The Modified Cumulant Expansion for Two-Dimensional Homogeneous Turbulence with a Mean Flow

Homogeneous two-dimensional turbulence in the presence of a mean flow is investigated using the modified zero-fourth-order-cumulant approximation. The homogeneity of turbulence is shown to be compatible with mean flows which have uniform velocity gradients, and among such flows the axisymmetric extension flow perpendicular to the plane of turbulent motions and coplanar flows with uniform velocity gradients are dealt with. In the former case, turbulence can be isotropic and a stationary state with a finite energy and enstrophy is shown to be possible. For the latter case, turbulence cannot be isotropic being distorted by the mean flow and no stationary state is possible since the enstrophy decays in time irrespective of the presence of the mean flow.