zbMATH — the first resource for mathematics

A multigrid solver for the vorticity-velocity Navier-Stokes equations. (English) Zbl 0724.76060
Summary: This paper provides a multirid incremental line-Gauss-Seidel method for solving the steady Navier-Stokes equations in two and three dimensions expressed in terms of the vorticity and velocity variables. The system of parabolic and Poisson equations governing the scalar components of the vector unknowns is solved using centred finite differences on a non- staggered grid. Numerical results for the two-dimensional driven cavity problem indicate that the spatial discretization of the equation defining the value of the vorticity on the boundary is extremely critical to obtaining accurate solutions. In fact, a standard one-sided three-point second-order-accurate approximation produces very inaccurate results for moderate-to-high values of the Reynolds number unless an exceedingly fine mesh is employed. On the other hand, a compact two-point second-order- accurate discretization is found to be always satisfactory and provides accurate solutions for Reynolds number up to 3200, a target impossible heretofore using this formulation and a non-staggered grid.

76M20 Finite difference methods applied to problems in fluid mechanics
76D05 Navier-Stokes equations for incompressible viscous fluids
Full Text: DOI
[1] Napolitano, Int. j. numer. methods fluids 4 pp 1101– (1984)
[2] Napolitano, AIAA J. 24 pp 770– (1986)
[3] Napolitano, AIAA J. 24 pp 2040– (1986)
[4] Morrison, Comput. Fluids 16 pp 119– (1988)
[5] Khosla, Comput. Fluids 2 pp 207– (1974)
[6] Beam, AIAA J. 16 pp 393– (1978)
[7] Napolitano, Comput. Fluids
[8] Fasel, J. Fluid Mech. 78 pp 355– (1976)
[9] Dennis, J. Comput. Phys. 33 pp 325– (1979)
[10] ’A third-order-accurate upwind scheme for Navier-Stokes solutions in three dimensions’, presented at ASME Winter Annual Meeting, Washington, DC, 15-20 November 1981.
[11] Farouk, Int. j. numer. methods fluids 5 pp 1017– (1985)
[12] Orlandi, Comput. Fluids 15 pp 137– (1987)
[13] and , ’A direct algorithm for solution of incompressible three-dimensional unsteady Navier-Stokes equations’, Proc. AIAA 8th Computational Fluid Dynamics Conf., Honolulu, HW, June 1987, AIAA, New York, 1987, pp. 408-421.
[14] Guj, Int. j. numer. methods fluids 8 pp 405– (1988)
[15] Gunzburger, Int. j. numer. methods fluids 8 pp 1229– (1988)
[16] and , ’Numerical solutions to the Navier-Stokes equations in vorticity-velocity form’, Proc. Third Italian Meeting of Computational Mechanics, Palermo, 7-10 June 1988, pp. 95-101.
[17] Napolitano, Notes Numer. Fluid Mech. 29 pp 430– (1990)
[18] Napolitano, Comput. Fluids.
[19] Burggraf, J. Fluid Mech. 24 pp 113– (1966)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.