Paul D. Orkwis

Discontinuous Galerkin Scheme Applied to Chimera Overset Viscous Meshes on Curved Geometries

The Chimera overset method is a powerful technique for modeling uid ow associated with complex engineering problems. The use of structured meshes has enabled engineers to develop a number of high-order schemes, such as the WENO and compact di erencing schemes. This paper demonstrates a Discontinuous Galerkin (DG) scheme with a Chimera overset method applied to viscous meshes on curved geometries. The small stencil of the DG scheme makes it particularly suitable for Chimera meshes. The small stencil simpli es the hole cutting and partitioning of grids that contain holes. In addition, because the DG scheme represents the solution as cell local polynomials, it does not require an interpolation scheme with a large stencil to establish the inter-grid communication in overlapping regions. Furthermore, the DG scheme is capable of using curved cells to represent geometric features. The curved cells resolve issues associated with linear non-co-located Chimera viscous meshes used for nite volume and nite di erence schemes. The DG-Chimera method is demonstrated on a set of viscous Chimera meshes, which would produce erroneous results for a nite volume or nite di erence scheme without corrections to the interpolation.

Extending the Discontinuous Galerkin Scheme to the Chimera Overset Method

The Chimera overset method is a powerful technique for modeling uid ow associated with complex engineering problems. The use of structured meshes has enabled engineers to develop a number of high-order schemes, such as the WENO and compact di erencing schemes. This paper demonstrates a methodology for using the Discontinuous Galerkin (DG) scheme with Chimera overset meshes. The small stencil of the DG scheme makes it particularly suitable for Chimera meshes as it simpli es the inter-grid communication scheme as well as hole cutting procedures for parallel computing. In addition, because the DG scheme represents the solution as cell local polynomials, the DG-Chimera scheme does not require a donor interpolation method with a large stencil. The DG-Chimera method also does not require the use of fringe points to maintain the interior stencil across inter-grid boundaries. Thus, inter-grid communication can be established so long as the receiving boundary is enclosed by or abuts to the donor mesh. This makes the inter-grid communication procedure applicable to both Chimera and zonal meshes. Details of DGChimera scheme are presented, and the method is demonstrated on a set of inviscid ow problems.

A implicit, discontinuous Galerkin Chimera solver using automatic differentiation

The development of a fully-implicit, discontinuous Galerkin solver based on Chimera, overset grids is presented. Newton’s method is used as a nonlinear solver for governing equation sets, enabled by an intrinsic automatic differentiation capability that computes the linearization of the spatial scheme. A GMRES matrix-solver is used to solve the linear system along with a block-ILU0 preconditioner. Results include solutions of the Euler equations for subsonic flow over a smooth bump as well as over a circular cylinder. The convergence rates for spatial orders of accuracy for the solver match the analytical rates. Up to 4th-order accuracy is shown for 3rd-order polynomials (P3). Additionally, quadratic convergence is demonstrated for the nonlinear solver; verifying that the automatic differentiation capability was implemented successfully. Solutions converged in 7 or fewer iterations. Results are presented utilizing the Chimera overset grid capability, demonstrating that quadratic convergence is maintained with the Chimera grid interfaces.

Modeling and Simulation of Bleed Holes in the Presence of Shock Wave/Boundary Layer Interaction

Bleed simulation and modeling are studied for shock wave boundary layer interactions using computational simulations obtained with the OVERFLOW Navier-Stokes equation solver for a mixed compression inlet geometry studied experimentally by the U.S. Air Force Research Laboratory and a single hole unit problem. The AFRL test rig is simulated with bleed using the Slater model using expected test bleed rates. A test of a modified bleed configuration was also conducted with the intent of affecting flow quality by controlling corner separation. The single hole unit problem was simulated with variations in plenum exit pressure, upstream Reynolds number and plenum depth and compared with a hole-only analysis developed previously by the authors. A full simulation of the AFRL rig with a row of discrete cooling holes was also conducted to demonstrate an application of the technique. Results show that the flow quality at the aerodynamic interface plane can be affected positively by side wall bleed. They also show that Reynolds number has only a minor effect on bleed mass flow but does alter the main flow/bleed hole interaction. Both plenum depth and exit pressure affect both bleed mass flow and main flow/bleed hole interaction much more significantly.

A Discontinuous Galerkin Chimera Scheme with Implicit Artificial Boundaries

The Chimera overset methods use of structured meshes has enabled engineers to utilize a number of high-order schemes, such as the WENO and compact di erencing schemes, for modeling uid ow associated with complex engineering problems. However, the large stencil of these high-order schemes leads to complications with establishing arti cial intergrid communication boundaries. This paper demonstrates a Discontinuous Galerkin (DG) scheme with a Chimera overset method that utilizes implicit arti cial boundaries to accelerate the convergence rate to a steady solution and reduces execution time for implicit time accurate calculations. The small stencil of the DG scheme makes it particularly suitable for Chimera meshes. The small stencil simpli es the hole cutting and partitioning of grids that contain holes. In addition, because the DG scheme represents the solution as cell local polynomials, it does not require an interpolation scheme with a large stencil to establish the inter-grid communication in overlapping regions. The convergence rate of the Chimera schemes is dramatically increased by including the linearization of the inter-grid communication in comparison with traditional Quasi-Newton Chimera schemes where only the right hand side vector is updated through the inter-grid communication.

Hole Cutting of Curved Discontinuous Galerkin Chimera Overset Meshes using a Direct Cut Method

The Chimera overset method has enabled the use of high-order nite di erence and nite volume approaches such as WENO and compact di erencing schemes, which require structured meshes for modeling uid ow associated with complex geometries. The exibility of the Chimera scheme is due, in large measure, to its ability to exclude regions of grids that lie outside the computational domain. The exclusion process is commonly known as hole cutting. However, the large stencils required by high-order nite di erence and nite volume schemes complicate the process of generating holes. Unlike the high-order nite di erence and nite volume schemes, the Discontinuous Galerkin (DG) method always retains a stencil that involves only the immediate neighbors of a given cell regardless of the order of approximation. The small stencil implies hole cutting can be performed without regard to maintaining a minimum stencil, which greatly simpli es the hole cutting process. However, DG schemes also require the use of curved cells to correctly represent curved geometries. This paper presents a form of the Direct Cut hole cutting method that is generalized for curved cells.

Automated Quadrature-free Discontinuous Galerkin Method Applied to Viscous Flows

The discontinuous Galerkin (DG) formulation has received much attention in recent years as a promising higher order method for solving the Navier-Stokes equations. However, it su ers from high operation count compared to traditional nite volume and nite di erence schemes. The high operation count can partially be attributed to the lack of a single set of numerical quadrature points that are capable of accurately integrating both volume and boundary integrals. In order to reduce operation counts, quadrature-free integration is employed and implemented through a pre-processor (PyDG) developed by the authors. The PyDG pre-processor relies on a freely available symbolic manipulation package to perform the analytical integration. The integrand is expressed as a matrix operator where all zero entries are automatically eliminated. This leads to a reduced operation count compared to traditional numerical quadrature implementations. In addition, this formulation allows numerical implementations of partial di erential equations to be expressed in a manner similar to the analytical equation. Using the PyDG pre-processor, the authors have also developed a C++ polynomial library (DGPoly++) that uses overloaded operators to allow expressions involving polynomials to be written as if they are scalars. This feature permits rapid numerical formulation of computer programs for solutions of partial di erential equations. The quadrature-free DG method is demonstrated here for a set of Euler and Navier-Stokes ows.

A 3-D Discontinuous Galerkin Chimera Overset Method

The Chimera overset method is a powerful technique for numerically solve partial differential equations associated with complex engineering problems using arbitrarily overlapping meshes. The method has allowed engineers to apply high-order schemes, such as the WENO and compact di erencing, which require structured meshes, to complex geometric con gurations. However, the large stencil associated with these high-order schemes can signi cantly complicate hole cutting procedures and the inter-grid communication scheme; particularly for three-dimensional geometries. This paper demonstrates a methodology for using the Discontinuous Galerkin (DG) scheme with Chimera overset meshes in threedimensions. The small stencil of the DG scheme makes it particularly suitable for Chimera meshes as it simpli es the inter-grid communication scheme. In addition, because the DG scheme represents the solution as cell local polynomials, the DG-Chimera scheme does not require a donor interpolation method with a large stencil. The DG-Chimera method also does not require the use of fringe points to maintain the interior stencil across inter-grid boundaries. Thus, inter-grid communication can be established so long as the receiving boundary is enclosed by or abuts with the donor mesh. This makes the inter-grid communication procedure applicable to both Chimera and zonal meshes. Details of threedimensional DG-Chimera scheme are presented, and the method is demonstrated on a set of inviscid ow problems.

Computational Fluid Dynamic (CFD) Analyses of the University of Michigan's Shock Boundary-Layer Interaction (SBLI) Experiments

This paper discusses computational fluid dynamic (CFD) analyses of the University of Michigan (UM) Shock/Boundary-Layer Interaction (SBLI) experiments, as an extension of the CFD SBLI Workshop held at the 48 th AIAA Aerospace Sciences Meeting in 2010. In particular, the UM Mach 2.75 Glass Tunnel with a semi-spanning 7.75 degree wedge was analyzed in attempts to explore key physics pertinent to SBLI’s, including thermodynamic and viscous boundary conditions. A fundamental exploration pertaining to the effects of particle image velocimetry (PIV) on post-processing data is also shown. Results showed an improvement in agreement with experimental data with key contributions by adding a laminar zone upstream of the wedge (the flow is considered transitional downstream of the nozzle throat) and the necessity of mimicking PIV for comparisons. All CFD analyses utilized the OVERFLOW solver.

An Implicit Harmonic Balance Method with a Discontinuous Galerkin Spatial Scheme

The Harmonic Balance method has the potential to reduce the computation cost of capturing unsteadiness for ows with a few dominant frequencies. A Fourier series representation is used to remove the temporal derivative term so that the governing equation is transformed into a series of steady equations. Each of these equations correspond to solutions at evenly spaced time steps. This paper details the implementation of the Harmonic Balance method with an implicit Quasi-Newton solver using a high-order Discontinuous Galerkin representation of the spatial uxes. The time steps are coupled in the implicit solve with the linearized Harmonic Balance operator, which reduces the number of Newton iterations for non-linear unsteady problems when compared to Harmonic Balance implementations without the linearized operator. For linear unsteady problems, the solution is obtained in one iteration. Five test cases are used to demonstrate the implementation of the Harmonic Balance method. Solutions to the linear convection equation and heat equation with a time varying boundary are obtained. The Harmonic Balance method with the non-linear Euler equations are solved for an acoustic source and a turbine cascade with a time-dependent inlet boundary. Lastly, a cylinder with natural shedding is simulated using the Harmonic Balance method. For each test case, simulations using time integration are completed and compared to the Harmonic Balance solutions.