Bulletin of the American Physical Society
68th Annual Meeting of the APS Division of Fluid Dynamics
Volume 60, Number 21
Sunday–Tuesday, November 22–24, 2015; Boston, Massachusetts
Session L8: CFD: High Order and Discontinuous Galerkin Methods |
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Chair: Walter Arias-Ramirez, UNICAMP-Univ de Campinas Room: 108 |
Monday, November 23, 2015 4:05PM - 4:18PM |
L8.00001: Combined Immersed-Boundary / High-Order Finite Difference Methods For Simulations of Acoustic Scattering Walter Arias-Ramirez, Britton Olson, William Wolf The suitability of a continuing forcing immersed boundary method (IBM) combined with a high-order finite difference method is examined on several acoustic scattering problems. A suite of two-dimensional numerical simulations of canonical cases are conducted with the aim of analyzing the error behavior associated with the IBM,~through wave reflection, wave diffraction, and the shock-boundary layer interaction phenomena. The compressible Navier-Stokes equations are solved using the Miranda code developed at Lawrence Livermore National Laboratory. Comparison of analytical solution against numerical results is shown for different flow parameters. Preliminary results indicate that the continuing forcing approach has the largest error in wave reflection compared to analytical solution.~ [Preview Abstract] |
Monday, November 23, 2015 4:18PM - 4:31PM |
L8.00002: A high-order solver for unsteady incompressible Navier-Stokes equations using the flux reconstruction method on unstructured grids with implicit dual time stepping Christopher Cox, Chunlei Liang, Michael Plesniak This paper reports development of a high-order compact method for solving unsteady incompressible flow on unstructured grids with implicit time stepping. The method falls under the class of methods now referred to as flux reconstruction/correction procedure via reconstruction. The governing equations employ the classical artificial compressibility treatment, where dual time stepping is needed to solve unsteady flow problems. An implicit non-linear lower-upper symmetric Gauss-Seidel scheme with backward Euler discretization is used to efficiently march the solution in pseudo time, while a second-order backward Euler discretization is used to march in physical time. We verify and validate implementation of the high-order method coupled with our implicit time-stepping scheme. Three-dimensional results computed on many processing elements will be presented. The high-order method is very suitable for parallel computing and can easily be extended to moving and deforming grids. The current implicit time stepping scheme is proven effective in satisfying the divergence-free constraint on the velocity field in the artificial compressibility formulation within the context of the high-order flux reconstruction method. [Preview Abstract] |
Monday, November 23, 2015 4:31PM - 4:44PM |
L8.00003: High-order boundary layer analysis using B-splines on hybrid unstructured meshes Alvin Zhang, Onkar Sahni Boundary layer flows are present in many engineering applications. In such flows, boundary layers span only a fraction of the characteristic length of the problem near the walls and possess large velocity gradients in the wall normal direction. This mandates use of a layered and graded mesh with a dense anisotropic h-resolution near the walls in order to accurately resolve the boundary layer. To account for complex geometries, a hybrid unstructured mesh approach is adopted. In this approach, the mesh is decomposed into wall normal and wall parallel directions. An alternative to an anisotropic h-resolution is to use a similar setting for both h- and p-resolution possibly with greater smoothness. For this purpose a mixed B-spline basis becomes attractive, where B-splines are used in the wall-normal direction and a C0 basis in the wall-parallel directions as well as the fully unstructured region of the mesh. A mixed B-spline basis offers several advantages over the traditional C0 basis utilized in finite element methods, which include greater accuracy per degree-of-freedom, ease of p-refinement as well as potential for k-refinement. In this study we demonstrate that the mixed B-spline basis, defined for the hybrid unstructured mesh, accurately models the boundary layer behavior. [Preview Abstract] |
Monday, November 23, 2015 4:44PM - 4:57PM |
L8.00004: The direct Discontinuous Galerkin method for the compressible Navier-Stokes equations on arbitrary grids Xiaoquan Yang, Jian Cheng, Tiegang Liu, Hong Luo The direct discontinuous Galerkin (DDG) method based on a traditional discontinuous Galerkin (DG) formulation is extended and implemented for solving the compressible Navier-Stokes equations on arbitrary grids. Compared to the widely used second Bassi-Rebay (BR2) scheme for the discretization of diffusive fluxes, the DDG method has two attractive features: first, it is simple to implement as it is directly based on the weak form, and therefore there is no need for any local or global lifting operator; second, it can deliver comparable results, if not better than BR2 scheme, in a more efficient way with much less CPU time. Two approaches to perform the DDG flux for the Navier- Stokes equations are presented in this work, one is based on conservative variables, the other is based on primitive variables. In the implementation of the DDG method for arbitrary grid, the definition of mesh size plays a critical role as the formation of viscous flux explicitly depends on the geometry. A variety of test cases are presented to demonstrate the accuracy and efficiency of the DDG method for discretizing the viscous fluxes in the compressible Navier-Stokes equations on arbitrary grids. [Preview Abstract] |
Monday, November 23, 2015 4:57PM - 5:10PM |
L8.00005: A New Reconstructed Discontinuous Galerkin Method for Compressible Flows on Unstructured Grids Jian Cheng, Tiegang Liu, Hong Luo A reconstructed discontinuous Galerkin method (rDG) has been developed for solving the compressible Euler equations on unstructured grids. The rDG method is designed not only to enhance the accuracy of the discontinuous Galerkin method, but also to avoid non-physical oscillations in the vicinity of discontinuities. In this work, a new hybrid least-squares reconstruction scheme is developed for the reconstructed discontinuous Galerkin method rDG(P1P2) for compressible flows on unstructured grids. The new hybrid least-squares reconstruction can be regards a combination of least-squares recovery method and least-square reconstruction method. Compared to Green-Gauss reconstruction and original least-squares reconstruction, the new hybrid least-squares reconstruction method can strictly satisfy 2-exact property when obtain a quadratic polynomial representation of the underlying discontinuous Galerkin linear polynomial solution on each cell. The numerical experiments for a variety of flow problems demonstrate that this new hybrid reconstruction method is more accurate than the Green-Gauss and the original least-squares reconstruction method, and is able to achieve the designed third-order of accuracy for both inviscid and viscous flow problems. [Preview Abstract] |
Monday, November 23, 2015 5:10PM - 5:23PM |
L8.00006: Boundary treatment for the Recovery discontinuous Galerkin method with application to the Navier-Stokes equations Philip Johnson, Eric Johnsen The Recovery discontinuous Galerkin (DG) method is a highly accurate approach to computing diffusion problems, which achieves up to 3p+2 convergence rates on Cartesian cells, where p is the order of the polynomial basis. Based on the construction of a unique and differentiable solution across cell interfaces, Recovery DG has mostly been investigated on periodic domains. However, whether such accuracy can be sustained for Dirichlet and Neumann boundary conditions has not been thoroughly explored. We present boundary treatments for Recovery DG on 2D Cartesian geometry that exhibit up to 3p+2 convergence rates and are stable. We demonstrate the efficiency of Recovery DG in context with other commonly used approaches using scalar shear diffusion problems and apply it to the compressible Navier-Stokes equations. The extension of the method to perturbed quadrilateral cells, rather than Cartesian, will also be discussed. [Preview Abstract] |
Monday, November 23, 2015 5:23PM - 5:36PM |
L8.00007: Quantifying numerical dissipation rate for discontinuous Galerkin methods Julian Domaradzki, Giacomo Castiglioni, Felix Schranner, Nico Krais, Andrea Beck, Claus-Dieter Munz The numerical dissipation quite often can be large for typical Finite Volume and Finite Difference schemes. In LES applications it inhibits the predictive capabilities if it is of the same order of magnitude or larger than the physical subgrid-scale dissipation. Because of that there is an increasing interest in CFD in using the discontinuous Galerkin (DG) methods because they are of high order and have the ability to handle complex domains. We present comparison between numerical dissipation rates computed for the DG method and for standard FV methods. The numerical dissipation is estimated following Schranner et al. (2015), allowing to compute the numerical dissipation rate for arbitrary sub-domains in a self-consistent way, using only information provided by the code in question. The specific flow considered is a 3D Taylor-Green vortex flow which is simulated with $64^3$ degrees of freedom and for different divisions of the computational domain into elements with polynomial orders inside elements varying from 3 to 31. We find that for low polynomial orders the numerical dissipation of the DG method is comparable to what is observed for the FV codes at the same resolution but it decreases by an order of magnitude for the polynomials of the highest order used. [Preview Abstract] |
Monday, November 23, 2015 5:36PM - 5:49PM |
L8.00008: Fully-Implicit Reconstructed Discontinuous Galerkin Method for Stiff Multiphysics Problems Robert Nourgaliev A new reconstructed Discontinuous Galerkin (rDG) method, based on orthogonal basis/test functions, is developed for fluid flows on unstructured meshes. Orthogonality of basis functions is essential for enabling robust and efficient fully-implicit Newton-Krylov based time integration. The method is designed for generic partial differential equations, including transient, hyperbolic, parabolic or elliptic operators, which are attributed to many multiphysics problems. We demonstrate the method's capabilities for solving compressible fluid-solid systems (in the low Mach number limit), with phase change (melting/solidification), as motivated by applications in Additive Manufacturing. We focus on the method's accuracy (in both space and time), as well as robustness and solvability of the system of linear equations involved in the linearization steps of Newton-based methods. The performance of the developed method is investigated for highly-stiff problems with melting/solidification, emphasizing the advantages from tight coupling of mass, momentum and energy conservation equations, as well as orthogonality of basis functions, which leads to better conditioning of the underlying (approximate) Jacobian matrices, and rapid convergence of the Krylov-based linear solver. [Preview Abstract] |
Monday, November 23, 2015 5:49PM - 6:02PM |
L8.00009: A new hybrid RANS/LES technique based on Reynolds stress reconstruction Michele Nini, Antonella Abba, Massimo Germano, Marco Restelli A new hybrid RANS/LES technique, based on the hybrid RANS/LES filter, has been studied. The novelty herein introduced is represented by the reconstruction of the Reynolds stress tensor. As a consequence, no explicit RANS model is needed. The model is implemented in a numerical code based on a high order Discontinuous Galerkin (DG) finite element formulation. The test case considered for numerical simulations is the turbulent channel flow at Mach = 0.2 and the simulations have been carried out for two friction Reynolds number, 180 and 395. The results show that, for coarse grid, the technique can give benefits with respect to the pure LES, confirming that the methodology herein proposed represents a promising approach to the numerical simulation of turbulent flows. [Preview Abstract] |
Monday, November 23, 2015 6:02PM - 6:15PM |
L8.00010: Implicit LES using the Embedded Discontinuous Galerkin method John Moore High order methods have been gaining greater traction in the CFD community recently, and are believed to be especially well-suited to vortex-dominated flows and Large Eddy Simulation (LES). However, realizing the theoretical performance of these methods has been difficult, in part due to the time step restrictions of explicit methods and the large number of coupled degrees of freedom arising from implicit high order schemes. In this presentation, the development and efficient implementation of an implicit high-order solver based on the Embedded Discontinuous Galerkin (EDG) method,\footnote{Peraire \textit{et al.} AIAA-2011-3228} which requires less coupled degrees of freedom than standard DG, is detailed. Results are presented for several external flow cases and validated against experimental results. [Preview Abstract] |
Monday, November 23, 2015 6:15PM - 6:28PM |
L8.00011: An Adaptive De-Aliasing Strategy for Discontinuous Galerkin methods Andrea Beck, David Flad, Hannes Frank, Claus-Dieter Munz Discontinuous Galerkin methods combine the accuracy of a local polynomial representation with the geometrical flexibility of an element-based discretization. In combination with their excellent parallel scalability, these methods are currently of great interest for DNS and LES. For high order schemes, the dissipation error approaches a cut-off behavior, which allows an efficient wave resolution per degree of freedom, but also reduces robustness against numerical errors. One important source of numerical error is the inconsistent discretization of the non-linear convective terms, which results in aliasing of kinetic energy and solver instability. Consistent evaluation of the inner products prevents this form of error, but is computationally very expensive. In this talk, we discuss the need for a consistent de-aliasing to achieve a neutrally stable scheme, and present a novel strategy for recovering a part of the incurred computational costs. By implementing the de-aliasing operation through a cell-local projection filter, we can perform adaptive de-aliasing in space and time, based on physically motivated indicators. We will present results for a homogeneous isotropic turbulence and the Taylor-Green vortex flow, and discuss implementation details, accuracy and efficiency. [Preview Abstract] |
Monday, November 23, 2015 6:28PM - 6:41PM |
L8.00012: Adaptive entropy-constrained discontinuous Galerkin method for simulation of turbulent flows Yu Lv, Matthias Ihme A robust and adaptive computational framework will be presented for high-fidelity simulations of turbulent flows based on the discontinuous Galerkin (DG) scheme. For this, an entropy-residual based adaptation indicator is proposed to enable adaptation in polynomial and physical space. The performance and generality of this entropy-residual indicator is evaluated through direct comparisons with classical indicators. In addition, a dynamic load balancing procedure is developed to improve computational efficiency. The adaptive framework is tested by considering a series of turbulent test cases, which include homogeneous isotropic turbulence, channel flow and flow-over-a-cylinder. The accuracy, performance and scalability are assessed, and the benefit of this adaptive high-order method is discussed. [Preview Abstract] |
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