A high-order spectral difference method for unstructured dynamic grids

Abstract A high-order spectral difference (SD) method has been further extended to solve the three dimensional compressible Navier–Stokes (N–S) equations on deformable dynamic meshes. In the SD method, the solution is approximated with piece-wise continuous polynomials. The elements are coupled with common Riemann fluxes at element interfaces. The extension to deformable elements necessitates a time-dependent geometric transformation. The Geometric Conservation Law (GCL), which is introduced in the time-dependent transformation from the physical domain to the computational domain, has been discussed and implemented for both explicit and implicit time marching methods. Accuracy studies are performed with a vortex propagation problem, demonstrating that the spectral difference method can preserve high-order accuracy on deformable meshes. Further applications of the method to several moving boundary problems including bio-inspired flow problems are shown in the paper to demonstrate the capability of the developed method.

[1]  Hui Hu,et al.  A Numerical Study of Vortex-Dominated Flow around an Oscillating Airfoil with High-Order Spectral Difference Method , 2010 .

[2]  R. Pletcher,et al.  Computational Fluid Mechanics and Heat Transfer. By D. A ANDERSON, J. C. TANNEHILL and R. H. PLETCHER. Hemisphere, 1984. 599 pp. $39.95. , 1986, Journal of Fluid Mechanics.

[3]  Marcel Vinokur,et al.  Discontinuous Spectral Difference Method for Conservation Laws on Unstructured Grids , 2004 .

[4]  Yongsheng Lian,et al.  Résumé of the AIAA FDTC Low Reynolds Number Discussion Group's Canonical Cases , 2010 .

[5]  Michael V. Ol,et al.  The High-Frequency, High-Amplitude Pitch Problem: Airfoils, Plates and Wings , 2009 .

[6]  Chris Lacor,et al.  On the Stability and Accuracy of the Spectral Difference Method , 2008, J. Sci. Comput..

[7]  Marcel Vinokur,et al.  Spectral difference method for unstructured grids I: Basic formulation , 2006, J. Comput. Phys..

[8]  Georg May,et al.  A Spectral Dierence Method for the Euler and Navier-Stokes Equations on Unstructured Meshes , 2006 .

[9]  C. M. Dohring,et al.  Experimental and Computational Investigation of the Knoller-Betz Effect , 1998 .

[10]  Miguel R. Visbal,et al.  High-Fidelity Simulation of Transitional Flows Past a Plunging Airfoil , 2009 .

[11]  Miguel R. Visbal,et al.  On the use of higher-order finite-difference schemes on curvilinear and deforming meshes , 2002 .

[12]  Zhi J. Wang,et al.  High-Order Multidomain Spectral Difference Method for the Navier-Stokes Equations , 2006 .

[13]  Z. J. Wang,et al.  Efficient Implicit Non-linear LU-SGS Approach for Compressible Flow Computation Using High-Order Spectral Difference Method , 2008 .

[14]  Antony Jameson,et al.  Spectral Difference Method for Unstructured Grids II: Extension to the Euler Equations , 2007, J. Sci. Comput..

[15]  Herman Deconinck,et al.  Algorithmic developments for a multiphysics framework , 2008 .

[16]  Antony Jameson,et al.  On the Temporal and Spatial Accuracy of Spectral Difference Method on Moving Deformable Grids and the Effect of Geometry Conservation Law , 2010 .

[17]  Jeff D. Eldredge,et al.  A Computational Study of a Canonical Pitch-Up, Pitch-Down Wing Maneuver , 2009 .

[18]  Ken Badcock,et al.  A grid deformation technique for unsteady flow computations , 2000 .

[19]  T. Tezduyar,et al.  Mesh Moving Techniques for Fluid-Structure Interactions With Large Displacements , 2003 .

[20]  John H. Kolias,et al.  A CONSERVATIVE STAGGERED-GRID CHEBYSHEV MULTIDOMAIN METHOD FOR COMPRESSIBLE FLOWS , 1995 .

[21]  H. T. Huynh,et al.  A Flux Reconstruction Approach to High-Order Schemes Including Discontinuous Galerkin Methods , 2007 .

[22]  K. V. Ellenrieder,et al.  PIV measurements of the asymmetric wake of a two dimensional heaving hydrofoil , 2008 .

[23]  Zhi J. Wang,et al.  Spectral (Finite) Volume Method for Conservation Laws on Unstructured Grids. Basic Formulation , 2002 .

[24]  Fang Q. Hu,et al.  Absorbing boundary conditions for nonlinear Euler and Navier-Stokes equations based on the perfectly matched layer technique , 2008, J. Comput. Phys..

[25]  G. Lauder,et al.  Hydrodynamics of a biologically inspired tandem flapping foil configuration , 2007 .

[26]  Chunlei Liang,et al.  High-Order Spectral Difference Method for the Navier-Stokes Equation on Unstructured Moving Deformable Grid , 2010 .

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

[28]  M. Platzer,et al.  Flapping Wing Aerodynamics - Progress and Challenges , 2006 .

[29]  Dimitri J. Mavriplis,et al.  On the geometric conservation law for high-order discontinuous Galerkin discretizations on dynamically deforming meshes , 2008, Journal of Computational Physics.

[30]  Manoochehr Koochesfahani,et al.  MTV measurements of the vortical field in the wake of an airfoil oscillating at high reduced frequency , 2009, Journal of Fluid Mechanics.

[31]  Z. Wang.  High-order methods for the Euler and Navier–Stokes equations on unstructured grids , 2007 .

[32]  P. Thomas,et al.  Geometric Conservation Law and Its Application to Flow Computations on Moving Grids , 1979 .

[33]  Jaime Peraire,et al.  Discontinuous Galerkin Solution of the Navier-Stokes Equations on Deformable Domains , 2007 .

[34]  Chunlei Liang,et al.  Extension of the SD Method to Viscous Flow on Unstructured Grids , 2009 .

[35]  A. Huerta,et al.  Arbitrary Lagrangian–Eulerian Methods , 2004 .

[36]  Antony Jameson,et al.  A Proof of the Stability of the Spectral Difference Method for All Orders of Accuracy , 2010, J. Sci. Comput..