Application of approximate dispersion-diffusion analyses to under-resolved Burgers turbulence using high resolution WENO and UWC schemes

Abstract

This paper presents a space-time approximate diffusion-dispersion analysis of high-order, finite volume Upwind Central (UWC) and Weighted Essentially Non-Oscillatory (WENO) schemes. We perform a thorough study of the numerical errors to find a-priori guidelines for the computation of under-resolved turbulent flows. In particular, we study the 3-rd, 5-th and 7-th order UWC and WENO reconstructions in space, and 3-rd and 4-th order Runge-Kutta time integrators. To do so, we use the approximate von Neumann analysis for non-linear schemes introduced by Pirozzoli. Moreover, we apply the “1% rule” for the dispersion-diffusion curves proposed by Moura et al. [41] to determine the range of wavenumbers that are accurately resolved by each scheme. The dispersion-diffusion errors estimated from these analyses agree with the numerical results for the forced Burgers' turbulence problem, which we use as a benchmark. The cut-off wavenumbers defined by the “1% rule” are evidenced to serve as a good estimator of the beginning of the dissipation region of the energy cascade and they are shown to be associated to a similar level of dissipation, with independence of the scheme.Finally, we show that WENO schemes are more diffusive than UWC schemes, leading to stable simulations at the price of more dissipative results. It is concluded both UWC and WENO schemes may be suitable schemes for iLES turbulence modeling, given their numerical dissipation level acting at the appropriate wavenumbers.