Riemann invariants

Riemann invariants - in gas dynamics - combined parameters for some particular flows of a gaseous medium.

Obtained as solutions of equations simulating a one-dimensional unsteady flow of an inviscid non-heat-conducting gas.

r_{1}=u+\int {\frac {c}{\rho }}d\rho ,

{\ displaystyle r_ {1} = u + \ int {\ frac {c} {\ rho}} d \ rho,}

{\ displaystyle r_ {1} = u + \ int {\ frac {c} {\ rho}} d \ rho,}

r_{2}=u-\int {\frac {c}{\rho }}d\rho ,

{\ displaystyle r_ {2} = u- \ int {\ frac {c} {\ rho}} d \ rho,}

{\ displaystyle r_ {2} = u- \ int {\ frac {c} {\ rho}} d \ rho,}

Where $u$ ${\ displaystyle u}$ $u$ - flow rate, ${\rho }$ ${\ displaystyle {\ rho}}$ ${\ displaystyle {\ rho}}$ - density

For barotropic gas, they remain constant along the characteristics , which explains their name.

Named in honor of the outstanding German scientist-mathematician and mechanic Bernhard Riemann , who first pointed them out in his work ^[1] .

Literature

Black G. G. Gas dynamics . - M .: Nauka , 1988 .-- 424 p. - ISBN 5-02-013814-2 .

↑ Riemann, Bernhard. Ueber die Fortpflanzung ebener Luftwellen von endlicher Schwingungsweite (German) // Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen: magazin. - 1860. - Bd. 8 .