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

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- Received:
- Accepted:

DOI: 10.1007/s00193-007-0096-8

- Cite this article as:
- Emanuel, G. & Hekiri, H. Shock Waves (2007) 17: 85. doi:10.1007/s00193-007-0096-8

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## 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