Study of the energy convergence of the Karhunen-Loeve decomposition applied to the large-eddy simulation of a high-Reynolds-number pressure-driven boundary layer

Abstract

We study the energy convergence of the Karhunen-Lo\`eve decomposition of the turbulent velocity field in a high-Reynolds-number pressure-driven boundary layer as a function of the number of modes. An energy-optimal Karhunen-Lo\`eve (KL) decomposition is obtained from wall-modeled large-eddy simulations at "infinite" Reynolds number. By explicitly using Fourier modes for the horizontal homogeneous directions, we are able to construct a basis of full rank, and we demonstrate that our results have reached statistical convergence. The KL dimension, corresponding to the number of modes per unit volume required to capture 90% of the total turbulent kinetic energy, is found to be $2.4 \times 10^5 |\Omega|/H^3$ (with $|\Omega|$ the domain volume and $H$ the boundary layer height). This is significantly higher than current estimates, which are mostly based on the method of snapshots. In our analysis, we carefully correct for the effect of subgrid scales on these estimates.