# Vorticity and its rate of change just downstream of a curved shock

## Abstract

A theory is developed for the vorticity and its substantial derivative just downstream of a curved shock wave, the resulting formulas are exact, algebraic, and explicit. Analysis is for a cylinder-wedge or sphere-cone body, at zero incidence, whose downstream half-angle is θ_{b}. Derived formulas directly depend only on the ratio of specific heats, γ, the freestream Mach number, *M*_{1}, the local slope and curvature of the shock, and the dimensionality parameter, σ, which is zero for a two-dimensional shock and unity for an axisymmetric shock. In turn, the slope and curvature depend on γ, *M*_{1}, and θ_{b}. Numerical results are provided for a bow shock in which θ_{b} is 5°, 10°, or 15°, *M*_{1} is 2, 4, or 6, and γ = 1.4. There is little dependence on the half angle but a strong dependence on the freestream Mach number and on dimensionality. For vorticity and its substantial derivative, the dimensionality dependence gradually decreases with increasing Mach number. In comparison to the two-dimensional case, an axisymmetric shock generates considerable vorticity in a region relatively close to the symmetry axis. Moreover, the magnitude of the vorticity, in this region, is further enhanced in the flow downstream of the shock. This dimensionality difference in vorticity and its substantial derivative is attributed to the three-dimensional relief effect in an axisymmetric flow.

### Keywords

Bow shock wave Vorticity Substantial derivative of vorticity### List of symbols

*D*ω/*Dt*substantial derivative of the vorticity

*F*,*E*elliptic integrals of the first and second kind, respectively

*g*_{j}defined in Appendix C

*G*_{j}defined in Appendix C

*m*_{1}\(= {{M}}^{2}_{1} {\sin}^{2} {{\beta}}\)

*M*Mach number

*n*coordinate normal to the shock

- \(\hat{{n}},\,{\hat{t}},\,{\hat{b}}\)
right-handed, orthonormal basis attached to the shock

*p*pressure

*r*= R̅

_{s}/ R̅_{b}*R*_{b}cylinder or sphere radius

*R*_{s}shock radius at the symmetry axis

*s*arc length along the shock measured from the symmetry axis

- \({{\tilde{s}}}\)
entropy

*t*Equation (B2)

*u*velocity component tangential to the shock

*v*velocity component normal to the shock, downstream of the shock

*w*flow speed

*x*coordinate parallel to the freestream velocity

*X*\(= {1} + {\left[\left({{{\gamma}} -- {1}} \right)/2 \right]}{{M}}^{2}_{1}{\sin}^{2} {{\beta}}\)

*y*transverse or radial coordinate

*Y*\(= {{\gamma}}\,{{M}}^{2}_{1}{\sin}^{2} {{\beta}} - \left({{{\gamma}} - {1}} \right)/2\)

*z*Equation (4)

*Z*\(= {{M}}^{2}_{1} {\sin}^{2} {\beta} - {1}\)

*β*shock wave slope

*β*_{∞}far-downstream shock-wave slope

*β′*shock-wave curvature

*γ*ratio of specific heats

*Δ*shock stand-off distance

*θ*velocity angle, relative to the freestream velocity, just downstream of the shock

*θ*_{b}cone or wedge half angle

*λ*Equation (A1)

*ρ*density

*σ*0 for two-dimensional flow, 1 for axisymmetric flow

- \(\phi\)
Equation (B5)

*χ*Equation (A2)

*ω*vorticity

### Subscripts

- 1
freestream

- s
scalar value

### Superscripts

- −
dimensional

- ′
derivative with respect to s

- *
sonic state just downstream of the shock

### PACS

47.40.Nm 47.32 C-## Preview

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### References

- 1.Truesdell C. (1954). The Kinematics of Vorticity. Indiana University Press, Bloomington MATHGoogle Scholar
- 2.Serrin, J.: Mathematical principles of classical fluid mechanics. In: Encyclopedia of Physics, vol. VIII/1, Springer, Heidelberg pp. 125–263 (1959)Google Scholar
- 3.Hornung H.G. (1998). Gradients at a curved shock in reacting flow.
*Shock Waves*8: 11–21 MATHCrossRefADSGoogle Scholar - 4.Kaneshige M.J. and Hornung H.G. (1999). Erratum.
*Gradients at a curved shock in reacting flow. Shock Waves*9: 219–220 Google Scholar - 5.Wu J.-Z., Ma H.-Y. and Zhou M.D. (2006). Vorticity and Vorticity Dynamics. Springer, New York Google Scholar
- 6.Emanuel, G.: Vorticity in unsteady, viscous, reacting flow and downstream of a curved shock. AIAA J. (2007) (in press)Google Scholar
- 7.Emanuel G. (2001). Analytical Fluid Dynamics, 2nd edn. CRC Press, Boca Raton Google Scholar
- 8.Billig F.S. (1967). Shock-wave shapes around spherical- and cylindrical-nosed bodies.
*J. Spacecraft Rockets*4: 822–823 CrossRefADSGoogle Scholar - 9.Ames Research Staff: Equations, tables, and charts for compressible flow. NACA Report 1135 (1953)Google Scholar
- 10.Gradshteyn I.S. and Ryzhik I.M. (1980). Tables of Integrals, Series and Products, p. 277, Eq. (13). Academic Press, New York Google Scholar
- 11.Milne-Thomson, L.M.: Elliptic integrals. In: Handbook of Mathematical Functions, National Bureau of Standards, Appl. Math. Ser. 55 (1972)Google Scholar