Discontinuous Galerkin solution of the Navier-Stokes equations on deformable domains Page: 1 of 16
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Discontinuous Galerkin Solution of the Navier-Stokes Equations on
Deformable Domains
P.-O. Persson a J. Bonet b and J. Peraire c,*
"Department of Mathematics, University of Cahfornia, Berkeley, Berkeley, CA 94720-3840, USA
"School of Engineering, Swansea University, Swansea SA2 8PP, UK
cDepartment of Aeronautics and Astronautics, Massachusetts Institute of Technology, Cambridge. MA 02139, USA
Abstract
We describe a method for computing time-dependent solutions to the compressible Navier-Stokes equations on
variable geometries. We introduce a continuous mapping between a fixed reference configuration and the time varying
domain. By writing the Navier-Stokes equations as a conservation law for the independent variables in the reference
configuration, the complexity introduced by variable geometry is reduced to solving a transformed conservation law in
a fixed reference configuration. The spatial discretization is carried out using the Discontinuous Galerkin method on
unstructured meshes of triangles, while the time integration is performed using an explicit Runge-Kutta method. For
general domain changes, the standard scheme fails to preserve exactly the free-stream solution which leads to some
accuracy degradation, especially for low order approximations. This situation is remedied by adding an additional
equation for the time evolution of the transformation Jacobian to the original conservation law and correcting for
the accumulated metric integration errors. A number of results are shown to illustrate the flexibility of the approach
to handle high order approximations on complex geometries.
Key words: Discontinuous Galerkin, Deformable domains, Navier-Stokes, Arbitrary Lagrangian-Eulerian, Geometric
Conservation
1. Introduction
There is a growing interest in high-order methods for fluid problems, largely because of their ability to
produce highly accurate solutions with minimum numerical dispersion. The Discontinuous Galerkin (DG)
method produces stable discretizations of the convective operator for any order discretization. Moreover, it
can be used with unstructured meshes of simplices, which appears to be a requirement for real-world complex
geometries. In this paper, we present a high order DG formulation for computing high order solutions to
problems with variable geometries.
Time varying geometries appear in a number of practical applications such us rotor-stator flows, flapping
flight or fluid-structure interactions. In such cases, it is necessary to properly account for the time variation
of the solution domain if accurate solutions are to be obtained. For the Navier-Stokes equations, there has
been a considerable effort in the development of Arbitrary Lagrangian Eulerian (ALE) methods to deal with
* Corresponding author. Tel.: +1-617-253-1981; Fax : +1-617-258-5143.
Emaal address. peraireemit .edu (J. Peraire).Preprint submitted to Computer Methods in Applied Mechanics and Engineerng
13 January 2009
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Persson, P.-O.; Bonet, J. & Peraire, J. Discontinuous Galerkin solution of the Navier-Stokes equations on deformable domains, article, January 13, 2009; Berkeley, California. (https://digital.library.unt.edu/ark:/67531/metadc893590/m1/1/: accessed April 25, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.