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.

Wall effects on pressure fluctuations in turbulent channel flow

“…Concerning the quasi-homogeneous part of the redistribution term, general tensorial representations are available (Ristorcelli et al 1995;Craft & Launder 1996;Jakirlić & Hanjalić 2002;Gerolymos et al 2012a) which in order to satisfy the two-component limit (tcl) realizability constraint (Shih & Lumley 1993, turbulence tends to tcl as the wall is approached because of the strong damping of velocity fluctuations normal to the wall) should use representation coefficients † that are function of the Reynolds-stress anisotropy tensor invariants (Lumley 1978).…”

Pressure–Strain Correlation Modeling: Towards Achieving Consistency with Rapid Distortion Theory

“…Except for a few notable exceptions, such as Ristorcelli et al [9] and Girimaji [8], these are considered constants, whose values are assigned from algebraic relations and numerical simulations.…”

“…Various pressure strain correlation models have been developed till date. Some of the notable examples include those by Launder, Reece and Rodi [3], Speziale et al [4], Shih and Lumley [5], Johansson and Hallback [6], Sjogren and Johansson [7], Girimaji [8] and Ristorcelli et al [9]. These models attempt to express the pressure strain correlation as a function of the Reynolds stress anisotropy and the mean velocity gradient tensors (Pope [10]) and are categorized as "classical" models [6,7].…”

Correcting Anisotropic Gradient Transport of k

“…The first term in the brackets is due to the anisotropy of the eddy diffusivity as given in the usual anisotropic GTH model (16). The second term is new and accounts for the fact that different components of k have different magnitudes and thus the gradient of each of the components of k is different.…”