AET lecture notes.pdf

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Introduction to Aerospace Technology

(MM3AET)

Aerodynamics

Professor K-S Choi

8, 15 Oct 2013

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MM3AET ( Aerodynamics)

Pressure distribution around a circular cylinder

The pressure drag of a circular cylinder per unit span with radius a is given by

d ) p-(pa-= D2

0

cos

whose drag coefficient is defined by AV

D C 2 D

21

, where . Therefore,

d C -=

d V

) p-(p -=

aV

d ) p-(pa -= C

p

2

0

2

2

0

2

0

D

cos2

1

cos2

1)2(

cos

21

2

1

where, the pressure coefficient C p is given by2

21 V

p pC

p

The coordinate system used for the integral of static pressure around a circular

cylinder to give the pressure drag. Here, p is the static pressure over the cylinder

surface and p∞ is the freestream pressure.

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There is a significant difference in the static pressure distribution between the

turbulent flow and the laminar flow (see below). While the static pressure forthe laminar flow stays at a near minimum value of C p = -1.0 in the rear of the

circular cylinder , the turbulent flow recovers to a much greater value of C p = -

0.4 after reaching the minimum value of C p = -2.1 at around θ = 75°. Thisreflects a small C D value of 0.3 for the turbulent flow as compared to C D = 1.2

for the laminar flow. In other words, the main reason for such a smaller C D value for the turbulent flow is that the flow separation takes place much further

downstream, therefore the wake is much narrower than that of laminar flow.

The inviscid (potential flow) theory gives a symmetric C p curve, on the other

hand, suggesting that the drag on a circular cylinder is zero for zero viscosity

fluids. Indeed the inviscid theory cannot impose the non-slip condition on the

wall, therefore there will be no boundary layer development or flow separation

over immersed bodies.

Non-dimensional pressure distribution over a circular cylinder, where the C p

curve for the laminar and turbulent flow are compared with the solution ofinviscid flow.

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Flow around a circular cylinder at different Reynolds numbers.

Laminar (left) and turbulent (right) flow separation over a sphere.

Re = O (1)

Re = O (10)

Re = O (1000)

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Kutta-Joukowski Theorem

The pressure distribution around a circular cylinder in a uniform velocity V with

a clockwise vortex with strength K (circulation Γ = 2π K ) can be given by theBernoulli equation as follow.

2 sin222

a

K V pV

2

21

sin21aV

K

V

p pC

p

Here, the term involving K is due to the added circumferential velocity of the

vortex. Therefore, the drag coefficient C D is given by

d C -=C p

2

0

D cos

2

1

d aV

K

o

cossin212

1 2 2

= 0

In other words, the drag on a circular cylinder is zero due to inviscid (potential

flow) theory. This is d’Alembert’s paradox.

The lift L on a circular cylinder per unit span can be calculated by

d ) p-(pa-= L2

0

sin

Therefore, the Lift coefficient AV

L C

2 L 2

1 is given by

d C C o

p L sin2

1 2

d aV

K

o

sinsin212

1 2 2

Using the flowing integrals

0sinsin2

0

32

0

d d and π θ d θ π

2

0

2sin

the lift coefficient is given by

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a

K

V C

L .2

Since the circulation is given by 2 (see a note below), the lift is given by

V L

This is the Kutta-Joukowski theorem on lift, which can apply to two-

dimensional objects of any size and shape. It states that the lift on a body is

proportional to the fluid density, the freestream velocity and the circulation

around the body.

The positive lift (upwards) is produced when the vortex is clockwise, while thenegative lift is given when the vortex is counter-clockwise.

The circulation Γ around a closed circuit is defined by

l d V

For a simple vortex with strength K , the circumferential velocity isr

K v

.

Therefore,

Γ = vθ (2πr ) = r

K (2πr ) = 2π K

This relates the circulation Γ to the vortex strength K.

V

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K

Increasing circulation Γ,

which gives geater lift L.

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Thin flat-plate aerofoil

A thin flat plate is the simplest form of aerofoil, which can be modelled by a

sheet of vortex with a distribution _

0 )1(sin2)( xC

U x . Here, U 0 is thefreestream velocity, α is the angle of attack, and C is the chord length.

The circulation around a thin flat plate can be obtained by integrating x over

the plate length, i.e. Γ = C

dx x0

)( . The lift on the aerofoil can then be obtained

from the Kutta-Joukowski theorem

L = ρ U 0 Γ = ρ U 0 C

xC dxU

0

1/2 _

0 )1(sin2 = ρ U 0

2 sinC

The lift coefficient on a thin flat plate can be given by

C U

LC

L 2

021

2s n2

While, the moment coefficient about the leading edge of the flat plate is given

by

22

sin22

021

C U

M C

M 4

LC

This means that the centre of moment is located at the one-quarter-chord point

(irrespective of the angle of attack) from the leading edge. Here, C M is positive

when the pitching moment is counter-clockwise.

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Aerofoils

Cessna 150 uses NACA 2412 aerofoil.

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-1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0

0

0.1

0.2

0.3

0.4

x/c

y / c

-1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 00

0.1

0.2

0.3

0.4

x/c

y / c

-1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0

0

0.1

0.2

0.3

0.4

x/c

y / c

-1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0

0

0.1

0.2

0.3

0.4

x/c

y / c

-1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0

0

0.1

0.2

0.3

0.4

x/c

y / c-1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0

-0.1

0

0.1

0.2

0.3

x/c

y / c

Aerofoil characteristics

This diagramme shows the lift coefficient C L as a function of the angle of attack

α, where α L=0 is the zero-lift angle of attack. Typically α L=0 is a small negative

value. The sensitivity of C L to the angle of attack is given by the lift slope a0.

The maximum lift coefficient C L, max is observed just before the stall, where

there is a flow separation from the surface of aerofoil.

NACA 0012 aerofoil (Re = 20,000)

α = 2°

α = 6°

α = 4°

α = 8°

α = 10°

α = 12°

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These figures show that the flow separation behaviour is different from aerofoil

to aerofoil. It also depends on the Reynolds number.

NACA 0015 aerofoil (Re = 160,000)

α = 13.4°

α = 14.4°

α = 14.8°

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Thin aerofoils

NACA 0012 aerofoil is a symmetric aerofoil (see an attached diagramme)

whose thickness is 12% of the chord length. The aerodynamic characteristics of

this relatively thin aerofoil are very similar to those of a thin flat plate. Here, the

lift coefficient and moment coefficient are given by

C L = 2π α and C M, C/4 = 0

NACA 2412 aerofoil is identical to NACA 0012 in shape except that it iscambered (curved). Here, the maximum camber is 2% of the chord length,

which is located at 40% from the leading edge. The lift coefficient C L of this

aerofoil is given by

C L = 2π (α - α L=0)

Indicating that there is a lift on the aerofoil even at zero angle of attack.

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The moment coefficient C M, C/4 at the quarter chord is not zero, i.e.

C M, C/4 ≠ 0

In other words, the quarter chord is not the centre of the pressure for cambered

aerofoils in general.

Pressure distribution over an aerofoil

The figure shows the pressure distribution over the upper (suction) surface of

NASA LS(1)-0417 subsonic (laminar flow) aerofoil at zero angle of attack. The

static pressure reduces sharply from the leading edge to about 10% chord. After

that the pressure remains constant up to 60% chord, although there is a small

but steady increase in the pressure. Then the pressure increases rapidly toward

the trailing edge, but the level of adverse (positive) pressure gradient is not

severe enough to bring about a flow separation.

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When the same aerofoil is set to an angle of attack of 18.4°, a flow separation

can be observed over the upper surface of the aerofoil as shown below. The

pressure distribution at this condition is shown in a figure, indicating that there

is a very sharp drop in surface pressure near the leading edge followed by a

rapid increase. In other words, there is a strong, adverse pressure gradient just

downstream of the leading edge, which leads to a flow separation.

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Finite-span wings

So far, our discussion has been restricted to two-dimensional aerofoils

(infinitely long wings). For finite-span wings, there is a development of a three-

dimensional vortical system as shown in a figure. A simplified horseshoe vortex

model for the vortex system consists of a bound vortex that is attached to the

wing with trailing vortices coming off the wing tips.

The development of a horseshoe vortex over finite-span wings can be explained

by the pressure difference between the lower and upper surfaces. A flow is

induced at wing tips from the lower surface where the pressure is higher to

upper surface with lower pressure, creating vortex roll-ups that trail in the

downstream. The horseshoe vortex will induce a downwash, a negative v-

velocity, to cause an extra drag to the wings.

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MM3AET ( Aerodynamics) 8, 15 Oct 2013

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Induced drag

As a result of trailing vortices from the wing tips, there is a downwash velocity

in the downstream of the wing, which can be given by

s

Γ w

4

This suggests that the magnitude of downwash w is increased with the

circulation Γ (therefore with the lift L), but is reduced with an increase in wingspan s.

We can apply the Kutta-Zhukovsky theorem to obtain the induced drag Di due

to the downwash velocity w as follows.

Di = ρ w Γ

Here, the induced drag will be normal to an induced velocity, i.e. parallel to the

freestream direction.

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The induced drag coefficient is given by

C Di = C L2/ (π AR)

Where, AR is the aspect ratio (the span divided by the chord) of the wing. As a

result of the downwash, the lift coefficient of finite span wings is reduced from

an infinite slope a0 to a, given by

a = a0 (α –

AR π

C L )

while the total drag coefficient is increased by an addition of C Di =AR π

C L .

AR 00

π

C C C C C L

D Di D D .

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MM3AET ( Aerodynamics) 8, 15 Oct 2013

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Flaps

The trailing edge flap effectively increases the camber of an aerofoil, thereby

creating a negative zero-lift angle (see the figure). As a result, the lift curve is

shifted to the left, increasing the lift at a given angle of attack. However, the

trailing edge flap does not change the lift slope. The maximum lift is increased

with a trailing edge flap, but the maximum lift angle is slightly reduced.

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Slats

Slats are leading-edge devices for boundary layer control of an aerofoil, which

can be represented by a vortex as shown in a figure. Without the device, the

velocity increases (pressure reduces) rapidly over the leading edge, which

creates a strong adverse pressure gradient. This causes a leading-edge

separation at a large angle of attack. With slats, velocity over the upper aerofoil

surface is reduced (increased over the lower surface) to reduce the severity of

the adverse pressure gradient, helping the flow remain attached over the

aerofoil.

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The maximum C L is increased by the leading-edge device as a result of this,

although there is no change in zero-lift angle.

The leading-edge device can be combined with trailing edge devices as shownin a figure below. During the cruise, no boundary-layer control devices are

used. During the take off, some of the high-lift devices are deployed. At

landing, all the devices are used.

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Stalling speed

The slowest speed at which an aeroplane can fly is the stalling speed, which is

important for a safe take-off and landing.

Since the lift L is given by

L AC U ρ L

2

21

We know that L = W and C L = C Lmax just before the aircraft stall, therefore the

stalling speed is given by

max

2

L AC ρ

W U

where, W is the weight of the aircraft and C Lmax is the maximum lift coefficient.

In order to reduce the stalling speed, therefore, we must either reduce the

aircraft weight, increase the lifting surface or to increase C Lmax.

All of these are difficult to achieve once the aircraft design is finalised. Without

any scope of changing the design, the Space Shuttle has an extremely high

stalling speed of nearly 100 m/s, making the manual landing nearly impossible.

8, 15 Oct 2013

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