AE3010

Aerodynamics and
Propulsion
Dr. Peter Barrington
RV219
p.barrington@kingston.ac.uk
Lecture Plan
1. Fundamentals
2. Thin Aerofoil Theory
3. Finite Wing Theory
4. Boundary Layers
5. Boundary Layers

Equations of Fluid Motion
Unknowns
Pressure (P)
Density ()
Temperature (T)
Velocity (u,v,w)
Continuum Assumption
We assume we can define: P,, T,
u,v,w at a point, however this is not
valid if we are dealing with length
scales that are similar to the
molecular spacing
Equations of Fluid Motion
Conservation of Mass
Momentum Equation (3
components)
Energy Equation
Equation of State (relates P,T,)
Conservation of Mass
The principle of conservation of mass
is often called the continuity equation.
We can derive this equation by
considering an infinitesimal control
volume.
Net mass flow out of CV + Rate of
change of mass in CV = 0
Conservation of Mass
The derivation is given in H+C 2.4
0 =
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w
y
v
x
u
z
w
y
v
x
u
t


Momentum Equation
The momentum equation is the
application of Newtons 2
nd
law to a
fluid domain. Since momentum is a
vector quantity there are three
equations
Momentum Equation (H+C 2.6)
Rate of increase of momentum within
CV + Net flow of momentum out of CV
= Force on CV
Fluid Forces
Gravitational Force
Pressure
Shear Stress
Pressure Forces
Pressure is the force per unit area
acting normal to any surface. At a
point in the fluid, the pressure force is
the same in all directions.
Pressure Forces
}
= dS n P F

We can use this expression to determine the

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=
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=
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+
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t
t
o

t t o

t
t
o

Constitutive Equation
In order to close the system of
equations we must relate the shear
stresses to the velocity. We usually
assume that the shear stress is
linearly related to the rate of strain. A
fluid that behaves in this way is known
as a Newtonian fluid
Constitutive Equation
If we assume that the fluid behaves
the same way in all directions
(anisotropic), we can show that the
viscous behaviour is characterized by
two coefficients of viscosity. We often
make the further simplifying
assumption that only the dynamic
viscosity, has a significant effect.
Newtonian Fluid
( ) ( )
( ) ( )
( ) ( )
(
(
(

+ +
+ +
+ +
=
(
(
(

c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
z
w
y
w
z
v
x
w
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u
y
w
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v
y
v
x
v
y
u
x
w
z
u
x
v
y
u
x
u
zz yz xz
yz yy yx
xz xy xx
2
1
2
1
2
1
2
1
2
1
2
1
2
o t t
t o t
t t o
Newtonian Fluid
In the simple case where there is only
an x component of velocity and no
changes in the x or z direction, eg flow
between to parallel plates
dy
du
t =
Navier-Stokes Equation for
incompressible flow
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=
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=
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+
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2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
z
w
y
w
x
w
z
P
g
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w
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u
t
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y
P
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x
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u
t
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u
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u
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u
x
P
g
z
u
w
y
u
v
x
u
u
t
u
z
y
x



Momentum Equation
This equation is also derived in
Aerodynamics for Engineering
Students by E.L. Houghton and P.W.
Carpenter (Section 2.6 and 2.8)
Energy Equation

We can do a similar analysis to
generate the energy equation.
Rate of increase of energy within CV
+ Net flow of energy out of CV = Work
done on CV + rate of heat transfer
into CV



Solving the Equations of
Motion
For some very simple geometries and
idealized fluids we can solve these
equation analytically (See
Aerodynamics for Engineering
Students 2.10 (H+C))
Solving the Equations of
Motion
typically we must solve them
numerically using Computational
Fluid Dynamics (CFD) software
or conduct experiments to model a
flow and measure quantities of
interest

Idealizations
Historically, applying these
equations to realistic geometries
requires idealizations
These idealizations do not always
give useful quantitative information
but they can aid our understanding

Idealizations
One common idealization is to treat
the flow as incompressible (constant
density)
This is a very good assumption for
the flow of most liquids but it is
questionable for flow around an
aircraft
Incompressible Flow
We can show that for speeds below
Mach 0.3 (30% of the speed of
sound) the changes in density are
negligible
We can extend our understanding of
incompressible flows to higher Mach
numbers but quantitative predictions
are not accurate
Incompressible Flow
Here we have four unknowns:
(P,u,v,w)

We determine these using the
momentum and conservation of
mass equations

Simplifications
At standard sea-level conditions:
= 1.7910
-5
N.s/m
2
.
The shear forces are usually small
relative to pressure forces unless
there is a large velocity gradient
Simplification
Shear stresses are always signficant
near a solid wall because they enforce
the no-slip condition. In many flows
of interest, there is a thin boundary
layer near the wall where shear
stresses are important but we can
often neglect shear forces outside the
boundary layer
Simplifications
Shear stresses are always significant
in the boundary layer, but we can
often determine the pressure outside
the boundary layer by neglecting them
Simplifications
This approach is only valid if the
boundary layer is thin and remains
attached to the wall
This is valid for high Reynolds
number flow around streamlined
bodies
Fluid Kinematics (H+C 2.7)
We can describe the motion of the
fluid in terms of:
Translation
Change in Volume (Dilatation)
Rotation
Change in shape at constant volume

Fluid Kinematics
We define the rotation of the fluid as
the average angular velocity of two
perpendicular axes
A more common approach is to
define the vorticity at a point as
twice the angular velocity
Vorticity
Vorticity, O, is a vector quantity
defined in a similar way to the angular
velocity vector directed along the axis
of rotation using the right hand rule
Vorticity
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c
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c
c
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c
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= V = O
y
u
x
v
x
w
z
u
z
v
y
w
V , ,

In a two dimensional flow, there
is only a z component of
vorticity and a counterclockwise
rotation is considered positive
Vorticity
The velocity at any point on a wheel
spinning with angular velocity, e is
given by:

Determine an expression for the
vorticity

x v y u e e = = ,
Vorticity
The previous equation allowed us to
determine the vorticity from the
velocity, for an incompressible flow we
can also determine the velocity from
the vorticity
Vorticity
x x r
x dV
r
r
x v
'
=
'
O
=
}}}

where
) (
4
1
) (
3
t
Irrotational Flow
Regions of flow where the vorticity is
zero are known as irrotational flow.
Since pressure forces act
perpendicular to a surface, an initially
irrotational flow will remain irrotational
unless shear stresses are present.
Aerodynamic Flows
For many aerodynamic flows of
interest, the rotational flow is confined
to the boundary layer and wake
region. If we know the vorticity in
these small regions we can determine
the velocity at any point in the flow
Vortex Filament
A vortex filament or vortex tube is a
long thin region of rotational flow,
examples include the trailing vortices
behind an aircraft or a tornado.
Trailing Vortex
Circulation
A key characteristic of a vortex is its
circulation.


We can show that this quantity is
constant along a vortex.
s d v

}
= I
Circulation
Here the contour of integration is the
perimeter of the area considered
previously. However, we can show
that the circulation does not change if
we include regions of irrotational flow
inside the contour
Velocity due to a vortex
filament
}
I
=
3
4
) (
r
s d r
x v

t
Infinite Straight Vortex Line
We can evaluate the integral if we
consider an infinite vortex filamant
aligned along the z axis. In 2
dimensions we call this a point vortex.
Point Vortex
The velocity due to a point vortex at
the origin is given by:

( )
( )
2 2
2 2
2
2
y x
x
v
y x
y
u
+
I
=
+
I
=
t
t
Exercise
Show that for a point vortex the
rotation is zero apart from at the origin
(0,0)
Point Vortex
We can transform this formula to polar
co-ordinates:
2 2 2
sin , cos
: where
cos sin
sin cos
y x r
r
y
r
x
v u v
v u v
r
+ =
= =
+ =
+ =
u u
u u
u u
u
Point Vortex
In polar co-ordinates

r
v
v
r
t
u
2
0
I
=
=
Point Vortex Exercise
Use the polar co-ordinate form to
evaluate the circulation around a
contour of radius R.
Point Vortex
A point vortex in a free stream can be
used to model the flow field around an
aerofoil in regions well away from the
aerofoil. The circulation is related to
the lift per unit span of the aerofoil by
the Kutta-Zhukovsky theorem (see
H+C pg167)
Kutta-Zhukovsky Theorem
n circulatio
velocity stream free V
density
span unit per lift
where
I
I =

l
V l
Flow around an aerofoil
=
Helmholtz Laws
The circulation along a vortex
filament is constant
A vortex filament cannot end in the
fluid
The circulation around any path that
moves with the fluid is constant