# Eddies Turbulent flows contain *eddies* or *vortices* of varying scale and energy, which deviate from the mean flow LES aims to resolve these eddies using a mesh ## How are Eddies Resolved? - 4 cells minimum are needed to resolve an eddy - one for each direction - velocity known only at cell centroids - ***sub-grid scale*** model used for eddies of smaller size - The mesh sets the minimum eddy size that can be resolved (implicit LES) - How size of eddies do we want to resolve --> How fine does the mesh need to be? ## Kolmogorov Turbulent Energy Cascade ### Wavenumber $(k)$ - Spatial frequency of the eddy $ k = \frac{2\pi}{d} $ - smaller eddies have a higher wavenumber Turbulent kinetic energy density of an eddy is proportional to its diameter, inversely proportional to wavenumber - area under curve gives turbulent kinetic energy, used in [[Reynolds Averaged Navier-Stokes (RANS)|RANS]] simulations Since the mesh doesn't resolve all eddies, not all of the turbulent kinetic energy is not resolved - A good LES mesh resolves >80% of the turbulent kinetic energy - Remaining ~20% modeled by the sub-grid model # Meshing - Eddy size/energy varies through out the domain - Determine the ***integral length scale*** - length of an eddy corresponding to the average kinetic energy of all the eddies - Representative of all eddies at a given location $ l_0=\frac{\int_0^\infty k^{-1}E(k)d(k)}{\int_0^\infty E(k)d(k)} $ - larger integral length scale --> higher turbulent kinetic energy --> larger eddies --> larger cells - Calculated from a previous RANS calculation $ \underbrace{l_0=\frac{k^{3/2}}{\epsilon}}_{k-\epsilon}\qquad \underbrace{l_0=\frac{k^{1/2}}{C_\mu\omega}}_{k-\omega} $ - good estimate - 5 cells across integral length scale likely to resolve 80% of the turbulent kinetic energy $ \Delta=\frac{l_0}{5} $ - can also rearrange the above $ f=\frac{l_0}{\Delta} $ - if $f < 5$ then mesh is too coarse and needs to be refined