A numerical method for solving the 3D unsteady incompressible Navier-Stokes equations in curvilinear domains with complex immersed boundaries

A novel numerical method is developed that integrates boundary-conforming grids with a sharp interface, immersed boundary methodology. The method is intended for simulating internal flows containing complex, moving immersed boundaries such as those encountered in several cardiovascular applications. The background domain (e.g the empty aorta) is discretized efficiently with a curvilinear boundary-fitted mesh while the complex moving immersed boundary (say a prosthetic heart valve) is treated with the sharp-interface, hybrid Cartesian/immersed-boundary approach of Gilmanov and Sotiropoulos [1]. To facilitate the implementation of this novel modeling paradigm in complex flow simulations, an accurate and efficient numerical method is developed for solving the unsteady, incompressible Navier-Stokes equations in generalized curvilinear coordinates. The method employs a novel, fully-curvilinear staggered grid discretization approach, which does not require either the explicit evaluation of the Christoffel symbols or the discretization of all three momentum equations at cell interfaces as done in previous formulations. The equations are integrated in time using an efficient, second-order accurate fractional step methodology coupled with a Jacobian-free, Newton-Krylov solver for the momentum equations and a GMRES solver enhanced with multigrid as preconditioner for the Poisson equation. Several numerical experiments are carried out on fine computational meshes to demonstrate the accuracy and efficiency of the proposed method for standard benchmark problems as well as for unsteady, pulsatile flow through a curved, pipe bend. To demonstrate the ability of the method to simulate flows with complex, moving immersed boundaries we apply it to calculate pulsatile, physiological flow through a mechanical, bileaflet heart valve mounted in a model straight aorta with an anatomical-like triple sinus.

[1]  H. S. Udaykumar,et al.  A Sharp Interface Cartesian Grid Methodfor Simulating Flows with ComplexMoving Boundaries , 2001 .

[2]  G. D. Raithby,et al.  A method for computing three dimensional flows using non‐orthogonal boundary‐fitted co‐ordinates , 1984 .

[3]  R. Verzicco,et al.  Combined Immersed-Boundary Finite-Difference Methods for Three-Dimensional Complex Flow Simulations , 2000 .

[4]  Martha Salcudean,et al.  Multigrid calculation of fluid flows in complex 3D geometries using curvilinear grids , 1996 .

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

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

[7]  Yousef Saad,et al.  Hybrid Krylov Methods for Nonlinear Systems of Equations , 1990, SIAM J. Sci. Comput..

[8]  Jungwoo Kim,et al.  An immersed-boundary finite-volume method for simulations of flow in complex geometries , 2001 .

[9]  S. Osher,et al.  A Non-oscillatory Eulerian Approach to Interfaces in Multimaterial Flows (the Ghost Fluid Method) , 1999 .

[10]  P. Moin,et al.  Eddies, streams, and convergence zones in turbulent flows , 1988 .

[11]  Wei Shyy,et al.  On the adoption of velocity variable and grid system for fluid flow computation in curvilinear coordinates , 1990 .

[12]  D. Kwak,et al.  A fractional step solution method for the unsteady incompressible Navier-Stokes equations in generalized coordinate systems , 1991 .

[13]  Timothy M. Healy.  Multi-block and overset-block domain decomposition techniques for cardiovascular flow simulation , 2001 .

[14]  Ajit P Yoganathan,et al.  Fluid mechanics of heart valves. , 2004, Annual review of biomedical engineering.

[15]  E. Balaras,et al.  A general reconstruction algorithm for simulating flows with complex 3D immersed boundaries on Cartesian grids , 2003 .

[16]  Harry A. Dwyer,et al.  Calculation of Unsteady Flows in Curved Pipes , 2001 .

[17]  Y. Saad,et al.  GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems , 1986 .

[18]  Cornelis W. Oosterlee,et al.  An Evaluation of Parallel Multigrid as a Solver and a Preconditioner for Singularly Perturbed Problems , 1998, SIAM J. Sci. Comput..

[19]  P. Swarztrauber.  A direct Method for the Discrete Solution of Separable Elliptic Equations , 1974 .

[20]  F. Sotiropoulos,et al.  A hybrid Cartesian/immersed boundary method for simulating flows with 3D, geometrically complex, moving bodies , 2005 .

[21]  Peter R. Voke,et al.  Large-eddy simulation of boundary-layer separation and transition at a change of surface curvature , 2001, Journal of Fluid Mechanics.

[22]  Henk A. van der Vorst,et al.  Bi-CGSTAB: A Fast and Smoothly Converging Variant of Bi-CG for the Solution of Nonsymmetric Linear Systems , 1992, SIAM J. Sci. Comput..

[23]  Hyung Jin Sung,et al.  Large-eddy simulation of turbulent flow inside a sudden-expansion cylindrical chamber , 2001 .

[24]  Elias Balaras,et al.  An embedded-boundary formulation for large-eddy simulation of turbulent flows interacting with moving boundaries , 2006, J. Comput. Phys..

[25]  Gene H. Golub,et al.  On direct methods for solving Poisson's equation , 1970, Milestones in Matrix Computation.

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

[27]  D. Keyes,et al.  Jacobian-free Newton-Krylov methods: a survey of approaches and applications , 2004 .

[28]  W. Shyy,et al.  Regular Article: An Accurate Cartesian Grid Method for Viscous Incompressible Flows with Complex Immersed Boundaries , 1999 .

[29]  G Gerrit Vossers,et al.  Unsteady entrance flow in a 90° curved tube , 1991, Journal of Fluid Mechanics.

[30]  J. Koseff,et al.  A non-staggered grid, fractional step method for time-dependent incompressible Navier-Stokes equations in curvilinear coordinates , 1994 .

[31]  B. P. Leonard,et al.  A stable and accurate convective modelling procedure based on quadratic upstream interpolation , 1990 .

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

[33]  Richard Barrett,et al.  Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods , 1994, Other Titles in Applied Mathematics.

[34]  Yousef Saad,et al.  A Flexible Inner-Outer Preconditioned GMRES Algorithm , 1993, SIAM J. Sci. Comput..

[35]  Pieter Wesseling,et al.  Computing Flows on General Three-Dimensional Nonsmooth Staggered Grids , 1999 .

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

[37]  R. F. Warming,et al.  An implicit finite-difference algorithm for hyperbolic systems in conservation-law form. [application to Eulerian gasdynamic equations , 1976 .