Immersed-boundary methods for general finite-difference and finite-volume Navier-Stokes solvers

We present an immersed-boundary algorithm for incompressible flows with complex boundaries, suitable for Cartesian or curvilinear grid system. The key stages of any immersed-boundary technique are the interpolation of a velocity field given on a mesh onto a general boundary (a line in 2D, a surface in 3D), and the spreading of a force field from the immersed boundary to the neighboring mesh points, to enforce the desired boundary conditions on the immersed-boundary points. We propose a technique that uses the Reproducing Kernel Particle Method [W.K. Liu, S. Jun, Y.F. Zhang, Reproducing kernel particle methods, Int. J. Numer. Methods Fluids 20(8) (1995) 1081-1106] for the interpolation and spreading. Unlike other methods presented in the literature, the one proposed here has the property that the integrals of the force field and of its moment on the grid are conserved, independent of the grid topology (uniform or non-uniform, Cartesian or curvilinear). The technique is easy to implement, and is able to maintain the order of the original underlying spatial discretization. Applications to two- and three-dimensional flows in Cartesian and non-Cartesian grid system, with uniform and non-uniform meshes are presented.

[1]  R. LeVeque,et al.  Analysis of a one-dimensional model for the immersed boundary method , 1992 .

[2]  Elias Balaras,et al.  A moving-least-squares reconstruction for embedded-boundary formulations , 2009, J. Comput. Phys..

[3]  N. Zhang,et al.  An improved direct-forcing immersed-boundary method for finite difference applications , 2007, J. Comput. Phys..

[4]  C. Rhie,et al.  Numerical Study of the Turbulent Flow Past an Airfoil with Trailing Edge Separation , 1983 .

[5]  S. Schwarzer,et al.  Navier-Stokes simulation with constraint forces: finite-difference method for particle-laden flows and complex geometries. , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

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

[7]  I. Babuska,et al.  The Partition of Unity Method , 1997 .

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

[9]  P. Queutey,et al.  A NUMERICAL SIMULATION OF VORTEX SHEDDING FROM AN OSCILLATING CIRCULAR CYLINDER , 2002 .

[10]  M. Lai,et al.  An Immersed Boundary Method with Formal Second-Order Accuracy and Reduced Numerical Viscosity , 2000 .

[11]  Wing Kam Liu,et al.  Reproducing kernel particle methods , 1995 .

[12]  U. Piomelli,et al.  Large Eddy Simulation of the Flow in an S-duct , 2003 .

[13]  Wing Kam Liu,et al.  Extended immersed boundary method using FEM and RKPM , 2004 .

[14]  David Farrell,et al.  Immersed finite element method and its applications to biological systems. , 2006, Computer methods in applied mechanics and engineering.

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

[16]  Parviz Moin,et al.  ADVANCES IN LARGE EDDY SIMULATION METHODOLOGY FOR COMPLEX FLOWS , 2002, Proceeding of Second Symposium on Turbulence and Shear Flow Phenomena.

[17]  P. Koumoutsakos,et al.  Simulations of the viscous flow normal to an impulsively started and uniformly accelerated flat plate , 1996, Journal of Fluid Mechanics.

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

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

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

[21]  J. Palma,et al.  Numerical simulation of isotropic turbulence using a collocated approach and a nonorthogonal grid system , 2002 .

[22]  Xiao-Yun Lu,et al.  Calculation of the Timing of Vortex Formation from AN Oscillating Cylinder , 1996 .

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

[24]  U. Piomelli,et al.  Wall-Modeled Large-Eddy Simulations of Flows With Curvature and Mild Separation , 2008 .

[25]  D. Tritton.  Experiments on the flow past a circular cylinder at low Reynolds numbers , 1959, Journal of Fluid Mechanics.

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

[27]  M. Uhlmann.  An immersed boundary method with direct forcing for the simulation of particulate flows , 2005, 1809.08170.

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

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

[30]  M. Mcpherson,et al.  Introduction to fluid mechanics , 1997 .

[31]  U. Piomelli,et al.  Reynolds-averaged and large-eddy simulations of turbulent non-equilibrium flows , 2006 .

[32]  T. H. van den Berg,et al.  Turbulent bubbly flow , 2004 .

[33]  C. Peskin.  Numerical analysis of blood flow in the heart , 1977 .

[34]  S. Biringen,et al.  Numerical Simulation of a Cylinder in Uniform Flow , 1996 .

[35]  Changhoon Lee,et al.  Stability characteristics of the virtual boundary method in three-dimensional applications , 2003 .

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

[37]  M. Berger,et al.  An Adaptive Version of the Immersed Boundary Method , 1999 .

[38]  V. Armenio,et al.  An improved immersed boundary method for curvilinear grids , 2009 .

[39]  R. A. Uras,et al.  Generalized multiple scale reproducing kernel particle methods , 1996 .

[40]  V. C. Patel,et al.  Flow past a sphere up to a Reynolds number of 300 , 1999, Journal of Fluid Mechanics.

[41]  C. Williamson.  Defining a Universal and Continuous Strouhal-Reynolds Number Relationship for the Laminar Vortex She , 1988 .

[42]  Z. Feng,et al.  The immersed boundary-lattice Boltzmann method for solving fluid-particles interaction problems , 2004 .

[43]  Rajat Mittal,et al.  A versatile sharp interface immersed boundary method for incompressible flows with complex boundaries , 2008, J. Comput. Phys..

[44]  R. Bouard,et al.  Experimental determination of the main features of the viscous flow in the wake of a circular cylinder in uniform translation. Part 1. Steady flow , 1977, Journal of Fluid Mechanics.

[45]  S. Taneda.  Experimental Investigation of the Wake behind a Sphere at Low Reynolds Numbers , 1956 .

[46]  Lucy T. Zhang,et al.  Immersed finite element method , 2004 .

[47]  Ted Belytschko,et al.  Immersed electrokinetic finite element method , 2007 .