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
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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.
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 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
 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