Leveraging unstructured grids for direct numerical simulations of wall turbulence

Abstract

Towards computational cost saving for direct numerical simulations (DNSs) of wall turbulence, we formulate an unstructured grid-generation framework, termed $\hat{I}\cdot$-grid, where the wall-normal ($y$) and spanwise ($z$) grid sizes are proportional to the local Kolmogorov scale $\hat{I}\cdot$. The framework consists of an inner layer, with a thickness $\sim 50$ viscous units, with viscous-scaled grid sizes similar to a conventional DNS grid: $0.3\lesssim \hat{I}”y^{+}\lesssim 4,\hat{I}”z^{+}\simeq 5$ over a smooth wall, and ${\ell }^{+}/30\lesssim \hat{I}”y^{+},\hat{I}”z^{+}\lesssim 4$ over uneven surfaces, where ${\ell }^{+}$ is the smallest surface wavelength. Above the inner layer, $\hat{I}”y^{+}\simeq \hat{I}”z^{+}\simeq 2\hat{I}{\cdot }^{+}$. We test $\hat{I}\cdot$-grid with finite volume and spectral element solvers, and conduct DNSs of turbulent channel flows and boundary layers over smooth wall and various streamwise-aligned riblets, up to friction Reynolds number $\hat{I}{´}_0^{+}=1000$. We assess the accuracy of $\hat{I}\cdot$-grid against the conventional Cartesian grids, through comparison with the reference DNS and experimental data. Results from $\hat{I}\cdot$-grid and the Cartesian grids differ by less than $1\%$, in terms of turbulence statistics up to second-order, and the energy spectra. For turbulent channel flows with $10^3\lesssim \hat{I}{´}_0^{+}\lesssim 10^4$, the number of grid points with $\hat{I}\cdot$-grid ($N_{\hat{I}}\cdot$) scales $\propto {\hat{I}{´}_0^{+}}^{2.47}$ over a smooth wall, and $\propto {\hat{I}{´}_0^{+}}^{2.0-2.47}$ over riblets, whereas the number of grid points with a Cartesian grid and hyperbolic-tangent $y$-grid ($N_{\mathrm{Tanh}}$) scales $\propto {\hat{I}{´}_0^{+}}^{3.0}$. By $\hat{I}{´}_0^{+}=6000$, $N_{\hat{I}}\cdot /N_{\mathrm{Tanh}}\simeq 0.1$ over a smooth wall, and $N_{\hat{I}}\cdot /N_{\mathrm{Tanh}}\simeq 0.04$ over typical drag-reducing riblets, with viscous-scaled spacing $15$.