aerodynamics part iii

1

AERODYNAMICS

Part III

SOLO HERMELIN

http://www.solohermelin.com

http://www.solohermelin.com/

2

Table of Content

AERODYNAMICS

Earth Atmosphere

Mathematical Notations

SOLO

Basic Laws in Fluid Dynamics

Conservation of Mass (C.M.)

Conservation of Linear Momentum (C.L.M.)

Conservation of Moment-of-Momentum (C.M.M.)

The First Law of Thermodynamics

The Second Law of Thermodynamics and Entropy Production

Constitutive Relations for Gases

Newtonian Fluid Definitions – Navier–Stokes Equations

State Equation

Thermally Perfect Gas and Calorically Perfect Gas

Boundary Conditions

Flow Description

Streamlines, Streaklines, and Pathlines

A

E

R

O

D

Y

N

A

M

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C

S

P

A

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http://1drv.ms/1kGuZKg

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Table of Content (continue – 1)

AERODYNAMICSSOLO

Circulation

Biot-Savart Formula

Helmholtz Vortex Theorems

2-D Inviscid Incompressible Flow

Stream Function ψ, Velocity Potential φ in 2-D Incompressible

Irrotational Flow

Aerodynamic Forces and Moments

Blasius Theorem

Kutta Condition

Kutta-Joukovsky Theorem

Joukovsky Airfoils

Theodorsen Airfoil Design Method

Profile Theory by the Method of Singularities

Airfoil Design

A

E

R

O

D

Y

N

A

M

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C

S

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AERODYNAMICSSOLO

Lifting-Line Theory

Subsonic Incompressible Flow (ρ∞ = const.) about Wings

of Finite Span (AR < ∞)

3D Lifting-Surface Theory through Vortex Lattice Method (VLM)

Incompressible Potential Flow Using Panel Methods

Dimensionless Equations

Boundary Layer and Reynolds Number

Wing Configurations

Wing Parameters

References

A

E

R

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D

Y

N

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M

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C

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AERODYNAMICSSOLO

Shock & Expansion Waves

Shock Wave Definition

Normal Shock Wave

Oblique Shock Wave

Prandtl-Meyer Expansion Waves

Movement of Shocks with Increasing Mach Number

Drag Variation with Mach Number

Swept Wings Drag Variation

Variation of Aerodynamic Efficiency with Mach Number

Analytic Theory and CFD

Transonic Area Rule

A

E

R

O

D

Y

N

A

M

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C

S

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AERODYNAMICSSOLO

Linearized Flow Equations

Cylindrical Coordinates

Small Perturbation Flow

Applications: Nonsteady One-Dimensional Flow

Applications: Two Dimensional Flow

Applications: Three Dimensional Flow (Thin Airfoil at Small Angles of Attack)

Applications: Two Dimensional Flow (Thin Airfoil at Small Angles of Attack)

Drag (d) and Lift (l) per Unit Span Computations for Subsonic Flow (M∞ < 1)

Prandtl-Glauert Compressibility Correction

Computations for Supersonic Flow (M∞ >1)

Ackeret Compressibility Correction

A

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7

SOLO

Table of Contents (continue – 5)

AERODYNAMICS

Wings of Finite Span at Supersonic Incident Flow

Theoretic Solutions for Pressure Distribution on a

Finite Span Wing in a Supersonic Flow (M∞ > 1)

1. Conical Flow Method

2. Singularity-Distribution Method

Theoretical Solutions for Compressible Supersonic Flow (M∞ >1)

Theoretic Solution for Pressure Distribution on a Semi-Infinite Triangular Wing

in a Supersonic Flow and Subsonic Leading Edge (tanΛ > β)

Theoretic Solution for Pressure Distribution on a Semi-Infinite Triangular Wing

in a Supersonic Flow and Supersonic Leading Edge (tanΛ < β)

Theoretic Solution for Pressure Distribution on a Delta Wing in a Supersonic

Flow and Subsonic Leading Edge (tanΛ > β)


Flow and Supersonic Leading Edge (tanΛ < β)

Arrowhead Wings with Double-Wedge Profile at Zero Incidence

Systematic Presentation of Wave Drag of Thin, Nonlifting Wings having

Straight Leading and Trailing Edges and the same dimensionless profile in

all chordwise plane [after Lawrence]

8


AERODYNAMICSSOLO

Aircraft Flight Control

References

CNα – Slope of the Normal Force Coefficient Computations of Swept Wings

Comparison of Experiment and Theory for Lift-Curve Slope of Un-swept Wings

Drag Coefficient

9

AERODYNAMICS

Continue from AERODYNAMICS – Part II

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10

SOLO Wings in Compressible Flow


The essential physical difference between Subsonic and Supersonic Flow is:

- Subsonic Flow: The disturbances of a sound point source propagates in all

directions.

- Supersonic Flow: The disturbance of a sound point propagates only within a

cone that lies downstream of the sound source. This so-called Mach-Cone has

the apex semi-angle μ

Supersonic

V > a

a t

V t

M

1sin 1

Sound

waves

Mach

waves

1

1tan

1sin

1/:

2

MM

aVM

11



Mach Cone

Wing

Leading Edge

Mach Cone

Wing

Leading Edge

Mach Cone

Wing

Leading Edge

Mach Cone

Wing

Leading Edge

If the Mach Line lies before

the Wing Edge, the component vn

of the incident Flow Velocity U∞

normal to the Wing Edge is

smaller than the Speed of Sound

a∞. Such a Wing Edge is called

Subsonic.

Conversely, if the Mach Line

lies behind the Wing Edge, the

component vn of the incident

Flow Velocity U∞ normal to the

Wing Edge is larger than the

Speed of Sound a∞. Such a Wing

Edge is called Supersonic.

Subsonic Edge vn<a∞ μ>γ m<1

Supersonic Edge vn>a∞ μ<γ m>1

12



Mach Line

Wing

Leading

Edge

Mach :Line

Wing

Trailing

Edge

Mach LineWing

Leading

Edge

Mach :Line

Wing

Trailing

Edge

Mach Line

Wing

Leading

Edge

Mach :Line

Wing

Trailing

Edge

Subsonic Leading EdgeSubsonic Trailing Edge

Subsonic Leading EdgeSupersonic Trailing Edge

Supersonic Leading EdgeSupersonic Trailing Edge

Subsonic Leading

Edge Flow

Subsonic Trailing

Edge Flow

Supersonic Leading

Edge Flow

Supersonic Trailing

Edge Flow

13



Mach Line

Wing

Leading Edge

Mach :Line

Influence

Range of A

Wing

Trailing Edge

Consider a point A’ (x,y,z) on a Wing in a

Supersonic Flow (V∞/a∞ > 1). The points

on the Wing that, by perturbing the Flow,

influence the Flow properties at A’ are

only downstream to A’, bounded by the

Wing Leading Edges and the Mach Lines

(ML) passing through A’ (see Figure).

Mach Line

Wing

Leading Edge

Mach :Line

Influence

Range of A

Wing

Trailing Edge

Subsonic Leading Edge

Supersonic Leading EdgeReturn to Table of Content

14


Theoretic Solutions for Pressure Distribution on a Finite Span Wing

in a Supersonic Flow (M∞ > 1)

We present here two solution methods for PDE equation:



1&1:0 2

2

2

2

2

2

2

MM

zyx

This method was proposed by Busemann in 1943 and was

extensively used before high speed computers were available.

A Conical Flow is defined by velocity, pressure , static

temperature, density constant along rays, through a common

vertex.

The Conical Flow can occur only at Supersonic Speeds.

Conical Flow are produced by passing over a conic body, but

It can be produced by small supersonic perturbations if the

Boundary Conditions satisfy the Conical Conditions.

In Supersonic Flow the disturbances are propagated only

downstream the Mach Cone.

Adolph

Busemann

(1901 – 1986).

This method is similar with that used in Incompressible Flow, but the

Singularities are Solutions of Supersonic Hyperbolic PDE.

Return to Table of Content

http://en.wikipedia.org/wiki/File:Adolf_Busemann_at_Langley.jpg


15


Theoretic Solution for Pressure Distribution on a Finite Span Wing



1&1:0 2

2

2

2

2

2

2

MM

zyx

Use for the Conical Flow the potential

Start with

x

z

x

y

fxzyx

:,:

,:,,

fff

x

2

222

2

2

2

2 1111

ff

x

f

x

ff

x

f

x

f

x

f

xx

Let compute

22

2

22

2

1,

1,

f

xz

f

z

f

xy

f

y

1&1:0/12/1 2

2

222

2

2

222

MM

fff

Mach Cone

16





Mach Cone

1&1:0/12/1 2

2

222

2

2

222

MM

fff

x

z

x

y

fxzyx

:,:

,:,,

Let compute

,,,,' fzyxx

u

The equation of a ray starting at the origin is given by 2121,, ccx

zc

x

yczyxr

We can see that for η = const., ζ = const., we have r (x,y,z) = const.

.,2

1,'.

,'2,

.,'

2 constCUpconstU

uC

constu

pp

.

,'

1

2,'

1

,','const

a

a

T

T

p

p

Isentropic Chain

17





Regions where Two-Dimensional Flow prevails

on Three-Dimensional Wings .

Shaded zones signify Two-Dimensional Flow.

Because in Supersonic Flows a perturbation is

felt only in the Mach cone downstream from the

source of disturbance, certain portion of the

Wings behave as though they were in the Two –

Dimensional Flow.

18



Inclined Rectangular Wing at

Supersonic Flow

(a) Planform

(b) Pressure disturbance at A-A Section

Conical Flow on Rectangular Wings

Propagations of Wing Edges (Leading and Side) on

the Supersonic Flow propagate over Mach Cones.

Looking at the Section A-E-A of the Wing, where E

is the intersection of Section A-A with the Mach

Line from the Wing Tip, we see that:

• Points on A-E (region II) are affected only by

the disturbances of the Wing Leading Edge. The

Flow is Conical and two dimensional on the

Wing, therefore the Pressure Coefficient is given

by22

1

4

2/

MU

ppcc plpp

• Points on E-A (region IV) are affected by

the disturbances of the Wing Leading Edge

and by the Side Edge. The Flow is Conical

and two dimensional on the Wing.

the Pressure Coefficient is given by

21

2121cos

1

4

Mx

ytt

Mcp

IIIV

A AE

EdgeLeadingt

EdgeSidet

Mx

ytt

c

c

plp

p

1

0

121cos 21

y

x

Area

Below Curve =

The mean value for is . 1,0t plpp cc 5.0

19



Mach ConeMach Cone

Wing

Leading Edge

Wing

Leading Edge

Wing

Leading Edge

Region II: AMNB

Region IV: ADM & BCN

Region II: ABE

Region IV: ADME & BCNE

Region V: MNE

Region II: ABE

Region IV: AME & BNE

Region V: MFNE


Propagations of Wing Edges (Leading

and Side) on the Supersonic Flow

propagate over Mach Cones.

Different Regions on the Wing are

affected by the Wing Edges.

Region II:

Flow over points on the Wing in

this region are affected only by

disturbances of Leading Edge.

Region IV:

Flow over points on the Wing in this

region are affected by disturbances of

both Leading Edge and one of Side

Edges.

Region V:

Flow over points on the Wing in this region are affected by disturbances of

Leading Edge and both Side Edges. The Flow is not Conical.

20




Aerodynamic Forces on Inclined Rectangular Wings of various Aspect Ratios at

Supersonic Incident Flow

(a) Lift Slope

(b) Neutral-point Position

(c) Drag Coefficient

21




Pressure Distribution over the Chord

and Lift Distribution over the Span for

the Inclined Rectangular Plate of

Aspect Ratio AR = 2.5 at Supersonic

Incident Flow

89.1;41 2 MMARa

13.1;3

41 2 MMARb

nMm

1:1tan

tan

tan 2

1

4

2/ 22

MU

ppc plp

tan

1:

:

x

yt

IRange

mtMx

yM

x

yt

IIIandIIRange

1tantan

1

tan

'tan1

:

22

10';sin11:'2/

0

22 EdmmE

Basic Solution for Pressure Distribution of the Inclined Flat Surface in Supersonic

Incident Flow (Cone-Symmetric Flow) for Ranges I, II, III and IV

SOLO Wings in Compressible FlowTheoretic Solution for Pressure Distribution on a Finite Span Wing





Conical Flow on Swept-Back Wings

Pressure Distribution over in the

Wing Chord (schematic) for a

section of an Inclined Swept-Back

Wing

(a) Subsonic Leading and

Trailing Edges.

(b) Subsonic Leading and

Supersonic Trailing Edge.

(c) Supersonic Leading and

Trailing Edges.




Pressure Distribution over in the

Wing Chord and Lift Distribution

over the Wing Span of Delta

Wings at Supersonic Incident

Flow

(a) Subsonic Leading Edge,

0 < m < 1.

(b) Supersonic Leading Edge,

m > 1.

nMm

1:1tan

tan

tan 2




Lift Distribution over the

Span of Delta Wings at

Supersonic Incident Flow for

several values of m:

• Subsonic Leading Edge,

0 < m < 1.

• Supersonic Leading Edge,

m > 1.

nMm

1:1tan

tan

tan 2


26

Linearized Flow Equations SOLO

Incompressible Flow (M∞ = 0)

Velocity Potential Equations:

A Particular Solution isR

Q

zyx

Q

44 222

That can be rewritten as

Q – Source Strength

Compressible Subsonic Flow (0 < M∞ < 1)

0

112

2

2

22

2

2

2

zMyMx

Potential Equation:

A Particular Solution is 2222 14 zyMx

Q


Q – Subsonic Compressible Source Strength

14/4/4/

222

Q

z

Q

y

Q

xSphere

1

4/1

14/

1

14/

2

2

2

2

2

Q

M

z

QM

y

Q

xEllipsoid of Revolution

02

2

2

2

2

2

zyx

Elliptic Second Order Linear

Partial Differential Equation.

Elliptic Second Order Linear

Partial Differential Equation.


27

Linearized Flow Equations SOLO

Compressible Supersonic Flow (M∞ >1)

1,0

112

2

2

22

2

2

2

i

zMiyMix

Velocity Potential Equation:

By analogy with the Subsonic Flow a Particular Solution is

2222 14 zyMx

Q


Q – Supersonic Compressible Source Strength

1

4/1

14/

1

14/

2

2

2

2

2

Q

M

z

QM

y

Q

x

Hyperboloid of Revolution

Only the part of the Flow lying downstream Mach Cone is physically significant.

Hyperbolic Second Order Linear Partial Differential Equation.


28


2. Singularity-Distribution Method for Supersonic Flow (M∞ >1)


1&1:0 2

2

2

2

2

2

2

MM

zyx

Flow is Linear even without the assumption of Small Disturbances. This allows to combine Elementary

Solutions similar to Subsonic Incompressible Flow (I.e. Source, Sink, Doublet, Vortex, etc.) to obtain

General Solution for Supersonic Flow. Those Elementary Solutions are spread on the Aerodynamic

Bodies in such a way that satisfy the Boundary Conditions.

Example of Supersonic Elementary Solutions are:

c

Sr

q

4 Source

Doublet

c

cV

r

vzq

4

Vortex

where

22

1

1

22

2/122

1

22

1

:

1:

:

zyy

xxv

M

zyyxxr

c

c

H. Lomax, M.A., Heaslet, F.B., Fuller, “Integrals and

Integral Equations in Linearized Wing Theory”,

Report 1054, NACA 1951zr

zq

c

D

3

2

4

29


2. Singularity-Distribution Method for Supersonic Flow (M∞ >1)

Four types of problems can be treated by the Singularity Distribution Method:

(a) Two Non-lifting Case (Symmetric Wing):

1. Given the Thickness Distribution and the Planform Shape, find the Pressure

Distribution on the Wing.

2. Given the Pressure Distribution on a Wing of Symmetrical Section, find the

Wing Shape (I.e. the Thickness Distribution and the Planform).

(b) Two Lifting Case (Non-Symmetric Wing):

4. A Lifting Surface, find the Pressure Distribution on it. In the Subsonic Case it is

necessary to satisfy the Kutta Condition at the Trailing Edge.

3. Given the Pressure Distribution on a Lifting Surface (Zero Thickness)

find the Slope of each point on the Surface.

Direct Problems: Cases 1 and 3, because they involve Integrals with known Integrands.

Indirect Problems: Cases 2 and 4, because the Unknown is inside the Integral Sign.

Cases 1 and 2 are more conveniently solved using Source or Doublet Distributions,

while Cases 3 and 4 are most often treated using Vortex Distributions.Return to Table of Content

30


Theoretical Solutions for Compressible Supersonic Flow (M∞ >1)

52&8.0012

2

2

2

2

22

MM

zyxM


1&1:0 2

2

2

2

2

2

2

MM

zyx

By analogy with the Subsonic Flow the influence of the

Point Source q located at (ξ’, η’, 0) is given by

2222''4

''0,',',,

zyx

ddqzyxd

The Point Source q must be such that whose boundary are defined by 2222'' zyx

This is a Mach Cone, with apex at (ξ’, η’, 0) and angle μ = cot-1β

1 2

10 2222''4

''0,',',,

zyx

ddqzyx

zx

zxy

zxy

1

222

2

222

1

/

/

0'' 2222 zyx

zw

yv

xu

zwUyvxuUu

',','

1'1'1'

Elementary Source

Of Strength q dξ dη

Elementary Source


Hyperbola (ξ, η) :

31

Elementary Source


Elementary Source


Hyperbola (ξ, η) :


Theoretical Solutions for Compressible Supersonic Flow (M∞ >1) (continue -1)

Let integrate for all Sources (ξ, η, 0) (on the Wing)

that are in the Front Mach Cone with the apex at

(x,y,z)

1 2

10 22224

0,,,,

zyx

ddqzyx

The boundary are defined by 222

2

222

11 /,/, zxyzxyzx

From this we can compute

1 2

10 2/32222

2

4

0,,,,,,

zyx

ddzq

z

zyxzyxw

We can see that w (x, y, z = 0) is zero everywhere, except at the source x = ξ, y = η where we have a

indeterminate value 0/0. This was solved by Puckett in his PhD Thesis at Caltech in 1946

For ϕ (x,y,z), integrate the second

integral by parts

222

1

2222

sin1

4

1

/4

,

zx

yvd

qud

zyxddvq

u

Note that

12

/

/

222

1 ,,8

1sin,

4

1

22212

22211

qqzx

yqvu

zxy

zxy

32


zxzx

dzx

yqddqq

0 222

1

012

2

1

sin4

1,,

8

1

1 2

10 22224

0,,,,

zyx

ddqzyx

we use LEIBNIZ THEOREM from CALCULUS:

)(

)(

)(

)(

),()),(()),((),(::

tb

ta

ChangeBoundariesthetodueChange

sb

sa

dxs

sxf

sd

sadssaf

sd

sbdssbfdxsxf

sd

dLEIBNITZ

To compute

zyxI

zx

zyxI

zx

dzx

yqd

zdqq

zzzyxw

,,

0 222

1

,,

012

2

2

1

1

sin4

1,,

8

1,,

zxzx

dqqz

yzxqdqqz

I

0

120

121 ,,8

1,

4

1,,

8

1

yzxyzx 222

12,11 /

zxzxy

zz

dqq

zx

zyzxqI

0 222

/

1

12

222

8

1,

4

1


33


We use again LEIBNIZ THEOREM from CALCULUS:

)(

)(

)(

)(

),()),(()),((),(::

tb

ta


sb

sa

dxs

sxf

sd

sadssaf

sd

sbdssbfdxsxf

sd

dLEIBNITZ

to compute

22

2

1

21

2

1

2

1

0 222

1

222

1

0 222

1

2

sin4

1sinlim

4

1

sin4

1

I

zx

I

zx

zx

dzx

yqd

zd

zx

yq

dzx

yqd

zI

024

1lim

24

1sinlim

4

1

max

/

12

max222

1

21

222

2

1

yyqq

dzx

yqI

finite

zxy

zx

finite

zx

Since we are interested in w (x,y, z=0) (the downwash in the Wing Plane)

x

zyxqd

qq

zx

zyxqzyxI

0

0

22201 ,

4

1lim

8

1,

4

10,,

12


34


We use again LEIBNIZ THEOREM from CALCULUS:

)(

)(

)(

)(

),()),(()),((),(::

tb

ta


sb

sa

dxs

sxf

sd

sadssaf

sd

sbdssbfdxsxf

sd

dLEIBNITZ

to compute

2

1

12

2

1

2222222

3

0

222

12

0222

12

0222

1

022

lim

sinlimlimsinlimlimsinlim:

dzyxzx

yzq

zx

yq

zzx

yq

zd

zx

yq

zIfrom

z

zzz

0sinlimlimlimsinlimlimlim

sinlimlimsinlimlim

2/

222

1

0

0

2220

2/

222

1

0

0

2220

/

222

12

0222

12

0

1

1

2

2

2222,1

12

zx

yq

zx

z

zx

yq

zx

z

zx

yq

zzx

yq

z

zzzz

zxy

zz

0lim

2

1 2222222

3

0

d

zyxzx

yzq

z

0limlimsinlim

4

1lim 22

021

00 222

1

02

0

2

1

IIdzx

yqd

zI

zz

zx

zz

Therefore:


35


Finally we obtained: yxqz

zyxw

z

,4

10,,

0

1 2

10 2/32222

2 ,,,,,

zyx

ddzU

z

zyxzyxw

Boundary Conditions:

U

xd

yxzdU

z

zyxzyxw S

CB

z

,,,0,,

..

0


where:

xd

yxzdyx S ,

:,

zx zxy

zxyzyx

ddUzyx

1222

12

222110

/

/ 2222

,,,

and:


SOLO Wings in Compressible FlowTheoretic Solution for Pressure Distribution on a Semi-Infinite Triangular Wing in a

Supersonic Flow and Subsonic Leading Edge (tanΛ > β)

Pressure Field for a Semi-Infinite Triangular Wing

with a Subsonic Leading Edge

Section aa

Mach ConeFrom P

Mach ConeFrom P

Consider the Point P (x,y,z=0) on a Single Wedge

Triangular Wing. The Mach Lines from P are PB

and PD.

The Parts of the Wing that influence the Flow at P

are located in the Area AEPBA.

CPB

AEPC

AEPBA

P

yx

dd

yx

ddU

yx

ddUzyx

222

222

2220,,

yxBPalong

ACBalong

:

tan:

tan,

tan

tan yxyxB

The Limits of Integrations are defined by the

points A, E, P, B, C. The Lines of Integrations are

37


CPB

yxyx

y

AEPC

yxy

P

yx

ddd

yx

ddd

Uzyx

tan 222

tan

tan 22200,,

yy

yxyxy

y

xd

U

y

xd

U

yx

ddd

U

0

1

0

1

0tan

1

tan 2220

tancosh1coshcosh

tan

0

1

0

1tan

tan

1

tan 222

tantan

cosh1coshcoshyxyx

y

yxyx

yx

y y

xd

U

y

xd

U

yx

ddd

U

Section aa

Mach ConeFrom P

Mach ConeFrom P

32

tan 1

0

1 tancosh

tancosh0,,

I

yx

y

I

y

Py

xd

y

xd

Uzyx

Let compute

x

zyxzyxu P

P

0,,0,,

tan

tancosh

tan

10,,0,,

21

22 yx

yxU

x

zyxzyxu P

P

We obtain

Theoretic Solution for Pressure Distribution on a Semi-Infinite Triangular Wing in a

Supersonic Flow and Subsonic Leading Edge (tanΛ > β) (continue – 1)

38


tn

tn

x

y

x

y

x

y

x

y

yx

yx

1tan1

tan

tantan

tan1

tan

tan

tan 2

2

2

tn

tn

n

U

x

zyxzyxuP

1cosh

1

10,,0,,

21

2

Therefore on the Wing ( t = 0 – Side Edge to t = 1 - Leading Edge)

11tan

tan 2

2

22

n

Define 1:1tan

:&tan: 2

Mnx

yt

y/x=t/tanΛ is the equation of a ray starting from Wing apex

t =0 (Side Edge), t = 1 (Leading Edge)

We found



tan

tancosh

tan

10,,0,,

21

22 yx

yxU

x

zyxzyxuP

Section aa

Mach ConeFrom P

Mach ConeFrom P

tn

tn

nU

zyxuCp

1cosh

1

120,,2

21

2

39


Let find how the disturbances of the Wing on the

Flow affect a point N (x,y,0) outside the Wing

between the Wing Side-Edge and the Mach Line

(see Figure). The Mach Line through N that

intersects The Wing between the points L and J

Determines the Wing area ALN that affects the Flow

at N.



Section aa

Mach ConeFrom N

Mach ConeFrom N

ALN

N

yx

ddUzyx

2220,,

yxNLJalong

AJalong

:

tan:

0,/

tan,

tan

tan

yxL

yxyxJ


points A, L,J. The Lines of Integrations are

tan

0

1

0

1tan

0tan

1

tan 222

tan

0

tancosh1coshcosh

yxyx yxyx

yx

Ny

xd

U

y

xd

U

yx

ddd

U

tan

0

1 tancosh0,,

yx

N dy

xUzyx

40



Flow affect a point N (x,y,0)



Section aa

Mach ConeFrom N

Mach ConeFrom N

tan

0

1 tancosh0,,

yx

N dy

xUzyx

tan

tancosh

tan

tan

tantan

tan

coshtan

1

tan

1

21

tan

0

22

22

22

2

1

22

tan

0 222

yx

yxU

yx

yx

U

yxd

U

xu

yx

yx

NN

tn

tn

x

y

x

y

x

y

x

y

yx

yx

1tan1

tan

tantan

tan1

tan

tan

tan 2

2

2

11

tantan 2

2

22

n

Define 1:1tan

:&tan: 2

Mnx

yt

y/x=t/tanΛ is the equation of a ray starting from Wing

apex , t = 0 (Side Edge), t =- n (Mach Line)

41



Section aa

Mach ConeFrom N

Mach ConeFrom N

Define 1:1tan

:&tan: 2

Mnx

yt


apex , t = 0 (Side Edge), t =- n (Mach Line)

tn

tn

n

U

x

zyxzyxuN

1cosh

1

10,,0,,

21

2

Therefore between t = 0 (Side Edge )

to t = -n (Mach Line)

tn

tn

nU

zyxuCp

1cosh

1

120,,2

21

2

42



Flow affect a point L (x,y,0) outside the Wing

between the Wing Leading-Edge and the Mach

Line(see Figure). The Mach Line through L that

intersects The Wing between the points J and G

Determines the Wing area AJG that affects the

Flow at L.



Section aa

Mach ConeFrom A

Mach ConeFrom A

AJG

L

yx

ddUzyx

2220,,

yxGJNalong

AJalong

:

tan:

0,

tan,

tan

tan

yxG

yxyxJ


points A, L,J. The Lines of Integrations are

tan

0

1

0

1tan

0tan

1

tan 222

tan

0

tancosh1coshcosh

yxyx yxyx

yx

y

xd

U

y

xd

U

yx

ddd

U

tan

0

1 tancosh0,,

yx

L dy

xUzyx

43



Section aa

Mach ConeFrom A

Mach ConeFrom A

tan

0

1 tancosh0,,

yx

L dy

xUzyx


Flow affect a point L (x,y,0)

xy

yxU

xy

yx

U

yxd

U

xu

yx

yx

NN

tan

tancosh

tan

tan

tantan

tan

coshtan

1

tan

1

21

tan

0

22

22

22

2

1

22

tan

0 222

Define 1:1tan

:&tan: 2

Mnx

yt


apex , t =1(Leading Edge) to t = n (Mach Line)

11tan

tan

tantan

1tan

tan

tan

tan 2

2

2

tn

tn

x

y

x

y

x

y

x

y

xy

yx

11

tantan 2

2

22

n

44



Section aa

Mach ConeFrom A

Mach ConeFrom A

Define 1:1tan

:&tan: 2

Mnx

yt


apex , t =1(Leading Edge) to t =+ n (Mach Line)

1cosh

1

10,,0,,

21

2 tn

tn

n

U

x

zyxzyxuL

Therefore between t = 1 (Leading Edge )

to t = +n (Mach Line)

1cosh

1

120,,2

21

2 tn

tn

nU

zyxuC L

pL

45



Pressure Field for a Semi-Infinite Triangular Wing with a Subsonic Leading Edge



Supersonic Flow and Supersonic Leading Edge (tanΛ < β)

Pressure Field for a Semi-Infinite Triangular

Wing with a Subsonic Leading Edge

Consider the Point L (x,y,z=0) on a Single Wedge

Triangular Wing. The Mach Lines from L are LB

and LC. B and C are on the Wing Leading Edge.

The Parts of the Wing that influence the Flow at L

are located in the Area LBC.

LBC

L

yx

ddUzyx

2220,,

tan:

:

:

CBalong

yxBLalong

yxCLalong

tan,

tan

tan

tan,

tan

tan

yxyxC

yxyxB


points C, L and B. The Lines of Integrations are

Mach Line

yxyx

y

yxy

yxL

yx

ddd

U

yx

ddd

U

tan 222

tan

tan 222tan

Section aa


Supersonic Flow and Supersonic Leading Edge (tanΛ < β) (continue – 1)

Pressure Field for a Semi-Infinite

Triangular Wing with a Subsonic Leading

Edge

Section aa


Triangular Wing.

y

x

y

x

y

x

y

d

yx

d

yx

yx yx

tancosh1coshcosh

1

1

0

1

tan

1

tan tan 2222

y

x

y

x

y

x

y

d

yx

d

yx

yx yx

tancosh1coshcosh

1

1

0

1

tan

1

tan tan 2222

tan

tan

1

tan

1

tan

1 tancosh

tancosh

tancosh

yx

yx

yx

y

y

yx

Ly

xU

y

xd

U

y

xd

U

yxyx

y

yxy

yxL

yx

ddd

U

yx

ddd

U

tan 222

tan

tan 222tan




Triangular Wing.

tan

tan

1 tancosh

yx

yx

LL d

y

x

x

U

xu

tan

tan

222

0

1

1

0

1

1

tan

tan

tantan

coshtan

1

tan

tantan

coshtan

1

yx

yx yx

dU

yxy

yxx

U

yxy

yxx

U

tan

tan

222tan

yx

yx

L

yx

dUu

tan

tan

1 tancosh

yx

yx

L dy

xUSection aa




Triangular Wing.

2

22

22

222

22

22

22222

tan

tan

tan

tantan

1

tan

tan

tan

tan

1

tan

yx

yx

yx

d

yx

d

tan

tan

22

22

222

1

22

tan

tan

tan

tantan

sintan

1

yx

yx

L yx

yx

Uu

2

1

1

1sin

uxd

udxu

xd

d

use

2/

1

22

22

222

1

22

2/

1

22

22

222

1

22

tan

tan

tan

tantan

tansin

tan

1

tan

tan

tan

tantan

tansin

tan

1

yx

yxyx

U

yx

yxyx

U

22 tan

UuL

tan

tan

222tan

yx

yx

L

yx

dUu

Section aa




Triangular Wing.

2

2221:,

tan:

1tan

Mn

n

UU

xu

L

L

Section aa


Supersonic Flow and Supersonic Leading Edge (tanΛ < β ) (continue – 5)



Section aa

Mach ConeFrom P

Mach ConeFrom P



and PD.


are located in the Area AEPBA.

AEDBPD

AEPBA

P

yx

dd

yx

ddU

yx

ddUzyx

222222

2220,,

yxDEPalong

DACBalong

:

tan:

0,

tan,

tan

tan

yxE

yxyxD


points A, E, P, B, C. The Lines of Integrations are

Mach Line

AED

yx

yx

BPD

P

yx

dddd

xUzyx

0

tan

tan22222 tan

0,,


Supersonic Flow and Supersonic Leading Edge (tanΛ < β) ) (continue – 6)



Edge

Section aa

Mach ConeFrom P

Mach ConeFrom P



and PD.

AED

yx

yx

BPD

P

yx

dddd

xUzyx

0

tan

tan22222 tan

0,,

y

x

y

x

y

x

y

d

yx

d

yx

yx yx

tancosh1coshcosh

1

1

0

1

tan

1

tan tan 2222

AED

yx

BPD

Py

xd

xUzyx

0

tan

1

22

tancosh

tan0,,

0

tan

222

0

1

1

22tan

tan

tantan

coshtan

1

tan

yx

PP

yx

d

yxy

yxx

U

xu





Edge

Section aa

Mach ConeFrom P

Mach ConeFrom P



and PD.

0

tan

22222tantan

yx

PP

yx

dU

xu

0

tan

22

22

222

1

22

tan

tan

tan

tantan

sintan

1

yx

L yx

yx

Uu

2/

1

22

22

222

12

1

22

tan

tan

tan

tantan

tansin

tan

tansin

tan

1

yx

yxyx

yx

yxU

tan

tansin

2tan

1 21

22 yx

yxUuL





Edge

Section aa

Mach ConeFrom P

Mach ConeFrom P



and PD.

tan

tansin

2tan

1 21

22 yx

yxUuL

tn

tn

x

y

x

y

x

y

x

y

yx

yx

1tan1

tan

tantan

tan1

tan

tan

tan 2

2

2

2

2

22 1tan

1tan n

Define 1:1tan

:&tan: 2

Mnx

yt


t = 0 (Side Edge), t = n (Leading Edge)

tn

tn

n

U

tn

tn

n

UuL

1cos

1

1

1sin

21

1 21

2

21

2





Edge

Section aa

Mach ConeFrom P

Mach ConeFrom N

Consider the Point N (x,y,z=0) between the Side

Edge of the Triangular and the Mach Lines from A

outside the Wing Planform. The f;ow disturbance on

N is due to Wing Surface AEC.

tan

tancos

tan

1 21

22 yx

yxU

xu N

N

Define 1:1tan

:&tan: 2

Mnx

yt


t -n (Mach Line), t = 0 (Side Edge)

tn

tn

n

U

tn

tn

n

UuL

1cos

1

1

1sin

21

1 21

2

21

2

ANC

N

yx

ddUzyx

2220,,

yxCEalong

ACalong

:

tan:

0,

tan,

tan

tan

yxE

yxyxC


points A, E, C. The Lines of Integrations are

tan

0 tan222

0,,

yx

yx

N

yx

dddd

Uzyx



Mach Line

Pressure Field for a Semi-Infinite Triangular Wing with a Supersonic Leading Edge


57


Section aa

Mach ConeFrom P

Mach ConeFrom P


Delta Wing. The Mach Lines from P are PB and PD.


are located in the Area ADPBA.

CPB

AEPC

ADE

ADPBA

yx

dd

yx

dd

yx

ddU

yx

ddUzyx

222

222

222

2220,,

The Limits of Integrations are defined by the points A, D, E, P, B, C. The Lines of Integrations are

yxDEPalong

yxBPalong

ACBalong

ADalong

:

:

tan:

tan:

tan,

tan

tan

tan,

tan

tan

yxyxD

yxyxB

Based on: A.E. Puckett, “Supersonic Wave Drag of Thin Airfoils”, 1949, Caltech PhD Thesis

http://thesis.library.caltech.edu/2697/1/Puckett_ae_1949.pdf


Flow and Subsonic Leading Edge (tanΛ > β)



58


Section aa

Mach ConeFrom P

Mach ConeFrom P


Flow and Subsonic Leading Edge (tanΛ > β) (continue – 1)

CPB

yxyx

y

AEPC

yxy

AED

yx

yx

yx

ddd

yx

ddd

yx

ddd

Uzyx

DE

AD

tan 222

tan

tan 2220

tan 222

0

tan

0,,

0

tan

1

0

10

tan tan

1

tan 222

0

tan

tancosh1coshcosh

yxyx

yxyx

yxy

xd

U

y

xd

U

yx

ddd

U

yy

yxyxy

y

xd

U

y

xd

U

yx

ddd

U

0

1

0

1

0tan

1

tan 2220

tancosh1coshcosh

tan

0

1

0

1tan

tan

1

tan 222

tantan

cosh1coshcoshyxyx

y

yxyx

yx

y y

xd

U

y

xd

U

yx

ddd

U

59


321

tan 1

0

10

tan

1 tancosh

tancosh

tancosh0,,

I

yx

y

I

y

I

yxy

xd

y

xd

y

xd

Uzyx

We want to compute

x

zyxzyxu

0,,0,,

We use LEIBNIZ THEOREM from CALCULUS:

)(

)(

)(

)(

),()),(()),((),(::

tb

ta


sb

sa

dxs

sxf

sd

sadssaf

sd

sbdssbfdxsxf

sd

dLEIBNITZ

yy

yxd

y

xd

xd

d

0 2220

1

tan

1tancosh

and 1

1cosh

2

1

uxd

udxu

xd

d

tan

222

0

1

1tan 1

tan

1

tan

tantan

coshtan

1tancosh

yx

y

yx

yyx

dyx

y

yxx

y

xd

xd

d

0

tan222

0

1

10

tan

1

tan

1

tan

tantan

coshtan

1tancosh

yxyx

yxd

yxy

yxx

y

xd

dx

d



60


2

22

2

22

22

2222

tan

tantan

tan

tantan

yxyxyx

1

tan

tan

tantan

tan

tan

tan

tan

tan

1

tan2

22

22

22

2

22

22

22222

yx

yx

yx

d

yx

d

0

tan

22

22

22

2

1

22

0

tan222

1

tan

tan

tantan

tan

coshtan

1

tan

1

yx

yxyx

yx

yxd

x

I

1

1cosh

2

1

uxd

udxu

xd

duse

0

tan222

1

tan

1

yx

yxd

x

I

EdgeLeadingSubsonictan



Start with

61


2

22

2

22

22

2222

tan

tantan

tan

tantan

yxyxyx

1

tan

tan

tantan

tan

tan

tan

tan

tan

1

tan2

22

22

22

2

22

22

22222

yx

yx

yx

d

yx

d

y

yxd

x

I

0 222

2

tan

1

y

y

yx

yx

yxd

x

I

0

22

22

22

2

1

220 222

2

tan

tan

tantan

tan

coshtan

1

tan

1

1

1cosh

2

1

uxd

udxu

xd

duse



62


2

22

2

22

22

2222

tan

tantan

tan

tantan

yxyxyx

1

tan

tan

tantan

tan

tan

tan

tan

tan

1

tan2

22

22

22

2

22

22

22222

yx

yx

yx

d

yx

d

tan

222

3

tan

1yx

yyx

dx

I

tan

22

22

22

2

1

22

tan

0 222

3

tan

tan

tantan

tan

coshtan

1

tan

1

yx

y

yx

yx

yx

yxd

x

I

1

1cosh

2

1

uxd

udxu

xd

duse



63


x

I

x

I

x

IU

x

zyxzyxu 3210,,

0,,

tan2221

0

2221

0

tan

2221

22

tan

tantancosh

tan

tantancosh

tan

tantancosh

tan

1

yx

y

y

yx

yx

yx

yx

yx

yx

yxU

tan

0

2221

0

tan

2221

22 tan

tantancosh

tan

tantancosh

tan

1yx

yx yx

yx

yx

yxU

tan

tancosh

tan

tantan

tan

cosh

tan

tantan

tan

coshtan

tancosh

tan

1

21

1

222

1

0

1

222

12

1

22

yx

yx

yx

yxyx

yx

yxyx

yx

yxU



64


tan

tancosh

tan

tancosh

tan

10,,0,,

21

21

22 yx

yx

yx

yxU

x

zyxzyxu

tn

tn

x

y

x

y

x

y

x

y

yx

yx

1tan1

tan

tantan

tan1

tan

tan

tan 2

2

2

tn

tn

x

y

x

y

x

y

x

y

yx

yx

1tan1

tan

tantan

tan1

tan

tan

tan 2

2

2

tn

tn

tn

tn

n

U

x

zyxzyxu

1cosh

1cosh

1

10,,0,,

21

21

2

Therefore

11tan

tan 2

2

22

n



Define 1:1tan

:&tan: 2

Mnx

yt


We found

65


tn

tn

tn

tn

n

U

x

zyxzyxu

1cosh

1cosh

1

10,,0,,

21

21

2

We want to prove that 2

221

21

21

1cosh2

1cosh

1cosh

t

tn

tn

tn

tn

tn

2

221

2 1cosh

1

120,,0,,

t

tn

n

U

x

zyxzyxu

tan

:&tan:,: nx

yt

xd

zd

S

Finally we obtain

tn

tn

tn

tn

1:cosh,

1:cosh

22

Define

Let compute2

sinh2

sinh2

cosh2

cosh2

cosh

tn

tnn

tn

tnn

tn

tnn

tn

tnn

12

12/1cosh

2sinh,

12

12/1cosh

2cosh

12

12/1cosh

2sinh,

12

12/1cosh

2cosh

2

22

2

22

2

22

112

1

12

1

2cosh

t

tn

t

tn

n

n

t

tn

n

n

q.e.d.



66


1&1

cosh1

120,,0,,

2

221

2

ttn

t

tn

n

U

x

zyxzyxu



Since the Pressure and Velocity are constant along

t = (y/x) tan Λ, i.e. along rays through the vertex of

the Delta Wing, the Solution is of Conical Flows.

For t = 1 we get the ray corresponding to the

Leading Edge. For t = n=tanΛ/β we get the ray

along the “Mach Line” from the vertex of the Delta

Wing.

1&

1cosh

1

140,,22

221

2

ttnt

tn

nU

zyxuCp

Pressure Coefficient

1:,tan

:&tan:,: 2

Mnx

yt

xd

zd

S

67Theoretical Solution for a Delta Wing

(a) Pressure Distribution for a Single-Wedge Delta Wing at α = 0 [From Puckett (1946)]


68

Theoretical Solution for a Delta Wing

(b) Thickness Drag of a Double-Wedge Delta Wing with a Supersonic Leading Edge

and a Supersonic Line of Maximum Thickness [From Puckett (1946)]


69

Theoretical Solution for a Delta Wing

(c) Thickness Drag of a Double-Wedge Delta Wing with a Supersonic Line of

Maximum Thickness [From Puckett (1946)]



70



Flow and Supersonic Leading Edge (tanΛ < β)

Consider first a Point P1 (x,y) on the Wing, lying between

the Wing Leading Edge and the Mach Line

(1 > t > n). This point have a Potential determined only by

the Sources lying in region (1) (Defined by Mach Lines P1A1

and P1C1 intersecting only the swept Trailing Edge OA1).

But this Potential must be the same as for an Infinite Sweep

(Λ) Wing, therefore is given by

2

21

21 1:,

tan:

1,

11

Mn

n

U

xPu

n

xUP

P

1

2

The Point P2 (x,y) on the Wing is lying behind the Mach

Lines from the Wing Tip. The Mach Line PA

intersects the Leading Edge OA and the Mach Line PC

intersects the Leading Edge OB. If only the Leading Edge

OA exists (no Leading Edge OB) than the Potential at P2

would be the Same as P1.

To consider the Leading Edge OB we must subtract the

disturbances in the area of region (2) OBC (no sources)

2

)2(2222

2 1:,tan

:,:,1

Mn

x

z

yx

ddU

n

xUP

S

2

71


ODB

y

CDB

y

y

yx

yx

ddd

U

yx

ddd

U

yx

ddUzyx

0 tan

tan 222tan 222

)2(222

2

2

1

0,,

tan1

:tan1

11

11

yxy

MLyyxx

OALEyx

tan2

:tan2

22

22

yxy

MLyyxx

OBLEyx

y

x

y

x

y

x

y

d

yx

dyx

yx yx tancosh1coshcosh

1

1

0

1

tan

1

tan tan 2222

y

x

y

x

y

x

y

x

y

d

yx

d tancosh

tancoshcosh

1

11

tan

tan

1tan

tan 2

tan

tan 222


Flow and Supersonic Leading Edge (tanΛ < β) (continue – 1)

2

72


)2(222

yx

ddU

0111

2

2

1

tancosh

tancosh

tancosh

y

y

yd

y

x

y

xUd

y

xU

01

01

21

tancosh

tancosh

yyd

y

xUd

y

xU

The u – velocity associated with this potential is given by

2

2

1

1

21

21

0

222

0

222

0

222

0

1

2

2120

222

0

1

1

111

01

01

tantan

tan

tancosh

tan

tancosh

tancosh

tancosh

I

y

I

y

yy

yy

yx

dU

yx

dU

yx

dU

yy

yx

x

yU

yx

dU

yy

yx

x

yU

dy

x

x

Ud

y

x

x

U

xu



2

73


0

2221

1 tany

yx

dUI

73

2

22

22

22

22

222tan

tan

tan

tan

tantan

yxyxyx

2

22

22

22

2

22

22

22222

tan

tan

tantan

tan

1

tan

tan

tan

tan

1

tan

yx

yx

yx

d

yx

d

0

tan

22

22

22

2

1

22

0

2221

1

1

tan

tan

tantan

tan

sintan

1

tan

yxy

y yx

yx

U

yx

dUI

2

1

1

1sin

uxd

udxu

xd

d

use

2/

1

21

22

21

22 tan

tantansin

tan

1

tan

tansin

tan

1

yx

yxyxU

yx

yxU



74


74

2

22

222

2

22

222

tan

tantan

tan

tantan

yxyxyx

2

22

22

222

22

22

22222

tan

tan

tan

tantan

1

tan

tan

tan

tan

1

tan

yx

yx

yx

d

yx

d

0

tan

22

22

222

1

22

0

2222

2

2

tan

tan

tan

tantan

sintan

1

tan

yxy

y yx

yx

U

yx

dUI

2

1

1

1sin

uxd

udxu

xd

d

use

0

2222

2 tany

yx

dUI

2/

1

21

22

21

22 tan

tantansin

tan

1

tan

tansin

tan

1

yx

yxyxU

yx

yxU



75


75



2

2

1

1

0

222

0

222tantan

I

y

I

yyx

dU

yx

dU

xu

21

22

21

2222

21

22 tan

1

2tan

tansin

tan

1

tan

1

2tan

tansin

tan

1

II

U

yx

yxUU

yx

yxU

Sx

z

yx

yx

yx

yxU:

tan

tansin

tan

tansin

tan

1 21

21

22

tn

tn

x

y

x

y

x

y

x

y

yx

yx

1tan1

tan

tantan

tan1

tan

tan

tan 2

2

2

tn

tn

x

y

x

y

x

y

x

y

yx

yx

1tan1

tan

tantan

tan1

tan

tan

tan 2

2

2

2

2

22 1tan

1tan n

Define 1:1tan

:&tan: 2

Mnx

yt


tn

tn

tn

tn

n

U

xu

1sin

1sin

1

1 21

21

2

76


76



tn

tn

tn

tn

n

U

xu

1sin

1sin

1

1 21

21

2

We want to prove that 2

221

21

21

1sin2

1sin

1sin

t

tn

tn

tn

tn

tn

2

221

2 1sin

1

12

t

tn

n

U

xu

2

221

222

1sin

21

1

1

1

1

2

t

tn

n

Uu

n

U

xPu

P

tn

tn

tn

tn

1:sin,

1:sin

22

Define

Let compute

2sin2

2cos

2sin2

2cos

2sin2

2sin

2cos

2sin

2cos

2sin

2cos

2sin

2cos

tn

tnn

tn

tnn

tn

tnn

tn

tnn

1

1sin1

2sin

2cos,

1

1sin1

2sin

2cos

12

1sin1

2sin

2cos,

1

1sin1

2sin

2cos

2

22

2

22

2

22

12

1

1

1

1

2sin2

t

tn

t

tn

n

n

t

tn

n

n

2

221

21

21

1sin2

1sin

1sin

t

tn

tn

tn

tn

tn

q.e.d.

77


77



2

221

22

221

222

1cos

1

12

1sin

21

12

1

1

2

t

tn

n

U

t

tn

n

Uu

n

U

xPu

P

1:,1tan

:&tan:,: 2

Mnx

yt

xd

zd

S

For an Un-swept Wing (Flat Surface) (Λ = 0)

we have t = 0 & n = 0

SPx

z

M

UU

xPu

122

2

2

221

22

221

2

22

1cos

1

14

1sin

21

1

12

2

t

tn

nt

tn

nU

PuPC p


Supersonic Flow


with a Supersonic Leading Edge



Mach Line

Summary

SOLO Wings in Compressible FlowTheoretic Solution for Pressure Distribution on a Delta Wing in a Supersonic Flow

Inclined Delta Wing with

Subsonic Leading Edge (0 < m < 1)

(a) Wing Planform (Triangular Wing)

(b) Pressure Distribution on a Section

Normal to the Flow Direction, m = 0.6.

22

21

2

1cos

1 tm

t

m

m

22

21

2

1cosh

1 tm

t

m

m

nMm /11tantan

tan: 2

mtMx

yM

x

yt 1tan

tan

11: 22

Inclined Delta Wing with

Supersonic Leading Edge ( m > 1)

(a) Wing Planform (Triangular Wing)

(b) Pressure Distribution on a Section

Normal to the Flow Direction, m = 1.5.

Summary

80


81

Delta wing vortices

Delta wing pressure distribution (suction effect at the tip)


http://tianyizf1.files.wordpress.com/2013/12/delta-wing-cp.png

http://tianyizf1.files.wordpress.com/2013/12/delta-wing-cp.png

82

(A)- Flow field in wing-tail plane, influence of angle of attack


83

(B)- Flow field in wing-tail plane, influence of

control deflection for pitch


84

(C)- Flow field in wing-tail plane, influence of

control deflection for roll



85


Arrowhead Wings with Double-Wedge Profile at Zero Incidence [after Puckett and Steward]

Nomenclature

86



Thickness Drag for e = 0

87



Thickness Drag for e = 0.5

88



Thickness Drag for b = 0.2Return to Table of Content

89


Arrowhead Wings with constant Chord, Biconvex Profile, and Subsonic Leading Edge [after Jones]

(a) Platform

(b) Pressure Distribution at various Spanwise Stations

(c) Thickness Drag Coefficient at various Spanwise Stations


90


Systematic Presentation of Wave Drag of Thin, Nonlifting Wings having Straight Leading and

Trailing Edges and the same dimensionless profile in all chordwise plane [after Lawrence]

(a) Nomenclature and Geometrical Relationships. Note that e is negative if C lies aft of B.

91



Trailing Edges and the same dimensionless profile in all chordwiseplane [after Lawrence]

(b), (c), (d), (e), (f) , (g) Wings with Double-Wedge Profiles

92





93





94





95





96





97


Systematic Presentation of Wave Drag of Thin, Nonlifting Wings having Straight Leanding and

Trailing Edges and the same dimensionless profile in all chordwise plane [after Lawrence]

(h) Wings with Biconvex Parabolic Arc Profile Return to Table of Content

98


λ – Taper Ratio, 12 M


ΛLE – Leading Edge Swept Angle

Wing Planform

S – Wing Area

2

12

12

bc

b

c

cc

bccS r

r

trtr

AR – Aspect Ratio

1

12

21

22

rr

c

b

bc

b

S

bAR

LE

bc tan

2

2/

0

2

3222/

0 2

222

2/

0

2

2/3

1

2/1

2/1

2

2/1

2/121

2/1

22

1b

rb

r

r

b

b

y

b

yy

b

cyd

b

y

b

yc

bcydyc

Sc

2

02/

112/

by

b

yc

b

ycccyc rrtr

1

1

3

212

3

111

1

21

621

22/1

2 222 rrr ccbbb

b

c

222

1

14

1

1

3

2

b

Scc r

99


λ – Taper Ratio,

12 M

ΛLE – Leading Edge Swept Angle

CNα – Slope Computation is done as follows:

1. Compute s = β/tan ΛLE.

If s<1 use the abscissa on the left side of the chart.

If s>1 use the right side of the chart with the

abscissa tanΛLE/β.

2. Pick the chart corresponding to the

Taper Ratio λ. If λ is between the values of

the given charts interpolation is needed.

3. Calculate AR tanΛLE for the given Airfoil.

This is the parameter in the charts. If λ is

between curves in the chart interpolation

is needed.

4. Read the corresponding value from the

ordinate; this value will correspond to

tanΛLE (CNα) if the left side of the chart is

used, and it will correspond to β(CNα)

if the right side of the charts is used.

5. Extract CNα by dividing the left ordinate

by tanΛLE , or by dividing the right ordinate

by β, as the case may be.

“USAF Stability and Control DATCOM Handbook” , Air Force

Flight Dynamics Lab. Wright-Patterson AFB, Ohio, 1965


100


λ = 0 – Taper Ratio

CNα – Slope of the Normal Force Coefficient, ΛLE – Leading Edge Swept Angle


101




λ = 1/5 – Taper Ratio

102





103





104





105




λ = 1 – Taper Ratio


106

Comparison of Experiment and Theory for Lift-Curve Slope of Un-swept Wings

(after Vincenti, 1950)


107Comparison of Experiment and Theory for Lift-Curve Slope of Swept Wings



108

Thickness plus Skin-Friction Drag as a function of Sweep Angle



109Thickness plus Skin-Friction Drag as a function of position of Maximum Thickness



110

Effect of radius of Subsonic Leading Edge on Pressure-Drag Ratio due to Lift



111

Effect of radius of Subsonic Leading Edge on Lift-to-Drag Ratio (after Vincenti, 1950)


112


Lift Slope of Swept-Back Wings (taper λ = 1) at Supersonic Incident Flow,

0 < m <1; Subsonic Leading Edge; Supersonic Leading Edge.

113


Drag Coefficient due to Lift versus Mach Number

for a Trapezoidal, a Swept-Back, and a Delta Wing

of Aspect Ratio Λ = 3.

Dashed curve: with suction force.

Solid curve: without suction force.

114


Drag Coefficient (Wave Drag) at Zero Lift for Delta Wing (Triangular Wing) versus

Mach Number.

Profile I: Double Wedge profile.

Profile II: Parabolic Profile,

0 < m < 1: Subsonic Leading Edge,

m > 1: Supersonic Leading EdgeReturn to Table of Content

115


Drag Coefficient (Wave Drag) at Zero Lift of Swept-Back Wings (tape λ = 1) at

Supersonic Incident Flow.

0 < m < 1: Subsonic Leading Edge,

m > 1: Supersonic Leading Edge

116


Lift Slope versus Mach Number for a

Trapezoidal, a Swept-Back, and a Delta Wing


Total Drag Coefficient (Wave Drag +

Friction Drag) versus Mach Number for a

Trapezoidal, a Swept-Back, and a Delta Wing


Double-Wedge profile t/c = 0.05, xt/c = 0.50

117

Lifting Properties of Three Planforms

(after Jones, 1946)


Induced Drag of Three Planforms

(after Jones, 1946)


118

Aircraft Flight ControlSOLO

119

centre stick ailerons

elevators

rudder


Generally, the primary cockpit flight controls are arranged as follows:

a control yoke (also known as a control column), centre stick or side-stick (the

latter two also colloquially known as a control or B joystick), governs the

aircraft's roll and pitch by moving the A ailerons (or activating wing warping

on some very early aircraft designs) when turned or deflected left and right,

and moves the C elevators when moved backwards or forwards

rudder pedals, or the earlier, pre-1919 "rudder bar", to control yaw, which move

the D rudder; left foot forward will move the rudder left for instance.

throttle controls to control engine speed or thrust for powered aircraft.

SOLO

//upload.wikimedia.org/wikipedia/commons/b/b8/ControlSurfaces.gif

//upload.wikimedia.org/wikipedia/commons/b/b8/ControlSurfaces.gif

120

Stick

Stick

Rudder

Pedals


An aircraft 'rolling', or

'banking', with its ailerons

Rudder Animation

SOLO

http://en.wikipedia.org/wiki/File:Aileron_roll.gif

http://en.wikipedia.org/wiki/File:Aileron_roll.gif

http://en.wikipedia.org/wiki/File:Aileron_yaw.gif

http://en.wikipedia.org/wiki/File:Aileron_yaw.gif

121

Stick

Stick

Rudder

Pedals


122

Stick

Stick

Rudder

Pedals


123


124

Differential ailerons


125


126


127


128


129


130


131

The effect of left rudder pressure Four common types of flaps

Leading edge high lift devices

The stabilator is a one-piece horizontal tail surface that

pivots up and down about a central hinge point.


SOLO

132

Flight Control


SOLO

133

Aerodynamics of Flight


SOLO

134


Specific Stabilizer/Tail Configurations

Tailplane

Fuselage mounted Cruciform T-tail Flying tailplane

The tailplane comprises the tail-mounted fixed horizontal stabiliser and movable elevator.

Besides its planform, it is characterised by:

• Number of tailplanes - from 0 (tailless or canard) to 3 (Roe triplane)

• Location of tailplane - mounted high, mid or low on the fuselage, fin or tail

booms.

• Fixed stabilizer and movable elevator surfaces, or a single combined stabilator or

(all) flying tail.[1] (General Dynamics F-111)

Some locations have been given special names:• Cruciform: mid-mounted on the fin (Hawker Sea Hawk, Sud Aviation Caravelle)

• T-tail: high-mounted on the fin (Gloster Javelin, Boeing 727)

Sud Aviation Caravelle

Gloster Javelin

http://en.wikipedia.org/wiki/File:Tail_fuselage_mounted.svg

http://en.wikipedia.org/wiki/File:Tail_fuselage_mounted.svg

http://en.wikipedia.org/wiki/File:Tail_cruciform.svg

http://en.wikipedia.org/wiki/File:Tail_cruciform.svg

http://en.wikipedia.org/wiki/File:Tail_T.svg

http://en.wikipedia.org/wiki/File:Tail_T.svg

http://en.wikipedia.org/wiki/File:Tail_plane_flying.svg

http://en.wikipedia.org/wiki/File:Tail_plane_flying.svg

http://en.wikipedia.org/wiki/Tailplane#cite_note-1

SOLO

135


Specific Stabilizer/Tail Configurations

Tailplane

Some locations have been given special names:

• V-tail: (sometimes called a Butterfly tail)

• Twin tail: specific type of vertical stabilizer arrangement found on the empennage of

some aircraft.

• Twin-boom tail: has two longitudinal booms fixed to the main wing on either side of

the center line.

The V-tail of a Belgian Air

Force Fouga Magisterde Havilland Vampire

T11, Twin-Boom TailA twin-tailed B-25 Mitchell

http://en.wikipedia.org/wiki/File:Fouga_magister.jpg

http://en.wikipedia.org/wiki/File:Fouga_magister.jpg

http://en.wikipedia.org/wiki/File:Vampire_t11_wz507_g-vtii_cotswoldairshow_2010_arp.jpg

http://en.wikipedia.org/wiki/File:Vampire_t11_wz507_g-vtii_cotswoldairshow_2010_arp.jpg

http://en.wikipedia.org/wiki/File:North_American_B-25C_Mitchell_(00910460_178).jpg

http://en.wikipedia.org/wiki/File:North_American_B-25C_Mitchell_(00910460_178).jpg

SOLO

136

Aircraft AvionicsAerodynamics of Flight


SOLO

137

Aircraft AvionicsAerodynamics of Flight

SOLO

138

Control Surfaces



139

I.H. Abbott, A.E. von Doenhoff

“Theory of Wing Section”, Dover,

1949, 1959

H.W.Liepmann, A. Roshko

“Elements of Gasdynamics”,

John Wiley & Sons, 1957

Jack Moran, “An Introduction to

Theoretical and Computational

Aerodynamics”

Barnes W. McComick, Jr.

“Aerodynamics of V/Stol Flight”,

Dover, 1967, 1999

H. Ashley, M. Landhal

“Aerodynamics of Wings

and Bodies”,

1965

Louis Melveille Milne-Thompson

“Theoretical Aerodynamics”,

Dover, 1988

E.L. Houghton, P.W. Carpenter

“Aerodynamics for Engineering

Students”, 5th Ed.

Butterworth-Heinemann, 2001

William Tyrrell Thomson

“Introduction to Space Dynamics”,

Dover

References

AERODYNAMICSSOLO

140

Holt Ashley

“Engineering Analysis of

Flight Vehicles”,

Addison-Wesley, 1974

J.J. Bertin, M.L. Smith

“Aerodynamics for Engineers”,

Prentice-Hall, 1979

R.L. Blisplinghoff, H. Ashley,

R.L. Halfman

“Aeroelasticity”,

Addison-Wesley, 1955

Barnes W. McCormick, Jr.

“Aerodynamics, Aeronautics,

And Flight Mechanics”,

W.Z. Stepniewski

“Rotary-Wing Aerodynamics”,

Dover, 1984

William F. Hughes

“Schaum’s Outline of

Fluid Dynamics”,

McGraw Hill, 1999

Theodore von Karman

“Aerodynamics: Selected

Topics in the Light of their

Historical Development”,

Prentice-Hall, 1979

L.J. Clancy

“Aerodynamics”,

John Wiley & Sons, 1975

References (continue – 1)

AERODYNAMICSSOLO

141

Frank G. Moore

“Approximate Methods

for Missile Aerodynamics”,

AIAA, 2000

Thomas J. Mueller, Ed.

“Fixed and Flapping Wing

Aerodynamics for Micro Air

Vehicle Applications”,

AIAA, 2002

Richard S. Shevell

“Fundamentals of Flight”,

Prentice Hall, 2nd Ed., 1988 Ascher H. Shapiro

“The Dynamics and Thermodynamics

of Compressible Fluid Flow”,

Wiley, 1953

Bernard Etkin, Lloyd Duff Reid

“Dynamics of Flight:

Stability and Control”,

Wiley 3d Ed., 1995

H. Schlichting, K. Gersten,

E. Kraus, K. Mayes

“Boundary Layer Theory”,

Springer Verlag, 1999


AERODYNAMICSSOLO

142

John D. Anderson

“Computational Fluid Dynamics”,

1995

John D. Anderson

“Fundamentals of Aeodynamics”,

2001

John D. Anderson

“Introduction to Flight”,

McGraw-Hill, 1978, 2004

John D. Anderson

“Introduction to Flight”,

1995

John D. Anderson

“A History of Aerodynamics”,

1995

John D. Anderson

“Modern Compressible Flow:

with Historical erspective”,

McGraw-Hill, 1982


AERODYNAMICSSOLO


February 11, 2015 143

SOLO

Technion

Israeli Institute of Technology

1964 – 1968 BSc EE

1968 – 1971 MSc EE

Israeli Air Force

1970 – 1974

RAFAEL

Israeli Armament Development Authority

1974 –2013

Stanford University

1983 – 1986 PhD AA

http://upload.wikimedia.org/wikipedia/en/a/ae/RAFAEL_logo.png

http://upload.wikimedia.org/wikipedia/en/a/ae/RAFAEL_logo.png

144

Ludwig Prandtl

(1875 – 1953)

University of Göttingen

Max Michael Munk

(1890—1986)[

also NACA

Theodor Meyer

(1882 - 1972

Adolph Busemann

(1901 – 1986)

also NACA &

Colorado U.

Theodore von Kármán

(1881 – 1963)

also USA

Hermann Schlichting

(1907-1982)Albert Betz

(1885 – 1968 ),

Jakob Ackeret

(1898–1981)

Irmgard Flügge-Lotz

(1903 - 1974)

also Stanford U.

Paul Richard Heinrich Blasius

(1883 – 1970)

//upload.wikimedia.org/wikipedia/commons/e/ea/Prandtl_portrait.jpg


http://en.wikipedia.org/wiki/University_of_G%C3%B6ttingen

http://en.wikipedia.org/wiki/File:Max_M._Munk_-_GPN-2000-001301.jpg

http://en.wikipedia.org/wiki/File:Max_M._Munk_-_GPN-2000-001301.jpg

http://en.wikipedia.org/wiki/File:Theodor-Meyer-at-Munsterlager.jpg

http://en.wikipedia.org/wiki/File:Theodor-Meyer-at-Munsterlager.jpg



//upload.wikimedia.org/wikipedia/commons/c/c7/Theodore_von_Karman_-_GPN-2000-001500.jpg

//upload.wikimedia.org/wikipedia/commons/c/c7/Theodore_von_Karman_-_GPN-2000-001500.jpg

http://en.wikipedia.org/wiki/File:Uni_gottingen_siegel.svg

http://en.wikipedia.org/wiki/File:Uni_gottingen_siegel.svg

http://en.wikipedia.org/wiki/File:Irmgard_Fl%C3%BCgge-Lotz.jpg

http://en.wikipedia.org/wiki/File:Irmgard_Fl%C3%BCgge-Lotz.jpg

145

Hermann Glauert

(1892-1934)Pierre-Henri Hugoniot

(1851 – 1887)

Gino Girolamo Fanno

(1888 – 1962)

Karl Gustaf Patrik

de Laval

(1845 - 1913)

Aurel Boleslav

Stodola

(1859 -1942)

Eastman Nixon Jacobs

(1902 –1987)

Michael Max Munk

(1890 – 1986)Sir Geoffrey Ingram

Taylor

(1886 – 1975)

ENRICO PISTOLESI

(1889 - 1968)Antonio Ferri

(1912 – 1975)

Osborne Reynolds

(1842 –1912)

//upload.wikimedia.org/wikipedia/commons/9/98/Gustaf_de_Laval_1875.jpg

//upload.wikimedia.org/wikipedia/commons/9/98/Gustaf_de_Laval_1875.jpg

//upload.wikimedia.org/wikipedia/commons/7/78/Auguste_Stodola.jpg

//upload.wikimedia.org/wikipedia/commons/7/78/Auguste_Stodola.jpg

//upload.wikimedia.org/wikipedia/en/f/f2/G_I_Taylor.jpg

//upload.wikimedia.org/wikipedia/en/f/f2/G_I_Taylor.jpg

//upload.wikimedia.org/wikipedia/commons/d/df/Osborne_Reynolds.jpg

//upload.wikimedia.org/wikipedia/commons/d/df/Osborne_Reynolds.jpg

146

Robert Thomas Jones

(1910–1999)

Gaetano Arturo Crocco

(1877 – 1968)Luigi Crocco

(1906-1986)

MAURICE MARIE

ALFRED COUETTE

(1858 -1943)

Hans Wolfgang Liepmann

(1914-2009)

Richard Edler

von Mises

(1883 – 1953)

Louis Melville

Milne-Thomson

(1891-1974)

William Frederick

Durand

(1858 – 1959)

Richard T. Whitcomb

(1921 – 2009)

Ascher H. Shapiro

(1916 — 2004)

http://en.wikipedia.org/wiki/File:Richard_von_Mises.jpeg

http://en.wikipedia.org/wiki/File:Richard_von_Mises.jpeg

http://en.wikipedia.org/wiki/File:William_Durand.jpg

http://en.wikipedia.org/wiki/File:William_Durand.jpg

http://www.google.com/url?sa=i&rct=j&q=richard+t.+whitcomb&source=images&cd=&cad=rja&docid=3rOUqTmflpLkFM&tbnid=UD22U1qvVZ3oXM:&ved=0CAUQjRw&url=http://www.nasa.gov/topics/aeronautics/features/whitcomb-hof_prt.htm&ei=VQ_JUt-RF4eU0QXygIH4Cg&bvm=bv.58187178,d.ZGU&psig=AFQjCNEdnft-Xn7-wyU9imWRMCM7OSuWhg&ust=1388994772543064

http://www.google.com/url?sa=i&rct=j&q=richard+t.+whitcomb&source=images&cd=&cad=rja&docid=3rOUqTmflpLkFM&tbnid=UD22U1qvVZ3oXM:&ved=0CAUQjRw&url=http://www.nasa.gov/topics/aeronautics/features/whitcomb-hof_prt.htm&ei=VQ_JUt-RF4eU0QXygIH4Cg&bvm=bv.58187178,d.ZGU&psig=AFQjCNEdnft-Xn7-wyU9imWRMCM7OSuWhg&ust=1388994772543064

http://www.google.com/url?sa=i&source=images&cd=&cad=rja&docid=5xwPGhXyFbHpRM&tbnid=4kB0r_rrMTRcMM:&ved=0CAgQjRw&url=http://web.mit.edu/newsoffice/2004/obit-shapiro.html&ei=FD7KUqZfpZ_sBqiggKgE&psig=AFQjCNFdwpBKmYP6DwjULswNVRCgcO_wtg&ust=1389072276131510

http://www.google.com/url?sa=i&source=images&cd=&cad=rja&docid=5xwPGhXyFbHpRM&tbnid=4kB0r_rrMTRcMM:&ved=0CAgQjRw&url=http://web.mit.edu/newsoffice/2004/obit-shapiro.html&ei=FD7KUqZfpZ_sBqiggKgE&psig=AFQjCNFdwpBKmYP6DwjULswNVRCgcO_wtg&ust=1389072276131510

147

John J. Bertin

(1928 – 2008)Barnes W. McCormick

(1926 - )

Antonio Filippone John D. Anderson, Jr. Holt Ashley

(1923 – 2006)

Milton Denman Van

Dyke

(1922 – 2010)

http://www.google.com/url?sa=i&source=images&cd=&cad=rja&docid=d9ia2b-E7N-_WM&tbnid=Fjj5GYr_QIEkXM:&ved=0CAgQjRw&url=http://www2.aero.psu.edu/barnes.html&ei=8RPIUorVKcee0wXV24HYDg&psig=AFQjCNFW4v_L-ANeBUhHiZLHRBFazAvSPg&ust=1388930417750202

http://www.google.com/url?sa=i&source=images&cd=&cad=rja&docid=d9ia2b-E7N-_WM&tbnid=Fjj5GYr_QIEkXM:&ved=0CAgQjRw&url=http://www2.aero.psu.edu/barnes.html&ei=8RPIUorVKcee0wXV24HYDg&psig=AFQjCNFW4v_L-ANeBUhHiZLHRBFazAvSPg&ust=1388930417750202

http://www.google.com/url?sa=i&source=images&cd=&cad=rja&docid=-fKtECHi3-mr6M&tbnid=LzmVSA8h1DVgZM:&ved=0CAgQjRw&url=http://www.mace.manchester.ac.uk/people/staff/academic-staff/profile/?staffId%3D102&ei=_BTIUoqfDeqW0AXr8ICgDw&psig=AFQjCNH0vFv2poqEXeaF2puVocvA2ulYRg&ust=1388930684271195

http://www.google.com/url?sa=i&source=images&cd=&cad=rja&docid=-fKtECHi3-mr6M&tbnid=LzmVSA8h1DVgZM:&ved=0CAgQjRw&url=http://www.mace.manchester.ac.uk/people/staff/academic-staff/profile/?staffId%3D102&ei=_BTIUoqfDeqW0AXr8ICgDw&psig=AFQjCNH0vFv2poqEXeaF2puVocvA2ulYRg&ust=1388930684271195

http://news.stanford.edu/news/2006/may24/gifs/ashley.jpg

http://news.stanford.edu/news/2006/may24/gifs/ashley.jpg

148

PERFECT GAS REAL GAS

FULL NAVIER-STOKES

OR “ZONAL APPROACH”

NAVIER-STOKES

POTENTIAL + B.L.

VISCOUS - INVISCID INTERACTION

EULER + B.L.

PANEL POTENTIAL (PANEL OR MARCHING)T.S.

EULER + B.L.

(REAL GAS)

POTENTIAL

EULER

N.S.

(REAL GAS)

MACH1 2 3 4 5

30

60

90

AOA

(deg)

APPLICABLE REGIONS OF DIFFERENT

COMPUTATIONAL METHODS

MISSILES

FIGHTER

AIRCRAFT

TRANSPORT

AIRCRAFT

153

Lockheed XFV1 Convair XFV1 Ryan X-13 Vertijet

//upload.wikimedia.org/wikipedia/commons/a/ac/Ryan_X-13.jpg

//upload.wikimedia.org/wikipedia/commons/a/ac/Ryan_X-13.jpg

159

NACA Airfoils

Profile geometry – 1: Zero lift line; 2: Leading edge; 3: Nose circle;

4: Camber; 5: Max. thickness; 6: Upper surface; 7: Trailing edge;

8: Camber mean-line; 9: Lower surface

Profile lines – 1: Chord, 2: Camber, 3: Length, 4: Midline

//upload.wikimedia.org/wikipedia/commons/8/84/Airfoil_geometry.svg

//upload.wikimedia.org/wikipedia/commons/8/84/Airfoil_geometry.svg

//upload.wikimedia.org/wikipedia/commons/1/1e/Airfoil_lines.png

//upload.wikimedia.org/wikipedia/commons/1/1e/Airfoil_lines.png

160

NACA Airfoils

Historical Overview of Airfoils Shapes

162

The Genealogical Tree of Aircraft

Pitot tubes are used on aircraft as a speedometer.

How does the Venturi Meter work?

2

22

2

11

1

2

2

1

21

222111

2

1

2

1:

,

:_

VpVpBernoulli

A

A

V

V

Thus

FlowibleIncompress

AVAV

11

2

2

2

1

121

1

12

2

2

2

12

1

12

2

1

2

22

1

12

2

2

2

1

_

:rate flow Compute

1

2

:Vfor Solve

21

21

2

1

2

1

AVrateFlow

A

A

ppV

pp

A

AV

pp

V

VV

ppVV

Giovanni Battista

Venturi

(1746 - 1822)

//upload.wikimedia.org/wikipedia/commons/2/20/Giovanni_Battista_Venturi.jpg

//upload.wikimedia.org/wikipedia/commons/2/20/Giovanni_Battista_Venturi.jpg

Characteristics of Cl vs.

Angle of Attack, in degrees

or radians

Cl

Slope= 2 if is in radians.

= 0

Angle of

zero lift

Stall

The angle of zero lift depends on

the camber of the airfoil

Angle of Attack, in degrees

or radians

Cl

= 0

Angle of

zero lift

Cambered airfoil

Symmetric Airfoil

Drag is caused by

• Skin Friction - the air molecules try to drag the airfoil with them. This effect is

due to viscosity.

• Form Drag - The flow separates near the trailing edge, due to the shape of the

body. This causes low pressures near the trailing edge compared to the leading

edge. The pressure forces push the airfoil back.

• Wave Drag: Shock waves form over the airfoil, converting momentum of the

flow into heat. The resulting rate of change of momentum causes drag.

Particles away

from the

airfoil move

unhindered.

Particles near the

airfoil stick to the

surface, and try to

slow down the

nearby particles.

A tug of war results - airfoil is dragged back with the flow.

Skin Friction

This region of low

speed flow is called

the boundary layer.

Laminar Flow

Streamlines move in an orderly fashion - layer by layer. The mixing between layers is due to

molecular motion. Laminar mixing takes place very slowly. Drag per unit area is proportional

to the slope of the velocity profile at the wall. In laminar flow, drag is small.

Airfoil Surface This slope

determines drag.

Airfoil Surface

Turbulent flow is highly unsteady, three-dimensional, and chaotic. It can still be viewed in a time-

averaged manner.

Turbulent Flow

• Laminar flows have a low drag.

• Turbulent flows have a high drag.

Achieving High Lift

One form of flaps, called Fowler

flaps increase the chord length as

the flap is deployed.

High energy air from the bottom side of the airfoil

flows through the gap to the upper side, energizes slow speed

molecules, and keeps the flow from stalling.

How do slats and flaps help?

1. They increase the camber as and when needed- during

take-off and landing.

Leading Edge Slats

Help avoid stall near the leading

edge

High Lift also Causes High Drag

177

Alexander Martin

Lippisch

(1894 – 1976)

Alexander Martin Lippisch (November 2,

1894 – February 11, 1976) was a German

pioneer of aerodynamics. He made important

contributions to the understanding of flying

wings, delta wings and the ground effect. His

most famous design is the Messerschmitt Me

163 rocket-powered interceptor.

GENERAL CHARACTERISTICS

Crew: 1

Length: 5.98 m (19 ft 7 in)

Wingspan: 9.33 m (30 ft 7 in)

Height: 2.75 m (9 ft 0 in)

Wing area: 18.5 m² (200 ft²)

Empty weight: 1,905 kg (4,200 lb)

Loaded weight: 3,950 kg (8,710 lb)

Max. takeoff weight: 4,310 kg (9,500 lb)

Powerplant: 1 × Walter HWK 109-509A-2 liquid-fuel

rocket, 17 kN (3,800 lbf)

http://www.google.co.il/url?sa=i&source=images&cd=&cad=rja&docid=oUxqwCK62QoxPM&tbnid=aeb4AyFBvJMNNM:&ved=0CAgQjRwwAA&url=http://militaryanalysis.blogspot.com/2010/10/x-113.html&ei=lqEQUtv9N4GshQffoIDwCA&psig=AFQjCNFdNdZ7PWLYh6Aq2sf0TL_wxrPlLw&ust=1376908055022542

http://www.google.co.il/url?sa=i&source=images&cd=&cad=rja&docid=oUxqwCK62QoxPM&tbnid=aeb4AyFBvJMNNM:&ved=0CAgQjRwwAA&url=http://militaryanalysis.blogspot.com/2010/10/x-113.html&ei=lqEQUtv9N4GshQffoIDwCA&psig=AFQjCNFdNdZ7PWLYh6Aq2sf0TL_wxrPlLw&ust=1376908055022542

http://en.wikipedia.org/wiki/File:Me163efJM.jpg

http://en.wikipedia.org/wiki/File:Me163efJM.jpg

http://en.wikipedia.org/wiki/File:Me-163.svg

http://en.wikipedia.org/wiki/File:Me-163.svg

181

CHORDWISE PRESSURE

DISTRIBUTION (DIFFERENTIAL

BETWEEN LOWER AND

UPPER SURFACE)

SPAN

CHORD

RELATIVE

AIRFLOW

AERODYNAMICS

182

Sir George Cayley is one of the most important people in the history of aeronautics. Many consider him the first true

scientific aerial investigator and the first person to understand the underlying principles and forces of flight. His

built his first aerial device in 1796, a model helicopter with contra-rotating propellers. Three years later, Cayley

inscribed a silver medallion (above) which clearly depicted the forces that apply in flight. On the other side of the

medallion Cayley sketched his design for a monoplane gliding machine

The Cayley Medallion, depicting (left) a Monoplane Glider

and (right) Lift and Drag - 1799

The following year Cayley discovered that dihedral (wings set lower at their center and higher at their outer ends)

improved lateral stability. He continued his research using models and by 1807 had come to understand that a

curved lifting surface would generate more lift than a flat surface of equal area. By 1810 Cayley had published his

now-classic three-part treatise "On Aerial Navigation" which stated that lift, propulsion and control were the three

requisite elelments to successful flight, apparently the first person to so realize and so state

The Cayley Model Monoplane Glider (reconstruction) - 1804

Sir George Cayley,

6th Baronet of Brompton

( 1773 – 1857)

George Cayley

183

Sir George Cayley,

6th Baronet of Brompton

(1773 – 1857)

Sir George Cayley, 6th Baronet of Brompton (27 December

1773 – 15 December 1857) was a prolific English engineer

and one of the most important people in the history of

aeronautics. Many consider him the first true scientific

aerial investigator and the first person to understand the

underlying principles and forces of flight.[

In 1799 he set forth the concept of the modern

aeroplane as a fixed-wing flying machine with

separate systems for lift, propulsion, and control.

He was a pioneer of aeronautical engineering

and is sometimes referred to as "the father of

aerodynamics." Designer of the first successful

glider to carry a human being aloft, he

discovered and identified the four aerodynamic

forces of flight: weight, lift, drag, and thrust,

which act on any flying vehicle. Modern

aeroplane design is based on those discoveries

including cambered wings

http://en.wikipedia.org/wiki/English_people

http://en.wikipedia.org/wiki/George_Cayley#cite_note-1

http://en.wikipedia.org/wiki/File:George_Cayley.jpg

http://en.wikipedia.org/wiki/File:George_Cayley.jpg

184

The Fifth Volta Congress, Roma,

October 6 1935

Gaetano Arturo Crocco

(1877 – 1968)

Theodore von Kármán

(1881 – 1963)

USA

Eastman Nixon Jacobs

(1902 –1987)

Subject: “High Velocities in Aviation”

Organized by General Arturo Crocco

Ludwig Prandtl

(1875 – 1953)

Adolph

Busemann

(1901 – 1986).

Prandtl – Compressible Flow General Introduction

and Survey Paper.

G.I. Taylor– Supersonic Conical Flow Theory

T. von Kármán – Minimum Wave Drag Shapes for

Axisymmetric Bodies

A. Busemann – Aerodynamic Forces at Supersonic

Speeds (Swept-Wing Concept)

E. Jacobs – New results for Compressibility Effects

obtained at Wind Tunnels at NACA

ENRICO PISTOLESI

(1889 - 1968)

E. Pistolesi – Derived again the

Prandtl-Glauert Relation

Sir Geoffrey Ingram

Taylor OM

(1886 – 1975)

http://www.google.com/url?sa=i&rct=j&q=&esrc=s&source=images&cd=&cad=rja&docid=Q5bSN5vblQKeeM&tbnid=7QvftQ0p7NXSQM:&ved=0CAUQjRw&url=http://www.aip.org/history/acap/biographies/bio.jsp?vonkarmant&ei=vS7IUpfoGoWe0QXc0YHQDg&bvm=bv.58187178,d.bGE&psig=AFQjCNEDDFmUhHtRsl7WB7UlAZuNCMmepA&ust=1388937266514512

http://www.google.com/url?sa=i&rct=j&q=&esrc=s&source=images&cd=&cad=rja&docid=Q5bSN5vblQKeeM&tbnid=7QvftQ0p7NXSQM:&ved=0CAUQjRw&url=http://www.aip.org/history/acap/biographies/bio.jsp?vonkarmant&ei=vS7IUpfoGoWe0QXc0YHQDg&bvm=bv.58187178,d.bGE&psig=AFQjCNEDDFmUhHtRsl7WB7UlAZuNCMmepA&ust=1388937266514512





http://en.wikipedia.org/wiki/Order_of_Merit

186The historical evolution of airfoil sections, 1908 1944. The last two shapes (N.A.C.A. 661 -212 and N.A.C.A. 74 7A315) are low-

drag sections designed to have laminar flow over 60 to 70 percent of chord on both the upper and the lower surface. Note that

the laminar flow sections are thickest near the center of their chords

187

ATR 72 propeller in flight

http://en.wikipedia.org/wiki/Propeller

http://www.princeton.edu/~stengel/AFDVirTex.html

http://en.wikipedia.org/wiki/File:Precision_air_ATR72_5423a.gif

http://en.wikipedia.org/wiki/File:Precision_air_ATR72_5423a.gif

188

Dutch roll is a type of aircraft motion, consisting of an out-of-phase combination of "tail-

wagging" and rocking from side to side. This yaw-roll coupling is one of the basic flight dynamic

modes (others include phugoid, short period, and spiral divergence). This motion is normally

well damped in most light aircraft, though some aircraft with well-damped Dutch roll modes can

experience a degradation in damping as airspeed decreases and altitude increases. Dutch roll

stability can be artificially increased by the installation of a yaw damper. Wings placed well above

the center of mass, sweepback (swept wings) and dihedral wings tend to increase the roll

restoring force, and therefore increase the Dutch roll tendencies; this is why high-winged

aircraft often are slightly anhedral, and transport-category swept-wing aircraft are equipped with

yaw dampers.

Scanned from U.S. Air Force flight manual

http://upload.wikimedia.org/wikipedia/commons/b/b2/DutchRollAnimGIF02.gif

http://upload.wikimedia.org/wikipedia/commons/b/b2/DutchRollAnimGIF02.gif

http://en.wikipedia.org/wiki/File:Dutch_Roll_Damping_Technique.jpg

http://en.wikipedia.org/wiki/File:Dutch_Roll_Damping_Technique.jpg

189Stengel, Aircraft Flight Dynamics, Princeton, MAE 331, Lecture 4

AERODYNAMICS


AERODYNAMICS

194

AERODYNAMICS

195

AERODYNAMICS

199

AERODYNAMICS

200

AERODYNAMICS

202

AERODYNAMICS

203

AERODYNAMICS

205

AERODYNAMICS

207

Ray Whitford, “Design for Air Combat”

aerodynamics part iii

Science

supersonic flow

subsonic flow

wing leading edge

semiinfinite triangular

compressible flow taper

single wedge triangular

mach lines

single wedge delta wing