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Calculation of the steady flow through a curved tube using a new finite-difference method

Published online by Cambridge University Press:  19 April 2006

S. C. R. Dennis
S. C. R. Dennis
Affiliation:
Department of Applied Mathematics, University of Western Ontario, London, Canada
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Abstract

A numerical method is described which is suitable for solving the equations governing the steady motion of a viscous fluid through a slightly curved tube of circular cross-section but which is also applicable to the solution of any problem governed by the steady two-dimensional Navier–Stokes equations in the plane polar co-ordinate system. The governing equations are approximated by a scheme which yields finite-difference equations which are of second-order accuracy with respect to the grid sizes but which have associated matrices which are diagonally dominant. This makes them generally more amenable to solution by iterative techniques than the approximations obtained using standard central differences, while preserving the same order of accuracy.

The main object of the investigation is to obtain numerical results for the problem of steady flow through a curved tube which corroborate previous numerical work on this problem in view of a recent paper (Van Dyke 1978) which tends to cast doubt on the accuracy of previous calculations at moderately high values of the Dean number; this is the appropriate Reynolds-number parameter in this problem. The present calculations tend to verify the accuracy of previous results for Dean numbers up to 5000, beyond which it is difficult to obtain accurate results. Calculated properties of the flow are compared with those obtained in previous numerical work, with the predictions of boundary-layer theory for large Dean numbers and with the predictions of Van Dyke (1978).

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 99 , Issue 3 , 14 August 1980 , pp. 449 - 467
Copyright
© 1980 Cambridge University Press

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References

Adler, M. 1934 Z. angew. Math. Mech. 14, 257.
Akiyama, M. & Cheng, K. C. 1971 Int. J. Heat Mass Transfer 14, 1659.
Austin, L. R. & Seader, J. D. 1973 A.I.Ch.E. J. 19, 85.
Barua, S. N. 1963 Quart. J. Mech. Appl. Math. 16, 61.
Collins, W. M. & Dennis, S. C. R. 1975 Quart. J. Mech. Appl. Math. 28, 133.
Collins, W. M. & Dennis, S. C. R. 1976a J. Fluid Mech. 76, 417.
Collins, W. M. & Dennis, S. C. R. 1976b Proc. Roy. Soc. A 352, 189.
Dean, W. R. 1927 Phil. Mag. 4, 208.
Dean, W. R. 1928 Phil. Mag. 5, 673.
Dennis, S. C. R. 1960 Quart. J. Mech. Appl. Math. 13, 487.
Dennis, S. C. R. & Hudson, J. D. 1978 Proc. 1st Int. Conf. on Numerical Methods in Laminar and Turbulent Flow, p. 69. London: Pentech Press.
Dennis, S. C. R. & Ng, M. 1980 22nd British Theoretical Mech. Coll., Univ. of Cambridge.
Greenspan, D. 1973 J. Fluid Mech. 57, 167.
Hasson, D. 1955 Res. Correspondence 1, 51.
Ito, H. 1959 Trans. A.S.M.E. D, J. Basic Engng 81, 123.
Ito, H. 1969 Z. angew. Math. Mech. 49, 653.
Larrain, J. & Bonilla, C. F. 1970 Trans. Soc. Reol. 14, 135.
McConalogue, D. J. & Srivastava, R. S. 1968 Proc. Roy. Soc. A 307, 37.
Mori, Y. & Nakayama, W. 1965 Int. J. Heat Mass Transfer 8, 67.
Patankar, S. V., Pratap, V. S. & Spalding, D. B. 1974 J. Fluid Mech. 62, 539.
Smith, F. T. 1975 J. Fluid Mech. 71, 15.
Smith, F. T. 1976 Proc. Roy. Soc. A 347, 345.
Taylor, G. I. 1929 Proc. Roy. Soc. A 124, 243.
Topakoglu, H. C. 1967 J. Math. Mech. 16, 1321.
Truesdell, L. C. & Adler, R. J. 1970 A.I.Ch.E.J. 16, 1010.
Van Dyke, M. 1978 J. Fluid Mech. 86, 129.
Varga, R. S. 1962 Matrix Iterative Analysis, Prentice-Hall.
White, C. M. 1929 Proc. Roy. Soc. A 123, 645.
Woods, L. C. 1954 Aero. Quart. 5, 176.