Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-22T18:52:20.178Z Has data issue: false hasContentIssue false
  • English
  • Français

A Divergence-Free Immersed Boundary Method and Its Finite Element Applications

Published online by Cambridge University Press:  06 August 2020

Chuan Zhou Open the ORCID record for Chuan Zhou [Opens in a new window] ,
Jianhua Li ,
Huaan Wang ,
Kailong Mu  and
Lanhao Zhao
Chuan Zhou*
Affiliation:
China Energy Engineering Group Guangdong Electric Power Design Institute Co., Ltd Guangzhou, China Guangdong Kenuo Surveying Engineering Co., Ltd Guangzhou, China
Jianhua Li
Affiliation:
China Energy Engineering Group Guangdong Electric Power Design Institute Co., Ltd Guangzhou, China Guangdong Kenuo Surveying Engineering Co., Ltd Guangzhou, China
Huaan Wang
Affiliation:
China Energy Engineering Group Guangdong Electric Power Design Institute Co., Ltd Guangzhou, China Guangdong Kenuo Surveying Engineering Co., Ltd Guangzhou, China
Kailong Mu
Affiliation:
College of Water Conservancy and Hydropower Engineering, Hohai University Nanjing, China
Lanhao Zhao
Affiliation:
College of Water Conservancy and Hydropower Engineering, Hohai University Nanjing, China
*
*Corresponding author ([email protected])
Get access

Abstract

In order to maintain the no-slip condition and the divergence-free property simultaneously, an iterative scheme of immersed boundary method in the finite element framework is presented. In this method, the Characteristic-based Split scheme is employed to solve the momentum equations and the formulation for the pressure and the extra body force is derived according to the no-slip condition. The extra body force is divided into two divisions, one is in relation to the pressure and the other is irrelevant. Two corresponding independent iterations are set to solve the two sections. The novelty of this method lies in that the correction of the velocity increment is included in the calculation of the extra body force which is relevant to the pressure and the update of the force is incorporated into the iteration of the pressure. Hence, the divergence-free properties and no-slip conditions are ensured concurrently. In addition, the current method is validated with well-known benchmarks.

Keywords

immersed boundary methoditerative schemedivergence-freefinite element framework
Type
Research Article
Information
Journal of Mechanics , Volume 36 , Issue 6 , December 2020 , pp. 901 - 914
Copyright
Copyright © 2020 The Society of Theoretical and Applied Mechanics

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

REFERENCES

Peskin, C. S., “Numerical analysis of blood flow in the heart,” Journal of Computational Physics, 25, pp. 220-252 (1977).10.1016/0021-9991(77)90100-0CrossRefGoogle Scholar
O’Brien, A., and Bussmann, M., “A moving immersed boundary method for simulating particle interactions at fluid-fluid interfaces,” Journal of Computational Physics, DOI: 10.1016/j.jcp.2019.109089 (2019).Google Scholar
Nestola, M. G. C., Becsek, B., Zolfaghari, H., Zulian, P., De Marinis, D., Krause, R., and Obrist, D., “An immersed boundary method for fluid-structure interaction based on variational transfer,” Journal of Computational Physics, DOI: 10.1016/j.jcp.2019.108884 (2019).10.1016/j.jcp.2019.108884CrossRefGoogle Scholar
Abdol Azis, M. H., Evrard, F., and van Wachem, B., “An immersed boundary method for incompressible flows in complex domains,” Journal of Computational Physics, 378, pp. 770-795 (2019).10.1016/j.jcp.2018.10.048CrossRefGoogle Scholar
Mao, J., Zhao, L., Di, Y., Liu, X., and Xu, W., “A resolved CFD-DEM approach for the simulation of landslides and impulse waves,” Computer Methods in Applied Mechanics and Engineering, 359, pp. 112750 (2020).10.1016/j.cma.2019.112750CrossRefGoogle Scholar
Mittal, R., and Iaccarino, G., “Immersed boundary methods,” Annual Review of Fluid Mechanics, 37, pp. 239-261 (2005).10.1146/annurev.fluid.37.061903.175743CrossRefGoogle Scholar
Kim, W., and Choi, H., “Immersed boundary methods for fluid-structure interaction: A review,” International Journal of Heat and Fluid Flow, 75, pp. 301-309 (2019).10.1016/j.ijheatfluidflow.2019.01.010CrossRefGoogle Scholar
Goldstein, D., Handler, R., and Sirovich, L., “Modeling a no-slip flow boundary with an external force field,” Journal of Computational Physics, 105, pp. 354-366 (1993).10.1006/jcph.1993.1081CrossRefGoogle Scholar
Lai, M. C., and Peskin, C. S., “An immersed boundary method with formal second-order accuracy and reduced numerical viscosity,” Journal of Computational Physics, 160, pp. 705-719 (2000).10.1006/jcph.2000.6483CrossRefGoogle Scholar
Mohd Yusof, J., “Interaction of massive particles with turbulence,” Cornell University, New York, America (1996).Google Scholar
Li, Z., Cao, W., and Le Touzé, D., “On the coupling of a direct-forcing immersed boundary method and the regularized lattice Boltzmann method for fluid- structure interaction,” Computers & Fluids, 190, pp. 470-484 (2019).10.1016/j.compfluid.2019.06.030CrossRefGoogle Scholar
Horng, T. L., Hsieh, P. W., Yang, S. Y., and You, C. S., “A simple direct-forcing immersed boundary projection method with prediction-correction for fluid-solid interaction problems,” Computers & Fluids, 176, pp. 135-152 (2018).10.1016/j.compfluid.2018.02.003CrossRefGoogle Scholar
Tao, S., He, Q., Wang, L., Huang, S., and Chen, B., “A non-iterative direct-forcing immersed boundary method for thermal discrete unified gas kinetic scheme with Dirichlet boundary conditions,” International Journal of Heat and Mass Transfer, 137, pp. 476-488 (2019).10.1016/j.ijheatmasstransfer.2019.03.147CrossRefGoogle Scholar
Luo, K., Wang, Z., Tan, J., and Fan, J., “An improved direct-forcing immersed boundary method with inward retraction of Lagrangian points for simulation of particle-laden flows,” Journal of Computational Physics, 376, pp. 210-227 (2019).10.1016/j.jcp.2018.09.037CrossRefGoogle Scholar
Posa, A., Vanella, M., and Balaras, E., “An adaptive reconstruction for Lagrangian, direct-forcing, immersed-boundary methods,” Journal of Computational Physics, 351, pp. 422-436 (2017).10.1016/j.jcp.2017.09.047CrossRefGoogle Scholar
Uhlmann, M., “An immersed boundary method with direct forcing for the simulation of particulate flows,” Journal of Computational Physics, 209, pp. 448-476 (2005).10.1016/j.jcp.2005.03.017CrossRefGoogle Scholar
Yang, J., and Stern, F., “A simple and efficient direct forcing immersed boundary framework for fluid- structure interactions,” Journal of Computational Physics, 231, pp. 5029-5061 (2012).10.1016/j.jcp.2012.04.012CrossRefGoogle Scholar
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).10.1016/j.jcp.2009.08.018CrossRefGoogle Scholar
Tschisgale, S., Kempe, T., and Frohlich, J., “A general implicit direct forcing immersed boundary method for rigid particles,” Computers & Fluids, 170, pp. 285-298 (2018).10.1016/j.compfluid.2018.04.008CrossRefGoogle Scholar
Wang, W. Q., Yan, Y., and Tian, F. B., “A simple and efficient implicit direct forcing immersed boundary model for simulations of complex flow,” Applied Mathematical Modelling, 43, pp. 287-305 (2017).10.1016/j.apm.2016.10.057CrossRefGoogle Scholar
Mori, Y., and Peskin, C. S., “Implicit second-order immersed boundary methods with boundary mass,” Computer Methods in Applied Mechanics and Engineering, 197, pp. 2049-2067 (2008).10.1016/j.cma.2007.05.028CrossRefGoogle Scholar
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).10.1006/jcph.2001.6778CrossRefGoogle Scholar
Ji, C., Munjiza, A., and Williams, J. J. R., “A novel iterative direct-forcing immersed boundary method and its finite volume applications,” Journal of Computational Physics, 231, pp. 1797-1821 (2012).10.1016/j.jcp.2011.11.010CrossRefGoogle Scholar
Fadlun, E., 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).10.1006/jcph.2000.6484CrossRefGoogle Scholar
Cheng, Y., and Zhang, H., “Immersed boundary method and lattice Boltzmann method coupled FSI simulation of mitral leaflet flow,” Computers & Fluids, 39, pp. 871-881 (2010).10.1016/j.compfluid.2010.01.003CrossRefGoogle Scholar
Chorin, A. J., “A numerical method for solving incompressible viscous flow problems,” Journal of Computational Physics, 135, pp. 118-125 (1997).10.1006/jcph.1997.5716CrossRefGoogle Scholar
Zienkiewicz, O. C., Nithiarasu, P., and Taylor, R. L., The finite element method for fluid dynamics, 7rd ed., Elsevier Butterworth-Heinemann, England, 2005.Google Scholar
Peskin, C. S., “The immersed boundary method,” Acta Numerica, 11, pp. 479-517 (2002).10.1017/S0962492902000077CrossRefGoogle Scholar
Su, S. W., Lai, M.-C., and Lin, C.-A., “An immersed boundary technique for simulating complex flows with rigid boundary,” Computers & Fluids, 36, pp. 313-324 (2007).10.1016/j.compfluid.2005.09.004CrossRefGoogle Scholar
Persillon, H., and Braza, M., “Physical analysis of the transition to turbulence in the wake of a circular cylinder by three-dimensional Navier-Stokes simulation,” Journal of Fluid Mechanics, 365, pp. 23-88 (1998).10.1017/S0022112098001116CrossRefGoogle Scholar
Liu, C., Zheng, X., and Sung, C. H., “Preconditioned multigrid methods for unsteady incompressible flows,” Journal of Computational Physics, 139, pp. 35-57 (1998).10.1006/jcph.1997.5859CrossRefGoogle Scholar
Russell, D. G., and Wang, Z. J., “A cartesian grid method for modeling multiple moving objects in 2D incompressible viscous flow,” Journal of Computational Physics, 191, pp. 177-205 (2003).10.1016/S0021-9991(03)00310-3CrossRefGoogle Scholar
Williamson, C. H. K., “Oblique and parallel modes of vortex shedding in the wake of circular cylinder at low reynolds numbers,” Journal of Fluid Mechanics, 206, pp. 579-627 (1989).10.1017/S0022112089002429CrossRefGoogle Scholar
Okajima, A., “Strouhal numbers of rectangular cylinders,” Journal of Fluid Mechanics, 123, pp. 379-398 (1982).10.1017/S0022112082003115CrossRefGoogle Scholar
Pavlov, A. N., Sazhin, S., Fedorenko, R. P., and Heikal, M., “A conservative finite difference method and its application for the analysis of a transient flow around a square prism,” International Journal of Numerical Methods for Heat & Fluid Flow, 10, pp. 6-46 (2000).10.1108/09615530010306894CrossRefGoogle Scholar
Pontaza, J. P., and Reddy, J. N., “Least-squares finite element formulations for viscous incompressible and compressible fluid flows,” Computer Methods in Applied Mechanics and Engineering, 195, pp. 2454-2494 (2006).10.1016/j.cma.2005.05.018CrossRefGoogle Scholar
Poux, A., Glockner, S., Ahusborde, E., and Azaiez, M., “Open boundary conditions for the velocity- correction scheme of the Navier-Stokes equations,” Computers & Fluids, 70, pp. 29-43 (2012).10.1016/j.compfluid.2012.08.028CrossRefGoogle Scholar
Sohankar, A., Norberg, C., and Davidson, L., “Low- Reynolds-number flow around a square cylinder at incidence: study of blockage, onset of vortex shedding and outlet boundary condition,” International Journal for Numerical Methods in Fluids, 26, pp. 39-56 (1998).10.1002/(SICI)1097-0363(19980115)26:1<39::AID-FLD623>3.0.CO;2-P3.0.CO;2-P>CrossRefGoogle Scholar
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).10.1016/j.cma.2007.08.008CrossRefGoogle Scholar
Russell, D., and Jane Wang, Z., “A cartesian grid method for modeling multiple moving objects in 2D incompressible viscous flow,” Journal of Computational Physics, 191, pp. 177-205 (2003).10.1016/S0021-9991(03)00310-3CrossRefGoogle Scholar
Liao, C. C., Chang, Y. W., Lin, C. A., and McDonough, J. M., “Simulating flows with moving rigid boundary using immersed-boundary method,” Computers & Fluids, 39, pp. 152-167 (2010).10.1016/j.compfluid.2009.07.011CrossRefGoogle Scholar
Xu, S., and Wang, Z. J., “An immersed interface method for simulating the interaction of a fluid with moving boundaries,” Journal of Computational Physics, 216, pp. 454-493 (2006).10.1016/j.jcp.2005.12.016CrossRefGoogle Scholar
Glowinski, R., Pan, T. W., Hesla, T. I., Joseph, D. D., and Périaux, 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).10.1006/jcph.2000.6542CrossRefGoogle Scholar
Hong, L., and Shengli, T., “Direct numerical simulation of sedimentation of rectangular particle with ALE method,” Journal of Chongqing University, 33, pp. 83-90 (2010).Google Scholar
Mao, J., Zhao, L., Liu, X., and Avital, E., “A resolved CFDEM method for the interaction between the fluid and the discontinuous solids with large movement,” International Journal for Numerical Methods in Engineering, DOI: 10.1002/nme.6285 (2019).Google Scholar