Transformation properties of dynamic subgrid-scale models in a frame of reference undergoing rotation

Abstract

Theoretical consideration is presented for the transformation properties of the subgrid-scale (SGS) models for the SGS stress tensor in a non-inertial frame of reference undergoing rotation. As was previously shown (Speziale, C.G., Geophys. Astrophys. Fluid Dynamics 33, 199 (1985)), an extra correction term is yielded for the SGS stress tensor in the transformation of a rotating frame relative to an inertial framing. We derived the exact expression for the correction term for the spherical Gaussian filter function. Certain transformation rules are imposed on the SGS stress by the derived correction term, namely the SGS stress is not indifferent to a frame rotation, but the divergence of the SGS stress is frame indifferent. Conformity of the modelled SGS stress tensor estimated using the previous dynamic SGS models (the dynamic Smagorinsky, dynamic mixed and nonlinear models) with these transformation rules is examined. It is shown that values for certain model parameters contained in the mixed models can be theoretically determined by imposing these rules. We have conducted the a priori and a postepriori numerical assessments of the SGS models in decaying homogeneous turbulence which is subjected to rotation. All of the previous dynamic models were found to violate the rules except for the nonlinear model. The nonlinear model is form invariant, but the result obtained using the nonlinear model showed significant deviation from the DNS data. Failure of previous models was attributable to insufficient accuracy in approximating the modified cross term in the decomposition of the SGS stress tensor. A dynamic mixed model is proposed to eliminate the truncation error for the modelled correction term, in which multilevel filtering of the velocity field was utilized. The proposed model obeyed the transformation rules when the level of the multifiltering operation was large. It was shown that the defiltered model is derived in the limit of the infinite level of multifiltering and that the defiltered model is form invariant.