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 ?
(1) A physical interpretation of Riemann Invariants is only possible for an isentropic flow. For this case, there are two families of characteristics $$ \frac{dx}{dt}=u+c\\ \frac{dx}{dt}=u-c\\ $$ where $u$ is the local gas speed and $c$ is the local sound speed. The two characteristics represent the left and right going waves.
The Riemann invariants for such a case are given by: $$ u+\frac{2c}{\gamma-1}=r \mbox{ along }\frac{dx}{dt}=u+c\\ u-\frac{2c}{\gamma-1}=s \mbox{ along }\frac{dx}{dt}=u-c\\ $$
$r$ and $s$ are the Riemann invariants. Now notice what this statment is saying. Consider a right going wave. The Riemann invariant tells you that $u$ and $c$ can't change arbitrarily. They have to change in such a way that when we add $u$ to $\frac{2c}{\gamma-1}$, we recover the constant $r$. That is why they are also called compatability conditions.
When entropy is involved for an non-isentropic flow, the Riemann invariants become complex and will no longer have a simple interpretation. To see this, you need to see the derivation of the Riemann Invariants in standard Gas Dynamics books like Anderson, Compressible Flow or Shapiro.
(2) could you clarify your second question and what you are trying to do? cheers, abiyo
