Bulletin of the American Physical Society
70th Annual Meeting of the APS Division of Fluid Dynamics
Volume 62, Number 14
Sunday–Tuesday, November 19–21, 2017; Denver, Colorado
Session A32: Computational Fluid Dynamics: Discontinuous Galerkin and ConservationCFD
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Chair: Arpiruk Hokpunna, Chiang Mai University Room: 104 |
Sunday, November 19, 2017 8:00AM - 8:13AM |
A32.00001: A novel family of DG methods for diffusion problems Philip Johnson, Eric Johnsen We describe and demonstrate a novel family of numerical schemes for handling elliptic/parabolic PDE behavior within the discontinuous Galerkin (DG) framework. Starting from the mixed-form approach commonly applied for handling diffusion (examples include Local DG and BR2), the new schemes apply the Recovery concept of Van Leer to handle cell interface terms. By applying recovery within the mixed-form approach, we have designed multiple schemes that show better accuracy than other mixed-form approaches while being more flexible and easier to implement than the Recovery DG schemes of Van Leer. While typical mixed-form approaches converge at rate 2p in the cell-average or functional error norms (where p is the order of the solution polynomial), many of our approaches achieve order 2p$+$2 convergence. In this talk, we will describe multiple schemes, including both compact and non-compact implementations; the compact approaches use only interface-connected neighbors to form the residual for each element, while the non-compact approaches add one extra layer to the stencil. In addition to testing the schemes on purely parabolic PDE problems, we apply them to handle the diffusive flux terms in advection-diffusion systems, such as the compressible Navier-Stokes equations. [Preview Abstract] |
Sunday, November 19, 2017 8:13AM - 8:26AM |
A32.00002: Shock capturing in discontinuous Galerkin spectral elements via the entropy viscosity method Jason Hackl, Mrugesh Shringarpure, Paul Fischer, Sivaramakrishnan Balachandar We present a 3D discontinuous Galerkin spectral element solver for compressible flows with shock waves using artificial viscosity to regularize the solution for representation by nested tensor products of high-order Lagrange polynomials. The viscosity is constructed from a smoothed evaluation of the residual of an entropy inequality, localizing the artificial viscosity around shock waves and other flow features that would otherwise not be representable in spectral elements without thermodynamic violations due to Gibbs oscillations. Applied to the Guermond-Popov (2014) stress tensor, this smoothed, continuous artificial viscosity is easily integrated with the non-symmetric numerical fluxes of Baumann and Oden (1999). The method is implemented on top of nek5000, leveraging an outstanding high-performance spectral element code to solve shocked flows over curved surfaces. The interaction of a Mach 3 shock with a sphere is shown to demonstrate this capability. [Preview Abstract] |
Sunday, November 19, 2017 8:26AM - 8:39AM |
A32.00003: High-Order, Stable, and Conservative Boundary Schemes for Central and Compact Finite Differences Peter Brady, Daniel Livescu Stable and conservative numerical boundary schemes are constructed such that they do not diminish the overall accuracy of the method for interior schemes of orders 4, 6, and 8 using both explicit (central) and compact finite differences. Previous attempts to develop stable numerical boundary schemes have resulted in schemes which significantly reduced the global accuracy or required numerical dissipation for stability when applied to the non-linear fluid equations. We discuss a general procedure for the construction of conservative boundary schemes of any order followed by a simple, yet novel, optimization strategy which focuses directly on the compressible Euler equations. The result of this non-linear optimization process is a set of high-order, stable, and conservative numerical boundary schemes which demonstrate excellent stability and convergence properties on an array of linear and non-linear hyperbolic problems. [Preview Abstract] |
Sunday, November 19, 2017 8:39AM - 8:52AM |
A32.00004: Symmetry preserving compact schemes for numerical solution of PDEs Ersin Ozbenli, Prakash Vedula In this study, a new approach for construction of invariant, high order accurate compact finite difference schemes that preserve Lie symmetry groups of underlying partial differential equations (PDEs) is presented. It is well known that compact numerical schemes based on Pade approximants achieve high order accuracy with a relatively small number of stencil points and are found to have good spectral-like resolution. Considering applicable Lie symmetry groups (such as translation, scaling, rotation, and projection groups) of underlying PDEs, invariant compact schemes are developed based on the use of equivariant moving frames and extended group transformations. This work represents an extension of the authors recent work on construction of invariant, high-order, non-compact, finite difference schemes based on the method of modified equations. Performance of the proposed symmetry preserving compact schemes is evaluated via consideration of some canonical PDEs like linear advection-diffusion equation, inviscid Burgers' equation, and viscous Burgers' equation. Effects on accuracy due to choice of subgroups used in construction of these schemes will be discussed. Generalization of the proposed framework to multidimensional problems and non-orthogonal grids will also be presented. [Preview Abstract] |
Sunday, November 19, 2017 8:52AM - 9:05AM |
A32.00005: Conservative DEC Discretization of Incompressible Navier-Stokes Equations on Arbitrary Surface Simplicial Meshes Mamdouh Mohamed, Anil Hirani, Ravi Samtaney A conservative discretization of incompressible Navier-Stokes equations over surfaces is developed using discrete exterior calculus (DEC). The mimetic character of many of the DEC operators provides exact conservation of both mass and vorticity, in addition to superior kinetic energy conservation. The employment of signed diagonal Hodge star operators, while using the circumcentric dual defined on arbitrary meshes, is shown to produce correct solutions even when many non-Delaunay triangles pairs exist. This allows the DEC discretization to admit arbitrary surface simplicial meshes, in contrast to the previously held notion that DEC was limited only to Delaunay meshes. The discretization scheme is presented along with several numerical test cases demonstrating its numerical convergence and conservation properties. Recent developments regarding the extension to conservative higher order methods are also presented. [Preview Abstract] |
Sunday, November 19, 2017 9:05AM - 9:18AM |
A32.00006: A mass-conserving mixed Fourier-Galerkin B-Spline-collocation method for Direct Numerical Simulation of the variable-density Navier-Stokes equations Bryan Reuter, Todd Oliver, M.K. Lee, Robert Moser We present an algorithm for a Direct Numerical Simulation of the variable-density Navier-Stokes equations based on the velocity-vorticity approach introduced by Kim, Moin, and Moser (1987). In the current work, a Helmholtz decomposition of the momentum is performed. Evolution equations for the curl and the Laplacian of the divergence-free portion are formulated by manipulation of the momentum equations and the curl-free portion is reconstructed by enforcing continuity. The solution is expanded in Fourier bases in the homogeneous directions and B-Spline bases in the inhomogeneous directions. Discrete equations are obtained through a mixed Fourier-Galerkin and collocation weighted residual method. The scheme is designed such that the numerical solution conserves mass locally and globally by ensuring the discrete divergence projection is exact through the use of higher order splines in the inhomogeneous directions. The formulation is tested on multiple variable-density flow problems. [Preview Abstract] |
Sunday, November 19, 2017 9:18AM - 9:31AM |
A32.00007: Long-time stability effects of quadrature and artificial viscosity on nodal discontinuous Galerkin methods for gas dynamics Bradford Durant, Jason Hackl, Sivaramakrishnan Balachandar Nodal discontinuous Galerkin schemes present an attractive approach to robust high-order solution of the equations of fluid mechanics, but remain accompanied by subtle challenges in their consistent stabilization. The effect of quadrature choices (full mass matrix vs spectral elements), over-integration to manage aliasing errors, and explicit artificial viscosity on the numerical solution of a steady homentropic vortex are assessed over a wide range of resolutions and polynomial orders using quadrilateral elements. In both stagnant and advected vortices in periodic and non-periodic domains the need arises for explicit stabilization beyond the numerical surface fluxes of discontinuous Galerkin spectral elements. Artificial viscosity via the entropy viscosity method is assessed as a stabilizing mechanism. It is shown that the regularity of the artificial viscosity field is essential to its use for long-time stabilization of small-scale features in nodal discontinuous Galerkin solutions of the Euler equations of gas dynamics. [Preview Abstract] |
Sunday, November 19, 2017 9:31AM - 9:44AM |
A32.00008: Lagrangian Particle Tracking in a Discontinuous Galerkin Method for Hypersonic Reentry Flows in Dusty Environments Eric Ching, Yu Lv, Matthias Ihme Recent interest in human-scale missions to Mars has sparked active research into high-fidelity simulations of reentry flows. A key feature of the Mars atmosphere is the high levels of suspended dust particles, which can not only enhance erosion of thermal protection systems but also transfer energy and momentum to the shock layer, increasing surface heat fluxes. Second-order finite-volume schemes are typically employed for hypersonic flow simulations, but such schemes suffer from a number of limitations. An attractive alternative is discontinuous Galerkin methods, which benefit from arbitrarily high spatial order of accuracy, geometric flexibility, and other advantages. As such, a Lagrangian particle method is developed in a discontinuous Galerkin framework to enable the computation of particle-laden hypersonic flows. Two-way coupling between the carrier and disperse phases is considered, and an efficient particle search algorithm compatible with unstructured curved meshes is proposed. In addition, variable thermodynamic properties are considered to accommodate high-temperature gases. The performance of the particle method is demonstrated in several test cases, with focus on the accurate prediction of particle trajectories and heating augmentation. [Preview Abstract] |
Sunday, November 19, 2017 9:44AM - 9:57AM |
A32.00009: Physics-Based Preconditioning of a Compressible Flow Solver for Large-Scale Simulations of Additive Manufacturing Processes Brian Weston, Robert Nourgaliev, Jean-Pierre Delplanque We present a new block-based Schur complement preconditioner for simulating all-speed compressible flow with phase change. The conservation equations are discretized with a reconstructed Discontinuous Galerkin method and integrated in time with fully implicit time discretization schemes. The resulting set of non-linear equations is converged using a robust Newton-Krylov framework. Due to the stiffness of the underlying physics associated with stiff acoustic waves and viscous material strength effects, we solve for the primitive-variables (pressure, velocity, and temperature). To enable convergence of the highly ill-conditioned linearized systems, we develop a physics-based preconditioner, utilizing approximate block factorization techniques to reduce the fully-coupled 3\texttimes 3 system to a pair of reduced 2\texttimes 2 systems. We demonstrate that our preconditioned Newton-Krylov framework converges on very stiff multi-physics problems, corresponding to large CFL and Fourier numbers, with excellent algorithmic and parallel scalability. Results are shown for the classic lid-driven cavity flow problem as well as for 3D laser-induced phase change. [Preview Abstract] |
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