Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-29T01:29:34.299Z Has data issue: false hasContentIssue false
  • English
  • Français

An Immersed Boundary Method Based on the Kinematic Relation of the Velocity-Vorticity Formulation

Published online by Cambridge University Press:  10 December 2014

I. Farahbakhsh ,
H. Ghassemi  and
F. Sabetghadam
I. Farahbakhsh
Affiliation:
Department of Ocean Engineering, Amirkabir University of Technology, Tehran, Iran
H. Ghassemi*
Affiliation:
Department of Ocean Engineering, Amirkabir University of Technology, Tehran, Iran
F. Sabetghadam
Affiliation:
Mechanical and Aerospace Engineering Department Science and Research Branch, Islamic Azad University, Tehran, Iran
*
*Corresponding author ([email protected])
Get access

Abstract

An immersed boundary method is proposed for the simulation of the interaction of an incompressible flow with rigid bodies. The method is based on a new interpretation of velocity-vorticity formulation and no longer includes the force term which is an essential issue of common immersed boundary methods. The system is considered in an Eulerian frame and retrieving the vorticity in this formulation enforces continuity at the fluid-solid interface and rigid motion of the solid. The method focuses on the mutual kinematic relations between the velocity and vorticity fields and with retrieving the vorticity field and recalculating the velocities yields the solenoidal velocity field. The method is applied to the two dimensional problems and the results show that the solenoidality is satisfied acceptably. The comparisons with 2D test cases are provided to illustrate the capabilities of the proposed method.

Keywords

Velocity-vorticity formulationImmersed boundary methodIncompressible flowSolenoidal velocity field
Type
Research Article
Information
Journal of Mechanics , Volume 31 , Issue 2 , April 2015 , pp. 171 - 181
Copyright
Copyright © The Society of Theoretical and Applied Mechanics, R.O.C. 2014 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Quartapelle, L., “Numerical Solution of the Incompressible Navier-Stokes Equations,” International Series of Numerical Mathematics, Birkhäuer-Verlag, Basel, 133 (1993).Google Scholar
2.Rempfer, D., “On Boundary Conditions for Incompressible Navier-Stokes Problems,” Applied Mechanics Reviews, 59, pp. 107–125 (2006).Google Scholar
3.Ferziger, J. H., “Simulation of Incompressible Turbulent Flows,” Journal of Computational Physics, 69, pp. 1–48 (1987).CrossRefGoogle Scholar
4.Peyret, R. and Taylor, T. D., Computational Methods for Fluid Flow, Springer-Verlag, Berlin (1983).CrossRefGoogle Scholar
5.Sabetghadam, F., Sharafatmandjoor, S. and Badri, M., “Construction of Solenoidal Immersed Velocity Vectors Using the Kinematic Velocity-Vorticity Relation,” arXiv: 1204.1916, http://arxiv.org/abs/1204. 1916 (2012).Google Scholar
6.Glowinski, R., Pan, T. W., Hesl, T. I. and Joseph, D. D., “A Distributed Lagrange Multiplier/Fictitious Domain Method for Particulate Flows,” International Journal of Multiphase Flow, 25, pp. 755–794 (1999).Google Scholar
7.Sussman, M., Smereka, P. and Osher, S., “A Level Set Approach for Computing Solutions to Incompressible Two-Phase Flows,” Journal of Computational Physics, 114, pp. 146–159 (1994).Google Scholar
8.Tryggvason, G., Bunner, B., Esmaeeli, A., Juric, D., Al-Rawahi, N., Tauber, W., Han, J., Nas, S. and Jan, Y. J., “A Front-Tracking Method for the Computations of Multiphase Flow,” Journal of Computational Physics, 169, pp. 708–759 (2001).CrossRefGoogle Scholar
9.Huang, W. X. and Sung, H. J., “An Immersed Boundary Method for Fluid-Flexible Structure Interaction,” Computer Methods in Applied Mechanics and Engineering, 198, pp. 2650–2661 (2009).Google Scholar
10.Wang, Z., Fan, J. and Cen, K., “Immersed Boundary Method for the Simulation of 2D Viscous Flow Based on Vorticity-Velocity Formulations,” Journal of Computational Physics, 228, pp. 1504–1520 (2009).Google Scholar
11.Peskin, C. S., “Flow Patterns Around Heart Valves: A Numerical Method,” Journal of Computational Physics, 10, pp. 252–271 (1972).Google Scholar
12.Fadlun, E. A., Verzicco, R., Orlandi, P. and Mohd-Yusof, J., “Combined Immersed-Boundary Finite-Difference Methods for Three-Dimensional Complex Flow Simulations,” Journal of Computational Physics, 161, pp. 35–60 (2000).CrossRefGoogle Scholar
13.Kim, J., Kim, D. and Choi, H., “An Immersed-Boundary Finite-Volume Method for Simulations of Flow in Complex Geometries,” Journal of Computational Physics, 171, pp. 132–150 (2001).Google Scholar
14.Griffith, B. E., “Simulating the Blood-Muscle-Valve Mechanics of the Heart by an Adaptive and Parallel Version of the Immersed Boundary Method,” Ph.D. Dissertation, Courant Institute of Mathematical Sciences, New York University, New York, US (2005).Google Scholar
15.Kim, Y. and Peskin, C. S., “Penalty Immersed Boundary Method for an Elastic Boundary with Mass,” Physics of Fluids, 19, pp. 1–18 (2007).CrossRefGoogle Scholar
16.Le, D. V., Khoo, B. C. and Lim, K. M., “An Implicit-Forcing Immersed Boundary Method for Simulating Viscous Flows in Irregular Domains,” Computer Methods in Applied Mechanics and Engineering, 197, pp. 2119–2130 (2008).Google Scholar
17.Poncet, P., “Analysis of an Immersed Boundary Method for Three-Dimensional Flows in Vorticity Formulation,” Journal of Computational Physics, 228, pp. 7268–7288 (2009).Google Scholar
18.Bonfigli, G., “High-Order Finite-Difference Implementation of the Immersed-Boundary Technique for Incompressible Flows,” Computer & Fluids, 46, pp. 2–11 (2011).Google Scholar
19.Ren, W. W., Wu, J., Shu, C. and Yang, W. M., “A Stream Function-Vorticity Formulation-Based Immersed Boundary Method and its Applications,” International Journal for Numerical Methods in Fluids, 70, pp. 627–645 (2012).Google Scholar
20.Wu, C. S. and Young, D. L., “Simulations of Free-Surface Flows with an Embedded Object by a Coupling Partitioned Approach,” Computer & Fluids, 89, pp. 66–77 (2014).CrossRefGoogle Scholar
21.Uhlmann, M., “An Immersed Boundary Method with Direct Forcing for the Simulation of Particulate Flows,” Journal of Computational Physics, 209, pp. 448–476 (2005).Google Scholar
22.Kim, D. and Choi, H., “Immersed Boundary Method for Flow Around an Arbitrarily Moving Body,” Journal of Computational Physics, 212, pp. 662–680 (2006).Google Scholar
23.Huang, W. X., Shin, S. J. and Sung, H. J., “Simulation of Flexible Filaments in a Uniform Flow by the Immersed Boundary Method,” Journal of Computational Physics, 226, pp. 2206–2228 (2007).Google Scholar
24.Zhao, H., Freund, J. B. and Moser, R. D., “A Fixed-Mesh Method for Incompressible Flow-Structure Systems with Finite Solid Deformations,” Journal of Computational Physics, 227, pp. 3114–3140 (2008).Google Scholar
25.Le, D. V., White, J., Peraire, J., Lim, K. M. and Khoo, B. C., “An Implicit Immersed Boundary Method for Three-Dimensional Fluid-Membrane Interactions,” Journal of Computational Physics, 228, pp. 8427–8445 (2009).CrossRefGoogle Scholar
26.Curet, O. M., AlAli, I. K., MacIver, M. A. and Patankar, N. A., “A Versatile Implicit Iterative Approach for Fully Resolved Simulation of Self-Propulsion,” Computer Methods in Applied Mechanics and Engineering, 199, pp. 2417–2424 (2010).Google Scholar
27.Herschlag, G. and Miller, L., “Reynolds Number Limits for Jet Propulsion: A Numerical Study of Simplified Jellyfish,” Journal of Theoretical Biology, 285, pp. 84–95 (2011).CrossRefGoogle ScholarPubMed
28.Chaudhuri, A., Hadjadj, A. and Chinnayya, A., “On the Use of Immersed Boundary Methods for Shock/Obstacle Interactions,” Journal of Computational Physics, 230, pp. 1731–1748 (2011).Google Scholar
29.Choi, J. I., Oberoi, R. C., Edwards, J. R. and Rosati, J. A., “An Immersed Boundary Method for Complex Incompressible Flows,” Journal of Computational Physics, 224, pp. 757–784 (2007).Google Scholar
30.Liao, C. C., Chang, Y. W., Lin, C. A. and McDonough, J. M., “Simulating Flows with Moving Rigid Boundary Using Immersed-Boundary Method,” Computer & Fluids, 39, pp. 152–167 (2010).CrossRefGoogle Scholar
31.Yang, J. and Balaras, E., “An Embedded-Boundary Formulation for Large-Eddy Simulation of Turbulent Flows Interacting with Moving Boundaries,” Journal of Computational Physics, 215, pp. 12–40 (2006).CrossRefGoogle Scholar
32.Dutsch, H., Durst, F., Becker, S. and Lienhart, H., “Low-Reynolds-Number Flow Around an Oscillating Circular Cylinder at Low Keulegan-Carpenter Numbers,” Journal of Fluid Mechanics, 360, pp. 249–271 (1998).CrossRefGoogle Scholar
33.Glowinski, R., Pan, T. W., Hesla, T. I., Joseph, D. D. and Périauxz, J., “A Fictitious Domain Approach to the Direct Numerical Simulation of Incompressible Viscous Flow Past Moving Rigid Bodies: Application to Particulate Flow,” Journal of Computational Physics, 169, pp. 363–426 (2001).CrossRefGoogle Scholar
34.Coquerelle, M. and Cottet, G. H., “A Vortex Level Set Method for the Two-Way Coupling of an Incompressible Fluid with Colliding Rigid Bodies,” Journal of Computational Physics, 227, pp. 9121–9137 (2008).Google Scholar
35.Bergmann, M. and Iollo, A., “Modeling and Simulation of Fish-Like Swimming,” Journal of Computational Physics, 230, pp. 329–348 (2011).Google Scholar
36.Horowitz, M. and Williamson, C. H. K., “Vortex-Induced Vibration of a Rising and Falling Cylinder,” Journal of Fluid Mechanics, 662, pp. 352–383 (2010).CrossRefGoogle Scholar
37.Ito, K., Lai, M. C. and Li, Z., “A Well-Conditioned Augmented System for Solving Navier-Stokes Equations in Irregular Domains,” Journal of Computational Physics, 228, pp. 2616–2628 (2009).CrossRefGoogle Scholar
38.Su, S. W., Lai, M. C. and Lin, C. A., “An Immersed Boundary Technique for Simulating Complex Flows with Rigid Boundary,” Computer & Fluids, 36, pp. 313–324 (2007).CrossRefGoogle Scholar