THE DRAG ON AN UNDULATING SURFACE INDUCED BY THE FLOW OF A TURBULENT BOUNDARY-LAYER

We investigate, using theoretical and computational techniques, the pr ocesses that lead to the drag force on a rigid surface that has two-di mensional undulations of length L and height H (with H/L much less tha n 1) caused by the flow of a turbulent boundary layer of thickness h. The recent asymptotic analyses of Sykes (1980) and Hunt, Leibovich & R ichards (1988) of the linear changes induced in a turbulent boundary l ayer that flows over an undulating surface are extended in order to ca lculate the leading-order contribution to the drag. It is assumed that L is much less than the natural lengthscale h = hU0/u* over which th e boundary layer evolves (u is the unperturbed friction velocity and U0 a mean velocity scale in the approach flow). At leading order, the perturbation to the drag force caused by the undulations arises from a pressure asymmetry at the surface that is produced by the thickening of the perturbed boundary layer in the lee of the undulation. This we term non-separated sheltering to distinguish it from the mechanism pro posed by Jeffreys (1925). Order of magnitude estimates are derived for the other mechanisms that contribute to the drag; the next largest is shown to be smaller than the non-separated sheltering effect by O(u/ U0). The theoretical value of the drag induced by the non-separated sh eltering effect is in good agreement with both the values obtained by numerical integration of the nonlinear equations with a second-order-c losure model and experiments. Although the analytical solution is deve loped using the mixing-length model for the Reynolds stresses, this mo del is used only in the inner region, where the perturbation shear str ess has a significant effect on the mean flow. The analytical perturba tion shear stresses are approximately equal to the results from a high er-order closure model, except where there is strong acceleration or d eceleration. The asymptotic theory and the results obtained using the numerical model show that the perturbations to the Reynolds stresses i n the outer region do not directly contribute a significant part of th e drag. This explains why several previous analyses and computations t hat use the mixing-length model inappropriately throughout the flow le ad to values of the drag force that are too large by up to 100%.