Algebraic/transcendental disturbance growth behind a row of roughness elements

Abstract

This paper is a continuation of the work begun in Goldstein et al. (J. Fluid Mech., vol. 644, 2010, p. 123), who constructed an asymptotic high-Reynolds-number solution for the flow over a spanwise periodic array of relatively small roughness elements with (spanwise) separation and plan form dimensions of the order of the local boundary-layer thickness. While that paper concentrated on the linear problem, here the focus is on the case where the flow is nonlinear in the immediate vicinity of the roughness with emphasis on the intermediate wake region corresponding to streamwise distances that are large in comparison with the roughness dimension, but small in comparison with the distance between the roughness array and the leading edge. An analytical O(h2) asymptotic solution is obtained for the limiting case of a small roughness height parameter h. These weakly nonlinear results show that the spanwise variable component of the wall-pressure perturbation decays as x -5/3 ln x when x →∞ (where x denotes the streamwise distance scaled on the roughness dimension), but the corresponding component of the streamwise velocity perturbation (i.e. the wake velocity) exhibits an O(x1/3 ln x) algebraic/transcendental growth in the main boundary layer. Numerical solutions for h = O(1) demonstrate that the wake velocity perturbation for the fully nonlinear case grows in the same manner as the weakly nonlinear prediction-which is considerably different from the strictly linear result obtained in Goldstein et al. (2010). © 2011 Cambridge University Press.