Skip to main content
Log in

Analytical extension of curved shock theory

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

Abstract

Curved shock theory (CST) is limited to shock waves in a steady, two-dimensional or axisymmetric (2-Ax) flow of a perfect gas. A unique feature of CST is its use of intrinsic coordinates that result in an elegant and useful formulation for flow properties just downstream of a shock. For instance, the downstream effect of upstream vorticity, shock wave curvature, and the upstream pressure gradient along a streamline is established. There have been several attempts to extend CST, as mentioned in the text. Removal of the steady, 2-Ax, and perfect gas limitations, singly or in combination, requires an appropriate formulation of the shock wave’s jump relations and the intrinsic coordinate Euler equations. Issues discussed include flow plane versus osculating plane, unsteady flow, vorticity, an imperfect gas, etc. The extension of CST utilizes concepts from differential geometry, such as the osculating plane, streamline torsion, and the Serret–Frenet equations.

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

Access this article

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

References

  1. Mölder, S.: Curved Aerodynamic Shock Waves. PhD Thesis, McGill University (2012)

  2. Mölder, S.: Curved shock theory. Shock Waves 26, 337–353 (2016). doi:10.1007/s00193-015-0589-9

    Article  Google Scholar 

  3. Uskov, V.N., Mostovykh, P.S.: The flow gradients in the vicinity of a shock wave for a thermodynamically imperfect gas. Shock Waves 26, 693–708 (2016). doi:10.1007/s00193-015-0606-z

    Article  Google Scholar 

  4. Emanuel, G.: Analytical Fluid Dynamics, 3rd edn. CRC Press, Boca Raton (2016)

  5. Emanuel, G.: Unsteady natural coordinates for a viscous compressible flow. Phys. Fluids A 5, 294–298 (1993). doi:10.1063/1.858694

    Article  MATH  Google Scholar 

  6. Struik, D.J.: Lectures on Classical Differential Geometry. Addison-Wesley Press, Cambridge (1950)

    MATH  Google Scholar 

  7. Serrin, J.: Mathematical principles of classical fluid mechanics. In: Flugge, S. (ed.) Encyclopedia of Physics, vol. VIII/1. Springer, Berlin (1959). doi:10.1007/978-3-642-45914-6_2

  8. Bonfiglioli, A., Paciorri, R.: Hypersonic Flow Computations on Unstructured Grids: Shock-Capturing vs. Shock-Fitting Approach. In: 40th Fluid Dynamics Conference and Exhibit, Chicago, 28 June-1 July, AIAA Paper 2010–4449 (2010). doi:10.2514/6.2010-4449

Download references

Acknowledgements

The author gratefully acknowledges the helpful comments of S. Mölder and B. Argrow of the Universities of Ryerson, Toronto, Canada, and Colorado, USA, respectively.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to G. Emanuel.

Additional information

Communicated by D. Zeitoun.

Appendix: Flow and shock parameters

Appendix: Flow and shock parameters

$$\begin{aligned}&{\vec V_1} = \sum {v_{1,i}}\left( {{x_i},t} \right) \hat{\vert }_{i} \end{aligned}$$
(76)
$$\begin{aligned}&\nabla F=\sum F_{x_j } \hat{\vert }_j \end{aligned}$$
(77)
$$\begin{aligned}&\theta =\sin ^{-1}\left( {\frac{\sum v_{1,j} F_{x_j } }{V_1 \left| {\nabla F} \right| }} \right) \end{aligned}$$
(78)
$$\begin{aligned}&K_1 =v_{1,3} F_{x_2 } -v_{1,2} F_{x_3 },\quad K_2 =v_{1,1} F_{x_3 } -v_{1,3} F_{x_1 } ,\nonumber \\&K_3 =v_{1,2} F_{x_1 } -v_{1,1} F_{x_2 } \end{aligned}$$
(79)
$$\begin{aligned}&L_1 =F_{x_3 } K_2 -F_{x_2 } K_3 ,\quad L_2 =F_{x_1 } K_3 -F_{x_3 } K_1 ,\nonumber \\&L_3 =F_{x_2 } K_1 -F_{x_1 } K_2 \end{aligned}$$
(80)
$$\begin{aligned}&\chi =\frac{1}{V_1 \left| {\nabla F} \right| \cos \theta } \end{aligned}$$
(81)
$$\begin{aligned}&\sum F_{x_j } K_j =\sum F_{x_j } L_j =\sum K_j L_j \nonumber \\&\qquad \qquad \qquad =\sum v_{1,j} K_j =0 \end{aligned}$$
(82)
$$\begin{aligned}&\sum K_j^2 =\frac{1}{\chi ^{2}},\quad \sum L_j^2 =\frac{\left| {\nabla F} \right| ^{2}}{\chi ^{2}} \end{aligned}$$
(83)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Emanuel, G. Analytical extension of curved shock theory. Shock Waves 28, 417–425 (2018). https://doi.org/10.1007/s00193-017-0735-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00193-017-0735-7

Keywords

Navigation