Discontinuous galerkin dg methods,, as a typical representative in the community of highorder methods, have been widely used in computational fluid dynamics, computational acoustics and computational magnetohydrodynamics. Discontinuous galerkin method for compressible viscous. So, we have presented a new finite element method for navier stokes, with i hdivconforming finite elements i hybrid discontinuous galerkin method for viscous terms i upwind ux in hdgsence for the convection term leading to solutions, which are i locally conservative i energystable d dt kuk2 l 2 c kfk2 l 2 i exactly incompressible i. Discontinuous galerkin methods for the navierstokes equations. A high order discontinuous galerkin method for compressible. This process is experimental and the keywords may be updated as the learning algorithm improves. Discontinuous galerkin methods for viscous incompressible flow guido kanschat auth.
Symmetric and nonsymmetric discontinuous galerkin methods for. Jun 21, 2012 discontinuous galerkin methods for computational aerodynamics 3d adaptive flow simulation with the dlr padge code aerospace science and technology, vol. Dibelius, editors, 2nd european conference on turbomachinery fluid dynamics and thermodynamics,pages 99108, antwerpen, belgium, march 57 1997. For solving reactive transport problems in porous media, we analyze three primal discontinuous galerkin dg methods with penalty, namely, symmetric interior penalty galerkin sipg, nonsymmetric interior penalty galerkin nipg, and incomplete interior penalty galerkin iipg. The advantages of the dg methods over classical continuous galerkin method, finite element, finite difference and finite volume methods are well. In this paper, firstly, discontinuous galerkin method for improved stokes equation is proposed. Peraire z massachusetts institute of technology, cambridge, ma 029, usa we are concerned with the numerical solution of the navierstokes and reynoldsaveraged navierstokes equations using the hybridizable discontinuous galerkin hdg. In this work the numerical discretization of the partial differential governing equations for compressible and incompressible flows is dealt within the discontinuous galerkin dg framework along spacetime adaptive meshes. Numerical experiments of the spectral volume method for viscous flows. Discontinuous galerkin and petrov galerkin methods for. Robust and efficient discontinuous galerkin methods for under. Analysis of hybrid discontinuous galerkin methods for incompressible flow problems christian waluga1 advised by prof.
Discontinuous galerkin dg finite element method for the. The discontinuous galerkin methods have been developed and studied for solving the navierstokes equations, e. Different formulations of the discontinuous galerkin method. The new method is based on a discontinuous galerkin.
Numerical methods for partial differential equations 30. A numerical investigation of finite volume fv and discontinuous galerkin dg finite element methods in the framework of the su2 software is presented. Discontinuous galerkin method for solving steady and. Pdf the discontinuous galerkin method for the numerical. Comparison of continuous and hybridizable discontinuous.
Wolfgang dahmen3 1aachen institute for advanced study in computational engineering science, rwth aachen university. A new interior penalty discontinuous galerkin ipmdg formulation is developed, leading to a. We present a robust and accurate discretization approach for incompressible turbulent flows based on highorder discontinuous galerkin methods. Galerkin dg methods for incompressible fluid flow and consider methods that. Discontinuous galerkin methods for the stokes equations using. In this paper, the spectral volume sv method is experimented for the navierstokes equations by treating the viscous terms with a mixed formulation named the local discontinuous galerkin approach. Time discretization is done fully implicit using bac. A fronttracking method for viscous, incompressible, multifluid flows. Navierstokes solution using hybridizable discontinuous galerkin methods. Comparison of continuous and discontinuous galerkin approaches for variableviscosity stokes flow ragnar s.
Galerkin and discontinuous galerkin spectralhp methods t. Introduction to the numerical analysis of incompressible viscous flow by layton. Research in finite element methods for the numerical solution of problems with. This diagram shows the relations of the proposed hdg method bold to other methods.
Discontinuous galerkin method dgm is one of the most potential highorder discretization method among the stateoftheart methods, such as finite difference methods and finite volume method fvm. Pdf discontinuous galerkin methods for compressible and incompressible flows on spacetime adaptive meshes, phd thesis by francesco fambri doctoral thesis 76mb. Pdf local discontinuous galerkin methods with implicit. Discontinuous galerkin method for compressible viscous reacting flow yu lv and matthias ihmey department of mechanical engineering, stanford university, stanford, ca, 94305, usa in the present study, a discontinuous galerkin dg framework is developed to simulate chemically reacting ows. Analysis of hybrid discontinuous galerkin methods for. Efficient discontinuous galerkin implementations and. Symmetric interior penalty discontinuous galerkin discretisations and block preconditioning for heterogeneous stokes flow authors. Lagrangian methods for the discretization of the nonlinear convective terms are.
The accuracy of different numerical variants is assessed with reference to the low mach double vortex pairing flow problem, which. Discontinuous galerkin methods for the navierstokes equations using solenoidal approximations. Discontinuous galerkin methods for viscous incompressible flow. A discontinuous galerkin method with divergencefree interpolation for the incompressible stokes equations, international journal for numerical methods in fluids, 57 9, 10711092 2008.
In this paper, highorder accuracy is added by using spectral. Hybrid discontinuous galerkin methods for incompressible flow. The subject of the book is the mathematical theory of the discontinuous galerkin method dgm, which is a relatively new technique for the. Discontinuous galerkin methods use concepts both from finite volume and finite element methodology. We are interested ill solving the following 2d time dependent incompressible euler equations in vorticity streamfimctioll fornmlation.
Subcell shock capturing for discontinuous galerkin methods. We present a provably stable discontinuous galerkin spectral element method for the incompressible navierstokes equations with artificial compressibility and variable density. Discontinuous galerkin methods for the incompressible flow. A discontinuous galerkin method for the navier stokes equations. An equivalent statement that implies incompressibility is that the divergence of the flow velocity is zero see the derivation below, which illustrates why.
Bassi universita degli studi di bergamo, dipartimento di ingegneria industriale, bergamo, italy. Different formulations of the discontinuous galerkin. Jul 25, 2006 2014 upwind discontinuous galerkin methods with mass conservation of both phases for incompressible twophase flow in porous media. Curved surface mesh is generated using a capri mesh parameterization tool for higherorder surface representations. Discontinuous galerkin methods for viscous incompressible. Special case of the generalized solution equation for linear and stationary improved stokes equations retrogresses into generalized solution equation for classical stokes equation. Incompressible magnetohydrodynamics is the area of physics that is concerned with the behaviour of electrically conducting, resistive, incompressible and viscous fluids in the presence of electromagnetic fields. While the inter face is explicitly tracked, it is not kept completely sharp but is rather given a finite thickness of the order of the mesh size. We formulate algorithms for both incompressible and compressible flows with emphasis on high reynolds number. Analysis and applications to compressible flow vit dolejsi, miloslav feistauer auth. Navierstokes solution using hybridizable discontinuous. A discontinuous galerkin method for the navier stokes. One major weakness of the dg methods is that more degree of freedom is needed.
The dg discretization of the incompressible navierstokes equations uses the local laxfriedrichs flux for the convective term, the symmetric interior penalty method for the viscous term, and central fluxes for the velocitypressure coupling. Entropystable discontinuous galerkin approximation with. Highorder hybridizable discontinuous galerkin method for. Popov2 1 max planck graduate center with the johannes gutenberguniversit. Sep 01, 2014 to this end, this work is concentrated on the development of highorder discretization methods, consisting of both discontinuous galerkin,6,18,9,20 and streamline upwindpetrov galerkin 2124 discretizations, to further expand the capability of highorder schemes in solving a wide range of viscous flow problems for complex geometries. We derive a discontinuous galerkin dg finite element formulation for the improved stokes equations.
Adaptive discontinuous galerkin methods for solving an. Galerkin and discontinuous galerkin spectralhp methods. Stability evaluation of highorder splitting method for. Discontinuous galerkin methods for compressible and. In particular, for incompressible flows we employ galerkin projections and combine a c spectralhp. I hybrid discontinuous galerkin method for viscous terms i upwind ux in hdgsence for the convection term leading to solutions, which are i locally conservative i energystable d dt kuk2 l 2 c kfk2 l 2 i exactly incompressible i static condensation i standard nite element assembly is possible i less matrix entries than for std. The discontinuous galerkin method for the numerical simulation of compressible viscous flow article pdf available in the european physical journal conferences 672014. Perairey massachusetts institute of technology, cambridge, ma 029, usa b. More recently, shu 19 summarized three diierent formulations of the discontinuous galerkin method for the diiusion equation, and zhang and shu 20 performed a fourier type analysis for these. A cutoff operator is introduced in dg to treat general kinetic chemistry. Discontinuous galerkin and petrov galerkin methods are investigated and developed for laminar and turbulent flows.
Nov 27, 2007 discontinuous galerkin methods for viscous incompressible flow by guido kanschat, 9783835040014, available at book depository with free delivery worldwide. However, the dg formulation is far less certain and advantageous for diffusion equations, such as the navierstokes equations, where viscous flux exists and requires the evaluation of the solution derivatives at the interfaces. Spectralhp element methods for computational fluid dynamics by karniadakis and sherwin, oxford, 2005. A fronttracking method for viscous, incompressible, multi. Otherwise, the full compressible navierstokes equations have to be employed. Setting out from nitsches method for weak boundary. Me 697f spring 2010 galerkin methods for fluid dynamics. A direct discontinuous galerkin method for the incompressible. Due to many advantages, recently, dg methods have been applied for solving variational inequalities.
The foundations of a new discontinuous galerkin method for simulating compressible viscous flows with shocks on standard unstructured grids are presented in. The spatial discretization of the unsteady incompressible navierstokes equations is stated as a system of differential algebraic equations daes, corresponding to the conservation of momentum equation plus the constraint due to the incompressibility condition. This phd thesis focuses on the development of an efficient and robust highorder hybridizable discontinuous galerkin hdg finite element method fem for compressible viscous flow computations. Dgm has attracted lots of attention from both academic and industry community for the easiness to achieve highorder spacial convergence rate. Discontinuous galerkin finite element method for the. To develop high order discontinuous galerkin \ud method for solving steady and unsteady incompressible flow \ud based on artificial compressibility method. This is a stable, highorder accurate and locally conservative nite element method whose approximate solution is discontinuous across interelement boundaries. Highorder discontinuous galerkin methods for incompressible. An extension of the simple based discontinuous galerkin. Stability proofs, which include boundary conditions, that follow a continuous entropy analysis are provided. Navierstokes solution using hybridizable discontinuous galerkin methods d. In fluid mechanics or more generally continuum mechanics, incompressible flow isochoric flow refers to a flow in which the material density is constant within a fluid parcelan infinitesimal volume that moves with the flow velocity. The compactness of dg methods, facilitate the parallelization and their elementbyelement discontinuous nature is also helpful for adaptivity. To this end, this work is concentrated on the development of highorder discretization methods, consisting of both discontinuous galerkin,6,18,9,20 and streamline upwindpetrov galerkin 2124 discretizations, to further expand the capability of highorder schemes in solving a wide range of viscous flow problems for complex geometries.
Interior mesh is deformed via a linear elasticity strategy to obtain valid highorder finite element meshes. A rungekutta discontinuous galerkin method for viscous flow equations. Several numerical examples, including viscous flow over a threedimensional cylinder and flow over an onera m6 swept wing are presented and compared with a discontinuous galerkin method. Cockburnz university of minnesota, minneapolis, mn 55455, usa we present a hybridizable discontinuous galerkin method for the numerical solution the incompressible navierstokes equations. Discontinuous galerkin methods, theory, computation. Discontinuous galerkin method for solving steady and unsteady.
Typical solutions of incompressible flow problems involve both fine and largescale phenomena, so that a uniform finite element mesh of sufficient granularity will at best be wasteful of computational resources, and at worst be infeasible because of resource. Discontinuous galerkin methods for computational aerodynamics 3d adaptive flow simulation with the dlr padge code aerospace science and technology, vol. A highorder accurate discontinuous finite element method for inviscid and viscous turbomachinery flows. This paper uses discontinuous galerkin finite element procedure which is based on the artificial compressibility technique in connection with a dual time stepping approach. A hybridizable discontinuous galerkin method for the.
Me 697f spring 2010 galerkin methods for fluid dynamics basics. The subject of the book is the mathematical theory of the discontinuous galerkin method dgm, which is a relatively new technique for the numerical solution of partial differential equations. Mixed discontinuous galerkin finite element methods for. A note on discontinuous galerkin divergencefree solutions of the.
Rungekutta methods applied to the solution of the resulting index2 dae system are analyzed, allowing a critical comparison. Symmetric and nonsymmetric discontinuous galerkin methods. Understanding and implementing the finite element method by gockenbach, siam 2006. Key words, incompressible flow, discontinuous galerkin, high order accuracy subject classification. Schnepp submitted on jul 2016 v1, last revised 3 mar 2017 this version, v2. Nodal discontinuous galerkin methods by hesthaven and warburton, springer 2008. Hybrizidable discontinuous galerkin methods hdg were then introduced to address some of the shortcomings of dg methods and they are nowadays an active area of. This work aims at employing the discontinuous galerkin dg methods for the incompressible flow with nonlinear leak boundary conditions of friction type, whose continuous variational problem is an inequality due to the subdifferential property of such boundary conditions. Peraire z massachusetts institute of technology, cambridge, ma 029, usa we are concerned with the numerical solution of the navierstokes and reynoldsaveraged navierstokes equations using the hybridizable discontinuous galerkin hdg methods recently introduced in ref. A discontinuous galerkin method for the navierstokes equations lomtev, igor. Hybrid discontinuous galerkin methods for solving incompressible flow problems.
Comparison of continuous and discontinuous galerkin. The dg discretization of the incompressible navierstokes equations uses the local laxfriedrichs flux for the convective term, the symmetric interior penalty method for the viscous term, and central fluxes for the velocitypressure. Comparison of the finite volume and discontinuous galerkin. For the linear problem of stokes, there is a large literature on the analysis of dg methods. Pdf hybrid discontinuous galerkin methods for solving. Pdf disclaimer disclaimer restricted to repository staff only until 9999. Hybrid discontinuous galerkin methods for incompressible. Pdf this thesis deals with a higher order discretization of incompressible flow. We develop and analyze mixed discontinuous galerkin finite element methods for the numerical approximation of incompressible magnetohydrodynamics problems. Robust and efficient discontinuous galerkin methods for.
Several numerical examples, including viscous flow over a threedimensional cylinder and flow over an onera m6 swept wing are presented and compared with a discontinuousgalerkin method. A parallel, reconstructed discontinuous galerkin method. A hybrid reconstructed discontinuous galerkin and continuous galerkin finite element method for incompressible flows on unstructured grids. Typical solutions of incompressible flow problems involve both fine and largescale phenomena, so that a uniform finite element mesh of sufficient granularity will at best be wasteful of computational resources, and at worst be infeasible because of resource limitations. The paper deals with the use of the discontinuous galerkin finite element method dgfem for the numerical solution of viscous compressible flows.
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