Two-Dimensional Analysis of Supersonic Gas Dynamics

1. Two-Dimensional Analysis of Supersonic Gas Dynamics P M V Subbarao Associate Professor Mechanical Engineering Department I I T Delhi More Realistic Modeling of Real Applications….

2. • Generalization of “ Differential form” of Prandtl-Meyer wave Using O.D.E. of Gas Dynamics Along characteristic lines:

4. Basic principle of Methods of Characteristics

5. Compatibility Equations • Compatibility Equations relate the velocity magnitude and direction along the characteristic line. • In 2-D and quasi 1-D flow, compatibility equations are Independent of spatial position, in 3-D methods, space becomes a player and complexity goes up considerably. • Computational Machinery for applying the method of Characteristics are the so-called “unit processes” • By repeated application of unit processes, flow field Can be solved in entirety.

6. Unit Process 1: Internal Flow Field • Conditions Known at Points {1, 2} • Point {3} is at intersection of {C+, C-} characteristics

7. “Method of Characteristics” • Basic principle of Methods of Characteristics -- If supersonic flow properties are known at two points in a flow field, -- There is one and only one set of properties compatible* with these at a third point, -- Determined by the intersection of characteristics, or machwaves, from the two original points.

8. ® q ® { 1 } { , } k n o w n P o i n t M 1 1 ì ü g + g - ï ï 1 1 ( ) - - n = - - - 1 2 1 2 t a n 1 t a n 1 M M í ý 1 1 1 g - g + 1 1 ï ï î þ ( ) { } n ® q + = = A l o n g C c o n s t K - - 1 1 1

9. ® q ® { 2 } { , } k n o w n P o i n t M 2 2 ì ü g + g - ï ï 1 1 ( ) - - n = - - - 1 2 1 2 t a n 1 t a n 1 M M í ý 2 2 2 g - g + 1 1 ï ï î þ ( ) { } n ® q - = = A l o n g C c o n s t K + + 2 2 2

10. ( ) ( ) ( ) ( ) n n é ù + q + + q - K K - + q = = 1 1 2 2 1 2 ê ú n n q + = q + é ù 3 2 2 ê ú 1 1 3 3 ® { 3 } P o i n t ê ú n n ( ) ( ) ( ) ( ) n n q - = q - ê ú - q + - q - ë û K K n 2 2 3 3 - + = = 1 1 2 2 1 2 ê ú 3 ë û 2 2 é ù ì ü g + g - ï ï 1 1 ( ) - - = n = - - - 1 2 1 2 ê t a n 1 t a n 1 ú M S o l v e M M í ý 3 3 3 3 g - g + 1 1 ï ï ê ú î þ ë û Mach and Flow Direction solved for at Point 3 ®

11. But where is Point {3} ? • {M,q} known at points {1,2,3} ---> {m1,m2,m3} known

12. • Slope of characteristics lines approximated by: Intersection locates point 3

14. Unit Process 2: Wall Point • Conditions Known at Points {4}, Wall boundary at point 5

15. • Iterative solution

16. • Pick q5 • Solve for

17. • Solve for Mach angle, C- slope

18. Unit Process 3: Shock Point • Conditions Known at Points {6}, Shock boundary at point 7 Freestream Mach Number Known • Along C+ characteristic • Iterative Solution

19. • Pick 7--->Oblique Shock wave solver M, q7 ---> M7 (behind shock) • Iterative solution Repeat Using new q7 until convergence

20. Initial Data Line • Unit Processes must start somewhere .. Need a datum from which too start process • Example nozzle flow … Throat

21. Supersonic Nozzle Design • Strategic contouring will “absorb” mach waves to give isentropic flow in divergent section

22. • Rocket Nozzle (Minimum Length) • Wind tunnel diffuser (gradual expansion) • Find minimum length nozzle with shock-free flow

23. Minimum Length Nozzle Design • Find minimum length nozzle with shock-free flow • Along C+ characteristic {b,c} C+ • Along C- characteristic {a,c} q=0 C-

24. • Find minimum length nozzle with shock-free flow • Along C- characteristic {a,c} at point a C+ • But from Prandtl-Meyer expansion C-

25. C+ C-

26. • Criterion for Minimum Length Nozzle • Length for a given expansion angle is more important than the precise shape of nozzle …

27. Physical Meaning of Characteristic Lines • Schlieren Photo of Supersonic nozzle flow with roughened wall