How Do Strong Shock Waves Behave in Adiabatic Conditions?

[Cantique]

Homework Statement

Derive the strong shock condition given the Rankine-Hugoniot conditions:

$\frac{ρ_{2}}{ρ_{1}}$ = $\frac{u_{1}}{u_{2}}$ = $\frac{γ+1}{γ-1}$

where u₁, ρ₁ are velocity and density upstream of the shock and u₂ ρ₂ are velocity and density downstream of the shock.


Two identical gas clouds collide with a relative velocity of 2V, where V is very much larger than the sound speed in the clouds. A strong, planar, adiabatic shock wave propagates into the clouds away from the surface of contact. Explain why, in the zero-momentum frame, the shocked gas must be stationary, while the unshocked gas continues to travel at its initial speed V.

How fast does the shock travel in the zero-momentum frame?

Without detailed calculation discuss what happens when the shock reaches the side of the cloud distant from the surface of contact.

Homework Equations

Rankine Hugoniot equations

The Attempt at a Solution

I have completed the first part and second part asking for the explanation. I am stuck on the problem of how fast the shock travels in the zero-momentum frame. In my opinion it should do one of two things:

Stay stationary at the centre of the two clouds - The reason being that the fact that the downstream speed is zero in the zero-momentum frame means that all the gas collides only at the centre where it then remains. However, this has the problem of creating an infinitely large pressure at the centre which would eventually result in the sound speed being higher than V (the speed of the unshocked gas) so the shocked gas would start to spread out under the thermal pressure and so the position of the shock would move out to each edge which leads to my second case:


The shock moves at speed V towards the edge. Following on from the reasoning of the other case.

I think the second case is more likely than the first, but don't believe it to be correct. Am I thinking about this problem incorrectly?

 

[Asim Riaz]

And how can I calculate the shock speed in the zero-momentum frame?For the third part, I would think that the shock will move outwards and eventually cause the two gas clouds to separate, with each cloud now travelling at a velocity of V in opposite directions. Is this correct?