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A Moving-Least-Square Immersed Boundary Method for Rigid and Deformable Boundaries in Viscous Flow

Abstract

We present a moving-least-square immersed boundary method for solving viscous incompressible flow involving deformable and rigid boundaries on a uniform Cartesian grid. For rigid boundaries, noslip conditions at the rigid interfaces are enforced using the immersed-boundary direct-forcing method. We propose a reconstruction approach that utilizes moving least squares (MLS) method to reconstruct the velocity at the forcing points in the vicinity of the rigid boundaries. For deformable boundaries, MLS method is employed to construct the interpolation and distribution operators for the immersed boundary points in the vicinity of the rigid boundaries instead of using discrete delta functions. The MLS approach allows us to avoid distributing the Lagrangian forces into the solid domains as well as to avoid using the velocity of points inside the solid domains to compute the velocity of the deformable boundaries. The present numerical technique has been validated by several examples including a Poiseuille flow in a tube, deformations of elastic capsules in shear flow and dynamics of red-blood cell in microfluidic devices.

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Corresponding author

*Corresponding author. Email addresses: ledv@ihpc.a-star.edu.sg (D.-V. Le), mpekbc@nus.edu.sg (B.-C. Khoo)

References

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[1] Peskin, C. S., Numerical analysis of blood flow in the heart, J. Comput. Phys. 25 (1977) 220252.
[2] Peskin, C. S., The immersed boundary method, Acta Numerica 11 (2) (2002) 479517.
[3] Fauci, L. J., Peskin, C. S., A computational model of aquatic animal locomotion, J. Comput. Phys 77 (1988) 85108.
[4] Eggleton, C. D., Popel, A. S., Large deformation of red blood cell ghosts in a simple shear flow, Phys. Fluids 10 (1998) 18341845.
[5] Dillon, R., Fauci, L. J., Graver, D., A microscale model of bacterial swimming, chemotaxis and substrate transport, J. Theor. Biol. 177 (1995) 325340.
[6] Wang, N. T., Fogelson, A. L., Computational methods for continuum models of platelet aggregation, J. Comput. Phys 151 (1999) 649675.
[7] Lai, M. C., Peskin, C. S., An immersed boundary method with formal second order accuracy and reduced numerical viscosity, J. Comput. Phys. 160 (2000) 707719.
[8] Silva, A. L. E., Silveira-Neto, A., Damasceno, J., Numerical simulation of two-dimensional flows over a circular cylinder using the immersed boundary method, J. Comput. Phys. 189 (2003) 351370.
[9] Le, D. V., Khoo, B. C., Lim, K. M., An implicit-forcing immersed boundary method for simulating viscous flows in irregular domains, Comput. Methods Appl. Mech. Engrg. 197 (2008) 21192130.
[10] Su, S. W., Lai, M. C., Lin, C. A., An immersed boundary technique for simulating complex flows with rigid boundary, Comput. Fluids 36 (2007) 313324.
[11] Mohd-Yusof, J., Combined immersed boundary/B-splines methods for simulations of flows in complex geometry, Annual Research Briefts, Center for Turbulence Research (1997) 317327.
[12] Fadlun, E. A., Verzicco, R., Orlandi, P., Combined immersed-boundary finite-difference methods for three-dimensional complex flows simulations, J. Comput. Phys. 161 (2000) 3560.
[13] Balaras, E., Modeling complex boundaries using an external force field on fixed Cartesian grids in large-eddy simulations, Computers & Fluids 33 (2004) 375404.
[14] Balaras, E., Yang, J., Nonboundary conforming methods for large-eddy simulations of biological flows, ASME J. Fluids Eng. 127 (2005) 851857.
[15] Yang, J., Balaras, E., An embedded-boundary formulation for large-eddy simulation of turbulent flows interacting with moving boundaries, J. Comput. Phys. 215 (2006) 1240.
[16] Yang, J., Stern, F., A simple and efficient direct forcing immersed boundary framework for fluid-structure interactions, J. Comput. Phys. 231 (2012) 50295061.
[17] Liu, C., Hu, C., An efficient immersed boundary treatment for complex moving object, J. Comput. Phys. 274 (2014) 654680.
[18] Uhlmann, M., An immersed boundary method with direct forcing for the simulation of particulate flows, J. Comput. Phys. 209 (2005) 448476.
[19] Udaykumar, H. S., Mittal, R., Rampunggoon, P., Khanna, A., A sharp interface Cartesian grid method for simulating flows with complex moving boundaries, J. Comput. Phys. 174 (2001) 345380.
[20] Ye, T., Mittal, R., Udaykumar, H. S., Shyy, W., An accurate Cartesian grid method for viscous incompressible flows with complex immersed boundary, J. Comput. Phys. 156 (1999) 209240.
[21] Hu, X., Khoo, B. C., Adams, N., Huang, F., A conservative interface method for compressible flows, J. Comput. Phys. 219 (2006) 553578.
[22] Tseng, Y. H., Ferziger, J. H., A ghost-cell immersed boundary method for flow in complex geometry, J. Comput. Phys. 192 (2003) 593623.
[23] Ge, L., Sotiropoulos, F., A numerical method for solving the 3D unsteady incompressible Navier-Stokes equations in curvilinear domains with complex immersed boundaries, J. Comput. Phys. 225 (2007) 17821809.
[24] Borazjani, I., Ge, L., Sotiropoulos, F., Curvilinear immersed boundary method for simulating fluid structure interaction with complex 3D rigid bodies, J. Comput. Phys. 227 (2008) 75877620.
[25] Roman, F., Napoli, E., Milici, B., Armenio, V., An improved immersed boundary method for curvilinear grids, Comput. Fluids 38 (2009) 15101527.
[26] Mittal, R., Iaccarino, G., Immersed boundary methods, Annu. Rev. Fluid Mech. 37 (2005) 239261.
[27] Sotiropoulos, F., Yang, X., Immersed boundary methods for simulating fluid-structure-interaction, Progress in Aerospace Sciences 65 (2014) 121.
[28] Liew, K., Cheng, Y., Kitipornchai, S., Boundary element-free method (BEFM) for two-dimensional elastodynamic analysis using Laplace transform, Int. J. Numer. Meth. Engng 64 (2005) 16101627.
[29] Vanella, M., Balaras, E., A moving-least-squares reconstruction for embedded-boundary formulations, J. Comput. Phys. 228 (2009) 66176628.
[30] Li, D., Wei, A., Luo, K., Fan, J., An improved moving-least-squares reconstruction for immersed boundary method, Int. J. Numer. Meth. Engng 104 (2015) 789804.
[31] Brown, D. L., Cortez, R., Minion, M. L., Accurate projection methods for the incompressible Navier-Stokes equations, J. Comput. Phys. 168 (2001) 464499.
[32] Adams, J., Swarztrauber, P., Sweet, R., FISHPACK: Efficient FORTRAN subprograms for the solution of separable elliptic partial differential equations, Available on the web at http://www.scd.ucar.edu/css/software/fishpack/.
[33] Schumann, U., Sweet, R. A., A direct method for the solution of Poisson's equation with Neumann boundary conditions on a staggered grid of arbitrary size, J. Comput. Phys. 20 (1976) 171182.
[34] Liu, G.-R., Gu, Y.-T., An introduction to meshfree methods and their programming, Springer, 2005.
[35] Knoll, D. A., Keyes, D. E., Jacobian-free Newton-Krylov methods: a survey of approaches and applications, J. Comput. Phys 193 (2004) 357397.
[36] Le, D. V., White, J., Peraire, J., Lim, K.M., Khoo, B. C., An implicit immersed boundary method for three-dimensional fluid-membrane interactions, J. Comput. Phys. 228 (2009) 84278445.
[37] Cirak, F., Ortiz, M., Schroder, P., Subdivision surfaces: a new paradigm for thin-shell finite-element analysis, Int. J. Numer. Meth. Engng. 47 (2000) 20392072.
[38] Le, D. V., Subdivision elements for large deformation of liquid capsules enclosed by thin shells, Comput. Methods Appl. Mech. Engrg. 199 (2010) 26222632.
[39] Le, D. V., Tan, Z., Hydrodynamic interaction of elastic capsules in bounded shear flow, Commun. Comput. Phys. 16 (2014) 10311055.
[40] Ramanujan, S., Pozrikidis, C., Deformation of liquid capsules enclosed by elastic membrane in simple shear flow: large deformations and the effect of fluid viscosities, J. Fluid Mech. 361 (1998) 117143.
[41] Barthès-Biesel, D., Rallison, J. M., The time-dependent deformation of a capsule freely suspended in a linear shear flow, J. Fluid Mech. 113 (1981) 251267.
[42] Huang, L. R., Cox, E. C., Austin, R. H., Sturm, J. C., Continuous particle separation through deterministic lateral displacement, Science 304 (2004) 987.
[43] Davis, J. A., Inglis, D. W., Morton, K. J., Lawrence, D. A., Huang, L. R., Chou, S. Y., Sturm, J. C., Austin, R. H., Deterministic hydrodynamics: Taking blood apart, PNAS 103 (2006) 1477914784.
[44] Quek, R., Le, D. V., Chiam, K. H., Separation of deformable particles in deterministic lateral displacement devices, Phys. Rev. E 83 (2011) 056301.
[45] Evans, E., Fung, Y. C., Improved measurements of the erythrocyte geometry, Microvasc. Res. 4 (1972) 335347.
[46] Skalak, R., Tozeren, A., Zarda, R. P., Chien, S., Strain energy function of red blood cell membranes, Biophys. J. 13 (1973) 245264.
[47] Zeming, K. K., Ranjan, S., Zhang, Y., Rotational separation of non-spherical bioparticles using I-shaped pillar arrays in a microfluidic device, Nature Communications 4 (2013) 1625.
[48] Skotheim, J.M., Secomb, T.W., Red blood cells and other nonspherical capsules in shear flow: Oscillatory dynamics and the tank-treading-to-tumbling transition, Phys. Rev. Lett. 98 (2007) 078301.
[49] Abkarian, M., Faivre, M., Viallat, A., Swinging of red blood cells under shear flow, Phys. Rev. Lett. 98 (2007) 188302.

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A Moving-Least-Square Immersed Boundary Method for Rigid and Deformable Boundaries in Viscous Flow

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