PDEs & Shock Waves: Understanding Inviscid Burgers' Equation

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Forgive me for the long post, but I'm in some desperate need of clarity on this matter. I just can't seem to grasp the whole shock wave concept, or at least the meaty part of it . I only have a couple of problems left to do to finish my HW I'm at an impasse until I dispel my confusion. I really hope someone here can help clear my confusion.

With the PDE we're working with (Inviscid Burgers' eqn u u_x + u_t = 0) the characteristics are lines and u(x, y) is constant along them. What I don't quite get is what happens exactly when a discontinuity is introduced and a shock wave develops.

Suppose we start at some x0 for which u(x0, 0) = ul is constant along some characteristic. At some (x*, t*), u(x, y) would be multi-valued and thus a shock develops. It makes sense to me that no value of u exists on the shock line and that physically u still propogates because of conservation laws. What I can't seem to grasp is how we find the value ur after the shock and what characteristic it is governed by.

I know that given the conservation law between rate of change of u and flux across the discontinuity (s-[q] = 0 where [x] defines xl - xr and s is the shock speed) we can find the jump conditions, but I don't understand what those mean or how to phrase them exactly. Also, if ul jumps to ur, what characteristic does it jump to? What confuses me is that other characteristics eventually collide as well for later times even though physically there may have only been one discontinuity. Do we no longer look to characteristics after the discontinuity? Maybe we ignore further collisions since we crossed the discontinuity already?

Very much thanks to anyone that can help me understand this better or point me somewhere I can find out. My textbooks suck and my prof is out of town.

 

[Clausius2]

I know that given the conservation law between rate of change of u and flux across the discontinuity (s-[q] = 0 where [x] defines xl - xr and s is the shock speed) we can find the jump conditions, but I don't understand what those mean or how to phrase them exactly. Also, if ul jumps to ur, what characteristic does it jump to? What confuses me is that other characteristics eventually collide as well for later times even though physically there may have only been one discontinuity. Do we no longer look to characteristics after the discontinuity? Maybe we ignore further collisions since we crossed the discontinuity already?

In my opinion, you're not too far of the real meaning of a shock wave. Physically, a characteristic line is a transporter of the information across the flow field. I'm going to show you a little example where more than one characteristic line is produced:

Imagine you have a closed pipeline of length L(closed at the two ends) filled with gas at pressure Po. The external pressure of the atmosphere is Pa. Let's consider Pa<<Po. At some instant (t=0), one end of the pipeline is opened so that the gas tends to flow out the pipeline into the atmosphere:

-----------------------!
Po-----> Pa;
----------L------------!

During the very first instants (at times of the order of L/a, where a=speed of the sound) there are produced an expansion fan of characteristic. This expansion fan contains some number of characteristics, which transports the information of the recent-opened section to the rest of the gas. They are who says to the rest of the gas an expansion is happening. Additionally this expansion fan runs to the left at a velocity (a-u) where u is the velocity of the accelerated flow. Through each characteristic line can be established some equations stating a conservation law: Continuity, Momentum and Energy. Surely the variables u, P or T are not defined just in the edge of the characteristic line, but the flux variables such $$\rho u; \rho u^2+P; \rho u T$$ do are defined at any point and through the characteristic line. These principal flux variables enhances us to solve the flow field without being necessary to pay attention to discontinuities. The expansion fan carries progressively through each characteristic line, the progressive expansion of the gas. This progressive gas expansion is made in a narrow thickness of the flow field, of the order of the înverse of the Reynolds Number.

On the other hand, a propagation of the shock wave into a flow field is something similar. The conservations laws are different because entropy is not constant across the shock, but there are again some flux variables well-defined, which erase totally the discontinuities of the equations. In this case, if you don't have any reflection effect against boundaries, only one characteristic line is found.

In my short experience about that, I have to tell you the unique mode of understanding the physics of this phenomena is solving a complete problem of shock waves or expansion fans. I said a complete problem. This only can be done by numeric techniques, but programming a well-posed problem of the Navier-Stokes equations will give you a wider idea of what is happening mathematically and physically inside the equations, and why you do not have any problems through the discontinuities of the flow.

Any questions?. I hope this could make you the things easier.

 

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Thanks for the reply.

You helped me to conceptualize it, so your post is much appreciated. :)

 

[Clausius2]

No thanks needed. Always happy of answering this kind of questions.