Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-27T12:30:35.406Z Has data issue: false hasContentIssue false
  • English
  • Français

Shock regularization in dense gases by viscous–inviscid interactions

Published online by Cambridge University Press:  11 February 2010

A. KLUWICK  and
G. MEYER
A. KLUWICK*
Affiliation:
Institute of Fluid Mechanics and Heat Transfer, Vienna University of Technology, Resselgasse 3/322, 1040 Vienna, Austria
G. MEYER
Affiliation:
Institute of Fluid Mechanics and Heat Transfer, Vienna University of Technology, Resselgasse 3/322, 1040 Vienna, Austria
*
†Email address for correspondence: alfred.kluwick@tuwien.ac.at
Get access Rights & Permissions [Opens in a new window]

Abstract

Transonic high-Reynolds-number flows through channels which are so narrow that the classical boundary-layer approach fails locally are considered in the presence of a weak stationary normal shock. As a consequence, the properties of the inviscid core and the viscosity-dominated boundary-layer region can no longer be determined in subsequent steps but have to be calculated simultaneously in a small interaction region. Under the requirement that the core-region flow should be considered to be one-dimensional to the leading order the resulting problem of shock–boundary-layer interaction is formulated by the means of matched asymptotic expansions for laminar flows of dense gases (Bethe–Zel'dovich–Thompson, or BZT, fluids). Such fluids have the distinguishing feature that the fundamental derivative of gas dynamics can become negative or even change sign under the thermodynamic conditions to be considered. The regularizing properties of the mechanism of viscous–inviscid interactions on the different anomalous shock forms possible in the flow of dense gases with mixed nonlinearity, namely rarefaction, sonic, double-sonic and split shocks, will be discussed. To this end we show the consistency of the resulting internal-shock profiles because of strong shock–boundary-layer interaction with a generalized shock admissibility criterion formulated for the case of purely inviscid flows. Representative solutions for the internal-shock structures are presented, and the importance of such flow phenomena in technical applications in the near future are shortly discussed by considering estimates of the actual dimensions of the interaction region for a specific representative situation in which the BZT fluid PP10 (C13F22) has been selected.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 644 , 10 February 2010 , pp. 473 - 507
Copyright
Copyright © Cambridge University Press 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Abramowitz, M. & Stegun, I. A. 1970 Handbook of Mathematical Functions, 7th edn. Dover.Google Scholar
Anderson, D. A., Tannehill, J. C. & Pletcher, R. H. 1997 Computational Fluid Mechanics and Heat Transfer, 2nd edn. Taylor & Francis.Google Scholar
Bethe, H. A. 1942 The theory of shock waves for an arbitrary equation of state. Report 545. Office of Scientific Research & Development.Google Scholar
Chandrasekar, D. & Prasad, P. 1991 Transonic flow of a fluid with positive and negative nonlinearity through a nozzle. Phys. Fluids A 3 (3), 427–438.CrossRefGoogle Scholar
Chung, T., Ajlan, M., Lee, L. L. & Starling, K. E. 1988 Generalized multiparameter correlation for nonpolar and polar fluid transport properties. Indus. Engng Chem. Res. 27, 671–678.CrossRefGoogle Scholar
Cinnella, P. & Congedo, P. M. 2007 Inviscid and viscous aerodynamics of dense gases. J. Fluid Mech. 580, 179–217.CrossRefGoogle Scholar
Colonna, P. & Guardone, A. 2006 Molecular interpretation of nonclassical gasdynamics of dense vapours under the van der Waals model. Phys. Fluids 18, 056101.CrossRefGoogle Scholar
Colonna, P., Guardone, A. & Nannan, N. R. 2007 Siloxanes: a new class of candidate Bethe–Zel'dovich–Thompson fluids. Phys. Fluids 19, 086102.CrossRefGoogle Scholar
Colonna, P. & Silva, P. 2003 Dense gas thermodynamic properties of single and multicomponent fluids for fluid dynamics simulations. J. Fluids Engng 125, 414–427.CrossRefGoogle Scholar
Cramer, M. & Kluwick, A. 1984 On the propagation of waves exhibiting both positive and negative nonlinearity. J. Fluid Mech. 142, 9–37.CrossRefGoogle Scholar
Cramer, M. S. 1989 Negative nonlinearity in selected fluorocarbons. Phys. Fluids A 1 (11), 1894–1897.CrossRefGoogle Scholar
Cramer, M. S. 1991 Nonclassical dynamics of classical gases. In Nonlinear Waves in Real Fluids, CISM Courses and Lectures No. 315 (ed. Kluwick, A.), pp. 91–145. Springer Wien.CrossRefGoogle Scholar
Cramer, M. S. & Crickenberger, A. B. 1991 The dissipative structure of shock waves in dense gases. J. Fluid Mech. 223, 325–355.CrossRefGoogle Scholar
Cramer, M. S. & Sen, R. 1986 Exact solutions for sonic shocks in van der Waals gases. Phys. Fluids 30 (2), 377–385.CrossRefGoogle Scholar
Fergason, S. H., Ho, T. L., Argrow, B. M. & Emanuel, G. 2001 Theory for producing a single-phase shock wave. J. Fluid Mech. 445, 37–54.CrossRefGoogle Scholar
Gittler, Ph. 1992 On similarity solutions occurring in the theory of interactive laminar boundary layers. J. Fluid Mech. 244, 131–147.CrossRefGoogle Scholar
Guardone, A. & Argrow, B. M. 2005 Nonclassical gasdynamic region of selected fluorocarbons. Phys. Fluids 17, 116102.CrossRefGoogle Scholar
Kluwick, A. 1993 Transonic nozzle flow of dense gases. J. Fluid Mech. 247, 661–688.CrossRefGoogle Scholar
Kluwick, A. 1994 Interacting laminar boundary layers of dense gases. Acta Mech. 4 (Suppl.), 335–349.Google Scholar
Kluwick, A. 1998 Interacting laminar and turbulent boundary layers. In Recent Advances in Boundary Layer Theory, CISM Courses and Lectures No. 390 (ed. Kluwick, A.), pp. 231–330. Springer Wien.CrossRefGoogle Scholar
Kluwick, A. 2004 Internal flows of dense gases. Acta Mech. 196, 123–143.CrossRefGoogle Scholar
Kluwick, A. & Bodonyi, R. J. 1979 Freely interacting laminar boundary layers in channels. ZAMM 59, T243–T245.Google Scholar
Kluwick, A., Exner, A., Cox, E. A. & Grinschgl, Ch. 2009 On the internal structure of weakly nonlinear bores in laminar high Reynolds number flow. Acta Mech. doi:10.1007/s00707-009-0188-x.CrossRefGoogle Scholar
Kluwick, A. & Scheichl, St. 1996 Unsteady transonic nozzle flow in dense gases. J. Fluid Mech. 310, 113–137.CrossRefGoogle Scholar
Kluwick, A., Scheichl, S. & Cox, E. A. 2007 Near-critical hydraulic flows in two-layer fluids. J. Fluid Mech. 575, 187–219.CrossRefGoogle Scholar
Lighthill, M. J. 1953 On boundary layers and upstream influence. Part II. Supersonic flows without separation. Proc. R. Soc. A 217, 478–504.Google Scholar
Martin, J. J. & Hou, Y. 1955 Development of an equation of state for gases. AIChE J. 1 (2), 142–151.CrossRefGoogle Scholar
Matsuo, K., Miyazato, Y. & Kim, H. 1999 Shock train and pseudo-shock phenomena in internal gas flows. Progr. Aerosp. Sci. 35 (1), 33–100.CrossRefGoogle Scholar
Menikoff, M. & Plohr, B. J. 1989 The Riemann problem for fluid flow of real materials. Rev. Mod. Phys. 61 (1), 75–130.CrossRefGoogle Scholar
Oleinik, O. A. 1957 Discontinuous solutions of nonlinear differential equations. Am. Math. Soc. Trans. 26, 95–172.Google Scholar
Reyhner, T. A. & Flügge-Lotz, I. 1968 Interaction of a shock wave with a laminar boundary layer. Intl J. Nonlin. Mech. 3, 173–199.CrossRefGoogle Scholar
Stewartson, K. 1974 Multistructured boundary layers on flat plates and related bodies. Adv. Appl. Mech. 14, 145–239.CrossRefGoogle Scholar
Stewartson, K. & Williams, P. G. 1969 Self-induced separation. Proc. R. Soc. A 312, 181–206.Google Scholar
Thompson, P. A. 1971 A fundamental derivative in gasdynamics. Phys. Fluids 14, 1843–1849CrossRefGoogle Scholar
Thompson, P. A., Chaves, H., Meier, G. E. A., Kim, Y. & Speckmann, H.-D. 1987 Wave splitting in a fluid of large heat capacity. J. Fluid Mech. 185, 385–414.CrossRefGoogle Scholar
Thompson, P. A. & Lambrakis, K. C. 1973 Negative shock waves. J. Fluid Mech. 60 (1), 187—208.CrossRefGoogle Scholar
Zamfirescu, C., Guardone, A. & Colonna, P. 2008 Admissibility region for rarefaction shock waves in dense gases. J. Fluid Mech. 599, 363–381.CrossRefGoogle Scholar
Zel'dovich, Y. B. 1946 On the possibility of rarefaction shock waves. Zh. Eksp. Teoret. Fiz. 4, 363–364.Google Scholar