An Angular Momentum Integral Equation for Turbulent Flows Over a Permeable Surfaces

The introduction of a porous substrate on an otherwise impermeable wall could offer potential advantages for passive control of wall-bounded turbulent flows. In this paper, an Angular Momentum Integral (AMI) equation is derived from first principles for turbulent flow over a porous substrate. The result is a concise, comprehensive relation for the total surface drag as the sum of the friction drag of a laminar flow over an impermeable surface, the effect of porosity and interfacial slip velocity, and the enhancement of drag due to turbulence. The AMI equation is applied to pore-resolved and volume-averaged direct numerical simulations of flow over permeable surfaces. The results highlight the quantitative impact of Kelvin-Helmholtz instabilities on drag coefficient as the permeability is increased. In the future, the AMI equation will provide a compact assessment tool for determining the accuracy of the volume-averaged models.