On the statistics of instantaneous wall-normal integrals

Wall-bounded turbulence is characterized by large (and very-large) scale motions with wall-normal extent comparable to the boundary layer thickness or channel half-height and with larger streamwise extent. These (V)LSMs are known to carry a substantial proportion of the turbulent kinetic energy and Reynolds stresses. Inspired by classical (RANS-based) integral methods for boundary layers, the statistics of instantaneous wall-normal integrals of velocity are analyzed. Spectral analysis is performed on turbulent channel flow data from Johns Hopkins Turbulence Database to assess the extent to which the use of instantaneous integrals could potentially enhance our ability to understand and model large scale motions. It is found that roughly 40-45% of the Reynolds shear stress is resolved by the instantaneously integrated velocity; a proportion that is made substantially larger when only long streamwise wavelengths are considered.