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We present a shock capturing method for large-eddy simulation of turbulent flows. The proposed method relies on physical mechanisms to resolve and smooth sharp unresolved flow features that may otherwise lead to numerical instability, such as shock waves and under-resolved thermal and shear layers. To that end, we devise various sensors to detect when and where the shear viscosity, bulk viscosity and thermal conductivity of the fluid do not suffice to stabilize the numerical solution. In such cases, the fluid viscosities are selectively increased to ensure the cell Peclet number is of order one so that these flow features can be well represented with the grid resolution. Although the shock capturing method is devised in the context of discontinuous Galerkin methods, it can be used with other discretization schemes. The performance of the method is illustrated through numerical simulation of external and internal flows in transonic, supersonic, and hypersonic regimes. For the problems considered, the shock capturing method performs robustly, provides sharp shock profiles, and has a small impact on the resolved turbulent structures. These three features are critical to enable robust and accurate large-eddy simulations of shock flows.

We present an implicit large eddy simulation (ILES) of hypersonic boundary-layer transition for a flared cone at Mach number 6.0 and Reynolds number $10.8 \times 10^6$. The simulation is performed using a matrix-free discontinuous Galerkin (DG) method and a diagonally implicit Runge-Kutta (DIRK) scheme on graphics processor units (GPUs). A Jacobian-free Newton-Krylov (JFNK) method is used to solve nonlinear systems arising from the discretization of the Navier-Stokes equations. The ILES simulation exhibits the onset of primary and second-mode instabilities, quite zone, followed by transition to turbulence breakdown. These distinct characteristics of hypersonic flows on cone-like geometries are also observed in experiments. The Stanton number distribution and the pressure fluctuation are studied for different freestream intensities and compared with the experimental data. The results suggest that the ILES method is capable of capturing the onset of transition and turbulence breakdown phenomena for hypersonic turbulent flows past a flared cone.

We present a high-order implicit large-eddy simulation (ILES) approach for simulating transitional turbulent flows. The approach consists of an Interior Embedded Discontinuous Galerkin (IEDG) method for the discretization of the compressible Navier-Stokes equations and a parallel preconditioned Newton-GMRES solver for the resulting nonlinear system of equations. The IEDG method arises from the marriage of the Embedded Discontinuous Galerkin (EDG) method and the Hybridizable Discontinuous Galerkin (HDG) method. As such, the IEDG method inherits the advantages of both the EDG method and the HDG method to make itself well-suited for turbulence simulations. We propose a minimal residual Newton algorithm for solving the nonlinear system arising from the IEDG discretization of the Navier-Stokes equations. The preconditioned GMRES algorithm is based on a restricted additive Schwarz (RAS) preconditioner in conjunction with a block incomplete LU factorization at the subdomain level. The proposed approach is applied to the ILES of transitional turbulent flows over a NACA 65-(18)10 compressor cascade at Reynolds number 250,000 in both design and off-design conditions. The high-order ILES results show good agreement with a subgrid-scale LES model discretized with a second-order finite volume code while using significantly less degrees of freedom. This work shows that high-order accuracy is key for predicting transitional turbulent flows without a SGS model.

We present an implicit large-eddy simulation of transonic buffet over the OAT15A supercritical airfoil at Mach number 0.73, angle of attack 3.5 degrees, and Reynolds number 3 millions. The simulation is performed using a matrix-free discontinuous Galerkin (DG) method and a diagonally implicit Runge-Kutta scheme on graphics processor units. We propose a Jacobian-free Newton-Krylov method to solve nonlinear systems arising from the discretization of the Navier?Stokes equations. The method successfully predicts the buffet onset, the buffet frequency, and turbulence statistics owing to the high-order DG discretization and an efficient mesh refinement for the laminar and turbulent boundary layers. A number of physical phenomena present in the experiment are captured in our simulation, including periodical low-frequency oscillations of shock wave in the streamwise direction, strong shear layer detached from the shock wave due to shock-wave/boundary-layer interaction and small-scale structures broken down by the shear-layer instability in the transition region, and shock-induced flow separation. The pressure coefficient, the root mean square of the fluctuating pressure, and the streamwise range of the shock wave oscillation agree well with the experimental data. The results suggest that the proposed method can accurately predict the onset of turbulence and buffet phenomena at high Reynolds numbers without a subgrid scale model or a wall model.

Over the past few years, high-order discontinuous Galerkin (DG) methods for Large-Eddy Simulation (LES) have emerged as a promising approach to solve complex turbulent flows. However, despite the significant research investment, the relation between the discretization scheme, the subgrid-scale (SGS) model and the resulting LES solver remains unclear. This paper aims to shed some light on this matter. To that end, we investigate the role of the Riemann solver, the SGS model, the time resolution, and the accuracy order in the ability to predict a variety of flow regimes, including transition to turbulence, wall-free turbulence, wall-bounded turbulence, and turbulence decay. The transitional flow over the Eppler 387 wing, the TaylorGreen vortex problem and the turbulent channel flow are considered to this end. The focus is placed on post-processing the LES results and providing with a rationale for the performance of the various approaches.

We present a high-order Implicit Large-Eddy Simulation (ILES) approach for transitional aerodynamic flows. The approach encompasses a hybridized Discontinuous Galerkin (DG) method for the discretization of the Navier?Stokes (NS) equations, and a parallel preconditioned Newton-GMRES solver for the resulting nonlinear system of equations. The combination of hybridized DG methods with an efficient solution procedure leads to a high-order accurate NS solver that is competitive to alternative approaches, such as finite volume and finite difference codes, in terms of computational cost. The proposed approach is applied to transitional flows over the NACA 65-(18)10 compressor cascade and the Eppler 387 wing at Reynolds numbers up to 460,000. Grid convergence studies are presented and the required resolution to capture transition at different Reynolds numbers is investigated. Numerical results show rapid convergence and excellent agreement with experimental data. In short, this work aims to demonstrate the potential of high-order ILES for simulating transitional aerodynamic flows. This is illustrated through numerical results and supported by theoretical considerations.

High-order discontinuous Galerkin (DG) methods have emerged as an attractive approach for large eddy simulation of turbulent flows owing to their high accuracy and implicit dissipation properties. However, the application of DG methods for hypersonic flows is still challenging due to the high-computational cost and the lack of robust shock capturing algorithms. In this paper, we adreess the efficiency and robustness of Discontinuous Galerkin methods. To that end we develop a high-order implicit discontinuous Galerkin method for the numerical simulation of hypersonic flows on graphics processors (GPUs). The main ingredients in our approach include i) implicit high-order DG approximation on unstructured/adapted meshes, ii) shock capturing for hypersonic flows, iii) iterative solution methods with CUDA/MPI implementation on GPU clusters, and iv) effective matrix-free preconditioner with reduced basis approximation of the Jacobian matrix. Numerical results on several test cases are presented to validate our method.