Analysis of Oblique Shocks

1. Analysis of Oblique Shocks P M V Subbarao Associate Professor Mechanical Engineering Department I I T Delhi A Mild, Efficient and Compact Compressor ….

2. Non Conical Inlets at Super Sonic Speeds High Mach Number>x Low Mach Number>x

3. High Angle Objects Sleek Bodies at supersonic Speeds Bluff Bodies at supersonic Speeds

4. Mach Waves, Revisited • • A ‘’point-mass’’ object moving with Supersonic velocity Generates an infinitesimally weak “mach wave”. • The direction of flow remains unchanged across Mach wave.

5. Oblique Shock Wave • When generating object is larger than a “point”, shockwave is stronger than mach wave …. Oblique shock wave •  -- shock angle •  -- turning or “wedge angle”

6. Tangential Normal Ahead wx, Mtx ux, Mnx Of Shock Behind wy, Mty uy, Mny Shock Oblique Shock Wave Geometry Vx & Mx Vy & My uy & Mny wy & Mty ux & Mnx wx & Mtx Vy & My Vx & Mx • Shock is A CV & Must satisfy i) continuity ii) momentum iii) energy

7. æ ö - > - > ò ò r · = = - r + r ® r = r 0 V d s u A u A u u è ø x x y y x x y y . . C S Continuity Equation wy wx Vy Vx uy ux x y • For Steady Flow

8. • For Steady Flow w/o Body Forces ( ) - r + r = 0 u w A u w A x x x y y y r = r u u = w w x x y y x y Momentum Equation • Tangential Component Tangential velocity is Constant across oblique Shock wave • But from continuity

9. ( ) - r + r = - ® 2 2 u A u A p p A x x y y y x + r = + r 2 2 p u p u x x x y y y • Normal Component Tangential velocity is Constant across oblique Shock wave

10. = + ® = + ® = 2 2 2 2 2 2 V u w V u w w w x x x x y y x y 2 2 u u 2 2 V V + = + x x h h x y + = + h h x x x y 2 2 2 2 Energy Equation • Write Velocity in terms of components • thus …

11. r = r u u x x y y b w x = w w x y b-q u + r = + r 2 2 p u p u x x x x y y y u w y y 2 2 u u b-q + = + x y c T c T q x y p p 2 2 Collected Oblique Shock Equations • Continuity • Momentum • Energy

12. Vx & Mx Vy & My • Defining: Mnx=Mxsin( Mtx=Mxcos( uy & Mny wy & Mty ux & Mnx wx & Mtx Vy & My • Then by similarity we can write the solution Vx & Mx

13. Letting • Similarity Solution Mnx= Mxsin(b)

14. • Properties across an Oblique Shock wave ~ f(Mx, b)

15. Total Mach Number Downstream of Oblique Shock Tangential velocity is Constant across oblique Shock wave

16. Tangential velocity is Constant across oblique Shock wave

17. Tangential velocity is Constant across oblique Shock wave • Or … More simply .. If we consider geometric arguments

18. Oblique Shock Wave Angle • Properties across Oblique Shock wave ~ f(M1, b) • q is the geometric angle that “forces” the flow • How do we relateq to b

19. • Since (from continuity)

20. Oblique Shock Wave Angle (cont’d) • from Momentum

21. Oblique Shock Wave Angle (cont’d) • Solving for the ratio u2/u1 Implicit relationship for shock angle in terms of Free stream mach number and “wedge angle”

22. ( ) ( ) ( ) é ù ( ) ( ) ( ) 2 2 b g + b - + g - b é ù é ù t a n 1 s i n 2 1 s i n M M ë û ë û ë û 1 1 ( ) q = t a n ( ) é ( ) ( ) ù ( ) ( ) ( ) 2 2 g + b + b + g - b é ù é ù 2 1 s i n t a n 2 1 s i n M M ë û ë û ë û 1 1 • Solve explicitly for tan(q)

23. Oblique Shock Wave Angle (cont’d) • Simplify Numerator

24. • Simplify Denominator

25. Oblique Shock Wave Angle (cont’d) • Collect terms • “Wedge Angle” Given explicitly as function of shock angle and freestream Mach number • Two Solutions “weak” and “strong” shock wave. In reality weak shock typically occurs; strong only occurs under very Specialized circumstances .e.g near stagnation point for a detached Shock.

26. “strong shock” M1=5.0 “weak shock” max curve M1=4.0 Oblique Shock Wave Angle M1=3.0 M1=2.5 M1=1.5 M1=2.0

27. Limiting Cases of Oblique Shock Wave

28. Maximum Turning Angle qmax

29. High Angle Objects q <qmax q >qmax

30. Weak And Strong Oblique Shock

31. Supersoinc Intakes

32. Solving for Oblique Shock Wave Angle in Terms of Wedge Angle • As derived • “Wedge Angle” Given explicitly as function of shock angle and freestream Mach number • For most practical applications, the geometric deflection angle (wedge angle) and Mach number are prescribed .. Need  in terms of  and M1 • Obvious Approach …. Numerical Solutionusing Newton’s method

33. Solving for Oblique Shock Wave Angle in Terms of Wedge Angle (cont’d) • Newton method

34. Solving for Oblique Shock Wave Angle in Terms of Wedge Angle (cont’d) • Newton method (continued) • Iterate until convergence

35. Solving for Oblique Shock Wave Angle in Terms of Wedge Angle (cont’d) Increasing Mach • “Flat spot” Causes potential Convergence Problems with Newton Method

36. Solving for Oblique Shock Wave Angle in Terms of Wedge Angle (cont’d) • Newton method … Convergence can often be slow (because of low derivative slope) • Converged solution

37. Solving for Oblique Shock Wave Angle in Terms of Wedge Angle (concluded) • Newton method … or can “toggle” to strong shock solution • Strong shock solution

38. Solving for Oblique Shock Wave Angle in Terms of Wedge Angle (improved solution) • Because of the slow convergence of Newton’s method for this implicit function… explicit solution … (if possible) .. Or better behaved .. Method very desirable Substitute

39. Solving for Oblique Shock Wave Angle in Terms of Wedge Angle (improved solution)(cont’d) • But, since

40. Solving for Oblique Shock Wave Angle in Terms of Wedge Angle (improved solution)(cont’d) • Simplify and collect terms

41. Solving for Oblique Shock Wave Angle in Terms of Wedge Angle (improved solution)(cont’d) • Again, Since

42. Solving for Oblique Shock Wave Angle in Terms of Wedge Angle (improved solution)(cont’d) • Regroup and collect terms

43. Solving for Oblique Shock Wave Angle in Terms of Wedge Angle (improved solution)(cont’d) • Finally • Regrouping in terms of powers of tan()

44. Solving for Oblique Shock Wave Angle in Terms of Wedge Angle (improved solution)(cont’d) • Letting • Result is a cubic equation of the form • Polynomial has 3 real roots i) weak shock ii) strong shock iii) meaningless solution ( < 0)

45. Solving for Oblique Shock Wave Angle in Terms of Wedge Angle (improved solution)(cont’d) • Numerical Solution of Cubic (Newton’s method)

46. Solving for Oblique Shock Wave Angle in Terms of Wedge Angle (improved solution)(cont’d) • Collecting terms

47. Solving for Oblique Shock Wave Angle in Terms of Wedge Angle (improved solution)(cont’d) • Solution Algorithm (iterate to convergence) • Where again

48. Solving for Oblique Shock Wave Angle in Terms of Wedge Angle (improved solution)(cont’d) • Properties of Solver algorithm are much improved Original Algorithm Improved Algorithm • Improved algorithm • Original algorithm

49. Solving for Oblique Shock Wave Angle in Terms of Wedge Angle (improved solution)(cont’d) • Three Solutions always returned depending on start condition Original Algorithm • Weak Shock Solution Improved Algorithm

50. Solving for Oblique Shock Wave Angle in Terms of Wedge Angle (improved solution)(cont’d) • Three Solutions always returned depending on start condition Improved Algorithm • Strong Shock Solution