An improved moving-least-squares reconstruction for immersed boundary method

Summary The current work presents an improved immersed boundary method based on the ideas proposed by Vanella and Balaras (M. Vanella, E. Balaras, A moving-least-squares reconstruction for embedded-boundary formulations, J. Comput. Phys. 228 (2009) 6617–6628). In the method, an improved moving-least-squares approximation is employed to build the transfer functions between the Lagrangian points and discrete Eulerian grid points. The main advantage of the improved method is that there is no need to obtain the inverse matrix, which effectively eliminates numerical instabilities caused by matrix inversion and reduces the computational cost significantly. Several different flow problems (Taylor-Green decaying vortices, flows past a stationary circular cylinder and a sphere, and the sedimentation of a free-falling sphere in viscous fluid) are simulated to validate the accuracy and efficiency of the method proposed in the present paper. The simulation results show good agreement with previous numerical and experimental results, indicating that the improved immersed boundary method is efficient and reliable in dealing with the fluid–solid interaction problems. Copyright © 2015 John Wiley & Sons, Ltd.

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

[2]  J. Damasceno,et al.  Numerical simulation of two-dimensional flows over a circular cylinder using the immersed boundary method , 2003 .

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

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

[5]  Howard H. Hu.  Direct simulation of flows of solid-liquid mixtures , 1996 .

[6]  Jinhee Jeong,et al.  On the identification of a vortex , 1995, Journal of Fluid Mechanics.

[7]  K. M. Liew,et al.  Boundary element‐free method (BEFM) and its application to two‐dimensional elasticity problems , 2006 .

[8]  J. Pinton,et al.  Velocity measurement of a settling sphere , 2000 .

[9]  Yanwen Ma,et al.  A High Order Accurate Difference Scheme for Complex Flow Fields , 1997 .

[10]  Z. J. Wang,et al.  A cartesian grid method for modeling multiple moving objects in 2D incompressible viscous flow , 2003 .

[11]  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.

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

[13]  J. F. Richardson,et al.  THE RESISTANCE TO MOTION OF A SOLID SPHERE IN A FLUID , 1987 .

[14]  K. M. Liew,et al.  Boundary element‐free method (BEFM) for two‐dimensional elastodynamic analysis using Laplace transform , 2005 .

[15]  Jianren Fan,et al.  Immersed boundary method for the simulation of 2D viscous flow based on vorticity-velocity formulations , 2009, J. Comput. Phys..

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

[17]  Richard Turton,et al.  An explicit relationship to predict spherical particle terminal velocity , 1987 .

[18]  A. Jameson,et al.  Some recent developments in numerical methods for transonic flows , 1985 .

[19]  M. Lai,et al.  An immersed boundary technique for simulating complex flows with rigid boundary , 2007 .

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

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

[22]  Howard H. Hu,et al.  Direct numerical simulations of fluid-solid systems using the arbitrary Langrangian-Eulerian technique , 2001 .

[23]  I. Orlanski.  A Simple Boundary Condition for Unbounded Hyperbolic Flows , 1976 .

[24]  Jianren Fan,et al.  A modified immersed boundary method for simulations of fluid–particle interactions , 2007 .

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

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

[27]  Z. Jane Wang,et al.  An immersed interface method for simulating the interaction of a fluid with moving boundaries , 2006, J. Comput. Phys..

[28]  S. Lele.  Compact finite difference schemes with spectral-like resolution , 1992 .

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

[30]  K. Squires,et al.  LES and DES investigations of turbulent flow over a sphere , 2000 .

[31]  Tony W. H. Sheu,et al.  A differentially interpolated direct forcing immersed boundary method for predicting incompressible Navier-Stokes equations in time-varying complex geometries , 2010, J. Comput. Phys..

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

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

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

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