Two-Dimensional Gas Dynamics

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

2. Geometrical Description of Wing Sweep Section A—A

3. Equivalent 2-D Flow on Swept Wing • Freestream Mach number resolved into 3 components i) vertical to wing … ii) in plane of wing, but tangent to leading edge iii) in plane of wing, but normal to leading edge Flow past a wing can be split into two independent 2D Flows.

4. An Approach to 2D Compressible Flow

5. Generalization of Prandtl-Meyer Expansion Fan • Consider flow expansion around an infinitesimal corner • From Law of Sines

6. - V V d V = p p æ ö æ ö - m + q - m s i n d s i n ç ÷ ç ÷ è ø è ø 2 2 Consider flow compression around an infinitesimal corner Mach Wave m V V V-dV dq dq

7. • Generalization of “ Differential form” of Prandtl-Meyer wave • For an infinitesimal disturbance (mach wave)

9. Characteristic Lines • Right running characteristic lines Slope: q  m • C- “right running” characteristic Line is a Generalization For infinitesimal expansion corner flow

10. • Left and Right running characteristic lines Slope: q + m • C+ “left running” characteristic Line is a Generalization infinitesimal compression corner flow

11. Characteristic Lines • Supersonic “compatibility” equations • Apply along “characteristic lines” in flow field

12. Regions of Influence and Domains of Dependence D strongly feels the influence of B,C A D

14. Regions of Influence and Domains of Dependence

15. Basic principle of Methods of Characteristics

16. 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.

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

18. “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.

19. ® 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

20. ® 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

21. ( ) ( ) ( ) ( ) 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 ®

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

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

24. Unit Process 1: Internal Flow Example

25. • Point 1, compute

26. • Point 2, compute

27. • Point 3 Solve for

28. • Point 3 Solve for M3 = 2.3419 ---> m = 25.2776o

29. • Locate Point 3 • Line Slope Angles

30. • Solve for {x3,y3}

31. • Solve for {x3,y3}

32. • Solve for {x3,y3} x3= =2.2794

33. • Solve for {x3,y3} y3= =1.726

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

36. • Iterative solution

37. • Iterative solution • Pick q5

38. • Pick q5 • Solve for

39. • Solve for Mach angle, C- slope • In Similar manner as before find intersection of C- and surface mold line .. Get new q5, repeat iteration

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

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

42. • Pick q7--->Oblique Shock wave solver ---> M7 • Iterative solution

43. • But • Iterative solution

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

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

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

47. 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-

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

49. C+ C-

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