Skip to main content
SpringerLink
Log in

Reflection of curved shock waves

  • Original Article
  • Published:
Shock Waves Aims and scope Submit manuscript

Abstract

Shock curvatures are related to pressure gradients, streamline curvatures and vorticity in flows with planar and axial symmetry. Explicit expressions, in an influence coefficient format, are used to relate post-shock pressure gradient, streamline curvature and vorticity to pre-shock gradients and shock curvature in steady flow. Using higher order, von Neumann-type, compatibility conditions, curved shock theory is applied to calculate the flow near singly and doubly curved shocks on curved surfaces, in regular shock reflection and in Mach reflection. Theoretical curved shock shapes are in good agreement with computational fluid dynamics calculations and experiment.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22
Fig. 23
Fig. 24
Fig. 25

Similar content being viewed by others

Notes

  1. This definition is unambiguous and does not depend on the chosen coordinate system.

  2. Also known as pseudo-stationary or pseudo-steady.

  3. Note that the shock angles are the same for the three cases, hence their zeroth-order solutions are identical and unable to reveal any effects of shock curvature. This is essentially why oblique shock equations, by themselves, do not expose any features of shock reflection at, and near, the centre line. These problems and differences from purely planar flow are strictly shock- and flow-induced effects, due to shock and streamline curvature.

  4. In axial flow, the Mach stem is called the Mach disk and the flow downstream is called the Mach stream.

  5. The assumption that the freestream is uniform and irrotational is made here for the sake of brevity. A non-uniform freestream is easily implemented by specifying appropriate nonzero values for \(P_{{\text {1}}}, D_{{\text {1}}}\) and \(\varGamma _{\text {1}}\) in the L-terms for both the i- and m-shocks.

  6. The equal pressure gradient condition is expressed as \((\partial p/\partial s)_{3}= (\partial p/\partial s)_{4}\) or \(\upgamma p_{3} M_{{\text {3}}}^{{\text {2}}}P_{{\text {3}}}= \gamma p_{{\text {4}}}M_{{\text {4}}}^{{\text {2}}}P_{{\text {4}}}\). Since \(p_{{\text {3}}}=p_{{\text {4}}}\), this becomes simply \(M_{{\text {3}}}^{{\text {2}}}P_{{\text {3}}}=M_{{\text {4}}}^{{\text {2}}}P_{{\text {4}}}\).

  7. These six equations can easily be simplified to four by eliminating any two of the \(P_{{\text {3}}}, D_{{\text {3}}}, P_{{\text {4}}},D_{{\text {4}}}\) terms. They have been left in the present format to make explicit the influence of the first-order closure conditions on pressure gradient and flow curvature at the slip layer.

  8. Some CFD results have shown that this is not necessarily so, especially for iMR.

  9. We will assume them to be equal in what follows.

  10. Such a constant curvature is possible only if the Mach shock lies in the fourth quadrant and is concave towards the upstream, dMR, or it lies in the third quadrant and is convex, iMR. If these conditions are not met, then the Mach shock must be inflected—and not of constant curvature (see Fig. 14). Calculations have indicated that the m-shock can have very large curvatures near the triple point so the constant curvature assumption should be treated with caution and should be used only when one of \(S_{a,\mathrm{m}}\) and \(y_\mathrm{m}\) is not available as inputs.

  11. The terms “strong” and “weak”, when applied to the incident shock in Mach reflection, designate shocks that produce either dMR or iMR, respectively. In both cases, the shocks are in the “weak shock family” where the post-shock Mach number is supersonic.

References

  1. Ames Research Staff: Equations, Tables and Charts for Compressible Flow. NACA Rep. 1135 (1953)

  2. Azevedo, D.J., Liu, C.S.: Engineering approach to the prediction of shock patterns in bounded high-speed flows. AIAA J. 31(1), 83–90 (1993)

    Article  MATH  Google Scholar 

  3. Ben-Dor, G.: Shock Wave Reflection Phenomena. Springer, Berlin (2007)

    MATH  Google Scholar 

  4. Courant, R., Friedrichs, K.O.: Supersonic Flow and Shock Waves. Interscience, New York (1948)

    MATH  Google Scholar 

  5. Crocco, L.: Singolarita della corrente gassosa iperacustica nell’ interno di una prora adiedro. Atti del Congr. dell Unione Mat. Ital. 15, 597–615 (1937)

    Google Scholar 

  6. Emanuel, G., Liu, M.-S.: Shock wave derivatives. Phys. Fluids 31(12), 3625–3633 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  7. Emanuel, G.: Analytical Fluid Dynamics. CRC Press, Boca Raton (1994)

    MATH  Google Scholar 

  8. Emanuel, G.: Shock Wave Dynamics, Derivatives and Related Topics. CRC Press, Boca Raton (2012)

    Book  MATH  Google Scholar 

  9. Hayes, W.D.: The vorticity jump across a gasdynamic discontinuity. J. Fluid Mech. 2, 595–600 (1957)

    Article  MATH  Google Scholar 

  10. Hayes, W.D., Probstein, R.F.: Hypersonic Flow Theory. Academic Press, Cambridge (1966)

    MATH  Google Scholar 

  11. Henderson, L.F.: Regions and boundaries for diffracting shock wave systems. Z. Angew. Math. Mech. 67(2), 73–86 (1987)

    Article  Google Scholar 

  12. Hornung, H.G.: Oblique shock reflection from an axis of symmetry. J. Fluid Mech. 409(5), 1–12 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  13. Hornung, H.G., Schwendeman, D.W.: Oblique shock reflection from an axis of symmetry: shock dynamics and relation to the Guderley singularity. J. Fluid Mech. 438, 231–245 (2001)

    Article  MATH  Google Scholar 

  14. Kaneshige, M.J., Hornung, H.G.: Erratum: gradients at a curved shock in reacting flow. Shock Waves 9, 219–220 (1999)

    Article  Google Scholar 

  15. Kanwal, R.P.: On curved shock waves in three-dimensional gas flows. Q. Appl. Math. 16, 361–372 (1958)

    Article  MathSciNet  MATH  Google Scholar 

  16. Kanwal, R.P.: Determination of vorticity and the gradients of flow parameters behind a three-dimensional unsteady curved shock wave. Arch. Ration. Mech. Anal. 1(1), 225–232 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  17. Li, H., Ben-Dor, G.: A parametric study of Mach reflection in steady flows. J. Fluid Mech. 341, 101–125 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  18. Lin, C.C., Rubinov, S.I.: On the flow behind curved shocks. J. Math. Phys. 27(1–4), 105–129 (1948)

    Article  MathSciNet  MATH  Google Scholar 

  19. Lozzi, A.: Double Mach Reflection of Shock waves. M. Eng. Sci. thesis, The University of Sydney, New South Wales, Australia (1971)

  20. Masterix. Software Package. Ver. 3.40.0.3018, RBT Consultants, Toronto, ON (2004-2015)

  21. Mölder, S.: Internal, axisymmetric, conical flow. AIAA J. 5(7), 1252–1255 (1967)

    Article  Google Scholar 

  22. Mölder, S.: Reflection of curved shock waves. In: \(7^{{\rm th}}\) International Congress of the Aeronautical Sciences, ICAS paper 70-11 (1970)

  23. Mölder, S.: Reflection of curved shock waves in steady supersonic flow. CASI Trans. 4(2), 73–80 (1971)

    Google Scholar 

  24. Mölder, S., Gulamhussein, A., Timofeev, E.V., Voinovich, P.: Focusing of conical shocks at the centre-line of symmetry. In: Shock Waves (ed. Houwing, A.F.P., Paull, A.) pp. 875-880. Panther Publishing and Printing, Canberra (1997)

  25. Mölder, S.: Curved shock theory. Shock Waves 26(4), 337–353 (2016)

    Article  Google Scholar 

  26. Rylov, A.I.: On the issue of impossibility of regular reflection from the axis of symmetry for stationary shock wave. Prikl. Math Mekh (Appl. Math. Mech.) 54(2), 245–249 (1990)

    MathSciNet  MATH  Google Scholar 

  27. Saito, T., Voinovich, P., Timofeev, E., Takayama, K.: Development and application of high-resolution adaptive numerical techniques in Shock Wave Research Center. In: Toro, E.F. (ed.) Godunov Methods: Theory and Applications, pp. 763–784. Kluwer Academic/Plenum Publishers, New York (2001)

  28. Thomas, T.Y.: On curved shock waves. J. Math. Phys. 26, 62–68 (1947)

    Article  MathSciNet  MATH  Google Scholar 

  29. Thomas, T.Y.: The consistency relations for shock waves. J. Math. Phys. 28(2), 62–90 (1949)

    Article  MathSciNet  MATH  Google Scholar 

  30. Thomas, T.Y.: On conditions with steady plane flow with shock waves. J. Math. Phys. 28(2), 91–98 (1949)

    Article  MathSciNet  MATH  Google Scholar 

  31. Thomas, T.Y.: The determination of pressure on curved bodies behind shocks. Commun. Pure Appl. Math. 3, 103–132 (1950)

    Article  MathSciNet  MATH  Google Scholar 

  32. Thomas, T.Y.: The extended compatibility conditions for the surfaces of discontinuity in continuum mechanics. J. Math. Mech. 6(3), 311–322 (1957). (See also correction in 6(6), 907-908)

  33. Thomas, T.Y.: On the propagation and decay of spherical blast waves. J. Math. Mech. 6(5), 607–619 (1957)

    MathSciNet  MATH  Google Scholar 

  34. Truesdell, C.: On curved shocks in steady plane flow of an ideal fluid. J. Aeronaut. Sci. 19, 826–828 (1952)

    Article  MathSciNet  MATH  Google Scholar 

  35. von Neumann, J.: Oblique reflection of shocks. Explos. Res. Rep. 12, Navy Department, Bureau of Ordinance, Washington, D.C, U.S.A. (1943)

Download references

Acknowledgements

Much insight of the analysis of curved shocks, especially the intricacies of vorticity, has been kindly provided by George Emanuel. Many CFD results were calculated by Evgeny Timofeev. Rabi Tahir provided very essential help with the Masterix CFD code. Experimental results are due to Beric Skews for Mach 3 and Julian Romeskie for Mach 8.33.

Author information

Authors and Affiliations

  1. Ryerson University, Toronto, Ontario, Canada

    S. Mölder

  2. 89 Joseph St., Port Carling, Ontario, P0B 1J0, Canada

    S. Mölder

Authors
  1. S. Mölder
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. Mölder.

Additional information

Communicated by B. W. Skews and E. Timofeev.

Appendix: The equation for vorticity behind a curved shock

Appendix: The equation for vorticity behind a curved shock

The change of vorticity across a curved shock wave has been treated in [7, 9, 16, 25]. The vorticity equation is,

$$\begin{aligned}&A_1^{{\prime \prime }}P_1 +B_1^{{\prime \prime }}D_1 +E_1 ^{{\prime \prime }}\Gamma _1\nonumber \\&\quad =A_2^{{\prime \prime }}P_2 +B_2^{{\prime \prime }} D_2 +E_2^{{\prime \prime }}\Gamma _2 +{C}''S_a +{G}''S_b \end{aligned}$$
(44)

where the vorticity behind the shock is \(\Gamma _{2}\) and the coefficients are,

$$\begin{aligned}&P_1 :{A}''_1 =\left( {M_1^2 -1} \right) \tan \theta \sin \theta +\cos \theta \nonumber \\&D_1 :{B}''_1 =-2\sin \theta \nonumber \\&\Gamma _1 :{E}''_1 =\sin \theta \nonumber \\&P_2 :{A}''_2 =\left( {M_2^2 -1} \right) \tan \left( {\theta -\delta } \right) \sin \left( {\theta -\delta } \right) +\cos \left( {\theta -\delta } \right) \nonumber \\&D_2 :{B}''_2 =-2\sin \left( {\theta -\delta } \right) \\&\Gamma _2 :{E}''_2=\sin \left( {\theta -\delta } \right) \nonumber \\&S_a :{C}''=\tan \left( {\theta -\delta } \right) -\tan \theta \nonumber \\&S_b :{G}''=-{F}''/\cos \left( {\theta +\delta _1 } \right) \nonumber \end{aligned}$$
(45)

Equation (44) can be written as,

$$\begin{aligned} {L}''={A}''_2 P_2 +{B}''_2 D_2 +{E}''_2 \Gamma _2 +{C}''S_a +{G}''S_b \end{aligned}$$
(46)

or

$$\begin{aligned} {L}''= & {} {A}''_2 P_2 +{B}''_2 D_2 +{E}''_2 \Gamma _2 +{C}''S_a +{F}''/y \end{aligned}$$
(47)

where

$$\begin{aligned} {L}''={A}''_1 P_1 +{B}''_1 D_1 +{E}''_1 \Gamma _1 \end{aligned}$$
(48)

Equation (44) can be used to solve for the post-shock vorticity, \(\Gamma _2\), in terms of the other variables. \(S_b\) and y are interchangeable through \(S_b =-\cos \left( {\theta +\delta _1} \right) /y\). Choosing (46) and solving (44) for \(\Gamma _2 \) gives the desired expression for the downstream vorticity,

$$\begin{aligned} \Gamma _2 =\left[ {{L}''-\left( {{A}''_2 P_2 +{B}''_2 D_2 +{C}''S_a +{G}''S_b } \right) } \right] /{E}''_2 \end{aligned}$$
(49)

This is the explicit, generalized vorticity equation in a rational form for \({\Gamma }_{2}\), the normalized vorticity behind a curved shock facing both non-uniform and divergent flow. Together with equations (24), it forms three equations for the three unknowns \(P_2 ,D_2 \hbox { and }\Gamma _2\) so as to completely define the non-uniform post-shock flow. For a uniform upstream, (49) reduces to,

$$\begin{aligned} \Gamma _2 =\left[ {\frac{{C}''}{{E}''_2 }+\frac{\left[ {BC} \right] }{\left[ {AB} \right] }\frac{{A}''_2 }{{E}''_2 }-\frac{\left[ {AC} \right] }{\left[ {AB} \right] }\frac{{B}''_2 }{{E}''_2 }} \right] S_a \end{aligned}$$
(50)

The coefficient multiplying \(S_{a}\) is identical to the coefficient of \(S_{a}\) in (32). Fortunately \(P_2\) and \(D_2\) are decoupled from \(\Gamma _2\), leading to explicit solutions for all unknowns. \(P_2\) and \(D_2\), appearing in the equations (44) and (49), are found from the two curved shock equations (18) [see also (32) and (33) in [25]] and repeated here:

$$\begin{aligned} \begin{aligned}&P_2 =\frac{B_2 \left( {{C}'S_a +{G}'S_b -{L}'} \right) -{B}'_2 \left( {CS_a +GS_b -L} \right) }{A_2 {B}'_2 -{A}'_2 B_2 }\\&D_2 =-\frac{A_2 \left( {{C}'S_a +{G}'S_b -{L}'} \right) -{A}'_2 \left( {CS_a +GS_b -L} \right) }{A_2 {B}'_2 -{A}'_2 B_2 } \end{aligned} \end{aligned}$$
(51)

where the L-terms above are given by,

$$\begin{aligned} \begin{aligned}&L=A_1 P_1 +B_1 D_1 +E_1 \Gamma _1 \\&{L}'={A}'_1 P_1 +{B}'_1 D_1 +{E}'_1 \Gamma _1 \end{aligned} \end{aligned}$$
(52)

Note that L and \({L}'\) contain the upstream gradients and that \(G\hbox { and }{G}'\) contain the upstream flow inclination \(\delta _1\). Substituting \(P_2 \hbox { and }D_2\) from (51) into (49) and collecting terms of the upstream gradients and the shock curvatures give the influence coefficient form of the vorticity equation (49),

$$\begin{aligned} \Gamma _2 =I_p P_1 +I_d D_1 +I_g \Gamma _1 +I_a S_a +I_b S_b \end{aligned}$$
(53)

where the I-coefficients, each multiplying their respective variables, appear in the full equation for \(\Gamma _2\) as,

$$\begin{aligned} \Gamma _2= & {} \left\{ \left[ {AB} \right] {A}''_1 +\left( {B_2 {A}'_1 -{B}'_2 A_1 } \right) {A}''_2\right. \nonumber \\&\left. -\left( {A_2 {A}'_1 -{A}'_2 A_1 } \right) {B}''_2 \right\} /\left\{ {\left[ {AB} \right] {E}''_2 } \right\} P_1\nonumber \\&+\left\{ \left[ {AB} \right] {B}''_1 +\left( {B_2 {B}'_{{1}} -{B}'_2 B_1} \right) {A}''_2 \right. \nonumber \\&\left. -\left( {A_2 {B}'_1 -{A}'_2 B_1 } \right) {B}''_2 \right\} /\left\{ {\left[ {AB} \right] {E}''_2 } \right\} D_1\nonumber \\&+\left\{ \left[ {AB} \right] {E}''_1 +\left( {B_2 {E}'_1 -{B}'_2 E_1}\right) {A}''_2 \right. \nonumber \\&\left. -\left( {A_2 {E}'_1 -{A}'_2 E_1 } \right) {B}''_2 \right\} /\left\{ {\left[ {AB} \right] {E}''_2 } \right\} \Gamma _1\nonumber \\&-\left\{ \left[ {AB} \right] {C}''+\left( {B_2 {C}'-{B}'_2 C} \right) {A}''_2\right. \nonumber \\&\left. -\left( {A_2 {C}'-{A}'_2 C} \right) {B}''_2 \right\} /\left\{ {\left[ {AB} \right] {E}''_2 } \right\} S_a \nonumber \\&-\left\{ \left[ {AB} \right] {G}''+\left( {B_2 {G}'-{B}'_2 G} \right) {A}''_2 \right. \nonumber \\&\left. -\left( {A_2 {G}'-{A}'_2 G} \right) {B}''_2 \right\} /\left\{ {\left[ {AB} \right] {E}''_2 } \right\} S_b \end{aligned}$$
(54)

The unprimed and single-primed coefficients, \(A{\ldots }G\), are listed as equations (19) and (20) (see also (34) and (35) in [25]); the double-primed are in (45). This equation shows clearly what the role is of each upstream non-uniformity \(P_1,\,D_1 \hbox { and }\Gamma _1\) and the shock curvatures \(S_a \hbox { and }S_b\) in determining the downstream vorticity, \(\Gamma _2\). Note that the above derivation for vorticity does not need Crocco’s thermodynamic relation between vorticity and entropy gradient and that the resulting equations account for upstream flow non-uniformity and vorticity as well as flow divergence. Derivation of the vorticity equation parallels those for the pressure gradient and streamline curvature but it is quite a bit simpler. The use of j to denote planar or axial symmetry has been dropped since the equations are uniformly valid for both geometries. For axial flow, y is the radius of the shock’s curvature in the transverse plane, so that the flow is sensitive to dimensionality through the parameter y. In the calculations for planar flow, y is set to a very large number. Figure 26 depicts the influence coefficients for vorticity plotted against shock angle. The blue curve shows the influence of pre-shock pressure gradient \(P_1\), and we see that a positive pressure gradient causes a positive vorticity contribution for an acute shock and a negative contribution for an obtuse shock. The green curve shows that a positive pre-shock flow curvature, \(D_1\), produces a positive contribution to vorticity. The red curve is for the effect of pre-shock vorticity itself. At the Mach wave limits, the influence coefficient has a value of 1, predicting that vorticity passes through Mach waves unchanged. All other curves are at zero so, at Mach wave conditions, there is no vorticity production due to pre-shock gradients or Mach wave curvatures. Stronger shocks tend to amplify and reverse the direction of vorticity. The cyan curve shows that positive vorticity is produced by a positive flow-plane shock curvature, \(S_a\), for an acute shock and that negative vorticity is produced by a positively curving obtuse shock. The black curve is for the effect of the transverse shock curvature, \(S_b\), and it shows that the influence coefficient for the transverse curvature is identically zero. This confirms the fact that the shock produces vorticity only by its flow-plane curvature and not by the transverse curvature so that flow behind a conical shock is irrotational. For flows with no pre-shock divergence/convergence, the \(I_b S_b\) term can be dropped from equations (53) and (54) since \(I_{b}\) is identically zero.

The situation becomes complicated when the pre-shock flow is diverging. The role played by flow divergence \(\delta _1\) and transverse shock curvature \(S_b\) is interactive and has to be carefully considered. From what we know of the behaviour of vorticity, it seems incorrect that post-shock vorticity is a function of the cross-stream curvature, \(S_b\), as evident from the last term of (54). So we examine the influence coefficient for \(S_b\),

$$\begin{aligned} \hbox { I}_b= & {} -\left\{ \left[ {AB} \right] {G}''+\left( {B_2 {G}'-{B}'_2 G} \right) {A}''_2 \right. \nonumber \\&\left. -\left( {A_2 {G}'-{A}'_2 G} \right) {B}''_2 \right\} /\left\{ {\left[ {AB} \right] {E}''_2 } \right\} \end{aligned}$$
(55)
Fig. 26
figure 26

Influence coefficients for vorticity behind doubly curved shock in non-uniform flow at Mach 3

Full size image

In particular, its numerator,

$$\begin{aligned} \hbox { N}_b = \left[ {AB} \right] {G}''+\left( {B_2 {G}'-{B}'_2 G} \right) {A}''_2 -\left( {A_2 {G}'-{A}'_2 G} \right) {B}''_2 \end{aligned}$$
(56)

must be zero when \(\delta _1\) is zero. The black curve in Fig. 26 shows that it is indeed so. Note that \(\delta _1\) is contained in \({G}'\hbox { and }{G}''\) only and not in G. This requires that the part of \(\hbox {N}_{b}\), not containing \(\delta _1\) equals zero, i.e.,

$$\begin{aligned} {A}'_2 {B}''_2 -{B}'_2 {A}''_2 =0. \end{aligned}$$
(57)

This has proven that there is no influence on post-shock vorticity from \(S_b\) when there is no pre-shock divergence. The proof of this is straightforward, with four lines of algebra, so that, without loss of generality, the influence coefficient for \(S_{b}\) can be simplified to,

$$\begin{aligned} \hbox { I}_b = -\left\{ {\left[ {AB} \right] {G}''+\left( {B_2 {G}'} \right) {A}''_2 -\left( {A_2 {G}'} \right) {B}''_2 } \right\} /\left\{ {\left[ {AB} \right] {E}''_2 } \right\} \end{aligned}$$
(58)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mölder, S. Reflection of curved shock waves. Shock Waves 27, 699–720 (2017). https://doi.org/10.1007/s00193-017-0710-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00193-017-0710-3

Keywords

Search

Navigation