Numerical modelling of the turbulent flow developing within and over a 3d building array, part ii: a mathematical foundation for a distributed drag force approach
 Fuesang Lien,
 Eugene Yee,
 John D. Wilson
 … show all 3 hide
Rent the article at a discount
Rent now* Final gross prices may vary according to local VAT.
Get AccessAbstract
In this paper, we lay the foundations of a systematic mathematical formulation for the governing equations for flow through an urban canopy (e.g., coarsescaled building array) where the effects of the unresolved obstacles on the flow are represented through a distributed meanmomentum sink. This, in turn, implies additional corresponding terms in the transport equations for the turbulence quantities. More specifically, a modified kε 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 NavierStokes equations, in which the effects of the obstacleatmosphere interaction are included through the introduction of a volumetric momentum sink (representing drag on the unresolved buildings in the array).The kε turbulence closure model requires two additional prognostic equations, namely one for the timeaveraged resolvedscale 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 selfconsistent manner from a parameterization of the sink term in the spatially averaged momentum equation. Towards this objective, the timeaveraged product of the spatially averaged velocity fluctuations and the distributed drag force fluctuations can be approximated systematically using a Taylor series expansion. A highorder 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 twoband spectral decomposition for parameterizing turbulence in a plant canopy is explored in detail.
 Ayotte, K. W., Finnigan, J. J., Raupach, M. R. (1999) A SecondOrder Closure forNeutrally Strati ed Vegetative Canopy Flows. BoundaryLayer Meteorol. 90: pp. 189216
 Belcher, S. E., Jerram, N., Hunt, J. C. R. (2003) Adjustment of a Turbulent Boundary Layer to a Canopy of Roughness Elements. J. Fluid Mech. 488: pp. 369398
 Brown, M. J., Lawson, R. E., DeCroix, D. S., Lee, R. L. (2001) Mass, Heat, and Momentum Exchange between Stands of Plants and their Atmospheric Environment. Quart. J. Roy. Meteorol. Soc. 94: pp. 318332
 Daly, B. J., Harlow, F. H. (1970) Transport Equations of Turbulence. Phys. Fluids 13: pp. 26342649
 DeCroix, D. S., Smith, W. S., Streit, G. E., Brown, M. J. (2000) LargeEddy and Gaussian Simulations of Downwind Dispersion from Large Building HVAC Exhaust. 11th Joint Conference on the Applications of Air Pollution Meteorology with the A&WMA. American Meteorological Society, Boston, MA, pp. 5358
 Finnigan, J. J. Turbulent Transport in Flexible Plant Canopies. In: Hutchison, B. A., Hicks, B. B. eds. (1985) The Forest–Atmosphere Interaction. D. Reidel Publishing Company, Boston, pp. 443480
 Getachew, D., Minkowycz, W. J., 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: pp. 29092915
 Ghosal, S., Moin, P. (1995) The Basic Equations for the LargeEddy Simulation of Turbulent Flows in Complex Geometry. J. Comput. Phys. 118: pp. 2437
 Green, S. R. (1992) Modelling Turbulent Air Flow in a Stand of WidelySpaced Trees. J. Comp. Fluid Dyn. Applic. 5: pp. 294312
 Greiner, W., Scha ¨fer, A. (1994) Quantum Chromodynamics. SpringerVerlag, Berlin
 Hanjalic, K., Launder, B. E., and Schiestel, R.: 1980, 'Multiple TimeScale 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, SpringerVerlag, Berlin, pp. 36–49.
 Howes, F. A., Whitaker, S. (1985) The Spatial Averaging Theorem Revisited. Chem. Eng. Sci. 40: pp. 13871392
 Ince, N. Z., Launder, B. E. (1989) On the Computation of BuoyancyDriven Turbulent Flows in Rectangular Enclosures. Int. J. Heat Mass Transfer 10: pp. 110117
 Inoue, E. (1963) On the Turbulent Structure of Air. ow within Crop Canopies. J. Meteorol. Soc. 41: pp. 317326
 Jaynes, E. T. (1982) On the Rationale of MaximumEntropy Methods. Proc. IEEE 70: pp. 939952
 Kleinert, H. (1990) Gauge Fields in Condensed Matter. World Scienti c Publishing Co., Teaneck, New Jersey
 Launder, B. E., Spalding, D. B. (1974) The Numerical Computation of Turbulent Flows. Comp. Meth. Appl. Mech. Eng. 3: pp. 269289
 Lien, F.S., Yee, E. (2004) Numerical Modelling of the Turbulent Flow Developing within and over a 3D Building Array, Part I: A HighResolution ReynoldsAveraged Navier– Stokes Approach. BoundaryLayer Meteorol 112: pp. 427466
 Lien, F.S., Yee, E. (2005) Numerical Modelling of the Turbulent Flow Developing within and over a 3D Building Array, Part III: A Distributed Drag Force Approach, its Implementation and Application. BoundaryLayer Meteorol 114: pp. 285311
 Lien, F.S., Yee, E., Cheng, Y. (2004) Simulation of Mean Flow and Turbulence over a 2D Building Array Using HighResolution CFD and a Distributed Drag Force Approach. J. Wind Eng. Ind. Aerodyn. 92: pp. 117158
 Liu, J., Chen, J. M., Black, T. A., Novak, M. D. (1996) EModelling of Turbulent Air Flow Downwind of a Model Forest Edge. BoundaryLayer Meteorol. 77: pp. 2144
 Matthews, J., Walker, R. L. (1970) Mathematical Methods of Physics. W. A. Benjamin, Inc., Menlo Park, CA
 Miguel, A. F., van de Braak, N. J., Silvia, A. M., Bot, G. P. A. (2001) WindInduced Airflow through Permeable Materials. Part 1: The Motion Equation. J. Wind Eng. Indust. Aero. 89: pp. 4557
 Raupach, M. R., Shaw, R. H. (1982) Averaging Procedures for Flow within Vegetation Canopies. BoundaryLayer Meteorol. 22: pp. 7990
 Raupach, M. R., Coppin, P. A., Legg, B. J. (1986) Experiments on Scalar Dispersion Within a Model Plane Canopy. Part I: The Turbulence Structure. BoundaryLayer Meteorol. 35: pp. 2152
 Roth, M. (2000) Review of Atmospheric Turbulence over Cities. Quart. J. Roy. Meteorol. Soc. 126: pp. 941990
 Sanz, C. (2003) A Note on k–Modelling of Vegetation Canopy AirFlows. BoundaryLayer Meteorol. 108: pp. 191197
 Scheidegger, A. E. (1974) The Physics of Flow through Porous Media. University of Toronto Press, Toronto, Ontario
 Schiestel, R. (1987) MultipleTimeScale Modelling of Turbulent Flows in One Point Closures. Phys. Fluids 30: pp. 722731
 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.
 Smith, W. S., Reisner, J. M., DeCroix, D. S., Brown, M. J., Lee, R. L., Chan, S. T., Stevens, D. E. (2000) A CFD Model Intercomparison and Validation Using High Resolution Wind Tunnel Data. 11th Joint Conference on the Applications of Air Pollution Meteorology with the A&WMA. American Meteorological Society, Boston, MA, pp. 4146
 Spanier, J., Oldham, K. B. (1987) An Atlas of Functions. Hemisphere Publishing Corporation, New York
 Spiegel, M. R. (1969) Theory and Problems of Real Variables: Lebesgue Measure and Integration with Applications to Fourier Series. McGrawHill Book Company, New York
 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: pp. 4455
 Uchijima, Z., Wright, J. L. (1964) An Experimental Study of Air Flow in a Corn PlantAir Layer. Bull. Natl. Inst. Agric. Sci. (Japan), Ser. A 11: pp. 1965
 Van Dyke, M. D. (1964) Perturbation Methods in Fluid Mechanics. Academic Press, New York
 Vasilyev, O., Lund, T. S., Moin, P. (1998) A General Class of Commutative Filters for LES in Complex Geometries. J. Comput. Phys. 146: pp. 82104
 Wang, H., Takle, E. S. (1995) BoundaryLayer Flow and Turbulence near Porous Obstacles: I. Derivation of a General Equation Set for a Porous Media. BoundaryLayer Meteorol. 74: pp. 7388
 Wang, H., Takle, E. S. (1995) A Numerical Simulation of BoundaryLayer Flows near Shelterbelts. BoundaryLayer Meteorol. 75: pp. 141173
 Wilson, J. D. (1985) Numerical Studies of Flow through a Windbreak. J. Wind Eng. Ind. Aerodyn. 21: pp. 119154
 Wilson, J. D. (1988) A SecondOrder Closure Model for Flow through Vegetation. BoundaryLayer Meteorol. 42: pp. 371392
 Wilson, J. D. and Mooney, C. J.: 1997, 'Comments on “A Numerical Simulation of BoundaryLayer Flows near Shelter Belts” by H. Wang and E. Takle', BoundaryLayer Meteorol. 85, 137–149.
 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.
 Wilson, J. D., Yee, E. (2003) Calculation of Winds Disturbed by an Array of Fences. Agric. For. Meteorol. 115: pp. 3150
 Wilson, J. D., Finnigan, J. J., Raupach, M. R. (1998) A FirstOrder Closure for Disturbed PlantCanopy Flows, and its Application to Winds in a Canopy on a Ridge. Quart. J. Roy. Meteorol. Soc. 124: pp. 705732
 Wilson, N. R., Shaw, R. H. (1977) A HigherOrder Closure Model for Canopy Flow. J. Appl. Meteorol. 16: pp. 11971205
 Yee, E., Kiel, D., Hilderman, T. (2001) Statistical Characteristics of Plume Dispersion from a Localized Source within an Obstacle Array in a Water Channel. Fifth GMU Transport and Dispersion Modelling Workshop. George Mason University, Fairfax, VA
 Title
 Numerical modelling of the turbulent flow developing within and over a 3d building array, part ii: a mathematical foundation for a distributed drag force approach
 Journal

BoundaryLayer Meteorology
Volume 114, Issue 2 , pp 245285
 Cover Date
 20050201
 DOI
 10.1007/s1054600492423
 Print ISSN
 00068314
 Online ISSN
 15731472
 Publisher
 Kluwer Academic Publishers
 Additional Links
 Topics
 Keywords

 Canopy flows
 Disturbed winds
 Drag coefficient
 Turbulence closure
 Urban winds
 Wind models
 Industry Sectors
 Authors

 Fuesang Lien ^{(1)}
 Eugene Yee ^{(2)}
 John D. Wilson ^{(2)}
 Author Affiliations

 1. Department of Mechanical Engineering, University of Waterloo, Waterloo, Ont, Canada
 2. Defence R&D Canada  Suffield, 4000, Medicine Hat, Alberta, T1A 8K6, Canada