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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.