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

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

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

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

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

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.

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

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

Disturbance Equations where Mean flow features are contained in coefficients A

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

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)

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

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

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

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

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

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

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

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

Transition locations of NACA0012 at zero angle of attack

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

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

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

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

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

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