The drag on an undulating surface induced by the flow of a turbulent boundary layer

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Abstract
We investigate, using theoretical and computational techniques, the processes that lead to the drag force on a rigid surface that has two—dimensional undulations of length L and height H (with H/L [Lt ] 1) caused by the flow of a turbulent boundary layer of thickness h. The recent asymptotic analyses of Sykes (1980) and Hunt, Leibovich & Richards (1988) of the linear changes induced in a turbulent boundary layer that flows over an undulating surface are extended in order to calculate the leading-order contribution to the drag. It is assumed that L is much less than the natural lengthscale h* = hU0/u* over which the 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 proposed 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 sheltering effect is in good agreement with both the values obtained by numerical integration of the nonlinear equations with a second-order-closure model and experiments. Although the analytical solution is developed using the mixing-length model for the Reynolds stresses, this model is used only in the inner region, where the perturbation shear stress has a significant effect on the mean flow. The analytical perturbation shear stresses are approximately equal to the results from a higher-order closure model, except where there is strong acceleration or deceleration. The asymptotic theory and the results obtained using the numerical model show that the perturbations to the Reynolds stresses in the outer region do not directly contribute a significant part of the drag. This explains why several previous analyses and computations that use the mixing-length model inappropriately throughout the flow lead to values of the drag force that are too large by up to 100%.

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