The statistical dynamics of homogeneous turbulence

The steady distribution function for homogeneous turbulence is studied starting from Liouville's equation, modified by the introduction of an instantaneously fluctuating external force, which acts as a random source of energy. A new technique for solving Liouville's equation is presented giving a systematic development of the concepts of turbulent diffusion and turbulent viscosity. It amounts to a consistent generalization of the random phase approximation. When the rate of input of energy into the kth Fourier component u k has a power form h | k | −α, the functional form of the mean value 〈 u k u − k 〉 can be determined exactly in the limit of large Reynolds number; it is Ah^2{3}| K|^ -{ {13} (5+2) }. Liouville's equation proves an inadequate basis for the steady time-dependent mean uₖ (t) u-₊ (t^) and a more general equation is derived. The new equation can be solved in a similar way and shows that the time-dependent correlation starts like a Gaussian in time, then passes through an exponentially decaying state, then eventually has a power dependence |t-t^|- ^ |k|.