Direct Numerical Simulation of Flow over Periodic Hills up to ReH=10,595\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}

We present fully resolved computations of flow over periodic hills at the hill-Reynolds numbers ReH=5,600$\text {Re}_{H}= 5{,}600$ and ReH=10,595$\text {Re}_{H}= 10{,}595$ with the highest fidelity to date. The calculations are performed using spectral incompressible discontinuous Galerkin schemes of 8th$8^{\text {th}}$ and 7th$7^{\text {th}}$ order spatial accuracy, 3rd$3^{\text {rd}}$ order temporal accuracy, as well as 34 and 180 million grid points, respectively. We show that the remaining discretization error is small by comparing the results to h- and p-coarsened simulations. We quantify the statistical averaging error of the reattachment length, as this quantity is widely used as an ‘error norm’ in comparing numerical schemes. The results exhibit good agreement with the experimental and numerical reference data, but the reattachment length at ReH=10,595$\text {Re}_{H}= 10{,}595$ is predicted slightly shorter than in the most widely used LES references. In the second part of this paper, we show the broad range of capabilities of the numerical method by assessing the scheme for underresolved simulations (implicit large-eddy simulation) of the higher Reynolds number in a detailed h/p convergence study.