No Access

A Transformation for Imposing the Rigid Bodies on the Solution of the Vorticity-Stream Function Formulation of Incompressible Navier–Stokes Equations

Mohammadali Badri

Aerospace Engineering Department, Science and Research Branch, Islamic Azad University, Tehran, Iran

E-mail Address: badri.nma@gmail.com

Search for more papers by this author

and

Fereidoun Sabetghadam

http://orcid.org/0000-0001-8412-8324

Aerospace Engineering Department, Science and Research Branch, Islamic Azad University, Tehran, Iran

E-mail Address: fsabet@srbiau.ac.ir

Corresponding author.

Search for more papers by this author

https://doi.org/10.1142/S0219876219500634Cited by:1 (Source: Crossref)

Abstract

A new penalization method is proposed for implementing the rigid bodies on the solution of the vorticity-stream function formulation of the incompressible Navier–Stokes equations. The method is based upon an active transformation of dependent variables. The transformation may be interpreted as time dilation. In this interpretation, the rigid body is considered as a region where the time is dilated infinitely, that is, time is stopped. The transformation is introduced in the vorticity and stream function equations to achieve a set of modified equations. The, in the modified equations, the time dilation of the solid region is approached to infinity. The mathematical and physical properties of the modified equations are investigated and implementation of the no-slip and no-penetration conditions are justified. Moreover, a suitable numerical method is presented for the solution of the modified equations. In the proposed numerical method, time integration is performed via the Crank–Nicolson method, and the semi-discrete equations are spatially discretized via second-order finite differencing on a uniform Cartesian grid. The method is applied to the fluid flow around a square obstacle placed in a channel, a sudden flow perpendicular to a thin flat plate, and the flow around a circular cylinder. The accuracy of the numerical solutions is evaluated.

Keywords:

Remember to check out the Most Cited Articles!
Check out these titles in finite element methods!

References

Adlam, J. H. [1986] “ Computation of two-dimensional time-dependent natural convection in a cavity where there are internal bodies,” Comput. Fluids 14(2), 141–157. Crossref, Web of Science, Google Scholar
Angot, P., Bruneau, C.-H. and Fabrie, P. [1999] “ A penalization method to take into account obstacles in viscous flows,” Numer. Math. 81, 497–520. Crossref, Web of Science, Google Scholar
Arquis, E. and Caltagirone, J. P. [1984] “ On the hydrodynamical boundary conditions along a fluid layer-porous medium interface, Application to the case of free convection,” Comptes Rendus de Acad. Sci. 299(1), Series II, 1–4. Google Scholar
Balaras, E. [2004] “ Modeling complex boundaries using an external force field on fixed Cartesian grid in large-eddy simulations,” Comput. Fluids 33, 375–404. Crossref, Web of Science, Google Scholar
Benamour, M., Liberge, E. and Béghein, C. [2015] “ Lattice Boltzmann method for fluid flow around bodies using volume penalization,” Int. J. Multiphys. 9(3), 299–315. Crossref, Google Scholar
Benamour, M., Liberge, E. and Béghein, C. [2018] “A volume penalization lattice Boltzmann method for simulating flows in the presence of obstacles,” arXiv:1812.05401v1 [physics.flu-dyn]. Google Scholar
Braza, M., Chassaing, P. and Ha Minh, H. [1986] “ Numerical study and physical analysis of the pressure and velocity fields in the near wake of a circular cylinder,” J. Fluid Mech. 165, 79–130. Crossref, Web of Science, Google Scholar
Breuer, M., Bernsdorf, J., Zeiser, T. and Durst, F. [2000] “ Accurate computations of the laminar flow past a square cylinder based on two different methods: Lattice-Boltzmann and finite-volume, Int. J. Heat Fluid Fl. 21, 186–196. Crossref, Web of Science, Google Scholar
Calhoun, D. [2002] “ A cartesian grid method for solving the two-dimensional stream function vorticity equations in irregular regions,” J. Comput. Phys. 176, 231–275. Crossref, Web of Science, Google Scholar
Carbou, G. and Fabrie, P. [2003] “ Boundary layer for a penalization method for viscous incompressible flow,” Adv. Differ. Equ. 8(12), 1453–1480. Google Scholar
Chen, Z., Shu, C. and Tan, D. [2018] “ Immersed boundary-simplified lattice Boltzmann method for incompressible viscous Flows,” Phys. Fluids 30, 053601. Crossref, Web of Science, Google Scholar
Coutanceau, M. and Bouard, R. [1977a] “ Experimental determination of the main feature of the viscous flow in the wake of a circular cylinder in uniform translation. Part 1. Steady flow,” J. Fluid Mech. 79, 231–256. Crossref, Web of Science, Google Scholar
Coutanceau, M. and Bouard, R. [1977b] “ Experimental determination of the main feature of the viscous flow in the wake of a circular cylinder in uniform translation. Part 2. Unsteady flow,” J. Fluid Mech. 79, 257–272. Crossref, Web of Science, Google Scholar
Cui, Z., Zixuan, Y., Jiang, H-Z., Huang, W-X. and Shen, L. [2017] “ A sharp-interface immersed boundary method for simulating incompressible flows with arbitrarily deforming smooth boundaries,” Int. J. Comput. Methods 14(2), 1750080. Google Scholar
Dennis, S. C. R. and Chang, G. [1970] “ Numerical solutions for steady flow past a circular cylinder at Reynolds number up to 100,” J. Fluid Mech. 42, 471–489. Crossref, Web of Science, Google Scholar
Dennis, S. C. R., Qiang, W., Coutanceau, M. and Launay, J.-L. [1993] “ Viscous flow normal to a flat plate at moderate Reynolds numbers,” J. Fluid Mech. 248, 605–635. Crossref, Web of Science, Google Scholar
Engels, T., Kolomenskiy, D., Schneider, K. and Sesterhenn, J. [2013] “ Two-dimensional simulation of the fluttering instability using a pseudospectral method with volume penalization,” Comput. Struct. 122, 101–112. Crossref, Web of Science, Google Scholar
Engels, T., Kolomenskiy, D., Schneider, K. and Sesterhenn, J. [2015] “ Numerical simulation of fluid-structure interaction with the volume penalization method,” J. Comput. Phys. 281, 96–115. Crossref, Web of Science, Google Scholar
Engels, T., Kolomenskiy, D., Schneider, K. and Sesterhenn, J. [2016] “ FluSI: A novel parallel simulation tool for flapping insect flight using a fourier method with volume penalization,” SIAM J. Sci. Comput. 38(5), S5–S24. Crossref, Web of Science, Google Scholar
Engels, T., Kolomenskiy, D., Schneider, K. and Sesterhenn, J. [2017] “ Numerical simulation of vortex-induced drag of elastic swimmer models,” Theor. Appli. Mech. Lett. 7(5), 280–285. Crossref, Web of Science, Google Scholar
Engels, T., Kolomenskiy, D., Schneider, K., Farge, M., Lehmann, F. and Sesterhenn, J. [2018] “ Helical vortices generated by flapping wings of bumblebees,” Fluid Dyn. Res. 50(1), 011419. Crossref, Web of Science, Google Scholar
Enright, D., Fedkiw, R., Ferziger, J. and Mitchell, I. [2002] “ A high particle level set method for improved interface capturing,” J. Comput. Phys. 183, 83–116. Crossref, Web of Science, Google Scholar
Fadlun, E. A., Verzicco, R., Orlandi, P. and Mohd-Yusof, J. [2000] “ Combined Immersed-Boundary Finite-Difference methods for Three-Dimensional complex flow simulations,” J. Comput. Phys. 161, 35–60. Crossref, Web of Science, Google Scholar
Favier, J., Revell, A. and Pinelli, A. [2014] “ A Lattice Boltzmann–Immersed Boundary method to simulate the fluid interaction with moving and slender flexible objects,” J. Comput. Phys. 261, 145–161. Crossref, Web of Science, Google Scholar
Griffith, B. E. and Peskin, C. S. [2005] “ On the order of accuracy of the immersed boundary method: Higher order convergence rates for sufficiently smooth problems,” J. Comput. Phys. 208, 75–105. Crossref, Web of Science, Google Scholar
Haeri, S. and Shrimpton, J. S. [2012] “ On the application of immersed boundary, fictitious domain and body-conformal mesh methods to many particle multiphase flows,” Int. J. Multiph. Flow 40, 38–55. Crossref, Web of Science, Google Scholar
Hejlesen, M. M., Koumoutsakos, P., Leonard, A. and Walther, J. H. [2015] “ Iterative Brinkman penalization for remeshed vortex methods,” J. Comput. Phys. 280, 547–562. Crossref, Web of Science, Google Scholar
Hudson, J. D. and Dennis, S. C. R. [1985] “ The flow of a viscous incompressible fluid past a normal flat plate at low and intermediate Reynolds numbers: The wake,” J. Fluid Mech. 160, 369–383. Crossref, Web of Science, Google Scholar
Ingham, D. B., Tang, T. and Morton, B. R. [1990] “ Steady two-dimensional flow through a row of normal flat plates,” J. Fluid Mech. 210, 281–302. Crossref, Web of Science, Google Scholar
Ingham, D. B. Tang, T. and Morton, B. R. [1991] “ Steady two-dimensional flow past a normal flat plates,” J. Appl. Math. Phys. 12, 584–604. Google Scholar
Introïni, C., Belliard, M. and Fournier, C. [2014] “ A second order penalized direct forcing for hybrid Cartesian/immersed boundary flow simulations,” Comput. Fluids 90, 21–41. Crossref, Web of Science, Google Scholar
Kadoch, B., Kolomenskiy, D., Angot, P. and Schneider, K. [2012] “ A volume penalization method for incompressible flows and scalar advection-diffusion with moving obstacles,” J. Comput. Phys. 131, 4365–438. Crossref, Web of Science, Google Scholar
Kolomenskiy, D., Moffatt, H. K., Farge, M. and Schneider, K. [2011] “ Two- and three-dimensional numerical simulations of the clap-fling-sweep of hovering insects,” J. Fluids Struct. 27, 784–791. Crossref, Web of Science, Google Scholar
Kolomenskiy, D., Nguyen van yen, R. and Schneider, K. [2015] “ Analysis and discretization of the volume penalized Laplace operator with Neumann boundary conditions,” Appl. Numer. Math. 95, 238–249. Crossref, Web of Science, Google Scholar
Kolomenskiy, D. and Schneider, K. [2009] “ A Fourier spectral method for the Navier–Stokes equations with volume penalization for moving solid obstacles,” J. Comput. Phys. 228, 5687–5709. Crossref, Web of Science, Google Scholar
Koumoutsakos, P. and Shield, D. [1996] “ Simulation of the viscous flow normal to an impulsively started and uniformly accelerated flat plate,” J. Fluid Mech. 328, 177–227. Crossref, Web of Science, Google Scholar
Linnick, M. N. and Fasel, H. F. [2005] “ A high-order immersed interface method for simulating unsteady incompressible flows on irregular domains,” J. Comput. Phys. 204, 157–192. Crossref, Web of Science, Google Scholar
Liu, C., Zheng, X. and Sung, C. H. [1998] “ Preconditioned multigrid methods for unsteady incompressible flows,” J. Comput. Phys. 139, 35–57. Crossref, Web of Science, Google Scholar
Mittal, R. and Iaccarino, G. [2005] “ Immersed Boundary Methods”, Annu. Rev. Fluid Mech. 37, 239–261. Crossref, Web of Science, Google Scholar
Najjar, F. M. and Vanka, S. P. L. [1995] “ Simulations of the unsteady separated flow past a normal flat plate,” Int. J. Numer. Methods Fluids 21, 525–547. Crossref, Web of Science, Google Scholar
Napolitano, M., Pascazioa, G. and Quartapelle, L. [1999] “ A review of vorticity conditions in the numerical solution of the $ς - ψ$ equations,” Comput. Fluids 28, 139–185. Crossref, Web of Science, Google Scholar
Okajima, A. [1982] “ Strouhal numbers of rectangular cylinders,” J. Fluid Mech. 123, 379–398. Crossref, Web of Science, Google Scholar
Orlanski, I. [1976] “ A simple boundary condition for unbounded hyperbolic flows,” J. Comput. Phys. 21, 251–269. Crossref, Web of Science, Google Scholar
Rahmati, A. R. and Niazi, S. [2014] “ Entropic lattice Boltzmann method for microflow simulation,” Nanomech. Sci. Technol. Int. J. 5(9) 153–167. Crossref, Google Scholar
Rahmati, A. R. and Niazi, S. [2015] “ Application and comparison of different lattice Boltzmann methods on non-uniform meshes for simulation of micro cavity and micro channel flow,” J. Comput. Methods Eng. 34(1), 97–118. Crossref, Google Scholar
Rahmati, A. R., Niazi, S. and Beni, M. N. [2012] “ Natural convection flow simulation of nanofluid in a square cavity using an incompressible generalized lattice Boltzmann method,” Defect Diffus. Forum 329, 69–79. Crossref, Google Scholar
Ren, W. W., Wu, J., Shu, C. and Yang, W. M. [2012] “ A stream function–vorticity formulation-based immersed boundary method and its applications,” Int. J. Numer. Methods Fluids 70, 627–645. Crossref, Web of Science, Google Scholar
Sabetghadam, F. [2014] An analytical framework for imposition of a rigid immersed surface on the incompressible Navier–Stokes equations, arXiv:1412.5130v1 [physics.flu-dyn]. Google Scholar
Sabetghadam, F. [2015] Exact imposition of the regular rigid immersed surfaces on the solution of the incompressible Navier–Stokes equations, doi:10.13140/ RG.2.1.4800.4001. Google Scholar
Sabetghadam, F. and Soltani, E. [2015] “ Simulation of solid body motion in a newtonian fluid using a vorticity-based Pseudo-spectral immersed boundary method augmented by the Radial Basis Functions,” Int. J. Mod. Phys. C 26, 1550053. Link, Web of Science, Google Scholar
Sabetghadam, F., Soltani, E. and Ghasemi, H. [2014] “ A fast immersed boundary Fourier Pseudo-spectral method for simulation of the incompressible flows,” Int. J. Eng. 27(9), 1457–1466. Google Scholar
Sabetghadam, F., Sharafatmandjoor, S. and Norouzi, F. [2009] “ Fourier spectral embedded boundary solution of the poisson’s and laplace equations with dirichlet boundary conditions,” J. Comput. Phys. 228(1), 55–74. Crossref, Web of Science, Google Scholar
Schneider, K. [2005] “ Numerical simulation of the transient flow behavior in chemical reactors using a penalisation method,” Comput. Fluids 34(10), 1223–1238. Crossref, Web of Science, Google Scholar
Schneider, K., Paget-Goy, M., Verga, A. and Farge, M. [2014] “ Numerical simulation of flows past flat plates using volume pealization,” Comput. Appl. Math. 33(2), 481–495. Crossref, Web of Science, Google Scholar
Sørensen, J. N. and Nygreen, P. J. [2001] “ Unsteady vorticity-streamfunction algorithm for external flows,” Comput. Fluids 30, 69–87. Crossref, Web of Science, Google Scholar
Taira, K. and Colonious, T. [2007] “ The immersed boundary method: A projection approach,” J. Comput. Phys. 225, 2118–2137. Crossref, Web of Science, Google Scholar
Tian, F., Luo, H., Zhu, L., Liao, J. C. and Lu, X. [2011] “ An efficient immersed boundary-lattice Boltzmann method for the hydrodynamic interaction of elastic filaments,” J. Comput. Phys. 230, 7266–7283. Crossref, Web of Science, Google Scholar
Trujilo, J. and Karniadakis, G. E. [1999] “ A penalty method for the vorticity-velocity formulation,” J. Comput. Phys. 149, 32–58. Crossref, Web of Science, Google Scholar
Turki, S., Abbassi, H. and Nasrallah, S. B. [2003] “ Effiect of the blockage ratio on the flow in a channel with a built-in square cylinder,” Comput. Mech., 33, 22–29. Crossref, Web of Science, Google Scholar
Verma, S., Abbati, G., Novati, G. and Koumoutsakos, P. [2017] “ Computing the force distribution on the surface of complex, deforming geometries using vortex methods and brinkman penalization,” Int. J. Numer. Methods Fluids 85(6), 1–18. Google Scholar
Wang, Z., Fan, J. and Cen, K. [2009] “ Immersed boundary method for the simulation of 2D viscous flow based on vorticity–velocity formulations,” J. Comput. Phys. 228, 1504–1520. Crossref, Web of Science, Google Scholar
Wu, J. and Shu, C. [2010] “ An improved immersed boundary-lattice Boltzmann method for simulating three-dimensional incompressible flows,” J. Comput. Phys. 229, 5022–5042. Crossref, Web of Science, Google Scholar
Yoon, D.-H., Yang, K.-S. and Choi, C.-B. [2010] “ Flow past a square cylinder with an angle of incidence,” J. Comput. Phys. 22, 043603. Google Scholar