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 $\eta$-grid, where the wall-normal ($y$) and spanwise ($z$) grid sizes are proportional to the local Kolmogorov scale $\eta$. 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: $0.3 \lesssim \Delta y^+ \lesssim 4, \Delta z^+ \simeq 5$ over a smooth wall, and $\ell^+/30 \lesssim \Delta y^+, \Delta z^+ \lesssim 4$ over uneven surfaces, where $\ell^+$ is the smallest surface wavelength. Above the inner layer, $\Delta y^+ \simeq \Delta z^+ \simeq 2\eta^+$. We test $\eta$-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 $\delta^+_0 = 1000$. We assess the accuracy of $\eta$-grid against the conventional Cartesian grids, through comparison with the reference DNS and experimental data. Results from $\eta$-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 $10^3 \lesssim \delta^+_0 \lesssim 10^4$, the number of grid points with $\eta$-grid ($N_\eta$) scales $\propto {\delta^+_0}^{2.47}$ over a smooth wall, and $\propto {\delta^+_0}^{2.0-2.47}$ over riblets, whereas the number of grid points with a Cartesian grid and hyperbolic-tangent $y$-grid ($N_\mathrm{Tanh}$) scales $\propto {\delta^+_0}^{3.0}$. By $\delta^+_0 = 6000$, $N_\eta/N_\mathrm{Tanh} \simeq 0.1$ over a smooth wall, and $N_\eta/N_\mathrm{Tanh} \simeq 0.04$ over typical drag-reducing riblets, with viscous-scaled spacing $15$.