, 24–37.Google Scholar
  9. Green, S. R.: 1992, 'Modelling Turbulent Air Flow in a Stand of Widely-Spaced Trees', J. Comp. Fluid Dyn. Applic. 5, 294–312.Google Scholar
  10. Greiner, W. and Scha ¨fer, A.: 1994, Quantum Chromodynamics, Springer-Verlag, Berlin, 414 pp.Google Scholar
  11. Hanjalic, K., Launder, B. E., and Schiestel, R.: 1980, 'Multiple Time-Scale Concepts in Turbulent Transport Modelling', in L. J. S. Bradbury, F. Durst, B. E. Launder, F. W. Schmidt, and J. H. Whitelaw (eds. ), Turbulent Shear Flows, Vol. II, Springer-Verlag, Berlin, pp. 36–49.Google Scholar
  12. Howes, F. A. and Whitaker, S.: 1985, 'The Spatial Averaging Theorem Revisited', Chem. Eng. Sci. 40, 1387–1392.Google Scholar
  13. Ince, N. Z. and Launder, B. E.: 1989, 'On the Computation of Buoyancy-Driven Turbulent Flows in Rectangular Enclosures', Int. J. Heat Mass Transfer 10, 110–117.Google Scholar
  14. Inoue, E.: 1963, 'On the Turbulent Structure of Air. ow within Crop Canopies', J. Meteorol. Soc. (Japan)41, 317–326.Google Scholar
  15. Jaynes, E. T.: 1982, 'On the Rationale of Maximum-Entropy Methods', Proc. IEEE 70, 939– 952.Google Scholar
  16. Kleinert, H.: 1990: Gauge Fields in Condensed Matter, World Scienti c Publishing Co., Tea-neck, New Jersey, 1456 pp.Google Scholar
  17. Launder, B. E. and Spalding, D. B.: 1974, 'The Numerical Computation of Turbulent Flows', Comp. Meth. Appl. Mech. Eng. 3, 269–289.Google Scholar
  18. 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
  19. Lien, F.-S. and Yee, E.: 2005, 'Numerical Modelling of the Turbulent Flow Developing within and over a 3-D Building Array, Part III: A Distributed Drag Force Approach, its Implementation and Application', Boundary-Layer Meteorol. 114, 285–311.Google Scholar
  20. Lien, F.-S., Yee, E., and Cheng, Y.: 2004, 'Simulation of Mean Flow and Turbulence over a 2-D Building Array Using High-Resolution CFD and a Distributed Drag Force Approach', J. Wind Eng. Ind. Aerodyn. 92, 117–158.Google Scholar
  21. Liu, J., Chen, J. M., Black, T. A., and Novak, M. D.: 1996, E-Modelling of Turbulent Air Flow Downwind of a Model Forest Edge', Boundary-Layer Meteorol. 77, 21–44.Google Scholar
  22. Matthews, J. and Walker, R. L.: 1970, Mathematical Methods of Physics, 2nd edn., W. A. Benjamin, Inc., Menlo Park, CA, 510.pp.Google Scholar
  23. Miguel, A. F., van de Braak, N. J., Silvia, A. M., and Bot, G. P. A.: 2001, 'Wind-Induced Airflow through Permeable Materials. Part 1: The Motion Equation', J. Wind Eng. Indust. Aero. 89, 45–57.Google Scholar
  24. Raupach, M. R. and Shaw, R. H.: 1982, 'Averaging Procedures for Flow within Vegetation Canopies', Boundary-Layer Meteorol. 22, 79–90.Google Scholar
  25. Raupach, M. R., Coppin, P. A., and Legg, B. J.: 1986, 'Experiments on Scalar Dispersion Within a Model Plane Canopy. Part I: The Turbulence Structure', Boundary-Layer Meteorol. 35, 21–52.Google Scholar
  26. Roth, M.: 2000, 'Review of Atmospheric Turbulence over Cities', Quart. J. Roy. Meteorol. Soc. 126, 941–990.Google Scholar
  27. Sanz, C.: 2003, 'A Note on k–Modelling of Vegetation Canopy Air-Flows', Boundary-Layer Meteorol. 108, 191–197.Google Scholar
  28. Scheidegger, A. E.: 1974, The Physics of Flow through Porous Media, 3rd edn., University of Toronto Press, Toronto, Ontario, 353 pp.Google Scholar
  29. Schiestel, R.: 1987, 'Multiple-Time-Scale Modelling of Turbulent Flows in One Point Clo-sures', Phys. Fluids 30, 722–731.Google Scholar
  30. Shaw, R. H. and Seginer, I.: 1985, 'The Dissipation of Turbulence in Plant Canopies', in 7th AMS Symposium on Turbulence and Diusion, Boulder, CO, pp. 200–203.Google Scholar
  31. Smith, W. S., Reisner, J. M., DeCroix, D. S., Brown, M. J., Lee, R. L., Chan, S. T., and Stevens, D. E.: 2000, 'A CFD Model Intercomparison and Validation Using High Reso-lution Wind Tunnel Data', in 11th Joint Conference on the Applications of Air Pollution Meteorology with the A&WMA, American Meteorological Society, Boston, MA, pp. 41– 46.Google Scholar
  32. Spanier, J. and Oldham, K. B.: 1987, An Atlas of Functions, Hemisphere Publishing Corpo-ration, New York, 700 pp.Google Scholar
  33. Spiegel, M. R.: 1969, Theory and Problems of Real Variables: Lebesgue Measure and Integration with Applications to Fourier Series, McGraw-Hill Book Company, New York, 194 pp.Google Scholar
  34. Thom, A. S.: 1968, 'The Exchange of Momentum, Mass, and Heat between an Artificial Leaf and Airflow in a Wind Tunnel', Quart. J. Roy. Meteorol. Soc. 94, 44–55.Google Scholar
  35. Uchijima, Z. and Wright, J. L.: 1964, 'An Experimental Study of Air Flow in a Corn Plant-Air Layer', Bull. Natl. Inst. Agric. Sci. (Japan), Ser. A 11, 19–65.Google Scholar
  36. Van Dyke, M. D.: 1964, Perturbation Methods in Fluid Mechanics, Academic Press, New York, 229 pp.Google Scholar
  37. Vasilyev, O., Lund, T. S., and Moin, P.: 1998, 'A General Class of Commutative Filters for LES in Complex Geometries', J. Comput. Phys. 146, 82–104.Google Scholar
  38. Wang, H. and Takle, E. S.: 1995 a, 'Boundary-Layer Flow and Turbulence near Porous Obstacles: I. Derivation of a General Equation Set for a Porous Media', Boundary-Layer Meteorol. 74, 73–88.Google Scholar
  39. Wang, H. and Takle, E. S.: 1995b, 'A Numerical Simulation of Boundary-Layer Flows near Shelterbelts', Boundary-Layer Meteorol. 75, 141–173.Google Scholar
  40. Wilson, J. D.: 1985, 'Numerical Studies of Flow through a Windbreak', J. Wind Eng. Ind. Aerodyn. 21, 119–154.Google Scholar
  41. Wilson, J. D.: 1988, 'A Second-Order Closure Model for Flow through Vegetation', Boundary-Layer Meteorol. 42, 371–392.Google Scholar
  42. Wilson, J. D. and Mooney, C. J.: 1997, 'Comments on “A Numerical Simulation of Boundary-Layer Flows near Shelter Belts” by H. Wang and E. Takle', Boundary-Layer Meteorol. 85, 137–149.Google Scholar
  43. Wilson, J. D. and Yee, E.: 2000, 'Wind Transport in an Idealized Urban Canopy' in 3rd Symposium on the Urban Environment, American Meteorological Society, Davis, CA, pp. 40–41.Google Scholar
  44. Wilson, J. D. and Yee, E.: 2003, 'Calculation of Winds Disturbed by an Array of Fences', Agric. For. Meteorol. 115, 31–50.Google Scholar
  45. Wilson, J. D., Finnigan, J. J., and Raupach, M. R.: 1998, 'A First-Order Closure for Disturbed Plant-Canopy Flows, and its Application to Winds in a Canopy on a Ridge', Quart. J. Roy. Meteorol. Soc. 124, 705–732.Google Scholar
  46. Wilson, N. R. and Shaw, R. H.: 1977, 'A Higher-Order Closure Model for Canopy Flow', J. Appl. Meteorol. 16, 1197–1205.Google Scholar
  47. Yee, E., Kiel, D., and Hilderman, T.: 2001, 'Statistical Characteristics of Plume Dispersion from a Localized Source within an Obstacle Array in a Water Channel', in abstracts and presentations, Fifth GMU Transport and Dispersion Modelling Workshop, George Mason University, Fairfax, VA.Google Scholar

Copyright information

© Springer 2005

Authors and Affiliations

  • Fue-sang Lien
    • 1
  • Eugene Yee
    • 2
  • John D. Wilson
    • 2
  1. 1.Department of Mechanical EngineeringUniversity of WaterlooWaterlooCanada
  2. 2.Defence R&D Canada -- Suffield, 4000, Medicine HatAlbertaCanada

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