19.4. Oblique shocks#

Jump conditions

19.4.1. Normal Component Analysis#

The oblique shock is treated as a normal shock acting only on the normal component of the Mach number:

  • \(M_{n1} = M_1 \sin \beta\)

  • \(M_{n2} = M_2 \sin(\beta - \theta)\)

The tangential component of velocity remains preserved: \(u_{t1} = u_{t2}\).

19.4.2. The \(\theta\)-\(\beta\)-\(M\) Relation#

By relating the geometry of the velocity vectors (\(\tan \beta = \frac{u_{n1}}{u_t}\) and \(\tan(\beta - \theta) = \frac{u_{n2}}{u_t}\)) and substituting the density ratio \(\rho_2/\rho_1\), we arrive at the «\(\theta\)-\(\beta\)-\(M\) equation»:

\[\tan \theta = 2 \cot \beta \left[ \frac{M_1^2 \sin^2 \beta - 1}{M_1^2 (\gamma + \cos 2\beta) + 2} \right]\]

19.4.3. Observations from the \(\theta\)-\(\beta\)-\(M\) function#

  1. Two Solutions: For any given \(M_1\) and \(\theta < \theta_{max}\), there are two possible shock angles \(\beta\):

    • Weak Shock: Smaller \(\beta\), usually results in \(M_2 > 1\). This is the one typically observed in open flow.

    • Strong Shock: Larger \(\beta\), results in \(M_2 < 1\). Usually occurs in confined flows or high back-pressure.

  2. Maximum Deflection: If \(\theta\) exceeds \(\theta_{max}\) for a given \(M_1\), no straight oblique shock solution exists, and the shock becomes detached (bow shock).

Property

Ratio

Trend (\(M_1 > 1\))

Pressure

\(\frac{p_2}{p_1} = 1 + \frac{2\gamma}{\gamma+1}(M_{n1}^2 - 1)\)

Increase

Density

\(\frac{\rho_2}{\rho_1} = \frac{(\gamma+1)M_{n1}^2}{(\gamma-1)M_{n1}^2 + 2}\)

Increase

Temperature

\(\frac{T_2}{T_1} = \frac{p_2}{p_1} \frac{\rho_1}{\rho_2}\)

Increase

Stagnation Pressure

\(\frac{p_{02}}{p_{01}} = e^{-\Delta s / R}\)

Decrease