Marshall C. Galbraith

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 Smooth Curvature-Defined Meanline Section Option for a General Turbomachinery Geometry Generator

The blade geometry design process is integral to the development and advancement of compressors and turbines in gas turbines or aeroengines. An airfoil section design feature has been added to a previously developed open source parametric 3D blade design tool. The second derivative of the mean-line (related to the curvature) is controlled using B-splines to create the airfoils. This is analytically integrated twice to obtain the mean-line. A smooth thickness distribution is then added to the airfoil with two options either the Wennerstrom distribution or a quartic B-spline thickness distribution. B-splines have also been implemented to achieve customized airfoil leading and trailing edges. Geometry for a turbine, compressor, and transonic fan are presented along with a demonstration of the importance of airfoil smoothness.Copyright

Error analysis of a modified discontinuous Galerkin recovery scheme for diffusion problems

Abstract A theoretical error analysis using standard Sobolev space energy arguments is furnished for a class of discontinuous Galerkin (DG) schemes that are modified versions of one of those introduced by van Leer and Nomura. These schemes, which use discontinuous piecewise polynomials of degree q , are applied to a family of one-dimensional elliptic boundary value problems. The modifications to the original method include definition of a recovery flux function via a symmetric L 2 -projection and the addition of a penalty or stabilization term. The method is found to have a convergence rate of O ( h q ) for the approximation of the first derivative and O ( h q +1 ) for the solution. Computational results for the original and modified DG recovery schemes are provided contrasting them as far as complexity and cost. Numerical examples are given which exhibit sub-optimal convergence rates when the stabilization terms are omitted.

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.

Design of a glass supersonic wind tunnel experiment for mixed compression inlet investigations

Mixed compression inlets offer a great increase in pressure recovery compared to conventional external compression inlets at Mach numbers above two. These inlets suffer from problems with shock wave boundary layer interactions (SBLI) which cause flow instabilities and severe performance reductions. Previous experiments conducted at the University of Michigan used a wind tunnel with glass side walls with an extensive test section to measure the SBLI associated with a single oblique shock. This work presents a redesign of the single oblique shock experimental setup, using computational fluid dynamics, to also include a downstream normal shock with a diffuser. The new experimental configuration will provide insights into the effects that combined oblique/normal shock boundary layer interactions have on the health of the boundary layer in the diffuser section of a mixed compression inlet. The extensive glass walls of the wind tunnel will allow direct access for optical measurements of the shock boundary layer interactions and the diffuser section.

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.

Evaluation of a Discontinuous Galerkin Implementation of RANS and Spalart Allmaras Turbulence Model

This paper presents the results of an initial evaluation of an implementation of the Spalart Allmaras (SA) turbulence model within the Discontinuous Galerkin (DG) solver XDG developed by Galbraith. The model is applied to three 2D test cases of increasing complexity: a zero-pressure gradient flat plate, a bump in a channel flow, and a backward facing step. Solutions are demonstrated on both single block and overset configurations utilizing second, third, and fourth order accurate schemes. The results show good agreement with those published in the NASA Langley Research Center’s Turbulence Modeling Resource database for the SA model utilizing standard finite volume solution techniques. Stability of the method was observed to be significantly dependent upon the grid spacing within the boundary layer, particularly in regions of high gradients in eddy viscosity.