12.2.2.4: Given Two Angles, $$\delta$$ and $$\theta$$

It is sometimes useful to obtain a relationship where the two angles are known. The first upstream Mach number, $$M_1$$ is

Mach Number Angles Relationship

$\label{2Dgd:eq:OM1} {M_1}^2 = \dfrac{ 2 \,( \cot \theta + \tan \delta ) } { \sin 2 \theta - (\tan \delta)\, ( k + \cos 2 \theta) } \tag{59}$

The reduced pressure difference is
$\dfrac{2\,(P_2 - P_1) }{ \rho\, U^2} = \dfrac{2 \,\sin\theta \,\sin \delta }{ \cos(\theta - \delta)} \label{2Dgd:eq:OreducedPressure} \tag{60}$
The reduced density is
$\dfrac{\rho_ 2 -\rho_1 }{ \rho_2} = \dfrac{\sin \delta }{ \sin \theta\, \cos (\theta -\delta)} \label{2Dgd:eq:OreducedDensity} \tag{61}$
For a large upstream Mach number $$M_1$$ and a small shock angle (yet not approaching zero), $$\theta$$, the deflection angle, $$\delta$$ must also be small as well. Equation (51) can be simplified into
$\theta \cong {k +1 \over 2} \delta \label{2Dgd:eq:OlargeM1theta} \tag{62}$
The results are consistent with the initial assumption which shows that it was an appropriate assumption.

Fig. 12.9 Color-schlieren image of a two dimensional flow over a wedge. The total deflection angel (two sides) is $$20^\circ$$ and upper and lower Mach angel are $$\sim 28^\circ$$ and $$\sim 30^\circ$$, respectively. The image show the end–effects as it has thick (not sharp transition) compare to shock over a cone. The image was taken by Dr.~Gary Settles at Gas Dynamics laboratory, Penn State University.

Contributors

• Dr. Genick Bar-Meir. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or later or Potto license.