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Transonic flows of Bethe—Zel'dovich—Thompson fluids

Published online by Cambridge University Press:  26 April 2006

M. S. Cramer  and
G. M. Tarkenton
M. S. Cramer
Affiliation:
Department of Engineering Science and Mechanics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061-0219, USA
G. M. Tarkenton
Affiliation:
Department of Physics, University of Texas at Austin, Austin, TX 78741, USA
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Abstract

Bethe–Zel'dovich–Thompson fluids are ordinary, single-phase fluids in which the fundamental derivative of gasdynamics is negative over a finite range of temperatures and pressures. We examine the steady transonic flow of these fluids over two-dimensional thin wings and turbine blades. The free-stream state is taken to be in the neighbourhood of one of the zeros of the fundamental derivative. It is shown that a modified form of the classical transonic small-disturbance equation governs such flows. Critical Mach number estimates are provided which take into account the non-monotone variation of the Mach number with pressure predicted in previous investigations. Critical Mach numbers well over 0.95 are predicted for conventional airfoil sections. Numerical solutions reveal substantial reductions in the strength of the compression shocks occurring in supercritical flows. Further new results include the prediction of expansion and compression shocks in the same flow and compression bow shocks in flows with subsonic free streams.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 240 , July 1992 , pp. 197 - 228
Copyright
© 1992 Cambridge University Press

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References

Abbott, R. H. & Doenhoff, A. E. 1959 Theory of Wing Sections. Dover.
Anderson, J. D. 1990 Modern Compressible Flow with Historical Perspective. McGraw-Hill.
Bethe, H. A. 1942 The theory of shock waves for an arbitrary equation of state. Office Sci. Res. & Dev. Rep. 545.Google Scholar
Chandrasekar, D. & Prasad, P. 1991 Transonic flow of a fluid with positive and negative nonlinearity through a nozzle. Phys. Fluids A3, 427–438.Google Scholar
Cole, J. D. & Cook, L. P. 1986 Transonic Aerodynamics. North Holland.
Cramer, M. S. 1987 Structure of weak shocks in fluids having embedded regions of negative nonlinearity. Phys. Fluids 30, 3034–3044.Google Scholar
Cramer, M. S. 1989a Negative nonlinearity in selected fluorocarbons. Phys. Fluids A1, 1894–1897.Google Scholar
Cramer, M. S. 1989b Shock splitting in single-phase gases. J. Fluid Mech. 199, 281–296.Google Scholar
Cramer, M. S. 1991 Nonclassical dynamics of classical gases. In Nonlinear Waves in Real Fluids (ed. A. Kluwick), pp. 91–145. Springer.
Cramer, M. S. & Best, L. M. 1991 Steady isentropic flows of dense gases. Phys. Fluids A3, 219–226.Google Scholar
Cramer, M. S. & Crickenberger, A. B. 1991 The dissipative structure of shock waves in dense gases. J. Fluid Mech. 223, 325–355.Google Scholar
Cramer, M. S. & Kluwick, A. 1984 On the propagation of waves exhibiting both positive and negative nonlinearity. J. Fluid Mech. 142, 9–37.Google Scholar
Cramer, M. S., Kluwick, A., Watson, L. T. & Pelz, W. 1986 Dissipative waves in fluids having both positive and negative nonlinearity. J. Fluid Mech. 169, 323–336.Google Scholar
Cramer, M. S. & Sen, R. 1987 Exact solutions for sonic shocks in van der Waals gases. Phys. Fluids 30, 377–385.Google Scholar
Duhem, P. 1909 Sur la propagation der ondes de choc au sein des fluides. Z. Phys. Chem, Leipzig 69, 169–186.Google Scholar
Fletcher, C. A. G. 1988 Computational Techniques for Fluid Dynamics. Springer.
Hayes, W. D. 1966 La seconde approximation pour les écoulements transsoniques non visqueux. J. Méc. 5, 163–206.Google Scholar
Kevorkian, J. & Cole, J. D. 1981 Perturbation Methods in Applied Mathematics. Springer.
Kluwick, A. & Koller, F. 1988 Ausbrietung periodischer Wellen kleiner Amplitude in Gasen mit großen specifischen Wärmen. Z. Angew. Math. Mech. 68, T306–T307.Google Scholar
Lambrakis, K. & Thompson, P. A. 1972 Existence of real fluids with a negative fundamental derivative Phys. Fluids 5, 933–935.Google Scholar
Landau, L. D. & Lifshitz, E. M. 1959 Fluid Mechanics. Addison-Wesley.
Leidner, P. 1990 Realgaseinflüsse in der Gasdynamik. Diplomarbeit, Universtät Karlsruhe.
Martin, J. J. & Hou, Y. C. 1955 Development of an equation of state for gases. AIChE J. 1, 142–151.Google Scholar
Menikoff, R. & Plohr, B. 1989 Riemann problem for fluid flow of real materials. Rev. Mod. Phys. 61, 75–130.Google Scholar
Morren, S. H. 1991 Transonic aerodynamics of dense gases. NASA TM 103732.Google Scholar
Murman, E. M. 1973 Analysis of embedded shock waves calculated by relaxation methods. Proc. 1st AIAA Comp. Fluid. Dyn. Conf., pp. 27–40.Google Scholar
Murman, E. M. & Cole, J. D. 1971 Calculation of plane steady transonic flow. AIAA J. 9, 114–121.Google Scholar
Nieuwland, G. Y. & Spee, B. M. 1973 Transonic airfoils: Recent development in theory, experiment, and design. Ann. Rev. Fluid Mech. 5, 119–150.Google Scholar
Pego, R. L. 1986 Nonexistence of a shock layer in gas dynamics with a nonconvex equation of state. Arch. Rat. Mech. Anal. 94, 165–178.Google Scholar
Reid, R. C., Prausnitz, J. M. & Poling, B. E. 1987 The Properties of Gases and Liquids, 4th Ed, Wiley.
Sobieczky, H. & Seebass, A. R. 1984 Supercritical airfoil and wing design. Ann. Rev. Fluid Mech. 16, 337–363.Google Scholar
Thompson, P. A. 1971 A fundamental derivative in gasdynamics. Phys. Fluids 14, 1843–1849.Google Scholar
Thompson, P. A. & Lambrakis, K. 1973 Negative shock waves. J. Fluid Mech. 60, 187–208.Google Scholar
Warner, S. M. 1990 Steady non-isentropic flows of dense gases, Virginia Polytechnic Institute and State University Engineering Rep. VPI-E-90–24.
Wendroff, B. 1972 The Riemann problem for materials with nonconvex equations of state II: General flow. J. Math. Anal. Applics 38, 640–658.Google Scholar
Zel'Dovich, Ya. B. 1946 On the possibility of rarefaction shock waves. Zh. Eksp. Teor. Fiz. 4, 363–364.Google Scholar