Couette flow in channels with wavy walls

Abstract

Three–dimensional Couette flows enclosed by a plane and by a wavy wall are addressed; the wave amplitude is proportional to the mean clearance of the channel multiplied by a small dimensionless parameter ɛ. A perturbation expansion in terms of the powers of ɛ of the full steady Navier–Stokes equations yields a cascade of boundary value problems which are solved at each step in closed form. The supremum value of ɛ for which the expansion converges, is determined as a function of the Reynolds number $$\mathcal{R}e$$ The analytical-numerical algorithm is applied to compute the velocity in the channel to O(ɛ4). Even in the first order approximation O(ɛ), new results are obtained which complement the triple deck theory and its modifications. In particular, the incipient separation–detachment is discussed using the Prandtl-Schlichting criterion of starting eddies. The value ɛ e for which eddies start in the channel, is analytically deduced as a function of $$\mathcal{R}e$$ as well as analytical formulas for the coordinates of the separation points. These analytical formulas show that ɛ e in 3D channels is always less than ɛ e in 2D channels. For non-smooth channels, a criterion of infinitesimally small ɛ e is deduced. The critical value of ɛ up to which bifurcation of the solutions can occur is estimated.