Linear dynamics of wind waves in coupled turbulent air–water flow. Part 2. Numerical model

Abstract

We develop a numerical model of the interaction between wind and a small-amplitude water wave. The model first calculates the turbulent flows in both the air and water that would be obtained with a flat interface, and then calculates linear perturbations to this base flow caused by a travelling surface wave. Turbulent stresses in the base flow are parameterized using an eddy viscosity derived from a low-turbulent-Reynolds-number κ – ε model. Turbulent stresses in the perturbed flow are parameterized using a new damped eddy viscosity model, in which the eddy viscosity model is used only in inner regions, and is damped exponentially to zero outside these inner regions. This approach is consistent with previously developed physical scaling arguments. Even on the ocean the interface can be aerodynamically smooth, transitional or rough, so the new model parameterizes the interface with a roughness Reynolds number and retains effects of molecular stresses (on both mean and turbulent parts of the flow).The damped eddy viscosity model has a free constant that is calibrated by comparing with results from a second-order closure model. The new model is then used to calculate the variation of form drag on a stationary rigid wave with Reynolds number, R. The form drag increases by a factor of almost two as R drops from 2 × 104 to 2 × 103 and shows remarkably good agreement with the value measured by Zilker & Hanratty (1979). These calculations show that the damped eddy viscosity model captures the physical processes that produce the asymmetric pressure that leads to form drag and also wave growth.Results from the numerical model show reasonable agreement with profiles measured over travelling water waves by Hsu & Hsu (1983), particularly for slower moving waves. The model suggests that the wave-induced flow in the water is irrotational except in an extremely thin interface layer, where viscous stresses are as likely to be important as turbulent stresses. Thus our study reinforces previous suggestions that the region very close to the interface is crucial to wind-wave interaction and shows that scales down to the viscous length may have an order-one effect on the development of the wave.The energy budget and growth rate of the wave motions, including effects of the sheared current and Reynolds number, will be examined in a subsequent paper.