Download or read book Energy Stable High order Methods for Simulating Unsteady Viscous Compressible Flows on Unstructured Grids written by David Michael Williams and published by . This book was released on 2013 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: High-order methods have the potential to dramatically improve the accuracy and efficiency of flow simulations in the field of computational fluid dynamics (CFD). However, there remain questions regarding the stability and robustness of high-order methods for practical problems on unstructured triangular and tetrahedral grids. In this work, a new class of 'energy stable' high-order methods is identified. This class of schemes (referred to as the 'Energy Stable Flux Reconstruction' class of schemes) is proven to be stable for linear advection-diffusion problems, for all orders of accuracy on unstructured triangular grids in 2D and unstructured tetrahedral grids in 3D. Furthermore, this class of schemes is shown to be capable of recovering the well-known collocation-based nodal discontinuous Galerkin scheme, along with new schemes that possess explicit time-step limits which are (in some cases) more than 2x larger than those of the discontinuous Galerkin scheme. In addition, the stability of the Energy Stable Flux Reconstruction schemes is examined for nonlinear problems, and it is shown that stability depends on the degree of nonlinearity in the flux and on the placement of solution and flux points in each element. In particular, it is shown that choosing the solution and flux point locations to coincide with the locations of quadrature points promotes nonlinear stability by minimizing (or eliminating) nonlinear aliasing errors. A new class of symmetric quadrature points is identified on triangles and tetrahedra for this purpose. Finally, the Energy Stable Flux Reconstruction schemes and the new quadrature points are applied to several nonlinear problems with the aim of assessing how well the schemes perform in practice.

Download or read book Towards High Order Compact Discretization of Unsteady Navier Stokes Equations for Incompressible Flows on Unstructured Grids written by Hashim Ibrahim Mohamed Elzaabalawy and published by . This book was released on 2020 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: A high-order energy-stable method for solving the incompressible Navier-Stokes equations based on hybrid discontinuous Galerkin method is presented for which the mass and momentum are conserved. The formulation computes exactly pointwise divergence-free velocity fields for standard element types without post-processing operators nor using textit{H}(div)-conforming spaces. This is achieved by proposing a simple and novel definition to the functional space of the pressure, such that it contains the divergence of the approximate velocity. Specific focus is given on applying this method on different element shapes by introducing the concept of reduced-order elements for all standard shapes in 2D and 3D. Further, the incompressibility constraint is handled via the static condensation to solve the saddle point problem. Furthermore, with the aim to simulate high Reynolds numbers flows, the significance of the diffusion stabilization in the hybridizable discontinuous Galerkin framework is analyzed. Referring to literature, the diffusion stabilization term is directly proportional to the diffusivity or the viscosity for the Navier-Stokes equations. In this work, a new expression for the diffusion stabilization term is mathematically derived, where the term is inversely proportional to the diffusivity or viscosity. Its importance for convection dominated flows is emphasized and supported by numerical examples.Moreover, the proposed formulation for the incompressible Navier-Stokes is extended to solve the RANSE for the TNT, BSL, and SST k- omega models for Reynolds numbers up to 10^9.Solving RANSE is a resilient task for high-order methods, due to the non-smooth profiles of the turbulence quantities. In the discontinuous Galerkin framework, the polynomial approximation for these quantities leads to large oscillations that obstruct the non-linear solver. Taking into account the complexity with high-order methods and the fairly large modeling errors of the RANS modeling, low-order methods are believed to be more pragmatic. However, it is illustrated that solving RANSE with high-order methods leads to significantly smaller error magnitudes compared with second-order finite volume based solvers. Additionally, there is a remarkable improvement regarding the number of iterations to obtain a converged solution. Attention is given to the treatment of the specific rate of turbulence dissipation omega in the high-order framework. The possibilities and limitations of simulating industrial incompressible flows using discontinuous Galerkin based methods are assessed in order to draw some general conclusions for industrial applications.

Download or read book High order Methods for Unsteady Flows on Unstructured Dynamic Meshes written by Kui Ou and published by . This book was released on 2012 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: A comprehensive study of discontinuous finite element based high-order methods has been performed in this thesis, addressing a wide range of important issues related to high-order methods. The thesis starts with a detailed discussion of nodal based high-order methods and careful analysis of their stability properties. In particular, the formulations of nodal Discontinuous Galerkin method, Spectral Difference method, and Flux Reconstruction method for the scalar conservation laws are discussed first. The differences and similarities among these high-order schemes are carefully examined and effectively used to establish the linear stability of these methods. Stability proofs of nodal Discontinuous Galerkin method, Spectral Difference method, and Flux Reconstruction method subsequently lead to a new type of energy stable high-order scheme called Energy Stable Flux Reconstruction scheme. The extension of this new scheme from linear advection equation to the diffusion equation is formulated and discussed. The fundamental study of the high-order methods for scalar conservation laws lays the theoretical foundation for the subsequent extension to include conservation laws for fluid dynamics. The formulation of spectral difference method for the Navier-Stokes equations is first discussed. Validation tests to verify the resulting flow solver are presented. The extension of the spectral difference based Navier-Stokes flow solver from static fixed computational mesh to include dynamic moving deforming mesh is discussed next. An efficient mesh deformation algorithm that can handle substantial boundary movement is proposed and examined. The invariance of conservation laws mapping between coordinate systems allows the high-order scheme to be formulated on dynamic deforming meshes without deteriorating the formal order of accuracy of the underlying scheme. Detailed formulation, analysis, and validation results are presented. As a result of mesh deformation, the issue of geometric conservation needs to be addressed. The definition and origin of the geometric conservation law are discussed. The differential form of the geometric conservation law is derived from first principles for both the scalar conservation law and the fluid dynamic conservation laws. Subsequently a geometric conservative high-order scheme is formulated. The significance of geometric conservation on the stability and accuracy of the flow solution is examined. Finally a wide range of interesting fluid dynamic phenomena have been studied using the resulting high-order flow solver based on dynamic unstructured meshes. The representative test cases cover fluid dynamic phenomena ranging from completely laminar flows, to unsteady vortex dominated flows, and to flows exhibiting mixed regions of laminar, transitional, and turbulent structures. Other work that has been completed in this thesis is included in the appendix. In particular, continuous unsteady adjoint equations for advection and Burger's equations have been derived and solved using the high-order methods. The method of mesh deformation is reformulated as an optimization problem and used to achieve adaptive mesh refinement.

Download or read book High order Energy Stable Flux Reconstruction Schemes for Fluid Flow Simulations on Unstructured Grids written by Patrice Castonguay and published by . This book was released on 2012 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: Nowadays, most commercial CFD software relies exclusively on low-order methods (methods for which the spatial order of accuracy is at most two) for the simulation of flows over complex geometries. Although these methods are extremely robust and efficient, they are often inadequate when simulations require a very high level of accuracy. As desired error tolerances continue to decrease, high-order methods for unstructured grids will continue to grow in popularity. In the context of fluid flow simulations, applications that require high levels of accuracy (which could therefore benefit greatly from the use of high-order methods) include Direct Numerical Simulations (DNS), Large-Eddy Simulations (LES), Computational Aero-Acoustics (CAA) and vortex dominated flows. In 2007, Huynh presented the Flux Reconstruction (FR) approach to high-order methods. The method is attractive because it is intuitive, straightforward to implement, unifying (in the sense that it can recover various well known high-order methods), computationally efficient (since it does not require the use of numerical integration), and easily parallelizable. However, there remained questions regarding the stability of FR schemes and their applicability to practical fluid flow problems. In this thesis, a new class of FR schemes is identified, and applied to solve the 3D Navier-Stokes equations on mixed unstructured grids. The new schemes are referred to as Energy Stable Flux Reconstruction (ESFR) schemes. Their implementation in 1D, on triangular and quadrilateral elements in 2D, and on prismatic and hexahedral elements in 3D is discussed. The stability and accuracy properties of ESFR schemes are studied thoroughly. An energy stability proof is used to show the linear stability of the schemes for all orders of accuracy, in 1D and on triangular elements in 2D. Various numerical experiments in the field of fluid dynamics are then used to demonstrate their effectiveness. The implementation of ESFR schemes on Graphics Processing Units (GPUs) is also discussed. Because of their high arithmetic intensity and their element-local nature, the schemes are well suited for new massively parallel hardware architectures such as GPUs. The efficient implementation of ESFR schemes on GPUs results in speedups of at least one order of magnitude relative to traditional high-order methods for unstructured grids running on conventional computer hardware.

Download or read book Nodal Discontinuous Galerkin Methods written by Jan S. Hesthaven and published by Springer Science & Business Media. This book was released on 2007-12-18 with total page 507 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book offers an introduction to the key ideas, basic analysis, and efficient implementation of discontinuous Galerkin finite element methods (DG-FEM) for the solution of partial differential equations. It covers all key theoretical results, including an overview of relevant results from approximation theory, convergence theory for numerical PDE’s, and orthogonal polynomials. Through embedded Matlab codes, coverage discusses and implements the algorithms for a number of classic systems of PDE’s: Maxwell’s equations, Euler equations, incompressible Navier-Stokes equations, and Poisson- and Helmholtz equations.

Download or read book Handbook of Numerical Methods for Hyperbolic Problems written by Remi Abgrall and published by Elsevier. This book was released on 2016-11-17 with total page 668 pages. Available in PDF, EPUB and Kindle. Book excerpt: Handbook of Numerical Methods for Hyperbolic Problems explores the changes that have taken place in the past few decades regarding literature in the design, analysis and application of various numerical algorithms for solving hyperbolic equations. This volume provides concise summaries from experts in different types of algorithms, so that readers can find a variety of algorithms under different situations and readily understand their relative advantages and limitations. - Provides detailed, cutting-edge background explanations of existing algorithms and their analysis - Ideal for readers working on the theoretical aspects of algorithm development and its numerical analysis - Presents a method of different algorithms for specific applications and the relative advantages and limitations of different algorithms for engineers or readers involved in applications - Written by leading subject experts in each field who provide breadth and depth of content coverage

Download or read book An Unstructured Grid Generation and Adaptive Solution Technique for High Reynolds number Compressible Flows written by Gregory Allan Ashford and published by . This book was released on 1996 with total page 500 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book High Order Numerical Methods for Inviscid and Viscous Flows on Unstructured Grids written by Yuzhi Sun and published by . This book was released on 2004 with total page 280 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Computations of Unsteady Viscous Compressible Flows Using Adaptive Mesh Refinement in Curvilinear Body fitted Grid Systems written by Erlendur Steinthorsson and published by . This book was released on 1994 with total page 18 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book A high order discontinuous Galerkin method for unsteady compressible flows with immersed boundaries written by Stephan Krämer-Eis and published by Cuvillier Verlag. This book was released on 2017-10-20 with total page 128 pages. Available in PDF, EPUB and Kindle. Book excerpt: Um die komplexe Physik in kompressiblen Strömungen genauer zu verstehen, kommen vermehrt Simulationen zum Einsatz. Jedoch können weit verbreitete kommerzielle Softwarepakete die Physik aufgrund ihrer niedrigen Genauigkeit oft nicht korrekt erfassen. In dieser Arbeit wird eine diskontinuierliche Galerkin Methode mit hoher Ordnung entwickelt, welche eine hohe Genauigkeit erzielt. Dabei werden insbesondere zwei Probleme, die im Kontext von Verfahren mit hoher Ordnung auftreten, behandelt. Zum einen wird die Gittergenerierung durch das Verwenden einer Immersed Boundary Methode deutlich vereinfacht. Dies bedeutet, dass die Problemgeometrie aus einem deutlich einfacheren Hintergrundgitter herausgeschnitten wird. Die Geometrie wird mit Hilfe einer Level-Set Funktion dargestellt, und die Integration auf den entstehenden geschnittenen Zellen wird mittels einer hierarchischen Moment-Fitting Quadratur durchgeführt. Das Problem der sehr kleinen oder stark gekrümmten Zellen wird durch Zellagglomeration gelöst. Zum zweiten wird die starke Zeitschrittbeschränkung durch anisotrope Gitter mit Hilfe eines lokalen Zeitschrittverfahrens behoben. Diverse numerische Experimente bestätigen die hohe Genauigkeit, Effizienz und geometrische Flexibilität der vorgestellten Methode.

Download or read book Scientific and Technical Aerospace Reports written by and published by . This book was released on 1995 with total page 704 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book A Hybrid Explicit implicit Second order TVD Method on Adaptive Unstructured Grids for Unsteady Compressible Flows written by Farhang Norouzi and published by . This book was released on 2015 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: " This work is devoted to the development of a hybrid, explicit-implicit, scheme for simulation of unsteady compressible flows with shock waves. The proposed scheme is of the second-order accuracy in space and time for both explicit and implicit modes, while satisfying the TVD (Total Variation Diminishing) property. The scheme is designed for simulation of the compressible flows with temporal stiffness. In this situation,the numerical time step of explicit schemes is severely limited by particular conditions in a relatively small part of the computational domain, while the rest of the domain admits much higher time steps. The hybrid scheme is designed to operate in its implicit mode in the small areas causing temporal stiffness, thus allowing to proceed with higher time steps and reduce the computational time. In this study, a new hybridization approach is suggested. On its basis, the hybrid scheme is first introduced for hyperbolic conservation laws in one dimension. In order to satisfy the TVD property and obtain monotone solutions in the presence ofdiscontinuities, TVD limiters are applied to both spatial and temporal reconstructions. The second-order accuracy in time for the implicit mode, which is the main distinction of the proposed hybrid scheme in comparison with the existing methods, is achieved through a reconstruction of the solution in time. To make the reconstruction TVD preserving, a novel time limiter is derived. The stability condition and the relation to the hybridization factor of the new scheme are obtained. Moreover, the relationship of the proposed scheme with another existing hybrid method is revealed and analyzed. A set of one-dimensional test problems is used to demonstrate the accuracy and efficiency of the new hybrid scheme as well as to show its advantagesin comparison with the existing hybrid schemes. For two-dimensional problems, the hybrid scheme is generalized for the Euler and Navier-Stokes equations on unstructured grids. Similar to one-dimensional problems, reconstructions in space and time are applied with the corresponding TVD limiters. The newly proposed time limiter of the hybrid scheme is further generalized for unstructured grids. The non-linear system of discretized equations is solved using the Lower-Upper-Symmetric-Gauss-Seidel (LU-SGS) approximate factorization method for unstructured grids. In order to eliminate the factorization and linearization errors, internal iterations are introduced at each time step. The lower and upper matrices in the LU-SGS scheme are formed via reordering of grid nodes at each time step. In addition, local transient grid adaptation is applied near solution peculiarities, such as shock waves and contact surfaces. Viscous terms of the Navier-Stokes equations are also approximated with the second order of accuracy both in space and time in both explicit and implicit modes. The new hybrid scheme is applied to a number of demonstrative problems chosen to represent typical classes of gasdynamic problems with specific source of stiffness. The numerical results demonstrate the ability of the proposed hybrid scheme to produce the same accuracy as the purely explicit scheme (the well-known MUSCLHancock scheme), while reducing the computational time." --

Download or read book An Implicit Numerical Scheme for the Simulation of Internal Viscous Flows on Unstructured Grids written by and published by . This book was released on 1994 with total page 18 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Fluid Mechanics and Fluid Power Vol 3 written by Suvanjan Bhattacharyya and published by Springer Nature. This book was released on 2023-04-17 with total page 628 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents the select proceedings of the 48th National Conference on Fluid Mechanics and Fluid Power (FMFP 2021) held at BITS Pilani in December 2021. It covers the topics such as fluid mechanics, measurement techniques in fluid flows, computational fluid dynamics, instability, transition and turbulence, fluid‐structure interaction, multiphase flows, micro- and nanoscale transport, bio-fluid mechanics, aerodynamics, turbomachinery, propulsion and power. The book will be useful for researchers and professionals interested in the broad field of mechanics.

Download or read book Higher Order Finite Element Methods written by Pavel Solin and published by CRC Press. This book was released on 2003-07-28 with total page 404 pages. Available in PDF, EPUB and Kindle. Book excerpt: The finite element method has always been a mainstay for solving engineering problems numerically. The most recent developments in the field clearly indicate that its future lies in higher-order methods, particularly in higher-order hp-adaptive schemes. These techniques respond well to the increasing complexity of engineering simulations and

Download or read book An Efficient Semi implicit Pressure based Scheme Employing a High resolution Finite Element Method for Simulating Transient and Steady Inviscid and Viscous Compressible Flows on Unstructured Grids written by Richard C. Martineau and published by . This book was released on 2002 with total page 438 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book A Reconstructed Discontinuous Galerkin Method for the Compressible Flows on Unstructured Tetrahedral Grids written by and published by . This book was released on 2011 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: A reconstruction-based discontinuous Galerkin (RDG) method is presented for the solution of the compressible Navier-Stokes equations on unstructured tetrahedral grids. The RDG method, originally developed for the compressible Euler equations, is extended to discretize viscous and heat fluxes in the Navier-Stokes equations using a so-called inter-cell reconstruction, where a smooth solution is locally reconstructed using a least-squares method from the underlying discontinuous DG solution. Similar to the recovery-based DG (rDG) methods, this reconstructed DG method eliminates the introduction of ad hoc penalty or coupling terms commonly found in traditional DG methods. Unlike rDG methods, this RDG method does not need to judiciously choose a proper form of a recovered polynomial, thus is simple, flexible, and robust, and can be used on unstructured grids. The preliminary results indicate that this RDG method is stable on unstructured tetrahedral grids, and provides a viable and attractive alternative for the discretization of the viscous and heat fluxes in the Navier-Stokes equations.