An Approximate Riemann Solver for Second-Moment Closures

Abstract

The hyperbolic convective subset of a second-moment turbulence closure for the Favreaveraged compressible Navier-Stokes equations can be written as [1]where ρ stands for the mean density, U is the density weighted mean velocity vector, R the Reynolds stress tensor with components R i j = u i u j , E the mean specific total energy, and p the mean pressure which can be expressed via the ideal gas law (with γ being the ratio of specific heats), viz.,For simplicity we will restrict the following presentation to flows with statistically two space dimensions, i.e., a variable vector W = (ρ, ρU, ρV, ρ E, ρ R 11 , ρ R 22 , ρ R 33 , ρ R 12 ) t , such that we can write the system in matrix-vector notation W ,t + (F i (W)) ,i = H(W, ∇W), i = 1, 2,990

A Numerical Tool to Compute Hybrid Euler-Lagrange Compressible Models

“…4 For practical reasons, we need to extend the notion of upwinding schemes developed within the conservative framework to the frame of non-conservative schemes. For all considerations refereing to the latter item, we thus refer to 6 for instance, and to, 4,5 which provide details of the computation of Reynolds stress models in an unsteady framework, and also to, 3 . 1 For sake of simplicity, the approximate Godunov scheme introduced in 7 has been used here to predict solutions in the evolution step.…”

“…The starting point is the following. When focusing on second-moment closures, it has already been checked (see 4,5 but also the recent work by Audebert and Coquel 1,2 ) that standard Euler solvers may fail in some standard situations corresponding to real complex flows. More precisely, the fan of waves in second-moment closures is richer than the one associated with Euler type equations.…”

“…As a result, wall boundary conditions for instance may generate time-space oscillations, which is due to the fact that some among these waves are actually "ghost waves" for upwinding Euler-type solvers. If one turns then to the work, 4,5 it occurs that an obvious way to stabilize the whole process simply consists in accounting for the true solution of the exact Riemann problem associated with the governing set of equations for both first and second-order moments. Hence, approximate Riemann solvers may be easily implemented, and these enable to compute approximations of full Reynolds stress models.…”

Hybrid low-diffusion approximate Riemann solvers for Reynolds-stress transport

“…The so-called isotropic effective pressure p + 2 3 ρk was included [28] in the convective fluxes, while the anisotropic part of the Reynolds-stresses ρu i u j − 2 3 ρkδ i j appearing in the mean-flow momentum and energy equations, was included in the diffusive fluxes (centered discretization in both the predictor and corrector sweeps of MacCormack's scheme [29]). Vandromme and Ha Minh [28] included only the isotropic part of the Reynolds-stresses in the convective flux, because of difficulties, which have since been identified with the fact that the convective part of the RSM-RANS equations (without the nonconservative products [32] associated with Reynolds-stress production by mean-flow velocity-gradients, P i j := −ρu i u ∂ x ũ j − ρu j u ∂ x ũi ) is not hyperbolic [33] because its Jacobian matrix does not have a complete system of eigenvectors [34,35,36]. Morrison [37] used an implicit O(∆ 2 ) MUSCL [38] scheme with Roe fluxes [18], which are contact-discontinuity-resolving [26], with ẽt as mean-flow energy variable.…”

“…Computational examples with these approaches [44] include complex 3-D subsonic flows on structured grids [55], but to the author's knowledge they have not been applied to flows with shock-waves. The mathematics of the construction of Roe fluxes [18] for the simplified RST (Reynolds-stress transport) model-system of Rautaheimo and Siikonen [34] were revisited by Brun et al [35] but without application to actual Reynolds-stress models or flows.…”