Conformal Mapping of Rotating Cylinder into Jowkowski Airfoil

Use Of Conformal Mapping To Theoretically Analyze the Flow Over Airfoil From Lifting

Cylinder for FADS

U. N. Mughal*1

Assistant Professor, Mechanical Engineering Department, NED University of Engineering and Technology, Karachi, PakistanEmail : mughal@neduet.edu.pk, umair.mughal@gmail.com

J. Masud*2

Associate Professor,Institute of Avionics and Aeronautics Engineering, Air University, Islamabad, Pakistan,Email : jehanzeb.masud@mail.au.edu.pk, polymech@gmail.com

Outline Flush Port Air Data System (FADS),

Theme/Scope of this Research Exercise

Various Techniques to Analyze Pressure Distributions on Airfoil

Conformal Mapping

Importance of this Technique over others

Mathematical Formulation

Analysis of 20% Thick Symmetrical Airfoil

Analysis of 20% Thick Cambered Airfoil

Analysis of 6% Thick Symmetrical Airfoil

Analysis of 6% Thick Cambered Airfoil

Room for Further Research

Conclusion

Flush Air Data SystemNeed of Flush Airdata System:

Pitot Tube.Critical Parameters (Dynamic Pressure, Angle of Attack).Development of Flush Airdata System (FADS)

Function of FADS Matrix of Flush PortsPressure DifferentialOnline computation Potential flow.

contd. Flush Air Data System Advantages:

Over determined system Robustness. Error Minimization Accuracy

Port Location depends upon Large angle of attack sensitivity Good linearity with angle of attack, Minimum variation of the parameter with Mach number

Used Hypersonic (X-15) Stealth, (B-2, A-12) Research aircrafts Hybrid System on F-22

Complications High-fidelity vehicle simulations, Pressure ports locations

Theme/Scope of this Research

To theoretically analyze the effects of Angle of Attack on Pressure Difference on airfoil.

To suggest the best port location on different airfoils, in order to install Pressure Differential Angle of Attack measuring instrument on them.

Techniques For Airfoil Analysis

Methods are available for predicting the pressure distribution around airfoils.

SURFACE PANEL METHODS

CONFORMAL MAPPING

THIN AIRFOIL THEORY

I have used conformal mapping because,

Conformal Mapping

A conformal mapping or angle-preserving transformation

Complex analysis

Importance in Fluid Mechanics

Velocity Potential and Stream Function

Simplest flow is about a cylinder of infinite AR.

Flow Around Infinite Span Cylinder

Streamline Pattern

Pressure Distribution

Velocity Vectors

Lift Distribution

Background of Transformation

Basic transformation by Nikolai Egorovich Joukowsky as ‘Joukowski transformation’

‘Joukowski Profiles’.

Basic analytic function w=f(z) is given as,

Generic Joukowski airfoil

2

( )b

w z zz

Jowkowski Transformation

4 plane transformation

Stream function equation of cylinder with circulation,

Stream function after Conformal Mapping,

2

2( , ) sin( ) 1 2 sin( ) ln

R rr U r RU

r R

2

2sin( ) 1 ln

2

R rU r

r R

Flow Around Jowkowski Airfoil

Streamline Pattern

Pressure Distribution

Velocity Vectors

Lift Distribution

Contd. Flow Around Jowkowski Airfoil

Vectorial sum of all the elementary forces = Lift Force

Kutta Condition

Airfoils Analyzed

. These four types of airfoils are generated using different values of R0 and 0. They are,

For 20% thick Cambered Airfoil R0=0.4, 0=1200 ,

For 20% thick Symmetrical Airfoil R0=0.3, 0=1800,

For 6% thick Cambered Airfoil R0=0.15, 0=1200,

For 6% thick Symmetrical Airfoil R0=0.06, 0=1800,

Conformal Transformation

Flow Around Cylinder with Circulation

Flow Oriented at Angle

Displaced Center of Cylinder

Conformal Shape Obtained

Ro

b

Various Shapes Obtained During Conformal Mapping of Cylinder

Ro=0

Ro=0.4

Ro=0.4

Ro=0.4

Analysis of 20% Thick Symmetrical Airfoil

Flow around 20% Thick Symmetric Airfoil

Cp Upstream and Cp Downstream

P at Free-stream Velocity

Analysis of 20% Thick Cambered Airfoil

Flow Around 20% Thick Cambered Airfoil

Cp Upstream and Cp Downstream

P at Free-stream Velocity

Analysis of 6% Thick Symmetrical Airfoil

Flow around 6% Thick Symmetric Airfoil

Cp Upstream and Cp Downstream

P at Free-stream Velocity

Analysis of 6% Thick Cambered Airfoil

Flow around 6% Thick Cambered Airfoil

Cp Upstream and Cp Downstream

P at Free-stream Velocity

Room for Further Research

Wind Tunnel Testing of 4 Airfoils which I have used in this Exercise may verify the results.

Using the same Conformal Mapping Technique we can also analyze many other Self Generated Aerodynamic Profiles.

I have used Conformal Mapping to analyze flow on Airfoil, but Surface Panel Method can also be used in show results in the same direction.

Conclusions

Airfoil Best Port Location 0.3 to 0.5 x/c

P Varies Linearly With Angle of Attack and Mach Number on all airfoils analyzed

Conformal Mapping is an excellent technique to analyze pressure distribution on any aerodynamic profile using our Personnel Computers.

Results shows that there is a Lift Force on all 4 Jowkowski Airfoils but there is 0 Drag on them equal to

Conformal Mapping Technique may be added in the Syllabus of Undergraduate Engineering Students which can help them a lot in analyzing the flow on different aerodynamic profiles without the use of high performance computers and softwares.

QUESTIONS???