×

Some uses of Green’s theorem in solving the Navier-Stokes equations. (English) Zbl 0695.76017

Summary: This paper gives a review of methods where Green’s theorem may be employed in solving numerically the Navier-Stokes equations for incompressible fluid motion. They are based on the concept of using the theorem to transform local boundary conditions given on the boundary of a closed region in the solution domain into global, or integral, conditions taken over it. Two formulations of the Navier-Stokes equations are considered: that in terms of the streamfunction and vorticity for two- dimensional motion and that in terms of the primitive variables of the velocity components and the pressure. In the first formulation overspecification of conditions for the streamfunction is utilized to obtain conditions of integral type for the vorticity and in the second formulation integral conditions for the pressure are found. Some illustrations of the principle of the method are given in one space dimension, including some derived from two-dimensional flows using the series truncation method. In particular, an illustration is given of the calculation of surface vorticity for two-dimensional flow normal to a flat plate. An account is also given of the implementation of these methods for general two-dimensional flows in both of the mentioned formulations and a numerical illustration is given.

MSC:

76D05 Navier-Stokes equations for incompressible viscous fluids
35Q30 Navier-Stokes equations
76M99 Basic methods in fluid mechanics
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] and , ’Boundary integral equation analyses of singular, potential and biharmonic problems’. Lecture Notes in Engineering, Vol. 7, Springer-Verlag, Berlin, 1984. · Zbl 0553.76001
[2] and , ’Numerical integration of the Navier-Stokes equations’, Mathematics Research Center, University of Wisconsin, Technical Summary Report 859, 1969.
[3] Woods, Aero. Q. 5 pp 176– (1954)
[4] Dennis, J. Comput. Phys. 52 pp 448– (1983)
[5] ’Compact explicit finite-difference approximations to the Navier-Stokes equations’, i. Lecture Notes in Physics, Vol. 218, Springer-Verlag, Berlin, 1985, pp. 23-36.
[6] and , ’The steady flow of a viscous fluid past a circular cylinder’, Brit. Aero. Res. Council, Current Paper 797, 1965.
[7] Dennis, J. Fluid Mech. 24 pp 577– (1966)
[8] and , ’Numerical integration of the Navier-Stokes equations for steady two-dimensional flows’, Phys. Fluids, Suppl. II, 88-93 (1969). · Zbl 0208.55009
[9] Dennis, J. Fluid Mech. 42 pp 471– (1970)
[10] Dennis, J. Fluid Mech. 48 pp 771– (1971)
[11] Dennis, J. Comput. Phys. 28 pp 297– (1978)
[12] Dennis, J. Fluid Mech. 101 pp 257– (1980)
[13] Dennis, Q. J. Mech. Appl. Math. 35 pp 305– (1982)
[14] Dennis, J. Eng. Math. 5 pp 263– (1971)
[15] Dennis, Phys. Fluids 15 pp 517– (1972)
[16] Collins, Q. J. Mech. Appl. Math. 26 pp 53– (1973)
[17] Collins, J. Fluid Mech. 60 pp 105– (1973)
[18] Badr, J. Fluid Mech. 158 pp 447– (1985)
[19] Anwar, Comput. Fluids 16 pp 1– (1988)
[20] Quartapelle, J. Comput. Phys. 40 pp 453– (1981)
[21] Quartapelle, Int. j. numer. methods fluids 1 pp 129– (1981)
[22] Dennis, J. Comput. Phys. 61 pp 218– (1985)
[23] Glowinski, SIAM Rev. 12 pp 167– (1979)
[24] Quartapelle, Int. j. numer. methods fluids 4 pp 109– (1984)
[25] Dennis, Comput. Fluids 12 pp 77– (1984)
[26] and , ’Rigorous numerical treatment of the no-slip condition in a vorticity formulation’, Numerical Boundary Condition Procedures, NASA Report, Moffett Field, CA, 1981, pp. 367-373.
[27] Davis, J. Fluid Mech. 51 pp 417– (1971)
[28] Campion-Renson, Int. j. numer. methods eng. 12 pp 1809– (1978)
[29] McLaurin, SIAM J. Numer. Anal. 11 pp 14– (1974)
[30] and , ’Treatment of incompressibility and boundary conditions in 3-D numerical spectral simulations of plane channel flows’, in Notes on Numerical Fluid Mechanics, Vieweg, Braunschweig, 1980, pp. 165-173.
[31] Quartapelle, J. Comput. Phys. 62 pp 340– (1986)
[32] Schumann, Z. Angew. Math. Mech. 64 pp t227– (1984)
[33] Patera, J. Comput. Phys. 54 pp 468– (1984)
[34] Glowinski, C. R. Acad. Sci. Paris Sér. A 286 pp 181– (1978)
[35] and , ’Steady flow normal to a flat plate at moderate Reynolds numbers’, to be published (1989).
[36] Loc, J. Fluid Mech. 100 pp 111– (1980)
[37] Loc, J. Fluid Mech. 160 pp 93– (1985)
[38] (ed.). Laminar Boundary Layers, Clarendon Press, Oxford, 1963.
[39] Dennis, Proc. Roy. Soc. 372 pp 257– (1980) · Zbl 0455.76037 · doi:10.1098/rspa.1980.0119
[40] Mills, J. Fluid Mech. 79 pp 609– (1977)
[41] Wang, AIAA J. 24 pp 1305– (1986)
[42] ’Application of the series truncation method to two-dimensional internal flows’, i. Lecture Notes in Physics, Vol. 35, Springer-Verlag, Berlin, 1975, pp. 138-143.
[43] Cerutti, Int. j. numer. methods fluids 6 pp 715– (1986)
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.