Calc Oblique Shock Angle for Supersonic Flow 0-5 Degrees

[Harry Fry]

Is there an equation to calculate the oblique shock angle for supersonic flow when the given angle of attack is greater than 0, but less than the half angle?

In my particular practical experiment, the half angle of the aerofoil is 5 degrees, so want to get a variety of figures between 5 and -5 as I know expansion waves begin when the angle is increased to greater than the half angle.

Thanks in advance

Harry

 

[boneh3ad]

What shape are we talking about here? There are ways to do this analytically for a wedge, but not just a general airfoil shape. Further, the rounded leading edge of a typical airfoil is going to cause a bow shock, not an oblique shock.

 

[Harry Fry]

Apologies, I meant a diamond shaped wedge.

 

[boneh3ad]

Ah, well in that case it is pretty easy as long as you know the angle the surface makes to the free-stream direction and the Mach number. Basically, there is an equation sometimes called the $\theta$-$\beta$-$M$ equation (you can find it on the Wikipedia page for oblique shocks). It is
$$\tan\theta = 2\cot\beta \dfrac{M_{\infty}^2\sin^2\beta -1}{M_{\infty}^2\left[ \gamma + \cos (2\beta) \right] + 2},$$
where $\theta$ is the angle your wedge surface makes to the free-stream direction (or turning angle since the flow must turn that much), $\beta$ is the angle that the shock makes with the free-stream direction, $M_{\infty}$ is the free-stream Mach number, and $\gamma$ is the ratio of specific heats. You can find some online calculators that will do this math for you, but it isn't too bad to just program a solver yourself to do it.

 

[Harry Fry]

Ah thanks.

I found one online but that is for when the angle of attack is 0. Should I work out each half of the wedge individually, relative to the free stream direction as if the half angle was greater than 5 in my example?

 

[boneh3ad]

Yes, do each side individually. That's one of the nice things about supersonic flows. The formula can handle any angle to the flow so long as it isn't so large as to cause a detached shock, at which point the solution breaks down. Also, if you have a wedge angled at greater that its half-angle, you can still use this formula on the surface that is angled away from the flow, but on the other surface you will have an expansion and so you have to use the Prandtl-Meyer function. It's still pretty easy to do, though.

 

[Harry Fry]

Thank you very much! You've been a huge help.