Finite-volume solution of two-dimensional compressible flows over dynamic adaptive grids

A novel Finite Volume (FV) technique for solving the compressible unsteady Euler equations is presented for two-dimensional adaptive grids over time dependent geometries. The interpretation of the grid modifications as continuous deformations of the underlying discrete finite volumes allows to determine the solution over the new grid by direct integration of the governing equations within the Arbitrary Lagrangian-Eulerian (ALE) framework, without any explicit interpolation step. The grid adaptation is performed using a suitable mix of grid deformation, edge-swapping, node insertion and node removal techniques in order to comply with the displacement of the boundaries of the computational domain and to preserve the quality of the grid elements. Both steady and unsteady simulations over adaptive grids are presented that demonstrate the validity of the proposed approach. The adaptive ALE scheme is used to perform high-resolution computations of the steady flow past a translating airfoil and of the unsteady flow of a pitching airfoil in both the airfoil and the laboratory reference, with airfoil displacement as large as 200 airfoil chords. Grid adaptation is found to be of paramount importance to preserve the grid quality in the considered problems.

[1]  Marco Fossati,et al.  A Node-Pair Finite Element / Finite Volume Mesh Adaptation Technique for Compressible Flows , 2010 .

[2]  Charbel Farhat,et al.  Design and analysis of robust ALE time-integrators for the solution of unsteady flow problems on moving grids , 2004 .

[3]  Rémi Abgrall,et al.  A Simple Method for Compressible Multifluid Flows , 1999, SIAM J. Sci. Comput..

[4]  P. George,et al.  Mesh Generation: Application to Finite Elements , 2007 .

[5]  Raphaël Loubère,et al.  ReALE: A reconnection-based arbitrary-Lagrangian-Eulerian method , 2010, J. Comput. Phys..

[6]  P. Mantegazza,et al.  Application of Navier-Stokes simulations for aeroelastic stability assessment in transonic regime , 2007 .

[7]  B. V. Leer,et al.  Towards the ultimate conservative difference scheme. II. Monotonicity and conservation combined in a second-order scheme , 1974 .

[8]  Len G. Margolin,et al.  Second-order sign-preserving conservative interpolation (remapping) on general grids , 2003 .

[9]  Jonathan Richard Shewchuk,et al.  What is a Good Linear Element? Interpolation, Conditioning, and Quality Measures , 2002, IMR.

[10]  Pavel Váchal,et al.  Sequential Flux-Corrected Remapping for ALE Methods , 2006 .

[11]  V. Selmin,et al.  The node-centred finite volume approach: bridge between finite differences and finite elements , 1993 .

[12]  Giuseppe Quaranta,et al.  Arbitrary Lagrangian Eulerian formulation for two-dimensional flows using dynamic meshes with edge swapping , 2011, J. Comput. Phys..

[13]  Oubay Hassan,et al.  Unstructured mesh methods for the solution of the unsteady compressible flow equations with moving boundary components , 2007, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[14]  R B Pember,et al.  A Dynamically Adaptive Arbitrary Lagrangian-Eulerian Method for Solution of the Euler Equations , 2003 .

[15]  Wei Shyy,et al.  Computational Fluid Dynamics with Moving Boundaries , 1995 .

[16]  Rong Wang,et al.  A comparison of adaptive software for 1D parabolic PDEs , 2004 .

[17]  Charbel Farhat,et al.  Design and analysis of ALE schemes with provable second-order time-accuracy for inviscid and viscous flow simulations , 2003 .

[18]  R. Garimella,et al.  Untangling of 2D meshes in ALE simulations , 2004 .

[19]  R. Abgrall.  How to Prevent Pressure Oscillations in Multicomponent Flow Calculations , 1996 .

[20]  P. Raviart,et al.  Numerical Approximation of Hyperbolic Systems of Conservation Laws , 1996, Applied Mathematical Sciences.

[21]  Fabio Nobile,et al.  Numerical approximation of fluid-structure interaction problems with application to haemodynamics , 2001 .

[22]  Fabio Nobile,et al.  A Stability Analysis for the Arbitrary Lagrangian Eulerian Formulation with Finite Elements , 1999 .

[23]  Mikhail Shashkov,et al.  An efficient linearity and bound preserving conservative interpolation (remapping) on polyhedral meshes , 2007 .

[24]  D. Isola.  An interpolation-free two-dimensional conservative ALE scheme over adaptive unstructured grids for rotorcraft aerodynamics , 2012 .

[25]  Marc R.J. Charest,et al.  A three-dimensional finite element arbitrary Lagrangian–Eulerian method for shock hydrodynamics on unstructured grids☆ , 2014 .

[26]  P. Roe.  Approximate Riemann Solvers, Parameter Vectors, and Difference Schemes , 1997 .

[27]  Dominique Pelletier,et al.  Perspective on the geometric conservation law and finite element methods for ALE simulations of incompressible flow , 2009, J. Comput. Phys..

[28]  Charbel Farhat,et al.  Partitioned procedures for the transient solution of coupled aeroelastic problems , 2001 .

[29]  Carl Ollivier-Gooch,et al.  Tetrahedral mesh improvement using swapping and smoothing , 1997 .

[30]  Dimitri J. Mavriplis,et al.  Construction of the discrete geometric conservation law for high-order time-accurate simulations on dynamic meshes , 2006, J. Comput. Phys..

[31]  Patrick Jenny,et al.  Correction of Conservative Euler Solvers for Gas Mixtures , 1997 .

[32]  V. Venkatakrishnan,et al.  IMPLICIT METHOD FOR THE COMPUTATION OF UNSTEADY FLOWS ON UNSTRUCTURED GRIDS , 1995 .

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

[34]  M. Shashkov,et al.  An efficient linearity-and-bound-preserving remapping method , 2003 .

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

[36]  Miguel A. Fernández,et al.  Deriving Adequate Formulations for Fluid-Structure Interaction Problems: from ALE to Transpiration , 2000 .

[37]  Randall J. LeVeque.  Finite Volume Methods for Hyperbolic Problems: Nonlinear Systems of Conservation Laws , 2002 .

[38]  Jérôme Breil,et al.  Two-step hybrid conservative remapping for multimaterial arbitrary Lagrangian-Eulerian methods , 2011, J. Comput. Phys..

[39]  H. Deconinck,et al.  An arbitrary Lagrangian Eulerian formulation for residual distribution schemes on moving grids , 2003 .

[40]  J. Lambert.  Numerical Methods for Ordinary Differential Systems: The Initial Value Problem , 1991 .

[41]  Charbel Farhat,et al.  The discrete geometric conservation law and the nonlinear stability of ALE schemes for the solution of flow problems on moving grids , 2001 .

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

[43]  Joe F. Thompson,et al.  Numerical grid generation: Foundations and applications , 1985 .