Toward a new Reynolds stress model for rotating turbulent flows

A new approach for modeling turbulent rotating flows is presented. The main objective of this model is to take into account the complex effects of background rotation, which have been elucidated by homogeneous spectral analysis. The model is based on the following elements. First, it is shown, or recalled, that the use of the deviatoric part a_ij = R _ij/R_ll - δ_ij/3 of the Reynolds stress tensor R_ij as the unique tensorial argument for closing unknown terms (a basic principle in actual models) is no longer valid in the presence of solid body rotation. A decomposition of this tensor into two parts (a _ij = a_ij^e + a_ij^z), namely a_ij^e (directional dependence) and a_ij ^z (polarization), is suggested from pure inviscid and homogeneous rapid distortion theory. It is shown that the first part a_ij ^e is preserved while the second a_ij^z is damped by the pressure-velocity correlations through the action of the Coriolis force. This decomposition appears to provide the minimum amount of information in physical space needed to deal with the complexity of the problem. A system of equations for a_ij^e and a_ij^z is then given for predicting in an "ad hoc" way some phenomena consistent with those identified by spectral models, recent direct numerical simulations (DNS), and experimental results. These phenomena are: (1) damping of any term attached to nonlinear interactions at very low Rossby numbers, in accordance with a convenient modification (for rotation) in the ε (dissipation rate) equation, (2) linear "rapid" reorganization of an initial anisotropy, as shown by pure inviscid rapid distortion theory, and (3) nonlinear "Taylor-Proudman structuring" at intermediate Rossby numbers. Although the present model is restricted to the case of pure solid body rotation, developments that could lead to a more general version that takes into account any kind of mean velocity gradient are discussed at the end of the paper. For this purpose, a consistent closure is provided for the "rapid" pressure rate of the strain correlation tensor Φ_ij² = eΦ_ij² + ^zΦ_ij² (involved in the R_ij equation), and for the related contribution involved in the equation for R_ij^e = R_lla _ij^e, without introducing any adjustable constant. Finally, we present a general two-tensor model (R_ij,R_ij ^e;ε) of which we hope will encourage new efforts in second-order modeling.