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12.2: Oblique Shock

The shock occurs in reality in situations where the shock has three–dimensional effects. The three–dimensional effects of the shock make it appear as a curved plane. However, one–dimensional shock can be considered a representation for a chosen arbitrary accuracy with a specific small area. In such a case, the change of the orientation makes the shock considerations two–dimensional. Alternately, using an infinite (or a two–dimensional) object produces a two–dimensional shock. The two–dimensional effects occur when the flow is affected from the ``side,'' i.e., the change is in the flow direction. An example of such case is creation of shock from the side by deflection shown in Figure . To match the boundary conditions, the flow turns after the shock to be parallel to the inclination angle schematicly shown in Figure 12.3. The deflection angle, \(\delta\), is the direction of the flow after the shock (parallel to the wall). The normal shock analysis dictates that after the shock, the flow is always subsonic. The total flow after the oblique shock can also be supersonic, which depends on the boundary layer and the deflection angle. The velocity has two components (with respect to the shock plane/surface). Only the oblique shock's normal component undergoes the ``shock.'' The tangent component does not change because it does not ``move'' across the shock line. Hence, the mass balance reads

\[
    \rho_1\, {U_1}_n    = \rho_2\, {U_2}_n    
    \label{2Dgd:eq:Omass} \tag{1}
\]

The momentum equation reads

\[
    P_1 + \rho_1 \,

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    \label{2Dgd:eq:OM1n} \tag{7}
\]

and in the downstream side reads

\[
    \sin (\theta - \delta ) = \dfrac

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    \label{2Dgd:eq:OM2n} \tag{8}
\]
Equation (8) alternatively also can be expressed as
\[
    \cos \theta = \dfrac
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    \label{2Dgd:eq:OM1t} \tag{9}
\]
And equation (9) alternatively also can be expressed as
\[
    \cos\, \left(\theta - \delta \right) = \dfrac
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    \label{2Dgd:eq:OM2t} \tag{10}
\]

The total energy across a stationary oblique shock wave is constant, and it follows that the speed of sound is constant across the (oblique) shock. It should be noted that although, \({U_1}_t = {U_2}_t\) the Mach number is \({M_1}_t  \neq {M_2}_t  \) because the temperatures on both sides of the shock are different, \(T_1 \neq T_2\). As opposed to the normal shock, here angles (the second dimension) have to be determined. (8) through (??), is a function of four unknowns of \(M_1\), \(M_2\), \(\theta\), and \(\delta\). Rearranging this set utilizing geometrical identities such as \(\sin\alpha = 2\sin\alpha\cos\alpha\) results in

Angle Relationship

\[
    \label {2Dgd:eq:Osol}
    \tan \delta = 2\, \cot \theta\,  
        \left[\dfrac{{M_1}^{2}\, \sin^2 \theta - 1 }{
         {M_1}^{2} \, \left(k + \cos\, 2 \theta \right) +2 }\right] \tag{11}
\]

The relationship between the properties can be determined by substituting \(M_1 \sin \theta\) for of \(M_1\) into the normal shock relationship, which results in

Pressure Ratio

\[
    \label{2Dgd:eq:OPbar}
    \dfrac{P_2 }{ P_1} = \dfrac{2\,k\, {M_1 }^{2} \sin^2 \theta - (k-1) }{ k + 1}   \tag{12}
\]

The density and normal velocity ratio can be determined by the following equation

Density Ratio

\[
    \label{2Dgd:eq:OrhoBar}
    \dfrac{\rho_2 }{\rho_1} = \dfrac{{U_1}_n }{ {U_2}_n} =
    \dfrac{ (k+1) {M_1}^{2} \sin^2\theta}  
        {(k-1) {M_1}^2 \sin^2\theta + 2}   \tag{13}
\]  

The temperature ratio is expressed as

Temperature Ratio

\[
    \label{2Dgd:eq:OTbar}
    \dfrac{T_2 }{ T_1} = \dfrac{{2\,k\, {M_1}^2  \sin^2\theta - (k-1)  
            \left[(k-1) {M_1}^2 + 2 \right] } }
        {{(k+1)^2 \,{M_1}} } \tag{14}
\]

Prandtl's relation for oblique shock is
\[
    U_{n_1}U_{n_2} = c^{2} - \dfrac{k -1 }{ k+1} \, {U_t}^2  
    \label{2Dgd:eq:Oprandtl} \tag{15}
\]
The Rankine-Hugoniot relations are the same as the relationship for the normal shock
\[
    \dfrac{P_2 - P_1 }{ \rho_2 - \rho_1} = k \,\dfrac{ P_2 - P_1 }{ \rho_2 - \rho_1}
    \label{2Dgd:eq:ORankineHugoniot} \tag{17}
\]

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.