Skip to main content
SpringerLink
Log in

An implicit non-staggered Cartesian grid method for incompressible viscous flows in complex geometries

  • Published:
Sadhana Aims and scope Submit manuscript

Abstract

A discrete forcing based Cartesian grid method is presented. The non-staggered arrangement of velocity and pressure is considered. The pressure gradient in localized discrete form is added separately with the velocity making them explicitly coupled. The governing equation is time-integrated implicitly with both linearized and non-linear forms are investigated. Both linear and bi-linear reconstruction techniques are tested for extrapolation of velocity near a complex boundary. The present method is tested for vortical flow in an inclined cavity, flow past circular and inclined square cylinder. Both homogeneous and non-homogeneous Dirichlet forcing problems are tested. The parallelized version of the method is applied to 2D-to-3D transitional flow behind a single and multiple circular cylinders. The present numerical results compare well with the previously documented results.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Figure 1
Figure 2
Figure 3
Figure 4
Figure 5
Figure 6
Figure 7
Figure 8
Figure 9
Figure 10
Figure 11
Figure 12
Figure 13
Figure 14
Figure 15
Figure 16
Figure 17

Similar content being viewed by others

References

  • Angot P, Bruneau C H, and Frabrie P 1999 A penalization method to take into account obstacles in viscous flows. Numer. Math. 81: 497–520

  • Armfield S W 1991 Finite difference solutions of the Navier-Stokes equations on staggered and non-staggered grids. Comput. Fluids 20: 1–17

  • Arthurs K M, Moore L C, Peskin C S, Pitman E B, and Layton H E 1998 Modeling arteriolar flow and mass transport using the immersed boundary method. J. Comput. Phys. 147: 402–440

  • Balaras E 2004 Modeling complex boundaries using an external force field on fixed Cartesian grids in large-eddy simulations. Comput. Fluids 33 (3): 375–404

  • Barton I E and Kirby R 2000 Finite difference scheme for the solution of fluid flow problems on non-staggered grids. Int. J. Numer. Methods Fluids 33: 939–959

  • Berthelsen P A and Ytrehus T 2007 Stratified smooth two-phase flow using the immersed interface method. Comput. Fluids 36 (7): 1273–1289

  • Borazjani I, Ge L, and Sotiropoulos F 2008 Curvilinear immersed boundary method for simulating fluid structure interaction with complex 3d rigid bodies. J. Comput. Phys. 227 (16): 7587–7620

  • Bottino D C 1998 Modeling viscoelastic networks and cell deformation in the context of the immersed boundary method. J. Comput. Phys. 147: 86

  • Demirdžić I, Lilek ž, and Perić M 1992 Fluid flow and heat transfer test problems for non-orthogonal grids: benchmark solutions. Int. J. Numer. Meth. Fluids 15: 329–354

  • Denaro F M and Sarghini F 2002 2-d transmitral flows simulation by means of the immersed boundary method on unstructured grids. Int. J. Numer. Methods Fluids 38: 1133–1158

  • der Wijngaart R J F V 1990 Composite grid techniques and adaptive mesh refinement in computational fluid dynamics. Report CLaSSiC-90-97

  • 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 (1): 35–60

  • Fauci L J 1990 Interaction of oscillating filamentsa computational study. J. Comput. Phys. 86: 294

  • Ferziger J H and Perić M 1996 Computational methods for fluid dynamics. Springer 95–106

  • Fogelson A L and Peskin C S 1988 A fast numerical method for solving three-dimensional stokes equations in the presence of suspended particles. J. Comput. Phys. 79: 50

  • Ge L and Sotiropoulos F 2007 A numerical method for solving the 3d unsteady incompressible Navier-Stokes equations in curvilinear domains with complex immersed boundaries. J. Comput. Phys. 225 (2): 1782–1809

  • Gilmanov A and Acharya S 2008 A hybrid immersed boundary and material point method for simulating 3d fluidstructure interaction problems. Int. J. Numer. Methods Fluids 56: 2151–2177

  • Goldstein D, Handler R, and Sirovich L 1993 Modeling a no-slip flow boundary with an external force field. J. Comput. Phys. 105: 354–366

  • Huang W -X and Sung H J 2009 An immersed boundary method for fluid-flexible structure interaction. Comput. Methods Appl. Mech. Engrg. 198 (33–36): 2650–2661

  • Iaccarino G and Verzicco R 2003 Immersed boundary technique for turbulent flow simulations. Appl. Mech. Rev. 56: 331–347

  • Kim D and Choi J 2000 A second-order time-accurate finite volume method for unsteady incompressible flow on hybrid unstructured grids. J. Comput. Phys. 162: 411–428

  • Kim J, Kim D, and Choi H 2001 An immersed-boundary finite-volume method for simulations of flow in complex geometries. J. Comput. Phys. 171 (1): 132–150

  • Kim J and Moin P 1985 Application of a fractional-step method to incompressible Navier-Stokes equations. J. Comput. Phys. 59: 308–323

  • Kirkpatrick M P, Armfield S W, and Kent J H 2003 A representation of curved boundaries for the solution of the Navier-Stokes equations on a staggered three-dimensional Cartesian grid. J. Comput. Phys. 184: 1–36

  • Labbé D F L and Wilson P A 2003 A numerical investigation of the effects of the spanwise length on the 3-d wake of a circular cylinder. J. Fluid Mech., 476: 303–334

  • Lai M -C and Peskin C S 2000 An immersed boundary method with formal second- order accuracy and reduced numerical viscosity. J. Comput. Phys. 160: 705–719

  • Li C W and Wang L L 2004 An immersed boundary finite difference method for les of flow around bluff shapes. Int. J. Numer. Methods Fluids 46: 85–107

  • Majumdar S, Iaccarino G, and Durbin P 2001 Rans solvers with adaptive structured boundary non-conforming grids. Annual Research Briefs353–366

  • Maruoka A 2003 Finite element analysis for flow around a rotating body using chimera method. Int. J. Comput. Fluid Dyn. 17: 289–297

  • McQueen D M and Peskin C S 1997 Shared-memory parallel vector implementation of the immersed boundary method for the computation of blood flow in the beating mammalian heart. J. Supercomput. 11: 213

  • Mittal R and Iaccarino G 2005 Immersed boundary methods. Annu. Rev. Fluid Mech. 37: 239–261

  • Mittal S and Kumar B 2003 Flow past a rotating cylinder. J. Fluid Mech. 476: 303–334

  • Mohd-Yosuf J 1997 Combined immersed boundary/b-spline methods for simulation of flow in complex geometries. Annual Research Briefs317–328

  • Peskin C S 1972 Flow patterns around heart valves: A numerical method. Journal of Computational Physics 10 (2): 252–271

  • Rhie C M and Chow W L 1983 A numerical study of the turbulent flow past an isolated airfoil with trailing edge separation. AIAA J. 21: 1525–1532

  • Saiki E M and Biringen S 1996 Numerical simulation of a cylinder in uniform flow: Application of a virtual boundary method. J. Comput. Phys. 123: 450–465

  • Sohankar A, Norberg C, and Davidson L 1998 Low-reynolds-number flow around a square cylinder at incidence: Study of blockage, onset of vortex shedding and outlet boundary condition. Int. J. Numer. Meth. Fluids 26: 39

  • Stone H L 1968 Iterative solution of implicit approximation of multidimensional partial differential equations. SIAM J. Numer. Anal. 5: 530–558

  • Tai C, Zhao Y, and Liew K 2005 Parallel computation of unsteady incompressible viscous flows around moving rigid bodies using an immersed object method with overlapping grids. J. Comput. Phys. 207 (1): 151–172

  • Tau E Y 1994 A 2nd-order projection method for the incompressible Navier- Stokes equations in arbitrary domains. J. Comput. Phys. 115: 147–152

  • Tseng Y -H and Ferziger J H 2003 A ghost-cell immersed boundary method for flow in complex geometry. J. Comput. Phys. 192 (2): 593–623

  • Tucker P G and Pan Z 2000 A Cartesian cut cell method for incompressible viscous flow. Appl. Math. Modell. 24: 591–606

  • Tyagi M and Acharya S 2005 Large eddy simulation of turbulent flows in complex and moving rigid geometries using the immersed boundary method. Int. J. Numer. Methods Fluids 48: 691–722

  • Unverdi S O and Tryggvason G 1992 A front-tracking method for viscous, incompressible, multi-fluid flows. J. Comput. Phys. 100: 250–37

  • Verma A K and Eswaran V 1999 An overlapping control volume method for Navier-Stokes equations on nonstaggered grids. Int. J. Numer. Meth. Fluids 30: 279–308

  • Verzicco R, Iaccarino G, Fatica M, and Orlandi P 2000 Flow in an impeller stirred tank using an immersed boundary method. Annual Research Briefs251–261

  • Williamson C 1988 The existence of two stages in the transition to three-dimensionality of a cylinder wake. Phys. Fluids 31: 3165–3168

  • Williamson C 1996 Vortex dynamics in the cylinder wake. Annu. Rev. Fluid Mech. 28: 477–539

  • Xia G, Zhao Y, and Yeo J 2009 Parallel unstructured multigrid simulation of 3d unsteady flows and fluid-structure interaction in mechanical heart valve using immersed membrane method. Comput. Fluids 38 (1): 71–79

  • Yang J and Stern F 2009 Sharp interface immersed-boundary/level-set method for wave-body interactions. J. Comput. Phys. 228 (17): 6590–6616

  • Ye T, Mittal R, Udaykumar H S, and Shyy W 1999 An accurate Cartesian grid method for viscous incompressible flows with complex immersed boundaries. J. Comput. Phys. 156: 209–240

  • Yoon D -H, Yang K -S, and Choi C -B 2010 Flow past a square cylinder with an angle of incidence. Phys. Fluids 22: 043603(1–11)

  • Zhang S -L 1997 Gpbi-cg: Generalized product-type methods based on bi-cg for solving nonsymmetric linear systems. SIAM J. Sci. Comput. 18: 537–551

  • Zhang X, Ni S, and He G 2008 A pressure-correction method and its applications on an unstructured chimera grid. Comput. Fluids 37: 993–1010

  • Zhu L and Peskin C 2003 Interaction of two filaments in a flowing soap film. Phys. Fluids 15: 128–136

Download references

Acknowledgements

The present research was carried out through the funds available from the institute start-up grant R&D/07/SG/ME/P/ARKD/1/2009-2010. The author thanks Prof. Vinayak Eswaran for his invaluable inputs.

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, Indian Institute of Technology Guwahati, Guwahati, 781 039, India

    A K DE

Authors
  1. A K DE
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A K DE.

Appendix A

Appendix A

The discrete linear equations (A ϕ=b) arising from the governing equations can be cast into a septa-diagonal form

$$ a_{B} \phi_{B}+a_{S} \phi_{S}+a_{W} \phi_{W}+a_{P} \phi_{P}+a_{E} \phi_{E}+a_{N} \phi_{N}+a_{T} \phi_{T}=b, $$
(23)

where a B =a T =0 in the two-dimensional form. Diagonals of the lower (L W,L S,L P,L B) and upper (U E,U N,U T) triangular factors, obtained by the normal LU factorization procedure along with the implicit relations proposed by Stone (1968), are given below

$$\begin{array}{@{}rcl@{}} LS_{P} &=& a_{S}/(1+\gamma(UT_{S}+UE_{S})) \\ LB_{P} &=& a_{B}/(1+\gamma(UE_{B}+UN_{B})) \\ LW_{P} &=& a_{W}/(1+\gamma(UN_{W}+UT_{W})) \\ LP_{P} &=& a_{P}+\gamma(LW_{P}~UN_{W} + LW_{P}~UT_{W} + LS_{P}~UT_{S} + LS_{P}~UE_{S} \\ &&+ LB_{P}~UE_{B} + LB_{P}~UN_{B}) - LW_{P}~UE_{W} - LS_{P}~UN_{S} - LB_{P}~UT_{B} \\ UE_{P} &=& (a_{E} - \gamma(LS_{P}~UE_{S} + LB_{P}~UE_{B})/LP_{P} \\ UT_{P} &=& (a_{T} - \gamma(LS_{P}~UT_{S} + LW_{P}~UT_{W}))/LP_{P} \\ UN_{P} &=& (a_{N} - \gamma(LW_{P}~UN_{W} + LB_{P}~UN_{B}))LP_{P} \end{array} $$

with γ being a implicit factor and γ≈0.9 gives the best convergence for a range of problems. The above factors reduce the original linear system into a two-step substitution procedure

$$ LU \phi =b \Longrightarrow LM=b~~\text{and}~~U\phi=M, $$
(24)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

DE, A.K. An implicit non-staggered Cartesian grid method for incompressible viscous flows in complex geometries. Sadhana 39, 1071–1094 (2014). https://doi.org/10.1007/s12046-014-0269-y

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12046-014-0269-y

Keywords

Search

Navigation