Analysis of mesh effects on turbulent flow statistics

Turbulence models, such as the Smagorinsky model herein, are used to represent the energy lost from resolved to under-resolved scales due to the energy cascade (i.e. non-linearity). Analytic estimates of the energy dissipation rates of a few turbulence models have recently appeared, but none (yet) study the energy dissipation restricted to resolved scales, i.e. after spatial discretization with h > micro scale. We do so herein for the Smagorinsky model. Upper bounds are derived on the com,- puted time-averaged energy dissipation rate, <epsilon(u(h))>, for an under-resolved mesh h for turbulent shear flow. For coarse mesh size O(Re-1) < h < L, it is proven, <epsilon(u(h))> <= [(C-s delta/h)(2) +L-5/(C-s delta)(4)h(+L)5/2(/(C)(s)(delta))4( h)3/2(] U)3(/L,) where U and L are global velocity and length scale and C-s and delta are model parameters. This upper bound being independent of the viscosity at high Reynolds number, is in accord with the equilibrium dissipation law (Kolmogrov's conventional turbulence theory). This estimate suggests over-dissipation for any of C-s > 0 and delta > 0, consistent with numerical evidence on the effects of model viscosity (without wall damping function). Moreover, the analysis indicates that the turbulent boundary layer is a more important length scale for shear flow than the Kolmogorov microscale. (C) 2019 Elsevier Inc. All rights reserved.