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FDeg Year 2 FDeg Year 2 Aerodynamics Aerodynamics 2009/2010 2009/2010 Prof Andrew Rae

Aerodynamics Course Notes v3

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Page 1: Aerodynamics Course Notes v3

FDeg Year 2FDeg Year 2AerodynamicsAerodynamics

2009/20102009/2010Prof Andrew Rae

Page 2: Aerodynamics Course Notes v3

Aerodynamics 2009/2010 Page 2

Learning Outcomes and Assessment Methods

Learning Outcomes - On successful completion of this module, the student will be able to:

Assessment Methods

1. Identify and analyse the aerodynamic forces on an aircraft. Explain the effects of airflow at subsonic, transonic and supersonic speeds.

Exam

2. Discuss the different types of aerodynamic experimental methods and the advantages and disadvantages of each method.

Assignment

3. Describe the factors leading to flow separation and solve simple boundary layer and skin friction problems, using standard basic results.

Exam

4. Discuss the relative merits of standard wing planforms and explain the use and benefits of lift augmentation devices.

Exam/Assignment

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Aerodynamics 2009/2010 Page 3

Content (1)

Fundamentals:• Static dynamic and total pressure; Bernoulli’s principle; Speed of

sound and Mach number; ISA tables.

Lift generation:• Circulation, lift and downwash; Kutta condition; Kutta-Joukowski

theorem; spanwise lift distribution, loads and bending moment.

Subsonic Flows: • Contributions to subsonic drag; zero-lift drag, skin-friction;

‘Horseshoe’ vortex system; wing planforms in subsonic flow; induced drag; span efficiency; tip devices; wing design through twist and camber including “wash-out” and “wash-in”.

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Content (2)

Viscosity: • Definition; boundary layers; Reynolds number; velocity profiles; no-

slip condition; effect of surface roughness on skin friction; laminar and turbulent flows; local and global skin friction calculations; boundary layer thickness definition; momentum and displacement thickness; equivalent body in inviscid flow; transition and flow separation.

Aerodynamic methods;• History of aerodynamic testing; Wind tunnel types; low speed and

high speed testing; Open and closed circuit (Eiffel/Goettingen) type tunnels; Open, closed and slotted/porous working section type tunnels; Flight testing; Model mounting systems; upwash, buoyancy and blockage correction methods; Mach similarity; Methods of increasing Reynolds number; Powered wind tunnel models; Pressurised, cryogenic, heavy gas and water tunnels; Introduction to CFD; Description of CFD; Advantages and disadvantages of CFD; Examples/demonstration of CFD usage.

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Content (3)

Lift augmentation and flow control devices;• The need for high lift; history of high lift; slats, flaps and other high lift

devices; the effects of slots; Coanda effects and blown devices; powered high lift devices; vortex generators.

Supersonic Flows: • Critical Mach Number; formation of shockwaves; Normal and oblique

shockwaves; Effect of wing thickness and camber; Wave drag and methods of reducing wave drag (Wing Sweep, Transonic Area Ruling, Supercritical Aerofoil design, Wing design); Shockwave control and the Shock-induced separation.

Swept wings:• Swept wing flows; Effect of spanwise and normal velocity components;

qualitative description of 3D boundary layers on swept wings; Forward, rearward and variable sweep wings; control surface effects; delta wings and vortical flows; vortex flap; aerodynamics of aircraft at high incidences.

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Aerodynamics 2009/2010 Page 6

• Static dynamic and total pressure; Bernoulli’s principle; Speed of sound and mach number; ISA tables.

Fundamentals

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• Circulation, lift and downwash; Kutta condition; Kutta-Joukowski theorem; spanwise lift distribution, loads and bending moment.

Lift Generation

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Lift generation (1)Circulation

– A term meaning rotation, which in aerodynamics is usually associated with vorticity.

– An commonly seen example is a type of forced circulation called the Magnus effect

• If a cylinder or sphere is made to rotate as it travels through air, friction causes:

– the air on the forward moving side to slow down

– the air on the rearward moving side to speed up

– a differential pressure (Bernoulli) and a lift force

The object moves sideways

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A David Beckham free kick

Lift generation (2)

Slice on a golf ball

Some you might have seen…..

A ‘curve’ ball

Spin on a tennis ball

Purposely ignoring cricket - polishing, seam, boot studs, etc. are all separation control (dimples on a golf ball)….see later….

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• An aerofoil is a body that induces the same effect through shape only.

• Consider first inviscid flow (no friction)

Lift generation (3)

• The flow around a body produces changes in velocity and thus changes in pressure– but the pressure variations are symmetrical, i.e. no lift

Note: Pictures from ‘An Album of Fluid Motion’ (Parabolic Press)

Dye injection shows the streamlines in water flowing at 1mm/s between glass plates spaced 1mm apart. It is interesting that the best way of showing the unseparated patterns of inviscid flow (which would be spoiled by separation in a real fluid of even the slightest viscosity) is to go to the extreme of creeping flow in a narrow gap, which is dominated by viscous forces, i.e. boundary layers.

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• So, in inviscid flow, the pressures on the upper and lower surfaces of an aerofoil are equal and thus so are the velocities:

• In real life (air, water) the flow is viscous:– The flow on the lower surface will not traverse the sharp trailing edge –

there is a limiting curvature (pressure gradient) round which a viscous fluid will flow (spoon under a tap)

– By not doing so it creates a creates a partial vacuum (low pressure) on the trailing-edge upper surface

– This draws the upper-surface flow down to the trailing edge too– Both upper and lower-surface flows leave smoothly at the aerofoil at the

trailing-edge– the Kutta Condition (M.Wilhelm Kutta, Germany, 1902)

Lift generation (4)

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• So, the combination of pressure gradients and the Kutta condition results in higher velocity air over the upper surface and lower velocity air over the bottom surface and hence lift

• A mathematical way of representing this is to take the inviscid flow…

Lift generation (5)

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• So, the combination of pressure gradients and the Kutta condition results in higher velocity air over the upper surface and lower velocity air over the bottom surface and hence lift

• …and add circulation (rotation, a vortex)

Lift generation (6)

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• As with the Magnus effect, the result of the circulation (vortex) is lift:

Lift generation (7)

=

• This circulation is also known as the ‘bound’ vortex and forms part of the horseshoe vortex system (see later)

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• The vorticity is not just a mathematical device, it is a real effect and can be seen most obviously when it is shed from a wing tip.

• In addition, the velocity gradients in the boundary layer produce vorticity that is thus distributed along the aerofoil surface.

Lift generation (8)

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• The circulation (vortex) has a strength () and the lift generated by a 2-D aerofoil (or per unit span for a 3-D wing) is given by:

(the Kutta-Joukowski Theorum)

where is the density of the air and U is the velocity of the aerofoil.– Thus for a given speed and altitude, higher lift means stronger

vorticity.

• So, could there be lift without friction?– Lift is a result of the surface pressure distribution

• an inviscid phenomenon– But the differential pressure between upper and lower surfaces is a

result of the Kutta condition• a viscous phenomenon

UL

Lift generation (9)

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• For a finite wing, the combination of bound and shed (free) vorticity is called the horseshoe vortex system

• A simple, finite, rectangular wing can be represented as a single bound vortex of constant strength, and a pair of semi-infinite trailing vortices

• The bound vortex is located at the centre of pressure (~c/4)

Lift generation (10)

Anderson ‘Fundamentals of Aerodynamics’

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• The starting vortex– When an aerofoil starts

moving the flow tried to curl around the trailing edge

– In so doing the flow velocities there become very large

– Consequently a thin region of very large velocity gradient (and thus high vorticity) is formed at the trailing edge

– Once the flow is established, the flow leaves the trailing edge smoothly (Kutta condition) and the velocity gradients disappear

– The shed vorticity rolls up into a starting vortex

Lift generation (11)

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• Trailing vorticity – an aside (#1)

– Crow instability

– Condensation trails from a B-47 taken at 15s intervals after its passage

Lift generation (12)

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• Trailing vorticity – an aside (#2)

– Wake vorticity and wake encounters

– Especially on descent ( U )

Lift generation (13)

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• The bound and trailing vortices are examples of vortex filaments:– Lines of constant strength (point) vorticity

– They can be curved but for current purposes we will consider only straight filaments

• The vortex filament will induce flow around it, depending on its strength and direction of circulation– E.g. the bound vortex….

– ….and the trailing vortices too

Lift generation (14)

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• Based on the Biot-Savart Law (see Anderson, ‘Fundamentals of Aerodynamics’ Section 5.2), the magnitude of the velocity (V) at a point (P) that is at a perpendicular distance (h) from a vortex filament of strength Γ is given by the equation:

• The magnitude of the induced velocity decreases with increasing distance from the vortex

hV

4

h

─∞V

∞Γ

Lift generation (15)

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• So from before, with the horseshoe vortex system, the downwash (w) induced along the span of the wing by the trailing vortices can be shown as:

Anderson ‘Fundamentals of Aerodynamics’

Lift generation (16)

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• The angle of attack (α) between the chord line and the free stream (V∞) is the geometric angle of attack

Lift generation (17)

• The downward component of velocity generated by downwash at the wing is w, producing a local relative wind inclined from the below V∞ the induced angle of attack αi. This has two effects:

– The angle of attack seen by the aerofoil is less than the geometric angle of attack and is known as the effective angle of attack

αeff = α – αi

– The local lift vector is perpendicular to the local relative wind and thus is now inclined behind the vertical by the angle αi and thus has a longitudinal component which contributes to drag, Di

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• But what is wrong with this picture?– Only applicable to rectangular wings

• No taper, no twist, no sweep

– Not even brilliant for that:• Downwash of infinite value at the wing tip?

• Consider the concept of lift distribution– The variation of lift along the span

• For the horseshoe vortex system, the bound vortex is of constant strength and the lift is thus constant across the span

BUT• The equalisation of pressure at the wing tip (bleed between upp an

lower surfaces) means that in reality there is zero lift at the wing tip

Lift generation (18)

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• The lift distribution of a simple horseshoe vortex system would thus be draw as:

• Whereas it should be:

(The local lift is the local height of the lift distribution and the total lift is the area under the lift distribution curve)

y

Lift generation (19)

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• To model reality better we introduce a concept called the lifting line– Ludwig Prandtl (Göttingen, 1911-1918)

• What happens if we take a single horseshoe vortex and add another of smaller span on same chordline?

∞2

b

2

bΓ1

Γ1

Γ 1

Γ1

Γ1

Γ2

Γ2

Γ 1 + Γ 2

Γ2

Γ2

Lift generation (20)

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• By playing games with the number, span and strength of the horseshoe vortices we can achieve any lift distribution to define any planform– Still used today for preliminary calculations, bearing in mind its

limitations (inviscid, incompressible)

Lift generation (21)

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• Wing loading– The mass of an aircraft divided by the area of its wing planform

• A useful indicator of an aircraft’s handling and performance

&

• Take-off and landing performance– Rearranging the equation for lift coefficient gives

and therefore

– So for aircraft with the same lift coefficient at take-off and landing (and in the same atmospheric conditions) the aircraft with the bigger wing will need less speed at take-off and landing, or less lift coefficient at the same speed

SV

LCL

2

21

gWS

MgCV

S

LSL 2

2

1

CL = lift coefficient

ρ = pressure

V = velocity

S = wing area

M = aircraft mass

g = acceleration due to gravity

WS = wing loading

L

S

C

WgV

2

SWαV

Lift generation (22)

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• Initial climb rate– Newton’s Second Law; F = Ma– At rotation, the lift generated is greater than that needed to balance the aircraft

weight otherwise the aircraft would not get airborne.– The vertical force due to the difference between the lift generated and the

aircraft weight is thus

Lift – Weight = L – Mg

=

– And the vertical acceleration can then be found from

– So for the same lift coefficient the aircraft with the lower wing loading will have the greater initial climb

MgCSV L 2

2

1

CL = lift coefficient

ρ = pressure

V = velocity

S = wing area

M = aircraft mass

g = acceleration due to gravity

WS = wing loading

av = vertical acceleration

gMCSVaM Lv 2

2

1 gCVW

a LS

v 2

2

1

Lift generation (23)

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• Turning performance– An aircraft performing a constant-speed,

constant radius turn obeys the mathematical rules of circular motion

• The velocity (rate of change of position) is the distance travelled around the circle (the circumference) divided by the time taken to complete a rotation. i.e.

• The acceleration is rate of change of velocity divided by the time taken to complete a rotation. The velocity rotates by 2π in time T.

• Rearranging and substituting gives

CL = lift coefficient

ρ = pressure

V = velocity

S = wing area

M = aircraft mass

g = acceleration due to gravity

WS = wing loading

av = vertical acceleration

ac = centripetal acceleration

T

RV

2

T

Vac

2

R

V

ac

R

Vac

2

Lift generation (24)

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• Turning performance (continued)– The centripetal force is given by

Newton’s 2nd Law and is equal to the horizontal component of lift, so

CL = lift coefficient

ρ = pressure

V = velocity

S = wing area

M = aircraft mass

g = acceleration due to gravity

WS = wing loading

av = vertical acceleration

ac = centripetal acceleration

θ = bank angle

L

θ

sin

2

sin

2

sin2

1 22

L

S

L

L

C

W

CS

MR

SCVR

VM

– And thus turn radius (R) is proportional to wing loading

Lift generation (25)

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• Gust response

– The acceleration due to a gust (a transient increase or decrease in lift) is, from Newton’s 2nd Law:

– Thus an aircraft with higher wing loading has better ‘ride quality’

CL = lift coefficient

ρ = pressure

V = velocity

S = wing area

M = aircraft mass

g = acceleration due to gravity

WS = wing loading

av = vertical acceleration

ac = centripetal acceleration

θ = bank angle

Sv W

L

M

SLa

Lift generation (26)

N.B. – These analyses are good indicators of aircraft performance and handling but should be treated with care as they are essentially static assessments of what are, in reality, dynamic manoeuvres

– Mean wind direction, turbulence, aircraft pitch, roll, yaw, etc.

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• A lot of mathematical analysis, but what does it mean in practice?

• Landing and take-off performance– Airbus aircraft have lower wing loading than Boeing aircraft and thus can

trade this for lower CL at take-off and landing

• Simpler and lighter high-lift devices (see later)

Lift generation (27)

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• Initial climb rate– The original A340 is underpowered but gets way with it because of low

wing loading– It can achieve a certifiable climb rate even with relatively small engines

• CFM56 on A340-200 and -300 (34,000 lbf each)• RR Trent 500 on A340-500 and -600 (60,000 lbf each)

Lift generation (28)

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• Turning performance– Fighter aircraft

• The difference between fighter and interceptor

Lift generation (29)

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• Turning performance– Fighter aircraft

• The Spitfire had much lower wing loading

BUT

• The Bf109 had automatic leading-edge slats

Higher CL

Lift generation (30)

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• Gust response– Airbus (low wing loading)

• Lower landing and take-off speeds, better initial climb

vs• Boeing (high wing loading)

• Smoother cruise

Lift generation (31)

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• Gust responseLow-level bomber vs high-level bomber

Lift generation (32)

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• Returning to lift distribution– The load resulting from the lift generated by a wing requires a structure strong enough to sustain it– The lift can be idealised as acting at the centroid of the lift distribution:

– So the wing tries to bend upwards when it generates lift• This load must be absorbed by the wing spar• Bending moment is dependent on the magnitude of the lift and how far it acts from the wing root

Lift generation (33)

Moment arm

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• Wing root bending moment– 0g to 1g flight

– 2.5g gust and manoeuvre load

Lift generation (34)

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• Ways to alleviate wing root bending moment– Engines

– Winglets

Lift generation (35)

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• Contributions to subsonic drag; zero-lift drag, skin-friction; ‘Horseshoe’ vortex system; wing planforms in subsonic flow; induced drag; span efficiency; tip devices; wing design through twist and camber including “wash-out” and “wash-in”.

Subsonic Flows

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• Definition; boundary layers; Reynolds number; velocity profiles; no-slip condition; effect of surface roughness on skin friction; laminar and turbulent flows; local and global skin friction calculations; boundary layer thickness definition; momentum and displacement thickness; equivalent body in inviscid flow; transition and flow separation.

Viscosity

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• What is viscosity?– The material property that measures a fluid's resistance to flowing

• It concerns the transport of mass, momentum and energy when the molecules move.

– It results in friction between air and any surface over which it flows

Viscosity (1)

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• Boundary Layers– A thin region of the flow adjacent to a surface, where the flow is retarded

by the influence of friction between a solid surface and the fluid (Anderson, ‘Fundamentals of Aerodynamics’)

– Its characteristics (size, composition, etc.) are determined by a variety of things

• viscosity (friction)

• Shape (pressure gradient)

• Reynolds number (density, velocity)

– Extremely difficult to measure

– Still not fully understood

• Transition mechanisms

• The limit on many numerical

methods

Viscosity (2)

y (v)

x (u)

u=U∞

y=0, u=0, v=0

U∞

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• Boundary Layers

– Inviscid vs Viscous

Viscosity (3)

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• Reynolds number– A non-dimensional measure of the ratio of inertia forces (U2) to

viscous forces (μU/d)

• Where = densityU = velocityd = reference lengthμ = viscosity

– Which gives dU

Re

Viscosity (4)

– One of the most powerful parameters in fluid dynamics

• Helps assess the similarity or equivalence of differing flow conditions

• Scale effect– High Re flows approach inviscid conditions

(thin boundary layers)

Stokes

Reynolds

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• Reynolds number

Viscosity (5)

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• Reynolds number– Some examples

Viscosity (6)Gliders

Re <20000

X15Re = 6 million

A380Re = 80 million

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• Reynolds number– Wind tunnel coverage

• (see ‘Aerodynamics Methods’)

EWA Wind Tunnel Performance Envelopes

0

5

10

15

20

25

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50

Mach Number

Rey

no

lds

nu

mb

er (

mil

lio

ns

per

met

re)

QinetiQ 5m

ONERA S1MA

CIRA IWT

VZLU 3 m

ONERA F1

FOI LT1

DNW-LLF

EWA Wind Tunnel Performance Envelopes

0

10

20

30

40

50

60

70

80

90

100

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40

Mach Number

Rey

no

lds

nu

mb

er (

mil

lio

ns

per

met

re)

ONERA S2MA DNW-HSTONERA S1MAARA-TWTCIRA PT-1VZLU A1CIRA IWTFOI T1500FOI S4FOI S5FOI TVM500

EWA Wind Tunnel Performance Envelopes

0

20

40

60

80

100

120

140

1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00

Mach Number

Rey

no

lds

nu

mb

er (

mil

lio

ns

per

met

re)

ONERA S2MA

DNW-SST

FOI T1500

FOI S4

FOI S5

FOI TVM500

Low-speed w/t

Transonic w/t

Supersonic w/t

European Wind Tunnels

ETW (Cryogenic)

Viscosity (7)

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• Velocity profiles

– The rate at which the velocity increases from zero at the wall to freestream velocity at the edge of the boundary layer

– The edge of the boundary layer can be difficult to define so a (sometimes artificial) definition is imposed:

u = 0.99 U∞

– The boundary-layer depth (the distance from the surface where u = 0.99 U∞) is defined as δ

Viscosity (8)

y (v)

x (u)

u=0.99U∞

y=0, u=0, v=0

U∞

Boundary-layer velocity profile

δ

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Viscosity (9)

• Shear Stress

– The assertion that the velocity at the surface is zero

– The action of viscosity tugs at the surface (rubbing hands together)

• Generates shear stress (τxy)

dy

du

dy

duxy

– You can imagine this as two adjacent layers of fluid, each at different velocities rubbing against each other

– The shear stress is thus related to the difference in velocity between the two layers and that is defined by the velocity gradient (du/dy), so

• In reality there is a semi-infinite number of adjacent layers (solid boundary)

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Viscosity (10)

• No-slip condition

– The no-slip condition maintains that the flow at the surface is stationary

• i.e. that u = 0 at y = 0

– This is difficult to justify theoretically and is demonstrably not true in many cases

– But it is close enough to the truth (and the convenience of it as a boundary condition so large) that its consequences are accepted

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Viscosity (11)

• Laminar and Turbulent Boundary Layers

– When a boundary layer starts on a surface it is laminar, i.e. smooth with the stream lines roughly parallel to the surface

– At some point a transition occurs (due to roughness, contamination, pressure gradients, etc.) to a turbulent boundary layer

– There is a general mean motion roughly parallel to the surface, but in addition there are local rapid, random fluctuations in velocity direction and magnitude

– These fluctuations provide a powerful mechanism for mixing within the layer

– Just as viscosity give rise to shear stress, the turbulent fluctuations give rise to eddy shear stresses

– Consequently there a important differences between the characteristics of laminar and turbulent boundary layers

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Viscosity (12)

• Laminar and Turbulent Boundary Layers

– Boundary-layer profile

Steeper profile

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Viscosity (13)

• Laminar and Turbulent Boundary Layers

– Boundary layer on a flat plate

– Boundary layers on a wing combine to form the wake (profile drag, CDo)

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Viscosity (14)

• Laminar and Turbulent Boundary Layers (see also later)

– Transition

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Viscosity (15)

• Laminar and Turbulent Boundary Layers

– Characteristics which encourage transition

• Increased surface roughness

– Boundary-layer tripping on wind tunnel models

– Dimples on a golf ball

• Increased freestream turbulence

– Wind tunnel comparisons

• Adverse pressure gradients

– Amplification of instabilities

– Sailplane wing profiles

• Heating of the fluid by the surface

– Amplification of instabilities

– The inverse of all these encourage laminar boundary layers

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Viscosity (16)

• Reynolds Experiment

– Classic experiment examining laminar and turbulent flow in pipes

• Flow through a pipe metered by a stopcock

• Dye injection at the centreline of the tube mouth

• Reynolds noted that low speed the dye filament remained smooth and narrow

• At higher speed the filament broke up and diffused throughout the cross section

• The speed at which it occurred was different for pipes of different diameter

– Relationship to U∞ and d, i.e Re

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Viscosity (17)

• Reynolds Experiment

a & b – laminar

c - turbulent

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Viscosity (18)

• Reynolds Experiment Results

– Recreation at the University of Manchester using Reynolds original apparatus a century later

Laminar

Transitional

Turbulent

Turbulent

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• Blasius’ Equation

– Uses the boundary layer equations

• Reduction of the Navier-Stokes equations to simpler forms which apply to boundary layers

– Continuity

– x momentum

– y momentum

– Uses a function to turn a set of partial differential equations into a single ordinary differential equation

– See Anderson ‘Fundamentals of Aerodynamics’, Chapter 18.2

02 ''''' fff

V

uf

x

Vy

'

Viscosity (19)

0

y

v

x

u

2

2

y

u

y

vv

x

uu

0

y

p

Where υ is the kinematic viscosity , defined as

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Viscosity (20)

• Blasius’ Equation

– The important result is that the solution of the equation is a velocity profile and that it is a function of η only

– This form of the velocity profile is independent of the distance along a surface (x)

– Self-similar solutions

– If f’=u/Uo, the b.l. edge is at f’=0.99 and η =5.0

x

x

x

V

x

Vy

Re

0.5

0.5

– The reduction of the boundary layer equations to an ODE is only valid for certain conditions

• E.g. flow on a flat plate

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Viscosity (21)• Local Skin Friction

– Remember that shear stress within the boundary layer was defined as

– So shear stress at the wall (skin friction) is given by

– And the local coefficient of skin friction (cf), the skin friction at a point x along a

surface, is given by

– And is the local coefficient of drag due to viscosity

dy

duxy

0

y

w dy

du

221

V

c wf

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Viscosity (22)

• Local Skin Friction

– From the boundary layer and Blasius equations1 it can be shown that

– And so the local shear stress is given by

– Reference [2] describes a numerical solution of the Blasius equation and tabulates the results, giving f’’(0) = 0.4696, so

– Where Rex is the local Reynolds

number

02

''

0

fx

VV

dy

du

y

02

2 ''22

21

fx

VV

VVc w

f

x

f

ff

xVc

Re

0

2

20

2

2 ''''

x

fcRe

664.0

1 Anderson ‘Fundamentals of Aerodynamics’, 3rd Edition, Chapter 18.2

2 Schlichting, ‘Boundary Layer Theory’, 8th Revised and Enlarged Edition, Page 158 and Table 6.1

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Viscosity (23)

• Global Skin Friction

– If the local skin friction drag coefficient is cf, the global or total drag due to

friction on that surface (say the chord of a wing, c) is found by integrating the local skin friction over the length of that surface. i.e.

– Substituting the previous expression for local skin friction coefficient gives

– And

– Where Rec is the Reynolds number based on the wing chord

c

ff dxcc

C0

1

c

fCRe

328.1

V

c

cdxx

VcC

c

f

328.1

)664.0(1

0

21

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Viscosity (24)

• Boundary layer thickness and displacement thickness

– Remember we described the boundary layer thickness, δ, as being the distance from the surface where u = 0.99U∞

– We now introduce the concept of displacement thickness, δ*

y (v)

x (u)

u=0.99U∞

y=0, u=0, v=0

U∞

δ

y (v)

x (u)

u=0.99U∞δ u=0.99U∞

Shaded regions have equal area

δ*

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Viscosity (25)

• Boundary layer displacement thickness

– The displacement thickness can be thought of in two ways:

– (a) The thickness representing the missing mass flow if it were crammed into a flow with the free stream characteristics (cf. inviscid)

– (b) The boundary layer displaces the flow around an object by acting as additional volume

y (v)

x (u)

u=0.99U∞δ u=0.99U∞

Shaded regions have equal area

δ*

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Viscosity (26)

• Boundary layer displacement thickness derivations

– (a) Missing mass flow

1

0

ydyu– The actual mass flow between y=0 and y=y1 is

– The hypothetical mass flow between y=0 and y=y1 if the boundary layer were not present is

– The difference between the two is the missing mass flow

– And this can be expressed in terms of δ*

1

0

y

ee dyu

1

0

y

ee dyuu

* ee u

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Viscosity (27)

• Boundary layer displacement thickness derivations

– (a) Missing mass flow (continued)

– So

– Or

1

0

* y

eeee dyuuu

1

0

* 1y

ee

dyu

u

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Viscosity (28)

• Boundary layer displacement thickness derivations

– (b) Displacement of external streamlines

1

0

. y

ee dyum

1

0

* 1y

ee

dyu

u

– The mass flow at Station 1

– At Station 2, the mass flow between the surface and the same streamline is

– Since the surface and the streamline form the boundaries of a stream tube, the mass flow must be constant, i.e.

Or

*

0

.1

y

ee udyum

*

00

11 y

ee

y

ee udyudyu

as before

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Viscosity (29)

• Boundary layer displacement thickness

– The displacement of external streamlines raises the idea of simulating a viscous flow using inviscid methods by modelling an effective or equivalent body

– The viscous flow is modelled by expanding the shape by the displacement thickness

• Should be an iterative process!

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Viscosity (30)

• Boundary layer momentum thickness

– While the displacement thickness accounted for the ‘missing mass flow’ another important boundary-layer characteristic accounts for loss of momentum within the boundary layer

– The mass flow across dy (dm) = u dy

– Now, momentum flow across dy in the b.l. = dm u = u2 dy

– Momentum across dy if it were in the freestream = dm ue = ( u dy)ue

– Therefore, the loss in momentum associated with dm = u(ue –u)dy

– So the total momentum deficit from y=0 to y=y1 =

1

0

y

e dyuuu

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Viscosity (31)

• Boundary layer momentum thickness

– We can now introduce a thickness (θ) representing the missing momentum if it were crammed into a flow with the free stream characteristics, i.e.

– The missing momentum flow = eue2θ =

– And so

– This momentum thickness (θ) is the height of a hypothetical streamtube carrying the missing momentum flow at freestream conditions

– The momentum thickness can be used to generate a similar effective or equivalent body, this time representing a body exhibiting an equivalent momentum loss

1

0

y

e dyuuu

1

01

y

eee

dyu

u

u

u

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Viscosity (32)• Methods of measuring boundary layer state

– Hot wires

– Hot films• Measurement element is one arm of a Wheatstone

bridge

• Measure the voltage changes required to keep a constant current (CCA), or the current changes require to keep a constant temperature (CTA)

– Sublimation/Evapouration

– Heat transfer (IR)

– All have difficulties• Intrusion, chemicals, temperature gradients, viewing

angle, calibration, interpretation

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Viscosity (33)

• Boundary layer transition

– From Reynolds’ pipe flow experiment a critical Reynolds number was observed, below which the flow was laminar and above which it was turbulent.

– The discovery that transition occurred on surfaces did not come until much later

– Transition on a flat plate

2300Re

critcrit

du

65 10105.3Re

critcritx

xU

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Viscosity (34)

Factors affecting transition

• Pressure gradient

– Positive pressure gradients suppress turbulence

– Adverse pressure gradients amplify them

Flow direction

Disturbances suppressed by positive pressure gradient

Disturbances amplified by negative pressure gradient

Shape of the profile

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Viscosity (35)Factors affecting transition

• Pressure Gradient

– On a wing transition generally occurs at or just after CPmin where the

pressure gradient changes from +ve to –ve (e.g. laminar separation bubble): 5% is a rule of thumb

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Viscosity (36)Factors affecting transition

• Pressure Gradient

– Turbulent vs Laminar

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Viscosity (37)

• Factors affecting transition

– Surface roughness

• Imperfections or roughness elements act like little bluff bodies, shedding eddies which disturb a laminar boundary layer and can induce transition

• It is possible to reduce drag by inducing transition through roughness

– Only if separation is normally of a laminar boundary layer

– A turbulent boundary layer is more resistant to separation than a laminar one

– Triggering transition before separation means the boundary layer will separate later

Golf Ball

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Viscosity (38)

• Factors affecting transition

– Surface roughness

– Addition of a trip wire

Laminar separation

Transition at the wire

Turbulent separation

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Viscosity (39)• Factors affecting transition

– Surface roughness– Ice – Liquids (rain & de-icing fluid)

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Viscosity (40)• Factors affecting transition

– Freestream turbulence (and noise)

• E.g. on a flat plate, transition starts with the formation of Tollmien-Schlichting (T-S) waves

• External excitation (especially of matching frequencies) can amplify the waves and hasten transition

– 3-D geometries (e.g. swept wings)

• In addition to chordwise disturbances we now have spanwise flow and spanwise disturbance.

• Traditional civil aircraft-type wings are fully turbulent

– Attachment line transition

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Viscosity (41)Boundary layer separation

– Once the pressure gradient on a surface becomes positive the pressure rises with distance

– The effect of which is shown [top right]

• Loss of kinetic energy which is only partially compensated for by mixing within the boundary layer

• The velocity profile becomes less full with the inner part of the layer slowing down w.r.t. the outer

– The shear stress at the wall reduces

– With a sufficiently large pressure gradient a point where the shear stress becomes zero and the flow on the surface is on the point of reversing

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Viscosity (42)Boundary layer separation

• The reversed flow forms a large eddy under the outer part of the boundary layer (wakes)

• Open separations are generally unstable and highly dynamic

• Closed separations exist (laminar bubbles) but even these are unsteady

• Classic wind tunnel surface flow visualisation can indicate these regions– Oil flow (time averaged)

– Tufts (point data)

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Viscosity (43)Boundary layer separation

• Bluff body separations

– E.g. delta vortices

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Viscosity (44)

• Boundary layer modelling

– It is important to capture the boundary layer with sufficient fidelity in viscous CFD methods

• Concentration of mesh points close to the surface

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• History of aerodynamic testing; Wind tunnel types; low speed and high speed testing; Open and closed circuit (Eiffel/Goettingen) type tunnels; Open, closed and slotted/porous working section type tunnels; Flight testing; Model mounting systems; upwash, buoyancy and blockage correction methods; Mach similarity; Methods of increasing Reynolds number; Powered wind tunnel models; Pressurised, cryogenic, heavy gas and water tunnels; Introduction to CFD; Description of CFD; Advantages and disadvantages of CFD; Examples/demonstration of CFD usage.

Aerodynamic Methods

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• The need for high lift; history of high lift; slats, flaps and other high lift devices; the effects of slots; Coanda effects and blown devices; powered high lift devices; vortex generators.

Lift Augmentation and Flow Control Devices

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• Early (propellor-driven) aircraft had low wing loadings and thus low take-off and landing speeds– High-lift systems were either not needed or were relatively simple

• Leading-edge devices, at least on civil aircraft, were rare• Trailing-edge devices were primarily plain or split flaps

Lift augmentation and flow control devices (1)

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• The advent of the jet engine saw an increase in cruise speed and wing loading– High-lift systems became necessary for take-off and landing (improved CL/CD and CLmax)

• Slotted leading-edge slats and trailing-edge flaps became commonplace and some were extremely complex.

– But the technology of slotted devices was born around 1920• A mixture of careful design and pure accident

Lift augmentation and flow control devices (2)

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• Handley-Page– In 1911 Sir Frederick Handley-Page noted that square wings (AR=1)

maintained lift to a much higher incidence than more conventional rectangular wings (AR≈6)

– In 1917 he and his aerodynamicist (R.O.Boswell) tried to combine the low drag characteristics of high aspect ratio with the delayed stall of low aspect ratio by incorporating chordwise slots in a conventional wing

Lift augmentation and flow control devices (3)

– Wind tunnel test results were disappointing

– Despite many variations in shape, gap and proportion the idea could not be made to work

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• Handley-Page– At some point someone (whether Handley-Page, Boswell or one of the

carpenters, it is not clear who) had the idea of cutting spanwise slots• Parallel to the leading edge, at about c/4 and sloping upwards and rearwards

Lift augmentation and flow control devices (4)

– The initial tests on a RAE15 aerofoil gave a spectacular 25% increase in maximum lift

– An improved slot shape in a RAE6 aerofoil gave a 50% increase, with only a slight increase in drag

– Various test throughout 1918 and 1920 showed that chordwise location was crucial

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• Lachmann– Independently, a parallel investigation was being conducted by a German engineer-

pilot, Gus Lachmann.• He transferred to the flying corps from the cavalry in 1917 but stalled and spun-in during an

early training flight, breaking his jaw

– In hospital he pondered the cause of his accident and how stall could be prevented, concluding that a cascade of small aerofoils within a normal wing profile might be better

Lift augmentation and flow control devices (4)

– He took his idea to the German patent office in February 1918 but this was rejected unless he could prove experimentally that the idea would work

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• Lachmann– He approached Prof Ludwig Prandtl at Gottingen who agreed to do the

tests for £50

– Lachmann had no money and so borrowed it from his mother

– The results convinced the patent office to grant his application

Lift augmentation and flow control devices (5)

– Lachmann ended up working for Handley-Page after the end of WW2

-

– The consensus was that the slot behaves as a boundary-layer control device

• The jet through the slot

– It was not until 1972 (A.M.O. Smith) that the correct physical principles underlying its operation were finally understood

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• Types of high-lift device - trailing edge

• Take-off– Predominantly Fowler motion

• Landing– Fowler motion plus deflection

Lift augmentation and flow control devices (6)

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• Types of high-lift device - leading edge

Lift augmentation and flow control devices (7)

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Lift augmentation and flow control devices (8)

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Lift augmentation and flow control devices (9)

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Lift augmentation and flow control devices (10)

HIGH-LIFT SYSTEM TERMINOLOGY

SlatCove

Shroud

Fixed leading edge (D-nose)

Vane

Main flap

Main element

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Lift augmentation and flow control devices (11)

• Take-off– Slat deployment and take-off flap setting (large Fowler motion and a little

deflection)

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Lift augmentation and flow control devices (12)

• Landing– Slat deployment, landing flap setting (large Fowler motion and large

deflection) and spoilers (shroud) after weight on wheels

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Lift augmentation and flow control devices (13)

BOEING TRAILING-EDGE SYSTEM TERMINOLOGY

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Lift augmentation and flow control devices (14)

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Lift augmentation and flow control devices (15)

So what effects do high-lift systems have?

• Flaps shift the CL-α curve

– Greater lift at a given incidence and greater CLmax, but

– Reduction in maximum α

• Slats extend the CL-α curve, increased αmax

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Lift augmentation and flow control devices (16)

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Lift augmentation and flow control devices (17)

• An example of how certification requirements lead to design choices– e.g. Airbus A340

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Lift augmentation and flow control devices (18)

• An example of how each design choice affects others– e.g. fuselage length, fuselage shape and undercarriage height

(γ = climb angle)

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Lift augmentation and flow control devices (19)

• The National High-Lift Programme (NHLP)– Instigated by the RAE in the

late 1960s in response to a perceived American lead in high-lift system design

• Current UK designs (BAC1-11, Trident, VC-10) had much simpler leading and trailing-edge devices than their US equivalents (707, 727, 737, 747, DC-8, DC-9, DC10)

– It was considered that more complex meant more powerful

– UK industry had been combined into BAC and HAS which was contemplating the next generation civil transport

Boeing 737

HS Trident

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Lift augmentation and flow control devices (20)

• The National High-Lift Programme (NHLP)– Lasted through to the late 1970s

and its legacy continues today

– Combinations and permutations of 8 different leading-edge devices and 11 different trailing-edge devices

– Defined the design philosophy for the Airbus A320 and all subsequent Airbus aircraft

• Flap chord, slat chord, shroud length, deflections, etc.

– Identified Reynolds number (scale effect) and testing fidelity as important design parameters

• RAE (QinetiQ) 5m Pressurised Wind Tunnel

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Lift augmentation and flow control devices (21)

• So how do high-lift systems really work?– A.M.O. Smith (Douglas) illuminated the aerodynamics community in 1972

• The low-speed aerodynamicists equivalent of Darwin’s ‘Origin of the Species’ but nowhere near as easy to read!

– Prior to 1972:• Fresh momentum through the

slot• High energy air from lower

surface to upper surface

– A.M.O. Smith• Leading-Edge Slat Effect• Circulation Effect• Dumping Velocity• Off-Surface Recovery• Fresh Start for the boundary

layer on each element

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Lift augmentation and flow control devices (22)

• Leading-Edge Slat Effect

– The leading-edge of a downstream element benefits from the circulation of an upstream element

– The velocity induced by the upstream element runs counter to that of the downstream element

Reduction of the pressure peak on the downstream element and a resilience to high angle of attack

– Works mainly at high angle of attack when the slat is generating a lot of lift and where the main element is highly loaded

– The main element has a similar effect on the flap Load on the main element is reduced, but

the combination of slat and main element is positive

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Lift augmentation and flow control devices (23)

• Circulation (Flap) Effect– The trailing-edge of an upstream element benefits from the circulation of

a downstream element

– The velocity induced by the downstream element reinforces that of the downstream element

– Increased resistance to main element trailing-edge flow separation and a resilience to high angle of attack

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Lift augmentation and flow control devices (24)

• Dumping Effect– The circulation effect not only improves the performance of the upstream

element– The increased trailing-edge velocity due to the circulation of the

downstream element means that the boundary layer (wake) from the upstream element is accelerated

Relieves the pressure rise on the trailing-edge of the downstream element and improves its ability to resist flow separation

• Off Surface Pressure Recovery• The boundary layer of the upstream

element is dumped at higher velocity and impinges upon the boundary layer of the downstream element

• The deceleration of the upstream wake is thus more efficient than if it were in contact with a solid boundary

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Lift augmentation and flow control devices (25)

• Fresh Boundary Layer Effect– Each element starts with a fresh boundary layer at the leading edge– Thin boundary layer can withstand stronger adverse pressure gradients

than thick ones

• Viscous Effects and Separation– Confluent boundary layers modify pressure gradients and boundary

layer velocity gradients

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Lift augmentation and flow control devices (26)

• High-Lift Optimisation • All of the above means that each element has an optimum position relative to its

neighbour• That position will change with the position of further upstream or downstream

elements• In practice, for a 3-element system (slat, main element, flap) the flap position is

optimised using a (e.g. 9 point) optimisation matrix, then the slat is optimised in a similar way. The flap is then re-optimised with the slat in its new position and ditto for the slat

• The optimisation of a system with a triple-slotted flap is not trivial!

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Lift augmentation and flow control devices (27)

• Coanda Effect– The tendency of a fluid jet to be attracted to a nearby surface– Used to enhance lift usually either by using the engine exhaust or

dedicated blowing of engine bleed air through slots

• Blown devices (boundary layer control)• Counteracts the deceleration and growth of the boundary layer by

injection of momentum• Over blowing (jet effect) can produce lift enhancement over and above

potential flow theory• (F-104 Starfighter, F-4 Phantom, A-5 Vigilante, Buccaneer, TSR.2. etc)

• Engine exhaust• (Antonov 72/74, YC, Boeing Globemaster III)

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Lift augmentation and flow control devices (28)

• Blown devices (boundary layer control)– Reduces engine performance– Requires internal ducting

• Weight• Volume• Maintenance

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Lift augmentation and flow control devices (29)

• Flow control– Which for high lift usually means separation control

• Entraining or redistributing higher momentum air close to the flap surface and delay flow separation

– Vortex generators• Small blades (rectangular or triangular) that create vortices close to the surface

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Lift augmentation and flow control devices (30)Flow control – An example of Aerodynamics Research

– Conventional vortex generators can cause a significant drag penalty

– Hence the concept of sub-boundary-layer vortex generators (SBVGs)

2-D separation control using SBVGs

-400 -300 -200 -100 0 100x(mm)

100

200

300

400

z(m

m)

10mmWheelerWedgewith1hgapx=52hfromseparationspacing=12h

-400 -300 -200 -100 0 100x(mm)

100

200

300

400

z(m

m)

10mmWheelerWedgex=52hfromseparationspacing=12h

-0.4 -0.3 -0.2 -0.1 0 0.1x(m)

0.05

0.1

0.15

0.2

0.25

0.3

0.35

z(m

)

5mmwedgesx=104hfromseparationspacing=12h

-400 -300 -200 -100 0 100x(mm)

100

200

300

z(m

m)

10mmwedgesx=52hfromseparationspacing=12h

-300 -200 -100 0 100x(mm)

50

100

150

200

250

300

z(m

m)

Umean: -5.0 5.0 15.0 25.0 35.0

NoVG'sa) Basic flow (no control)

c) Joined Counter-rotating vanes

b) Forwards wedges

d) Counter-rotating vanes spaced apart by 1h

a) Basic flow (no control)

b) Forwards wedges

c) Joined Counter-rotating vanes

d) Counter-rotating vanes spaced apart by 1h

Regions of constant streamwise velocity ‘above’ the bump in vertical plane of symmetry.

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Lift augmentation and flow control devices (31)

Extension to 2.75D (sweep and taper)

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Lift augmentation and flow control devices (32)

Extension to fully 3D and to high Reynolds number

340 flap flow separation (AWIATOR)

No VGs

VGs on

A380 in QinetiQ 5m Wind Tunnel

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Lift augmentation and flow control devices (33)

Flow control• Alternative flow control devices

– Air jet, Synthetic jets

• Thousands of devices– Networked, Sequenced– Rapid response

• Manufacturing– Embedding in composite/metal structures– Power supply– Calibration

• Maintenance– Robustness

• Certification– Consequences of failure

• Flow control• Loads control

– Performance degradation

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Lift augmentation and flow control devices (34)

Flow control• Also used for as separation

control on:– Pylon/slat junction

• Douglas invention• Big effect on CLmax

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Lift augmentation and flow control devices (35)

Flow control• Pylon/slat junction

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Lift augmentation and flow control devices (36)

Flow control• Also used for as separation control on:

– Flight control devices

Gloster Javelin

Boeing 727

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Lift augmentation and flow control devices (37)

Flow control• Also used for as separation control on:

– Afterbody

Rockwell B-1 Lancer

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Lift augmentation and flow control devices (38)

Flow control• Also used for as separation control on:

– Shock/boundary layer interaction

Boeing 737

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• Factors affecting high-lift performance– Surface roughness– Ice – Liquids (rain & de-icing fluid)

Lift augmentation and flow control devices (39)

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• Abnormal use of high-lift devices

– British Airways Boeing 777 crash at Heathrow (January 2008)

– Glide approach due to loss of power

– Correct selection of take-off setting

– Maximum L/D

Lift augmentation and flow control devices (40)

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Lift augmentation and flow control devices (41)

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• Critical Mach Number; formation of shockwaves; Normal and oblique shockwaves; Effect of wing thickness and camber; Wave drag and methods of reducing wave drag (Wing Sweep, Transonic Area Ruling, Supercritical Aerofoil design, Wing design); Shockwave control and the Shock-induced separation.

Supersonic Flows

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• Incompressible flow– Most of our analysis so far has assumed that the flow is incompressible

– We do this because it allows us to simplify things in two important ways• The density is known and can be treated as a constant in, for example, the

continuity equation

1u1A1 = 2u2A2

• The interaction between mechanical and thermal energy is weak which permits use of a simplified version of the energy equation

– This assumption does not, however, match reality• All fluids are compressible, even liquids

– If the pressure changes in a flow are sufficient to cause significant density changes we have to abandon the incompressible flow assumption

– It is more likely to be of concern in a gas than in a liquid

• A pressure change of 500kPa (~72psi) causes a density change of 0.024% in water but 250% in air

Supersonic flows (1)

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• Forward influence– In subsonic flow pressure changes are

propagated through the fluid at the speed of sound through pressure waves

– The pressure changes caused by a body moving through a fluid are thus transmitted through the fluid

– The air ahead of a subsonic aircraft, for example, therefore ‘knows’ that it is coming because the pressure changes are transmitted forward

– The degree to which the flow ahead of body is altered depends greatly on the pressure changes on the body itself and these are dependent on its shape and speed

– This effect is known as forward influence and can be a very important consideration

Supersonic flows (2)

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• Compressible flow– As an aircraft approaches the

speed of sound, the difference between the speed at which the pressure waves are transmitted ahead of it and that of the aircraft reduces

– So the time between the pressure wave and aircraft passing through the same point reduces

– When the aircraft reaches the speed of sound (Mach 1) the air receives no ‘warning’ and so has to react instantly to the presence of the aircraft

– This instantaneous change can lead to shock waves

Supersonic flows (3)

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• Compressible flow– An aircraft does not have to be travelling at the speed of sound to

generate shock waves

– The wings are designed to accelerate the air to produce lift so shock waves will form at relatively low Mach number

Supersonic flows (4)

– Shock waves can be a problem even at landing speed

• the acceleration generated by a slat can be so severe as to cause near sonic flow on its upper surface as is a design limit

• CL for Cp-10

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• Critical Mach number– The speed at which sonic flow occurs on a body is called the critical

Mach number

– Shock waves cause drag (wave drag) (see later) and so the speed at which they start to form is important

– For example, the economy of a civil aircraft reduces dramatically once strong shock waves form its wings and the point at which this occurs is the drag rise Mach number

• the wing section must be designed so that the design cruise speed is below the drag rise Mach number

Supersonic flows (5)

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• Mach wave– If we imagine a stationary disturbance emitting pressure waves (e.g.

sound) the waves will propagate uniformly from the source• Ripples in the surface of water

– The waves will travel at the speed of sound (c) so that after a time interval (Δt) the waves will have travelled a distance of cΔt

Supersonic flows (6)

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• Mach wave

– Now suppose that there is a flow over the disturbance travelling at Mach 0.5, i.e. V = ½c– In addition to spreading into the fluid, the waves will also be swept downstream by the flow

– As a result the waves bunch up on the upstream side and spread out on the downstream side and the rate at which the fluid experiences the disturbances is greater on the upstream side than the downstream

• If the disturbance was sound an upstream listener would here a different frequency to one downstream

Doppler effect

Supersonic flows (7)

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• Mach wave– Now suppose that the flow past the disturbance is sonic, i.e. at a speed

exactly equal to the speed of sound (M =1, V = c)

– In this case the waves are swept downstream at exactly the same speed at which they spread

– The waves cannot propagate upstream and the fluid is not affected by the disturbance until it arrives at it

Supersonic flows (8)

– The upstream waves will sit on top of each other forming an envelope

– The fluid experiences the total effect of all of the waves at once upon crossing the envelope

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• Mach wave– Finally suppose that the fluid flows past the disturbance at supersonic

speed, i.e. a speed greater than the speed of sound (M > 1, V > c)

– At supersonic speed the disturbances are swept downstream faster than they can spread

– All the waves are confined to a triangular (in 2-D) or conical (in 3-D) region extending downstream from the disturbance

– The waves form an envelope of half angle μ

Supersonic flows (9)

– Only the fluid inside the envelope is affected by the disturbance

• If it were sound it would be silent outside the envelope

– The envelope is called the Mach wave (in 2-D) or the Mach cone (in 3-D)

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Supersonic flows (10)• Mach angle

– The half angle of the envelope can be calculated

– So the Mach angle (μ) is given by:

– Because the fluid experiences the combined effects of a disturbance almost instantaneously, the fluid property and velocity variations may be discontinuous

– E.g. density changes• Schlieren technique

MV

c

tV

tc 1sin

M

1sin 1-

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Supersonic flows (11)• Shock waves

– A finite strength disturbance (e.g. a sharp wedge or cone) in supersonic flow creates a finite strength wave which is stationary with respect to the disturbance

– These finite strength disturbances are known as shock waves

– They are extremely thin and fluid properties change dramatically across them

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Supersonic flows (12)

Shock wave on an A320 in cruiseSuper-critical wing design (see later)

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Supersonic flows (13)• Normal and oblique shock waves

– A shock wave that is perpendicular to the upstream flow is a normal shock

– One that is inclined at a constant angle to the upstream flow is an oblique shock

– A curved shock has a varying angle between it and the upstream flow

Page 147: Aerodynamics Course Notes v3

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Supersonic flows (14)• Normal shock waves

– Consider a normal shock wave of zero thickness as shown in the picture

– The continuity equation along a streamline is 1u1A1 = 2u2A2

– For a section of the shock wave, the area before and after the shock will be equal, i.e. A1 = A2 = A, so

1u1 = 2u2

– The ideal gas law is

– And we can write the velocity as

– So 1u1 = 2u2 becomes

→ or TR

p

TRMMcu

222

211

1

1 TRMTR

pTRM

TR

p

1

2

2

1

1

2

2

22

1

11

T

T

p

p

M

M

T

Mp

T

Mp

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Supersonic flows (15)• Normal shock waves

– Now consider the forces acting across the shock wave

– The momentum equation gives

– The force is equal to the pressure difference across the shock wave

= pressure x area = p1 A = p2 A

– Therefore

or

and

AuAuFx211

222

AuAuApAp 211

22221

TR

p

TRMMcu

)1(

)1(22

21

1

2

M

M

p

p

)1()1( 222

211 MpMp

211

22221 uupp

Page 149: Aerodynamics Course Notes v3

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Supersonic flows (16)• Normal shock waves

– We also need to make use of the energy equation

– The state of a gas is defined by several properties including the temperature, pressure, and the volume which the gas occupies.

– From the first law of thermodynamics (conservation of energy) we find that the internal energy of a gas is also a state variable

• That is, a variable which depends only on the state of the gas and not on any process that produced that state

– We are free to define additional state variables which are combinations of existing state variables

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Supersonic flows (17)• Normal shock waves

– The new variables often make the analysis of a system much simpler

– For a gas, a useful additional state variable is the enthalpy (H) which is defined to be the sum of the internal energy E plus the product of the pressure p and volume V, i.e. H = E + pV

– The enthalpy can be made into a specific variable ( ) by dividing by the mass

– Propulsion engineers use the specific enthalpy (or more often the change in specific enthalpy) in engine analysis more than the enthalpy itself

h~

Page 151: Aerodynamics Course Notes v3

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Supersonic flows (18)• Normal shock waves

– How do we use this new variable called enthalpy?

– Let's consider the first law of thermodynamics for a gas

– For a system with heat transfer Q and work W, the change in internal energy E from State 1 to State 2 is equal to the difference in the heat transfer into the system and the work done by the system:

E2 - E1 = Q - W

– The work and heat transfer depend on the process used to change the state.

– For the special case of a constant pressure process, the work done by the gas is given as the constant pressure p times the change in volume V. i.e.

W = p (V2 - V1 )

– Substituting into the first equation, we have:

E2 - E1 = Q - p (V2 - V1 )

Page 152: Aerodynamics Course Notes v3

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Supersonic flows (19)• Normal shock waves

– Let's group the conditions at State 2 and the conditions at State 1 together:

E2 - E1 = Q – p (V2 - V1 )

becomes

(E2 + p V2) - (E1 + p V1) = Q

– The (E + pV) can be replaced by the enthalpy H

H2 - H1 = Q

– From the definition of the heat transfer, we can represent Q by some heat capacity coefficient Cp times the temperature T

H2 - H1 = Cp (T2 - T1)

Page 153: Aerodynamics Course Notes v3

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Supersonic flows (19)• Normal shock waves

– We have previously divided by the mass of gas to produce the specific enthalpy equation version

– The specific heat capacity (cp) is called the specific heat at constant

pressure

– This final equation is used to determine values of specific enthalpy for a given temperature

– Across shock waves, the total enthalpy of the gas remains a constant

221212

~~~TchTTchh pp

Page 154: Aerodynamics Course Notes v3

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Supersonic flows (20)• Normal shock waves

– The energy equation for this situation is:

where is the heat transfer rate,

is power (rate of work),

is mass flow rate,

and is specific enthalpy.

– Work is the energy transfer by the action of a force through a distance

– Because we have defined the thickness of the shock wave to be zero, there can be neither heat transfer or work as they require finite volumes

– So the equation reduces to:

22

~~ 21

22

12

uuhhmWQ s

h

m

W

Q

s

~

2

~

2

~ 22

2

21

1

uh

uh

Page 155: Aerodynamics Course Notes v3

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Supersonic flows (21)• Normal shock waves

– So we now have five equations:

Continuity 1u1 = 2u2 (1)

Momentum (2)

Energy (3)

Enthalpy (4)

Equation of state p2 = 2 R T (5)

– And five unknowns, the flow conditions after the shock wave:

2 , u2 , p2 , , and T2

– These equations then are sufficient to calculate these unknown conditions in an ideal gas

2

~

2

~ 22

2

21

1

uh

uh

211

22221 uupp

22

~Tch p

2

~h

Page 156: Aerodynamics Course Notes v3

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Supersonic flows (22)• Normal shock waves

– Using the continuity and momentum equations

– We introduce a characteristic Mach number

where a* is the value of the speed of sound at sonic conditions, not the

actual local value, and

222

21

11

1 uu

pu

u

p

1222

2

11

1 uuu

p

u

p

122

22

1

21 uu

u

a

u

a

pa

a

uM

TRa

Page 157: Aerodynamics Course Notes v3

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Supersonic flows (23)• Normal shock waves

– The energy equation we had as

where u1 and u2 are velocities at any two points along a 3-D streamline

– We had that for a perfect gas, , so

– Also for a perfect gas, cp – cv = R

– Which we can modify by dividing through by cp to give

or

2

~

2

~ 22

2

21

1

uh

uh

Tch p~

22

22

2

21

1

uTc

uTc pp cp - specific heat at constant pressure

cv - specific heat at constant volume

pv

p

pp

p

c

R

c

c

c

R

c

c

1

11 soand

1

R

cp

Page 158: Aerodynamics Course Notes v3

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Supersonic flows (24)• Normal shock waves

– So

remember

so

– If we make the Point 2 on the streamline represent sonic flow, then u = a* so that

or

2121becomes

22

222

211

22

2

21

1

uTRuTRuTc

uTc pp

TRa

2121

22

22

21

21 uaua

2121

2*2*22 aaua

2*22

(2

1a

ua

1)21

Page 159: Aerodynamics Course Notes v3

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Supersonic flows (25)• Normal shock waves

– Rearranging the equation and applying it

first ahead of the shock wave and then behind it, we get

a* is the same constant value because the flow is adiabatic, i.e. one in which no heat is added or removed from the system

– Substituting this pair into

– Gives

2*22

1)(2

1

21a

ua

22

2*22

21

2*21 2

1

2

1and

2

1

2

1uaauaa

122

22

1

21 uu

u

a

u

a

1222

2*

11

2*

2

1

2

1

2

1

2

1uuu

u

au

u

a

Page 160: Aerodynamics Course Notes v3

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Supersonic flows (26)• Normal shock waves

– Rearranging

– Gives

– Dividing by u2 - u1 gives

– Which can be rearranged and solved for a* to give

a* = u1u2

– This is called the Prandtl relation and is a useful intermediate relation for shock waves

1222

2*

11

2*

2

1

2

1

2

1

2

1uuu

u

au

u

a

1212

2*12

21

)(2

1)(

2

1uuuuauu

uu

12

2*

21 2

1

2

1uua

uu

Page 161: Aerodynamics Course Notes v3

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Supersonic flows (27)• Normal shock waves

– The usefulness of the Prandtl relation is shown if we recall the equation

– Dividing through by u2 gives

– And converting to Mach number

– And rearranging gives

2*22

1)(2

1

21a

ua

2*2

1)(2

1

2

1

1

)/(

u

aua

2

11

1)(2

1

1

)/1(2

*

2

M

M

2

22*

2*

2

(2

)1(

()/)1((

2

M

MM

MM

1)or

1)

Page 162: Aerodynamics Course Notes v3

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Supersonic flows (28)• Normal shock waves

– We now take the Prandtl relation and incorporate the characteristic Mach

number (M* = u/a*)

a* = u1u2 becomes → or

– On the previous page we derived the equation

– Substituting this into

– Gives

– And solving for M22

a

u

a

u 211*1

*2

*2

*1

11

MMMM

2

22*

(2

)1(

M

MM

1)

*1

*2

1

MM

1

21

21

22

22

)1(2

)1(

)1(2

)1(

M

M

M

M

2/)1(

2/)1(12

1

212

2

M

MM

Page 163: Aerodynamics Course Notes v3

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Supersonic flows (29)• Normal shock waves

– The equation is an important result

– It shows that the Mach number after a normal shock wave is dependent only upon the Mach number before it

– If M1=1, then M2=1 and this is an infinitely weak shock wave, or Mach wave

– If M1>1, then M2<1, i.e. the flow after the shock wave will be subsonic

– As M1 increases above 1 the shock wave becomes progressively stronger

and M2 becomes progressively less than 1

– As M1 → , M2 approaches a finite minimum value

which for air ( = 1.4) is 0.378

2/)1(

2/)1(12

1

212

2

M

MM

2/)1(2 M

Page 164: Aerodynamics Course Notes v3

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Supersonic flows (30)• Normal shock waves

– Now we have a way of determining the relationship between the Mach numbers before and after a normal shock wave

– We also need to determine the relationships between the other flow parameters 2 /1, p2 /p1, and T2 /T1

– Using the continuity equation ( 1 u1 = 2 u2) and the Prandtl relation

(a* = u1u2) we get

– Substituting into the above equation

– Gives

2*12*

2

21

21

2

1

1

2 Ma

u

uu

u

u

u

2

22*

(2

)1(

M

MM

1)

21

21

2

1

1

2

(2

)1(

M

M

u

u

1)

Page 165: Aerodynamics Course Notes v3

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Supersonic flows (31)• Normal shock waves

– To obtain the pressure ratio we combine the continuity equation with the momentum equation

1u1 = 2u2 and

– To give

– Dividing by p1 and recalling that

– For u2/u1 in this equation we can substitute

1

22112111

222

21112 1)(

u

uuuuuuupp

211

22221 uupp

21

21

2

1

1

2

(2

)1(

M

M

u

u

1)

1121111 papa or

1

221

1

221

21

1

2

1

211

1

12 111u

uM

u

u

a

u

u

u

p

u

p

pp

Page 166: Aerodynamics Course Notes v3

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Supersonic flows (32)• Normal shock waves

– The substitution gives

– Which simplifies to

– To get the temperature ratio we use the gas equation p = R T, i.e.

– Which gives

2

1

212

11

12

)1(

(21

M

MM

p

pp

1)

)1(1

21 2

11

2

Mp

p

2

1

1

2

1

2

p

p

T

T

21

212

11

2

1

2

)1(

(2)1(

1

21

M

MM

h

h

T

T

1)

Page 167: Aerodynamics Course Notes v3

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Supersonic flows (33)• Normal shock waves

– So now we have

)1(1

21 2

11

2

Mp

p

21

212

11

2

1

2

)1(

(2)1(

1

21

M

MM

h

h

T

T

1)

2/)1(

2/)1(12

1

212

2

M

MM

21

21

2

1

1

2

(2

)1(

M

M

u

u

1)

Page 168: Aerodynamics Course Notes v3

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Supersonic flows (34)• Normal shock waves

– Note that all these relationships are in terms of upstream Mach number (M1) only

– M1 is the determining parameter for changes across a normal shock wave

in a perfect gas

– This is a good example of the power of the Mach number as a governing factor in compressible flow

– As before, at M=1 p1=p2, 1=2, and T1=T2 and we have a normal shock

wave of vanishing strength; a Mach wave

– As M1 increases above 1, p2, 2 and T2 all progressively increase

Page 169: Aerodynamics Course Notes v3

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Supersonic flows (35)• Normal shock waves

– In the limiting case, where M1 →

1

2

1

2

1

22

11

11

lim1

1lim

lim2

1lim

p

p

p

pM

MM

MM

6

0.378

– So as the upstream Mach number increases towards infinity

• the downstream Mach number decreases to a finite value

• density increases to a finite number

• but temperature and pressure can increase without bound

Page 170: Aerodynamics Course Notes v3

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Supersonic flows (36)• Measurement of velocity in compressible flow

– In low-speed, incompressible flow, the velocity can be measured using a Pitot-static tube

• The total pressure is measured by the Pitot tube and the static pressure from a static pressure orifice

• Bernoulli gives the dynamic pressure as the difference between the two

Page 171: Aerodynamics Course Notes v3

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Supersonic flows (37)• Measurement of velocity in compressible flow

– The same is true in high-speed, compressible flow if we use Mach number instead of velocity, although the formulae are different for each Mach-number regime

• Subsonic compressible

• Supersonic compressible

– For Region 1 the isentropic flow relationships hold

– (see slides ‘A’ at the end of the section for derivation if desired)

– Solving for M12 gives

)1(2

11

1,0

2

11

Mp

p

11

2)1(

1

1,021

p

pM

Page 172: Aerodynamics Course Notes v3

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Supersonic flows (38)• Measurement of velocity in compressible flow

– Using the relationship for Mach number

– becomes

– So, unlike incompressible flow, a knowledge of the total and static pressures is not enough

– We also need to know the freestream speed of sound, a1

11

2)1(

1

1,021

p

pM

11

2)1(

1

1,0212

1

p

pau

Page 173: Aerodynamics Course Notes v3

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Supersonic flows (39)• Measurement of velocity in compressible flow

– Now in supersonic flow the Pitot tube creates a stagnation region and the flow is brought to rest at the throat

– However, because the upstream flow is supersonic and the Pitot tube is an obstruction, there will be a bow wave ahead of it

– The centreline streamline crosses the normal portion of the bow shock

– So the flow is decelerated to subsonic speed (non-isentropically) through the shock wave

– And then (isentropically) to zero velocity at the throat

– The total pressure measured by the Pitot tube is (p0,1) but of the flow behind

a normal shock wave (p0,2)

Page 174: Aerodynamics Course Notes v3

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Supersonic flows (40)• Measurement of velocity in compressible flow

– However, knowing the freestream static pressure (p1) and the throat total

pressure (p0,2) is still enough to calculate the freestream Mach number (M)

– Where p0,2/p2 is the ratio of total and

static pressure after the shock and p2/p1

is the static pressure ratio across the shock

1

2

2

2,0

1

2,0

p

p

p

p

p

p

)1(22

2

2,0

2

11

Mp

p

]2/)1[(

]2/)1[(12

1

212

2

M

MM

)1(1

21 2

11

2

Mp

p

Page 175: Aerodynamics Course Notes v3

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Supersonic flows (41)• Measurement of velocity in compressible flow

– Substituting rearranging and simplifying gives

– This is called the Rayleigh Pitot tube formula

– It relates the Pitot pressure measured by the tube (p0,2) and the freestream static pressure

(p1) to the freestream Mach number (M1)

1

21

)1(24

)1( 21

)1(

21

21

2

1

2,0

M

M

M

p

p

Page 176: Aerodynamics Course Notes v3

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Supersonic flows (42)• Oblique shock waves

– Most shock waves form an oblique angle with the upstream flow

– Normal shock waves are just a special case of oblique shock wave where the angle is 90°

– In addition to oblique compression waves where the pressure increases discontinuously across the shock wave, there are expansion waves where the pressure decreases continuously across the shock wave

Page 177: Aerodynamics Course Notes v3

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Supersonic flows (43)• Oblique shock waves

– Consider supersonic flow encountering a concave corner

– The wall is turned upwards at an angle θ– An oblique shock wave will form at the corner where the streamlines before

and after the shock are all parallel, deflected through the angle θ

– The Mach number suddenly (discontinuously) decreases while the pressure, density and temperature all suddenly increase

Page 178: Aerodynamics Course Notes v3

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Supersonic flows (44)• Oblique shock waves

– Now consider the same supersonic flow encountering a convex corner

– The wall is turned downwards at an angle θ– A series oblique shock waves form an expansion fan will form at the

corner

– The fan opens continuously away from the corner

– Again the streamlines before and after are all parallel, deflected continuously and smoothly through the expansion angle θ

– The Mach number smoothly (continuously) increases while the pressure, density and temperature all smoothly decrease

– In contrast to essentially 1-D normal shock waves, oblique shock and expansion waves are inherently 2-D

• i.e. the flow field properties are a

function of x and y

Page 179: Aerodynamics Course Notes v3

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Supersonic flows (45)• Oblique shock waves

– If we return to the oblique compression wave, this is representative of supersonic flow past a wedge

– The wedge semi-angle is now θ and the angle of the oblique shock wave is β

– The relationship between the shock angle (β), the wedge angle (θ) and the upstream Mach number (M1) is given by

2

1

21 1

1

2

1

11

M

sin

cossintan

– (See slides ‘B’ at the end of the section for derivation if desired)

– This is the θ-β-M relation and it specifies θ as a unique function of M1

and β– The results from the equation are plotted graphically on the next slide

and the graph is used to solve oblique shock problems

Page 180: Aerodynamics Course Notes v3

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Supersonic flows (46)Oblique shock wavesThe wedge angle (θ) plotted against shock angle ( β) for varying upstream Mach number (M1)

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Supersonic flows (47)• Oblique shock waves

– The graph illustrates a lot of physical phenomena associated with oblique shock waves:

– For any given upstream Mach number M1, there is a maximum deflection angle, θmax

– If the physical geometry is such that θ > θmax then no solution exists for a straight oblique shock wave

– Nature establishes a curved shock wave detached from the corner or the nose of a body

– Note that as the freestream Mach number increases, θmax also increases

• straight oblique shock waves can exist at higher deflection angles at higher speed

– But there is a limit (for gamma = 1.4)

• θmax → 45.5° as M1 →

Page 182: Aerodynamics Course Notes v3

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Supersonic flows (48)• Oblique shock waves

– For any given θ less than θmax there are two straight oblique shock solutions for a given upstream Mach number

– E.g. if M1=2.0 and θ =15° then β can equal either 45.3 or 79.8°

– The smaller value is called the weak shock solution and the larger value is the strong shock solution

– The terms “weak” and “strong” derive from the fact that the for a given upstream Mach number (M1), the larger the wave angle the larger the normal component of upstream Mach number (Mn,1) and thus the larger the pressure ratio p2/p1

– That is, the higher-angle shock wave will compress the air more than the lower-angle shock wave

– In nature the weak shock solution usually prevails

Page 183: Aerodynamics Course Notes v3

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Supersonic flows (49)• Oblique shock waves

– Whenever you see straight, attached oblique shock waves (as shown in the bottom picture) they are almost always the weak shock solution

– It is safe to make this assumption unless you have information to the contrary

– Note that the locus of points connecting all the values of θmax divides the weak and strong shock solutions

• Above the curve the strong shock prevails

• Below the curve the weak shock prevails

– There is another curve just below this one– This is the dividing line above which the

downstream Mach number is subsonic (M2 < 1) and below which it is supersonic (M2 > 1)

– For the strong solution the downstream Mach number is always subsonic

– For the majority of weak shock solutions the downstream Mach number is supersonic

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Supersonic flows (50)• Use of oblique shock waves

– Engine inlets

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Supersonic flows (51)• Critical Mach number

– We saw before that the freestream Mach number at which the flow on an aerofoil first becomes sonic is the critical Mach number, Mcr

– If we define the static pressure in the freestream as p and that at a point A on an aerofoil as pA, we can use the isentropic pressure ratio (slide 169) to give us

)1(

2

2

0

0

]2/)1[(1

]2/)1[(1

A

AA

M

M

pp

pp

p

p

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Supersonic flows (52)• Critical Mach number

– Recall that pressure coefficient is given by

where

– So

q

ppC p

221

Vq

)(2

2

2

1

2

1

22

2

22

paMp

Vp

p

Vp

pVq

12

2 p

p

MC p

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Supersonic flows (53)• Critical pressure coefficient

– So the pressure coefficient at point A is given by

– This is the compressible equivalent of the Bernoulli equation, relating local pressure to the local Mach number

– The critical pressure coefficient (Cp,cr) is the pressure coefficient at the point where the flow on the aerofoil first becomes sonic, i.e. MA=1

– This equation allows us to calculate the pressure coefficient at any point where the local Mach number is 1 (i.e. along the sonic line)

1]2/)1[(1

]2/)1[(12)1(

2

2

2,

AAp M

M

MC

1]2/)1[(1

]2/)1[(12)1(2

2,

M

MC crp

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Supersonic flows (54)• Critical pressure coefficient

– When the freestream Mach number is precisely equal to the critical Mach number, there is only one point on the aerofoil where M=1, namely point A

– In this case M = Mcr and

– This equation has no connection with aerofoil shape and is thus a universal relationship which can be used for all aerofoils

– The Prandtl-Glauert rule relates the incompressible pressure coefficient (Cp,0) to a compressible one:

– (Other approximations exist, but this is the simplest and most common)

1]2/)1[(1

]2/)1[(12)1(2

2,

cr

crcrp

M

MC

2

0,

1

M

CC p

p

Page 189: Aerodynamics Course Notes v3

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Supersonic flows (55)• Critical pressure coefficient

– To estimate the critical Mach number we need to:

• By some means (either experimental or theoretical) obtain the low-speed,

incompressible value of Cp,0 at the minimum pressure point on the aerofoil

• Using a compressibility correction (e.g. Prandtl-

Glauert) plot the variation of Cp with M (curve B)

• The point where curve B crosses the line representing

• Is the point where sonic flow occurs at the minimum pressure location on the aerofoil

• The value of M at this intersection is thus the

critical Mach number Mcr

1]2/)1[(1

]2/)1[(12)1(2

2,

cr

crcrp

M

MC

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Supersonic flows (56)• Critical pressure coefficient

– The graph is not an exact determination of Mcr

• The curve for Cp.cr is exact, but curve B is only an approximation

– Hence the value of Mcr obtained is only approximate

– However, such an estimate is useful for preliminary design and the results are accurate enough for most applications

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Supersonic flows (57)• Effect of thickness

– Thicker aerofoils perturb the airflow more and create greater suction on the top surface than thinner aerofoils

– That is, on a thick aerofoil the value of the pressure coefficient at the minimum pressure location will be a larger negative number than the equivalent value on a thin aerofoil

– Plotting this results shows immediately that a thick aerofoil has a lower critical Mach number than a thin one

– For high-speed aircraft it is desirable to have a high value of Mcr and this drives the designer towards a thinner wing

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Supersonic flows (58)• Effect of thickness

– For example a Lear Jet has a 9% thick aerofoil, while the Piper Aztec has one that is 14% thick

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Supersonic flows (59)• Drag-divergence Mach number

– As we increase the freestream Mach number (M), from a to b, for a given aerofoil the drag remains virtually constant

– We then encounter the critical Mach number where the flow on the aerofoil first becomes sonic, point c

– As we increase M to slightly above Mcr (to point d) a finite region of supersonic flow appears on the aerofoil

– As we nudge M still higher we encounter point e where the drag suddenly starts to increase.

– The value of M where this sudden increase in drag starts is called the drag-divergence Mach number

Douglas dCd/dM > 0.1

Boeing ΔCd = 0.002

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Supersonic flows (60)• Drag-divergence Mach number

– Beyond the drag-divergence Mach number the drag coefficient can be very large, typically increasing by a factor of 10 or more

– This drag increase is associated with an extensive region of supersonic flow over the aerofoil terminating in a shock wave

– For an aerofoil design for low-speed application, the local Mach number can reach 1.2 or higher and the terminating shock can be very strong

– These shocks generally cause severe flow separation, with an attendant increase in drag (wave drag)

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Supersonic flows (61)• Drag rise reduction

– Research since 1945 has focused on reducing the large drag rise– Instead of a factor of 10 increase in drag at Mach 1, can reduce it to 2

or 3?– Several design ploys have been utilised to achieve this– The first was the use of thin aerofoils– We have already seen that thinner aerofoils have higher critical Mach

numbers than thicker ones

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Supersonic flows (62)• Drag rise reduction

Variation of thickness-to-chord ratio for a representative selection of different aircraft

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Supersonic flows (63)• Drag rise reduction

– The second design ploy was to use swept wings

– Imagine a straight wing with a thickness-to-chord ratio (t/c) of 0.15

– If we sweep the same wing at 45° the flow sees the same physical thickness but the chord has extended

and the thickness-to-chord ratio has reduced

– Thus by sweeping the wing the flow behaves as if the aerofoil is thinner and it has a higher critical Mach number

cc

c 1.41cos

2

North American F-86 Sabre

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Supersonic flows (64)• Area rule

– Beside using thin, swept aerofoils to reduce the drag rise at Mach 1, two other concepts have been developed

– The first of these is the area rule

– The early jets did not have enough thrust to overcome the massive drag rise near Mach 1

– Even the early “century” series aircraft designed to provide the UASF with supersonic fighters in the early 1950s (e.g. the Convair F-102 Delta Dagger) could not at first penetrate the sound barrier in level flight

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Supersonic flows (65)• Area rule

– The picture shows the area distribution (the variation of cross-sectional area with distance along the aircraft axis) of a typical US aircraft of that period

• Note the discontinuities in the distribution

– Ballisticians had known for almost a century that bullets and shells with smoothly varying cross-sections were faster than those with those with abrupt or discontinuous shape changes

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Supersonic flows (66)• Area rule

– The NACA engineer Richard Whitcomb applied this knowledge to the problem of transonic aircraft

– He reasoned that the area distribution should be as smooth as possible– This meant that, in the region of the wing and tail, the fuselage cross-

sectional area had to reduce to compensate for the additional area of these structures

– This led to the “coke bottle” shape and the design philosophy is called the area rule

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Supersonic flows (67)• Area rule

– The F102 was redesigned and rebuilt in 118 days and achieved M1.22

F-102

Straight-sided fuselage

F-102A

Coke bottle fuselage

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Supersonic flows (68)• Area rule

– Other ways of area ruling

Boeing 747-100

Boeing 747-400

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Supersonic flows (69)• Area rule

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Supersonic flows (70)• Area rule

– Some aircraft cannot change the fuselage shape so extra volume is added to smooth the area distribution

• Kuchemann “carrots”

Convair CV-990

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Supersonic flows (71)• The supercritical aerofoil

– While thinner aerofoils help reduce the drag rise near Mach 1, there is a practical limit on how thin an aerofoil can be

• Spar depths and fuel volume

– So is there a way we can delay the drag rise to higher Mach numbers for an aerofoil of given thickness?

– Increasing Mcr is one way, but another is to increase the increment between the critical Mach number and the drag-divergence Mach number

• i.e. increase the gap between point c and point e

– An aerofoil which does this is known as a supercritical aerofoil

Page 206: Aerodynamics Course Notes v3

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Supersonic flows (72)• The supercritical aerofoil

– The shape of a supercritical aerofoil is compared to a common NACA 64-series aerofoil in the picture

– The supercritical aerofoil has a relatively flat top leading to a region of supersonic flow with lower Mach number than the NACA 64-series

– In turn, the terminating shock is much weaker and thus creates less drag

– The picture shows the NACA section at a lower Mach number but the supersonic region is taller, the local Mach numbers higher and the terminating shock stronger than the supercritical aerofoil at a higher speed

Page 207: Aerodynamics Course Notes v3

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Supersonic flows (73)• The supercritical aerofoil

– The picture shows experimental data from the two aerofoils– The drag divergence Mach number for the NACA 64-series aaerofoil is

0.67 and for the supercritical aerofoil is 0.79

– The relatively flat upper surface is achieved through negative camber for the forward 60% of the aerofoil

– This lowers the lift which is compensated by extreme positive camber on the rearward 30%

– This produces the cusp-like shape of the lower surface near the trailing edge

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Supersonic flows (74)• The supercritical aerofoil

Handley Page Victor

Airbus A300

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Supersonic flows (75)• Shock-induced separation

– The onset of transonic shock-induced flow separation is not confined to large increases in drag

– It can also trigger a variety of aeroelastic instability and response phenomena including flutter, oscillations and control surface buzz, shock-induced oscillations

– That is, the shock is not always static

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Supersonic flows (76)• Shock-induced separation

– We might wish, therefore, to consider some form of shock wave control

– This will consist of either• Modification of the geometry at the foot of the shock wave to either

smear a single shock into a series of weaker ones or fix its location (shock bump)

• Modification of the boundary layer to withstand the pressure gradient across the shock (passive or active vortex generators, suction or blowing)

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Supersonic flows (77)• Shock-induced separation

– Shock bodies

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Supersonic flows (77)• Shock-induced separation

– To maintain their operational ceiling of 70,000 feet (21,000 m), the U-2A and U-2C flew very near their maximum speed

– However, the aircraft's stall speed at that altitude is only 10 knots less than its maximum speed

– There was a danger when turning that the inner wing stalled because it was going too slow and the outer wing stalled because of shock formation and shock-induced separation

– This point of the flight enveloped was referred to by the pilots as "coffin corner"

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Supersonic flows (78)• Total aircraft drag

Trim Drag

Wave (Wing) Drag

Parasitic Drag

Nacelle Interaction Drag

Unaccounted Drag

Vortex (Induced) Drag

Profile drag

Drag Breakdown of a Representative 4 Engine Civil Transport Aircraft

At Design Cruise Mach Number & Lift Coefficient

Page 214: Aerodynamics Course Notes v3

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Supersonic flows (79)

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Supersonic flows (A1)• Isentropic flow (derivations)

– The basic equations:

Continuity 1u1A1 = 2u2A2

Momentum

Energy

Enthalpy

i.e.

Equation of state

Speed of sound

2

~

2

~ 22

2

21

1

uh

uh

0))(( 122121

221122221

211 AAppApApAuAu

2211

~,

~TchTch pp

222

12

212

11 uTcuTc pp

22

2

11

1

T

p

T

p

M

uTcTR

pa p )1(

Page 216: Aerodynamics Course Notes v3

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Supersonic flows (A2)• Isentropic flow (derivations)

– The energy equation

– Thus becomes

– Or

– If we take p1 as p0 (stagnation conditions) and p2 as p (local conditions) and

rearrange the equations, we get

2

222

1

121

)1(2)1(2

pupu

)1(20

2

11

Mp

p

222

12

212

11 uTcuTc pp

1212

22

22

21

21

auau

Page 217: Aerodynamics Course Notes v3

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• Oblique shock angle (derivation)– The basic equations (note the resolution of the velocity vector into components normal to and

tangential to the shock wave):

Continuity 1V1n = 2V2n

Momentum (normal to the shock wave)

Momentum (parralel to the shock wave)• i.e. no change in pressure)

Energy (from slide ‘Supersonic flows (25)’)

(a* is the value of the speed of sound at sonic conditions and is constant)

211

22221 nn VVpp

tntn VVVV 1112220

2*22

1)(2

1

21a

ua

Supersonic flows (B1)

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• Oblique shock angle (derivation)– Applying the energy equation before and after the shock gives

– From 1V1n = 2V2n and it follows that V1t=V2t

– i.e. the tangential velocity is the same on both sides of the shock

– Since the tangential velocity doesn’t change we just need to determine the normal velocity after the shock

– Again using continuity we get

– Where p2/2 and p1/1 can be eliminated using the equations at the top of the slidennnn

VVV

p

V

p21

11

1

22

2

tntn VVVV 1112220

2*

1

12

12

1

(2

1

2a

pVV tn

1)1-

Supersonic flows (B2)

2*

2

22

22

2

(2

1

2a

pVV tn

1)1-

Page 219: Aerodynamics Course Notes v3

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• Oblique shock angle (derivation)– Dropping the station subscript (1 or 2) for Vt (because V1t = V2t) gives

– Which we arrange to the form

– This equation is satisfied when either factor is zero

– The solution that the second factor is zero (i.e. V1n – V2n = 0, or V1n = V2n) corresponds to a shock wave of zero intensity, or a Mach wave

– Setting the first factor to zero gives a non-trivial solution:

22

21 1

1tnn VaVV

nnn

tn

n

tn

nn

VVV

VV

V

VV

VVa 21

2

2

21

2

112

2*

2

111

2

1

Supersonic flows (B3)

0)(2

1

2

1

2

121

21

2

21

2*

nn

nn

t

nn

VVVV

V

VV

a

Page 220: Aerodynamics Course Notes v3

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• Oblique shock angle (derivation)

– From the picture we get V1n = V1 sinβ and Vt = V1 cosβ and we substitute

these into the previous equation to get

– Which can be written as

– And we can replace a*/a0 and a0 /V1 for terms involving and M to give

sin

cos

1sin

2

11

2

2

1V

V

aV n

Supersonic flows (B4)

2

2

1

0

0

12

1cos

1sin V

a

a

aVV n

2

1

212

121

M

VV n 1

sin1sin

Page 221: Aerodynamics Course Notes v3

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Supersonic flows (B4)• Oblique shock angle (derivation)

– From the picture we can see that V2n is related to the wave angle( β)and

deflection angle (θ) by

V2n = Vt tan(β - θ) = V1 cosβ tan(β - θ)

– Which may be equated to the equation at the bottom of the previous slide to give

– Or

2

1

2 1

1

2

1

11)(

M

sin

cossintan

2

1

21 1

1

2

1

11

M

sin

cossintan

Page 222: Aerodynamics Course Notes v3

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• Swept wing flows; Effect of spanwise and normal velocity components; qualitative description of 3D boundary layers on swept wings; Forward, rearward and variable sweep wings; control surface effects; delta wings and vortical flows; vortex flap; aerodynamics of aircraft at high incidences.

Swept Wings

Page 223: Aerodynamics Course Notes v3

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• Why sweep the wings

– We’ve already seen that wing sweep increases the effective t/c

– But it also moves the wings behind the bow shock

Swept wings (1)

Page 224: Aerodynamics Course Notes v3

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• Incidence of a shock on a wing

– Sweeping the wing back alters the onset Mach number, i.e. that normal to the leading edge, M n

– If the wing is swept at the angle of the Mach line

M n = M cos (90°-μ)

= M sin μ

=1

– That is, the normal Mach number is unity

– Sweeping it back further reduces the normal Mach number to subsonic

Swept wings (2)

Page 225: Aerodynamics Course Notes v3

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• Bell X-1 and X-2

Swept wings (3)

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• Effect of sweep on pressure

– Recall that the pressure coefficient is dependent on onset Mach number only

– So the normal pressure distribution is thus related to the normal Mach number

– And thus varies with sweep

Swept wings (4)

12

2 p

p

MC p

12

2 p

p

MC

npn

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• Velocity components

– Sweep introduces a spanwise component of the freestream velocity on the wing

– Transition of the boundary layer can now occur in each direction

– The surface flow is different to the freestream

– This creates shear stresses and an additional transition mechanism

Swept wings (5)

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• Cross-flow instabilities

– The shear causes waves and vorticity in a spanwise direction

– In addition to 2-D transition (Tollmien-Schlichting waves) there is additional 3-D transition mechanism due to these spanwise disturbances or cross-flow instabilities

Swept wings (6)

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• Stall characteristics

– Rectangular wings have larger downwash angles at the tip than at the root

– The effective angle of attack at the tip is therefore lower at the tip and it will stall last

– However, rectangular wings are not very efficient

• They have more induced drag than the ideal elliptical planform

– A compromise is to taper the wing

– But with a small tip chord come reduced local Reynolds number, increased effective angle of attack and thicker boundary layers (due to spanwise flow)

– This generally leads to swept, tapered wings being prone to tip stall

Swept wings (7)

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• Tip stall

– Tip stall is not good!

– Ailerons become ineffective

– Loss of lift means the aerodynamic centre moves forward

– Pitch up

Swept wings (8)

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• Methods to prevent pitch-up

– Wing twist

– By twisting the wing tip nose downwards (wash-out), the local angle of attack is reduced

Swept wings (9)

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• Methods to prevent pitch-up

– Wing fences

– Reduces or stops the spanwise flow

Swept wings (10)

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• Methods to prevent pitch-up

– Wing snags, saw teeth, dog teeth

– Generate discrete, strong vorticity that helps the flow remain attached

Swept wings (11)

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• Methods to prevent pitch-up

– Forward sweep

– The spanwise velocity is in the other direction – tip to root

– Other advantages include:

• Better pilot vision as the wing root is relatively far aft

• Wing spars can be placed behind a weapons bay rather than through it

• Controllability to much higher angle of attack (67° for the X-29)

Swept wings (12)

Grumman X-29

Schleicher ASK 13

Page 235: Aerodynamics Course Notes v3

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• Methods to prevent pitch-up

– Forward sweep

– The main disadvantage is structural

– The wing tip tends to twist up, increasing the local load and thus increasing the twist even more

– An unfortunate tendency that can be countered by strengthening the structure of a metal wing or using cunning layups of carbon fibre

– Aeroelastic tailoring

Swept wings (13)Junkers Ju 287

Hansa HFB320

Sukhoi Su-47

Page 236: Aerodynamics Course Notes v3

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• Variable sweep wings

– We have seen that a swept wing is more suitable for high speeds

– An unswept wing is suitable for lower speeds

– A variable-sweep wing allows a pilot (or flight control system) to select the correct wing configuration for the plane's intended speed

– The variable-sweep wing is most useful for those aircraft that are expected to function at both low and high speed, and for this reason it has been used primarily in military aircraft

Swept wings (14)

Messerschmitt Me P.1101

Bell X-5

Grumman XF10F Jaguar

Page 237: Aerodynamics Course Notes v3

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• Variable sweep wings

– But the extra mechanisms are heavy

Swept wings (15)

Tornado

Page 238: Aerodynamics Course Notes v3

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• Variable sweep wings

Swept wings (16)Sukhoi Su-17

Rockwell B-1

Tupolev Tu-160

Grumman F-14

General Dynamics F-111

MiG-23

Page 239: Aerodynamics Course Notes v3

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• Delta wings

– Delta wings are a special form of swept wing pioneered by Lippisch

– The wing leading edge remains behind the shock wave generated by the nose of the aircraft when flying at supersonic speeds

– While this is also true of ordinary swept wings, the delta's planform carries across the entire aircraft which has structural advantages

– Another advantage is vortex lift

– Beyond a certain angle of attack, the wing leading edge generates a stable vortex which remains attached to the upper surface of the wing

– This gives delta wings a relatively high stall angle

Swept wings (17)

Convair XF-92

Lippisch P.13

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• Delta wings

– Types of delta wing

Swept wings (18)

Page 241: Aerodynamics Course Notes v3

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• Vortex lift

– Leading edge boundary layer rolls up into a vortex

Swept wings (19)

Page 242: Aerodynamics Course Notes v3

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• Vortex lift

– The wing generates lift like a conventional aerofoil at low angles of attack

– The leading edge vortices form at increasing angle of attack and contribute significant lift and enable stability and control at relatively high angles of attack

Swept wings (20)

Page 243: Aerodynamics Course Notes v3

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• Vortex lift

– The solution for low-speed, high-lift performance of Concorde (Kuchemann at RAE)

Swept wings (21)

Page 244: Aerodynamics Course Notes v3

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• Vortex lift

– Aircraft that wish to operate at high angles of attack tend to generate and utilise vortex lift

– This can be from areas other than the wing, e.g. leading-edge extensions (LEX)

• N.B. Fin Buffet

Swept wings (22)

Page 245: Aerodynamics Course Notes v3

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• Vortex lift

– The limiting factor is a phenomenon called vortex burst

– Vortex bursting is a phenomenon in which the structured character of the vortex is destroyed resulting in a loss of most of the vortex lift

Swept wings (23)

– It occurs due to adverse pressure gradients acting on the vortex

– When the vortex burst occurs on the wing (as opposed to downstream of the wing) the lift drops substantially.

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• Vortex flap– The concept of the vortex flap was to reposition the leading-edge vortex

system which normally develops over a delta (or high-sweep) wing at high angles of attack onto a forward facing flap surface

– This results in a reduction of induced drag due to a thrust component derived from the low pressure on the flap.

Swept wings (24)

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• Vortex flap– NASA Langley conducted flight tests on a modified NF-106B Delta

Dagger

Swept wings (25)

Page 248: Aerodynamics Course Notes v3

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• Vortex flap– NASA Langley conducted flight tests on a modified NF-106B Delta

Dagger

Swept wings (26)

α =9°, 30° vortex flap

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• Vortex flap

Swept wings (27)

α =13°, 30° vortex flap

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• Vortex flap

Swept wings (28)

α =13°, 40° vortex flap

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• Vortex flap

– The flow visualisation shows that the flowfield does not behave as anticipated

– The vortex flap generates lots of weak vortices rather than a single strong one

– The next vortex along the span is triggered by the secondary vortex of the previous one

Swept wings (29)

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• Vortex flap

– Never really adopted operationally

– Typhoon uses a drooped nose device not dissimilar to the A380

Swept wings (30)

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