1. Prediction of Transition over Airfoils by using Parabolized Stability Equation
APISAT 2010
Sep. 13~15, 2010, Xi’An, China
Dong-hun Park , Seung O Park Korea Advanced Institute of Science and Technology

2. Outline Motivation Brief Introduction to PSE-eN Method
Computational Procedure
Results & Discussion
Concluding Remarks

3. Motivation Prediction of Transition - Important in Aerodynamic Design
(Low Re Airfoils, MAV, UAV,…)
- However, reliable prediction method is not yet available
- For practical purpose, empirical or semi-empirical methods are utilized

4. Motivation of the present work
Transition Prediction Methods
Correlation
LST & eN
PSE
DNS, LES
Turbulent Model Equations
 Want to see how ‘Linear PSE + eN’ works for airfoil flows

5. Fundamental Concepts: Transition Scenario
Typical Boundary Layer Transition Process
Receptivity
Linear Amplification
Nonlinear Amplification

6. Concept of Boundary Layer Stability Analysis
Mean Flow + Fluctuation
Stability Characteristic
solve
Stability Equation
(LST , PSE)
Governing Equation
Growth Rates
Disturbance Profile
Equation for disturbance amplitude profile
Transition Prediction
Subtract Mean Flow Eqn.

7. Subtract Mean flow Eqns.
Review on PSE
Disturbance
Governing Equations
Subtract Mean flow Eqns.
Disturbance Eqns. involving

8. Review on PSE Assume Disturbance Equations
Density fluctuation in terms of pressure and temperature fluctuations
Disturbance Equations
where

9. Disturbance Equations
where
Mean flow features are contained in coefficients A

10. Curvilinear Coordinate System
Neglect the non-linear terms with the assumption of infinitesimally small disturbance
Transformation into general curvilinear coordinate system
Linear Disturbance Equation in General Curvilinear Coordinate System

11. Linear PSE Wave Form Assumption for Disturbance where
Assumption : Disturbances are in wave form which is periodic in time and space
Shape function
Wave part ( fast oscillating part )
Apply the above expression to disturbance equation
Take the solution which satisfies that change of ψ in streamwise direction is small O(Re-1)
Neglect the terms which are O(Re-2) and smaller : Parabolization
where
PSE : Parabolized linear governing equation for disturbance (stability equation)

12. Advantages of PSE
Growth of Boundary layer is naturally taken into account
(No parallel flow assumption)
Eigenvalue and shape function are determined simultaneously by marching PSE with local iteration

13. Conditions for PSE solution
Boundary Condition
Initial Condition
The shape of initial disturbances can be obtained from LST theory ( for a given frequency disturbance)
Normalization Condition
Normalization condition is needed to deal with the un-uniqueness for expression of disturbance which has separation of shape function and wave part
Local iteration for updating alpha

14. Growth Rate of Disturbance
Growth Rate of Variable φ along s
By using α and shape function ψ from PSE solution
- This may change by the choice of ψ
- Integrated disturbance kinetic energy concept is most widely used :
: Complex conjugate

15. The eN method Trnasition prediction by eN -Method
Stream-wise Integration of growth rate :
N-factor
N reaches a specified value at a certain location (semi-empirical method)  transition point
Example : Transition Predition with N=10

16. Computational Procedure
Geometry & Flow Condition
Inviscid Solver
Boundary Layer Edge Information
Boundary Layer Code
Laminar Boundary Layer Profiles
PSE Code
Boundary Layer
Stability Data
eN -method
Transition Location
Prediction

17. Validation of the present code
Comparison of growth rates of F=112 disturbance (Incompressible flat plate boundary layer)

18. Validation of the code -continued
N-factor curves for F=40 and 60 disturbance in incompressible flat plate boundary layer

19. Results
N-factor curves (F=35∼135) for NACA 0012 airfoil (α=0º, Ue*=30m/s, c*=0.5m )

20. Transition locations of NACA0012
at zero angle of attack

21. NLF(1)-0414F
Pressure distribution at 2o angle of attack

22. Transition locations on the upper surface of
NLF(1)-0414F airfoil (Rc≈10× 106 )

23. NLF (1)-0416
Pressure distribution at zero angle of attack

24. Transition locations on the upper surface of NLF(1)-0416 airfoil ( Rc=2×106)

25. Concluding Remarks
Prediction of Transition over Airfoil surfaces by PSE and eN method
N= 11 or 12 worked very well to locate transition points
* Acknowledgement: Financial support by ADD and NRF of Korea are gratefully acknowledged

26. Thank you very much for your attention !
Hope to see you again at APISAT 2011