# Advanced scientific computing (B673) by Meglicki Z.

By Meglicki Z.

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C, (50) , we can obtain p, u, Using the isentropic law we obtain p = Using equation = Ap^ p,p7p^ ^^') • we obtain, by solving for c ('50) c = c^ + ^^ (u^- u) (52) . By substitution of (52) into {^9) and solving for u we obtain o u By substitution of of ( 52) 2P Ax / 1 \ 7" ( 53) into ( into the definition of 52) c c is obtained; by substitution and solving for p we obtain 27 P = If Ij^Ax lies to the right of the left rarefaction wave (3) we obtain p = p^, u = u^, II. The sample point (I^Ax > n p and p = p^.

The sample point (I^Ax > n p and p = p^. Ax lies to the right of the slip line ^ u^At/2). (a) i (53) • (f^)'^"'"' If the right wave is a shock wave (p^ > p and (l) if ) dy Ax lies to the left of the shockline defined by -^ = U we have dt r = p#j u = u^, and p = p^, where p^ is obtained from (15) , -M ^ '* (2) ^= U^, (b) If ^ r_ u^ '* - U (5M r Ax lies to the right of the shockline defined by we have p = p^, u = u^, and p = p^, If the right wave is a rarefaction wave "^ (p^ P The ) . rarefaction wave is bounded on the left by the line defined by dx -TT- = u^ + c^, i^P* where c^ = j and p^ can be obtained from the isentropic law P„p"^ = P*p;^ = A Then we obtain from (55 ; tz> p i = p^, u = u^, dx = u + ) (56 ) /^Pr "'dtrr'rVp and on the right by the line defined by J If (55 ) P* = (-^) (1) .

Rarefaction wave is bounded on the left by the line defined by dx -TT- = u^ + c^, i^P* where c^ = j and p^ can be obtained from the isentropic law P„p"^ = P*p;^ = A Then we obtain from (55 ; tz> p i = p^, u = u^, dx = u + ) (56 ) /^Pr "'dtrr'rVp and on the right by the line defined by J If (55 ) P* = (-^) (1) . -rrr c , c = / , Ax lies to the left of the rarefaction wave, then and p = p^ o 28 (2) If i Ax lies inside the right rarefaction wave, we equate the slope of the characteristic dx = ^ + ^ to the slope of -g^ the line through the origin and (I^^Ax, At/2 ), obtaining Ax 2i u+c=-^.