Scaling laws for homogeneous turbulent shear flows in a rotating frame

Abstract

The scaling properties of plane homogeneous turbulent shear flows in a rotating frame are examined mathematically by a direct analysis of the Navier–Stokes equations. It is proved that two such shear flows are dynamically similar if and only if their initial dimensionless energy spectrum E*(k*,0), initial dimensionless shear rate SK0/ε0, initial Reynolds number K20/νε0, and the ratio of the rotation rate to the shear rate Ω/S are identical. Consequently, if universal equilibrium states exist at high Reynolds numbers, they will only depend on the single parameter Ω/S. The commonly assumed dependence of such equilibrium states on Ω/S through the Richardson number Ri=−2(Ω/S)(1−2Ω/S) is proved to be inconsistent with the full Navier–Stokes equations and to constitute no more than a weak approximation. To be more specific, Richardson number similarity is shown to only rigorously apply to certain low-order truncations of the Navier–Stokes equations (i.e., to certain second-order closure models) wherein closure is achieved at the second-moment level by assuming that the higher-order moments are a small perturbation of their isotropic states. The physical dependence of rotating turbulent shear flows on Ω/S is discussed in detail, along with the implications for turbulence modeling.