Leveraging unstructured grids for direct numerical simulations of wall turbulence

We formulate an unstructured grid-generation framework for direct numerical simulations (DNSs) of wall turbulence, termed {\eta}-grid, based on setting the wall-normal (y) and spanwise (z) grid sizes proportional to the local Kolmogorov scale {\eta}. The framework consists of an inner layer, with a thickness ~50 viscous units, with viscous-scaled grid sizes similar to a conventional DNS grid; 0.3 < {\Delta}y+ < 4, {\Delta}z+ ~ 5 over a smooth wall, and l+/30 < {\Delta}y+, {\Delta}z+ < 4 over a non-smooth surface, where l+ is the smallest surface wavelength. Above the inner layer, {\Delta}y+~ {\Delta}z+ ~ 2{\eta}+. We test {\eta}-grid with a finite volume method (FVM) code, as well as a spectral element method (SEM) code, and conduct a campaign of DNSs of turbulent channel flow and turbulent boundary layer over smooth wall and various riblet geometries (as streamwise-aligned microgrooves), up to friction Reynolds number {\delta}+0= 1000. We assess the accuracy of the {\eta}-grid against the conventional Cartesian grids, as well as the reference DNS and experimental data. We obtain less than 1% difference between the {\eta}-grid and the Cartesian grids, in terms of skin-friction coefficient, mean velocity, turbulent stresses, and their spectrograms. Up to {\delta}+0 ~ 104, the number of grid points with the {\eta} -grid (N{\eta}) scales proportional to {\delta}+02.5 over smooth wall, and proportional to {\delta}+02.0 over riblets, whereas the number of grid points with a Cartesian grid and hyperbolic tangent y-gird (NTanh) scales proportional to {\delta}+03.0. This leads to an enormous grid saving with the {\eta}-grid; by {\delta}+0 = 6000, N{\eta} / NTanh ~ 0.1 over smooth wall, and N{\eta} / NTanh ~ 0.03 over typical drag-reducing triangular riblets with tip angle 60o, and viscous-scaled spacing 15.