An efficient method for the incompressible Navier-Stokes equations on irregular domains with no-slip boundary conditions, high order up to the boundary

Common efficient schemes for the incompressible Navier-Stokes equations, such as projection or fractional step methods, have limited temporal accuracy as a result of matrix splitting errors, or introduce errors near the domain boundaries (which destroy uniform convergence to the solution). In this paper we recast the incompressible (constant density) Navier-Stokes equations (with the velocity prescribed at the boundary) as an equivalent system, for the primary variables velocity and pressure. equation for the pressure. The key difference from the usual approaches occurs at the boundaries, where we use boundary conditions that unequivocally allow the pressure to be recovered from knowledge of the velocity at any fixed time. This avoids the common difficulty of an, apparently, over-determined Poisson problem. Since in this alternative formulation the pressure can be accurately and efficiently recovered from the velocity, the recast equations are ideal for numerical marching methods. The new system can be discretized using a variety of methods, including semi-implicit treatments of viscosity, and in principle to any desired order of accuracy. In this work we illustrate the approach with a 2-D second order finite difference scheme on a Cartesian grid, and devise an algorithm to solve the equations on domains with curved (non-conforming) boundaries, including a case with a non-trivial topology (a circular obstruction inside the domain). This algorithm achieves second order accuracy in the L^~ norm, for both the velocity and the pressure. The scheme has a natural extension to 3-D.

[1]  A. Prohl Projection and quasi-compressibility methods for solving the incompressible navier-stokes equations , 1997 .

[2]  S. Orszag,et al.  High-order splitting methods for the incompressible Navier-Stokes equations , 1991 .

[3]  Jacques Periaux,et al.  Numerical simulation and optimal shape for viscous flow by a fictitious domain method , 1995 .

[4]  Olivier Pironneau,et al.  Pressure boundary condition for the time‐dependent incompressible Navier–Stokes equations , 2006 .

[5]  E Weinan,et al.  GAUGE METHOD FOR VISCOUS INCOMPRESSIBLE FLOWS , 2003 .

[6]  Gianluca Iaccarino,et al.  IMMERSED BOUNDARY METHODS , 2005 .

[7]  Tim Colonius,et al.  The immersed boundary method: A projection approach , 2007, J. Comput. Phys..

[8]  Claudio Canuto,et al.  Spectral Methods: Evolution to Complex Geometries and Applications to Fluid Dynamics (Scientific Computation) , 2007 .

[9]  D. Gottlieb,et al.  Numerical analysis of spectral methods : theory and applications , 1977 .

[10]  L. Sirovich,et al.  Modeling a no-slip flow boundary with an external force field , 1993 .

[11]  R. Temam Sur l'approximation de la solution des équations de Navier-Stokes par la méthode des pas fractionnaires (II) , 1969 .

[12]  F. Wubs Notes on numerical fluid mechanics , 1985 .

[13]  Don Redmond,et al.  Introduction to Numerical Analysis , 1994 .

[14]  U. Ghia,et al.  High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method , 1982 .

[15]  R. Sani,et al.  On pressure boundary conditions for the incompressible Navier‐Stokes equations , 1987 .

[16]  N A Petersson,et al.  A Split-Step Scheme for the Incompressible Navier-Stokes , 2001 .

[17]  J. Kan A second-order accurate pressure correction scheme for viscous incompressible flow , 1986 .

[18]  U. Schumann,et al.  Treatment of incompressibility and boundary conditions in 3-D numerical spectral simulations of plane channel flows , 1980 .

[19]  R. Temam Sur l'approximation de la solution des équations de Navier-Stokes par la méthode des pas fractionnaires (I) , 1969 .

[20]  F. Harlow,et al.  Numerical Calculation of Time‐Dependent Viscous Incompressible Flow of Fluid with Free Surface , 1965 .

[21]  N. Anders Petersson,et al.  Stability of pressure boundary conditions for Stokes and Navier-Stokes equations , 2001 .

[22]  T. A. S. Jackson Orthogonal Curvilinear Coordinates , 1966 .

[23]  M. Minion,et al.  Accurate projection methods for the incompressible Navier—Stokes equations , 2001 .

[24]  R. Rannacher On chorin's projection method for the incompressible navier-stokes equations , 1992 .

[25]  Jian-Guo Liu,et al.  Stable and accurate pressure approximation for unsteady incompressible viscous flow , 2010, J. Comput. Phys..

[26]  J. Strikwerda Finite Difference Schemes and Partial Differential Equations , 1989 .

[27]  J. B. Perot,et al.  An analysis of the fractional step method , 1993 .

[28]  P. Moin,et al.  Application of a Fractional-Step Method to Incompressible Navier-Stokes Equations , 1984 .

[29]  H. Fasel,et al.  A high-order immersed interface method for simulating unsteady incompressible flows on irregular domains , 2005 .

[30]  Philippe Angot,et al.  A penalization method to take into account obstacles in incompressible viscous flows , 1999, Numerische Mathematik.

[31]  D. Rempfer On Boundary Conditions for Incompressible Navier-Stokes Problems , 2006 .

[32]  C. Peskin Flow patterns around heart valves: A numerical method , 1972 .

[33]  A. Chorin Numerical solution of the Navier-Stokes equations , 1968 .

[34]  L. Quartapelle,et al.  A review of vorticity conditions in the numerical solution of the ζ–ψ equations , 1999 .

[35]  R. D. Richtmyer,et al.  Survey of the stability of linear finite difference equations , 1956 .

[36]  L. Quartapelle,et al.  Numerical solution of the incompressible Navier-Stokes equations , 1993, International series of numerical mathematics.

[37]  Hans Johnston,et al.  Accurate, stable and efficient Navier-Stokes solvers based on explicit treatment of the pressure term , 2004 .

[38]  A. Hammouti Simulation numérique directe en différence finie de l'écoulement d'un fluide incompressible en présence d'interfaces rigides , 2009 .

[39]  Jie Shen,et al.  An overview of projection methods for incompressible flows , 2006 .

[40]  D. Calhoun A Cartesian Grid Method for Solving the Two-Dimensional Streamfunction-Vorticity Equations in Irregular Regions , 2002 .

[41]  Heinz-Otto Kreiss,et al.  A fourth-order-accurate difference approximation for the incompressible Navier-Stokes equations☆ , 1994 .

[42]  P. Colella,et al.  A second-order projection method for the incompressible navier-stokes equations , 1989 .

[43]  Joel H. Ferziger,et al.  Computational methods for fluid dynamics , 1996 .

[44]  Jie Shen On error estimates of projection methods for Navier-Stokes equations: first-order schemes , 1992 .

[45]  Hans Johnston,et al.  Finite Difference Schemes for Incompressible Flow Based on Local Pressure Boundary Conditions , 2002 .

[46]  M. Ben-Artzi,et al.  A pure-compact scheme for the streamfunction formulation of Navier-Stokes equations , 2005 .

[47]  S. Orszag,et al.  Boundary conditions for incompressible flows , 1986 .

[48]  Khodor Khadra,et al.  Fictitious domain approach for numerical modelling of Navier–Stokes equations , 2000 .

[49]  C. Peskin The immersed boundary method , 2002, Acta Numerica.

[50]  William D. Henshaw,et al.  A Fourth-Order Accurate Method for the Incompressible Navier-Stokes Equations on Overlapping Grids , 1994 .