Nearly Normal Expansion for Forced and Inhomogeneous Turbulence.

An extension of the Cameron-Martin-Wiener method of "nearly Gaussian" expansion to determine the energy spectrum of stationary, incompressible, forced, inhomogeneous real turbulence. Previous analyses have used an unrealistic model of "homogeneous turbulence" for simplicity. Here it is assumed that the energy and inertial ranges are nearly Gaussian, and that convergence is fast enough for the random part of the field to be represented by just the first two terms in the nearly normal expansion. Difficulties with the time transformation of the white-noise process are avoided by using the representation at only one instant. By dealing with the time derivative of the energy-transfer term, a formulation is obtained similar in some ways to the zero-fourth-cumulant approximation. The energy spectrum is found to be about k exp (-2) in the inertial range; similar results have been obtained by a kinematic argument and have been observed empirically for flows with random intermittencies. Using the intermittency hypothesis, and assuming an eddy viscosity to account for the nonlinear transfer of energy, a correction is obtained for the empirical mixing-length theory of turbulence. 33 pp. Ref.