A modified-Leray-α subgrid scale model of turbulence

Inspired by the remarkable performance of the Leray-α (and the Navier–Stokes alpha (NS-α), also known as the viscous Camassa–Holm) subgrid scale model of turbulence as a closure model to Reynolds averaged equations (RANS) for flows in turbulent channels and pipes, we introduce in this paper another subgrid scale model of turbulence, the modified Leray-α (ML-α) subgrid scale model of turbulence. The application of the ML-α to infinite channels and pipes gives, due to symmetry, similar reduced equations as Leray-α and NS-α. As a result the reduced ML-α model in infinite channels and pipes is equally impressive as a closure model to RANS equations as NS-α and all the other alpha subgrid scale models of turbulence (Leray-α and Clark-α). Motivated by this, we present an analytical study of the ML-α model in this paper. Specifically, we will show the global well-posedness of the ML-α equation and establish an upper bound for the dimension of its global attractor. Similarly to the analytical study of the NS-α and Leray-α subgrid scale models of turbulence we show that the ML-α model will follow the usual k−5/3 Kolmogorov power law for the energy spectrum for wavenumbers in the inertial range that are smaller than 1/α and then have a steeper power law for wavenumbers greater than 1/α (where α > 0 is the length scale associated with the width of the filter). This result essentially shows that there is some sort of parametrization of the large wavenumbers (larger than 1/α) in terms of the smaller wavenumbers. Therefore, the ML-α model can provide us another computationally sound analytical subgrid large eddy simulation model of turbulence.