Convergence and scaling of large-eddy simulations of a turbulent free jet flow

A large set of large-eddy simulations (LES) is performed for a turbulent free jet flow with Reynolds number 21,000 to investigate systemically the convergence and scaling of the LES results with respect to the turbulence resolution scale (the filter width $\Delta$) and the grid size $h$. Four convergence problems are considered: (a) convergence of the numerical error with $h$, for the Smagorinsky model with fixed $\Delta$; (b) convergence of the Smagorinsky model error with $\Delta$, for fixed $h$ ($h\le\Delta$); (c) convergence of the Smagorinsky model with $\Delta=h$; (d) convergence of the dynamic Smagorinsky model with $\Delta=h$. The convergence results are analyzed for the different LES quantities: the sub- filter eddy viscosity, the sub-filter shear stress, the resolved first, second and third order statistics. The scaling laws of the different LES quantities are analyzed based on the Kolmogorov energy-spectrum, and the LES convergence results agree with the scaling very well.