Building proper invariants for eddy-viscosity subgrid-scale models

Abstract

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 = ▸ū. 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 PGGT, QGGT, and RGGT of the tensor GGT, and imposing the proper cubic near-wall behavior, i.e., νe = O(y3), we deduce that the eddy-viscosity is given by νe = (Cs3pqrΔ)2PpGGTQ-(p+1)GGT R(p+5/2)/3GGT. Moreover, only RGGT-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 Cs3pqr = √ 3CVr ≈ 0.458; this guarantees both numerical stability and that the models have less or equal dissipation than Vreman's model, i.e., 0 ≤ νe ≤ νVre. 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, Cs3pqr, should be higher than 0.458 to obtain the right amount of subgrid-scale dissipation, i.e., Cs3pq = 0.572 (p = -5/2), Cs3pr = 0.709 (p = -1), and Cs3qr = 0.762 (p = 0). C 2015 AIP Publishing LLC.