[Farouk]

Hi there,
I am really new to the CFD simulation, and started some simple algorithms recently. I then got introduced to the Riemann Invariants. Can any one provide some physical interpretation?
Also, why is it the case, that when we have an open tube, and the flow is entering with a subsonic speed, then at this point, only one characteristic exist dx/dt=u+a ?

Thank you in advance.

[FMDenaro]

Originally Posted by Farouk

Hi there,
I am really new to the CFD simulation, and started some simple algorithms recently. I then got introduced to the Riemann Invariants. Can any one provide some physical interpretation?
Also, why is it the case, that when we have an open tube, and the flow is entering with a subsonic speed, then at this point, only one characteristic exist dx/dt=u+a ?

Thank you in advance.

Riemann invariants are a combination of convective and sound velocity (multiplied by a gas-dependent constant) that remains constant along particular curves of the space-time domain. Therefore, you have a physical global quantity that propagates with the same initial value in some directions and at some velocity. I dont think that some further physical meaning exist...

Furthermore at subsonic speed you have u<a, thus

dx/dt = u+a= a*(M+1) >0 for C+
dx/dt = u-a= a*(M-1) <0 for C-

You can see that for subsonic flows two characteristic curves exist but having opposite direction

[Farouk]

Originally Posted by FMDenaro

Riemann invariants are a combination of convective and sound velocity (multiplied by a gas-dependent constant) that remains constant along particular curves of the space-time domain. Therefore, you have a physical global quantity that propagates with the same initial value in some directions and at some velocity. I dont think that some further physical meaning exist...

Furthermore at subsonic speed you have u<a, thus

dx/dt = u+a= a*(M+1) >0 for C+
dx/dt = u-a= a*(M-1) <0 for C-

You can see that for subsonic flows two characteristic curves exist but having opposite direction

Hi FMDenaro and thanks for the quick reply. Actually, I see that there are most of the times three characteristics, two which you already mentioned, and the third one, for dx/dt=u.

I see that for subsonic case, with subsonic flow exiting the tube, the two characteristics which exist are dx/dt=u+a and dx/dt=u. The question now is why is it so? what did cancel the third characteristic C-?

Thank you for your help.

[FMDenaro]

For homoentropic flows, the entropy is constant everywhere in the domain and the characteristic C0, that is dx/dt = u (trajectory) is not relevant to define an invariant property.

The third characteristic becomes relevant for isoentropic flows where s is constant only along the trajectory dx/dt=u. However, Riemann invariants do not exist for such case.

In a subsonic flow, at inlet you have two characteristic curves (u, u+a) entering in the domain and one leaving (u-a) while at an outlet you have two characteristics leaving (u, u+a) and one entering (u-a) from outlet.

This fact must be respected in prescribing the correct BCs.