From high-fidelity high-order to reduced-order modeling for unsteady shock wave/boundary layer interactions

In transonic turbomachinery flows, the shock wave/boundary layer interaction is the primary aerodynamic performance-limiting factor. For the next-generation engines, improvements in efficiency can be achieved, provided that the computational fluid dynamics tools used for the design are able to properly capture this phenomenon. While high-fidelity simulations would be required, their cost is still prohibitive, and industry therefore relies mostly on low-fidelity methods. Besides (unsteady) Reynolds-Averaged Navier-Stokes simulations, frequency-domain approaches, such as the Non-Linear Harmonic method, are also employed. Special care must, however, be taken for the harmonic turbulence closure assumption, in particular in shock-induced separated flows. In this context, the present work intends to carry out high-fidelity simulations shock wave/boundary layer interactions, using a high-order solver, to obtain an accurate database of harmonic turbulence and improve its modeling in frequency-domain approaches for turbomachinery applications. The first milestone of this dissertation is the development of a robust high-order solver, able to perform high-fidelity simulations of shock wave/boundary layer interactions. A scheme based on the flux reconstruction approach is chosen for the spatial discretization. To handle shock waves but avoid damping turbulent fluctuations within the boundary layer, the solver is supplemented with an artificial viscosity method combined with the Ducros sensor. Further stabilization is obtained thanks to a positivity-preserving limiter. A digital filtering technique is adopted to provide a realistic turbulent boundary layer within a reasonable distance from the inlet. For validation purposes, the capabilities of the high-order solver are demonstrated for the canonical case of an oblique shock reflection on a turbulent boundary layer. The results from a wall-resolved implicit large-eddy simulation, performed at the experimental Reynolds number, are thoroughly compared to the abundant literature, and an excellent agreement is reported. Especially, the typical low-frequency broadband unsteadiness of the reflected shock is captured. Conditional averaging is put in place and allows to identify coherent structures of turbulence kinetic energy. This successful experience gives confidence in the use of the high-order solver. The second stage is the study of harmonic turbulence in shock-induced separated flows, for which the investigation is led on the transonic flow over a bump, using wall-resolved implicit large-eddy simulations. To replicate rotor/stator interactions occurring in turbomachinery, harmonic forcing of the back pressure is imposed at the outlet. Various perturbation frequencies are prescribed and encompass different regimes, from a fully locked configuration to a decoupling between the unperturbed and forced flows. The mean solution is, however, found to be independent of the perturbation. In a triple decomposition framework, the coherent component of the flow is extracted by phase-averaging. Organized structures of streamwise velocity and turbulence kinetic energy are highlighted. Whereas of similar shapes beneath the shock system, their extent in the downstream boundary layer is controlled by the frequency of the perturbation. Mean and harmonic turbulent stress budgets are presented. A typical three-peaks distribution of mean turbulent diffusion is reported, which is also found to appear for the coherent turbulent diffusion. Harmonic production arises mainly from the mean shear and its modulation. Finally, the Non-Linear Harmonic method is employed on the same bump configuration. Its inaccurate predictions of the harmonic content of the flow are emphasized and are attributed to the freezing of turbulence, or the neglect of harmonic turbulence. In an attempt to address this issue, the findings related to harmonic production are exploited to derive a simple and analytical model for a harmonic eddy viscosity. Its a priori performance is assessed and a satisfactory quantitative and qualitative match is reported with respect to the reference at the lowest forcing frequencies. These encouraging results give credibility to the methodology developed and applied throughout this work to eventually overcome the frozen turbulence assumption of the Non-Linear Harmonic method.