a Viscoelastic Model for Turbulent Flow Over Undulating Topography and Progressive Waves.

In Chapter I, A viscoelastic model is developed to investigate the mean flow perturbation and the drag -force induced by turbulent shear flow over an undulating surface. A relaxation term is added to Miles's (1993) eddy-viscosity model to incorporate the effect of turbulent advection. This term is proportional to the ratio of eddy "turn-over" time to traveling time; accordingly, near the surface, the relaxation model reduces to an eddy-viscosity model or a mixing-length model, while far from the surface it reduces to a rapid-distortion model. Nonlinear terms are included in the Reynolds-averaged equations or motion. Transformed into streamline coordinates, the equations are solved through matched asymptotic expansions, and the solutions are compared with previous theoretical and experimental results. According to order-of-magnitude estimates in Belcher et al. (1993), the drag-force contributed by nonlinear shear stress is of the same order as that contributed by asymmetric pressure arising from the leeward thickening of the perturbed boundary layer, which suggests that its quantitative calculation is necessary. The present model confirms this estimate in most cases. Our analytical results show a dip in shear stress at the interface between the inner and outer layers and provides evidence that this feature is related to eddy-advection. In addition, the present modeling provides a better prediction of perturbation shear stress where there is strong acceleration or deceleration and eddy "memory" and nonlinearity are significant. Numerical calculation using a shooting method gives results that compare well with the analytical ones. The model is then extended to turbulent flow over a traveling water waves where the effect of the critical layer is taken account of. In contrast to most previous models, we find the critical layer plays an important role in the growth of ocean waves in an old sea. In Chapter II, Whitham's average-Lagrangian method is used to derive the evolution equations for the envelope of a gravity-wave train in water of variable depth h(x). An asymptotic solution of these equations reveals that, in addition to the well known long-wave component, locked to the envelope of the wave train and traveling at the group velocity C_{g}, a long-wave component is induced that depends on the derivatives h_{x}, h_ {xx}, .... If the depth change is limited to a finite region, the latter component radiates away from this region as a free wave with an amplitude that depends primarily on the discontinuity of h_{x} at the junction.