Wave-Induced Turbulent Flow Near a Rough Bed: Implications of a Time-Varying Eddy Viscosity

摘要

Approximate equations governing the turbulent flow produced near a fixed, rough bed by plane, progressive, monochromatic, small amplitude surface waves are derived in detail. The equations involve two approximations: the boundary layer approximation and the Stokes expansion in powers of the wave steepness. An order-of-magnitude analysis of the flow very near the bed provides the basis for two time-varying eddy viscosity models with different vertical structures. In both models, the temporal variation of the viscosity is related to the first few Fourier components of a shear velocity based on the instantaneous bed shear stress. Approximate analytical expressions are derived for the near-bed flow produced by first and second order Stokes waves. Comparison of the first-order solution with available measurements indicates that the theoretical models are physically sound. At first order, the solution is insensitive to the assumed vertical structure of the time-varying part of the viscosity, and temporal variation of the viscosity has a noticeable but relatively small effect on quantities of practical interest. The most dramatic result of the analysis is a predicted reversal of the second-order steady streaming produced by long waves. The prediction of the steady streaming is quite sensitive to the assumed vertical structure of the time-varying part of the eddy viscosity. Qualitative support for the reversal of the steady streaming is found in the available experimental data. In contrast to the steady streaming, the second-order bed shear stress is found to be insensitive to assumptions regarding the temporal variation of the eddy viscosity.