Application of Novel Conserving Immersed Boundary Method to Moving Boundary Problem

Authors: S. N. Hosseini, S. M. H. Karimian

Abstract:

A new conserving approach in the context of Immersed Boundary Method (IBM) is presented to simulate one dimensional, incompressible flow in a moving boundary problem. The method employs control volume scheme to simulate the flow field. The concept of ghost node is used at the boundaries to conserve the mass and momentum equations. The Present method implements the conservation laws in all cells including boundary control volumes. Application of the method is studied in a test case with moving boundary. Comparison between the results of this new method and a sharp interface (Image Point Method) IBM algorithm shows a well distinguished improvement in both pressure and velocity fields of the present method. Fluctuations in pressure field are fully resolved in this proposed method. This approach expands the IBM capability to simulate flow field for variety of problems by implementing conservation laws in a fully Cartesian grid compared to other conserving methods.

Keywords: Immersed Boundary Method, conservation of mass and momentum laws, moving boundary, boundary condition.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1334009

References:

[1] Peskin, C. S. Flow patterns around heart valves: A numerical method. Computational Physics, v. 10, p. 252-71, 1972. Peskin, C. S. The fluid dynamics of heart valves: experimental, theoretical and computational methods. Annual Review. Fluid Mechanic, v. 14, p. 235-59, 1981.
[2] Berger, M.; Aftosmis, M. Aspects (and aspect ratios) of cartesian mesh methods. (S.l.): Proc. 16th Int. Conf. Numerical Methods Fluid. 1998.
[3] Lai, M.; Peskin, C. An immersed boundary method with formal secondorder accuracy and reduced numerical viscosity. Computational Physics, v. 160, p. 705-19, 2000.
[4] Saiki, E.; Biringen, S. Numerical simulation of a cylinder in uniform flow: application of a virtual boundary method, v. 123, p. 450-65, 1996.
[5] Goldstein, D.; Handler, R.; Sirovich, L. Modeling a no-slip flowboundary with an external force field. Computational Physics, v. 105, p. 354-66, 1993.
[6] Iaccarino, G.; Verzicco, R. Immersed boundary technique for turbulent flow simulations. Mechanical Review, v. 56, p. 331-47, 2003.
[7] Angot, P.; Bruneau, C.; Frabrie, P. Apenalization method to take into account obstacles in viscous flows. Numericall Mathematics, v. 81, p. 497-520, 1999.
[8] Khadra, K. et al. Fictitious domain approach for numerical modeling of Navier-Stokes equations. Numerical Methods Fluids, v. 34, p. 651-84, 2000.
[9] Majumdar, S.; Iaccarino, G.; Durbin, P. Rans solver with adaptive structured boundary non-conforming grids. Cent. Turbul. Res. , n. 2001, , p. 353-64, 2001.
[10] Ghias, R.; Mittal, R.; Lund, T. A non-body conformal grid method for simulation of compressible flows with complex immersed boundaries. AIAA, p. 2004-0080, 2004.
[11] Ghias, R.; Mittal, R.; Dong, H. A sharp interface immersed boundary method for compressible viscous flows. computational phusics, v. 225, p. 528-53, 2007.
[12] Clarke, D.; M., S.; H., H. Euler calculations for multi-element airfoils using Cartesian grids. AIAA, v. 24, p. 1128-35, 1986.
[13] Zeeuw, D.; Powell, K. An adaptively refined cartesian mesh solver for the Euler equations. AIAA, v. 1991-1542, 1991.
[14] Udaykumar, H.; Shyy, W.; Rao, M. Elafint: A mixed Eulerian- Lagrangian method for fluid flows with complex and moving boundaries. Numerical Methods, v. 22, p. 691-705, 1996.
[15] Udaykumar, H. et al. A sharp interface cartesian grid method for simulating flows with complex moving boundaries. computational physics, v. 174, p. 345-80, 2001.
[16] Udaykumar, H.; Mittal, R.; Rampunggoon. Interface tracking finite volume method for complex solid-fluid interactions on fixed meshes. Commun. Numerical Methods Engineering, v. 18, p. 89-97, 2002.
[17] Ye, T. et al. An accurate cartesian grid method for viscous incompressible flows with complex immersed boundaries. Computational Physics, v. 156, p. 209-40, 1999.
[18] Karimian, S. M. H.; Amoli, A.; Mazaheri, K. Control-volume finiteelement method for the solution of 2D euler equations on unstructured moving grids. Iranian Journal of Science & Technology, v. 26, p. 465-76 , 2002.
[19] Karimian, S. M. H.; Schneider, G. E. Pressure-Based comptational method for compressible and incompressibleflows. AIAA, Journal of Thermodynamics and heat transfer, v. 8, p. 267-74, 1994.
[20] Mittal, R.; Iaccarino, G. Immersed Boundary Methods. Fluid Mechanics , v. 37, p. 239-61, 2005.