Asymptotic behaviour of commutation errors and the divergence of the Reynolds stress tensor near the wall in the turbulent channel flow

The derivation of the space averaged Navier-Stokes equations for the large eddy simulation (LES) of turbulent incompressible flows introduces two groups of terms which do not depend only on the space averaged flow field variables: the divergence of the Reynolds stress tensor and commutation errors. Whereas the former is studied intensively in the literature, the latter terms are usually neglected. This note studies the asymptotic behaviour of these terms for the turbulent channel flow at a wall in the case that the commutation errors arise from the application of a non-uniform box filter. To perform analytical calculations, the unknown flow field is modelled by a wall law (Reichardt law and 1/a th power law) for the mean velocity profile and highly oscillating functions model the turbulent fluctuations. The asymptotics show that near the wall, the commutation errors are at least as important as the divergence of the Reynolds stress tensor.