Skip to main content
SpringerLink
Log in

A unified description of oblique waves in ideal and non-ideal steady supersonic flows around compressive and rarefactive corners

  • Original Paper
  • Published:
Acta Mechanica Aims and scope Submit manuscript

Abstract

According to classical gas dynamic theory, if a steady supersonic parallel flow encounters a sudden change in the wall slope, two very different phenomena may occur. If the flow expands around a sharp corner, the well-known isentropic Prandtl–Meyer fan is observed. Conversely, a shock wave occurs if the flow is compressed: for wedge angles smaller than the detachment value, which depends on the uniform upstream state, an oblique shock originates at the corner; at larger deviation angles, a detached shock is formed. A unified description of these flows is presented here to extend the validity of the common \(\beta \)–\(\vartheta \) (shock angle–deflection angle) diagram for shocked non-isentropic flows into the realm of isentropic expansions. The new graph allows for a straightforward identification of the wave angles for self-similar flow fields around compressive and rarefactive corners. Besides, it clarifies the relation between shock waves and rarefaction fans in the neighbourhood of the \({\vartheta =0}\) axis, where shock waves are weak enough to be fairly well approximated by isentropic compressions. At \({\vartheta =0}\), indeed, shock and rarefaction curves are demonstrated to be first order continuous. This result is interpreted in view of the bisector rule for oblique shock waves. Exemplary diagrams are reported for both ideal-gas flows, dilute-gas flows and non-ideal flows of dense vapours in the close proximity of the liquid–vapour saturation curve and critical point. The application of the new diagram is illustrated for the textbook case of the supersonic flow past a diamond-shaped airfoil.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bethe, H.A.: The theory of shock waves for an arbitrary equation of state. Technical paper 545, Office Sci. Res. & Dev. (1942)

  2. Colonna, P., Guardone, A.: Molecular interpretation of nonclassical gasdynamics of dense vapors under the van der Waals model. Phys. Fluids 18(5), 056101-1–056101-14 (2006)

    Article  Google Scholar 

  3. Colonna, P., Guardone, A., Nannan, N.R.: Siloxanes: a new class of candidate Bethe–Zel’dovich–Thompson fluids. Phys. Fluids 19(10), 086102-1–086102-12 (2007)

    MATH  Google Scholar 

  4. Cramer, M.S.: Negative nonlinearity in selected fluorocarbons. Phys. Fluids A 1(11), 1894–1897 (1989)

    Article  Google Scholar 

  5. Cramer, M.S.: Nonclassical dynamics of classical gases. In: Kluwick, A. (ed.) Nonlinear Waves in Real Fluids, pp. 91–145. Springer, New York (1991)

    Chapter  Google Scholar 

  6. Cramer, M.S., Best, L.M.: Steady, isentropic flows of dense gases. Phys. Fluids A 3(4), 219–226 (1991)

    Article  MATH  Google Scholar 

  7. Cramer, M.S., Crickenberger, A.B.: Prandtl–Meyer function for dense gases. AIAA J. 30(2), 561–564 (1992)

    Article  MATH  Google Scholar 

  8. Cramer, M.S., Fry, N.R.: Nozzle flows of dense gases. Phys. Fluids A 5(5), 1246–1259 (1993)

    Article  Google Scholar 

  9. Cramer, M.S., Kluwick, A.: On the propagation of waves exhibiting both positive and negative nonlinearity. J. Fluid Mech. 142, 9–37 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  10. Cramer, M.S., Sen, R.: Shock formation in fluids having embedded regions of negative nonlinearity. Phys. Fluids 29, 2181–2191 (1986)

    Article  MATH  Google Scholar 

  11. D’yakov, S.P.: On the stability of shock waves. Zh. Eksp. Teor. Fiz 27(3), 288–295 (1954)

    MathSciNet  Google Scholar 

  12. Fowles, G.R.: Stimulated and spontaneous emission of acoustic waves from shock fronts. Phys. Fluids 24(2), 220–227 (1981)

    Article  MATH  Google Scholar 

  13. Gilbarg, D.: The existence and limit behavior of the one-dimensional shock layer. Am. J. Math. 73(2), 256–274 (1951)

    Article  MathSciNet  MATH  Google Scholar 

  14. Guardone, A., Vimercati, D.: Exact solutions to non-classical steady nozzle flows of Bethe–Zel’dovich–Thompson fluids. J. Fluid Mech. 800, 278–306 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  15. Guardone, A., Zamfirescu, C., Colonna, P.: Maximum intensity of rarefaction shock waves for dense gases. J. Fluid Mech. 642, 127–146 (2010)

    Article  MATH  Google Scholar 

  16. Hayes, W.D.: The basic theory of gasdynamic discontinuities. In: Emmons, H.W. (ed.) Fundamentals of Gasdynamics, High Speed Aerodynamics and Jet Propulsion, vol. 3, pp. 416–481. Princeton University Press, Princeton (1958)

    Google Scholar 

  17. Kluwick, A.: Zur ausbreitung schwacher Stöße in dreidimensionalen instationären Strömungen. ZAMM 51(3), 225–232 (1971)

    Article  MATH  Google Scholar 

  18. Kluwick, A.: Transonic nozzle flow of dense gases. J. Fluid Mech. 247, 661–688 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  19. Kluwick, A.: Handbook of Shock Waves, chap. 3.4. Rarefaction shocks, pp. 339–411. Academic Press, New York (2001)

  20. Kontorovich, V.M.: Concerning the stability of shock waves. Soviet Phys. JETP 6, 1179–1180 (1958)

    Google Scholar 

  21. Lambrakis, K.C., Thompson, P.A.: Existence of real fluids with a negative fundamental derivative \(\Gamma \). Phys. Fluids 15(5), 933–935 (1972)

    Article  Google Scholar 

  22. Landau, L.D., Lifshitz, E.M.: Fluid Mechanics, 2nd edn. Pergamon Press, Oxford (1987)

    MATH  Google Scholar 

  23. Lemmon, E.W., Huber, M.L., McLinden, M.O.: NIST reference database 23: reference fluid thermodynamic and transport properties—REFPROP, version 9.1. Standard Reference Data Program (2013)

  24. Menikoff, R., Plohr, B.J.: The Riemann problem for fluid flow of real material. Rev. Mod. Phys. 61(1), 75–130 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  25. Nannan, N.R., Guardone, A., Colonna, P.: Critical point anomalies include expansion shock waves. Phys. Fluids 26(2), 021701 (2014)

    Article  Google Scholar 

  26. Nannan, N.R., Sirianni, C., Mathijssen, T., Guardone, A., Colonna, P.: 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 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  27. Setzmann, U., Wagner, W.: A new equation of state and tables of thermodynamic properties for methane covering the range from the melting line to 625 k at pressures up to 100 MPa. J. Phys. Chem. Ref. Data 20(6), 1061–1155 (1991)

    Article  Google Scholar 

  28. Thompson, P.A.: A fundamental derivative in gasdynamics. Phys. Fluids 14(9), 1843–1849 (1971)

    Article  MATH  Google Scholar 

  29. Thompson, P.A.: Compressible Fluid Dynamics. McGraw-Hill, New York (1988)

    Google Scholar 

  30. Thompson, P.A., Lambrakis, K.C.: Negative shock waves. J. Fluid Mech. 60, 187–208 (1973)

    Article  MATH  Google Scholar 

  31. Weyl, H.: Shock waves in arbitrary fluids. Commun. Pure Appl. Math. 2(2–3), 103–122 (1949)

    Article  MathSciNet  MATH  Google Scholar 

  32. Zamfirescu, C., Guardone, A., Colonna, P.: Admissibility region for rarefaction shock waves in dense gases. J. Fluid Mech. 599, 363–381 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  33. Zucrow, M.J., Hoffman, J.D.: Gas Dynamics, vol. 1976. Wiley, New York (1976)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Aerospace Science and Technology, Politecnico di Milano, Via La Masa 34, 20156, Milan, Italy

    Stefano D’Angelo, Davide Vimercati & Alberto Guardone

Authors
  1. Stefano D’Angelo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Davide Vimercati
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Alberto Guardone
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alberto Guardone.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

D’Angelo, S., Vimercati, D. & Guardone, A. A unified description of oblique waves in ideal and non-ideal steady supersonic flows around compressive and rarefactive corners. Acta Mech 229, 2585–2595 (2018). https://doi.org/10.1007/s00707-018-2130-6

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00707-018-2130-6

Search

Navigation