An oblique shock wave is a shock wave that, unlike a normal shock, is inclined with respect to the direction of incoming air. It occurs 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.
Wave theory
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.
The θ-β-M equation 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.
tan θ = 2 cot β M 1 2 sin 2 β − 1 M 1 2 ( γ + cos 2 β ) + 2 {\displaystyle \tan \theta =2\cot \beta \ {\frac {M_{1}^{2}\sin ^{2}\!\beta -1}{M_{1}^{2}(\gamma +\cos 2\beta )+2}}}
It is more intuitive to want to solve for β as a function of M1 and θ, but this approach is more complicated, the results of which are often contained in tables or calculated through a numerical method.
Maximum deflection angle Within the θ-β-M equation, a maximum corner angle, θMAX, exists for any upstream Mach number. When θ > θMAX, the oblique shock wave is no longer attached to the corner and is replaced by a detached bow shock. A θ-β-M diagram, common in most compressible flow textbooks, shows a series of curves that will indicate θMAX for each Mach number. The θ-β-M relationship will produce two β angles for a given θ and M1, with the larger angle called a strong shock and the smaller called a weak shock. The weak shock is almost always seen experimentally. The rise in pressure, density, and temperature after an oblique shock can be calculated as follows:
… excerpt ends here. Continue reading the full article.
