ECINADS Deliverable D30-anisotropic turbulence modelling URANS/OES/DDES Tensorial eddy-viscosity modelling

IMFT has contributed in the context of URANS modelling with a directional anisotropy tensor transport modelling in the Organised Eddy Simulation (OES), (Braza, Perrin, Hoarau, (2006)), by means of URANS/DRSM. A tensorial eddyviscosity concept for the eddy-viscosity has been derived, directionally sensitized to capture non-equilibrium turbulence physics. The turbulence anisotropy tensor in the DRSM model has been projected according to the principal directions of the strain-rate, by transporting a directional criterion of stress-strain misalignment, based on the DESIDER-EU program experiment “The IMFT circular cylinder’, Perrin et al (Exps. in Fluids 2007), and modelling development by Bourguet et al, (AIAA J 2007, J Fluids & Struct. 2008). This modelling is also useful in modifying the RANS turbulence length scale in DES. 1.1 Tensorial eddy-viscosity concept in the turbulence behaviour law This modelling takes into account modification of the turbulence spectrum in the inertial range, due to the non-linear interaction between the coherent and random turbulence processes (Perrin et al, Flomania book, 2006). Therefore, modified turbulence scales are required for flow physics modelling of the turbulent stresses. The equations of motion in the time-domain are the phase-averaged (or ensemble – averaged) Navier-Stokes equations, (collected works in the book by Dervieux, Braza, Dussauge, (1998)-ETMA EU program, Jin & Braza, (1994), Cantwell & Coles, (1983)). It is recalled that by using Differential Reynolds Stress transport Modelling (DRSM) the eddy-diffusion coefficient C used in OES two-equation modelling (Bourdet et al., 2007) was evaluated, by adopting the Boussinesq behaviour law as a first approximation. However, this tends to reinforce isotropic character especially in the near wall region. The first order turbulence behaviour law (in the sense of the strain rate expressions) has been reconsidered by means of a directional eddy viscosity tensor that aims at reinforcing turbulence anisotropy within non-equilibrium regions. This anisotropic constitutive law involves the elements of a spectral decomposition applied to the mean strain-rate tensor, whose respective ‘weights’ are the components of the eddy-viscosity tensor. The IMFT’s circular cylinder test-case allowed quantification of the afore mentioned anisotropy character in the near region of a strongly detached turbulent flow, thanks to an ensemble of advanced measurements technique involving especially time-resolved 3C-PIV (Perrin et al, 2009 in Notes on Num. Fluid Mech. Vol. 103). These results allowed quantification of a directional misalignment between the turbulence anisotropy tensor and the strain rate. This has been used as a criterion for the projection of DRSM anisotropy tensor on the principal directions of the strain rate. This approach is complementary to modifications of turbulence consitutive laws by means of scalar eddy-viscosity (Non Linear Eddy-Viscosity Models, NLEVM, (Gatski and Speziale, (1993), among other) and to the Cas model, (Revell et al, 2005, 2007) that suggested a scalar criterion of stress-strain misalignment derived from DRSM. 1.2 Summary of the OES anisotropic first-order model Three transport equations have been derived from DRSM to close the anisotropic constitutive law previously described. The correponding anisotropic first-order closure scheme thus involves five equations in addition to the three momentum equations in the general three-dimensional case: two transport equations for k and as well asthree corresponding to each component of directional misalignment coefficient, CVi. The scalar eddy-viscosity is replaced by the tensorial one in the whole system. The OES turbulence damping function (Jin & Braza, 1994) is used. The modelling consists therefore of solving the following set of equations: