In a hyperbolic problem having 2 or more discontinuous initial data, is it possible for two shocks or two rarefactions or one shock and one rarefaction to run into each other?
In that case what happens?
In my opinion, if 2 shocks face the same direction, and the one behind has higher speed, then they would run into each other, forming a new shock.
But then what is the speed of this new shock?
Please, share your experience.
I suppose you can compute it using the two states on the extremities of both colliding shocks, ie. if shock 1 is between u1 and u2, and shock 2 between u2 and u3, the resulting shock will be between u1 and u3; the total pressure ratio (not jump) is then the product of both ratios.
Two shock fronts meeting together form new wave, like Mach front, when blast wave meets its reflection from the ground.
Shock dynamics is driven by the pressure. I'm not sure, but in your case probably velocity of resultant wave after will be higher then velocity of faster wave. A reason is higher total pressure in joined fronts.
You can evaluate it by considering that the total pressure jump across the shock will be the product of the two smaller ones.
Try George Rudinger in Flow energetics and Joseph Foa in Flow propulsion
They have done amazing things.
Try also younger researchers
I can think of at least one situation in 'nature' in which this phenomenon occurs - a shock diamond or Mach disk. A shock diamond is formed when a flow exits a nozzle at supersonic speeds and at a pressure that is different from that of the external atmosphere. In this situation one will observe a series of compression and expansion waves which intersect (i.e., 'run into') each other in a characteristic 'diamond pattern.' This is commonly observed in the exhaust of the Space shuttle boosters, SR-71, as well as many fighter jets etc.
For further reading and theory I suggest taking a look at Modern Compressible Flow by John Anderson.
I suppose you can compute it using the two states on the extremities of both colliding shocks, ie. if shock 1 is between u1 and u2, and shock 2 between u2 and u3, the resulting shock will be between u1 and u3; the total pressure ratio (not jump) is then the product of both ratios.
Please read chapter 7.6 in vol. II of Handbook on Shock Waves by Ben-Dor, Igra & Elperin
The following work contains Rudinger and Foa in the references and gives enough information for the traveling waves.
Article Analysis of industrial hydraulic systems and components
Here http://www.ifm.umich.mx/guzman/Grupo/Educacion_3.pdf we show that case in a spherical blast wave. There is a blast (shock) wave intially, then a different wave reflects back (crosses the center) from the center and chases the original shoch. The result is a reverse shock.
Take the simple case of a scalar non-linear conservation equation with two jumps in the initial data. Assume the flux function is either concave or convex. Let's take the case of two shockwaves. Each jump in the data results in a discontinuity that travels through space-time at a constant slope (a shockwave). The speed of each shockwave depends on the conditions on either side of the discontinuity. Let's assume that the "state" is described by a density "u" and a flux "f = f(u)", so that the initial data comprise three "states" {(u1, f1), (u2,f2), (u3,f3)}. What happens when the two waves meet is that at that moment (and thereafter), the state (u2,f2) seizes to exist; the speed of the new shockwave depends now on the two states (u1,f1) and (u3,f3).
I thought it this way but I was wondering if it was right. So, one needs to apply the jump condition to these states?
Thanks.
It is right. And, yes, the new shock speed is calculated from the remaining two states. For the simple case of scalar non-linear conservation laws, you can find more info in "Numerical Methods for Conservation Laws" by Randall LeVeque
@Koen yes, you did .. I didn't read previous responses before typing mine. Thanks.
As I have already mentioned on June 6 please read chapter 7.6 in Vol. II of Handbook on Shock Waves by Ben-Dor, Igra & Elperin
Here is the exact solution to the Riemann problem.... http://www.ifm.umich.mx/guzman/Grupo/Educacion_4.pdf
Here is a reference that may provide some assistance to you.
http://www.jetp.ac.ru/cgi-bin/dn/e_061_02_0275.pdf
This older report (1954) seems to be exactly what will help you.
http://aerade.cranfield.ac.uk/ara/1954/naca-report-1165.pdf
A simple way of looking at this, similar to what Koen Hillewaert said, is that since a shock propagates at a velocity that is about the average of the speed of sound ahead (c1) and the speed of sound behind the shock (c2) , v = (c1 + c2)/2, with c2 > c1. If there were a second shock behind the first shock there could be three regions with increasing sound speeds c3 > c2 > c1, so the second shock propagates faster than the first v_23 > v_12, and should catch up and merge.
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