Calculation of the pressure distribution on a pitching airfoil with application to the Darrieus Rotor

119 Pages·2012·2.97 MB·English

by  Ghodoosian, Nader

AN ABSTRACT OF THE THESIS OF

Nader Ghodoosian for the degree of Master of Science in

Mechanical Engineering presented on September 28, 1983

Title: Calculation of the Pressure Distribution on a Pitching

Airfoil with Application to the Darrieus Rotor

Redacted for Privacy
Abstract approved:

Robert E. Wilson

An analytical model leading to the pressure distribution on

the cross section of a Darrieus Rotor Blade (airfoil) was con­
structed. The model was based on the inviscid flow theory and the

contribution of the nonsteady wake vortices was neglected. The

analytical model was translated into a computer code in order to

study a variety of boundary conditions encountered by the rotating

blades of the Darrieus Rotor.

The results of the program indicate that for a pitching air­
foil, lift can be adequately approximated by the Kutta-Joukowski

forces, despite notable deviations in the pressure distribution on

the airfoil. These deviations are most significant at the upwind

half of the Darrieus Rotor where higher lift is accompanied by

increased adverse pressure gradients. It was also found that the
effect of pitching on lift can be approximated by a linear shift in

the angle of attack proportional to the blade angular velocity.

Finally, the fluid velocity about the pitching-only NACA 0015

was tabulated; thus allowing the principle of superposition to be

used in order to determine the fluid velocity about a translating

and pitching airfoil.
Calculation of the Pressure Distribution on a

Pitching Airfoil with Application to the Darrieus Rotor

by

Nader Ghodoosian

A THESIS

submitted to

Oregon State University

in partial fulfillment of

the requirements for the

degree of

Master of Science

Completed September 28, 1983

Commencement June 1984
APPROVED:

Redacted for Privacy
Professor o Mechanical Engineering in charge of major

Redacted for Privacy
Head of Depart t of Mechanical Engi eering

Redacted for Privacy
Dean of Graduat hool
Date thesis is presented September 28, 1983
Typed by Nancy Platz for Nader Ghodoosian
1
ACKNOWLEDGEMENT

am indebted to many people for their help, support, and

I
encouragement during the course of my schooling. I would like to

take this opportunity to express my gratitude to the following

individuals:

Dr. Robert E. Wilson, my major professor, whose financial

support made this research possible. I am grateful for

the freedom under which he allowed me to operate.

Dr. Ronald L. Adams, my minor professor, with whom
I
directly worked. His concern for me and my work has been

overwhelming. It has indeed been a pleasure to associate

with Dr. Adams.

Finally, I would like to extend my love and appreciation

to my parents whose many sacrifices made my education

possible.
11

TABLE OF CONTENTS

Chapter Page

I. Introduction
1

I.1 Darrieus Rotor
1

II. Model Description 5

II.1 Review of Literature 5

11.2 General Approach 7

11.2.1 The Ouasi-Steady Problem 7

11.2.2 The Unsteady Wake Contribution 8

11.3 Adopted Model 9

III. Analytical Development 11

III.1 Governing Equations 11

111.2 Conformal Transformations 13

111.2.1 Karman-Trefftz Transformation 14

111.2.2 Pseudo-Circle to Unit Circle

Transformation 16

111.3 Complex Potential 25

111.3.1 Boundary Function 27

111.3.2 Velocity Potential at the Boundary

Before Adding Circulation 29

111.4 Velocity 30

111.4.1 Circulation and Kutta Condition 30

111.5 Pressure and Forces 32

111.6 Closure 34

IV. Results 35

IV.1 Accuracy 35

IV.2 Coefficients of Lift, Pressure, and Thrust for

NACA 0015 40

IV.3 Alternative Approach 48

IV.3.1 Superposition 48

IV.3.2 Lift 56

IV.4 Closure 57

V. Summary and Conclusions 58

Bibliography 60

Appendices:

A. Poisson Integral Formulas 63

B. Computer Code 65
111
List of Tables
Table Page
IV.1 Input Parameters 35
IV.2 Aerodynamic Characteristics of NACA 0015 54
iv
List of Figures

Figures Page

I.1 The Darrieus Rotor Located in Newport, Oregon 2
1.2 Darrieus Rotor Aerodynamics 3
III.la Pitching Airfoil Subject to Relative Wind 12
III.lb Pitching and Translating Airfoil 12
111.2 Mapping Sequence and Order 15
111.3 Mapping of the NACA 0015 to a Pseudo-Circle Using
the Karman-Trefftz Transformation 17
111.4 Mapping of the NACA 2412 to a Pseudo-Circle Using
the Karman-Trefftz Transformation 18
111.5 Magnification Factor for the NACA 2412 along the
Perimeter 26
111.6 Normal Velocity Component of the Airfoil and Fluid 28
IV.1 Pressure Distribution on the NACA 0015 Executing
Translatory Motion in the Undisturbed Fluid 37
IV.2 Pressure Distribution on the Airfoil M6 Executing
Translatory Motion in the Undistributed Fluid 38
IV.3 Pressure Distribution on a 22.3 Percent Thick
Ellipse Translating and Pitching in the Undisturbed
Fluid 39
IV.4 Circulation and Lift for the Translating and Pitching
NACA 0015 at Various Angular Velocities 41
IV.5 The Effect of Clockwise Pitch on the Pressure
Distribution 43
IV.6 The Effect of Counterclockwise Pitch on the
Pressure Distribution 44
IV.7 Deviation of the Pressure Distribution on a
Pitching Airfoil (Qc/W = 0.07) from that of a
Non-pitching Airfoil Which Produces the Same Lift 45
1V.8 Deviation of the Pressure Distribution on a
Pitching Airfoil (cc /W = - 0.07) from that of a
Non-pitching Airfoil Which Produces the Same Lift 46
IV.9 Deviation of the Pressure Distribution on a
Pitching Airfoil (Qc/W = 0.105) from that of a
Non-pitching Airfoil Which Produces the Same Lift 47
IV.10 Additional Velocity 51
IV.11 Nondimensional Relative Fluid Velocity About the
NACA 0015 Executing Pitching Motion About Point
0.38c 52
V
List of Figures (Continued)
Figures Page
IV.12 Nondimensional Relative Fluid Velocity About the
NACA 0015 Executing Pitching Motion About Point
0.5c 53
vi
Nomenclature

Symbol Description

Ak, Bk Variables associated with differentiation

c Chord length

C Coefficient of lift

Cp Coefficient of pressure

C Coefficient of thrust

t

e Base of the natural logarithm 2.71828...

f Real part of F

F A function

F Mapping function

1

F Complex potential on the boundary before

2

applying circulation

F Force along the positive real axis of airfoil

F Force along the positive imaginary axis of

n
airfoil

g Imaginary part of F

i Unit imaginary number, ,/ -1

Summation index

j
L Lift/Number of airfoil nodes

L Lift for the translating airfoil

T

M Number of arcs the unit circle is divided into

p Pressure

P.
Pressure of the undisturbed fluid

r Radial coordinate in the unit circle plane

r' Radial coordinate in the pseudo-circle plane