Volume Averaging for Urban Canopies

Vertical profiles of urban canopy flows are often obtained by averaging flow quantities in horizontal direction. This averaging has to account for the fact that a significant part of the averaging volume may be occupied by solid obstacles, resulting in a reduced fluid volume fraction within the canopy layer. As a consequence, the averaging operation can be defined in two different ways depending on whether the average is normalized by the total volume or by the fluid volume only.

We discuss how the method of volume averaging, as developed in the context of flow through porous media, applies to urban flow profiles. We show how it can be employed to obtain one-dimensional budget equations from their three-dimensional counterparts for either form of averaging. We discuss the relative merits of the two averaging operations and show how the averaging interacts with the surface geometry to affect mean profiles in a distinct way. Based on these results, we argue that the averaging operation and its interaction with the surface geometry should be considered a first-order effect on the shape of mean profiles within urban canopies and similar rough surfaces.