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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.

Oblique Shock 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. M₂ is always less than M₁. Unlike after a normal shock, M₂ can still be supersonic. 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 and β. It was believed that it is complicated and difficult to solve for β as a function of M₁ and θ. In the past, the results were obtained from tables or graphes. More recently, several computers program are available on-line for example Potto-GDC . In fact, today there two textbooks with the partial solution and one book with the full solution. The interesting part of the analytical solution is not that it provides easy solution but beter inside and explanation to why things happens the way they happen.

The geometrical representation of the physical situation is

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

Within the θ-β-M equation, a maximum corner angle, θ_MAX, exists for any upstream Mach number. When θ > θ_MAX, the oblique shock wave cannot exist and is replaced with normal wave. A one possibility of θ-β-M diagram, shows a series of curves that will indicate θ_MAX for each Mach number. The θ-β-M relationship will produce three β angles for a given θ and M₁ (but only two are shown), with the larger angle called a strong shock and the smaller called a weak shock. The boundary conditions determine what kind of shock that appear after the oblique shock.

The rise in pressure, density, and temperature after an oblique shock can be calculated as follows:

${\displaystyle {\frac {p_{2}}{p_{1}}}=1+{\frac {2\gamma }{\gamma +1}}(M_{1}^{2}\sin ^{2}(\beta )-1)}$

${\displaystyle {\frac {\rho _{2}}{\rho _{1}}}={\frac {(\gamma +1)M_{1}^{2}\sin ^{2}(\beta )}{(\gamma -1)M_{1}^{2}\sin ^{2}(\beta )+2}}}$

${\displaystyle {\frac {T_{2}}{T_{1}}}={\frac {p_{2}}{p_{1}}}{\frac {\rho _{1}}{\rho _{2}}}}$

M₂ is solved for as follows:

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

Oblique Shock Wave Applications

Oblique shock waves are used predominantly in engineering applications when compared with normal shock waves. This can be attributed to the fact that using one or a combination of oblique shock waves results in more favorable post-shock conditions (lower post-shock temperature, etc.) when compared to utilizing a single normal shock. An example of this technique can be seen in the design of supersonic aircraft engine inlets, which are wedge-shaped to compress air flow into the combustion chamber while minimizing thermodynamic losses. Early supersonic aircraft jet engine inlets were designed using compression from a single normal shock, but this approach caps the maximum achievable Mach number to roughly 1.6. The wedge-shaped inlets are clearly visible on the sides of the F-14 Tomcat, which has a maximum speed of Mach 2.34.

Many supersonic aircraft wings are designed around a thin diamond shape. Placing a diamond-shaped object at an angle of attack relative to the supersonic flow streamlines will result in two oblique shocks propagating from the front tip over the top and bottom of the wing, with Prandtl-Meyer expansion fans created at the two corners of the diamond closest to the front tip. When correctly designed, this generates lift.

Oblique Shock Waves and the Hypersonic Limit

As the Mach number of the upstream flow becomes hypersonic, the equations for the pressure, density, and temperature after the oblique shock wave reach a limit (mathematics). The pressure and density ratios can then be expressed as:

${\displaystyle {\frac {p_{2}}{p_{1}}}\approx {\frac {2\gamma }{\gamma +1}}M_{1}^{2}\sin ^{2}(\beta )}$

${\displaystyle {\frac {\rho _{2}}{\rho _{1}}}\approx {\frac {\gamma +1}{\gamma -1}}}$

For a perfect atmospheric gas approximation using γ = 1.4, the hypersonic limit for the density ratio is 6. However, hypersonic post-shock dissociation of O₂ and N₂ into O and N lowers γ, allowing for higher density ratios in nature. The hypersonic temperature ratio is:

${\displaystyle {\frac {T_{2}}{T_{1}}}\approx {\frac {2\gamma (\gamma -1)}{(\gamma +1)^{2}}}M_{1}^{2}\sin ^{2}(\beta )}$

See also

• Gas dynamics
• Bow shock (aerodynamics)
• Shock wave
• Prandtl-Meyer expansion fan

External links

• NASA oblique shock wave calculator (Java applet).

• Fluid dynamics
• Aerodynamics