Capturing shocks and turbulence spectra in compressible flows. Part 2: A new hybrid PPM/WENO method

In the Part 1 of the present paper the performance of several different low and high-order finite-volume methods were assessed by investigating how well they can capture the turbulent spectra of a compressible flow where small smooth turbulent structures interact with shocks and discontinuities. The comparisons showed that a second-order Godunov method with PPM interpolation provides results virtually the same as a fourth-order WENO scheme but at a significant lower cost. However, it is shown that the PPM method fails to provide an accurate representation in the high-frequency range of the spectra. In the present paper we show that this specific issue comes from the slope-limiting procedure and a novel hybrid PPM/WENO method is developed, which has the ability to capture the turbulent spectra with the accuracy of a formally high-order method, but at the cost of the second-order Godunov method. Overall, it is shown that virtually the same physical solution can be obtained much faster by refining a simulation with the second-order method and carefully chosen numerical procedures, rather than running a coarse high-order simulation.