CFD Online Discussion Forums - Riemann Invariants

The Riemann invariants are defined only for one-dimensional flows, in two-dimensional flows they are called the Charactersitics of the flow and are the analogs to the 1D Riemann invariants. The Riemann invariants are really about what and how is propagated through the flow? Any information can propagates in the flow at 3 different velocities. If v is the velocity of the fluid, c is the sound velocity there, then any information can propagates at the speeds:

v

v+c

v-c

these are the characteristics speeds of the propagation of the information. What propagates are called (Riemann) invariants and so you have in the present case 3 of them. say R1, R2 and R3. In a subsonic regime v<c .

R1 propagates at speed v with the flow (down stream)

R2 propagates at supersonic speed v+c with the flow (down stream)

R3 propagates at subsonic speed v-c against the flow (up stream).

THese 3 quantitties (R1, R2, R3) are the solution of an algebraic equations, they are the eigen vectors of that system of equations with eigen values v, v+c, and v-c.

When one consider a boundary and wants to know what flows through that boundary, one needs to consider the flow normal (perpendicular) to that boundary and negelct any space derivative in the dimension parallel (tangent) to that boundary, as only the component of the velocity normal to that boundary contributes to the passing of flow and information through that boundary. Then one writes the flow equations at that boundary (neglecting parallel derivatives) and one solves for the linearized equations and obtains the Riemann Invariants R1, R2, R3 or characteristics of the flow.

For a complete treatment see e.g. Landau Lifshitz.