Simulation of turbulent flow over idealized water waves

Turbulent flow over idealized water waves with varying wave slope ak and wave age c/u* is investigated using direct numerical simulations at a bulk Reynolds number Re = 8000. In the present idealization, the shape of the water wave and the associated orbital velocities are prescribed and do not evolve dynamically under the action of the wind. The results show that the imposed waves significantly influence the mean flow, vertical momentum fluxes, velocity variances, pressure, and form stress (drag). Compared to a stationary wave, slow (fast) moving waves increase (decrease) the form stress. At small c/u*, waves act similarly to increasing surface roughness z(o) resulting in mean vertical velocity profiles with shorter buffer and longer logarithmic regions. With increasing wave age, z(o) decreases so that the wavy lower surface is nearly as smooth as a flat lower boundary. Vertical profiles of turbulence statistics show that the wave effects depend on wave age and wave slope but are confined to a region kz < 1 (where k is the wavenumber of the surface undulation and z is the vertical coordinate). The turbulent momentum flux can be altered by as much as 40% by the waves. A region of closed streamlines (or cat's-eye pattern) centred about the critical layer height was found to be dynamically important at low to moderate values of c/u*. The wave-correlated velocity and flux fields are strongly dependent on the variation of the critical layer height and to a lesser extent the surface orbital velocities. Above the critical layer z(cr) the positions of the maximum and minimum wave-correlated vertical velocity w(w) occur upwind and downwind of the peak in z(cr), like a stationary surface. The wave-correlated flux u(w)w(w) is positive (negative) above (below) the critical layer height.