Regular Article: A Discontinuous Galerkin ALE Method for Compressible Viscous Flows in Moving Domains

We present a matrix-free discontinuous Galerkin method for simulating compressible viscous flows in two- and three-dimensional moving domains. To this end, we solve the Navier-Stokes equations in an arbitrary Lagrangian Eulerian (ALE) framework. Spatial discretization is based on standard structured and unstructured grids but using an orthogonal hierarchical spectral basis. The method is third-order accurate in time and converges exponentially fast in space for smooth solutions. A novelty of the method is the use of a force-directed algorithm from graph theory that requires no matrix inversion to efficiently update the grid while minimizing distortions. We present several simulations using the new method, including validation with published results from a pitching airfoil, and new results for flow past a three-dimensional wing subject to large flapping insect-like motion.

[1]  Juhani Pitkäranta,et al.  An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation , 1986 .

[2]  F. J. Blom,et al.  Analysis of Fluid-Structure Interaction by Means of Dynamic Unstructured Meshes , 1998 .

[3]  Sherwin,et al.  Tetrahedral hp Finite Elements : Algorithms and Flow Simulations , 1996 .

[4]  R. Stephenson.  A and V , 1962, The British journal of ophthalmology.

[5]  D. Gottlieb,et al.  Numerical analysis of spectral methods , 1977 .

[6]  Ioannis Gregorios Giannakouros.  Spectral element/flux-corrected methods for unsteady compressible viscous flows , 1994 .

[7]  J. Halleux,et al.  An arbitrary lagrangian-eulerian finite element method for transient dynamic fluid-structure interactions , 1982 .

[8]  Spencer J. Sherwin,et al.  Hierarchical hp finite elements in hybrid domains , 1997 .

[9]  Chi-Wang Shu,et al.  TVB Runge-Kutta local projection discontinuous galerkin finite element method for conservation laws. II: General framework , 1989 .

[10]  Moshe Dubiner.  Spectral methods on triangles and other domains , 1991 .

[11]  Chi-Wang Shu,et al.  On a cell entropy inequality for discontinuous Galerkin methods , 1994 .

[12]  J. Batina.  Unsteady Euler airfoil solutions using unstructured dynamic meshes , 1989 .

[13]  J. Hyvärinen,et al.  An Arbitrary Lagrangian-Eulerian finite element method , 1998 .

[14]  George Em Karniadakis,et al.  Spectral/hp Methods for Viscous Compressible Flows on Unstructured 2D Meshes , 1998 .

[15]  Wing Kam Liu,et al.  Lagrangian-Eulerian finite element formulation for incompressible viscous flows☆ , 1981 .

[16]  Claes Johnson.  Numerical solution of partial differential equations by the finite element method , 1988 .

[17]  J. Oden,et al.  A discontinuous hp finite element method for convection—diffusion problems , 1999 .

[18]  Chi-Wang Shu,et al.  TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: one-dimensional systems , 1989 .

[19]  Russ D. Rausch,et al.  Three-dimensional time-marching aeroelastic analyses using an unstructured-grid Euler method , 1992 .

[20]  M. Visbal,et al.  Investigation of the flow structure around a rapidly pitching airfoil , 1989 .

[21]  S. Rebay,et al.  A High-Order Accurate Discontinuous Finite Element Method for the Numerical Solution of the Compressible Navier-Stokes Equations , 1997 .

[22]  Chittur S. Venkatasubban.  A new finite element formulation for ALE (Arbitrary Lagrangian Eulerian) compressible fluid mechanics , 1995 .

[23]  Jérôme Jaffré,et al.  CONVERGENCE OF THE DISCONTINUOUS GALERKIN FINITE ELEMENT METHOD FOR HYPERBOLIC CONSERVATION LAWS , 1995 .

[24]  Chi-Wang Shu,et al.  The Local Discontinuous Galerkin Method for Time-Dependent Convection-Diffusion Systems , 1998 .

[25]  George Em Karniadakis,et al.  Unstructured spectral element methods for simulation of turbulent flows , 1995 .

[26]  T. Tezduyar,et al.  A new strategy for finite element computations involving moving boundaries and interfaces—the deforming-spatial-domain/space-time procedure. I: The concept and the preliminary numerical tests , 1992 .

[27]  Rainald Löhner,et al.  Improved ALE mesh velocities for moving bodies , 1996 .

[28]  P. Tallec,et al.  Load and motion transfer algorithms for fluid/structure interaction problems with non-matching discrete interfaces: Momentum and energy conservation, optimal discretization and application to aeroelasticity , 1998 .

[29]  R. F. Warming,et al.  An Implicit Factored Scheme for the Compressible Navier-Stokes Equations , 1977 .

[30]  Thomas J. R. Hughes,et al.  An arbitrary Lagrangian-Eulerian finite rigid element method for interaction of fluid and a rigid body , 1992 .

[31]  J. Oden,et al.  hp-Version discontinuous Galerkin methods for hyperbolic conservation laws , 1996 .

[32]  J. Tinsley Oden,et al.  hp-version discontinuous Galerkin methods for hyperbolic conservation laws: A parallel adaptive strategy , 1995 .

[33]  K. Kawachi,et al.  A Numerical Study of Insect Flight , 1998 .

[34]  C. W. Hirt,et al.  An Arbitrary Lagrangian-Eulerian Computing Method for All Flow Speeds , 1997 .

[35]  Bernardo Cockburn,et al.  The P(1-) RKDG Method for Two-Dimensional Euler Equations of Gas Dynamics , 1991 .

[36]  Chi-Wang Shu,et al.  The Runge-Kutta Discontinuous Galerkin Method for Conservation Laws V , 1998 .

[37]  Charbel Farhat,et al.  On the significance of the GCL for flow computations on moving meshes , 1999 .

[38]  J. Tinsley Oden,et al.  A discontinuous hp finite element method for the Euler and Navier–Stokes equations , 1999 .

[39]  Chi-Wang Shu,et al.  The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. IV. The multidimensional case , 1990 .

[40]  George Em Karniadakis,et al.  TetrahedralhpFinite Elements , 1996 .