INCOMPRESSIBLE AERODYNAMICS

Theme 1
Basic Concepts

Dr Mostafa Nabawy

1
INCOMPRESSIBLE AERODYNAMICS

Topic 1
Definitions

2
Learning Outcomes
By the end of this topic you should be able to:

Describe aerofoil geometric characteristics

Extract aerofoil characteristics for NACA 4-digit
and 5-digit series
Explain how lift is generated on an aerofoil section
Define the 2D aerofoil aerodynamic coefficients

3
Introduction
Development of aviation in the last century

Space shuttle, 1981,

M~25,
19000miles/hour
Concorde, 1976, M~2
~ 1500miles/hour

Spitfire, 1938,
357miles/hour

Wright flyer, 1903

30miles/hour 4
Introduction
• Aerospace engineering is a multidisciplinary subject

• Progresses in aerodynamics has made fundamental

contributions to the developments in aeronautics
5
Introduction
• Basic aircraft components

• The aerodynamics of aerofoils & wings will be the

focus of this course.
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Introduction
• Aerodynamics is a branch of fluid dynamics
concerned with the external motion of air when it
interacts with a solid object, such as an
aircraft wing.
• This course is about the behaviour of aerofoil and
wings in low-speed flows.
• It teaches the fundamental knowledge and theories
required for carrying out engineering analyses of
the forces and moments acting on aerofoil and
wings.
• We will study mainly the inviscid flow theories,
which provide adequate predictions of pressure
distributions and lift on aerodynamic bodies.
2D Aerofoil
A 2D aerofoil is:
The cross-sectional shape of a wing
Often referred to as a wing section with an infinite span

Wing

Aerofoil section

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Milestones
1884: First aerofoil shape
patented

Double surface aerofoil sections by

Horatio Philips. These were patented
by Philips in 1884

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Milestones
1884: First aerofoil shape Wright brothers’ first flight (1903)
patented
1903: First flight of Wright
brothers

Some aerofoils tested by the

Wright brothers 10
Milestones
1884: First aerofoil shape
patented
1903: First flight of Wright
brothers
1917: Development of
thick aerofoil
Fokker Dr –I (1917)

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Milestones
1884: First aerofoil shape
patented
1903: First flight of Wright
brothers
1917: Development of
thick aerofoil
1930s: Systematic
testing and development
of aerofoil shapes by
NACA

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Aerofoils of Early Years

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Milestones
1884: First aerofoil shape
patented
1903: First flight of Wright
brothers
1917: Development of
thick aerofoil
Developed by Richard Whitcomb in
1965 at NASA Langley
1930: Systematic testing
and development of
aerofoil shapes by
NACA
1965: Supercritical A Boeing 747-8 with supercritical
wings
aerofoils
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2D Aerofoil Nomenclature
The aerofoil geometry is
defined by the camber
distribution and the thickness
distribution

The camber line is the curve

half-way between upper and
lower surfaces

The angle of attack (AoA) is

the angle between the inflow
direction and the aerofoil chord
cc Olivier Clynen
line (Is there a more
convincing definition for AoA?)
zero camber = symmetric aerofoil
The aerofoil shape & angle of
attack are the main controllers
of the aerofoil aerodynamic
behaviour
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Camber and Thickness

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Aerofoil Nomenclature

Important aerofoil parameters:

Chord length, c
Maximum thickness and its location along the chord
Maximum camber and its location along the chord

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NACA Aerofoil Numbering
Four digit series: e.g. NACA 4415

Can you sketch the shape of an NACA 0012 aerofoil?

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NACA Aerofoil Numbering
Five-digit series: e.g. NACA 23012

NACA 23012
Thickness:
Location of max 12%c
Design cl= 0.3
camber: 15%C
(1st digit * 3/20)
(2nd and 3rd digit /2)

Refer to “The theory of wing sections” by Abbot & Von Doenhoff

for definition and experimental data.
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How is Lift Created?
For mechanical forces to be Lift Fluid force
created, the wing has to be moving
in a fluid
According to the shape of aerofoil
and angle of attack, the
Drag
momentum (mass times velocity) V
varies around the aerofoil
The pressure varies around the
aerofoil in a moving fluid because it
is related to the fluid momentum Note that most of the pressure
forces are due to the suction force
A net fluid force is generated by on the upper surface.
the pressure acting over the entire
surface of the aerofoil What is the other contributor to the
The component of fluid force net fluidic force?
normal to flow direction is the Lift
force and the component in flow For a comprehensive explanation:
direction is the Drag force See “How do wings work” by H
Babinsky, 2003 Phys. Educ. 38 497
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Aerodynamic Forces
The aerodynamic forces and moments on a body
placed in a fluid stream are due to two basic sources:

Pressure distribution, p, acting normally to surface

Shear stress distribution, t, acting tangentially along surface

The net effect of p and t distributions integrated over the

complete body surface is a resultant aerodynamic force R
and a moment M.
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 AERODYNAMIC FORCES
The resultant force R can be split into:

A component, N, perpendicular to c
A component, A, parallel to c
or
The lift, L, perpendicular to V∞
The drag, D, parallel to V∞

The relationship between

N & A and L & D:
L = N cos  − A sin 
D = N sin  + A cos 
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2D vs 3D Nomenclature

For a 2D aerofoil
Lift, drag and moment are forces/moment acting on a unit
span
In this case, the area S = c x 1!
They are denoted as L’, D’ and M’

For a 3D finite wing

Lift, drag and moment are forces/moment acting on the
whole wing
They are denoted as L, D and M

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2D vs 3D Nomenclature
As a convention in the literature

For a 2D aerofoil
Lift, drag and moment coefficient: cl, cd and cm
L' D' M'
cl = 1 , cd = 1 , cm = 1
 V
2  
2
 c  V
2  
2
 c  V
2  
2
 c 2

For a 3D finite wing

Lift, drag and moment coefficient: CL, CD and CM.
L' D' M'
CL = 1 , CD = 1 , CM = 1
 V
2  
2
 S  V
2  
2
 S 
2   S c
V 2

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2D Aerofoil Coefficients
For aerodynamic coefficients for an aerofoil.
L' D' M'
cl = , cd = , cm =
q c q c q c 2
N' A'
cn = , ca = ,
q c q c

where q = 12  V2 , is dynamic pressure.

By dividing the below equations

L' = N ' cos  − A' sin 
D' = N ' sin  + A' cos 
by q c we have
cl = cn cos  − ca sin 
cd = cn sin  + ca cos 
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INCOMPRESSIBLE AERODYNAMICS

Topic 2
Reynolds Number

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Learning Outcomes
By the end of this topic you should be able to:

Recognise the importance of Reynolds number

Explain the differences between laminar and
turbulent flows
Construct the drag polar of a 2D aerofoil

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Dimensional Analysis
What physical quantities determine the variation of
aerodynamic forces and moments?
Consider a body of a given shape at a given angle of
attack. The resultant aerodynamic force R will depend
on the following parameters:

𝑅 = 𝑓(𝜌∞ , 𝑉∞ , 𝜇∞ , 𝑎∞ , 𝑐)
Freestream velocity, V∞
Freestream density, ∞
Viscosity of the fluid, m∞
Compressibility of the fluid or the speed of sound, a∞
Characteristic length of the body, c

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Dimensional Analysis
Based on the Buckingham p theorem
𝑅
𝑅 = 𝑓(𝜌∞ , 𝑉∞ , 𝜇∞ , 𝑎∞ , 𝑐) 𝐶𝑅 = 1 2 2
= 𝑓(𝑅𝑒, 𝑀∞ )
2 𝜌∞ 𝑉∞ 𝑐
where
 V c V
Re = , M = (Similarity parameters)
m a

If the angle of attack is allowed to vary, then we have:

𝐶𝑅 = 𝑓1 (𝑅𝑒, 𝑀∞ , 𝛼)
𝐶𝐿 = 𝑓2 (𝑅𝑒, 𝑀∞ , 𝛼)
𝐶𝐷 = 𝑓3 (𝑅𝑒, 𝑀∞ , 𝛼)
𝐶𝑀 = 𝑓4 (𝑅𝑒, 𝑀∞ , 𝛼)

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Dynamic Similarity
When a scaled model of flight vehicle is tested in a wind
tunnel, CR , CL , CD and CM at a given  will be identical
to the free-flight as long as Re and M∞ are kept the
same as those in the free-flight case.

At M∞ <0.3, the flow is regarded as incompressible. As

such in this case, only Re has to be kept the same.
Note that, there may be other parameters that can
influence the measurements such as:
Wind tunnel wall effect (tunnel blockage)
Turbulence level
Surface roughness of the model
Presence of heat transfer
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Reynolds Number

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Reynolds Number
Typical Reynolds numbers of flight

 V L
Re =
m

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Reynolds Number
The Reynolds number is a measure of the relative
importance of inertial force to viscous force, i.e.
 V c Inertial _ force
Re = 
m Viscous _ force

Reynolds number has a significant impact on propulsion

mechanisms

Micro-organisms Jelly fish Jet aircraft

Re<<1 Re=200-3000 Re>108

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Reynolds Number
The Reynolds number determines the state of the boundary
layer, i.e. fully laminar, fully turbulent or transitional.

As a rule of thumb, a boundary layer developing along a flat

plate undergoes transition from laminar to turbulent flow at a
distance x from the plate’s leading edge where Rex=5x105.

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Laminar-Turbulent Transition
Other parameters also affects the location of transition.

Moving upstream due to moving downstream due to

• Adverse pressure gradient (dp/dx>0) • Favorable pressure gradient
• Higher surface roughness (dp/dx<0)
• Higher freestream turbulence intensity • Increased compressibility

An adverse pressure gradient hastens

the onset of transition.

Transition often occurs right after the

point of minimum surface pressure. Point of Minimum
pressure 35
Flow Separation
Consider 2D flow over an aerofoil section
Pre-stall
y

dt
t+
dy
A B dp Post-stall
p p+ x
dx
D
t
C

 u 
  = 0
 y  w

Flow separation occurs when the flow in a boundary layer

(either laminar or turbulent) does not have sufficient kinetic
energy to accommodate the pressure rise in the flow direction.
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Laminar vs Turbulent Flows
Instantaneous streamlines

Turbulent flows are

Laminar flows are characterised by
characterised by Vigorous transverse
smooth streamlines motion of fluid particles
hence a limited amount bringing higher
of mixing within momentum fluid to the
near wall region

Velocity profiles

In a turbulent boundary layer, the

velocity in the near wall region is higher
than in a laminar boundary layer.
Hence a turbulent boundary layer is
more capable of resisting flow
separation.

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Reynolds Number – Circular Cylinder
The Reynolds number determines how flow separation
occurs via affecting the state of boundary layer.

Laminar flow separation

(D)

Turbulent flow separation

(E)

The step change in CD from point D to E is

caused by the flow becoming turbulent before
the point of separation
The flow over a rough cylinder encourages the
boundary layer to become turbulent
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Reynolds Number - Aerofoils
Reynolds number effect is most pronounced at stall
regions

.
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Drag Components of a 2D Aerofoil
Aerofoil profile drag = Skin friction drag + Pressure drag

Friction drag due to surface friction

A strong function of incoming
velocity and viscosity
Proportional to the surface area

Pressure drag due to flow separation

Proportional to the pressure difference
between the front and back sides of
the body
Proportional to the frontal area

Note that for an aircraft at cruise, 50% of the total drag is due to
surface friction drag!
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Drag Components of a 2D Aerofoil
Aerofoil profile drag = Skin friction drag + Pressure drag

Affected by
laminar-turbulent High when flow
transition separation occurs

To reduce skin friction drag, it is helpful to maintain laminar

flow for a longer distance.

However, laminar flow is less able to resist flow separation

than turbulent flow and hence it may lead to a higher
pressure drag

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Drag Components of a 2D Aerofoil
Drag Considerations:
Cd = Cd 0 + k1 (Cl − Cl 0 )
2

Minimum drag coefficient for the 2d Pressure drag

aerofoil ( C d 0 ). This represents the
skin friction drag contribution

Lift coefficient at minimum drag

coefficient for the 2d aerofoil ( Cl 0 ).
This becomes zero for a
symmetric aerofoil Cl 0
Skin friction drag

Fitting coefficient for the aerofoil

polar ( k1 ). This represents the
pressure drag contribution 0

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Note on Skin Friction on a Flat Plate
Skin friction drag coefficient, Cf, for a flat plate at zero
incidence (hence no pressure contribution) could be
obtained for both laminar and turbulent conditions
Cf is strong function of Re, it decreases with the increase of
Re, and is higher for a turbulent flow
Skin friction drag of other aerodynamic bodies is usually
expressed in terms of the flat plate coefficient, hence its
importance

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INCOMPRESSIBLE AERODYNAMICS

Topic 3
Aerodynamic Centre
&
Centre of Pressure

44
Learning Outcomes
By the end of this topic you should be able to:

Describe the concept of equivalent physical

systems
Define the centre of pressure and aerodynamic
centre
Determine the relative position for the centre of
pressure and aerodynamic centre

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Remember: Aerodynamic Forces
The aerodynamic forces and moments on a body
placed in a fluid stream are due to two basic sources:

Pressure distribution, p(s), acting normally to surface

Shear stress distribution, t(s), acting tangentially along
surface

The net effect of p(s) and t(s) distributions integrated over the
complete body surface is a resultant aerodynamic force R’ and a
moment M’.

Where should we put R’ and M’ ? 46

Equivalent Physical Systems
Where should we put R’ and M’?
R’ can be placed at any position along the chord line,
as long as an additional moment M’ is also added
such that they produce the same effect around this
point as the distributed load does.
Equivalent physical systems (EPS)
A B
R’
M’

= •
x

R’ is put at an arbitrary
Actual distributed load point along the chord, x
B is an equivalent physical system of A if B produces
the same R’ and M’ around the same point as A does
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Equivalent Physical Systems
If R’ and M’ are put at the following special positions

M’le R’
R’ is put at the Additional moment, M’LE
leading edge, should be added
xle. •
R’
R’ is put at the
Centre of Pressure, • Additional moment is 0
xCP xCP

M’AC R’
R’ is put at the Additional moment, M’AC
Aerodynamic Centre, • should be added. However,
dM ' AC dcm , AC
xCA xAC = 0 or =0
dcl dcl

How do we find XCP and XAC?

48
Equivalent Physical Systems
If  is small, we can assume:
L’ ┴ chord and L’=N’
D’ // chord and D’=A’.

Hence we have
R’ L’
M’le
R’ is put at the
leading edge, • M’le
•
xle.
R’ L’
R’ is put at the Centre
of Pressure, xCP • • D’
xCP xCP

M’AC L’
R’ M’AC
R’ is put at the
Aerodynamic Centre, • • D’
xCA xAC xAC
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Centre of Pressure
Let MLE’ be the moment generated by the distributed load
around LE

Taking the moment of L’ about LE gives

Hence

A negative sign is added since

M '
LE = − xCP L '
M’ LE is defined as positive in
the pitch up sense.
'
M LE L’ & M’ LE can be found if
xCP =− surface pressure and shear
L' stress distributions are known.

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 Aerodynamic Centre
The aerodynamic centre is a point on the aerofoil around
which cm is independent of cl or  (see figure below) i.e.
dcm , AC
=0 cl
dcl
For a flat or curved plate in
inviscid incompressible flow,
AC is theoretically located at x=c/4.

Effects of aerofoil thickness,

viscosity & compressibility cm,c/4

xAC=c/4

moves forward due moves backwards

to thickness, and/or due to compressibility
viscosity
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Aerodynamic Centre
Consider the two equivalent physical systems below

M’x x AC x M’dcm , x AC

= −
c x c dcl
x AC

Resultant force at an Resultant force at the

arbitrary point aerodynamic centre

Taking the moment around le for both systems,

M x' − L'  x = M AC
'
− L'  x AC
[1]
M =M
'
x
'
AC − L  x AC + L  x
' '

Dividing Eq.1 by 1
2  V2 c 2 gives
 x AC x 
cm , x = cm, AC − cl  −  [2]
 c c 52
Aerodynamic Centre
How to find cm,AC experimentally?
From
 x AC x  when cl =0, cm,AC is equal
cm , x = cm , AC − cl  −  [2] to the (constant) pitching
 c c moment measured at
When cl =0, we have cm , x = cm , AC. any point x.

How to find xAC experimentally?

Differentiating Eqn. 2 wrt cl gives: M’x

dcm, x dcm, AC  x AC x 
= −  − 
dcl dcl  c c x

x AC x dcm , x xAC can be found if the slope

= −
c c dcl of cm,x versus cl is obtained

Take the measurement of cl and cm,x at several incidence angles and

then find dcm,x /dcl by best fitting your data with a straight line 53
Relation Between xCP and xAC
From
 x AC x 
cm, x = cm, AC − cl  −  [2]
 c c

If x=xCP, we have cm, x =0, substituting into Eq. 2

 x AC xCP  xCP x AC cm , AC
0 = cm, AC − cl  −  = −
 c c  c c cl

Since cm , AC is often negative, xCP tends to be behind xAC.

M’AC

xAC

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Relation Between xCP and xAC
Relative positions between xCP and xAC

xCP x AC cm , AC
= −
c c cl

xCP
c

xCP goes asymptotically to xAC as cl increases and to infinity

as cl approaches to 0
Due to large excursion of the CP position,
cm about AC (or c/4 point) are given in aerofoil database
In flight dynamics analysis, wing lift, drag and moment are
frequently shown to act at AC 55
Relation Between CP and CG
Centre of pressure moves forward as the aerofoil
incidence angle increases.

Is this good or bad? Why?

56
Reading Materials
Lecture notes

Fundamentals of §1.5, 1.6, 1.7,

Aerodynamics
by J Anderson, 6th edition, 1.11, 1.12
McGraw-Hill §4.2, 4.9, 4.14

Aerodynamics for §1.4.1

Engineering Students
by E L Houghton & P W §3.1
carpenter, 7th edition, Elsevier §3.3

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