1. Section 11 Lecture 2: Analysis of Supersonic Conical Flows
• Anderson,
Chapter 10 pp

2. Physical Aspects of Conical Flow
• Flow is circumferentially symmetric
• “Axi-symmetric flow”

3. Physical Aspects of Conical Flow (cont’d)
• Flow Properties are constant along a ray from the vertex

4. Physical Aspects of Conical Flow (cont’d)
• Look at shock wave
“straight”
• Shock strength is the same at points 1 and 2
“irrotational flow”

5. Rotational and Irrotational Flow
• Condition for Irrotational flow
• At every point in flow field

6. Spherical Coordinate System
• Axi-symmetric flow
is best represented
using spherical
coordinate system
(Anderson conventions)

7. Spherical Coordinate System (cont’d)
• Velocity vector
• Gradient vector
• Divergence

8. Continuity Equation for Conical Flow
• Steady flow
• Axi-symmetric flow -- use spherical coordinates
• Evaluating derivatives and letting

9. Continuity Equation for Conical Flow (concluded)
• Continuity Equation for Axi-symmetric Conical Flow

10. Crocco’s Relation for Axi-symmetric Conical Flow
• Irrotational flow
• Crocco’s relationship solves both momentum and
energy equation
… thus for conical flow all we need to satisfy is:

11. Crocco’s Relation for Axi-symmetric Conical Flow (cont’d)
• For Arbitrary V
• Curl Operation in Spherical Coordinates

12. Crocco’s Relation for Axi-symmetric Conical Flow (cont’d)
• For Arbitrary V
• Curl Operation in Spherical Coordinates

13. Crocco’s Relation for Axi-symmetric Conical Flow (cont’d)
• Apply
• Crocco Relation reduces to
• Now apply

14. Crocco’s Relation for Axi-symmetric Conical Flow (concluded)
• Crocco Relation reduces to (wow!)
• Irrotationality
Condition for
Axi-symmetric conical flow

15. Euler’s Equation for Axi-symmetric Conical Flow
• Steady Axi-symmetric conical flow
• Flow is irrotational … Apply along stream line direction

16. Euler’s Equation for Axi-symmetric Conical Flow (cont’d)
• Steady Axi-symmetric conical flow
Divide by dr
• Thus behind the shock wave

17. Euler’s Equation for Axi-symmetric Conical Flow (cont’d)
• But from enthalpy equation

18. Euler’s Equation for Axi-symmetric Conical Flow (concluded)
• Sub into Euler’s equation

19. Collected Equations for for Axi-symmetric Conical Flow
• Only q is independent variable
Can re-write partials at total derivatives

20. Collected Equations for for Axi-symmetric Conical Flow (cont’d)

21. Collected Equations for for Axi-symmetric Conical Flow (cont’d)
• Substituting in

22. Collected Equations for for Axi-symmetric Conical Flow (cont’d)
• CLASSICAL FORM OF TAYLOR-MACCOLL EQUATION
… BUT DIFFICULT TO SOLVE NUMERICALLY IN THIS FORM

23. Taylor-Maccoll Equation
• O.D.E for Vr in terms of q
Vr --> dependent variable,
q --> independent variable
• Solve for Vr --->
Then

24. Taylor-Maccoll Equation (cont’d)
• BETTER SOLUTION …Nondimensionalize by
• let
• and the Taylor-Maccol
equation reduces to

25. Taylor-Maccoll Equation (cont’d)

26. Taylor-Maccoll Equation (cont’d)
• Take a closer look at:
• Once we find we can calculate M
• So how do we find ?

27. Numerical Procedure for Solving Axi-symmetric Conical Flow Field
• Re-write
• As
• But
• Substitute in

28. Numerical Procedure for Solving Axi-symmetric Conical Flow Field
• Re-write
• As
• But
• Substitute in

29. Numerical Procedure for Solving Axi-symmetric Conical Flow Field (cont’d)
• Solve for

30. Numerical Procedure for Solving Axi-symmetric Conical Flow Field (cont’d)
• System of first order ordinary differential equations

31. Numerical Procedure for Solving Axi-symmetric Conical Flow Field (cont’d)
• Can be written in vector form as

32. Numerical Procedure for Solving Axi-symmetric Conical Flow Field (cont’d)

33. Boundary Value Problem
• Inverse solution method
• Given M … Start with assumed
bshock (initial condition)
• Properties immediately behind
Shock given by Oblique
Shock relations
• On cone Jq=0
Boundary Condition

34. Boundary Value Problem (cont’d)
• Starting Conditions
Given M … Start with assumed bshock (initial condition)
Calculate M2 (immediately behind shock)

35. Boundary Value Problem (cont’d)
• Starting Conditions

36. Boundary Value Problem (cont’d)
• Using Starting Jr, Jq
Integrate the equations of motion over increments of q
Until Jq =0 … this angle corresponds t the solid
Cone boundary corresponding to M and the assumed
bshock.
• So How do we integrate
This?

37. Integration of Equations of Motion
• The Integral starts at bshock and proceeds
towards the cone where q=qcone
• Look at small segment dq for the integral

38. Integration of Equations of Motion (cont’d)
• Look at
• Approximate are
under curve =

39. Integration of Equations of Motion (cont’d)
• The the integral is approximated by
• And
“trapezoidal rule”

40. Integration of Equations of Motion (cont’d)
• Or we perform the same argument using “finite differences”
• Basic differencing scheme

41. Predictor / Corrector • But at step (j) we don’t know
• So we predict it …
• and then correct the prediction
• Be very careful
About units on Dq

42. Predictor / Corrector
• Where

43. Collected Algorithm Compute Initial Conditions,
a) Given M … Start with assumed bshock
b) Calculate M2 (immediately behind shock)

44. Collected Algorithm Compute Initial Conditions,
c) Calculate p2,T2 ratios (immediately behind shock)

45. Collected Algorithm (cont’d)
Compute Initial Conditions,
c) Compute starting non-dimensional velocities

46. Collected Algorithm (cont’d)
Compute q decrement
Integration Loop, j=1,…N
a) Predictor
b) Corrector
• Be very careful
About units on Dq ;

47. Collected Algorithm (cont’d)
3) Integration Loop, j=1,…N (cont’d)
d) Test for convergence
if Jq > 0 you have a problem
Why?

48. Collected Algorithm (cont’d)
4) Post-process data
Compute total J
Compute Mcone
• Why?

49. Collected Algorithm (cont’d)
4) Post-process data
c) Calculate pcone,Tcone ratios (flow behind shock is isentropic)

50. Physical Aspects of Cone Flow
• Compare cone flow to wedge
• Cone flow supports a
much larger wedge angle
before shock wave detaches
Cone Flow
M=2.0
Wedge Flow
M=1.5
M=2.0
M=1.5
M=4.0
M=8.0
M=8.0
M=4.0

51. Physical Aspects of Cone Flow (cont’d)
• Three-dimensional
“relieving” effect
• Cone shock wave is
Effectively weaker
Than shock wave for
Corresponding wedge angle

52. Isentropic Deceleration on Cone with shock wave near detachment angle
• Flow goes from supersonic
to subsonic with out shock wave
M < 1

53. • Flow is getting “Squeezed”
Isentropic Deceleration on Cone with shock wave near detachment angle (cont’d)
• Flow is getting “Squeezed”

54. Qualitative Aspects of Conical Supersonic Flow Fields at Angles of Attack
• Look at Zero-a conical flow field projected onto spherical surface
• Really a special case
Of a 1-dimensional
Flow field
• Shock strength is
uniform … irrotational
flow field
• Simplifying conditions
result in flow field
That is “easy” to solve
Crocco’s Theorem

55. Qualitative Aspects of Conical Supersonic Flow Fields at Angles of Attack (cont’d)
• Look at Non Zero-a conical flow field
• Shock Strength is
no longer uniformly
strong
• Tds is no longer
constant
• Flow field is “rotational”

56. Qualitative Aspects of Conical Supersonic Flow Fields at Angles of Attack (cont’d)
• Look at Non Zero-a conical flow field
Flow field is a function of
two independent variables
q, f
 hock wave angle bs is different for
each meridional plane ()
Stream lines about body are curved
From windward to leeward surface
Stream lines that pass thru different
Point on shock wave experience different
Entropy changes

57. Qualitative Aspects of Conical Supersonic Flow Fields at Angles of Attack (cont’d)
• “Vortical Singularity”
• Stream lines with different
entropy levels converge on a
single line on leeward side
• Ray along cone surface at 180∞
degrees to freestream wind has
“multi-valued” entropy level
• Referred to as “Vortical
Singularity”

58. Qualitative Aspects of Conical Supersonic Flow Fields at Angles of Attack (cont’d)
• When a < qcone … singularity lies along cone surface

59. Qualitative Aspects of Conical Supersonic Flow Fields at Angles of Attack (cont’d)
• When a > qcone … singularity lies above cone surface

60. Qualitative Aspects of Conical Supersonic Flow Fields at Angles of Attack (cont’d)
• Cross Flow Sonic Lines, cross flow Mach number > 1

61. Qualitative Aspects of Conical Supersonic Flow Fields at Angles of Attack (cont’d)
• Cross Flow Sonic Lines, cross flow Mach number > 1

62. Qualitative Aspects of Conical Supersonic Flow Fields at Angles of Attack (cont’d)
• “Small angle” approximation
• Solution formulated as “corrections” to zero-a solution
Zero alpha condition
“Correction”

63. • Boundary conditions (behind shock wave)
Qualitative Aspects of Conical Supersonic Flow Fields at Angles of Attack (cont’d)
• “Taylor-Maccoll” type of equation for alpha perturbations
• Boundary conditions (behind shock wave)

64. Link to paper in appendix to this section
Qualitative Aspects of Conical Supersonic Flow Fields at Angles of Attack (cont’d)
• Solution is similar (but more complex) than zero alpha case
Starts at shock wave and works toward surface
• Solution tables for right cones
NASA SP 3007 "Tables for Flow Around
Right Circular Cones at Small Angle of Attack"
Link to paper in appendix to this section

65. Homework 12 • Code Taylor-Maccoll algorithm for cone flow
• Solve for flow conditions on surface of Cone
at freestream Mach 2.0 with 15 half angle
p = kPa
T = 288
r = 1.23 kg/m3
15

66. Part 1 Solution
11.1

67. Homework 12 (Continued) • Define • Derive an expression
• Hint: You’ll have to do trial
And error for each mach number to get the
Shock angle correct
• Derive an expression
for the cone wave drag
as a function of the
cone surface pressure
(pcone) and the base
pressure (pbase)
• Assume pbase = p
plot CDcone versus Mach over range from 1.5 to 7

68. Part 2 Solution
11.2

69. Part 2 Solution (cont’d)

70. Part 2 Solution (concluded)
Cone Drag Coefficient versus Mach Number