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

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 mach waves, from the two original points.

19. ( ) { } ( ) P o i n t { 1 } ® M , q k w n = g + - t a M ì í ï î ü ý þ2 ( ) ì í ï î ü ý þ A l o n g C - { } q 1 + = c s t K ( )

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

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

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

23. Intersection locates point 3
• 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 =
---> m = o

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. • Iterative solution Repeat
• 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 …

51. Minimum Length Nozzle: Construction Example
• Use Method of characteristics, compute and graph contour for
two-D minimum length nozzle for a design exit mach
number of 2.0
Mexit = 2.0

52. = o
= o

54. Point K- = K+ =  =  = M  =
#  +   - v 1/2(K-+K+) 1/2(K--K+)
(deg), (deg), (deg), (deg). (deg),

55. Point K- = K+ =  =  = M  =
#  +   - v 1/2(K-+K+) 1/2(K--K+)
(deg), (deg), (deg), (deg). (deg),

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