A stabilized edge-based implicit incompressible flow formulation

Abstract An edge-based implementation of an implicit, monolithic, finite element (FE) scheme for the solution of the incompressible Navier–Stokes (NS) equations is presented. The original element formulation is based on the pressure stability properties of an implicit second-order in time fractional step (FS) method, which is conditionally stable. The final monolithic scheme preserves the second-order accuracy of the FS method, and is unconditionally stable. Furthermore, it can be demonstrated that the final pressure stabilizing term is practically the same fourth-order pressure term added by some authors (but following different arguments) to obtain high order accurate results, and that the final discretized convective terms are formally a second-order discretization of the respective continuous one. The development of the edge implementation is supported by two criteria: the properties of the element based one, which has already been extensively tested and for which convergence and stability analysis has already been presented, and on the enforcement of global conservation and symmetry at the discrete level. A monotonicity preserving term which decreases the discretization order in sharp gradient regions to avoid localized oscillations (overshoots and undershoots), is formulated and tested. Some numerical examples and experimental comparisons are presented.

[1]  L. Franca,et al.  Stabilized finite element methods. II: The incompressible Navier-Stokes equations , 1992 .

[2]  Caskey,et al.  GENERAL CIRCULATION EXPERIMENTS WITH THE PRIMITIVE EQUATIONS I . THE BASIC EXPERIMENT , 1962 .

[3]  Wayne A. Smith,et al.  Preconditioning Applied to Variable and Constant Density Flows , 1995 .

[4]  Alexandre J. Chorin,et al.  On the Convergence of Discrete Approximations to the Navier-Stokes Equations , 1969 .

[5]  Rainald Löhner,et al.  A parallel implicit incompressible flow solver using unstructured meshes , 1993 .

[6]  J. Smagorinsky,et al.  GENERAL CIRCULATION EXPERIMENTS WITH THE PRIMITIVE EQUATIONS , 1963 .

[7]  R. Codina,et al.  A numerical model to track two‐fluid interfaces based on a stabilized finite element method and the level set technique , 2002 .

[8]  Juan R. Cebral,et al.  An Implicit Monolithic Time Accurate Finite Element Scheme for Incompressible Flow Problems , 2001 .

[9]  Rainald Loehner,et al.  An unstructured grid-based, parallel free surface solver , 1997 .

[10]  B. Armaly,et al.  Experimental and theoretical investigation of backward-facing step flow , 1983, Journal of Fluid Mechanics.

[11]  R. Codina.  Pressure Stability in Fractional Step Finite Element Methods for Incompressible Flows , 2001 .

[12]  C. L. Merkle,et al.  The application of preconditioning in viscous flows , 1993 .

[13]  Takanori Hino,et al.  An unstructued grid method for incompressible viscous flows with a free surface , 1997 .

[14]  Ramon Codina,et al.  A nodal-based implementation of a stabilized finite element method for incompressible flow problems , 2000 .

[15]  M. Hafez,et al.  Frontiers of Computational Fluid Dynamics 2002 , 2001 .

[16]  R. Codina.  A stabilized finite element method for generalized stationary incompressible flows , 2001 .

[17]  Marek Behr,et al.  Finite element solution strategies for large-scale flow simulations☆ , 1994 .

[18]  E. Turkel,et al.  Preconditioned methods for solving the incompressible low speed compressible equations , 1987 .

[19]  Arthur Rizzi,et al.  Computation of inviscid incompressible flow with rotation , 1985, Journal of Fluid Mechanics.

[20]  Rainald Löhner,et al.  An unstructured grid-based, parallel free surface solver , 1997 .

[21]  Jonathan Weiss,et al.  Preconditioning Applied to Variable and Constant Density Time-Accurate Flows on Unstructured Meshes , 1994 .

[22]  Rainald Löhner,et al.  CFD SHAPE OPTIMIZATION USING AN INCOMPLETE-GRADIENT ADJOINT FORMULATION , 2001 .

[23]  Dorothee Martin,et al.  An implicit, linelet-based solver for incompressible flows , 1992 .

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

[25]  Ravi Ramamurti,et al.  Unstructured Grids for Ship Unsteady Airwakes: A Successful Validation , 2002 .

[26]  R. Codina.  Stabilization of incompressibility and convection through orthogonal sub-scales in finite element methods , 2000 .

[27]  Rainald Löhner,et al.  A linelet preconditioner for incompressible flow solvers , 2003 .

[28]  Yousef Saad,et al.  Iterative methods for sparse linear systems , 2003 .

[29]  T. Hughes,et al.  Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations , 1990 .

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

[31]  R. Codina,et al.  Finite element solution of the Stokes problem with dominating Coriolis force , 1997 .

[32]  Ravi Ramamurti,et al.  LPD-17 Topside Aerodynamic Study: FEFLO , 2000 .