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Viscous–inviscid interactions in transonic flows through slender nozzles

Published online by Cambridge University Press:  18 February 2011

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
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Abstract

Considering the miniaturization trend in technical applications, the need of a slender nozzle theory for such conventional, that is ideal-gas-like, fluids, which accounts for a strong boundary-layer interaction with the core region, arises in quite a natural way as the dimensions of the flow device are successively reduced. Moreover, a number of modern technological processes (e.g. organic Rankine cycles) involve fluids with high molecular complexity, some of which are expected to exhibit embedded regions with negative values of the fundamental derivative in the vapour phase commonly termed Bethe–Zel'dovich–Thompson (BZT) fluids. Linked to it, unconventional Laval nozzle geometries are needed to transform subsonic to supersonic internal flows. In the present paper, the transonic flows through nozzles of short length scales located in a channel of constant cross-section so slender that the flow in the inviscid core region is one-dimensional are considered. Rapid streamwise changes of the flow field caused by the nozzle then lead to a local breakdown of the classical hierarchical boundary-layer approach, which is overcome by the triple-deck concept. Consequently, the properties of the inviscid core and the near-wall (laminar) boundary-layer regions have to be calculated simultaneously. The resulting problem is formulated for both regular (ideal-gas-like) fluids and dense gases. Differences and similarities of the resulting flow pattern with respect to the well-known classical Laval nozzle flow are presented for perfect gases, and the regularizing influence of viscous–inviscid interactions, is examined. Furthermore, the analogous problem is considered for BZT fluids in detail as well. The results indicate that the passage through the sonic point in the inviscid core is strongly affected by the combined influence of nozzle geometry and boundary-layer displacement effects suggesting in turn an inverse Laval nozzle design in order to obtain the desired flow behaviour.

JFM classification

Boundary Layers: Boundary Layers Complex Fluids: Complex Fluids Compressible Flows: Compressible Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 672 , 10 April 2011 , pp. 487 - 520
Copyright
Copyright © Cambridge University Press 2011

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References

REFERENCES

Brown, B. P. & Argrow, B. M. 2000 Application of Bethe–Zel'dovich–Thompson fluids in organic Rankine cycle engines. J. Propul. Power 16 (6), 1118–1124.CrossRefGoogle Scholar
Cinnella, P. & Congedo, P. M. 2007 Inviscid and viscous aerodynamics of dense gases. J. Fluid Mech. 580, 179–217.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.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
Guardone, A., Zamfirescu, C. & Colonna, P. 2010 Maximum intensity of rarefaction shock waves for dense gases. J. Fluid Mech. 642, 127–146.CrossRefGoogle Scholar
Hao, P.-F., Ding, Y.-T., Yao, Z.-H., Feng, H. & Zhu, K.-Q. 2005 Size effect on gas flow in micro nozzles. J. Micromech. Microengng 15, 2069–2073.CrossRefGoogle Scholar
Kluwick, A. 1993 Transonic nozzle flow of dense gases. J. Fluid Mech. 247, 661–688.CrossRefGoogle 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.CrossRefGoogle Scholar
Kluwick, A. 2004 Internal flows of dense gases. Acta Mechanica 196, 123–143.CrossRefGoogle Scholar
Kluwick, A. 2009 Transonic flows in narrow channels. J. Therm. Sci. (invited) 18 (2), 99–108. Dedicated to Prof. Dr.-Ing. Dr.techn. E.h. Jürgen Zierep on the occasion of his 80th birthday.CrossRefGoogle Scholar
Kluwick, A., Exner, A., Cox, E. A. & Grinschgl, Ch. 2010 On the internal structure of weakly nonlinear bores in laminar high Reynolds number flow. Acta Mechanica 210 (1–2), 135–157.CrossRefGoogle Scholar
Kluwick, A. & Meyer, G. 2010 Shock regularization in dense gases by viscous–inviscid interactions. J. Fluid Mech. 644, 473–507.CrossRefGoogle Scholar
Lighthill, M. J. 2000 Upstream influence in boundary layers 45 years ago. Phil. Trans. R. Soc.: Math. Phys. Engng Sci. 358 (1777), 3047–3062.CrossRefGoogle Scholar
Matsuo, K., Miyazato, Y. & Kim, H. 1999 Shock train and pseudo-shock phenomena in internal gas flows. Prog. Aerosp. Sci. 35 (1), 33–100.CrossRefGoogle Scholar
Oswatitsch, K. 1952 Gas Dynamics. Academic Press.Google Scholar
Oswatitsch, K. & Wieghardt, K. 1946 Theoretical investigations on steady potential flows and boundary layers at high speed. Rep. 10378. Aeronaut. Res. Council, London.Google Scholar
Powell, M. 1970 A hybrid method for nonlinear equations. In Numerical Methods for Nonlinear Algebraic Equations (ed. Rabinowitz, P.), chap. 6, pp. 115–161. Gordon and Breach.Google Scholar
Prandtl, L. 1938 Zur Berechnung der Grenzschichten. Z. Angew. Math. Mech. 18, 77–82.CrossRefGoogle Scholar
Reyhner, T. A. & Flügge-Lotz, I. 1968 Interaction of a shock wave with a laminar boundary layer. Intl J. Nonlinear Mech. 3, 173–199.CrossRefGoogle Scholar
Schenk, O. & Gärtner, K. 2004 Solving unsymmetric sparse systems of linear equations with PARDISO. J. Future Gener. Comput. Syst. 20 (3), 475–487.CrossRefGoogle Scholar
Schenk, O. & Gärtner, K. 2006 On fast factorization pivoting methods for symmetric indefinite systems. Elec. Trans. Numer. Anal. 23, 158–179.Google Scholar
Schenk, O., Gärtner, K. & Fichtner, W. 2000 Efficient sparse LU factorisation with left–right looking strategy on shared multiprocessors. BIT 40 (1), 158–176.CrossRefGoogle Scholar
Seydel, R. & Hlavacek, V. 1987 Role of continuation in engineering analysis. Chem. Engng Sci. 42 (6), 1281–1295.CrossRefGoogle Scholar
Stewartson, K. 1974 Multistructured boundary layers on flat plates and related bodies. Adv. Appl. Mech. 14, 145–239.CrossRefGoogle Scholar
Stewartson, K. 1981 D'Alembert's paradox. SIAM Rev. 23 (3), 308–343.CrossRefGoogle Scholar
Stoer, J. & Bulirsch, R. 2002 Introduction to Numerical Analysis, 3rd edn. Texts in Applied Mathematics, vol. 12. Springer.CrossRefGoogle Scholar
Williams, J. C. III 1963 Viscous compressible and incompressible flow in slender channels. AAIA J. 1 (1), 186–195.CrossRefGoogle Scholar