I have started the simulation of 1D unsteady multiphase flow. This flow can be described by the hyperbolic equations of the gas dynamics with complicated equation of state. The problem I have is with the boundary conditions. I have a chamber connected with an atmosphere by a pipe. Initially the exit pressure is atmospheric. Then after the diaphragm removing the flow develops. After some time when the chamber pressure decreases the flow becomes subsonic. It is clear that after a relaxation process governed by the outer flow which I cannot calculate (it is 2D and govered by different set of equations), pressure on the end of the pipe must become atmospheric.

Could anyone suggest the boundary condition which can describe the smooth transition from supersonic flow to subsonic in this case?

Dorogoi Oleg,

Could you please state your problem more clearly, in more details? What is the initial state of the flow in the chamber and in the pipe. What is the transitional state of these later on, and what is the final outcome of the flow. When is the flow subsonic and when is it supersonic? Do you expect to reach a steady state and eventually all the flow will be stationary? One way of coming around the problem of the boundary condition on the velocities is to treat the boundary conditions on the characteristics of the flow. Vsievo Xoroshievo. PG.

Dear Patrick, Thanks a lot for reply!

Initially velocity is 0 everywhere, pressure at the end and in a part of the pipe is atmospheric and higher in the chamber and other part of the pipe. Then I remove the diaphragm and rarefaction wave propagates into the pipe. Shock wave propagates in the opposite direction. After it reaches the end of the pipe the supersonic flow region develops near the end of the pipe. But as the chamber pressure decreases as it empties the discharge rate decreases and flow becomes subsonic everywhere. Steady state solution is when pressure in the chamber is equal to atmospheric and discharge rate is zero. I have a problem with transition from supersonic flow to subsonic. It is clear that I don't need any BC when the flow is supersonic and I have to assume continuity of parameters at the end of the pipe when flow is subsonic. The problem is that I cannot calculate the flow in the atmosphere with the same set of equations. It is also clear that after some time exit pressure will become atmospheric.

I hope I have clarified the problem. As for treating boundary conditions on the characteristics of the flow, I have one incoming characteristic that keeps the information from the atmosphere for subsonic flow.

It was nice to see a bit of Russian in your reply. Do you have any connections with Russia?

Best regards, Oleg

Hi,

When the flow is supersonic all the characteristics of the flow are outgoing at velocities v, v+c, v-c. So as you state there is no need for imposing conditions from the outisde on the boundary. When it is subsonic you probably have only one incoming at velocity v-c.

I am not sure I understand when you mentioned that you cannot calculate the atmosphere with the same set of equations. I though that the atmosphere is just a condition at the end of the pipe (boundary condition). And continuity of parameters is expected with the ambient atmosphere only when the steady state is reached, before that the flow at the boundary adjust itself to a combination of imposed and computed values through the values taken by the characteristics. When the flow is supersonic you don't really worry about bondary conditions and it is OK. Once the flow becomes subsonic you solve for the characteristics of the flow. The incoming characteristics take value from the outside ambient atmosphere (p=atmospheric pressure, v=0). The ougoing characteristics take computed values (eiter by extrapolating the computed value from inside the computational domain through the boundary; or by solving for the charateristics itself). Then v and P are obtained by solving v and P as a function of the characteristics. So v and P (and rho, all at the boundary) are a combination of computed and exact values of the characteristics of the flow through the boundary. So their values change with time ('adapt') and eventually settle to the steady state value when this one is reached.

Sorry, no connection, just a few words. PG.

i, The problem is that really I have a jet in the atmosphere (which is mixture of air and multiphase mixture from the chamber) and I need to take information not from ambient atmosphere (p= atmospheric pressure, v=0), but from the point of the jet located near the pipe exit (on distance (c -V)*dt). There velocity and pressure are not equal to ambient until everything slows down significantly.

I have tried artificial method: if the flow is subsonic and pressure greater then atmospheric I assume that v=c and calculate density using the discharge rate (rho*v) from the nearest point inside the pipe. It gives monothonic decrease of pressure to atmospheric value in sonic regime. Then I keep pressure equalled to atmospheric.

As it is very artificial I want to know you opinion.

Regards, Oleg

I understand that you are imposing the boundary conditions directly on the primitive variables (v, P and rho) and so the results you get at the boundary is a direct consequence of the 'input'. Whether it is a good approximation or not I don't know. But you could check your results with the experiments and also you could try to change the conditions to see how they affect the results. A good approximation would give the same results no matter if you change the conditions. For example putting v=c (when v is actually subsonic) means that you are 'pumping' the flow outside.

If you impose the boundary conditions using the characteristics method, then the values of v, P and rho at the boundary will not be the atmospheric values (unless the flow has reached steady state). The values there will be close to the exact physical solution if the treatment is made properly. You might get a transient solution at the boundary very different than the ambient atmosphere and the characteristics method is able to treat it correctly (in a similar manner to the supersonic shock escaping from the pipe; it is clear that the atmosphere is not supersonic, and though you have no problem to get a shock at the exit - this is because when the flow is supersonic, imposing the conditions at the boundary on the primitive variables is completely equivalent to imposing them on the chracteristics, and the values imposed are just the computed one from the simulations. So no matter if you don't put bc when the flow is supersonic, it is equivalent to solving the bc correctly and therefor you don't have any problem). This treatment is not perfect but it is usually better than simply 'forcing' the conditions on the primitive variables at the boundary.

PG

Ok! I can write these equations in characteristic form and moreover can find Riemann invariants, but it is still not clear how to calculate the value of Riemann invariant which is coming from the outside (what velocity and pressure function to use).

Oleg

From the outside you should put the values expected from the atmosphere: v=0, P=atmospheric pressure, rho=atmospheric density. As you mentioned in 1D the characteristics of the flow are the Riemann invariants, and their exact form depends on the equation of state actually. But assuming an ideal gas, or a polytropic relation or another barotropic equation of state might be enough to approximate the flow in order to find the characteristics. So on the outside you use the atmospheric values of P, v and rho to compute the incoming characteristics. On the inside you use the computed values (obained by extrapolating wisely for example) of P, v and rho to computed the outgoing characteristics. Then you solve back for P, v and rho using the 'mixture' of the outgoing and incoming characteristics (let me know if what I write makes sens to you). I don't know how much you know about the method, so I am not sure how much I should go into the details. The values so obtained for P, v and rho are not the atmospheric ones as I said, but rather a correction of them due to the propagation of the flow there.

Cheers, Patrick.

Hi, Patrik! Thank you for your time. I know the method of characteristics and will think carefully about what you suggested.

Regards, Oleg