I am really new to the CFD simulation, and started some simple algorithms recently. I then got introduced to the Riemann Invariants.

  1. Can any one provide some physical interpretation?

  2. 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 Answer 1


(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

  • $\begingroup$ Thanks for the comprehensive and simple clarification! Conceserning my second question, I am actually trying to understand the application of the Method of Characterestics to treat the boundary conditions, and in a given reference, the author states that when in an open tube, the flow is entering with a subsonic speed, he considers only one characterestic, that is to say, one riemann invariant. The author does not provide any explanations, so...My question is, why only one caracterestic, i.e , one riemann invariant? $\endgroup$ Commented Jul 18, 2013 at 22:21
  • $\begingroup$ if you specify one Riemann invariant and also specify the gas speed or the local sound speed, the other Riemann invariant is determined immediately. For example if a left going wave hits a wall, $u=0$ and if you know the $s$ for such a wave, you can determine $c$ which in turn implies that you now know $r$. Could you tell me what book it is and its page number? I could take a look and provide a detailed explanation. $\endgroup$
    – felasfa
    Commented Jul 19, 2013 at 22:22
  • $\begingroup$ @felasfa Wonder if you could recommend any books, web pages on this subject (Riemann invariants, characteristics) for someone just starting out in this area? $\endgroup$
    – jim
    Commented Mar 15, 2016 at 21:28
  • $\begingroup$ @ jim An excellent reference is "Modern Compressible Flow: With Historical Perspective" by Anderson. The classic "Supersonic flow and shock waves" can be hard read but is very comprehensive. There is another brief but good treatment on "Physics of Shock Waves and High-Temperature" by Zeldovich. I did quite a work on this so if you need specific references, let me know. I hope this helps. $\endgroup$
    – felasfa
    Commented Mar 29, 2016 at 21:44

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