On a Non-local Parameterisation for Shear Turbulence and the Uniqueness of its Solutions

A non-local parameterisation of shear turbulence is proposed, which includes a dimensionless multiplicative constant as the sole tunable parameter. Analytical and numerical solutions in the case of plane Couette flow exhibit sheared velocity profiles with logarithmic behaviour near the boundaries, and the classical logarithmic flow profile is reproduced for a semi-infinite domain. We also prove that the families of analytical solutions obtained are locally unique: if the velocity is a strictly-increasing function of the distance from the boundary, a small perturbation of the velocity profile must be of the same functional form as the basic flow.