TURBULENT SHEAR-FLOW OVER SLOWLY MOVING WAVES

We investigate the changes to a fully developed turbulent boundary lay er caused by the presence of a two-dimensional moving wave of waveleng th L = 2-pi/k and amplitude a. Attention is focused on small slopes, a k, and small wave speeds, c, so that the linear perturbations are calc ulated as asymptotic sequences in the limit (u + c)/U(B)(L) --> 0 (u* is the unperturbed friction velocity and U(B)(L) is the approach-flow mean velocity at height L). The perturbations can then be described b y an extension of the four-layer asymptotic structure developed by Hun t, Leibovich & Richards (1988) to calculate the changes to a boundary layer passing over a low hill. When (U + C)/U(B)(L) is small, the mat ched height, z(m) (the height where U(B) equals c), lies within an inn er surface layer, where the perturbation Reynolds shear stress varies only slowly. Solutions across the matched height are then constructed by considering an equation for the shear stress. The importance of the shear-stress perturbation at the matched height implies that the invi scid theory of Miles (1957) is inappropriate in this parameter range. The perturbations above the inner surface layer are not directly influ enced by the matched height and the region of reversed flow below z(m) : they are similar to the perturbations due to a static undulation, bu t the 'effective roughness length' that determines the shape of the un perturbed velocity profile is modified to z(m) = z0 exp(kappac/u). Th e solutions for the perturbations to the boundary layer are used to ca lculate the growth rate of waves, which is determined at leading order by the asymmetric pressure perturbation induced by the thickening of the perturbed boundary layer on the leeside of the wave crest. At firs t order in (u + c)/U(B)(L), however, there are three new effects whic h, numerically, contribute significantly to the growth rate, namely: t he asymmetries in both the normal and shear Reynolds stresses associat ed with the leeside thickening of the boundary layer, and asymmetric p erturbations induced by the varying surface velocity associated with t he fluid motion in the wave; further asymmetries induced by the variat ion in the surface roughness along the wave may also be important.