Mathematical Derivation of Spatially-Averaged Momentum Equations for an Urban Canopy Model Using Underlying Concepts of the Immersed Boundary Method

摘要

We propose a new approach to derive spatially-averaged momentum equations for an urban canopy model that resolves buildings vertically and not horizontally. First, in order to mathematically describe the actual momentum field as a completely continuous field, the underling concepts of the immersed boundary method are employed, where we assume that (i) the entire simulation space, including that occupied by buildings, is filled with a fluid, and (ii) an external body force field exists that reduces the wind speed to zero at all positions coinciding with the space occupied by the buildings. Then, in order to obtain the required spatially-averaged momentum equations in a self-consistent manner, a spatial-averaging operation is applied to the Navier-Stokes equations that include a term representing the external force field. The applied spatial-averaging operation is equivalent to the conventional spatial filtering operation used in large-eddy simulations. To examine the significance of the subgrid-scale (SGS) stresses of the spatially-averaged momentum equations, a numerical simulation is performed for a flow around a regular array of cubical blocks with a grid resolution that is sufficient to resolve the blocks. By estimating the individual terms in the spatially-averaged momentum equations using the simulation results, we show that the SGS stresses contribute significantly to the spatially-averaged momentum budget, and therefore they should not be neglected in urban canopy modelling.