A mixing-length model for shallow turbulent wakes

A three-dimensional boundary-layer model of shallow-water flows assuming hydrostatic pressure with negligible numerical diffusion and wave damping has been extended to turbulent flow. A standard two-layer mixing-length model determines vertical length scales. The horizontal mixing length is made a multiple $\beta$ of the vertical value and $\beta$ is determined from comparison with experiment. Eddy viscosity is of a general three-dimensional form where, for example, the horizontal mixing length and associated strain rates determine the magnitude of eddy viscosity and hence vertical mixing (and vice versa). Direct comparison is made with previous experiments for subcritical flow around a conical island of small side slope which exhibits the transition from a vigorous vortex-shedding wake to a steady recirculating wake as the stability parameter, $St$, is increased. The value of $\beta$ influences wake structure, particularly for stability parameters close to the critical (the value at which the wake becomes steady or stable). The critical value in the experiments was 0.4 and this was reproduced in the model with $\beta \,{=}\,6$. Vortex shedding patterns with $\hbox{\it St} \,{=}\, 0.26$ and 0.36 were qualitatively reproduced. The flows were subcritical with an onset Froude number of about 0.2, with values approaching 0.6–0.7 in areas where depth-averaged vorticity magnitude was also greatest, at a small distance from the wet/dry intersection. At this intersection, depth-averaged vorticity approached zero while potential vorticity (depth-averaged vorticity/depth) was at a maximum, indicating the importance of the intersection as an origin for vorticity.