An efficient method for the incompressible Navier-Stokes equations on irregular domains with no-slip boundary conditions, high order up to the boundary

Journal of Computational Physics

Volumn 230, Issue 23, 2011, Pages 8619-8646

  Shirokoff, D ^a   Rosales, R R ^a  

^a MASSACHUSETTS INSTITUTE OF TECHNOLOGY   (United States)

Author keywords

Incompressible flow; Navier Stokes; Poisson boundary conditions; Pressure Poisson equation; Projection methods; Staggered grid

Indexed keywords

BOUNDARY CONDITIONS; FINITE DIFFERENCE METHOD; NAVIER STOKES EQUATIONS; NUMERICAL METHODS; POISSON EQUATION; VISCOUS FLOW;

HIGH-ORDER; INCOMPRESSIBLE (NAVIER )STOKES EQUATION; INCOMPRESSIBLE NAVIER STOKES EQUATIONS; IRREGULAR DOMAINS; NAVIER STOKES; NO-SLIP BOUNDARY CONDITIONS; POISSON BOUNDARY CONDITION; PRESSURE POISSON EQUATION; PROJECTION METHOD; STAGGERED-GRIDS;

INCOMPRESSIBLE FLOW;

EID: 80053133014     PISSN: 00219991     EISSN: 10902716     Source Type: Journal    
DOI: 10.1016/j.jcp.2011.08.011     Document Type: Article

References (47)

1. Angot P., Bruneau C.-H., Fabrie P. A penalization method to take into account obstacles in viscous flows. Numer. Math. 1999, 81:497-520.
2. Bell J.B., Collela P., Glaz H.M. A second-order projection method for the incompressible Navier-Stokes equations. J. Comput. Phys. 1989, 85:257-283.
3. Ben-Artzi M., Croisille J.-P., Fishelov D., Trachtenberg S. A pure-compact scheme for the streamfunction formulation of the Navier-Stokes equations. J. Comput. Phys. 2005, 205:640-644.
4. Brown D.L., Cortez R., Minion M.L. Accurate projection methods for the incompressible Navier-Stokes equations. J. Comput. Phys. 2001, 168:464-499.
5. Calhoun D. A Cartesian grid method for solving the two-dimensional stream function-vorticity equations in irregular regions. J. Comput. Phys. 2002, 176:231-275.
6. Chorin A.J. Numerical solutions of the Navier-Stokes equations. Math. Comput. 1968, 22:745-762.
7. E W., Liu J.-G. Gauge method for viscous incompressible flows. Commun. Math. Sci. 2003, 1:317-332.
8. Ghia U., Ghia K.N., Shin C.T. High-resolutions for incompressible flow using the Navier-Stokes equations and a multigrid method. J. Comput. Phys. 1982, 48:387-411.
9. Glowinski R., Pan T.-W., Kearsley A.J., Periaux J. Numerical simulation and optimal shape for viscous flow by a fictitious domain method. Int. J. Numer. Methods Fluids 1995, 20:695-711.
10. Goldstein D., Handler R., Serovich L. Modeling a no-slip flow boundary with an external force field. J. Comput. Phys. 1993, 105:354-366.
11. Greenspan D. Introduction to Numerical Analysis of Elliptic Value Problems 1965, Harper, New York, pp. 39-42. international ed.
12. Gresho P.M., Sani R.L. On pressure boundary conditions for the incompressible Navier-Stokes equations. Int. J. Numer. Methods Fluids 1987, 7:1111-1145.
13. Guermond J.L., Minev P., Shen Jie An overview of projection methods for incompressible flows. Comput. Methods Appl. Mech. Eng. 2006, 195:6011-6045.
14. A. Hammouti, Simulation Numérique Directe en différence finie de l'écoulement d'un fluide incompressible en présence d'une interface rigide, docteur és sciences theses, Ecole Nationale des Ponts et Chaussés, Paris, France, December 2009.
15. Harlow F.H., Welch J.E. Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface. Phys. Fluids 1965, 8:2182-2189.
16. Henshaw W.D. A fourth-order accurate method for the incompressible Navier-Stokes equations on overlapping grids. J. Comput. Phys. 1994, 113:13-25.
17. Henshaw W.D., Kreiss H.-O., Reyna L.G.M. A fourth-order accurate difference approximation for the incompressible Navier-Stokes equations. J. Comput. Fluids 1994, 23:575-593.
18. Henshaw W.D., Anders Petersson N. A split-step scheme for the incompressible Navier-Stokes equations. Numerical Simulation of Incompressible Flows 2003, 108-125. World Scientific. M.M. Hafez (Ed.).
19. Jackson T.A.S. Orthogonal curvilinear coordinates. Math. Gaz. 1966, 50(371):28-30.
20. Johnston H., Liu J.-G. A finite difference method for incompressible flow based on local pressure boundary conditions. J. Comput. Phys. 2002, 180:120-154.
21. Johnston H., Liu J.-G. Accurate, stable and efficient Navier-Stokes solvers based on explicit treatment of the pressure term. J. Comput. Phys. 2004, 199:221-259.
22. Karniadakis G., Israeli M., Orszag S.A. High-order splitting methods for the incompressible Navier-Stokes equations. J. Comput. Phys. 1991, 97:414-443.
23. Khadra K., Angot P., Parneix S., Caltagirone J.-P. Fictitious domain approach for numerical modelling of Navier-Stokes equations. Int. J. Numer. Methods Fluids 2000, 34:651-684.
24. Kim J., Moin P. Application of a fractional step method to incompressible Navier-Stokes equations. J. Comput. Phys. 1985, 59:308-323.
25. Kleiser L., Schumann U. Treatment of the incompressibility and boundary conditions in 3-D numerical spectral simulation of plane channel flows. Notes on Numerical Fluid Mechanics 1980, 165-173. Vieweg, Braunschweig. E.H. Hirschel (Ed.).
26. Krzywicki A., Ladyzhenskaya O.A. A grid method for the Navier-Stokes equations. Dokl. Akad. Nauk SSSR 1966, 167:309-311. English translation, Soviet Physics Dokl. 1 (1966) 212-213.
27. Lax P.D., Richtmyer R.D. Survey of the stability of linear finite difference equations. Commun. Pure Appl. Math. 1956, 9:267-293.
28. Linnick M., Fasel H. A high-order immersed interface method for simulating unsteady incompressible flows on irregular domains. J. Comput. Phys. 2005, 204:157-192.
29. Liu J.-G., Liu J., Pego R. Stable and accurate pressure approximation for unsteady incompressible viscous flow. J. Comput. Phys. 2010, 229:3428-3453.
30. Mittal R., Iaccarino G. Immersed boundary methods. Annu. Rev. Fluid Mech. 2005, 37:39-261.
31. Napolitano M., Pascazio G., Quartapelle L. A review of vorticity conditions in the numerical solution of the ζ-ψ equations. J. Comput. Fluids 1999, 28:139-185.
32. Orszag S.A., Israeli M., Deville M. Boundary conditions for incompressible flows. J. Sci. Comput. 1986, 1:75-111.
33. Perot J.B. An analysis of the fractional step method. J. Comput. Phys. 1993, 108:51-58.
34. Peskin C.S. Flow patterns around heart valves: a numerical method. J. Comput. Phys. 1972, 10:252-271.
35. Peskin C.S. The immersed boundary method. Acta Numer. 2002, 11:479-517.
36. Petersson N.A. Stability of pressure boundary conditions for Stokes and Navier-Stokes equations. J. Comput. Phys. 2001, 172:40-70.
37. Prohl A. Projection and quasi-compressibility methods for solving the incompressible Navier-Stokes equations. Wiley-Teubner Series 1997, B.G. Teubner, Stuttgart.
38. Quartapelle L. Numerical Solution of the Incompressible Navier-Stokes Equations 1993, Birkhauser Verlag.
39. Rannacher R. On Chorin's projection method for the incompressible Navier-Stokes equations. Lect. Notes Math. 1991, 1530:167-183.
40. Rempfer D. On boundary conditions for the incompressible Navier-Stokes problems. Appl. Mech. Rev. 2006, 59:107-125.
41. Sani R., Shen J., Pironneau O., Gresho P. Pressure boundary condition for the time-dependent incompressible Navier-Stokes equations. Int. J. Numer. Methods Fluids 2006, 50:673-682.
42. Shen J. On error estimates of the projection methods for the Navier-Stokes equations: first-order schemes. SIAM J. Numer. Anal. 1992, 29:57-77.
43. Strikwerda J.C. Finite Difference Schemes and Partial Differential Equations 1989, Wadsworth and Brooks Cole.
44. Taira K., Colonius T. The Immersed boundary method: a projection approach. J. Comput. Phys. 2007, 225:2118-2137.
45. Temam R. Sur l'approximation de la solution des equations de Navier-Stokes par la methode des pas fractionnaries, ii. Arch. Ration. Mech. Anal. 1969, 33:377-385.
46. Taylor M. Partial Differential Equations III, Nonlinear Equations 1996, Springer, New York.
47. van Kan J. A second-order accurate pressure-correction scheme for viscous incompressible flow. SIAM J. Sci. Stat. Comput. 1986, 7:870-891.