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

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-epsilon 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-epsilon turbulence closure model requires two additional prognostic equations, namely one for the time-averaged resolved-scale kinetic energy of turbulence, (kappa) over bar, and another for the dissipation rate, epsilon, of (kappa) over bar. The transport equation for (kappa) over bar 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 (kappa) over bar 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 (kappa) over bar. The dissipation rate (epsilon) equation is simply obtained as a dimensionally consistent analogue of the (kappa) over bar 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.