A direct numerical simulation is performed for a supersonic turbulent boundary layer interacting with a compression/expansion ramp at an angle
$\alpha =24^{\circ }$
, matching the same operating conditions of the direct numerical simulation by Priebe & Martín (J. Fluid Mech., vol. 699, 2012, pp. 1–49). The adopted numerical method relies on the high-order spectral difference scheme coupled with a bulk-based, low-dissipative, artificial viscosity for shock-capturing purposes (Tonicello et al., Comput. Fluids, vol. 197, 2020, 104357). Filtered and averaged fields are evaluated to study total kinetic energy transfers in the presence of non-negligible compressibility effects. The compression motions are shown to promote forward transfer of kinetic energy down the energy cascade, whereas expansion regions are more likely to experience backscatter of kinetic energy. A standard decomposition of the subgrid scale tensor in deviatoric and spherical parts is proposed to study the compressible and incompressible contributions in the total kinetic energy transfers across scales. On average, the correlation between subgrid scale dissipation and large-scale dilatation is shown to be caused entirely by the spherical part of the Reynolds stresses (i.e. the turbulent kinetic energy). On the other hand, subtracting the spherical contribution, a mild correlation is still noticeable in the filtered fields. For compressible flows, it seems reasonable to assume that the eddy-viscosity approximation can be a suitable model for the deviatoric part of the subgrid scale tensor, which is exclusively causing forward kinetic energy cascade on average. Instead, more complex models are likely to be needed for the spherical part, which, even in statistical average, provides an important mechanism for backscatter.
This study presents a comprehensive spatial eigenanalysis of fully-discrete discontinuous spectral element methods, now generalizing previous spatial eigenanalysis that did not include time integration errors. The influence of discrete time integration is discussed in detail for different explicit Runge-Kutta (1st to 4th order accurate) schemes combined with either Discontinuous Galerkin (DG) or Spectral Difference (SD) methods, both here recovered from the Flux Reconstruction (FR) scheme. Selected numerical experiments using the improved SD method by Liang and Jameson [1, 2, 3] are performed to quantify the influence of time integration errors on actual simulations. These involve test cases of varied complexity, from one-dimensional linear advection equation studies to well-resolved and under-resolved inviscid vortical flows. The effect of mesh regularity is also considered, where time integration errors are found to be, in the case of irregular grids, less pronounced than those of the spatial discretisation. Still, when simulations are well-resolved, the overall order of accuracy of the (fully-discrete) method of choice is limited to that of the time integration scheme. Moreover, it is shown that, while both well-resolved and under-resolved simulations of linear problems correlate well with the eigenanalysis prediction of time integration errors, the correlation can be much worse for under-resolved nonlinear problems.In fact, for the under-resolved vortical flows considered, the predominance of spatial errors made it practically impossible for time integration errors to be distinctly identified. Nevertheless, for well-resolved nonlinear simulations, the effect of time integration errors could still be recognized. As a result, the eigenanalysis predictions are expected to hold (even if partially) in direct numerical simulations of turbulence. This highlights that the interaction between space and time discretisation errors is more complex than otherwise anticipated, contributing to the current understanding about when eigenanalysis can effectively predict the behavior of numerical errors in practical under-resolved nonlinear problems, including under-resolved turbulence computations.
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