Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-29T05:11:12.056Z Has data issue: false hasContentIssue false
  • English
  • Français

Shock interactions in two-dimensional steady flows of Bethe–Zel’dovich–Thompson fluids

Published online by Cambridge University Press:  23 January 2020

Davide Vimercati ,
Alfred Kluwick  and
Alberto Guardone Open the ORCID record for Alberto Guardone [Opens in a new window]
Davide Vimercati
Affiliation:
Department of Aerospace Science and Technology, Politecnico di Milano, Via La Masa 34, 20156Milano, Italy
Alfred Kluwick
Affiliation:
Institute of Fluid Mechanics and Heat Transfer, Vienna University of Technology, Getreidemarkt 9, 1060Vienna, Austria
Alberto Guardone*
Affiliation:
Department of Aerospace Science and Technology, Politecnico di Milano, Via La Masa 34, 20156Milano, Italy
*
†Email address for correspondence: alberto.guardone@polimi.it
Get access Rights & Permissions [Opens in a new window]

Abstract

The morphology of nodes generated by the interaction of discontinuities in steady two-dimensional inviscid flows is examined. The fluids considered are Bethe–Zel’dovich–Thompson (BZT) fluids, which feature negative values of the fundamental derivative of gas dynamics in the vapour phase. The operating conditions correspond to the non-classical gas-dynamic regime where expansion shocks, compression fans and composite waves are admissible in addition to the classical compression shocks and expansion fans. Interactions caused by the crossing, overtaking and splitting of compression/expansion shocks, along with the refraction of these through a contact discontinuity, are analysed here. The well-established method of wave curves is applied to non-classical wave curves, revealing a variety of interaction patterns that are simply not admissible in classical gas dynamics. It is shown that shock waves can be reflected, transmitted and refracted as Prandtl–Meyer fans or composite waves. Based on numerical evidence, the splitting (and consequently the Mach reflection) of an expansion shock seems to be disallowed. Theoretical considerations on the admissibility of such configurations are also provided. The present analysis is relevant to applications potentially involving supersonic flows of BZT fluids, e.g. organic Rankine cycle power systems, and can also be used in front-tracking algorithms for general equations of state.

JFM classification

Compressible Flows: Gas dynamics Compressible Flows: Shock waves
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 887 , 25 March 2020 , A12
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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

Alferez, N. & Touber, E. 2017 One-dimensional refraction properties of compression shocks in non-ideal gases. J. Fluid Mech. 814, 185–221.CrossRefGoogle Scholar
Ben-Dor, G. 2007 Shock Wave Reflection Phenomena. Springer.Google Scholar
Callen, H. B. 1985 Thermodynamics and An Introduction to Thermostatistics, 2nd edn. Wiley.Google Scholar
Colonna, P. & Guardone, A. 2006 Molecular interpretation of nonclassical gasdynamics of dense vapors under the van der Waals model. Phys. Fluids 18 (5), 056101-1-14.CrossRefGoogle Scholar
Colonna, P., Guardone, A. & Nannan, N. R. 2007 Siloxanes: a new class of candidate Bethe–Zel’dovich–Thompson fluids. Phys. Fluids 19 (10), 086102.CrossRefGoogle Scholar
Colonna, P., Nannan, N. R. & Guardone, A. 2008 Multiparameter equations of state for siloxanes: [(CH3)3-Si-O1/2]2-[O-Si-(CH3)2]i=1, …, 3 , and [O-Si-(CH3)2]6. Fluid Phase Equilib. 263 (2), 115–130.CrossRefGoogle Scholar
Courant, R. & Friedrichs, K. 1948 Supersonic Flow and Shock Waves. Interscience.Google Scholar
Cramer, M. S. 1989a Negative nonlinearity in selected fluorocarbons. Phys. Fluids A 1 (11), 1894–1897.CrossRefGoogle Scholar
Cramer, M. S. 1989b Shock splitting in single-phase gases. J. Fluid Mech. 199, 281–296.CrossRefGoogle Scholar
Cramer, M. S. & 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. & Sen, R. 1987 Exact solutions for sonic shocks in van der Waals gases. Phys. Fluids 30, 377–385.CrossRefGoogle Scholar
Edney, B. E. 1968 Effects of shock impingement on the heat transfer around blunt bodies. AIAA J. 6 (1), 15–21.CrossRefGoogle Scholar
Fowles, G. R. 1981 Stimulated and spontaneous emission of acoustic waves from shock fronts. Phys. Fluids 24 (2), 220–227.CrossRefGoogle Scholar
Glimm, J., Klingenberg, C., McBryan, O., Plohr, B., Sharp, D. & Yaniv, S. 1985 Front tracking and two-dimensional Riemann problems. Adv. Appl. Maths 6 (3), 259–290.CrossRefGoogle Scholar
Glimm, J. & Sharp, D. H. 1986 An S matrix theory for classical nonlinear physics. Found. Phys. 16, 125–141.CrossRefGoogle Scholar
Godlewski, E. & Raviart, P. A. 2013 Numerical Approximation of Hyperbolic Systems of Conservation Laws. Springer.Google Scholar
Grove, J. 1989 The interaction of shock waves with fluid interfaces. Adv. Appl. Maths 10 (2), 201–227.CrossRefGoogle Scholar
Grove, J. W. & Menikoff, R. 1990 Anomalous reflection of a shock wave at a fluid interface. J. Fluid Mech. 219, 313–336.CrossRefGoogle Scholar
Guderley, K. G. 1962 The Theory of Transonic Flow. Pergamon Press.Google Scholar
Henderson, L. F. & Atkinson, J. D. 1976 Multi-valued solutions of steady-state supersonic flow. J. Fluid Mech. 75 (4), 751–764.CrossRefGoogle Scholar
Henderson, L. F. & Menikoff, R. 1998 Triple-shock entropy theorem and its consequences. J. Fluid Mech. 366, 179–210.CrossRefGoogle Scholar
Hornung, H. 1986 Regular and Mach reflection of shock waves. Annu. Rev. Fluid Mech. 18 (1), 33–58.CrossRefGoogle Scholar
Ivanov, M. S., Gimelshein, S. F. & Beylich, A. E. 1995 Hysteresis effect in stationary reflection of shock waves. Phys. Fluids 7 (4), 685–687.CrossRefGoogle Scholar
Kluwick, A. 2001 Rarefaction shocks. In Handbook of Shock Waves (ed. Ben-Dor, G., Igra, O. & Elperin, T.), pp. 339–411. Academic.CrossRefGoogle Scholar
Kluwick, A. & Cox, E. A. 2018 Steady small-disturbance transonic dense gas flow past two-dimensional compression/expansion ramps. J. Fluid Mech. 848, 756–787.CrossRefGoogle Scholar
Kluwick, A. & Cox, E. A. 2019a Steady transonic dense gas flow past two-dimensional compression/expansion ramp revisited. Proc. Appl. Math. Mech. 19, e201900060.CrossRefGoogle Scholar
Kluwick, A. & Cox, E. A. 2019b Weak shock reflection in channel flows for dense gases. J. Fluid Mech. 874, 131–157.CrossRefGoogle Scholar
Lambrakis, K. C. & Thompson, P. A. 1972 Existence of real fluids with a negative fundamental derivative 𝛤. Phys. Fluids 15 (5), 933–935.CrossRefGoogle Scholar
Landau, L. D. & Lifshitz, E. M. 1987 Fluid Mechanics, 2nd edn. Pergamon Press.Google Scholar
Lemmon, E. W., Huber, M. L. & McLinden, M. O.2013 NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties REFPROP, Version 9.1. National Institute of Standards and Technology.Google Scholar
Martin, J. J. & Hou, Y. C. 1955 Development of an equation of state for gases. AIChE J. 1 (2), 142–151.CrossRefGoogle Scholar
Martin, J. J., Kapoor, R. M. & De Nevers, N. 1959 An improved equation of state for gases. AIChE J. 5 (2), 159–160.CrossRefGoogle Scholar
Menikoff, R. 1989 Analogies between Riemann problem for 1-D fluid dynamics and 2-D steady supersonic flow. In Contemporary Mathematics Proc. 1988 Joint Research Conference on Current Progress in Hyperbolic Systems: Riemann Problems and Computations, pp. 225–240. American Mathematical Society.Google Scholar
Menikoff, R. & Plohr, B. J. 1989 The Riemann problem for fluid flow of real materials. Rev. Mod. Phys. 61, 75–130.CrossRefGoogle Scholar
Nannan, N. R., Guardone, A. & Colonna, P. 2014 Critical point anomalies include expansion shock waves. Phys. Fluids 26, 021701.CrossRefGoogle Scholar
Nannan, N. R., Sirianni, C., Mathijssen, T., Guardone, A. & Colonna, P. 2016 The admissibility domain of rarefaction shock waves in the near-critical vapour-liquid equilibrium region of pure typical fluids. J. Fluid Mech. 795, 241–261.CrossRefGoogle Scholar
Sanderson, S. R. 2004 Gasdynamic wave interaction in two spatial dimensions. J. Fluid Mech. 506, 187–205.CrossRefGoogle Scholar
Serre, D. 2007 Shock reflection in gas dynamics. In Handbook of Mathematical Fluid Dynamics, pp. 39–122. Elsevier.CrossRefGoogle Scholar
Span, R. & Wagner, W. 2003a Equations of state for technical applications. I. Simultaneously optimized functional forms for nonpolar and polar fluids. Intl J. Thermophys. 24 (1), 1–39.CrossRefGoogle Scholar
Span, R. & Wagner, W. 2003b Equations of state for technical applications. II. Results for nonpolar fluids. Intl J. Thermophys. 24 (1), 41–109.CrossRefGoogle Scholar
Stryjek, R. & Vera, J. H. 1986 PRSV: An improved Peng–Robinson equation of state for pure compounds and mixtures. Can. J. Chem. Engng 64 (2), 323–333.CrossRefGoogle Scholar
Teshukov, V. M. 1989 Stability of the regular reflection of shock waves. PMTF Zh. Prikl. Mekhan. Tekhn. Fiz. 2, 26–33.Google Scholar
Thol, M., Javed, M. A., Baumhögger, E., Span, R. & Vrabec, J. 2019 Thermodynamic properties of dodecamethylpentasiloxane, tetradecamethylhexasiloxane, and decamethylcyclopentasiloxane. Ind. Engng Chem. Res. 58 (22), 9617–9635.CrossRefGoogle Scholar
Thompson, P. A. 1971 A fundamental derivative in gasdynamics. Phys. Fluids 14 (9), 1843–1849.CrossRefGoogle Scholar
Thompson, P. A. 1988 Compressilbe Fluid Dynamics. McGraw-Hill.Google Scholar
Thompson, P. A. & Lambrakis, K. C. 1973 Negative shock waves. J. Fluid Mech. 60, 187–208.CrossRefGoogle Scholar
Touber, E. & Alferez, N. 2019 Shock-induced energy conversion of entropy in non-ideal fluids. J. Fluid Mech. 864, 807–847.CrossRefGoogle Scholar
Vimercati, D., Kluwick, A. & Guardone, A. 2018 Oblique waves in steady supersonic flows of Bethe–Zel’dovich–Thompson fluids. J. Fluid Mech. 855, 445–468.CrossRefGoogle Scholar
van der Waals, J. D.1873 Over de Continuiteit van den Gas- en Vloeistoftoestand (on the continuity of the gas and liquid state). PhD thesis, Leiden University.Google 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