Building proper invariants for eddy-viscosity subgrid-scale models

Direct simulations of the incompressible Navier-Stokes equations are limited to relatively low-Reynolds numbers. Hence, dynamically less complex mathematical formulations are necessary for coarse-grain simulations. Eddy-viscosity models for large-eddy simulation is probably the most popular example thereof: they rely on differential operators that should properly detect different flow configurations (laminar and 2D flows, near-wall behavior, transitional regime, etc.). Most of them are based on the combination of invariants of a symmetric tensor that depends on the gradient of the resolved velocity field, G = del(u) over bar. In this work, models are presented within a framework consisting of a 5D phase space of invariants. In this way, new models can be constructed by imposing appropriate restrictions in this space. For instance, considering the three invariants P-GGT, Q(GGT), and R-GGT of the tensor T-GG, and imposing the proper cubic near-wall behavior, i.e., nu(e) = O(y(3)), we deduce that the eddy-viscosity is given by nu(e) = (C-s3pqr Delta)(2)P(GGT)(p)Q(GGT)(-(p+1))R(GGT)((p+5/2)/3). Moreover, only R-GGT-dependent models, i.e., p > -5/2, switch off for 2D flows. Finally, the model constant may be related with the Vreman's model constant via C-s3pqr = root 3CV(r) approximate to 0.458; this guarantees both numerical stability and that the models have less or equal dissipation than Vreman's model, i. e., 0 <= nu(e) <= nu(Vr)(e). The performance of the proposed models is successfully tested for decaying isotropic turbulence and a turbulent channel flow. The former test-case has revealed that the model constant, C-s3pqr, should be higher than 0.458 to obtain the right amount of subgrid-scale dissipation, i. e., C-s3pq = 0.572 (p = -5/2), C-s3pr = 0.709 (p = -1), and C-s3qr = 0.762 (p = 0). (C) 2015 AIP Publishing LLC.