This document summarizes simulations of supersonic airfoil modeling and ultrasonic non-destructive evaluation. It describes CFD modeling of flow over a diamond wedge airfoil and biconvex airfoil using Star CCM+, comparing results to analytical models. It also details ultrasonic simulations using the UTSim tool to model submerged non-destructive evaluation, analyzing the effect of grid resolution. The goal is to use fast surrogate models for optimization and inversion analysis of airfoil design and non-destructive evaluation.

1. Supersonic Airfoil Modeling and
Ultrasonic NDE Simulations
Jacob Siegler
December 7, 2015

2. Contents
• Supersonic airfoil modeling
• Diamond wedge
• Biconvex airfoil
• Ultrasonic non-destructive evaluation simulations
2

3. Diamond Wedge
• Supersonic flow past a diamond wedge airfoil
• Six regions of interest
• 1: The upstream flow
• 2: The shock wave
• 3-6: The four panels of the airfoil
1
5 6
2
3 4
3

4. Diamond Wedge: Validation Case
Nomenclature:
• M = free-stream Mach
• α = angle of attack
• θ = shock angle
• μ = mach wave angle
• δ = wedge angle
∞
Aerodynamics for Engineers1 4

5. Diamond Wedge: CFD Model
• Star CCM+ CFD
• 15km altitude
• Steady Time
• Euler Equations
• Ideal Gas Behavior
• Fine mesh
5

6. 0 0.5 1 1.5 2 2.5 3 3.5 4
x 10
5
0.412
0.413
0.414
0.415
0.416
0.417
0.418
0.419
0.42
0.421
CL
Convergence
Number Grid Points
C
L
Value
Diamond Wedge: Grid Convergence
0 0.5 1 1.5 2 2.5 3 3.5 4
x 10
5
0.137
0.138
0.139
0.14
0.141
0.142
0.143
0.144
0.145
0.146
CD
Convergence
Number Grid Points
C
D
Value 6

7. Diamond Wedge: Pressure Contours
7

8. 0 0.2 0.4 0.6 0.8 1 1.2 1.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
X/C
C
P
CP
Plot
Airfoil Top
Airfoil Bottom
Diamond Wedge: Pressure Distribution
Panel 1
Panel 2
Panel 3
Panel 4
8

9. Diamond Wedge: Analytical Models
1. Linearized Theory (Appendix A)
2. Second-Order Busemann Theory (Appendix B)
3. Shock Expansion Theory (Appendix C)
9

10. Diamond Wedge: Analytical vs. CFD
ε CP1 ε CP2 ε CP3 ε CP4 Average Runtime
0.00009 0.00010 0.0020 0.00078 0.000738 ~1 Hr
0% 0.0386% 0.3008% 7.2605% 1.9000% 0. 52024 sec
Error ε between shock expansion and CFD; Runtime
10

11. Biconvex Airfoil
Images from Tracy, Sturzda, and Chase2 11

12. Biconvex Airfoil: Validation Case
Nomenclature:
• t/c = max thickness to
chord length ratio
• α = angle of attack
• M = free-stream Mach
t/c=10%
α=5°
M =2∞
∞
12

13. Biconvex Airfoil: Pressure Contours
13

14. Biconvex Airfoil: CFD vs. Analytical
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
CP
Comparison of Star CCM+ Models and Shock Expansion
x/c
CP Shock Expansion Model
Inviscid Model
Turbulent Model
Top
Bottom
Shock Expansion
CFD Turbulent
CFD Inviscid
14

15. Supersonic Airfoil Modeling: Conclusion
• The 2D CFD and analytical models compare well
• Further testing is required
• Will consider cases where viscous effects are important
• The shock expansion model seems to be a good candidate for
surrogate-based modeling and optimization
• In particular, variable-fidelity physics modeling and optimization
15

16. Ultrasonic Nondestructive Evaluation (NDE)
Typical computer/transducer setup3
16

17. Ultrasonic Simulation with the UTSim Tool
• UTSim simulates submerged ultrasonic NDE
• Two models
• Asymptotic Green’s Function
• High-fidelity model
• Gaussian Hermite Series Expansion
• Low-fidelity model
Setup for submerged ultrasonic NDE3
17

18. Ultrasonic Simulation: Test Case
• Fluid = water
• Solid = aluminum
• Water path distance = 0.1m
• #z points = 51*n
n = scaling integer
 n = 2
Test case overlaid on ultrasonic NDE setup3
WP = 0.1m
Aluminum
+Z
18

19. Ultrasonic Simulation: Fluid Only
19

20. Ultrasonic Simulation: Fluid and Solid
20
Fluid Solid

21. Ultrasonic Simulation: Effect of Grid
(Zoomed)
Z Points: 806
Z Points: 13056
21
Solid

22. Ultrasonic Simulations: Conclusion
• UTSim offers a quick analysis of NDE
• UTSim is still under development
• Near future versions capable of analyzing more complex geometries
• The plan is to use the models to create fast surrogates to accelerate
inversion analysis
22

23. References
1. Bertin, John J., and Russell M. Cummings. Aerodynamics for Engineers.
Upper Saddle River, NJ: Prentice Hall, 2010.
2. Tracy, Richard; Sturzda, Peter; Chase, James. Laminar Flow Optimized for
Supersonic Cruise Aircraft. Aerion Corporation. Published 28 April 2011.
3. NDT Resource Center. Iowa State University, 2014. Web. <https://www.nde-
ed.org/>.
4.Isentropic Flow Equations. NASA, 5 May 2015. Web.
<https://www.grc.nasa.gov/www/k-12/airplane/isentrop.html>.
23

24. Linearized Supersonic Thin Airfoil Theory
Equations obtained from Fundamentals of Aerodynamics1
Appendix A: Linear Theory
2
4
1
lC
M




2
, 2
4
1
d liftC
M




 2 2
, 2
2
1
d thickness u lC
M
 

 

   2 2 2 2
u l w w     
0
0
2
4 1
21
xm
x
C
cM


  
  
 
where

25. Second-Order Busemann Theory
Equations obtained from Fundamentals of Aerodynamics1
Appendix B: Second-Order Busemann Theory
   
2
1 2 3 4 1 2 3 4
tan1
8 8
w
m p p p p p p p pC C C C C C C C C

        
1
cos
2cos
i
n
p i
i
l
w
C
C




 1
sin
2cos
i
n
p i
i
d
w
C
C





4 2
2
2 22
( 1) 4 42
2( 1)1
np n n
M M
C
MM

  

     
    
    
where is the number of panels,
and is the turning angle𝛿 𝑤
𝑛
where is the current
orientation angle
𝜃 𝑛

26. Shock Expansion Theory
Equations obtained from Fundamentals of Aerodynamics1
Appendix C: Shock Expansion Theory
where is obtained via isentropic relations
𝑝 𝑛
𝑝∞
1
cos
2cos
i
n
p i
i
l
w
C
C




 1
sin
2cos
i
n
p i
i
d
w
C
C





2
2
1n
n
p
p
C
M p  
 
  
 

27. Appendix C: Shock Expansion Theory
Isentropic Relations
Equations obtained form NASA’s website4
 2 21 1
arctan 1 arctan 1
1 1
M M
 

 
  
       
1
1
21
1
2
t t tT p
M
T p


  



    
      
  
1
arcsin
M

 
  
 
where is the Prandtl-Meyer angle𝜈

28. Appendix C: Shock Expansion Theory
Shock Relations
Equations obtained from Fundamentals of Aerodynamics1
 
     
2 2
1
2 2 2 2
1
1 sin 2
2 sin 1 sin w
M
M
M
 
    
 

  
 2 2
12
1
2 sin 1
1
Mp
p
  

 


  2 2
12
1
1 sin
( 1)
M 
 



     
 
2 2 2 2
1 12
2 2
1 1
2 sin 1 1 sin 2
1 sin
M MT
T M
    
 
   


1
1
12
1
1 1
1t
t
p
p p





  
  
 
where is shock angle𝛽

29. Newton Raphson Application
PM angle, shock angle equations obtained from Fundamentals of Aerodynamics1
Appendix C: Shock Expansion Theory
 
1
'( )
n
n n
n
f u
u u
f u
   where represents the quantity being solved for𝑢
   2 21 1
arctan 1 arctan 1
1 1
M M
 

 
  
       
Solve for when is known𝜈𝑀
 
 
2
2 2
1
cot tan
2 sin 1 1
M
M

 



 
Solve for when and are known𝛽 𝑀 𝜃

30. Pressure Relations
Pressure coefficient equation obtained from Fundamentals of Aerodynamics1
Appendix C: Shock Expansion Theory
2
2
1n
n
p
p
C
M p  
 
  
 
relates the current pressure to free stream to find
11
1
n
n
tn n n
t n
pp p p
p p p p

  
where
𝐶 𝑝 𝑛
(terms cannot cancel)