Mean momentum balance in moderately favourable pressure gradient turbulent boundary layers

Abstract

Moderately favourable pressure gradient turbulent boundary layers are investigated within a theoretical framework based on the unintegrated two-dimensional mean momentum equation. The present theory stems from an observed exchange of balance between terms in the mean momentum equation across different regions of the boundary layer. This exchange of balance leads to the identification of distinct physical layers, unambiguously defined by the predominant mean dynamics active in each layer. Scaling domains congruent with the physical layers are obtained from a multi-scale analysis of the mean momentum equation. Scaling behaviours predicted by the present theory are evaluated using direct measurements of all of the terms in the mean momentum balance for the case of a sink-flow pressure gradient generated in a wind tunnel with a long development length. Measurements also captured the evolution of the turbulent boundary layers from a non-equilibrium state near the wind tunnel entrance towards an equilibrium state further downstream. Salient features of the present multi-scale theory were reproduced in all the experimental data. Under equilibrium conditions, a universal function was found to describe the decay of the Reynolds stress profile in the outer region of the boundary layer. Non-equilibrium effects appeared to be manifest primarily in the outer region, whereas differences in the inner region were attributed solely to Reynolds number effects.