Hello,
Yes, it has already been done in Hannover university, including the scour. A constant and very small diffusion coefficients on the horizontal is probably the better, and on the vertical I do not know (mixing-length ?). As a matter of fact the k-epsilon model should not work properly, as it is a RANS model, supposed to give you the average flow, so it raises in this case a philosophical problem. Practically k-epsilon could give you a diffusion that kills the eddies (however in practice it does not with the mesh size we use).
An important thing is the numerical diffusion in the advection of velocities, so the method of characteristics will perform well with a Courant number in the order of 1 or even more. If you are starting your work, in the while you will get our version 6.3, with the weak form of characteristics that is even better for von Karman eddies.
With best regards,
Jean-Michel Hervouet