A direct numerical simulation-based re-examination of coefficients in the pressure–strain models in second-moment closures

The most challenging task in closing the Reynolds-averaged Navier?Stokes equations at the second-moment closure (SMC) level is to model the pressure?rate-of-strain correlation in the transport equation for the Reynolds-stress tensor. The accurate modelling of this term, commonly denoted as ?ij, is the key prerequisite for the correct capturing of the stress anisotropy, which potentially gives SMCs a decisive advantage over the ?anisotropy-blind? eddy-viscosity models. A variety of models for ?ij proposed in the literature can all be expressed as a function of the stress-anisotropy-, rate-of-strain- and rate-of-rotation second-rank tensors, so that the modelling task is reduced to determining the model coefficients. It is, thus, the coefficients, associated with various terms in the expression, which differ from one model to another. The model coefficients have been traditionally determined with reference to the available data for sets of generic flows while being forced to satisfying the known values at flow boundaries. We evaluated the coefficients up to the second-order terms (in stress-anisotropy aij) directly from the DNS database for ?ij and the turbulence variables involved in its modelling. The variations of the coefficients across the flow in a plane channel over a range of Reynolds numbers are compared with several popular models. The analysis provided a reasonable support for the common tensor-expansion representation of both the slow and rapid terms. Apart from the near-wall region and the channel centre, most coefficients for higher Re numbers showed themselves to be reasonably uniform, with the values closest to those proposed by Sarkar et al (1991 J.?Fluid Mech. 227 245?72). An illustration of the coefficient variation for the ?quasi-linear? model is also presented for flow over a backward-facing step.