Problem of Steady-State Shear-Flow Turbulence

Some steady-state solutions of two-point nonlinear correlation equations for shear flow turbulence are obtained by expansion in power series in the space variables. The correlation equations, which are for inhomogeneous turbulence, are constructed from the Navier-Stokes equations. To make the problem determinate, the weak-turbulence approximation is used. Because only the low-order terms are retained in the power series used in the expansions, the solutions are accurate only for small values of the space variables. The forms of the solutions show that critical values of a parameter (similar to Reynolds number) exist below which the turbulent fluctuations are zero. The presence of pressure-velocity correlations and of nonlinear production terms in the equations was found to be indispensable if steady-state shear-flow turbulence is to exist. It is concluded that a system of correlation equations for turbulent shear flow which is closed by neglecting the highest order correlations can yield reasonable steady-state solutions.