Analytical theory of homogeneous mean shear turbulence

Abstract

AbstractA compact nonlinear expression for the velocity spectra of homogeneous mean shear flow is derived by means of a simplified two-point closure. It applies to all scales and times. The derived equation can be viewed as a nonlinear extension of the linear, rapid-distortion-theory (RDT) equation. The principal simplification is to model the nonlinear pressure–strain rate as first-order in the spectral anisotropy: a spectral Rotta-equation. This simplified equation and its solution are expressed in terms of the RDT solution. That solution helps reveal the role of nonlinearity. An equation for the velocity spectrum is then obtained at all scales and times. A dominant characteristic predicted for nonlinear behaviour is that the turbulence energy grows exponentially with time, with the spectrum simultaneously moving to smaller and smaller wavenumbers. The nonlinear growth rate is determined. Other analytical predictions of the derived equation include: the conditions for self-similarity; local isotropy; various properties of mean shear flow, including characteristic energy, length and temporal growth scales; and a critique of perturbation theory. Comparisons are made with laboratory experiments and direct numerical simulations. Although the theory applies to all scales and times, including an exact expression for RDT, the calculations are focused on nonlinear behaviour at large times. Several approximations used in this work are examined.