Oblique shock

An oblique shock wave, unlike a normal shock, is inclined with respect to the incident upstream flow direction. It will occur when a supersonic flow encounters a corner that effectively turns the flow into itself and compresses. The upstream streamlines are uniformly deflected after the shock wave. The most common way to produce an oblique shock wave is to place a wedge into supersonic, compressible flow. Similar to a normal shock wave, the oblique shock wave consists of a very thin region across which nearly discontinuous changes in the thermodynamic properties of a gas occur. While the upstream and downstream flow directions are unchanged across a normal shock, they are different for flow across an oblique shock wave.

It is always possible to convert an oblique shock into a normal shock by a Galilean transformation.

For a given Mach number, M₁, and corner angle, θ, the oblique shock angle, β, and the downstream Mach number, M₂, can be calculated. Unlike after a normal shock where M₂ must always be less than 1, in oblique shock M₂ can be supersonic (weak shock wave) or subsonic (strong shock wave). Weak solutions are often observed in flow geometries open to atmosphere (such as on the outside of a flight vehicle). Strong solutions may be observed in confined geometries (such as inside a nozzle intake). Strong solutions are required when the flow needs to match the downstream high pressure condition. Discontinuous changes also occur in the pressure, density and temperature, which all rise downstream of the oblique shock wave.

Using the continuity equation and the fact that the tangential velocity component does not change across the shock, trigonometric relations eventually lead to the θ-β-M equation which shows θ as a function of M₁ β, and ɣ, where ɣ is the Heat capacity ratio.

${\displaystyle \tan \theta =2\cot \beta {\frac {M_{1}^{2}\sin ^{2}\beta -1}{M_{1}^{2}(\gamma +\cos 2\beta )+2}}}$

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