Aeronautical Engineer Data Book Episode 6 doc

Aircraft -general 0.012 M = 2.5 0.016 Airship 0.020–0.025 Helicopter download 0.4–1.2 II Turbulent flat plate Airfoil section, minimum Airfoil section, at stall 2-element airfoil

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5.7 Normal shock waves

5.7.1 1D flow

A shock wave is a pressure front which travels

at speed through a gas Shock waves cause an increase in pressure, temperature, density and entropy and a decrease in normal velocity Equations of state and equations of conser-vation applied to a unit area of shock wave give (see Figure 5.10):

State p1/1T1= p2/2T2

Mass flow m = 1u1= 2u2

Basic fluid mechanics 91

u

u1

p1

1

p2 2

Shock wave travels into area of stationary gas

Fig 5.10(a) 1-D shock waves

u

u1

p1 1 p2 2

Shock wave becomes a stationary discontinuity

Fig 5.10(b) Aircraft shock waves

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92 Aeronautical Engineer’s Data Book

2

Momentum p1+ p1u12 = p2+ 2u2

Energy c T p 1+ = c p T2+ = c p T0

Pressure and density relationships across the

shock are given by the Rankine-Hugoniot

equations:

+ 1 2

– 1

p 2 – 1 1

=

p 1 + 1 2

– – 1 1

( + 1)p

+ 1

2 ( – 1)p1

= 1  + 1 p2

+  – 1 p1

Static pressure ratio across the shock is given

by:

2 – ( – 1)

=

Temperature ratio across the shock is given by:

=

T1 p1/1

1 – ( + 1) 2 + ( – 1)M2

1

=

T1 � + 1 �� ( + 1)M2 1 �

Velocity ratio across the shock is given by:

From continuity: u2/u1= 1/2

u2 2 + ( – 1)M2

u1 ( + 1)M2

In axisymmetric flow the variables are indepen

dent of so the continuity equation can be

expressed as:

1 ∂(R2 q R) 1 ∂(sin q)

Similarly in terms of stream function :

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�

93 Basic fluid mechanics

R2 sin ∂

R sin ∂R

Additional shock wave data is given in Appen dix 5 Figure 5.10(b) shows the practical effect

of shock waves as they form around a super sonic aircraft

5.7.2 The pitot tube equation

An important criterion is the Rayleigh super sonic pitot tube equation (see Figure 5.11)

M

� �/( – 1)

2

+ 1

Pressure ratio: =

p1

2

M1 1 p1 u1 p2 p02

M2

Fig 5.11 Pitot tube relations

2M2

1 – ( – 1)

+ 1

5.8 Axisymmetric flows

Axisymmetric potential flows occur when bodies such as cones and spheres are aligned

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y

x

z

R

r

qR

q θ

q ϕ

θ

ϕ

∂

Fig 5.12 Spherical co-ordinates for axisymmetric flows

∂R

into a fluid flow Figure 5.12 shows the layout

of spherical co-ordinates used to analyse these types of flow

Relationships between the velocity compo nents and potential are given by:

1

r

∂

5.9 Drag coefficients

Figures 5.13(a) and (b) show drag types and

‘rule of thumb’ coefficient values

U

Shape Pressure drag Friction drag

DP(%) Df(%)

Fig 5.13(a) Relationship between pressure and fraction

drag: ‘rule of thumb’

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95 Basic fluid mechanics

d

l

d

l

U

Cylinder (flow direction)

Shape Dimensional

ratio

Datum area, A Approximate drag coefficient, CD

Cylinder (right angles to flow)

Hemisphere (bottomless)

Cone

d

I

I/d = 1 0.91

I/d = 1 0.63

a = 60˚ 0.51

a = 30˚ 0.34

1.2

π – d 2

4

π – d 2

4

dl

π – d 2

4

π – d 2

4

Bluff bodies

Rough Sphere (Re = 10 6 ) 0.40 Smooth Sphere (Re = 10 6 ) 0.10 Hollow semi-sphere opposite stream 1.42 Hollow semi-sphere facing stream 0.38 Hollow semi-cylinder opposite stream 1.20 Hollow semi-cylinder facing stream 2.30 Squared flat plate at 90 ° 1.17 Long flat plate at 90 ° 1.98 Open wheel, rotating, h/D = 0.28 0.58

Streamlined bodies

Laminar flat plate (Re = 10 6 ) 0.001

Re = 10 6 ) 0.005

0.006 0.025 0.025 0.05 0.05 0.16 0.005 0.09 n.a

Aircraft -general

0.012

M = 2.5 0.016 Airship 0.020–0.025 Helicopter download 0.4–1.2

II

Turbulent flat plate (

Airfoil section, minimum

Airfoil section, at stall

2-element airfoil

4-element airfoil

Subsonic aircraft wing, minimum

Subsonic aircraft wing, at stall

Subsonic aircraft wing, minimum

Subsonic aircraft wing, at stall

Aircraft wing (supersonic)

Subsonic transport aircraft

Supersonic fighter,

Fig 5.13(b) Drag coefficients for standard shapes

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Section 6

Basic aerodynamics

6.1 General airfoil theory

When an airfoil is located in an airstream, the flow divides at the leading edge, the stagna tion point The camber of the airfoil section means that the air passing over the top surface has further to travel to reach the trail ing edge than that travelling along the lower surface In accordance with Bernoulli’s equation the higher velocity along the upper airfoil surface results in a lower pressure, producing a lift force The net result of the velocity differences produces an effect equiv alent to that of a parallel air stream and a rotational velocity (‘vortex’) see Figures 6.1 and 6.2

For the case of a theoretical finite airfoil section, the pressure on the upper and lower surface tries to equalize by flowing round the tips This rotation persists downstream of the wing resulting in a long U-shaped vortex (see Figure 6.1) The generation of these vortices needs the input of a continuous supply of energy; the net result being to increase the drag

of the wing, i.e by the addition of so-called

induced drag

6.2 Airfoil coefficients

Lift, drag and moment (L, D, M) acting on an

aircraft wing are expressed by the equations:

U Lift (L) per unit width = C L l 2

2

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97 Basic aerodynamics

An effective rotational velocity (vortex) superimposed on the parallel airstream

+ + + + +

– – – –

– (a)

Pressures equalize by flows (b) around the tip

– – – – – – – –

+ + + + + + + +

Tip Midspan

Tip

Core of vortex (c)

Finite airfoil

‘Horse-shoe’ vortex

persists downstream

Fig 6.1 Flows around a finite 3-D airfoil

Camber line

edge

Chord Camber

Thickness

Leading

edge

l

L

D

a

U

General airfoil section

Trailing

Profile of an asymmetrical airfoil section

Centre line Chord line

x

t

c

Fig 6.2 Airfoil sections: general layout

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U Drag (D) per unit width = C D l2

2

Moment (M) about LE or

U 1/4 chord = C M l2

2 per unit width

C L , C D and C M are the lift, drag and moment coefficients, respectively Figure 6.3 shows typical values plotted against the angle of attack, or incidence, () The value of C D is

small so a value of 10 C D is often used for the

characteristic curve C L rises towards stall point and then falls off dramatically, as the wing

enters the stalled condition C D rises gradually, increasing dramatically after the stall point Other general relationships are:

• As a rule of thumb, a Reynolds number of

Re 106 is considered a general flight condition

• Maximum C L increases steadily for Reynolds numbers between 105 and 107

• C D decreases rapidly up to Reynolds numbers of about 106, beyond which the rate of change reduces

• Thickness and camber both affect the

maximum C L that can be achieved As a

general rule, C L increases with thickness and then reduces again as the airfoil

becomes even thicker C L generally increases as camber increases The

minimum C D achievable increases fairly steadily with section thickness

6.3 Pressure distributions

The pressure distribution across an airfoil section varies with the angle of attack () Figure 6.4 shows the effect as increases, and

the notation used The pressure coefficient C p

reduces towards the trailing edge

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Characteristics for an asymmetrical ‘infinite-span 2D airfoil’

75

50

25

0

–25

1.5

1.0

0.5

0

–0.5

10 CD

10˚

5˚

α

L/D

CL

CD

CL= 0 at the no-lift angle (– α)

Stall point

Characteristic curves of a practical wing

CL

CD

CM1/4

CL

CM

CD

–0.08

–0.12

–8˚ –4˚ 0˚ 4˚ 8˚ 12˚ 16˚ 20˚

α

Fig 6.3 Airfoil coefficients

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Arrow length represents the magnitude of pressure coefficient Cp

P∞= upstream

pressure

S

Stagnation point (S)

moves backwards on

the airfoil

lower surface

(p – p∞)

α 5˚

S

Pressure coefficient C = p

2

Fig 6.4 Airfoil pressure coefficient (Cp)

6.4 Aerodynamic centre

The aerodynamic centre (AC) is defined as the point in the section about which the pitching

moment coefficient (C M) is constant, i.e does

not vary with lift coefficient (C L) Its theoreti cal positions are indicated in Table 6.1

Table 6.1 Position of aerodynamic centre

< 10° At approx 1/4 chord

somewhere near the chord line Section with high

Flat or curved plate:

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Using common approximations, the following equations can be derived:

d (C Ma)

dC L

= –

xAC

c

9

c

where C Ma = pitching moment coefficient at

distance a back from LE

xAC = position of AC back from LE

c = chord length

6.5 Centre of pressure

The centre of pressure (CP) is defined as the point in the section about which there is no pitching moment, i.e the aerodynamic forces

on the entire section can be represented by lift and drag forces acting at this point The CP does not have to lie within the airfoil profile and can change location, depending on the

magnitude of the lift coefficient C L The CP is

conventionally shown at distance kCP back from the section leading edge (see Figure 6.5) Using

Lift and drag only cut at the CP

C

xAC

MAC

MLE

Lift

Drag

Aerodynamic centre

Lift

kCP

Centre of pressure (CP)

Fig 6.5 Aerodynamic centre and centre of pressure

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the principle of moments the following expres

sion can be derived for kCP:

c C L cos + C D sin

Assuming that cos 1 and C D sin 0 gives:

6.6 Supersonic conditions

As an aircraft is accelerated to approach super sonic speed the equations of motion which describe the flow change in character In order

to predict the behaviour of airfoil sections in upper subsonic and supersonic regions, compressible flow equations are required

6.6.1 Basic definitions

M Mach number

M∞ Free stream Mach number

Mc Critical Mach number, i.e the value of which results in flow of M∞ = 1 at some location on the airfoil surface

Figure 6.6 shows approximate forms of the pressure distribution on a two-dimensional airfoil around the critical region Owing to the complex non-linear form of the equations of motion which describe high speed flow, two popular simplifica

tions are used: the small perturbation approxima tion and the so-called exact approximation

6.6.2 Supersonic effects on drag

In the supersonic region, induced drag (due to lift) increases in relation to the parameter

M2 – 1function of the plan form geometry of the wing

6.6.3 Supersonic effects on aerodynamic centre

Figure 6.7 shows the location of wing aerody namic centre for several values of tip chord/root chord ratio () These are empirically based results which can be used as a ‘rule of thumb’

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103

0

Basic aerodynamics

M1 (local)

M∞ > Mcrit M∞ > Mcrit

0.4

0.8

1.2

0 0.2 0.4 0.6 0.8 1.0 0 0.2 0.4 0.6 0.8 1.0

M∞ >> Mcrit

Supersonic regions

–0.8

–0.4

–1.2

–Cp

0

0.4

0.8

1.2

0

Fig 6.6 Variation of pressure deterioration (2-D airfoil)

6.7 Wing loading: semi-ellipse assumption

The simplest general loading condition assump tion for symmetric flight is that of the semi-ellipse The equivalent equations for lift, downwash and induced drag become:

For lift:

VK0πs

2

1

replacing L by C L /2V 2S gives:

C L VS

K0= πs

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2.0

1.8

1.6

1.4

Xa.c

C

1.0

0.8

0.6

0.4

0.2

1.4

1.2

1.0

Xa.c

C

0.6

0.4

1.2

1.0

Xa.c 0.8

Cr

0.6

0.4

0.2

0

tanΛ LE β β tanΛ LE

λ = C t /CR = 1.0

Ct/CR = 0.5

Ct/CR = 0.25

AR tanΛ LE

6

5

4

3

2

1

6

5

4

3

2

1

AR tanΛ LE

6

5

4

3

2

1

Subsonic Supersonic

Unswept T.E

Sonic T E

Taper ratio

β tanΛLE tanΛLE β

Fig 6.7 Wing aerodynamic centre location: subsonic/

supersonic flight Originally published in The AIAA

Aerospace Engineers Design Guide, 4th Edition Copyright

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For downwash velocity (w):

w = K0

4S , i.e it is constant along the span

For induced drag (vortex):

C L 2

πAR

D D =

V

where aspect ratio (AR) =

2 span 4s2

Hence, C D

V falls (theoretically) to zero as aspect ratio increases At zero lift in symmetric flight,

C D = 0

V

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Section 7

Principles of flight dynamics

7.1 Flight dynamics – conceptual breakdown

Flight dynamics is a multi-disciplinary subject consisting of a framework of fundamental mathematical and physical relationships Figure 7.1 shows a conceptual breakdown of the subject relationships A central tenet of the framework are the equations of motion, which provide a mathematical description of the physical response of an aircraft to its controls

7.2 Axes notation

Motions can only be properly described in relation to a chosen system of axes Two of the

most common systems are earth axes and aircraft body axes

The equations of motion

and handling

properties

Aerodynamic characteristics

Common aerodynamic parameters

Stability and

control

derivatives

Stability and control parameters Aircraft flying

of the airframe

Fig 7.1 Flight dynamics – the conceptual breakdown

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107 Principles of flight dynamics

Conventional earth axes are used as a reference frame for

‘short-term’ aircraft motion

S

N

y0

z0

o0

x0

yE zE

xE

oE

• The horizontal plane oE, E, E, lies parallel to the plane o0, 0, 0, on the earth’s surface

• The axis oE, zE, points vertically downwards

Fig 7.2 Conventional earth axes

7.2.1 Earth axes

Aircraft motion is measured with reference to a fixed earth framework (see Figure 7.2) The system assumes that the earth is flat, an assump tion which is adequate for short distance flights

7.2.2 Aircraft body axes

Aircraft motion is measured with reference to

an orthogonal axes system (Ox b , y b , z b) fixed on the aircraft, i.e the axes move as the aircraft moves (see Figure 7.3)

7.2.3 Wind or ‘stability’ axes

This is similar to section 7.2.2 in that the axes

system is fixed in the aircraft, but with the Ox-axis orientated parallel to the velocity vector V0

(see Figure 7.3)

7.2.4 Motion variables

The important motion and ‘perturbation’ variables are force, moment, linear velocity,

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angular velocity and attitude Figure 7.4 and Table 7.1 show the common notation used

7.2.5 Axes transformation

It is possible to connect between axes

refer-ences: e.g if Ox0, y0, z0 are wind axes and components in body axes and , , are the angles with respect to each other in roll, pitch and yaw, it can be shown that for linear quanti-ties in matrix format:

� �= D � �Ox0

Oy0

Oz0

Ox3

Oy3

Oz3

x b

x w

z b

z w

Conventional body axis system.

O x b is parallel to the ‘fuselage horizontal’ datum

O z b is ‘vertically downwards’

O

Conventional wind (or‘stability’) axis

system: O x w is parallel to the velocity vector V o

Roll L,p, φ Pitch

M,q, θ

Yaw N,r, ψ

X,U e ,U,u

Z,W e ,W,w Y,V e ,V,v

Fig 7.3 Aircraft body axes

Fig 7.4 Motion variables: common notation

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