Volume averaging for urban canopies

When canopy flows are horizontally averaged to obtain mean profiles, the averaging operation can be defined either as an intrinsic average, normalized by the variable fluid volume, or as a superficial average, normalized by the total volume including solid canopy elements. Properties of spatial averages have been explored extensively in the context of flow through plant canopies, albeit with the assumption that the solid volume fraction is negligible. Without this simplification, properties relevant for non-linear terms apply to intrinsic averages while properties of gradients apply to superficial averages. To avoid inconsistencies and inaccuracies the impact of a non-negligible solid volume fraction should be considered carefully when interpreting mean profiles, when deriving mathematical relations for averaged quantities, and when introducing modelling assumptions for such terms. On this basis, we review the definitions and properties of the method of volume averaging, as developed in the more general context of flow through porous media, and discuss its application to urban canopy flows. We illustrate the properties of intrinsic and superficial averages and their effect on mean profiles with example data from a simulation of flow over constant-height cubes.