A rapid-pressure covariance representation consistent with the Taylor—Proudman theorem materially frame indifferent in the two-dimensional limit

Abstract

A nonlinear variable-coefficient representation for the rapid-pressure covariance appearing in the Reynolds stress and heat-flux equations, consistent with the Taylor–Proudman theorem, is presented. The representation ensures that the modelled second-order equations are frame indifferent with respect to rotation in a number of different flows for which such an invariance is required. The model coefficients are functions of the state of the turbulence; they are valid for all states of a mechanical turbulence, attaining their limiting values only when the limit state is achieved. This is accomplished by a special ansatz that is used to obtain – analytically – the coefficients valid away from the realizability limit. Unlike other rapid-pressure representations in which extreme states are used to set model constants, here the coefficients are variable functions asymptotically consisted with – not fixed by – the limit states of the turbulence field. The mathematical principles invoked do not specify all the coefficients in the model; undetermined coefficients appear as free parameters which are used to ensure that the representation is asymptotically consistent with an experimentally determined equilibrium state of homogeneous sheared turbulence. This is done by ensuring that the modelled evolution equations have the same fixed points as those obtained from numerical and laboratory experiments for the homogeneous shear. Results of computations of homogeneous shear, with rotation and with curvature, are shown. Results are better, in a wide class of planar flows for which the model has not been calibrated, than those of other nonlinear models.