Laslo T. Diosady

Higher-Order Methods for Compressible Turbulent Flows Using Entropy Variables

A higher-order space-time discontinuous Galerkin method is presented for the simulation of compressible ows. The eect of the discrete formulation on the nonlinear stability of the scheme is assessed through numerical simulations. For marginally resolved turbulent simulations at moderate Reynolds number, polynomial dealiasing is shown to be necessary in order to maintain stability at high order. With increasing Reynolds number, the formulation using conservative variables is shown to be unstable at high order even when using polynomial dealiasing. Using an entropy variable formulation consistent with established entropy stability theory ensures nonlinear stability at high and innite Reynolds number. The eect of the numerical ux for underresolved turbulent simulations is investigated. A low-Mach modied ux term is presented to suppress the biased pressure-dilatation term seen with other upwind numerical uxes. Subgrid-scale modeling eects of dierent numerical ux functions on the kinetic energy spectrum are examined in the limit of innite Reynolds number.

DNS of Flows over Periodic Hills using a Discontinuous-Galerkin Spectral-Element Method

ow in a channel. The higher-order method is shown to provide increased accuracy relative to low-order methods at a given number of degrees of freedom. The turbulent ow over a periodic array of hills in a channel is simulated at Reynolds number 10,595 using an 8th-order scheme in space and a 4th-order scheme in time. These results are validated against experiment and compared with previous large eddy simulation (LES) results. A preliminary analysis provides insight into how these detailed simulations can be used to improve Reynolds-averaged Navier-Stokes (RANS) modeling.

Tensor-product preconditioners for higher-order space-time discontinuous Galerkin methods

A space-time discontinuous-Galerkin spectral-element discretization is presented for direct numerical simulation of the compressible Navier-Stokes equations. An efficient solution technique based on a matrix-free Newton-Krylov method is developed in order to overcome the stiffness associated with high solution order. The use of tensor-product basis functions is key to maintaining efficiency at high-order. Efficient preconditioning methods are presented which can take advantage of the tensor-product formulation. A diagonalized Alternating-Direction-Implicit (ADI) scheme is extended to the space-time discontinuous Galerkin discretization. A new preconditioner for the compressible Euler/Navier-Stokes equations based on the fast-diagonalization method is also presented. Numerical results demonstrate the effectiveness of these preconditioners for the direct numerical simulation of subsonic turbulent flows.

A Space-Time Discontinuous-Galerkin Approach for Separated Flows

The motivation and goals for developing a space-time spectral-element Discontinuous-Galerkin solver for complex separated flows are discussed. The desire for spectral elements in space and time to leverage current and next-generation computing hardware and enable the development of novel subgrid-scale physical models for scaleresolving simulations at practical engineering Reynolds numbers is discussed. Timing results for hardware-optimized kernels are presented and demonstrate the ability of space-time spectral-elements to utilize a significant fraction of the available computing power of an Intel Xeon processor. A dynamic Variational Multiscale Method is developed and applied to the simulation of channel flow at Re� = 544 .

DNS of Flow in a Low-Pressure Turbine Cascade Using a Discontinuous-Galerkin Spectral-Element Method

A new computational capability under development for accurate and efficient high-fidelity direct numerical simulation (DNS) and large eddy simulation (LES) of turbomachinery is described. This capability is based on an entropy-stable Discontinuous-Galerkin spectral-element approach that extends to arbitrarily high orders of spatial and temporal accuracy and is implemented in a computationally efficient manner on a modern high performance computer architecture. A validation study using this method to perform DNS of flow in a low-pressure turbine airfoil cascade are presented. Preliminary results indicate that the method captures the main features of the flow. Discrepancies between the predicted results and the experiments are likely due to the effects of freestream turbulence not being included in the simulation and will be addressed in the final paper.

Design of a Variational Multiscale Method for High Reynolds Number Compressible Flows

The ultimate goal of this work is the simulation of separated flow about the threedimensional FAITH bump to support Reynolds-averaged modeling efforts. Given the relatively high Reynolds number of this experimental configuration, numerical efficiency becomes paramount to enable practical simulations. This work describes the progress in the design of a VMM for high-Reynold-number separated flows, focusing on performance vs. accuracy trade-offs of the numerical schemes.

Development of a High-Order Space-Time Matrix-Free Adjoint Solver

The growth in computational power and algorithm development in the past few decades has granted the science and engineering community the ability to simulate flows over complex geometries, thus making Computational Fluid Dynamics (CFD) tools indispensable in analysis and design. Currently, one of the pacing items limiting the utility of CFD for general problems is the prediction of unsteady turbulent ows.1{3 Reynolds-averaged Navier-Stokes (RANS) methods, which predict a time-invariant mean flowfield, struggle to provide consistent predictions when encountering even mild separation, such as the side-of-body separation at a wing-body junction. NASAs Transformative Tools and Technologies project is developing both numerical methods and physical modeling approaches to improve the prediction of separated flows. A major focus of this e ort is efficient methods for resolving the unsteady fluctuations occurring in these flows to provide valuable engineering data of the time-accurate flow field for buffet analysis, vortex shedding, etc. This approach encompasses unsteady RANS (URANS), large-eddy simulations (LES), and hybrid LES-RANS approaches such as Detached Eddy Simulations (DES). These unsteady approaches are inherently more expensive than traditional engineering RANS approaches, hence every e ort to mitigate this cost must be leveraged. Arguably, the most cost-effective approach to improve the efficiency of unsteady methods is the optimal placement of the spatial and temporal degrees of freedom (DOF) using solution-adaptive methods.

Simulation of a Hammerhead Payload Fairing in the Transonic Regime

Detached-eddy simulations of unsteady buffet in the transonic regime about a hammerhead payload fairing using the OVERFLOW solver are presented. The computed results are compared to mean and unsteady pressures measured by Coe and Nute. Details of the transonic flow physics provide understanding in regions where the simulation and experimental results mismatch. Spectral and proper orthogonal decompositions of the unsteady surface pressure field are presented to further aid the analysis. A sensitivity study for temporal resolution, spatial operator, and turbulence model is included.

A Linear-Elasticity Solver for Higher-Order Space-Time Mesh Deformation

A linear-elasticity approach is presented for the generation of meshes appropriate for a higher-order space-time discontinuous finite-element method. The equations of linearelasticity are discretized using a higher-order, spatially-continuous, finite-element method. Given an initial finite-element mesh, and a specified boundary displacement, we solve for the mesh displacements to obtain a higher-order curvilinear mesh. Alternatively, for moving-domain problems we use the linear-elasticity approach to solve for a temporally discontinuous mesh velocity on each time-slab and recover a continuous mesh deformation by integrating the velocity. The applicability of this methodology is presented for several benchmark test cases.

DNS of Low-Pressure Turbine Cascade Flows With Elevated Inflow Turbulence Using a Discontinuous-Galerkin Spectral-Element Method

Recent progress towards developing a new computational capability for accurate and efficient high–fidelity direct numerical simulation (DNS) and large–eddy simulation (LES) of turbomachinery is described. This capability is based on an entropy– stable Discontinuous–Galerkin spectral–element approach that extends to arbitrarily high orders of spatial and temporal accuracy, and is implemented in a computationally efficient manner on a modern high performance computer architecture. An inflow turbulence generation procedure based on a linear forcing approach has been incorporated in this framework and DNS conducted to study the effect of inflow turbulence on the suction– side separation bubble in low–pressure turbine (LPT) cascades. The T106 series of airfoil cascades in both lightly (T106A) and highly loaded (T106C) configurations at exit isentropic Reynolds numbers of 60,000 and 80,000, respectively, are considered. The numerical simulations are performed using 8th–order accurate spatial and 4th–order accurate temporal discretization. The changes in separation bubble topology due to elevated inflow turbulence is captured by the present method and the physical mechanisms leading to the changes are explained. The present results are in good agreement with prior numerical simulations but some expected discrepancies with the experimental data for the T106C case are noted and discussed.