[CFDDmitry]

Tell me please how to make a numerical simulation of tee or junction of two pipes for the equations

[adrin]

Can you explain the variables please? The notation, if it's in the standard format, doesn't make physical sense to me. At the very least, there are a list of assumptions above that you'd need to elaborate on.

adrin

[CFDDmitry]

This is ordinary 1d euler equations.
rho - density
v - velocity
P - pressure
eik - internal energy

I did like in this article http://www.hindawi.com/journals/jam/2010/407648/
But I need a solution for shock tube.

[adrin]

Ah, I see; a tee would be at least a 2-D problem, and the notation v for velocity (instead of u) threw me off into thinking that you're trying to solve a 2D problem with a (wrong) 1D equation.

So, basically, with a tee you have to break up the geometry into 3 "pipe" segments, each of which would solve the 1D partial differential equation with its own boundary and initial conditions; and the three solutions connected at the "tee point", by making sure they share the same density, pressure, energy, etc. By the way, you are missing a constitutive relationship between density and pressure (or something similar). There are 4 variables and 3 equations.

For each pipe section "i", you'd have to discretize the pipe length L_i into N_i discrete points and write the differential equation in its finite difference form. The discretization is not that difficult and can be found in any basic applied numerical analysis. You have to sit down and write out the formulation; collect the like terms; combine them at the tee section to end up with a system of equations to solve. That's more than is warranted on this forum

adrin

[CFDDmitry]

Big thanks!