General Mean Velocity Distribution Law for Smooth-Wall Plane Couette Flow

Abstract

Plane Couette flow between two parallel smooth walls is one of the classic wall-bounded shear flows. Analytical description of this flow is still limited to the linear law for laminar flow, the classic law of the wall, and the velocity defect law for fully turbulent flow, although extensive direct numerical simulations (DNS) and laboratory experiments are available. This paper integrates the existing knowledge of mean velocity distribution from theory, experiments, and DNS into a single velocity distribution law by introducing a rational eddy viscosity model. Specifically, the eddy viscosity distribution is approximated by an even rational function which is cubic near the wall, linear in the log-law overlap, and symmetrical about the channel centerline. The rational eddy viscosity model leads to a general velocity distribution law in terms of four inverse hyperbolic tangent functions. This law reduces to the linear law for laminar flow, agrees with the classic van Driest law in the inner region, and is antisymmetrical about the channel centerline. Particularly, it well reproduces DNS and laboratory data for transitional and turbulent flows. Furthermore, this general velocity distribution law results in a general friction law. Finally, the rational eddy viscosity model has clear implications for other wall-bounded flows in future studies.