Summary
The one-dimensional equations for transient two-phase flow are a system of nonlinear hyperbolic partial differential equations, expressible, under certain assumptions, in conservation form. Inasmuch as the use of the method of characteristics becomes complicated if shock waves are present, it is easier to follow a gas-dynamics approach and employ one of the available procedures for solving one-dimensional systems of conservation equations. A recently introduced technique, due to McGuire and Morris [1, see also 12] and known as an Explicit-Implicit method, is used here for a simple boundary-value problem of wave propagation in bubbly two-phase mixtures, and is found to be simple and versatile. A comparison of this method with the well-known Lax-Wendroff (two-step) scheme demonstrates that shock fronts are simulated better, oscillations behind the shocks are smoothable by parameter adjustment, and computation time is reduced when the Explicit-Implicit method is employed.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
G. R. McGuire and J. L. Morris, Explicit-implicit schemes for the numerical solution of nonlinear hyperbolic systems, Math. of Comp. 29, 130 (1975) 407–424.
A. Prosperetti and L. van Wijngaarden, On the characteristics of the equation of motion for bubbly flow and the related problem of critical flow, J. Eng. Math., 10, 1976.
R. W. Lyczokowski, D. Gidaspow, C. W. Solbrig and F. D. Hughes, Characteristics and stability analyses of transient one-dimensional two-phase flow equations, Am. Soc. Mech. Engrs., Paper No. 75-WA/HT-23, 1975.
C. Kranenburg, Gas release during transient cavitation, J. Hyd. Div. Am. Soc. Civil Engrs., 100, 10 (1974) 1383–1398.
C. S. Martin, M. Padmanabhan and D. C. Wiggert, Pressure wave propagation in two-phase bubbly air-water mixtures, Second International Conference on Pressure Surges, City Univ., London, 1976, Paper C1.
C. S. Martin and M. Padmanabhan, The effect of free gases on pressure transients, Report submitted to NSF, Research Grant GK-38570, June, 1976.
P. D. Lax and B. Wendroff, Systems of conservation laws, Comm. Pure and Appl. Math., XIII (1960), 217–237.
A. C. Vliegenthart, The Shuman filtering operator and the numerical computation of shock waves, J. Eng. Math., 4, 4 (1970) 341–348.
G. R. McGuire and J. L. Morris, A class of second-order accurate methods for the solution of systems of conservation laws, J. Comp. Phys., 11 (1973) 531–549.
G. R. McGuire and J. L. Morris, A class of implicit, second-order accurate, dissipative schemes for solving systems of conservation laws, J. Comp. Phys., 14 (1974) 126-pp147.
W. F. Ames, Nonlinear partial differential equations in engineering, Academic Press, New York, 1965.
W. F. Ames, Numerical methods for partial differential equations, 2nd Edition, Thomas Nelson-London; Academic Press-New York, 1977.
R. D. Richtmyer and K. W. Morton, Difference methods for initial value problems, Interscience Publications, New York, 1967.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Padmanabhan, M., Ames, W.F. & Martin, C.S. Numerical analysis of pressure transients in bubbly two-phase mixtures by explicit-implicit methods. J Eng Math 12, 83–93 (1978). https://doi.org/10.1007/BF00042806
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00042806