A hybrid implicit-explicit method for the LES of compressible flows on unstructured grids

Compressible turbulent flows are ubiquitous in many engineering applications such as gas turbine engines, scramjets, and supersonic aircraft. Large Eddy Simulation (LES) is an accurate and efficient mean to study such flows and to provide insight into improved designs. Practical engineering applications of LES demand cost-effective numerical methods, as well as broad geometrical flexibility offered by unstructured grids. In this work, an adaptive implicit-explicit scheme for the LES of compressible turbulent flows on unstructured grids is developed.The methodology uses a node-based finite-volume discretization with Summation-by-Parts (SBP) property, which, in conjunction with Simultaneous Approximation Terms (SAT) for imposing boundary conditions, leads to a linearly stable semi-discrete scheme for smooth solutions. In the presence of shock waves, the scheme is augmented with a bulk viscosity based method to capture discontinuities. A method is proposed to compute the artificial bulk viscosity on a general unstructured mesh. It is shown that the proposed scheme results in a proper scaling of the artificial dissipation on skewed and on unstructured meshes. The solution is marched in time using a hybrid Implicit-Explicit Runge-Kutta (IMEX-RK) time-advancement scheme. An algorithm for splitting the system into implicit and explicit parts is developed. The splitting method adapts in time and advances only those phenomena and/or regions implicitly in time that are stiff. It is shown that the splitting does not introduce any eigenvalues with positive real parts, which would have introduced instability in the split system. For multi-dimensional problems, a simple change of the algorithm is proposed to further reduce the size of the implicit part. Implemented in a parallel computational framework, the hybrid scheme achieves proper load balance using a dual-constraint, domain decomposition algorithm. The scalability of the method is confirmed up to 8,192 processors. Furthermore, the computational efficiency of the method is investigated. Memory savings compared with a fully implicit scheme is demonstrated and it is shown that the hybrid method can reduce the minimum number of required processors by a factor of two. A notable reduction of computational costs of the hybrid method compared to both fully implicit and fully explicit schemes is also demonstrated.The methodology is validated with various canonical laminar/turbulent shock-free/with shock cases. Each test case is chosen to verify a specific property of the method. Second-order convergence of the scheme is verified for a 2D propagating vortex on prisms. Further tests for which the method has been validated are: a 2D acoustic impulse a boundary layer transition with TS waves a 2D laminar flow over a cylinder on a mixed unstructured grid supersonic flow over a blunt body 2D shock-vorticity/entropy interaction DNS of homogeneous isotropic turbulence on both hexahedral and tetrahedral elements and LES of the flow over a cylinder with a turbulent wake at several Reynolds and Mach numbers.In addition, the LES of subsonic and supersonic turbulent jets, with nozzle geometry included, are presented. A transonic jet with a Mach number 0.89 and a temperature ratio 0.84 (cold) and two supersonic jets with Mach number 1.4 and temperature ratios 1.0 (isothermal) and 1.76 (heated), all issuing from axisymmetric nozzles are considered. The far-field noise is computed from the near-field LES data using the integral solution to the Ffowcs Williams-Hawkings (FWH) equations. The methodology is extensively validated by comparing both the near field flow and the far-field sound with the experimental data obtained at NASA Glenn Research Center. For all the three jets, near field data including axial and radial profiles of mean velocities and turbulent intensities, are in reasonable agreement with the corresponding experimental data. Some differences between the simulation results and the experimental data in the very near nozzle regions have been observed due to laminar boundary layer of the simulations inside the nozzle. This boundary layer is turbulent in the experiment. For the subsonic jet, power spectra densities (PSD) along the nozzle lip-line, and space-time correlations at different axial locations are computed and they were in good agreement with experimental data. For all three jets at most angles, the predicted far-field noise spectra at different angles are all within 3dB of the measured experimental data for Strouhal ranging from 0.05 to 4. Finally, the extensive comparisons of the near-field flow and the far-field sound with the experimental data demonstrate the predictive capability of the methodology.