aerodynamics part i

1

AERODYNAMICSPart I

SOLO HERMELIN

http://www.solohermelin.com

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2

Table of Content

AERODYNAMICS

Earth AtmosphereMathematical 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

Dimensionless Equations

Boundary Layer and Reynolds Number

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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

Flow Description

Streamlines, Streaklines, and Pathlines

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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

Wing Configurations

Wing Parameters

References

5


AERODYNAMICSSOLO

Linearized Flow Equations

Cylindrical Coordinates

Small Perturbation Flow

Applications: Nonsteady One-Dimensional Flow

Applications: Two Dimensional Flow

Shock & Expansion Waves

Shock Wave Definition

Normal Shock Wave

Oblique Shock Wave

Prandtl-Meyer Expansion WavesMovement of Shocks with Increasing Mach Number

Drag Variation with Mach Number

Swept Wings Drag Variation

Variation of Aerodynamic Efficiency with Mach Number

AERODYNAMICSPARTII

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AERODYNAMICSSOLO

Analytic Theory and CFD

Transonic Area Rule

Aircraft Flight Control

AERODYNAMICSPARTII

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7Wright Brothers First Flight

AERODYNAMICSSOLO

SOLO

Atmosphere

Continuum FlowLow-density and

Free-molecular Flow

Viscous Flow Inviscid Flow

Incompressible Flow

Compressible Flow

Subsonic Flow

Transonic Flow

Supersonic Flow

Hypersonic Flow

AERODYNAMICS

AERODYNAMICS

9

Percent composition of dry atmosphere, by volume

ppmv: parts per million by volume

Gas Volume

Nitrogen (N2) 78.084%

Oxygen (O2) 20.946%

Argon (Ar) 0.9340%

Carbon dioxide (CO2) 365 ppmv

Neon (Ne) 18.18 ppmv

Helium (He) 5.24 ppmv

Methane (CH4) 1.745 ppmv

Krypton (Kr) 1.14 ppmv

Hydrogen (H2) 0.55 ppmv

Not included in above dry atmosphere:

Water vapor (highly variable) typically 1%

Gas Volume

nitrous oxide 0.5 ppmv

xenon 0.09 ppmv

ozone 0.0 to 0.07 ppmv (0.0 to 0.02 ppmv in winter)

nitrogen dioxide 0.02 ppmv

iodine 0.01 ppmv

carbon monoxide trace

ammonia trace

•The mean molecular mass of air is 28.97 g/mol.

Minor components of air not listed above include:

Composition of Earth's atmosphere. The lower pie represents the trace gases which together compose 0.039% of the atmosphere. Values normalized for illustration. The numbers are from a variety of years (mainly 1987, with CO2 and methane from 2009) and do not represent any single source

Earth AtmosphereSOLO

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http://en.wikipedia.org/wiki/Image:Atmosphere_gas_proportions.gif

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smb://upload.wikimedia.org/wikipedia/commons/9/91/Atmosphere_composition_diagram.jpg

The Earth Atmosphere might be described as a Thermodynamic Medium in a Gravitational Field and in Hydrostatic Equilibrium set by Solar Radiation. Since Solar Radiation and Atmospheric Reradiation varies diurnally and annually and with latitude and longitude, the Standard Atmosphere is only an approximation.

SOLO

12

The purpose of the Standard Atmosphere has been defined by the World Metheorological Organization (WMO). The accepted standards are the COESA (Committee on Extension to the Standard Atmosphere) US Standard Atmosphere 1962, updated by US Standard Atmosphere 1976.

Earth Atmosphere

The basic variables representing the thermodynamics state of the gas are the Density, ρ, Temperature, T and Pressure, p.

SOLO

13

The Density, ρ, is defined as the mass, m, per unit volume, v, and has units of kg/m3.

v

mv ∆

∆=→∆ 0

limρ

The Temperature, T, with units in degrees Kelvin ( K). Is a measure of the average kinetic energy of gas particles.

The Pressure, p, exerted by a gas on a solid surface is defined as the rate of change of normal momentum of the gas particles striking per unit area.

It has units of N/m2. Other pressure units are millibar (mbar), Pascal (Pa), millimeter of mercury height (mHg)

S

fp n

S ∆∆=

→∆ 0lim

kPamNbar 100/101 25 ==

( ) mmHginHgkPamkNmbar 00.7609213.29/325.10125.1013 2 === The Atmospheric Pressure at Sea Level is:

Earth Atmosphere

14

Physical Foundations of Atmospheric Model

The Atmospheric Model contains the values of Density, Temperature and Pressure as function of Altitude.

Atmospheric Equilibrium (Barometric) Equation

In figure we see an atmospheric element under equilibrium under pressure and gravitational forces

( )[ ] 0=⋅+−+⋅⋅⋅− APdPPHdAg gρ

or ( ) gg HdHgPd ⋅⋅=− ρ

In addition, we assume the atmosphere to be a thermodynamic fluid. At altitude bellow 100 km we assume the Equation of an Ideal Gas

where V – is the volume of the gas N – is the number of moles in the volume V m – the mass of gas in the volume VR* - Universal gas constant

TRNVP ⋅⋅=⋅ *

V

m

M

mN == ρ&

MTRP /* ⋅⋅= ρ


( ) mmHginHgkPamkNmbar 00.7609213.29/325.10125.1013 2 ===


We must make a distinction between:- Kinetic Temperature (T): measures the molecular kinetic energy and for all practical purposes is identical to thermometer measurements at low altitudes. - Molecular Temperature (TM): assumes (not true) that the Molecular Weight at any altitude (M) remains constant and is given by sea-level value (M0)

SOLO

16

TM

MTM ⋅= 0

To simplify the computation let introduce:- Geopotential Altitude H- Geometric Altitude Hg

Newton Gravitational Law implies: ( )2

0

+

⋅=gE

Eg HR

RgHg

The Barometric Equation is ( ) gg HdHgPd ⋅⋅=− ρ

The Geopotential Equation is defined as HdgPd ⋅⋅=− 0ρ

This means thatg

gE

Eg Hd

HR

RHd

g

gHd ⋅

+

=⋅=2

0

Integrating we obtaing

gE

E HHR

RH ⋅

+

=

Earth Atmosphere

17

Atmospheric Constants

Definition Symbol Value Units

Sea-level pressure P0 1.013250 x 105 N/m2

Sea-level temperature T0 288.15 K

Sea-level density ρ0 1.225 kg/m3

Avogadro’s Number Na 6.0220978 x 1023 /kg-mole

Universal Gas Constant R* 8.31432 x 103 J/kg-mole - K

Gas constant (air) Ra=R*/M0 287.0 J/kg--K

Adiabatic polytropic constant γ 1.405

Sea-level molecular weight M0 28.96643

Sea-level gravity acceleration g0 9.80665 m/s2

Radius of Earth (Equator) Re 6.3781 x 106 m

Thermal Constant β 1.458 x 10-6 Kg/(m-s- K1/2)

Sutherland’s Constant S 110.4 K

Collision diameter σ 3.65 x 10-10 m


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Atmospheric Equilibrium Equation

HdgPd ⋅⋅=− 0ρAt altitude bellow 100 km we assume t6he Equation of an Ideal Gas

TRMTRP a

MRR

a

aa

⋅⋅=⋅⋅==

ρρ/

**

/

HdTR

g

P

Pd

a

⋅=− 0

Combining those two equations we obtain

Assume that T = T (H), i.e. function of Geopotential Altitude only. The Standard Model defines the variation of T with altitude based on experimental data. The 1976 Standard Model for altitudes between 0.0 to 86.0 km is divided in 7 layers. In each layer dT/d H = Lapse-rate is constant.


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Layer Index

GeopotentialAltitude Z,

km

GeometricAltitude Z;

km

MolecularTemperature T,

K

0 0.0 0.0 288.150

1 11.0 11.0102 216.650

2 20.0 20.0631 216.650

3 32.0 32.1619 228.650

4 47.0 47.3501 270.650

5 51.0 51.4125 270.650

6 71.0 71.8020 214.650

7 84.8420 86.0 186.946

1976 Standard Atmosphere : Seven-Layer Atmosphere

Lapse RateLh;

K/km

-6.3

0.0

+1.0

+2.8

0.0

-2.8

-2.0


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• Troposphere (0.0 km to 11.0 km). We have ρ (6.7 km)/ρ (0) = 1/e=0.3679, meaning that 63% of the atmosphere lies below an altitude of 6.7 km.

( ) HdHLTR

gHd

TR

g

P

Pd

aa

⋅⋅+

=⋅=−0

00

kmKLHLTT /3.60−=⋅+=

Integrating this equation we obtain

( )∫∫ ⋅⋅+

=−H

a

P

P

HdHLTR

g

P

PdS

S 0 0

0 1

0

( )0

00 lnln0

T

HLT

RL

g

P

P

aS

S ⋅+⋅⋅

−=

HenceaRL

g

SS HT

LPP

⋅−

⋅+⋅=

0

0

0

1

and

−

⋅=

⋅

10

0

0g

RL

S

S

a

P

P

L

TH


21


HdTR

g

P

Pd

Ta

⋅=− *0


( )T

TaS

S HHTR

g

P

P

T

−⋅⋅

−= *0ln

Hence( )T

Ta

T

HHTR

g

SS ePP−⋅

⋅−

⋅=*

0

andS

STTaT P

P

g

TRHH ln

0

*

⋅⋅+=

∫∫ =−H

HTa

P

P T

S

TS

HdTR

g

P

Pd*

0

• Stratosphere Region (HT=11.0 km to 20.0 km). Temperature T = 216.65 K = TT* is constant (isothermal layer), PST=22632 Pa


22


( )[ ] HdHHLTR

gHd

TR

g

P

Pd

SSTaa

⋅−⋅+⋅

=⋅=− *00

( ) ( ) PaPHPkmKLHHLTT SSSSSST 5474.9,/0.1* ===−⋅−=


( )[ ]∫∫ ⋅−⋅+

=−H

H SSTa

P

P S

S

SS

HdHHLTR

g

P

Pd*

0 1

( )[ ]*

*0 lnln

T

ST

aSSS

S

T

HHLT

RL

g

P

P −⋅+⋅⋅

=

Hence ( ) aRL

g

S

T

SSSS HH

T

LPP

⋅−

−⋅+⋅=

0

*1

and

−

⋅+=

⋅

10

* g

RL

SS

S

S

TS

aS

P

P

L

THH

Stratosphere Region (HS=20.0 km to 32.0 km).


23

1962 Standard Atmosphere from 86 km to 700 km

Layer Index GeometricAltitude

km

MolecularYemperature

,K

KineticTemperature

K

MolecularWeight

LapseRateK/km

7 86.0 186.946 186.946 28.9644 +1.6481

8 100.0 210.65 210.02 28.88 +5.0

9 110.0 260.65 257.00 28.56 +10.0

10 120.0 360.65 349.49 28.08 +20.0

11 150.0 960.65 892.79 26.92 +15.0

12 160.0 1110.65 1022.20 26.66 +10.0

13 170.0 1210.65 1103.40 26.49 +7.0

14 190.0 1350.65 1205.40 25.85 +5.0

15 230.0 1550.65 132230 24.70 +4.0

16 300.0 1830.65 1432.10 22.65 +3.3

17 400.0 2160.65 1487.40 19.94 +2.6

18 500.0 2420.65 1506.10 16.84 +1.7

19 600.0 2590.65 1506.10 16.84 +1.1

20 700.0 2700.65 1507.60 16.70


24

1976 Standard Atmosphere from 86 km to 1000 km

Geometric Altitude Range: from 86.0 km to 91.0 km (index 7 – 8)

78

/0.0

TT

kmKZd

Td

=

=


2/12

8

2

8

2/12

8

1

1

−

−−

−⋅−=

−−⋅+=

a

ZZ

a

ZZ

a

A

Zd

Td

a

ZZATT C

kma

KA

KTC

9429.19

3232.76

1902.263

−=−=

=

Geometric Altitude Range: from 110.0 km to 120.0 km (index 9 – 10)( )

kmKZd

Td

ZZLTT Z

/0.12

99

+=

−⋅+=


( ) ( )

( )

( )

+

+⋅−=

+

+⋅−⋅=

⋅−⋅−−=

∞

∞∞

ZR

ZRZZ

kmKZR

ZRTT

Zd

Td

TTTT

E

E

E

E

1010

1010

10

/

exp

ξ

λ

ξλ

KT

kmR

km

E

1000

10356766.6

/01875.03

=

×=

=

∞

λ


25

Sea Level Values

Pressure p0 = 101,325 N/m2

Density ρ0 = 1.225 kg/m3

Temperature = 288.15 K (15 C)Acceleration of gravity g0 = 9.807 m/sec2

Speed of Sound a0 = 340.294 m/sec


26


27

Winds Winds represents the relative motion of the Atmosphere

Earth Atmosphere

Although in the standard atmosphere the air is motionless with respect to the Earth, it is known that the air mass through which an airplane flies is constantly in a state of motion with respect to the surface of the Earth. Its motion is variable both in time and space and is exceedingly complex. The motion may be divided into two classes: (1) large-scale motions and (2) small-scale motions. Large-scale motions of the atmosphere (or winds) affect the navigation and the performance of an aircraft.

SOLO

Return to Table of Content

28

FLUID DYNAMICS

1. MATHEMATICAL NOTATIONS

VECTOR NOTATION CARTESIAN TENSOR NOTATION

1.1 VECTOR

1.2 SCALAR PRODUCT

1.3 VECTOR PRODUCT

u kk = 1 2 3, , u u e u e u e= + +1 1 2 2 3 3

u v u v u v u v⋅ = + +1 1 2 2 3 3 u v kk k = 1 2 3, ,

u v

u u

u u

u u

v

v

v

× =−

−−

0

0

0

3 2

3 1

2 1

1

2

3

=−+

±=−=

ji

permutjiodd

permutjieven

CevittaLevi

vu

ij

jiij

0

.,

.,1

ε

ε

SOLO

29

FLUID DYNAMICS

1. MATHEMATICAL NOTATIONS (CONTINUE)


1.5 ROTOR OF A VECTOR

1.4 DIVERGENCE OF A VECTOR

1.6 GRADIENT OF A SCALAR

∇ ⋅ = + +u

u

x

u

x

u

x

∂∂

∂∂

∂∂

1

1

2

2

3

3 i

i

x

u

∂∂

∇× = −

+ −

+ −

uu

x

u

xe

u

x

u

xe

u

x

u

xe

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

3

2

2

31

1

3

3

12

1

2

2

13

u u

uu u×∇× =∇

− ⋅∇

2

2

∂∂

∂∂

u

x

u

xi

k

k

i

−

i

kj

k

ii x

uu

x

uu

∂∂

∂∂

−

∇ = + +

=

φ∂ φ∂

∂ φ∂

∂ φ∂

∂ φ∂

∂ φ∂

∂ φ∂

xe

xe

xe

x x x

11

22

313

1 2 3

∂ φ∂ xk

SOLO

30

FLUID DYNAMICS



1.7 GRADIENT OF A VECTOR

∇ = ∇ + ∇ + ∇ u u e u e u e1 1 2 2 3 3

∇ =

u

u

x

u

x

u

x

u

x

u

x

u

x

u

x

u

x

u

x

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

1

1

1

2

1

3

2

1

2

2

2

3

3

1

3

2

3

3

∇ =

+ + +

+ + +

+ + +

u

u

x

u

x

u

x

u

x

u

x

u

x

u

x

u

x

u

x

u

x

u

x

u

x

u

x

u

x

u

x

u

x

u

x

u

x

Dik

1

2

1

1

1

1

1

2

2

1

1

3

3

1

2

1

1

2

2

2

2

2

2

3

3

1

3

1

1

3

3

2

2

3

3

3

3

3

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

+

ik

x

u

x

u

x

u

x

u

x

u

x

u

x

u

x

u

x

u

x

u

x

u

x

u

Ω

−−

−−

−−

+

0

0

0

2

1

3

2

2

3

3

1

1

3

1

3

3

2

2

1

1

2

1

3

3

1

1

2

2

1

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

u

xi

k

∂∂

∂∂

∂∂

∂∂

∂∂

u

x

u

x

u

x

u

x

u

xi

k

i

k

k

i

i

k

k

i

= +

+ −

1

2

1

2

Du

x

u

xiki

k

k

i

= +

∆ 1

2

∂∂

∂∂

Ω∆

iki

k

k

i

u

x

u

x= −

1

2

∂∂

∂∂

SOLO

31

FLUID DYNAMICS



1.8 GAUSS’ THEOREMS

d s

A

V

∇⋅A analytic in V

↓ = = A C C const vectorη .

( ) ∫∫ ∫∫∫∇=S V

dvsdGAUSS ηη 2 ∇η analytic in V ∫∫ ∫∫∫=

S k

k

V

dvs

ds∂

η∂η

SOLO

Johann Carl Friederich Gauss 1777-1855

( ) ∫∫ ∫∫∫ ⋅∇=⋅S V

dvAsdAGAUSS

1

∫∫ ∫∫∫=S k

k

kk

V

dvx

AdsA

∂∂

32

FLUID DYNAMICS



1.8 GAUSS’ THEOREMS (CONTINUE)

( ) ( ) ( )∫∫ ∫∫∫ ⋅∇=⋅S V

dvAsdAGAUSS

ηη3

( )= ⋅∇ + ∇⋅∫∫∫ A A dvη η

η∇⋅∇ ,A

analytic in V

( )η∂ η

∂A ds

A

xdv

Vk k

k

kS

= ∫∫∫∫∫

∫∫∫

+=

V k

k

kk x

A

xA

∂∂η

∂η∂

↓ = + + B e e eη η η1 1 2 2 3 3

( ) ( ) ( )[ ]∫∫ ∫∫∫ ⋅∇+∇⋅=⋅S V

dvABBAsdABGAUSS

4 B A ds AB

xB

A

xdv

Vi k k k

i

k

ik

kS

= +

∫∫∫∫∫

∂∂

∂∂

∇ ×A analytic in V( ) ∫∫ ∫∫∫ ×∇=×

S VdvAAsdGAUSS

5 ( )ds A ds A

A

x

A

xdv

Vi j j i

j

i

i

jS

− = −

∫∫∫∫∫

∂∂

∂∂

SOLO

33

FLUID DYNAMICS



1.9 STOCKES’ THEOREM

A d r A d s

C S

⋅ = ∇ × ⋅∫ ∫∫ ∇ ×A analytic on S

A d rA

x

A

xd si i

C

j

i

i

j

k

S∫ ∫∫= −

∂∂

∂∂

Gauss’ and Stokes’ Theorems are generalizations of theFundamental Theorem Of CALCULUS

( )A b A a

d A x

d xd x

a

b

( ) ( )− = ∫

George Stokes 1819-1903

SOLO

SOLO

Variational Principles of Hydrodynamics

Joseph-Louis Lagrange

1736-1813 Leonhard Euler

1707-1783 FIXED IN SPACE

(CONSTANT VOLUME)

EULER

LAGRANGE

MOVING WITH THE FLUID(CONSTANT MASS)

1e

3e

2e

u

The phenomena considered in Hydrodynamics are macroscopic and the atomic or molecular nature of the fluid is neglected. The fluid is regarded as a continuous medium. Any small volume element is always supposed to be so large that it still contains a large number of molecules.

There are two representations normally employed in the study of Hydrodynamics:

- Euler representation: The fluid passes through a Constant Volume Fixed in Space

- Lagrange representation: The fluid Mass is kept constant during its motion in Space.

Hydrodynamic Field

SOLO

Variational Principles of Hydrodynamics

Material Derivatives (M.D.)

Vector Notation Cartesian Tensor Notation

1e

2e

3e

r

u

b

rd( ) Frddtt

FtrFd

∇⋅+=

∂∂

,

( )d

dtF r t

F

t

dr

d tF

, = + ⋅∇

∂∂

( )d

dtF r t

F

tb F

b

, = + ⋅∇

∂∂

rdanyfor ( )d F r t

F

tdt d r

F

xi ki

ki

k

, = +∂∂

∂∂

( )d

d tF r t

F

t

d r

d t

F

xi ki k i

k

, = +∂∂

∂∂

( )d

d tF r t

F

tb

F

xb

i ki

ki

k

, = +∂∂

∂∂

vectoranybbtdrd

=

( ) Fut

FF

tD

DtrF

td

d

u

∇⋅+=≡

∂∂

,( )

k

ik

iki

u x

Fu

t

FF

tD

DtrF

td

d

∂∂

∂∂

+=≡,velocityfluiduu

td

rdIf

=

uuu

t

u

uut

uu

tD

D

×∇×−

∇+=

∇⋅+=

2

2

∂∂

∂∂

⋅−⋅−

+=

+=

k

ik

i

jj

ji

i

k

ik

ii

x

uu

x

uu

uxt

u

x

uu

t

uu

tD

D

∂∂

∂∂∂∂

∂∂

∂∂

∂∂

2

2

1

Acceleration Of The Fluid

1e

2e

3e

r

u duu +

dr

Material Derivatives = = Derivative Along A Fluid Path (Streamline) tD

D

Hydrodynamic Field

36

FLUID DYNAMICS



1.10 MATERIAL DERIVATIVES (CONTINUE)

d uu

tdt dr u

= + ⋅∇∂∂

d uu

tdt d x

u

xii

ki

k

= + ⋅∂∂

∂∂

rdrdDtdt

u

xd

xd

xd

x

u

x

u

x

u

x

u

x

u

x

u

x

u

x

u

x

u

t

u

t

u

t

u

ud

ud

ud

ikik

Ω++=

=

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂∂

∂∂∂

3

2

1

3

3

2

3

1

3

3

2

2

2

1

2

3

1

2

1

1

1

3

2

1

3

2

1 d u

u

td t

u

x

u

xd x

u

x

u

xd x

ii

Translation

i

k

k

i

Dilation

k

i

k

k

i

Rotation

k

= +

+ +

+ −

∂∂

∂∂

∂∂

∂∂

∂∂

1

2

1

2

( )

( )

( )

( )[ ] Dilationrduu

rdurdu

urdrdurdu

rdurdurdD

T

u

u

ik

⇒⋅∇+∇=

⋅∇+⋅∇=

∇⋅−⋅∇+⋅∇=

××∇−⋅∇=

2

12

1

2

12

1

2

12

1

( )Ω ik dr u dr Rotation = ∇ × × ⇒

1

2

SOLO

37

REYNOLDS’ TRANSPORT THEOREM

- any system of coordinatesOxyz

- any continuous and differentiable functions in

( ) ( )trtr OO ,,, ,,

ηχ( )tandrO,

( )trO ,,

ρ - flow density at point

and time tOr,

SOLO

- mass flow through the element .mdsdV S

=⋅− ,ρ sd

- any control volume, changing shape, bounded by a closed surface S(t)v (t)

- flow velocity, relative to O, at point and time t( )trV OOflow ,,,

Or,

- position and velocity, relative to O, of an element of surface, part of the control surface S(t).

OSOS Vr ,, ,

- area of the opening i, in the control surface S(t).iopenS

- gradient operator in O frame.O,∇

- flow relative to the opening i, in the control surface S(t).OSiOflowSi VVV ,,,

−=

- differential of any vector , in O frame.O

td

d ζ

ζ

FLUID DYNAMICS

38

Start with LEIBNIZ THEOREM from CALCULUS:( ) ( )

ChangeBoundariesthetodueChange

tb

ta

tb

ta td

tadttaf

td

tbdttbfdx

t

txfdxtxf

td

dLEIBNITZ

−+= ∫∫ )),(()),((

),(),(::

)(

)(

)(

)( ∂∂

and generalized it for a 3 dimensional vector space on a volume v(t) bounded by thesurface S(t).

Using LEIBNIZ THEOREM followed by GAUSS THEOREM (GAUSS 4):

( ) ( )( )

( )∫∫∫∫∫

⋅∇+∇⋅+=⋅+

→

=tv

OSOOOSGAUSS

OpotolativedsofMovement

thetodueChage

tSOS

tvO

LEIBNITZ

Otv

vdVVt

GAUSSsdVvd

tvd

td

d,,,,)4(

intRe

)(,

χχ∂χ∂χ

∂χ∂χ

This is REYNOLDS’ TRANSPORT THEOREM

OSBORNEREYNOLDS

1842-1912

SOLO

GOTTFRIED WILHELMvon LEIBNIZ

1646-1716

REYNOLDS’ TRANSPORT THEOREM

FLUID DYNAMICS


39

FLUID DYNAMICS


1.11 REYNOLDS’ TRANSPORT THEOREM (CONTINUE)


( )

∫∫∫

∫∫∫∫∫∫∫∫

⋅∇+∇⋅+=

⋅+=

)(,,,,)4(

,)()()(

tvOSOOOS

OGAUSS

OStStv

O

LEIBNITZ

Otv

vdVVt

GAUSS

sdVvdt

vdtd

d

χχ∂

χ∂

χ∂

χ∂χ

∫∫∫

∫∫∫∫∫∫∫∫

++=

+=

)(

,

,)4(

,)()()(

tv k

kOS

i

k

i

kOSi

GAUSS

kkOStS

itv

iLEIBNITZ

tvi

vdx

V

xV

t

GAUSS

sdVvdt

vdtd

d

∂∂

χ∂

χ∂∂

χ∂

χ∂

χ∂χ

SOLO

40

FLUID DYNAMICS




O

OOS td

RduV

== ,,

CASE 1 (CONTROL VOLUME vF ATTACHED TO THE FLUID)

kkOS uV =,

( )

∫∫∫

∫∫∫∫∫∫∫∫

⋅∇+∇⋅+=

⋅+=

)(,,,)4(

,)()()(

tvOOO

OGAUSS

OtStv

OOtv

F

FFF

vduut

GAUSS

sduvdt

vdtd

d

χχ∂

χ∂

χ∂

χ∂χ

∫∫∫

∫∫∫∫∫∫∫∫

++=

+=

)()4(

)()()(

tv k

kI

k

Ik

I

GAUSS

kKtS

Itv

I

tvI

F

FFF

vdx

u

xu

t

GAUSS

sduvdt

vdtd

d

∂∂χ

∂χ∂

∂χ∂

χ∂χ∂χ

SOLO

41

FLUID DYNAMICS




1&, == χkkOS uV1&, == χuV OS

CASE 2 (CONTROL VOLUME vF ATTACHED TO THE FLUID AND )1=χ

∫∫∫∫∫∫∫∫ ⋅∇=⋅==)(

,,)(

,)(

)(

tvOO

tSO

tv

F

FFF

vdusduvdtd

d

td

tvd ∫∫∫∫∫∫∫∫ ===)()()(

)(

tv k

kk

tSk

tv

F

FFF

dvx

udsudv

td

d

td

tvd

∂∂

=⋅∇

→ td

tvd

tvu F

Ftv

OOF

)(

)(

1lim

0)(,,

=

→ td

tvd

tvx

u F

Ftv

k

k

F

)(

)(

1lim

0)(∂∂

EULER 1755

SOLO

42

FLUID DYNAMICS




CASE 3 (CONTROL VOLUME vF ATTACHED TO THE FLUID AND )

ρχ == &, kkOS uVρχ == &, uV OS

ρχ =

or, since this is true for any attached volume vF(t)

( )∫∫∫

∫∫∫∫∫ ∫∫∫

⋅∇+=

⋅+===

)(,,

)(,

)( )(

)(0

tvOO

tSO

tv tv

F

FF F

vdut

sduvdt

vdtd

d

td

tmd

ρ∂

ρ∂

ρ∂

ρ∂ρ

( )∫∫∫

∫∫∫∫∫ ∫∫∫

+=

+===

)(

)()( )(

)(0

tvk

k

tSkk

tv tv

F

FF F

vduxt

sduvdt

dvtd

d

td

tmd

ρ∂

∂∂

ρ∂

ρ∂

ρ∂ρ

Because the Control Volume vF is attached to the fluid and they are not sources or sinks, the mass is constant.

( ) OOOOOO uut

ut ,,,,,,0

⋅∇+∇⋅+=⋅∇+= ρρ

∂ρ∂ρ

∂ρ∂ ( )

k

k

k

k

k x

u

xu

tu

xt ∂∂ρ

∂ρ∂

∂ρ∂ρ

∂∂

∂ρ∂ ++=+=

0

SOLO

43

FLUID DYNAMICS




CASE 4 (CONTROL VOLUME WITH FIXED SHAPE C.V. )0,

=OSV

Define

∫∫∫∫∫∫ =.... VC

OOVC

vdt

vdtd

d

∂χ∂χ

∫∫∫∫∫∫ =.... VC

i

VCi vd

tvd

td

d

∂χ∂χ

( ) ( ) ( ) χ ρ ηr t r t r t, , ,≡ ( ) ( ) ( )χ ρ ηi k k i kx t x t x t, , ,≡

( )∫∫

∫∫∫∫∫∫

⋅+

+=

)(,

)()(

tSOS

tvOO

tv

sdV

vdtt

vdtd

d

ηρ

∂ρ∂η

∂η∂ρηρ

ktS

kOSi

tvi

i

tvi

sdV

vdtt

vdtd

d

FF

∫∫

∫∫∫∫∫∫

+

+=

)(,

)()(

ηρ∂

ρ∂η∂η∂ρηρ

We have

but

( ) ( )OOOO ut

ut ,,,, 0

ρη∂

ρ∂ηρ∂

ρ∂ ⋅∇−=⇒=⋅∇+ ( ) ( )k

k

iik

k

uxt

uxt

ρ∂

∂η∂

ρ∂ηρ∂

∂∂

ρ∂ −=⇒=+ 0

CASE 5 ( ) ( ) ( ) χ ρ ηr t r t r t, , ,≡

SOLO

44

FLUID DYNAMICS




We have

( )

( )

( ) ( )[ ]

( )

( )[ ]∫∫∫∫∫=

∫∫

∫∫∫

∫∫

∫∫∫∫∫∫

⋅−+

⋅+

⋅∇+∇⋅−

∇⋅+=

⋅+

⋅∇−=

+

+)(

,,)(

4

.

)(,

)(,,,,,,

)(,

)(,,

)(

tSOOS

tvO

MDG

DerMatGAUSS

tSOS

tvOOOOOO

O

tSOS

tvOO

OOtv

sduVvdtD

D

sdV

vduuut

sdV

vdut

vdtd

d

ρηρη

ρη

ρηηρη∂

η∂ρ

ρη

ρηρ∂

η∂ρη ( )

( )

( ) ( )

( )

( )[ ]∫∫∫∫∫=

∫∫

∫∫∫

∫∫

∫∫∫∫∫∫

−+

+

+−

+=

+

−=

+

+)(

,)(

4

.

)(,

)(

)(,

)()(

tSkkkOSi

tv

iMDG

DerMatGAUSS

tSkkOSi

tv k

ki

k

ik

k

ik

i

tSkkOSi

tv k

ki

i

tvi

sduVvdtD

D

sdV

vdx

u

xu

xu

t

sdV

vdx

u

tvd

td

d

ρηρη

ρη

∂ρ∂η

∂η∂ρ

∂η∂

∂η∂ρ

ρη

∂ρ∂ηρ

∂η∂ρη

CASE 5 ( ) ( ) ( ) χ ρ ηr t r t r t, , ,≡

SOLO

45

FLUID DYNAMICS




REYNOLDS 1

( )[ ]

⋅−+= ∫∫∫∫∫

∫∫∫

)(,,

)(

)(

tSOOS

tvO

Otv

sduVvdtD

D

vdtd

d

ρηρη

ρη

( )[ ]

−+= ∫∫∫∫∫

∫∫∫

)(,

)(

)(

tSkkkOSi

tv

i

tvi

sduVvdtD

D

dvtd

d

ρηρη

ρη

REYNOLDS 2

( )[ ]

=

⋅−+

∫∫∫

∫∫∫∫∫

)(

)(,,

)(

tvO

tSOSO

Otv

vdtD

D

sdVuvdtd

d

ρη

ρηρη

( )[ ]

=

−+

∫∫∫

∫∫∫∫∫

)(

)(,

)(

tv

i

tSkkOSki

tvi

vdtD

D

sdVuvdtd

d

ρη

ρηρη

CASE 5 ( ) ( ) ( ) χ ρ ηr t r t r t, , ,≡

SOLO

46

FLUID DYNAMICS




REYNOLDS 3

CASE 1 (CONTROL VOLUME ATTACHED TO THE FLUID vF(t) )

kkOS uV =,

∫∫∫∫∫∫ =)()( tv

OOtv FF

vdtD

Dvd

td

d ρηρη

∫∫∫∫∫∫ =

)()( tv

i

tvi

FF

vdtD

Dvd

td

d ρηρη

SOLO

O

OOS td

RduV

== ,,

( ) ( ) ( ) χ ρ ηr t r t r t, , ,≡

CASE 4 (CONTROL VOLUME WITH FIXED SHAPE C.V. )0,

=OSV

REYNOLDS 4

( )

⋅+= ∫∫∫∫∫

∫∫∫

..,

..

..

SCO

OVC

VCO

sduvdtd

d

vdtD

D

ρηρη

ρη

( )

+= ∫∫∫∫∫

∫∫∫

....

..

SCkki

VCi

VC

i

sduvdtd

d

vdtD

D

ρηρη

ρη


47

FLUID DYNAMICS

2. BASIC LAWS IN FLUID DYNAMICS

THE FLUID DYNAMICS IS DESCRIBED BY THE FOLLOWING FIVE LAWS:

SOLO

(1) CONSERVATION OF MASS (C.M.)

(2) CONSERVATION OF LINEAR MOMENTUM (C.L.M.)

(3) CONSERVATION OF MOMENT OF MOMENTUM (C.M.M.)

(4) THE FIRST LAW OF THERMODYNAMICS

(5) THE SECOND LAW OF THERMODYNAMICSReturn to Table of Content

48

FLUID DYNAMICS2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE)

(2.1) CONSERVATION OF MASS (C.M.)

Control Volume attached to the fluid (containing a constant mass m) bounded by

the Control Surface SF (t).

( )tvF

( )tr ,

ρ ( )3/ mkgFlow density

SOLO

Because vF(t) is attached to the fluid and there are no sources or sinks in this volume,the Conservation of Mass requires that:

d m t

d t

( ) = 0

( ) ( )trVtru OfluidO ,, ,,

= Flow Velocity relative to a predefined

Coordinate System O (Inertial orNot-Inertial) ( )sm /

49


(2.1) CONSERVATION OF MASS (CONTINUE - 1)


d m t

d t

( ) = 0

( )∫∫∫=

∫∫∫∫∫ ∫∫∫

⋅∇+

⋅+===

)(,,

1

)(,

)( )(

)(0

tvOO

GAUSS

tSO

tv tv

REYNOLDS

F

FF F

vdut

sduvdt

dvtd

d

td

tmd

ρ∂

ρ∂

ρ∂

ρ∂ρ

( )∫∫∫=

∫∫∫∫∫ ∫∫∫

+

+===

)(

1

)()( )(

)(0

tvk

k

GAUSS

tSkk

tv tv

REYNOLDS

F

FF F

vduxt

sduvdt

dvtd

d

td

tmd

ρ∂

∂∂

ρ∂

ρ∂

ρ∂ρ

The Control Volume mass rate is zero as long as vF(t) is attached to the fluid and therefore contains the same amount of mass.

0),,,()(

=∫∫∫tvF

vdtzyxtd

d ρ is true in any Coordinate System (O) and so is:

( ) ( ) ( ) ( )( ) 0,,,,,,,,,

,,,)(

,,)(

=

⋅∇+= ∫∫∫∫∫∫

tvOO

tv FF

vdtzyxutzyxt

tzyxvdtzyx

td

d ρ

∂ρ∂ρ

SOLO

50




For any Control Volume v (t) (not necessarily attached to the fluid)

The following is true for any Coordinate System (for points that are not sources orsinks – mathematically equivalent to analytic ) ( )OO u

t ,,,ρ

∂ρ∂ ⋅∇

( ) OOOOOO uut

ut ,,,,,,0

⋅∇+∇⋅+=⋅∇+= ρρ∂

ρ∂ρ∂

ρ∂ ( )k

k

k

kO

k x

u

xu

tu

xt ∂∂ρ

∂ρ∂

∂ρ∂ρ

∂∂

∂ρ∂ ++=+= ,0

( )

( ) 0)(

,,

4

).(

,

).()(

≠=

⋅∇+

⋅+

∫∫∫=

∫∫∫∫∫=∫∫∫

mvdVt

sdVvdt

vdtd

d

tv

OSO

GAUSS

tS

OS

tv

LEIBNITZ

tv

ρ∂

ρ∂

ρ∂

ρ∂ρ

( ) 0)(

,

4

).(

,

).()(

≠=

+

+

∫∫∫=

∫∫∫∫∫=∫∫∫

mvdVxt

sdVvdt

vdtd

d

tvkOS

k

GAUSS

tS

kkOS

tv

LEIBNITZ

tv

ρ∂

∂∂

ρ∂

ρ∂

ρ∂ρ

The integral above is not zero because the mass in v (t) is not constant.

SOLO

51



Material Derivative of vdmd ρ=

Let use EULER’s 1755 expression ( ) ( )( ) ( )vd

tD

D

vdtd

tvd

tvu F

Ftv

OOF

11lim

0,, =

=⋅∇

→

and the (C.M.):

to develop the following:

( ) 0,, =⋅∇+ OO ut

ρ∂ρ∂

( ) ( )

( ) 0,,,,,,

,,,,

=

⋅∇+

∂∂=

⋅∇+∇⋅+

∂∂=

⋅∇+

∇⋅+

∂∂=+==

vdut

vduut

uvdvdut

vdtD

Dvd

tD

Dvd

tD

D

tD

mD

OOOOOO

OOOO

ρρρρρ

ρρρρρρ

SOLO

52

FLUID DYNAMICS

∑+=openings

iiopenW SCSCSC ....

2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE)

(2.1) CONSERVATION OF MASS (CONTINUE – 4)

SOLO

Control Volume with fixed shape C.V. and boundary C.S. in O Coordinates ( ) 0,

=OSV

There are no sources or sinks in the volume C.V. The change in the mass of the system is due only to the flow through the surface openings C.Sopen i (i=1,2,…). The surface C.S. can be divided in:

• C.Sw the impermeable wall through which the fluid can not escape .

=−= 0

0

,,,

OSOs VuV

• C.Sopen i the openings (i=1,2,…) through which the fluid enters or exits .( )0>m ( )0<m

∑∑ ∫∫∫∫∫∫∫∫∫ =⋅−⋅−=⋅−==openings

ii

openings

i

m

SCO

SCO

SCO

VC

msdusdusduvdtd

d

td

md

i

iopenw

.,

.0

,..

,..

ρρρρ

Therefore

where is the flow rate entering through the opening Sopen i.∫∫ ⋅−=iopenSC

Oi sdum.

,

ρ


53

FLUID DYNAMICS


(2.2) CONSERVATION OF LINEAR MOMENTUM (C.L.M.)

- Fluid density at he point and time t ( )tr ,ρ

r ( )3/ mKg

- Fluid inertial velocity at the point and time t

( )tru I ,,

r

( )sec/m

- Surface Stress ( )2/ mNT

- Pressure (force per unit surface) of the surrounding on the control surface ( )2/ mN

p

- Stress tensor (force per unit surface) of the surrounding on the control surface ( )2/ mN

σ~

- Body forces acceleration-(gravitation, electromagnetic,..)

G ( )2sec/m

nnpnT ˆ~ˆˆ~ ⋅+−=⋅= τσ

Consider a volume vF(t) attached to the fluid, bounded by the closed surface SF(t).

SOLO

- unit vector normal to the surface S(t) and pointing outside the volume v (t)n

vF (t)

m

SF (t)

O

x

y

z

r u,O

np ˆ−

n~ ⋅τ

n~ ⋅σdSn

- Shear stress tensor (force per unit surface) of the surrounding on the control surface ( )2/ mN

τ~

54

FLUID DYNAMICS


(2.2) CONSERVATION OF LINEAR MOMENTUM (CONTINUE - 1)

Derivation From Integral Form

The LINEAR MOMENTUM of the Constant Mass in vF(t) is given by:

∫∫∫=)(

,tv

I

F

vduP

ρ

The External Forces acting on the mass are Body and Surface Forces:

( )

ForcesSurface

tS

ForcesBody

tvexternal

FF

sdTvdGF ∫∫∫∫∫∑ +=)(

ρ

According to NEWTON’s Second Law, for a constant mass in vF(t), we have:

I

external td

PdF

=∑

SOLO

55

FLUID DYNAMICS




( )

I

IMomentumLinear

tvI

REYNOLDS

tvI

I

ForcesSurface

tS

ForcesBody

tvexternal

Ptd

dvdu

td

dvd

tD

uD

sdvdGF

FF

FF

===

⋅+=

∫∫∫∫∫∫

∫∫∫∫∫∑

)(,

3

)(

,

)()(

~

ρρ

σρ

itv

i

REYNOLDS

tv

i

tSkik

tviiex

Ptd

dvdu

dt

dvd

tD

uD

dsvdGF

FF

FF

===

+=

∫∫∫∫∫∫

∫∫∫∫∫∑

)(

3

)(

)()(_

ρρ

σρ

C.L.M.-1

T ds n ds dsds n ds

= ⋅ ⋅=

=~ ~σ σ T ds n ds dsi ik k

ds n ds

ik k

k k

==

=σ σ

C.L.M.-2

( )∫∫∫

∫∫∫∫∫∫∫∫

⋅∇+=

⋅+=

)(,

)()()(

,

~

~

tvI

tStvtvI

I

F

FFF

vdG

sdvdGvdtD

uD

σρ

σρρ

∫∫∫

∫∫∫∫∫∫∫∫

+=

+=

)(

)()()(

tv i

iki

tS

kik

tv

i

tv

i

F

FFF

vdx

G

sdvdGvdtD

uD

∂σ∂ρ

σρρ

SOLO

Derivation From Integral Form (Continue)

56

FLUID DYNAMICS



VECTOR NOTATION CARTESIAN TENSOR NOTATIONC.L.M.-2

Since this is true for all volumes vF (t) attached to the fluid we can drop the volume integral.

[ ] [ ] [ ]τστρσρ

∂∂ρ

∂∂ρρ

~~

~~

2

1

,,,

,

2

,

,

.).(

+−=⋅∇+∇−=⋅∇+=

×∇×−

∇+=

∇⋅+=

Ip

pGG

uuut

u

uut

u

tD

uD

III

II

I

I

I

DM

I

ikikik

i

ik

ii

i

iki

k

ik

i

jjjj

i

i

k

ik

iDM

i

p

xx

pG

xG

x

uu

x

uuuu

xt

u

x

uu

t

u

tD

uD

τδσ∂τ∂

∂∂ρ

∂σ∂ρ

∂∂

∂∂

∂∂

∂∂ρ

∂∂

∂∂ρρ

+−=

+−=+=

⋅−⋅−

+=

⋅+=

2

1

.).(

SOLO

Derivation From Integral Form (Continue)

( )∫∫∫

∫∫∫∫∫∫∫∫

⋅∇+=

⋅+=

)(,

)()()(

,

~

~

tvI

tStvtvI

I

F

FFF

vdG

sdvdGvdtD

uD

σρ

σρρ

∫∫∫

∫∫∫∫∫∫∫∫

+=

+=

)(

)()()(

tv i

iki

tS

kik

tv

i

tv

i

F

FFF

vdx

G

sdvdGvdtD

uD

∂σ∂ρ

σρρ

57

FLUID DYNAMICS



Derivation From a Cartesian Differential Volume


σ∂ σ

∂xxxx

xdx+ 1

2

σ ∂ σ∂xx

xx

xdx− 1

2

τ∂ τ

∂yxyx

ydy+ 1

2

τ∂ τ

∂yzyz

ydy− 1

2

τ∂τ

∂zxzx

zdz+ 1

2

τ∂τ

∂zxzx

zdz− 1

2

τ∂ τ

∂xyxy

xdx+ 1

2

τ∂ τ

∂xyxy

xdx− 1

2σ

∂ σ∂yy

yy

ydy+ 1

2σ

∂ σ∂yy

yy

ydy− 1

2

τ∂τ

∂zyzy

zdz+ 1

2

τ∂τ

∂zy

zy

zdz− 1

2

τ∂ τ

∂xzxz

xdx− 1

2

τ∂ τ

∂yz

yz

ydy+ 1

2

τ∂ τ

∂yx

yx

ydy−

1

2

σ∂ σ

∂zzzz

zdz+ 1

2

σ∂ σ

∂zzzz

zdz− 1

2

z

y

xd y

d x

d z

O

τ∂ τ

∂xzxz

xdx+ 1

2

∂σ∂

∂τ∂

∂τ∂

ρ ρ

∂τ∂

∂σ∂

∂τ∂

ρ ρ

∂τ∂

∂τ∂

∂σ∂

ρ ρ

xx yx zxxB x

xy yy zyyB y

xz yz zzzB z

x y zG a

x y zG a

x y zG a

+ + + =

+ + + =

+ + + =

CAUCHY’s First Law of Motion

ItD

uDa

aG

=

=+⋅∇ ρρσ~

tD

uDa

aGx

ii

iii

ij

=

=+ ρρ∂σ∂

SOLO

AUGUSTIN LOUIS CAUCHY

)1789-1857(

58

FLUID DYNAMICS


(2.2) CONSERVATION OF LINEAR MOMENTUM (CONTINUE-5)

Derivation For Any Control Volume v (t) (the velocity of an element of surface is )d s

ISV ,

V (t)

bds

V*(t)

I

T d s= ⋅~σ

G

m

u Use REYNOLDS’ Transport Theorem (REYNOLDS 2)with and O = I, and then the Conservation

of Linear Momentum (C.L.M.)Iu,

=η


( )[ ]( ) ( ) ( )

∑∫∫∫∫∫

∫∫∫∫∫∫

∫∫∫∫∫

=⋅+=

⋅∇+==

⋅−+

iexternaltStv

tvI

MLC

tvI

REYNOLDS

tSISII

Itv

I

FsdvdG

vdGvdtD

uD

sdVuuvdutd

d

FF

FF

FF

)()(

)(,

...

)(

2

)(,,,

)(,

~

~

σρ

σρρ

ρρ ( )( ) ( )

∑∫∫∫∫∫

∫∫∫∫∫∫

∫∫∫∫∫

=+=

+==

−+

iexternaltS

kiktv

i

tv k

iki

MLC

tv

iREYNOLDS

tSkkISki

tvi

FsdvdG

vdx

GvdtD

uD

sdVuuvdutd

d

)()(

)(

...

)(

2

)(,

)(

σρ

∂σ∂ρρ

ρρ

SOLO


59

( ) ( ) PdRRvdVRRHd OOO

×−=×−= ρ,


(2.3) CONSERVATION OF MOMENT-OF-MOMENTUM (C.M.M.)

SOLO

The Absolute Angular Momentum, of the differential mass and Inertial Velocity ,relative to a reference point O is defined as

vdmd ρ=V

The Absolute Angular Momentum of the mass enclosed by C.V. is defined as

( ) ( )∫∫∫∫∫∫ ×−=×−=....

,VC

OVC

OOCV PdRRvdVRRH

ρ

Let differentiate the Absolute Angular Momentum and use Reynolds’ Transport Theorem

( ) ( ) ( ) ( )∫∫∫∫∫∫∫∫ ⋅−×−+×−

=×−=..

,....

,

SCmd

SOVC

I

OREYNOLDS

IVC

O

I

OCV sdVVRRvdtD

VRRDvdVRR

td

d

td

Hd

ρρρ

We have ( ) ( )

( ) ( ) ( ) VVtD

VDRRVVV

tD

VDRR

VtD

RD

tD

RD

tD

VDRR

tD

VRRD

O

I

OO

I

O

I

O

II

O

I

O

×−×−=×−+×−=

×

−+×−=×−

FLUID DYNAMICS

60

( ) ( ) ( ) int, : fdRRfdRRvdtD

VDRRMd OextO

I

OO

×−+×−=×−= ρ

( ) ( ) ( ) ( )∫∫∫∫∫∫∫∫∫∫∫ ⋅−×−+×−×−=×−=..

,......

,

SCmd

SO

P

VCO

VCI

O

REYNOLDS

IVC

O

I

OCV sdVVRRvdVVvdtD

VDRRvdVRR

td

d

td

Hd

CV

ρρρρ



SOLO

The Moment, of the differential mass dm = ρdv, relative to a reference point O is defined as

Therefore

Let integrate this equation over the control volume C.V.

( ) ( ) ( )

0

..int

...., ∫∫∫∫∫∫∫∫∫∑ ×−+×−=×−=

VCO

VCextO

VCI

OOCV fdRRfdRRvdtD

VDRRM ρ

Using the differential of Angular Momentum equation we obtain

( ) ( ) ( )∫∫∫∫∫∑∫∫∫ ⋅−×−+×−=×−=..

,..

,..

,

SCmd

SO

P

VCOOCV

IVC

O

I

OCV sdVVRRvdVVMvdVRRtd

d

td

Hd

CV

ρρρ

( ) ( ) ( ) ( ) ( ) ∑∑∫∫∫∫∫∫∫∫∑ +×−++−×−+×−=×−==⋅

kk

jjOj

SCsdTsd

OVC

OVC

extOOtCV MFRRsdtfnpRRvdgRRfdRRM

......, 11

σ

ρ

Also

( )∑ ×−j

jOj FRR

- Moment, relative to O, of discrete forces exerting by the surrounding at point jR

- Discrete Moments exerting by the surrounding.∑k

kM

FLUID DYNAMICS

61

( )( ) ( )

∑∑∫∫∫∫∫∫∫∫ +×+×+×=×+⋅−×−×k

kj

jOtv

Otv

extO

P

VCO

SCmd

SO

IVC

O MFrfdrfdrvdVVsdVVrvdVrtd

d

CV

,

0

int,,....

,,..

, ρρρ



SOLO

Let find the equation of moment around the turbomachine axis. We shall use polar coordinates , where z is the turbomachine axis.

zr ,,θ

zzrrrOˆˆ

, +=

zVVrVV zrˆˆˆ ++= θθ

zFFrFF zrˆˆˆ ++= θθ

( ) zVrVrVzrVz

VVV

zr

zr

Vr zrz

zr

Oˆˆ0

ˆˆˆ

, θ

θ

θ+−+−==×

( ) ( )

∑∑∫∫∫∫ ++=×+⋅−−k

kzj

jtv

extCVOSC

SVC

MFrdfrPVsdVVrvdVrtd

dθθθθ ρρ

0

..,

..

The moment of momentum equation around the turbomachine z axis.

Example

FLUID DYNAMICS

62



SOLO

( )( ) ( ) ( ) ( )

( )

systemoutsidefromexertedTorque

M

llz

jj

tvext

AVVrAVVr

SCS

statesteady

VC

zSnSn

MFrdfrsdVVrvdVrtd

d ∑∑∫∫∫∫ ++=⋅−−

+−−→

θθ

ρρ

θθ

θθ

ρρ

22,21111,122

..,

0

..

We obtain

( ) ( )[ ] zflow MQVrVr =− 111122 ρθθ

or

( ) ( ) ( ) ( ) ( ) ( )[ ] ( ) zSnSnSn MAVVrVrAVVrAVVr =−=− 11,1112211,11122,222 ρρρ θθθθ

Euler Turbine Equation

ρ1 - mean fluid density one inlet (1) of area A1. where

ρ2 - mean fluid density one outlet (2) of area A2.

(Vθ )1, r1 - mean fluid tangential velocity and radius one inlet (1) of area A1.

(Vθ )2, r2 - mean fluid tangential velocity and radius one outlet (2) of area A2.

(V,Sn )1 - mean fluid normal velocity (relative to A1) one inlet (1) of area A1.

(V,Sn )2 - mean fluid normal velocity (relative to A2) one outlet (2) of area A2.

- mean flow rate one outlet (1) of area A1.( ) 11,1 : AVQ Snflow =

FLUID DYNAMICS


63

FLUID DYNAMICS


(2.4) CONSERVATION OF ENERGY (C.E.) – THE FIRST LAW OF THERMODYNAMICS (DIFFERENTIAL FORM)

- Fluid mean velocity [m/sec[( ) u r t,

- Body Forces Acceleration- (gravitation, electromagnetic,..)

G

- Surface Stress [N/m2[T

nnpnT ˆ~ˆˆ~ ⋅+−=⋅= τσ

mV(t)

G

q

T n= ⋅~σ

d E

d t

∂∂

Q

t

uu

d s n ds=- Internal Energy of Fluid molecules (vibration, rotation, translation per

mass [W/kg[

e

- Rate of Heat transferred to the Control Volume (chemical, external sources of heat) [W/m3[

∂∂

Q

t

- Rate of Work change done on fluid by the surrounding (rotating shaft, others) positive for a compressor, negative for a turbine) [W[td

Ed

SOLO

Consider a volume vF(t) attached to the fluid, bounded by the closed surface SF(t).

- Rate of Conduction and Radiation of Heat from the Control Surface (per unit surface) [W/m3[

q

64

FLUID DYNAMICS


(2.4) CONSERVATION OF ENERGY (C.E.) – THE FIRST LAW OF THERMODYNAMICS (CONTINUE - 1)

- The Internal Energy of the molecules of the fluid plus the Kinetic Energy of the mass moving relative to an Inertial System (I)

The FIRST LAW OF THERMODYNAMICS

CHANGE OF INTERNAL ENERGY + KINETIC ENERGY =CHANGE DUE TO HEAT + WORK + ENERGY SUPPLIED BY SUROUNDING

SOLO

The energy of the constant mass m in the volume vF(t) attached to the fluid, bounded by the closed surface SF(t) is

This energy will change due to

- The Work done by the surrounding

- Absorption of Heat

- Other forms of energy supplied to the mass (electromagnetic, chemical,…)

65

FLUID DYNAMICS



VECTOR NOTATION CARTESIAN TENSOR NOTATIONC.E.-1

systementeringtd

Qd

tSv

systemontnmenenvirobydonetd

Wd

shaft

tSv

v

REYNOLDS

KineticInternal

tv

FF

FF

FF

sdqvdt

Q

td

Wd

ForcesSurface

sdTu

ForcesBody

vdGu

vduetD

Dvdue

td

d

∫∫∫∫∫

∫∫∫∫∫

∫∫∫∫∫∫

⋅−+

+⋅+⋅=

+=

+

+

)(

)(

2)3(

)(

2

2

1

2

1

∂∂

ρ

ρρ

systementeringtd

Qd

tSkk

tv

systemontnemnoenvirbydonetd

Wd

shaft

tSkk

tvkk

tv

REYNOLDS

KineticInternal

tv

FF

FF

FF

dsqvdt

Q

td

Wd

ForcesSurface

sdTu

ForcesBody

vdGu

vduetD

Dvdue

td

d

∫∫∫∫∫

∫∫∫∫∫

∫∫∫∫∫∫

−+

++=

+=

+

+

)()(

)()(

)(

2)3(

)(

2

2

1

2

1

∂∂

ρ

ρρ

SOLO

66

FLUID DYNAMICS




( ) ( )

∫∫∫∫∫∫

∫∫∫∫∫∫∫∫∫

∫∫∫∫∫

∫∫∫∫∫∫∫

∫∫∫

⋅∇−+

⋅⋅∇+⋅∇−⋅=

⋅−+

⋅⋅+⋅−⋅=+

+

)()(

)()()(

)1(

)()(

)()()(

)(

2

~

~

2

1

tvtv

tvtvtv

GAUSS

td

Qd

tStv

td

Wd

tStStv

tv

FF

FFF

FF

FFF

F

vdqvdt

Q

vduvdupvdGu

sdqvdt

Q

sdusdupvdGu

KineticInternal

vduetD

D

∂∂

τρ

∂∂

τρ

ρ

( ) ( )

∫∫∫∫∫∫

∫∫∫∫∫∫∫∫∫=

∫∫∫∫∫

∫∫∫∫∫∫∫

∫∫∫

−+

+−

−+

+−=

+

+

)()(

)()()(

)1(

)()(

)()()(

)(

2

2

1

tV s

s

tV

tV

kk

iki

tV

kk

k

tV

kk

GAUSS

td

Qd

tS

kk

tV

td

Wd

tS

kiki

tS

kk

tV

kk

KineticInternal

tV

vdx

qvd

t

Q

dsx

uds

x

upvdGu

dsqvdt

Q

dsudsupvdGu

vduetD

D

∂∂

∂∂

∂τ∂

∂∂ρ

∂∂

τρ

ρ

T n pn n ds n ds= ⋅ = − + ⋅ =~ ~ &σ τ0=

td

Wd shaftassume and use

SOLO

67

FLUID DYNAMICS


(2.4) CONSERVATION OF ENERGY (C.E.) – THE FIRST LAW OF THERMODYNAMICS (CONTINUE-4)


Since the last equation is valid for each vF(t) we can drop the integral and obtain:

( ) ( )

qt

Q

uGuupuetD

D

⋅∇−+

⋅+⋅⋅∇+⋅−∇=

+

∂∂

ρτρ ~2

1 2 ( ) ( )

k

k

kk

k

iik

k

k

x

q

t

Q

uGx

u

x

upue

tD

D

∂∂

∂∂

ρ∂τ∂

∂∂ρ

−+

++−=

+ 2

2

1

Multiply (C.L.M.-2) byu

τρρ ~⋅∇⋅+∇⋅−⋅=⋅ upuuGtD

uDu

( )

k

iki

k

kkki

i xu

x

puuGu

tD

D

tD

uDu

∂τ∂

∂∂ρρρ +−== 2

Subtract this equation from (C.E.-3)C.E.-4

( )[ ]ρ τ τ

∂∂

D e

D tp u u u

Q

tq

= − ∇⋅ + ∇⋅ ⋅ − ⋅∇⋅

+ −∇⋅

~ ~

Φ

ρ∂∂ τ

∂∂

∂∂

∂∂

D e

D tp

u

xu

u

x

Q

t

q

x

k

kik

i

k

k

k

=− +

+ −

Φ

( )Φ ≡ ∇ ⋅ ⋅ − ⋅ ∇ ⋅ >~ ~τ τ u u 0

Φ ≡ >τ ∂∂ik

i

k

u

x0

(Proof of inequality given later)

SOLO

68

FLUID DYNAMICS




Enthalpy

Use this result and (C.E.-4)

C.E.-5

ρp

eh +=:

( )tD

pDup

tD

hDu

p

tD

pD

tD

hD

tD

Dp

tD

pD

tD

hD

tD

pD

tD

hD

tD

eD

−⋅∇−=⋅∇−+−=

+−=

−=

ρρ

ρρ

ρρρ

ρρρ

ρρρρ

2

tD

pD

x

up

tD

hD

x

up

tD

pD

tD

hD

tD

pDp

tD

hD

tD

pD

tD

pD

tD

hD

tD

eD

k

k

k

k −−=

−+−=

+−=

−=

∂∂ρ

∂∂ρ

ρρ

ρρ

ρρρ

ρρρρ

2

Φ++⋅∇−=t

Qq

tD

pD

tD

hD

∂∂ρ

Φ++−=t

Q

x

q

tD

pD

tD

hD

k

k

∂∂

∂∂ρ

SOLO

( )Φ ≡ ∇ ⋅ ⋅ − ⋅ ∇ ⋅ >~ ~τ τ u u 0

Φ ≡ >τ ∂∂ik

i

k

u

x0

69

FLUID DYNAMICS




Total Enthalpy

Use this result and (C.E.-3)

C.E.-6

22

2

1

2

1: u

peuhH ++=+=

ρ

( )t

pup

tD

HD

tD

pDup

tD

HD

p

tD

D

tD

HDue

tD

D

∂∂ρρ

ρρρρ

−⋅∇−=−⋅∇−=

−=

+

2

2

1

( )t

pup

xtD

HD

tD

pD

x

up

tD

HD

p

tD

D

tD

HDue

tD

D

kk

k

∂∂

∂∂ρ

∂∂ρ

ρρρρ

−−=−−=

−=

+

2

2

1

( ) qt

QuGu

t

p

tD

HD ⋅∇−+⋅+⋅⋅∇+=

∂∂ρτ

∂∂ρ ~ ( )

k

kkk

k

iik

x

q

t

QuG

x

u

t

p

tD

HD

∂∂

∂∂ρ

∂τ∂

∂∂ρ −+++=

SOLO


70

FLUID DYNAMICS

2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE)SOLO

THERMODYNAMIC PROCESSES

1. ADIABATIC PROCESSES

2. REVERSIBLE PROCESSES

3. ISENTROPIC PROCESSES

No Heat is added or taken away from the System

No dissipative phenomena (viscosity, thermal, conductivity, mass diffusion, friction, etc)

Both adiabatic and reversible

(2.5) THE SECOND LAW OF THERMODYNAMICS AND ENTROPY PRODUCTION

71

FLUID DYNAMICS


(2.5) THE SECOND LAW OF THERMODYNAMICS AND ENTROPY PRODUCTION

2nd LAW OF THERMODYNAMICS

Using GAUSS’ THEOREM

0)()(

≥+ ∫∫∫∫∫tStv FF

AdT

qvds

td

d

ρ

00)(

)1(

)()(

≥

⋅∇+⇒≥+ ∫∫∫∫∫∫∫∫

tv

GAUSS

tStv FFF

vdT

q

tD

sDAd

T

qvd

tD

sD

ρρ

- Change in Entropy per unit volumed s

- Local TemperatureT [ ]K

- Fluid Densityρ [ ]3/ mKg

d e q w T ds p dv= + = −δ δ d sd e

T

p

Tdv= +

SOLO

For a Reversible Process

- Rate of Conduction and Radiation of Heat from the System per unit surface

q

[ ]2/ mW

72

FLUID DYNAMICS


(2.5) THE SECOND LAW OF THERMODYNAMICS (CONTINUE - 1)

d e q w T ds p dv= + = −δ δ d sd e

T

p

Tdv= +

uT

p

tD

eD

Tu

T

p

tD

eD

T

tD

D

T

p

tD

eD

TtD

D

T

p

tD

eD

TtD

vD

T

p

tD

eD

TtD

sD

utD

DMC

v

⋅∇+=

⋅∇+=

−+=

+=+=

⋅∇−=

=

ρρρ

ρρ

ρρ

ρρρ

ρρρρρ

ρρ

ρ

2

.).(

2

1

1

11

The Energy Equation (C.E.-4) is

( )k

iik x

uoruu

t

Qqup

tD

eD

∂∂τττ

∂∂ρ =Φ⋅∇⋅−⋅⋅∇=ΦΦ++⋅∇−⋅∇−= ~~

Tt

Q

TT

qup

tD

eD

TtD

sD Φ++⋅∇−=

⋅∇+=

∂∂ρ 11

or

Φ++⋅−∇=t

Qq

tD

sDT

∂∂ρ

SOLO

73

FLUID DYNAMICS



Define

ρ∂∂

TD s

D tq

Q

t= −∇ ⋅ + + Φ

Θ ≡ + ∇ ⋅ ≥ρ

D s

Dt

q

T

0 Entropy Production Rate per unit volume

Therefore

( )Θ

ΦΘ= −

∇ ⋅+ + + ∇ ⋅

≥∫∫∫

q

T T

Q

t T

q

Tdv

V t

10

∂∂

&

SOLO

or

01 ≥Φ++⋅∇⋅−=Θ

nDissipatio

Systemtoadded

Heat

Systemfrom

RadiationHeat

t

QTq

TT

∂∂

74

FLUID DYNAMICS



q q qq conduction rate per unit surface

q radiation rate per unit surfacec r

c

r

= +

q K T K FOURIER s Conduction Lawc = − ∇ > 0 '

( )−∇ ⋅

+ ∇ ⋅

= −

∇ ⋅+ ∇ ⋅ + ⋅ ∇

= ⋅ ∇

= − ∇ + ⋅ ∇

= − ∇ ⋅ − ∇

+ ⋅ ∇

=

∇

+ ⋅ ∇

q

T

q

T

q

T Tq q

Tq

TK T q

T

K TT

T qT

KT

Tq

T

r

r r

1 1 1 1

1 1 12

2

ΘΦ

Φ=∇

+ + + ⋅ ∇

>>>

KT

T T T

Q

tq

T

K

Tr

2 1 10

0

0

∂∂

ΘΦ

≡ + ∇⋅

=

∇

+ + + ⋅∇

≥ρ

∂∂

D s

D t

q

TK

T

T T T

Q

tq

Tr

2 1 10

SOLO

JEAN FOURIER1768-1830

75

FLUID DYNAMICS



SOLO

Gibbs Function

Helmholtz Function

sThG ⋅−=:

sTeH ⋅−=:

Josiah Willard Gibbs (1839-1903)

Hermann Ludwig Ferdinandvon Helmholtz(1821 – 1894)

Using the Relations

vdpsdTed ⋅−⋅=

( ) pdvsdTvpdedhd ⋅+⋅=⋅+=vpep

eh ⋅+=+=ρ

:

pdvTdssdTTdshdGd ⋅+⋅−=⋅−⋅−=

vdpTdsTdssdTedHd ⋅−⋅−=⋅−⋅−=

dvT

p

T

edsd +=

76

FLUID DYNAMICS



SOLO

Maxwell’s Relations

vdpsdTed ⋅−⋅=

pdvsdThd ⋅+⋅=

pdvTdsGd ⋅+⋅−=

vdpTdsHd ⋅−⋅−=

Ts

pv

v

Fp

v

e

s

hT

s

e

∂∂=−=

∂∂

∂∂==

∂∂

vp

Ts

T

Fs

T

G

p

Gv

p

h

∂∂=−=

∂∂

∂∂==

∂∂

ps

vs

s

v

p

T

s

p

v

T

∂∂=

∂∂

∂∂−=

∂∂

vT

pT

T

p

v

s

T

v

p

s

∂∂=

∂∂

∂∂−=

∂∂

James Clerk Maxwell(1831-1879)


77

FLUID DYNAMICS


(2.6) CONSTITUTIVE RELATIONS FOR GASES

(2.6.1) NEWTONIAN FLUID DEFINITION – NAVIER–STOKES EQUATIONS

[ ] τσ ~~ +−= Ip

Stress

NEWTONIAN FLUID:

The Shear Stress onA Surface ParallelTo the Flow =Distance Rate ofChange of Velocity

SOLO

CARTESIAN TENSOR NOTATION

ikikik p τδσ +−=

VECTOR NOTATION

- Stress tensor (force per unit surface) of the surrounding on the control surface ( )2/ mN

σ~

- Shear stress tensor (force per unit surface) of the surrounding on the control surface ( )2/ mN

τ~

78

FLUID DYNAMICS


(2.6) CONSTITUTIVE RELATIONS

(2.6.1) NEWTONIAN FLUID DEFINITION – NAVIER–STOKES EQUATIONS

M. NAVIER 1822INCOMPRESSIBLE FLUIDS

(MOLECULAR MODEL)

G.G. STOKES 1845COMPRESSIBLE FLUIDS(MACROSCOPIC MODEL)


[ ] [ ] ( )[ ] [ ]IuuuIpIp T ∇+∇+∇+−=+−= λµτσ ~~ik

k

k

i

k

k

iikikikik x

u

x

u

x

upp δ

∂∂λ

∂∂

∂∂µδτδσ +

++−=+−=

( )[ ] [ ]( ) ( ) ( ) µλλµλµτ3

232~0 −=⇒∇+∇=∇+∇+∇== utrutrIutruutrtr T ( ) µλ

∂∂λµδ

∂∂λ

∂∂µτ

3

20322 −=⇒=+=+=

i

iik

k

k

i

iii x

u

x

u

x

u

SOLO

STOKES ASSUMPTION µλ3

2−=0~ =τtrace

μ, λ - Lamé parameters from Elasticity

79

FLUID DYNAMICS



(2.6.1) NAVIER–STOKES EQUATIONS (CONTINUE)

(2.6.1.2) VECTORIAL DERIVATION

I

x

y

zT n= ⋅~σ

d s n ds=

r

druu + du( )unrdtd

t

uurdtd

t

uud

∇⋅+=∇⋅+= 1∂∂

∂∂

( ) ( ) ( ) rdnurdnuuntdt

uud

RotationnTranslatio

1

2

11

2

11 ××∇+

××∇−∇⋅+=

∂∂

OR

DEFINITION OF NEWTONIAN FLUID, NAVIER-STOKES EQUATION

( ) ( ) nnunuunnpT

nTranslatio

1~11

2

1121 ⋅=⋅∇+

××∇−∇⋅+−≡ σλµ

CONSERVATION OF LINEAR MOMENTUM EQUATIONS

SOLO

80

FLUID DYNAMICS




(2.6.1.2)VECTORIAL DERIVATION (CONTINUE) I

x

y

zT n= ⋅~σ

d s n ds=

r

druu + du

CONSERVATION OF LINEAR MOMENTUM EQUATIONS

( ) ( )

( ) ( ) ( )

( ) ( )

( )( )

( )( )

( )∫∫∫

∫∫∫∫∫∫∫∫∫∫∫

∫∫∫∫∫∫∫∫∫∫∫∫∫∫∫

⋅∇∇+×∇×∇+∇⋅∇+∇−=

=⋅∇+×∇×+∇⋅+−=

⋅∇+

××∇−∇⋅+⋅−=+=

)(

)()()()()(

)()()()()()(

251

2

2

2

112

1121

tV

GAUSS

tStStStStV

tStStVtStVtV

vd

GAUSS

u

GAUSS

u

GAUSS

u

GAUSS

pG

usdusdusdsdpvdG

sdnunuunsdnpvdGdsTvdGvdtD

uD

λµµρ

λµµρ

λµρρρ

BUT

( ) ( ) ( )∇× ∇× ≡ ∇ ∇⋅ − ∇⋅ ∇2 2 2µ µ µ u u u

( ) ( ) ( ) ( )∇⋅ ∇ + ∇× ∇× = ∇ ∇⋅ − ∇× ∇×2 2µ µ µ µ u u u u

THEN

SOLO

81

FLUID DYNAMICS


I

x

y

zT n= ⋅~σ

d s n ds=

r

druu + du

THEREFORE

( ) ( ) ( ) ∫∫∫∫∫∫ ⋅∇∇+×∇×∇−⋅∇∇+∇−=)()(

2tVtV

vduuupGvdtD

uD

λµµρρ

OR

( ) ( )[ ]uupGtD

uD

⋅∇+∇+×∇×∇−∇−= µλµρρ 2

SOLO



(2.6.1.2)VECTORIAL DERIVATION (CONTINUE)

82

FLUID DYNAMICS



CONSERVATION OF LINEAR MOMENTUM

( ) ( )[ ]∇ ⋅ = − ∇ − ∇ × ∇ × + ∇ + ∇ ⋅~σ µ µ λp u u

2 ( )

++

++−=

k

k

ii

k

k

i

iii

ik

x

u

xx

u

x

u

xx

p

x ∂∂λµ

∂∂

∂∂

∂∂µ

∂∂

∂∂

∂σ∂

2

( ) ( )[ ]ρ ρ σ

ρ µ µ λ

D u

DtG

G p u u

= + ∇ ⋅

= − ∇ − ∇ × ∇ × + ∇ + ∇ ⋅

~

2 ( )

++

++−=

+=

k

k

ii

k

k

i

iii

i

iki

i

x

u

xx

u

x

u

xx

pG

xG

tD

uD

∂∂λµ

∂∂

∂∂

∂∂µ

∂∂

∂∂ρ

∂σ∂ρρ

2

USING STOKES ASSUMPTION tr ~τ λ µ= ⇒ = −02

3

( )

⋅∇∇+×∇×∇−∇−=

⋅∇+=

uupG

GtD

uD

µµρ

σρρ

3

4

~

+

++−=

+=

k

k

ki

k

k

i

iii

i

iki

i

x

ui

xx

u

x

u

xx

pG

xG

tD

uD

∂∂µ

∂∂

∂∂

∂∂µ

∂∂

∂∂ρ

∂σ∂ρρ

3

4

SOLO



83

FLUID DYNAMICS



Euler Equations are obtained by assuming Inviscid Flow

03

20~ =−=⇒= µλτ

pGtD

uD ∇−=

ρρi

ii

x

pG

tD

uD

∂∂ρρ −=

SOLO


(2.6.2) EULER EQUATIONS

Leonhard Euler (1707-1783)

pGuut

u ∇−=

∇⋅+

∂∂

ρρi

ik

ik

i

x

pG

x

uu

t

u

∂∂ρρ −=

∂∂+

∂∂

or or

http://www.google.com/url?sa=i&source=images&cd=&docid=FGHTb099JUGsMM&tbnid=s-2w-SrvyvHiCM:&ved=0CAgQjRw&url=http%3A%2F%2Fwww.usna.edu%2FUsers%2Fmath%2Fmeh%2Feuler.html&ei=nJ_LUpekIqyp7QaXwICYDw&psig=AFQjCNFEbjQB_j7mSz810OBBTvl_2f5SeA&ust=1389162780662237

84

FLUID DYNAMICS


(2.6.1.3) COMPUTATION

BUT

Φ

Φ = = +

= +

=

=τ ∂

∂τ ∂

∂τ ∂

∂τ ∂

∂∂∂

ττ τ

iki

kik

i

kki

k

iik

i

k

k

iik ik

u

x

u

x

u

x

u

x

u

xD

ik ki1

2

1

2

τ µ λ δik ik kk ikD D= +2

HENCE ( )Φ = = +τ µ λ δik ik ik kk ik ikD D D D2

OR( )[ ] ( )[ ]

( )[ ] ( )Φ = + + + + + + +

+ + + + + + + + + + ⇒=

2 2

2 2

11 11 22 33 11 22 11 22 33 22

33 11 22 33 33 122

212

132

312

232

322

µ λ µ λ

µ λ µ

D D D D D D D D D D

D D D D D D D D D D DD Dij ji

( ) ( )Φ= + + + + + + + +2 2 2 2112

222

332

122

132

232

11 22 33

2µ λD D D D D D D D DOR

SOLO



85

FLUID DYNAMICS


(2.6.1.3) COMPUTATION (CONTINUE)

USING STOKES ASSUMPTION: tr ~τ λ µ= ⇒ = −02

3

Φ

( ) ( )Φ= + + + + + + + +2 2 2 2112

222

332

122

132

232

11 22 33

2µ λD D D D D D D D D

( ) ( ) ( )( )

( )

( )

Φ = + + − + + + + +

+ + + − + +

+ +

2

3

4

3

4

3

42

3

11 22 33

2

11 22 11 33 22 33 112

222

332

2

122

132

232

11 22 33

2

112

222

332

µ µ µ

µµ

λ

µ

D D D D D D D D D D D D

D D D D D D

D D D

OR

( ) ( ) ( )[ ] ( )Φ = − + − + − + + + >2

34 011 22

2

11 33

2

22 33

2

122

132

232µ µD D D D D D D D D

SOLO



86

FLUID DYNAMICS


(2.6.1.4) ENTROPY AND VORTICITY

From (C.L.M.)

or

( ) ( )[ ]D u

D t

u

t

uu u G p u u

= + ∇

− × ∇ × = − ∇ − ∇ × ∇ × + ∇ + ∇ ⋅

∂∂ ρ ρ

µρ

λ µ2

2

1 1 12

GIBBS EQUATION: T d s d hd p

= −ρ

∀

+⋅∇−

+⋅∇=

+⋅∇

→→→→tld

pd

tdt

pldp

hd

tdt

hldh

sd

tdt

sldsT &

1

∂∂

ρ∂∂

∂∂

Since this is true for all d l t→

&

T s hp

Ts

t

h

t

p

t∇ = ∇ −

∇= −

ρ∂∂

∂∂ ρ

∂∂

&1

SOLO



Josiah Willard Gibbs(1903 – 1839)

87

FLUID DYNAMICS


(2.6.1.4) ENTROPY AND VORTICITY

from (C.L.M.)

or

GIBBS EQUATION: T d s d hd p

= −ρ

∀

+⋅∇−

+⋅∇=

+⋅∇

→→→→tld

pd

tdt

pldp

hd

tdt

hldh

sd

tdt

sldsT &

1

∂∂

ρ∂∂

∂∂

Since this is true for all d l t→

&

T s hp

Ts

t

h

t

p

t∇ = ∇ −

∇= −

ρ∂∂

∂∂ ρ

∂∂

&1

SOLO

hsTGp

Guuut

uII

III

II

I

,,

,,,

,

2

,

~~

2

1 ∇−∇+⋅∇

+=⋅∇

+∇

−=

×∇×−

∇+

ρτ

ρτ

ρ∂∂

ρp

hsT

dlpdp

dlhdh

dlsds∇

−∇=∇→

⋅∇=

⋅∇=

⋅∇=

88

Luigi Crocco 1909-1986

FLUID DYNAMICS


(2.6.1.4) ENTROPY AND VORTICITY (CONTINUE)

Define

Let take the CURL of this equation

Vorticityu

×∇≡Ω

If , then from (C.L.M.) we get:

G = − ∇ Ψ

CRROCO’s EQUATION (1937)

( ) ( )

⋅∇×∇+

Ψ++∇×∇−∇×∇=×Ω×∇+×∇Ω

τρ∂

∂ ~1

0

2

2

uhsTuu

t

SOLO

ρτ

∂∂ ~

2

1 ,2

,,

⋅∇+

Ψ++∇−∇=×Ω+ I

II

I

uhsTut

u

hsTGuuut

uII

I

II

I

,,

,

,

2

,

~

2

1 ∇−∇+⋅∇

+=

×∇×−

∇+

ρτ

∂∂

From

89

FLUID DYNAMICS



( ) ( ) ( ) ( ) ( )∇ × × = ⋅ ∇ − ∇ ⋅ + ∇ ⋅ − ⋅ ∇ ← ∇ ⋅ = ∇ ⋅ ∇ × =

Ω Ω Ω Ω Ω Ωu u u u u u

0

0

( )∇ × ∇ = ∇ × ∇T s T s

τρ

τρ

τρ

~

0

1~1~1 ⋅∇×∇+⋅∇×

∇=

⋅∇×∇

Therefore ( ) ( ) ( ) τρ∂

∂ ~1 ⋅∇×

∇−∇×∇=∇⋅Ω−Ω⋅∇+Ω∇⋅+

ΩsTuuu

t

SOLO

( ) ( ) τρ

~1 ⋅∇×

∇−∇×∇+⋅∇Ω−∇⋅Ω=

ΩsTuu

tD

D

or

90

FLUID DYNAMICS





( ) ( ) τρ

~1 ⋅∇×

∇−∇×∇+⋅∇Ω−∇⋅Ω=

ΩsTuu

tD

D

FLUID WITHOUT VORTICITY WILL REMAIN FOREVER WITHOUTVORTICITY IN ABSENSE OF ENTROPY GRADIENTS OR VISCOUSFORCES

- FOR AN INVISCID FLUID ( )λ µ τ= = → =0 0~ ~

( ) ( ) sTuutD

DINVISCID

∇×∇+⋅∇Ω−∇⋅Ω=Ω =

0~~τ

- FOR AN HOMENTROPIC FLUID INITIALLY AT REST

s const everywhere i e ss

t. ; . . &∇ = =

0 0

∂∂( )( )

Ω0 0=

( )D

D ts

Ω

Ω= = = ∇ =0 0 0 0 0~ ~, ,τ

SOLO


91

FLUID DYNAMICS



(2.6.2) STATE EQUATION

p - PRESSURE (FORCE / SURFACE)

V - VOLUME OF GAS

M - MASS OF GAS

R - 8314

- 286.9

T - GAS TEMPERATURE

- GAS DENSITY

[ ]m3

[ ]kg

[ ]J kg mol Ko/ ( )⋅

[ ]J kg Ko/ ( )⋅R

[ ]kgmol /−η

[ ]oK

[ ]kg m/ 3ρ

[ ]2/ mN

IDEAL GAS

TRMVp η=

TMVp R=

DEFINE: ρρ

= = =∆ ∆M

Vv

V

M&

1

pv T= R

p T= ρ ROR

SOLO

92

FLUID DYNAMICS


IDEAL GAS TMVp R=

SOLO


(2.6.2) STATE EQUATION


a, 09/24/2005

Pictures from:Lee, Sears:"Thermodynamics", 2nd Edition, 1962

93

FLUID DYNAMICS



(2.6.3) THERMALLY PERFECT GAS AND CALORICALLLY PERFECT GAS

A THERMALLY PERFECT GAS IS DEFINED AS A GAS FOR WHICH THEINTERNAL ENERGY e IS A FUNCTION ONLY OF THE TEMPERATURE T.

( ) ( )h e T p e T RT h T= + = + =/ ( )ρ THERMALLY PERFECT GAS

DEFINE

C

C

v

V V

p

p p p p

e

T

q

T

h

T

de pdv v d p

d T

de pdv

d T

dq

d T

= =

= = = =

+ +

+

∆

∆

∂∂

∂∂

∂∂

A CALORICALLY PERFECT GAS IS DEFINED AS A GAS FOR WHICH Cv IS CONSTANT CALORICALLY PERFECT GAS e C Tv=

SOLO

94

FLUID DYNAMICS



(2.6.3) CALORICALLLY PERFECT GAS (CONTINUE)

A CALORICALLY PERFECT GAS IS DEFINED AS A GAS FOR WHICH Cv IS CONSTANT CALORICALLY PERFECT GAS e C Tv=

FOR A CALORICALLY PERFECT GAS

( )h C T RT C R T C T C C Rv v p p v= + = + = → = +

γγ

γ γ= ⇒ =

−⇒ =

−

= + = −∆ C

CC R C

Rp

v

C C R

p

R C C

v

p v p v

1 1

γ air = 14.

SOLO

95

FLUID DYNAMICS




(2.6.3.1) ENTROPY CALCULATIONS FOR A CALORICALLLY PERFECT GAS

pv T= R p T= ρ R IDEAL GAS

( )ds

de pdv

T

de pdv vdp vdp

T

dh vdp

T=

+=

+ + −=

−∆

ds CdT

TR

dv

vs s C

T

TR

v

vC

T

TRv v v= + → − = + = −2 1

2

1

2

1

2

1

2

1

ln ln ln lnρρ

1

2

1

212 lnln

p

pR

T

TCss

p

dpR

T

dTCds pp −=−→−=

s s Cp

pR C

p

pCv v p2 1

2

1

1

2

2

1

2

1

2

1

− = ⋅

− = −ln ln ln ln

ρρ

ρρ

ρρ

ENTROPY

SOLO

96

FLUID DYNAMICS




(2.6.3.1) ENTROPY CALCULATIONS FOR A CALORICALLLY PERFECT GAS

p

p

T

Te

T

Te

p

p

T

T

C

R s s

R

s s

R

isentropic

s sp

2

1

2

1

2

1

12

1

2

1

12 1 2 1 2 1

=

=

=

−− − −

− = −

⇒γ

γγ

γ

ρρ

ρρ

γγ γ

2

1

2

1

2

1

1

12

1

2

1

1

12 1 2 1 2 1

=

=

=

−− − −

− = −

⇒T

Te

T

Te

T

T

C

R s s

R

s s

R

isentropic

s sv

p

pe e

p

p

C

C s s

R

s s

R

isentropic

s sp

v2

1

2

1

2

1

2

1

2

1

2 1 2 1 2 1

=

=

=

−−

−− =

⇒ρρ

ρρ

ρρ

γ γ

T

T

h

h

p

pe

p

pe

T

T

h

h

p

p

s s

C

s s

C

isentropic

s sv p2

1

2

1

2

1

2

1

2

1

1

2

1

12

1

2

1

2

1

1

2

1

12 1 2 12 1

= = ⋅ =

=

= =

=

−−

− − −− = −

−

⇒ρρ

ρρ

ρρ

γγ

γγ

γγ

ISENTROPIC CHAIN

SOLO


97

FLUID DYNAMICS

BASIC LAWS IN FLUID DYNAMICS (CONTINUE)

BOUNDARY CONDITIONS

SOLO


98

SOLODimensionless Equations

Dimensionless Variables are:

0/~ ρρρ = 0/~

Uuu = gGG /~

= ( )200/~ Upp ρ=

0/~ lUtt =

20/

~UCTT p=( )2

00/~ Uρττ =2

0/~

UHH = 20/

~Uhh = 2

0/~ Uee = ( )200/~ Uqq ρ= ( )2/

~UQQ =

∇=∇ 0

~l

Field Equations

(C.M.): ( )00

00U

lu

t ρρ

∂ρ∂ =⋅∇+

( )200

0

~

3

4

U

luupGuu

t

u

ρµµρ

∂∂ρ

τ

⋅∇

⋅∇∇+×∇×∇−∇−=

∇⋅+(C.L.M.):

( ) ( )300

0~U

lTk

t

QuGu

t

pHu

t

H

qρ∂

∂ρτ∂∂ρ

∇⋅∇−+⋅+⋅⋅∇+=

∇⋅+

∂∂

(C.E.):

( )( ) ( ) 0

/

/

000

00

0 =

⋅∇+

U

ul

lUt

ρρ

∂ρρ∂

( ) ( ) ( )

( ) ( )

⋅∇∇

+

×∇×∇

−

∇−=

∇⋅+

000

000

0

00

00

000

0

200

020

0

000

000

0

0

3

4

/

/

U

ull

UlU

ull

Ul

U

pl

g

G

U

lg

U

ul

U

u

lUt

Uu

ρµ

µµ

ρµ

ρρρ

∂∂

ρρ

( ) ( ) ( ) ( ) ( ) ( )

∇⋅∇

−+⋅+

⋅⋅∇+

∂

∂=

2

0

00

00

0

000

02

000002

0

0

02

00

020000

200000 /

~

// U

CTl

k

kl

C

k

UlU

Q

lUtU

u

g

G

U

gl

U

u

Ul

U

p

lUtU

H

lUtD

D p

pµρµ

∂∂

ρρ

ρτ

ρρρρ

0/~ ρρρ = 0/~

Uuu = gGG /~

= ( )200/~ Upp ρ=

0/~ lUtt =

20/

~UCTT p=( )2

00/~ Uρττ =2

0/~

UHH = 20/

~Uhh = 2

0/~ Uee = ( )200/~ Uqq ρ= ( )2/

~UQQ =

∇=∇ 0

~l

0/~ µµµ =

0/~

kkk =

Reference Quantities: ρ0(density), U0(velocity), l0 (length), g (gravity), μ0 (viscosity), k0 (Fourier Constant), λ0 (mean free path)

0/~ λλλ =

99


Dimentionless Field Equations

(C.M.): ( ) 0~~~~

=⋅∇+ ut

ρ∂

ρ∂

( ) ( )uR

uR

pGF

uut

u

eer

~~~~1

3

4~~~~1~~~~1~~~~

~~

2

⋅∇∇+×∇×∇−∇−=

∇⋅+ µµρ

∂∂ρ(C.L.M.):

( ) ( )TkPRt

QuG

Fu

t

pHu

t

H

rer

∇⋅∇−+⋅+⋅⋅∇+=

∇⋅+

∂∂ 11

~

~~~~1~~~

~~~~~

~

~~

2 ∂∂ρτ

∂∂ρ

(C.E.):

Reynolds:0

000

µρ lU

Re = Prandtl:0

0

k

CP p

r

µ= Froude:

0

0

gl

UFr =

0/~ ρρρ = 0/~

Uuu = gGG /~

= ( )200/~ Upp ρ=

0/~ lUtt =

20/

~UCTT p=( )2

00/~ Uρττ =2

0/~

UHH = 20/

~Uhh = 2

0/~ Uee = ( )200/~ Uqq ρ= ( )2/

~UQQ =

∇=∇ 0

~l

0/~ ρρρ = 0/~

Uuu = gGG /~

= ( )200/~ Upp ρ=

0/~ lUtt =

20/

~UCTT p=( )2

00/~ Uρττ =2

0/~

UHH = 20/

~Uhh = 2

0/~ Uee = ( )200/~ Uqq ρ= ( )2/

~UQQ =

∇=∇ 0

~l

0/~ µµµ =

0/~

kkk =



0/~ λλλ =

Knudsenl

Kn0

0:λ=

100


Constitutive Relations

TRp ρ=

2

2

1uTCH p +=

Tkq ∇−=

TCh p=

−==200

200

200

1

U

TC

U

TC

C

R

U

p pp

p ρρ

γγ

ρρ

ρ

=

20

20 U

TC

U

h p

2

020

20 2

1

+

=

U

u

U

TC

U

H p

( )

∇

−=

20

000

0

000

0300 U

TCl

k

k

C

k

UlU

q p

pµρµ

ρ

( ) [ ]33

2~ Iuuu T ⋅∇−∇+∇= µµτ [ ]30

00000

0

00

00

0000

0

00 3

2~I

U

ul

UlU

ul

U

ul

UlU

T

⋅∇

−

∇+∇

=

µµ

ρµ

µµ

ρµ

ρτ

0/~ ρρρ = 0/~

Uuu = gGG /~

= ( )200/~ Upp ρ=

0/~ lUtt =

20/

~UCTT p=( )2

00/~ Uρττ =2

0/~

UHH = 20/

~Uhh = 2

0/~ Uee = ( )200/~ Uqq ρ= ( )2/

~UQQ =

∇=∇ 0

~l

0/~ ρρρ = 0/~

Uuu = gGG /~

= ( )200/~ Upp ρ=

0/~ lUtt =

20/

~UCTT p=( )2

00/~ Uρττ =2

0/~

UHH = 20/

~Uhh = 2

0/~ Uee = ( )200/~ Uqq ρ= ( )2/

~UQQ =

∇=∇ 0

~l

0/~ µµµ =

0/~

kkk =



0/~ λλλ =

101


Dimensionless Constitutive Relations

2~2

1~~uTH +=

Tp~~1~ ρ

γγ −= Ideal Gas

( ) [ ]3

~~~

3

2~~~~~~~ IuR

uuR e

T

e

⋅∇−∇+∇= µµτ Navier-Stokes

Th~~ = Calorically Perfect Gas

TkPR

qre

~~~11~∇−= Fourier Law

Reynolds:0

000

µρ lU

Re =

Prandtl:0

0

k

CP p

r

µ=

0/~ ρρρ = 0/~

Uuu = gGG /~

= ( )200/~ Upp ρ=

0/~ lUtt =

20/

~UCTT p=( )2

00/~ Uρττ =2

0/~

UHH = 20/

~Uhh = 2

0/~ Uee = ( )200/~ Uqq ρ= ( )2/

~UQQ =

∇=∇ 0

~l

0/~ ρρρ = 0/~

Uuu = gGG /~

= ( )200/~ Upp ρ=

0/~ lUtt =

20/

~UCTT p=( )2

00/~ Uρττ =2

0/~

UHH = 20/

~Uhh = 2

0/~ Uee = ( )200/~ Uqq ρ= ( )2/

~UQQ =

∇=∇ 0

~l

0/~ µµµ =

0/~

kkk =



0/~ λλλ =


102

SOLO

Mach Number

Mach number (M or Ma) / is a dimensionless quantity representing the ratio of speed of an object moving through a fluid and the local speed of sound.

• M is the Mach number,• U0 is the velocity of the source relative to the medium, and

• a0 is the speed of sound

Mach:0

0

a

UM =

The Mach number is named after Austrian physicist and philosopher Ernst Mach, a designation proposed by aeronautical engineer Jakob Ackeret.

Ernst Mach (1838–1916)

Jakob Ackeret (1898–1981)

m

Tk

Mo

TRa Bγγ ==0

• R is the Universal gas constant, (in SI, 8.314 47215 J K−1 mol−1), [M1 L2 T−2 θ−1 'mol'−1]

• γ is the rate of specific heat constants Cp/Cv and is dimensionless γair = 1.4.• T is the thermodynamic temperature [θ1]

• Mo is the molar mass, [M1 'mol'−1]

• m is the molecular mass, [M1]

AERODYNAMICS

http://en.wikipedia.org/wiki/Help:IPA_for_English

http://en.wikipedia.org/wiki/File:Ernst_Mach_01.jpg

103

SOLOMach Number – Flow Regimes

Regime Mach mph km/h m/s General plane characteristics

Subsonic <0.8 <610 <980 <270Most often propeller-driven and commercial turbofan aircraft with high aspect-ratio (slender) wings, and rounded features like the nose and leading edges.

Transonic 0.8-1.2 610-915

980-1,470 270-410Transonic aircraft nearly always have swept wings, delaying drag-divergence, and often feature design adhering to the principles of the Whitcomb Area rule.

Supersonic 1.2–5.0915-3,840

1,470–6,150 410–1,710

Aircraft designed to fly at supersonic speeds show large differences in their aerodynamic design because of the radical differences in the behaviour of flows above Mach 1. Sharp edges, thin aerofoil-sections, and all-moving tailplane/canards are common. Modern combat aircraft must compromise in order to maintain low-speed handling; "true" supersonic designs include the F-104 Starfighter, SR-71 Blackbird and BAC/Aérospatiale Concorde.

Hypersonic 5.0–10.03,840–7,680

6,150–12,300

1,710–3,415

Cooled nickel-titanium skin; highly integrated (due to domination of interference effects: non-linear behaviour means that superposition of results for separate components is invalid), small wings, such as those on the X-51A Waverider

High-hypersonic

10.0–25.07,680–16,250

12,300–30,740

3,415–8,465

Thermal control becomes a dominant design consideration. Structure must either be designed to operate hot, or be protected by special silicate tiles or similar. Chemically reacting flow can also cause corrosion of the vehicle's skin, with free-atomic oxygen featuring in very high-speed flows. Hypersonic designs are often forced into blunt configurations because of the aerodynamic heating rising with a reduced radius of curvature.

Re-entry speeds >25.0

>16,250 >30,740 >8,465 Ablative heat shield; small or no wings; blunt shape

104

SOLO

Different Regimes of Flow

Mach Number – Flow Regimes

AERODYNAMICS

105

whereρ = air densityV = true speedl = characteristic lengthμ = absolute (dynamic) viscosityυ = kinematic viscosity

Reynolds:υµ

ρ ρµυ

lVlVRe

=

==

Osborne Reynolds (1842 –1912)

It was observed by Reynolds in 1884 that a Fluid Flow changes from Laminar to Turbulent at approximately the same value of the dimensionless ratio (ρ V l/ μ) where l is the Characteristic Length for the object in the Flow. This ratio is called the Reynolds number, and is the governing parameter for Viscous Flow.

Reynolds Number and Boundary Layer

SOLO 1884AERODYNAMICS

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

106

Boundary Layer

SOLO 1904AERODYNAMICS

Ludwig Prandtl(1875 – 1953)

In 1904 at the Third Mathematical Congress, held at Heidelberg, Germany, Ludwig Prandtl (29 years old) introduced the concept of Boundary Layer. He theorized that the fluid friction was the cause of the fluid adjacent to surface to stick to surface – no slip condition, zero local velocity, at the surface – and the frictional effects were experienced only in the boundary layer a thin region near the surface. Outside the boundary layer the flow may be considered as inviscid (frictionless) flow. In the Boundary Layer on can calculate the •Boundary Layer width•Dynamic friction coefficient μ•Friction Drag Coefficient CDf

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

107

The flow within the Boundary Layer can be of two types:•The first one is Laminar Flow, consists of layers of flow sliding one over other in a regular fashion without mixing.•The second one is called Turbulent Flow and consists of particles of flow that moves in a random and irregular fashion with no clear individual path, In specifying the velocity profile within a Boundary Layer, one must look at the mean velocity distribution measured over a long period of time.There is usually a transition region between these two types of Boundary-Layer Flow

SOLO AERODYNAMICS

108

Normalized Velocity profiles within a Boundary-Layer, comparison betweenLaminar and Turbulent Flow.

SOLO

Boundary-Layer

AERODYNAMICS

109

Flow Characteristics around a Cylindrical Body as a Function of Reynolds Number (Viscosity)

AERODYNAMICSSOLO

110

Relative Drag Force as a Function of Reynolds Number (Viscosity)

AERODYNAMICS

Drag CD0 due toFlow Separation

SOLO

111


AERODYNAMICS

Drag due to Viscosity:1.Skin Friction2.Flow Separation (Drop in pressure behind body)

∫∫

∫∫

⋅+⋅−−=

⋅+⋅−=

∧∧∞

∧∧

W

W

S

S

fpD

dswtV

fwn

V

pp

S

dswtCwnCS

C

xx

xx

11

11

ˆ2/

ˆ2/

1

ˆˆ1

22 ρρ

SOLO

112

Parasite Drag

Pressure Differential,Viscous Shear Stress,and Separation

AERODYNAMICS


DragFrictionSkin

ET

EL

ll

ET

EL

uu

DragPressure

ET

EL

ll

ET

EL

uu

sdfsdf

sdpsdpD

∫∫

∫∫

++

+−=

..

..

..

..

..

..

..

..

coscos

sinsin

θθ

θθ

SOLO

113

AERODYNAMICS


• Blunt Body: Most of Drag is Pressure Drag.• Streamlined Body: Most of Drag is Skin Friction Drag.

SOLO

http://www.google.com/url?sa=i&source=images&cd=&docid=2WYTKZwy2NSPqM&tbnid=RHa4sH6r54Vk6M:&ved=0CAgQjRw&url=http%3A%2F%2Fwww.insideracingtechnology.com%2Ftech102drag.htm&ei=L6_LUqydLu3Y7Aa604CwCg&psig=AFQjCNG5meulIjYvBSZWt5c6C7jNpIlBKA&ust=1389166767807035

http://www.google.com/url?sa=i&source=images&cd=&cad=rja&docid=ijGxbanrpcQ2cM&tbnid=Hq5Ha8RqPJugNM:&ved=0CAgQjRw4rwI&url=http%3A%2F%2Fwww.vivaf1.com%2Fglossary.php&ei=rrHLUt-aMbOS7Aads4CwDg&psig=AFQjCNHpH89yojC9uUaI54i1kPU9d4JheQ&ust=1389167406850587

114

AERODYNAMICS


SOLO

115

AERODYNAMICS


SOLO

116

AERODYNAMICS


SOLO

Variation of total skin-friction coefficient with Reynolds number for a smooth, flat plate.[From Dommasch, et al. (1967).]

117

Typical Effect of Reynolds Number on Parasitic Drag

Flow may stay attachedfarther at high Re,reducing the drag

AERODYNAMICSSOLO


118

FluidsSOLO

Knudsen number (Kn) is a dimensionless number defined as the ratio of the molecular mean free path length to a representative physical length scale. This length scale could be, for example, the radius of the body in a fluid. The number is named after Danish physicist Martin Knudsen.

Knudsenl

Kn0

0:λ= Martin Knudsen

(1871–1949).

For a Boltzmann gas, the mean free path may be readily calculated as:

• kB is the Boltzmann constant (1.3806504(24) × 10−23 J/K in SI units), [M1 L2 T−2 θ−1]p

TkB20

2 σπλ =

• T is the thermodynamic temperature [θ1]

λ0 = mean free path [L1]

Knudsen Number

l0 = representative physical length scale [L1].

• σ is the particle hard shell diameter, [L1]

• p is the total pressure, [M1 L−1 T−2].

See “Kinetic Theory of Gases” Presentation

For particle dynamics in the atmosphere and assuming standard atmosphere pressure i.e. 25 °C and 1 atm, we have λ0 ≈ 8x10-8m.

http://en.wikipedia.org/wiki/File:Knudsen,Martin_1934_London.jpg

119

FluidsSOLO

Martin Knudsen (1871–1949).

Knudsen Number (continue – 1)

Relationship to Mach and Reynolds numbers

Dynamic viscosity,

Average molecule speed (from Maxwell–Boltzmann distribution),

thus the mean free path,

where

• kB is the Boltzmann constant (1.3806504(24) × 10−23 J/K in SI units), [M1 L2 T−2 θ−1]

• T is the thermodynamic temperature [θ1]

• ĉ is the average molecular speed from the Maxwell–Boltzmann distribution, [L1 T−1]

• μ is the dynamic viscosity, [M1 L−1 T−1]

• m is the molecular mass, [M1]

• ρ is the density, [M1 L−3].

02

1 λρµ c=

m

Tkc B

π8=

Tk

m

B20

πρµλ =


120

FluidsSOLO

Martin Knudsen (1871–1949).


Relationship to Mach and Reynolds numbers (continue – 1)

The dimensionless Reynolds number can be written:

Dividing the Mach number by the Reynolds number,

and by multiplying by

yields the Knudsen number.

The Mach, Reynolds and Knudsen numbers are therefore related by:

Reynolds:Re0

000

µρ lU=

Tk

m

lmTklallU

aUM

BB γρµ

γρµ

ρµ

µρ 00

0

00

0

000

0

0000

00

//

/

Re====

KnTk

m

lTk

m

l BB

==22 00

0

00

0 πρµπγ

γρµ

2Re

πγMKn =


121

FluidsSOLO


Relationship to Mach and Reynolds numbers (continue –2)

According to the Knudsen Number the Gas Flow can be divided in three regions:1.Free Molecular Flow (Kn >> 1): M/Re > 3 molecule-interface interaction negligible between incident and reflected particles2.Transition (from molecular to continuum flow) regime: 3 > M/Re and M/(Re)1/2 > 0.01 (Re >> 1). Both intermolecular and molecule-surface collision are important.3.Continuum Flow (Kn << 1): 0.01 > M/(Re)1/2. Dominated by intermolecular collisions.

2Re

πγMKn =

FluidsSOLO


InviscidLimit Free

MolecularLimitKnudsen Number

Boltzman EquationCollisionless

Boltzman Equation

DiscreteParticlemodel

Euler Equation

Navier-Stokes Equation

Continuummodel

Conservation Equationdo not form a closed set

Validity of conventional mathematical models as a function of localKnudsen Number


123

AERODYNAMICS

Fluid flow is characterized by a velocity vector field in three-dimensional space, within the framework of continuum mechanics. Streamlines, Streaklines and Pathlines are field lines resulting from this vector field description of the flow. They differ only when the flow changes with time: that is, when the flow is not steady.

• Streamlines are a family of curves that are instantaneously tangent to the velocity vector of the flow. These show the direction a fluid element will travel in at any point in time.

• Streaklines are the locus of points of all the fluid particles that have passed continuously through a particular spatial point in the past. Dye steadily injected into the fluid at a fixed point extends along a streakline

• Pathlines are the trajectories that individual fluid particles follow. These can be thought of as a "recording" of the path a fluid element in the flow takes over a certain period. The direction the path takes will be determined by the streamlines of the fluid at each moment in time.

• Timelines are the lines formed by a set of fluid particles that were marked at a previous instant in time, creating a line or a curve that is displaced in time as the particles move.

The red particle moves in a flowing fluid; its pathline is traced in red; the tip of the trail of blue ink released from the origin follows the particle, but unlike the static pathline (which records the earlier motion of the dot), ink released after the red dot departs continues to move up with the flow. (This is a streakline.) The dashed lines represent contours of the velocity field (streamlines), showing the motion of the whole field at the same time. (See high resolution version.

Flow DescriptionSOLO

http://en.wikipedia.org/wiki/File:Streaklines_and_pathlines_animation_(low).gif

124

3-D FlowFlow Description

SOLO

Steady Motion: If at various points of the flow field quantities (velocity, density, pressure)associated with the fluid flow remain unchanged with time, the motion is said to be steady.

( ) ( ) ( )zyxppzyxzyxuu ,,,,,,,, === ρρ

Unsteady Motion: If at various points of the flow field quantities (velocity, density, pressure)associated with the fluid flow change with time, the motion is said to be unsteady.

( ) ( ) ( )tzyxpptzyxtzyxuu ,,,,,,,,,,, === ρρ

Path Line: The curve described in space by a moving fluid element is known as its trajectoryor path line.

tt

tt ∆+

t

tt ∆+tt ∆+ 2

t

tt ∆+tt ∆+ 2

Path Line (steady flow)

t

tt ∆+

t

tt ∆+ 2

tt ∆+

t

Path Line (unsteady flow)

tt ∆+ 2

tt ∆+

t

125

3-D Flow

Flow Description

SOLO


ttt ∆+ tt ∆+ 2

Streamlines: The family of curves such that each curve is tangent at each point to the velocity direction at that point are called streamlines.

Consider the coordinate of a point P and the direction of the streamline passingthrough this point. If is the velocity vector of the flow passing through P at a time t,then and parallel, or:

r

rdu

u

rd

0=×urd

( )( )( )

0

1

1

1111

=

−

−

−

=

zdyudxv

ydxwdzu

xdzvdyw

wvu

dzdydx

zyx

w

zd

v

yd

u

xd==

Cartesian

t

u

r

rd

126

3-D Flow

Flow Description

SOLO



( ) ( ) ( )tzyxw

zd

tzyxv

yd

tzyxu

xd

,,,,,,,,,==

t

u

r

rd

Those are two independent differential equations for a streamline. Given a point the streamline is defined from those equations.( )0000 ,,, tzyxr

( ) ( ) ( )( ) ( ) ( )tzyxw

zd

tzyxv

yd

tzyxv

yd

tzyxu

xd

,,,,,,2

,,,,,,1

=

= ( ) ( ) ( )( ) ( ) ( ) 0,,,,,,,,,

0,,,,,,,,,

222

111

=++=++

zdtzyxcydtzyxbxdtzyxa

zdtzyxcydtzyxbxdtzyxa

( ) ( )( ) ( )21

21

22

11

•+••+•

βαβα

022

11 ≠βαβα

Pfaffian Differential Equations

For a given a point the solution of those equations is of the form:( )0000 ,,, tzyxr

( )( ) 2,,,

1,,,

02

01

consttzyx

consttzyx

==

ψψ

u

( )0tr

rd

0t

( ) 11 cr =ψ

( ) 22 cr =

ψStreamline Those are two surfaces, the

intersection of which is the streamline.

127

3-D Flow

Flow Description

SOLO



( ) ( ) ( )tzyxw

zd

tzyxv

yd

tzyxu

xd

,,,,,,,,,==

t

u

r

rd

For a given a point the solution of those equations is of the form:( )0000 ,,, tzyxr

( )( ) 2,,,

1,,,

02

01

consttzyx

consttzyx

==

ψψ

u

( )0tr

rd

0t

( ) 11 cr =

ψ

( ) 22 cr =

ψStreamline Those are two surfaces, the

intersection of which is the streamline.

The streamline is perpendicular to the gradients (normals) of those two surfaces.

( ) ( ) ( )0201 ,, trtrVr ψψµ ∇×∇=

where μ is a factor that must satisfy the following constraint.

( )( ) ( ) ( ) 0,, 0201 =∇×∇⋅∇=⋅∇ trtrVr ψψµ


128

AERODYNAMICS

Streamlines, Streaklines, and PathlinesMathematical description

Streamlines

If the components of the velocity are written and those of the streamline aswe deduce

which shows that the curves are parallel to the velocity vector

Pathlines

Streaklines

where, is the velocity of a particle P at location and time t . The parameter , parametrizes the streakline and 0 ≤ τP ≤ t0 , where t0 is a time of interest .

The suffix P indicates that we are following the motion of a fluid particle. Note that at point

the curve is parallel to the flow velocity vector where the velocity vector is evaluated at the position of the particle at that time t .

SOLO

129

∞V

Airfoil Pressure Field variation with α

AERODYNAMICS

Airfoil Velocity Field variation with αAirfoil Streamline variation with αAirfoil Streakline with α

Streamlines, Streaklines, and PathlinesSOLO

130

AERODYNAMICSStreamlines, Streaklines, and Pathlines

SOLO

131

AERODYNAMICSSOLO

132

AERODYNAMICSSOLO

133

AERODYNAMICSStreamlines, Streaklines, and Pathlines

SOLO


134


Circulation

SOLO

Circulation Definition:

tV ∆

( ) tVV ∆∆+

S∆

Sn ∆1

V

×∇

tr

∆

ttr ∆+∆

tC

ttC ∆+

∫ ⋅=ΓC

rdV

:

Material Derivative of the Circulation

( )∫∫∫ ⋅+⋅=

⋅=

Γ

CCC

rdtD

DVrd

tD

VDrdV

tD

D

tD

D

From the Figure we can see that:

( ) tVrtVVr ttt ∆+∆=∆∆++∆ ∆+

( ) VdrdtD

DV

t

rr tttt

=→∆=

∆∆−∆ →∆

∆+0

( ) 02

2

=

=⋅=⋅ ∫∫∫

CCC

VdVdVrd

tD

DV

Therefore:

∫ ⋅=Γ

C

rdtD

VD

tD

D

integral of an exact differential on a closed curve.

C – a closed curve

135

3-D Inviscid Incompressible FlowSOLO

tV ∆

( ) tVV ∆∆+

S∆

Sn ∆1

V

×∇

tr

∆

ttr ∆+∆

tC

ttC ∆+

S

∫ ⋅=ΓtC

rdV

:

Material Derivative of the Circulation (second derivation)

Subtract those equations:

tVrdSn t ∆×=∆

1

( )∫∆+

⋅∆+=Γ∆+ΓttC

rdVV

:

( ) ( )∫∫∫∫ ∆⋅×∇=⋅∆+−⋅=Γ∆−∆+ S

TheoremsStoke

CC

SnVrdVVrdVttt

1'

S is the surface bounded by the curves Ct and C t+Δ t

( ) ( ) ( ) tVVrdtVrdVSnVS

t

S

t

S

∆

×∇×⋅=∆×⋅×∇=∆⋅×∇=Γ∆− ∫∫∫∫∫∫

1

td

d

ttd

rd

tV

ttD

D rdd Γ+

∂Γ∂

=Γ∇⋅+∂

Γ∂=Γ∇⋅+

∂Γ∂

=Γ Γ∇⋅=Γ

Computation of: ∫ ⋅∂∂

=∂

Γ∂

tC

rdt

V

t

Computation of:td

d Γ

136


tV ∆

( ) tVV ∆∆+

S∆

Sn ∆1

V

×∇

tr

∆

ttr ∆+∆

tC

ttC ∆+

Material Derivative of the Circulation (second derivation)

( ) tVVrdS

t ∆

×∇×⋅=Γ∆− ∫∫

When Δ t → 0 the surface S shrinks to the curve C=Ct and the surface integral transforms to a curvilinear integral:

( ) ( ) ( )∫∫∫∫∫ ∇⋅⋅+

−=∇⋅⋅+

∇⋅−=×∇×⋅−=Γ

C

t

CC

t

C

t

C

t VVrdV

dVVrdV

rdVVrdtd

d

0

22

22

Computation of: (continue)td

d Γ

Finally we obtain:

( ) ∫∫∫ ⋅=∇⋅⋅+⋅∂∂

=Γ

+∂

Γ∂=

Γ

tt CC

t

C

rdtD

VDVVrdrd

t

V

td

d

ttD

D

137


tV ∆

( ) tVV ∆∆+

S∆

Sn ∆1

V

×∇

tr

∆

ttr ∆+∆

tC

ttC ∆+

Material Derivative of the Circulation

We obtained: ∫ ⋅=Γ

tC

rdtD

VD

tD

D

Use C.L.M.: hsTp

VVt

V

tD

VDII

I

G

II

II

,,

,

,,

~∇−∇+

⋅∇+Ψ∇=

∇⋅+=

τ∂∂

( ) ( )

0

,

,,

,

,

~~

∫∫∫∫ −Ψ+⋅

⋅∇+∇=⋅∇−Ψ∇+⋅

⋅∇+∇=

Γ

tttt CC

I

I

C

I

C

I

I

I

hddrdp

sTrdhrdp

sTtD

D ττ

to obtain:

∫ ⋅

⋅∇+∇=

Γ

tC

I

I

I

rdp

sTtD

D τ~,

,or:

Kelvin’s Theorem

William Thomson Lord Kelvin(1824-1907)

In an inviscid , isentropic flow d s = 0 with conservative body forces the circulation Γ around a closed fluid line remains constant with respect to time.

0~~ =τ

Ψ∇=G


1869

138


Circulation Definition: ∫ ⋅=ΓC

rdV

:


Biot-Savart Formula

1820

Jean-Baptiste Biot1774 - 1862

VorticityV

×∇≡Ω

∫ −Ω=

Space

dVsr

A

π4

1

( )lddSnsr

Ad

⋅−Ω=

π4

1

The contribution of a length dl of the Vortex Filament to isA

∫∫∫∫∫ ⋅Ω=⋅×∇=⋅=ΓSS

Stokes

C

SdnSdnVrdV

:

If the Flow is Incompressible 0=⋅∇ u

so we can write , where is the Vector Potential. We are free tochoose so we choose it to satisfy .

AV

×∇=A A

0=⋅∇ A

We obtain the Poisson Equation that defines the Vector Potential A

Ω−=∇

A2 Poisson Equation Solution( ) ∫ −Ω=

Space

dvsr

rA

π4

1

Félix Savart1791 - 1841

Biot-Savart Formula

139



rdV

:


Biot-Savart Formula (continue - 1)

1820


VorticityV

×∇≡Ω

( )lddSnsr

Ad

⋅−Ω=

π4

1We found

∫∫∫∫∫ ⋅Ω=⋅×∇=⋅=ΓSS

Stokes

C

SdnSdnVrdV

:

also we have dlldΩΩ=

( ) ( ) ∫∫∫∫∫ ×−

∇⋅Ω=⋅−Ω×∇=×∇=

Γ

ΩΩ=

ldsr

dSnlddSnsr

AdrV r

S

dlld

v

rr

1

4

1

4

1

ππ

( ) ( )∫ −

−×Γ= 34 sr

srldrV

π Biot-Savart Formula


Biot-Savart Formula

140


Circulation

SOLO


rdV

:



1820

Jean-Baptiste Biot1774 - 1862( ) ( )

∫ −−×Γ= 34 sr

srldrV

πBiot-Savart Formula

General 3D Vortex


141



rdV

:



1820



( ) ( )∫ −

−×Γ= 34 sr

srldrV

πBiot-Savart Formula

General 3D Vortex

For a 2 D Vortex:

( ) θθθθθd

hsr

dl

sr

srld sinˆˆsin23 =

−=

−−×

θθ

θ dh

dlhl2sin

cot =⇒=−

θsin/hsr =−

θπ

θθθπ

πˆ

2sinˆ

4 0 hd

hV

Γ=Γ= ∫ Biot-Savart Formula

General 2D Vortex

Biot-Savart Formula

142



rdV

:



1820

Jean-Baptiste Biot1774 - 1862( ) ( )

∫ −−×Γ= 34 sr

srldrV

πBiot-Savart Formula General 3D Vortex


Lifting-Line Theory

Biot-Savart Formula


143


Helmholtz Vortex Theorems

SOLO

Helmholtz : “Uber the Integrale der hydrodynamischen Gleichungen, welcheDen Wirbelbewegungen entsprechen”, (“On the Integrals of the Hydrodynamical Equations Corresponding to Vortex Motion”), in Journal fur die reine und angewandte, vol. 55, pp. 25-55. , 1858He introduced the potential of velocity φ.

Hermann Ludwig Ferdinandvon Helmholtz

1821 - 1894

Theorem 1: The circulation around a given vortex line (i.e., the strength of the vortex filament) is constant along its length.

Theorem 2: A vortex filament cannot end in a fluid. It mustform a closed path, end at a boundary, or go to infinity.

Theorem 3: No fluid particle can have rotation, if it did not originally rotate.Or, equivalently, in the absence of rotational external forces, a fluid that is initially irrotational remains irrotational. In general we can conclude that thevortex are preserved as time passes. They can disappear only through the action of viscosity (or some other dissipative mechanism).


1858

144

( ) ( ) ( )

MomentFriction

S

C

Momentessure

S

CCA

WW

dstRRfdsnRRppM ∫∫∫∫∑ ×−+×−−= ∞ 11

Pr

/

( )

FrictionSkinorFrictionViscous

S

essureNormal

S

A

WW

dstfdsnppF ∫∫∫∫∑ +−= ∞ 11

Pr

AERODYNAMIC FORCES AND MOMENTS.

SOLO

Aerodynamic Moments Relative to a point can be divided in Pressure Moments and Friction Moments Relative to this point.

CR

Aerodynamic Forces can be divided in Pressure Forces, normal to Body Surface , and Friction Forces, tangent to Body Surface .

dsn

1

dst

1

CR

AERODYNAMICS

145

SOLO

Body Coordinates (B)

The origin of the Body coordinate systemis located at the instantaneous center ofgravity CG of the vehicle, with xB pointedto the front of the Air Vehicle, yB pointedtoward the right wing and zB completingthe right-handed Cartesian reference frame.

ψθ

φBx

Lx

Bz

Ly

LzBy

Rotation Matrix from LLLN to B (Euler Angles):

[ ] [ ] [ ]

−++−

−==

θφψφψθφψφψθφθφψφψθφψφψθφ

θψθψθψθφ

cccssscsscsc

csccssssccss

ssccc

C BL 321

ψ - azimuth (yaw) angle

θ - pitch angle

φ - roll angle

AERODYNAMICS

146

SOLO

Wind Coordinates (W)

The origin of the Wind coordinate systemis located at the instantaneous center ofgravity CG of the vehicle, with xW pointedin the direction of the vehicle velocity vectorrelative to air .AV

[ ] [ ]

−−−=

−

−=−=

ααβαββαβαββα

αα

ααββββ

αβcos0sin

sinsincossincos

cossinsincoscos

cos0sin

010

sin0cos

100

0cossin

0sincos

23WBC

The Wind coordinate frame (W) is defined by the following two angles, relative toBody coordinates (B):

α - angle of attack

β - sideslip angle

AERODYNAMICS

( )BBB

BA zwyvxuV

111 ++=

[ ]

=

⇒

=

βαβ

βα

cossin

sin

coscos

0 V

w

v

u

o

V

C

w

v

uTW

B

147

LowerSurface

UpperSurface

AERODYNAMIC FORCES AND MOMENTS.

SOLO

∞V

Airfoil Pressure Field variation with α

Distribution of Pressure around an Airfoil causes Aerodynamic Forces and Moments

148

SOLO Linearized Flow Equations

Preasure Field

Stream Lines Streak Lines (α = 0º) Streak Lines (α = 15º)

Streak Lines (α = 30º) Forces in the Body

149

SOLO Linearized Flow Equations

Velocity Field

Sum of the elementary Forces on the Body

Lift as the Sum of the elementary Forces on the Body

150

SOLO

Aerodynamic Forces

( )

−−−

=L

C

D

F WA

ForceLiftL

ForceSideC

ForceDragD

−−−

L

C

D

CSVL

CSVC

CSVD

2

2

2

2

12

12

1

ρ

ρ

ρ

=

=

= ( )( )( ) tCoefficienLiftRMC

tCoefficienSideRMC

tCoefficienDragRMC

eL

eC

eD

−−−

βαβαβα

,,,

,,,

,,,

viscositydynamic

lengthsticcharacteril

soundofspeedHa

numberReynoldslVR

BodytoRelativeVelocityFlowV

numberMachaVM

e

−−−

−=−−=

µ

µρ)(

/

/

AERODYNAMICS

( )WWW

WA zLyCxDF

111 −−−=

151

SOLO

Aerodynamic Forces (continue -1)

∫∫

⋅+⋅−=

∫∫

⋅+⋅−=

∫∫

⋅+⋅−=

∧∧

∧∧

∧∧

W

W

W

SfpL

SfpC

SfpD

dswztCwznCS

C

dswytCwynCS

C

dswxtCwxnCS

C

1ˆ1ˆ1

1ˆ1ˆ1

1ˆ1ˆ1

Wf

Wp

SsurfacetheontcoefficienfrictionV

fC

SsurfacetheontcoefficienpressureV

ppC

−=

−−= ∞

2/

2/

2

2

ρ

ρ

ntonormalplanonVofprojectiont

dstonormaln

ˆˆ

ˆ

−

−

AERODYNAMICS

152

SOLO

( )

=

Y

P

RB

CA

M

M

M

M /

MomentYawM

MomentPitchM

MomentRollM

Y

P

R

−−−

YY

PP

RR

ClSVM

ClSVM

ClSVM

2

2

2

2

12

12

1

ρ

ρ

ρ

=

=

= ( )( )( ) tCoefficienMomentYawRMC

tCoefficienMomentPitchRMC

tCoefficienMomentRollRMC

eY

eP

eR

−−−

βαβαβα

,,,

,,,

,,,

viscositydynamic

lengthsticcharacteril

soundofspeedHa

numberReynoldslVR

BodytoRelativeVelocityFlowV

numberMachaVM

e

−−−

−=−−=

µ

µρ)(

/

/

AERODYNAMICS Aerodynamic Moments Relative to CR

CR

( )BYBPBR

BCA zMyMxMM

111/ ++=


( )∫

∫∫

−=

−=

==′

EdgeTrailing

EdgeLeading

sideupper sidelower

EdgeTrailing

EdgeLeading

sideupper

EdgeTrailing

EdgeLeading

sidelower

pp

pp

sideupper on Force-sidelower on the acting Forces

direction wind the tonormal Force

dx

dxdx

L

Relationship between Lift and Pressure on Airfoil

SOLO

( )

[ ] [ ]( )∫

∫

∞∞

=

−−−=

−=′

EdgeTrailing

EdgeLeading

sideupper sidelower

cos

EdgeTrailing

EdgeLeading

sideupper sidelower

pp

cospcosp

dxpp

dsL

sdxd

USLS

θ

θθ

Divide left and right sides of the first equation by cV 2

2

1∞ρ

∫

−

−−=′

∞

∞

∞

∞

∞

EdgeTrailing

EdgeLeading

upperlower

c

xd

V

pp

V

pp

cV

L

222

21

21

21 ρρρ

We get:

Relationship between Lift and Pressure on Airfoil (continue – 1)

LowerSurface

UpperSurface

( )∫ −=−EdgeTrailing

EdgeLeading

sideupper sidelower sinpsinp dsD USLS θθ

Lift – Aerodynamic component normal to VDrag – Aerodynamic component opposite to V

SOLO

From the previous slide,

∫

−

−−=′

∞

∞

∞

∞

∞

EdgeTrailing

EdgeLeading

upperlower

c

xd

V

pp

V

pp

cV

L

222

21

21

21 ρρρ

The left side was previously defined as the sectional lift coefficient C l.

The pressure coefficient is defined as: 2

21

∞

∞−=V

ppC p

ρThus, ( )∫ −=

edgeTrailing

edgeLeading

upperplowerpl c

xdCCC ,,

LowerSurface

UpperSurface

Relationship between Lift and Pressure on Airfoil (continue – 2)

SOLO


156


In 2-D the velocity vector

SOLO


x

y V

u

vru

θv

rθ

θθ 1111 vruyvxuV r +=+=

θθ

∂∂

++∂∂

=∂∂

+∂∂

=⋅∇v

r

u

r

u

y

v

x

uV rr

zu

r

v

z

ur

z

vz

y

u

x

vy

z

ux

z

vV rr 111111

∂∂

−∂∂

+∂∂

+∂∂

−=

∂∂

−∂∂

+∂∂

+∂∂

−=×∇θ

θ θθ

0

111

0

111

rr vu

zr

zr

vu

zyx

zyx

V∂∂

∂∂

∂∂

=∂∂

∂∂

∂∂

=×∇θ

θ

−

=

v

u

v

u r

θθθθ

θ cossin

sincos ( ) θθ

i

r eviuviu +=+( ) θ

θi

r eviuviu −+=+

157



SOLO


x

y V

u

vru

θv

rθ


−

=

v

u

v

u r

θθθθ

θ cossin

sincos ( ) θθ

i

r eviuviu +=+( ) θ

θi

r eviuviu −+=+

Continuity: 00 =⋅∇→=⋅∇+ uutD

D ρρ

( )

∂∂

−=∂∂

=×

∂∂

+∂∂

∂∂

−=∂∂

=×

∂∂

+∂∂

=×∇=×∇=

rv

ruz

rr

r

xv

yuzy

yx

xzzu

r

ψθψθ

θψψ

ψψψψ

ψψ

θ

111

11

111

11 22

Incompressible: 0=tD

D ρ

Irrotational:

∂∂

=∂∂

=

∂∂

=∂∂

==∇=

θφφ

φφ

φθ r

vr

u

yv

xu

u

r

12

0=×∇ u

rrv

rru

xyv

yxu

r ∂∂−=

∂∂=

∂∂=

∂∂=

∂∂−=

∂∂=

∂∂=

∂∂=

ψθφ

θψφ

ψφψφ

θ11

158



SOLO


x

y V

u

vru

θv

rθ


−

=

v

u

v

u r

θθθθ

θ cossin

sincos ( ) θθ

i

r eviuviu +=+( ) θ

θi

r eviuviu −+=+

00 222 =∇⋅∇→∇=+=⋅∇ φφuu

2-D Incompressible:

2-D Irrotational:

( )( ) ( ) ( )ψψψ

ψψ

222

0

222

222

1110

110

∇⋅∇−∇∇⋅=×∇×∇=

→×∇=×∇=+=×∇

zzz

zzuu

02

2

2

2 =∇=∇ ψφ

Complex Potential in 2-D Incompressible-Irrotational Flow:( ) ( ) ( )

yixz

yxiyxzw

+=+= ,,: ψφ

( )=

zd

zwdx

ix ∂

∂+

∂∂ ψφ

yyi

∂∂

+∂∂

−ψφ0=x

0=y

( )[ ] ( ) θθ

θθ

i

r

i

r eviueviuVviu −∗∗ −=+==−

zd

wdviu =− θ

θi

r ezd

wdviu =−

xyyx ∂∂

−=∂∂

∂∂

=∂∂ ψφψφ

Cauchy-Riemann Equations

We found:

159


Examples:

SOLO


αα sincos 00 UiUV +=Uniform Stream:

xyUv

yxUu

∂∂

−=∂∂

==

∂∂

=∂∂

==

ψφα

ψφα

sin

cos

0

0

( ) ( )( ) ( ) yUxU

yUxU

ααψααφ

cossin

sincos

00

00

+−=+=

( ) ( )zU

zUzUiw∗=

−=+=

0

00 sincos ααψφ

0U

α

160


Examples:

SOLO


∂∂

−=∂∂

==

∂∂

=∂∂

==

rrv

rrr

mu r

ψθφ

θψφ

π

θ

10:

1

2:

( )

==

+==

−

x

ymm

yxm

rm

1

22

tan22

ln2

ln2

πθ

πψ

ππφ

( ) ( ) zm

rem

irm

iw i ln2

ln2

ln2 ππ

θπ

ψφ θ ==+=+=

Definition:

Source , Sink :( )0>m ( )0<m

Sink 0<m

Source 0>m

The equation of a streamline is: constm == θπ

ψ2

161


Examples:

SOLO

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

( ) ( ) ( )r

Kvvr

rzuvr

rVu

rrr =→=

∂∂→=

∂∂−

∂∂=×∇→=

≠ θθθ θ0010:

02

( )

+Γ=Γ=

Γ−=Γ−= −

22

1

ln2

ln2

tan22

yxr

x

y

ππψ

πθ

πφ

( ) ( ) zi

rei

riiw i ln2

ln2

ln2 ππ

θπ

ψφ θ Γ=Γ=+−Γ=+=

Definition:

Infinite Line Vortex :

∂∂

−=∂∂

−=Γ−=

∂∂

=∂∂

−==

rrrv

rru r

ψθφ

π

θψφ

θ

1

2:

10:

( ) Γ−=Γ−=+⋅

Γ−=⋅ ∫∫∫ θπ

θθθπ

ddrrdrr

drV2

1112

Circulation

streamlines:

( )Λ=+

→+Γ=

/222

22ln2

ψπ

πψ

eyx

yx

Irotational

162


Examples:

SOLO


Definition: Let have a source and a sink of equal strength m = μ/ε situated at x = -εand x = ε such that

Doublet at the Origin with Axis Along x Axis :

m+ m−

ε+ε−

y

x

.lim0

constm ==→

µεε

( ) ( ) ( )

−+=

−+=

−−+=

z

zm

z

zm

zm

zm

zw

/1

/1ln

2ln

2

ln2

ln2

εε

πεε

π

επ

επ

.lim0

constm ==→

µεεwhen

( )

zz

m

zO

z

m

zO

zz

m

z

zmzw

m

πµε

πεε

π

εεεπε

επ

µε =

=≈

++≈

++

+≈

−+=

22

21ln2

11ln2/1

/1ln

2

2

2

2

2

163


Examples:

SOLO


( ) ( )θθπππ

sincos2

1

2ln

2: i

r

m

z

mz

m

zd

d

zd

Wdzw Source

Doublet −==

==

+==

+==

=

=

22

2/

22

2/

sin

cos

yx

y

r

yx

x

r

m

m

πµθ

πµψ

πµθ

πµφ

µ

µ

Definition:

Doublet at the Origin with Axis Along x Axis (continue):

2

1

2

1

2 z

m

z

m

zd

d

zd

wdviuV

ππ−=

==−=∗

The equation of a streamline is: .22

constyx

y=

+=

πµψ

22

2

22

=

++

ψµ

ψµ

yx

164

SOLO 2-D Inviscid Incompressible Flow

Stream Functions (φ), Potential Functions (ψ) for Elementary Flows

Flow W (z=reiθ)=φ+i ψ φ ψ

Uniform Flow θcosrU∞ θsinrU∞( )yixUzU += ∞∞

Source ( )θ

ππire

kz

kln

2ln

2= r

kln

2πθ

π2

k

Doubletθier

B

z

B = θcosr

B θsinr

B−

Vortex(with clockwise

Circulation)

( )θ

ππire

iz

iln

2ln

2

Γ=Γ θπ2

Γ−rln2πΓ

90 Corner Flow ( ) 22

22yix

Az

A += yxA( )22

2yx

A −


165

Wing Types Computations1. Subsonic Incompressible Flow (ρ∞ = constant)

1.1 Infinite Span (2-D, AR = ∞) (Profile Theory)

1.2 Finite Span (3-D, AR ≠ ∞) (Lifting Line Theory)

2. Supersonic Incompressible Flow (ρ∞ = constant)

1.1.1 Kutta-Joukowsky Lift Theorem

1.1.2 Profile Theory by the Method of Conformal Mapping

1.1.3 Profile Theory by the Method of Singularities

1.2.1 Wing Theory by the Method of Vortex Distribution (Prandtl Wing Theory)

1.2.2 Profile Theory by the Method of Conformal Mapping

1.1.3 Profile Theory by the Method of Singularities

SOLO


166


Paul Richard Heinrich Blasius(1883 – 1970)

Blasius was a PhD Student of Prandtl at Götingen University

x

y

xδyδ

βsd

M

−=

=−

∫

∫

C

C

zdzd

wdzM

zdzd

wdiiYX

2

2

2

2

ρ

ρ

Re

where-w (z) – Complex Potential of a Two-Dimensional Inviscid Flow -X, Y – Force Components in x and y directions of the Force per Unit Span on the Body-M – the anti-clockwise Moment per Unit Span about the point z=0-ρ – Air Density-C – Two Dimensional Body Boundary Curve

1911Blasius Theorem

167




−=

=−

∫

∫

C

C

zdzd

wdzM

zdzd

wdiiYX

2

2

2

2

ρ

ρ

Re

1911Blasius Theorem

Proof of Blasius Theorem

Consider the Small Element δs on the Boundary C

sysx δβδδβδ cos,sin =−=

xpspY

ypspX

δδβδδδβδ

⋅=⋅−=⋅−=⋅−=

sin

costhen

p = Normal Pressure to δs

The Total Force on the Body is given by

( ) ( )∫∫ −⋅−=+⋅−=−CC

ydixdpixdiydpYiX

Use Bernoulli’s Theorem .2

1 2constUp =+ ∞ρ

U∞ = Air Velocity far from Body

x

y

xδyδ

βsd

M

X

Y

168




−=

=−

∫

∫

C

C

zdzd

wdzM

zdzd

wdiiYX

2

2

2

2

ρ

ρ

Re

1911Blasius Theorem

Proof of Blasius Theorem (continue – 1)

( )∫ −⋅

−−=− ∞

C

ydixdUconstiYiX 2

2

1 ρ

but ( ) 00 =−⋅⇒== ∫∫∫CCC

ydixdconstydxd

( ) ( ) ( ) ( ) ( ) ( )( ) ( )( ) ( ) ( ) ( ) yduivuxduivvdyixdviu

dyuixdvdyixdvu

dyvuidyixdvudyixdvudyixdU

+−+++−=

−++−=

+−++=−+=−∞

22

22

2

2

2222

2222222

viuU +=∞and

x

y

xδyδ

βsd

M

X

Y

169




−=

=−

∫

∫

C

C

zdzd

wdzM

zdzd

wdiiYX

2

2

2

2

ρ

ρ

Re

1911Blasius Theorem


( ) ∫∫ ⋅

=−⋅=− ∞

CC

zdzd

wdiydixdU

iYiX

22

22ρρ

( ) ( ) ( ) zdzd

wddyixdviudyixdU

2

22

=+−=−∞

( ) ( ) 00 =−⇒+×+=×= ∞ xdvyduviuydixdUsd

Since the Flow around C is on a Streamline defined by

therefore ( ) ( ) yduivuxduivv +=+ 22

( ) ( ) ( )yixz

yxiyxzw

+=+= ,,: ψφ

and

xyv

yxu

∂∂−=

∂∂=

∂∂=

∂∂= ψφψφ

,where

Completes the Proof for the Force Equation

viuzd

wd −=

x

y

xδyδ

βsd

M

X

Y

170




−=

=−

∫

∫

C

C

zdzd

wdzM

zdzd

wdiiYX

2

2

2

2

ρ

ρ

Re

1911Blasius Theorem


( ) ( ) ( ) ( ) ( )ydxixdyiydyxdxvuivudyixdviuyixzdzd

wdz ++−−−=+−+=

2222

2

The Moment around the point z=0 is defined by

( ) ( )∫∫ +⋅−=+⋅= ∞CC

ydyxdxUydyxdxpM2

2

ρ

since 2

2 ∞−= UconstpBernoulli ρ

and ( ) 0=+⋅∫C

ydyxdxconst

hence( ) ( ) ( )xdyydxvuydyxdxvuzd

zd

wdz ++−−=

222

2

Re

x

y

xδyδ

βsd

M

X

Y

171




−=

=−

∫

∫

C

C

zdzd

wdzM

zdzd

wdiiYX

2

2

2

2

ρ

ρ

Re

1911Blasius Theorem


( ) ( ) ( )

−=+⋅+−=+⋅= ∫∫∫

CCC

zdzd

wdzydyxdxvuydyxdxpM

2

22

22

ρρRe

hence

( ) ( ) ( )xdyydxvuydyxdxvuzdzd

wdz ++−−=

222

2

Re

Since the Flow around C is on a Streamline we found that u dy = v dx

( ) ( ) ( ) ydyuxdxvxdvyuyduxvxdyydxvu 22 22222 +=+=+

( ) ( )ydyxdxvuzdzd

wdz ++=

22

2

2Re

Completes the Proof for the Moment Equation

x

y

xδyδ

βsd

M

X

Y

172

SOLO 2-D Inviscid Incompressible FlowBlasius Theorem Example

Circular Cylinder with Circulation

Let apply Blasius Theorem

Assume a Cylinder of Radius a in a Flow of Velocity U∞ at an Angle of Attack αand Circulation Γ.The Flow is simulated by:-A Uniform Stream of Velocity U∞

-A Doublet of Strength U∞ a2.-A Vortex of Strength Γ at the origin.

Since the Closed Loop Integral is nonzero only for 1/z component, we have

viuz

i

z

eaUeU

zd

wd ii −=Γ−−=

+∞−

∞ π

αα

22

2

∫∫ ⋅

Γ−−=⋅

=−

+∞−

∞C

ii

C

zdz

i

z

eaUeU

izd

zd

wdiYiX

2

2

22

222 πρρ

αα

ααα

ρπ

ρπ

ρ ii

C

i

eUiz

eUResiduezd

z

eUiiYiX −

∞

−∞

−∞ Γ=

Γ=⋅

Γ−=− ∫ 22

02 =

==⋅∫ zenclosesCif

z

AResidueAizd

z

A

C

πwhere we used:

α

αX

Y−L

∞U

x

y

( ) ( )αα

α

πi

ii ez

i

ez

aUezUzw −

−∞−

∞Γ−+= ln2

2

173


Circular Cylinder with Circulation (continue – 1)

Γ−−−=

−= ∫∫

+∞−

∞C

ii

C

zdz

i

z

eaUeUzzd

zd

wdzM

2

2

22

222 πρρ α

αReRe

Since the Closed Loop Integral is nonzero only for 1/z component, we have

=≠

==

=

=⋅∫0'10

012

zenclosendoesCornif

zenclosesCandnifz

AResidueAi

zdz

A

Cn

πwe used:

04

2224

2

2 2

222

2

222

=

Γ−−=

Γ−−= ∞∞∫ π

πρπ

ρaUizd

zz

aUM

C

ReRe

αρ ieUiYiX −∞Γ=−

( )

Γ==

⇒Γ=−=+∞

∞−

UL

DUieYiXiLD i

ρρα 0

:

α

αX

Y−L

∞U

x

y

Zero Moment around the Origin.

174



On the Cylinder z = a e iθ

We found: viuz

i

z

eaUeU

zd

wd ii −=Γ−−=

+∞−

∞ π

αα

22

2

( )

( )

Γ−−=

=Γ−−==−=−

∞

−+∞

−∞

aUi

a

ieeUeeUe

zd

Wdeviuviv iiiiii

r

παθ

πθαθαθθ

θ

2sin2

2

Stagnation Points are the Points on the Cylinder for which vθ = 0:

( ) 02

sin2 =Γ−−=− ∞ aUv

παθθ

Γ+=∞

−

Uastagnation παθ

4sin 1

175


176

The Flow Pattern Around a Spinning Cylinderwith Different Circulations Γ Strengths


177



The Pressure Coefficient on the Cylinder Surface is given by:

( )2

2

2

22

2

2sin2

11

2

1∞

∞

∞∞

∞

Γ−−−=+−=

−=

U

aU

U

vv

U

ppC rSurface

Surfacep

παθ

ρθ

Using Bernoulli’s Law:

22

2

1

2

1∞∞ +=+ UpUp SurfaceSurface ρρ

( ) ( )

Γ−+

Γ−−−=∞∞ UaUa

CSurfacep π

αθπ

αθ4

sin84

4sin412

2

178


179

SOLO

Stream Lines

Flow Around a Cylinder

Streak Lines (α = 0º)

Preasure Field

Streak Lines (α = 5º)

Streak Lines (α = 10º) Forces in the Body

http://www.diam.unige.it/~irro/cilindro_e.html


180

SOLO

Velocity Field

http://www.diam.unige.it/~irro/cilindro_e.html

University of Genua, Faculty of Engineering,



181


C

'C

''C '''C

Corollary to Blasius Theorem

−=

−=

=

=−

∫∫

∫∫

'

22

'

22

22

22

CC

CC

zdzd

wdzzd

zd

wdzM

zdzd

wdizd

zd

wdiiYX

ρρ

ρρ

ReRe

C – Two Dimensional Curve defining Body BoundaryC’ – Any Other Two Dimensional Curve inclosing C such that No Singularity exist between C and C’

Proof of Corollary to Blasius Theorem

Add two Close Paths C” and C”’ , connecting C and C’, in opposite direction, s.t.

∫∫ −=''''' CC

then, since there are No Singularities between C and C’, according to Cauchy:

0'

0

'''''

=−++ ∫∫∫∫CCCC

q.e.d.∫∫ ='CC

therefore

182


183


Louis Melville Milne-Thomson

(1891-1974)

SOLO

184


185


186


187


AERODYNAMICSSubsonic Incompressible Flow (ρ∞ = const.) about Wings of Infinite Span (AR = ∞)

SOLO


188

Kutta Condition

We want to obtain an analogy between a Flow around an Airfoil and that around a Spinning Cylinder. For the Spinning Cylinder we have seen that when a Vortex isSuperimposed with a Doublet on an Uniform Flow, a Lifting Flow is generated.The Doublet and Uniform Flow don’t generate Lift. The generation of Lift is alwaysassociated with Circulation. Suppose that is possible to use Vortices to generate Circulation, and thereforeLift, for the Flow around an Airfoil. • Figure (a) shows the pure non-circulatory Flow around an Airfoil at an Angle of Attack. We can see the Fore SF and Aft SA Stagnation Points.•Figure (b) shows a Flow with a Small Circulation added. The Aft Stagnation Point Remains on the Upper Surface.•Figure (c) shows a Flow with Higher Circulation, so that the Aft Stagnation Point moves to Lower Surface. The Flow has to pass around the Trailing Edge. For an Inviscid Flow this implies an Infinite Speed at the Trailing Edge.•Figure (d) shows the only possible position for the Aft Stagnation Point, on the Trailing Edge. This is the Kutta Condition, introduced by Wilhelm Kutta in 1902, “Lift Forces in Flowing Fluids” (German), Ill. Aeronaut. Mitt. 6, 133.

Martin Wilhelm Kutta

(1867 – 1944)


1902

SOLO

Definition: A Stagnation Point is a point in a flow field where the local velocity of the fluid is zero

smb://upload.wikimedia.org/wikipedia/commons/3/32/Martin_Wilhelm_Kutta.jpg

189

Effect of Circulation on the Flow around an Airfoil at an Angle of Attack


Definition: A Stagnation Point is a point in a flow field where the local velocity of the fluid is zero

SF – Forward Stagnation Point SA – Aft Stagnation Point

Kutta Condition:SA on the Trailing Edge


190

Martin Wilhelm Kutta (1867 – 1944)

Nikolay Yegorovich Joukovsky (1847-1921

Kutta-Joukovsky Theorem

The Kutta–Joukowsky Theorem is a Fundamental Theorem of Aerodynamics. The theorem relates the Lift generated by a right cylinder to the speed of the cylinder through the fluid, the density of the fluid, and the Circulation. The Circulation is defined as the line integral, around a closed loop enclosing the cylinder or airfoil, of the component of the velocity of the fluid tangent to the loop. The magnitude and direction of the fluid velocity change along the path.

The force per unit length acting on a right cylinder of any cross section whatsoever is equal to ρ∞V ∞Γ, and is perpendicular to the direction of V ∞.

Kutta–Joukowsky Theorem:


19061902

Γ= ∞∞UL ρKutta–Joukowsky Theorem:

LCUL 2

2

1∞∞= ρLift:

Kutta in 1902 and Joukowsky in 1906, independently, arrived to this result.

Circulation ∫∫ =⋅=Γ θcos: ldVldV

SOLO


https://en.wikipedia.org/wiki/File:NikolaiYegorovichZhukovsky.jpg

191


General Proof of Kutta-Joukovsky TheoremUsing the Corollary to Blasius Theorem

Suppose that we wish to determine theAerodynamic Force on a Body of Any Shape.Use Corollary to Blasius Theorem, integratingRound a Circle Contour with a Large Radius andCenter on the Body

( ) zi

z

aUzUzw ln

2

2

πΓ−+= ∞

∞

The proof is identical to development in the Example ofFlow around a Two Dimensional Cylinder using

According to Corollary to Blasius Theorem we use C’ instead of C for Integration

z

i

z

aUU

zd

wd 1

22

2

πΓ−−= ∞

∞

LiftiDragUiUi

ii

z

UiResidue

i

zdz

Uiizd

z

i

z

aUU

izd

zd

wdizd

zd

wdiiYX

CCCC

+=Γ=

Γ−=

Γ−=

Γ−=

Γ−−=

=

=−

∞∞∞

∞∞∞ ∫∫∫∫

ρπ

πρπ

ρ

πρ

πρρρ

22

1

2

1

2

1

2222 ''

2

2

2

'

22

Therefore 0& =Γ== ∞ DragULLift ρq.e.d.

02 =

==⋅∫ zenclosesCif

z

AResidueAizd

z

A

C

πwhere we used:

C

'C

∞UL

D

192


D’Alembert Paradox

The fact that the Inviscid Flow Theories give Drag = 0 is called D’Alembert Paradox.

In 1768 d’Alembert enunciated his famous paradox in the following words:

“Thus I do not see, I admit, how one can satisfactorily explain by theory the resistance of fluids. On the contrary, it seems to me that the theory, developed in all possible rigor, gives, at least in several cases, a strictly vanishing resistance; a singular paradox which I leave to future geometers for elucidation.”

Jean-Baptiste le Rond d'Alembert

(1717 – 1783)

The resistance (Drag) experienced by a Real Airfoil is do to a combination of Skin-Friction and Pressure Distribution Distortions due to displacements effects of its Boundary Layers, which are not considered in the Inviscid Flow Theories.

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

193

The Kutta-Joukowsky Theory can be used to design Wings of Infinite Span that flow at Subsonic Speeds (Incompressible Flows). The design methods for such wings are called methods of “Profile Theory”.

AERODYNAMICS

Subsonic Incompressible Flow (ρ∞ = const.) about Wings of Infinite Span (AR = ∞)

Profile (of Airfoil) Theory can be treated in two different ways:

1.Conformal Mapping This Method is limited to 2 – dimensional problems. The Flow about a given body is obtained by using Conformal Mapping to transform it into a known Flow about another body (usually Circular Cylinder)

2.Method of Singularities The body in the Flow Field is substitute by Sources, Sinks, and Vortices, the so called Singularities.

For practical purposes the Method of Singularities is much simpler than Conformal Mapping. But, the Method of Singularities produces, in general, only ApproximateSolution, whereas Conformal Mapping leads to Exact Solutions, although these often require considerable effort.

SOLO

194

Joukovsky Airfoils

Joukovsky transform, named after Nikolai Joukovsky is a conformal map historically used to understand some principles of airfoil design.


Profile Theory Using Conformal Mapping

It is applied on a Circle of Radius R and Center at cx, cy. The radius to the Point (a,0) make an angle β to x axis. Velocity U∞ makes an angle αwith x axis.

β

xcyc

∞U

R

αx

y

( )0,a

The transform isz

az

2

+=ζ

( ) ββ sincosˆ RiRacicc yx +−=+=For α=0 we have

( ) ( ) ( )czi

cz

RczUzw ˆln

2ˆˆ

2

−Γ+

−

+−= ∞ πFor any α we have

( ) ( ) ( )cezi

cez

RcezUzw i

ii ˆln

2ˆˆ

2

−Γ+

−

+−= −−

−∞

αα

α

π

AERODYNAMICSSOLO


smb://upload.wikimedia.org/wikipedia/commons/d/d5/Jouktrans.png

195

Kutta-Joukovsky


( ) ( ) ( )cezi

cez

RcezUzw i

ii ˆln

2ˆˆ

2

−Γ+

−

+−= −−

−∞

αα

α

π

( ) viucez

i

cez

RUe

zd

wdii

i −=

−Γ+

−−= −−∞

−

ˆ1

2ˆ1 2

2

ααα

π

we have

Kutta Condition: The Flow Leaves Smoothly from the Trailing Edge.This is an Empirical Observation that results from the tendency ofViscous Boundary Layer to Separate at Trailing Edge.

Martin Wilhelm Kutta (1867 – 1944)

( ) ( )

( ) yxi

ii

i

azaz

caBcaABiA

i

BiA

RUe

cea

i

cea

RUe

zd

wdivu

+=−=

−Γ+

−

−=

−Γ+

−−===−

∞−

−−∞−

==

ααπ

π

α

ααα

sin:,cos:1

21

ˆ1

2ˆ10

2

2

2

2

( ) ( )[ ] ( ) ( )( )

+

Γ++−+

Γ+−−−+=

∞∞−

222

22222222222

22

2

BA

BAAURBAiBABBARBAU

e i ππα

Profile Theory Using Conformal MappingAERODYNAMICSSOLO



196

we have

( ) ( )[ ] ( ) ( )( )

+

Γ++−+

Γ+−−−+==

∞∞−

=222

22222222222

22

20

BA

BAAURBAiBABBARBAU

ezd

wd i

az

ππα

( ) βααβαα sinsinsin:,coscoscos: RacaBRaacaA yx +=+=−−=−=

( ) ( )( ) ( )[ ] 222

2222

coscos2cos12

sinsincoscos

RRaRa

RaaRaBA

≈−−++−=++−+=+

ββααβαβα

( )π2

20 222 Γ++−= ∞ BAAURBA ( )βαπππ sinsin44422

2

RaUUBUBBA

R +=≈+

=Γ ∞∞∞

( ) ( )[ ] ( ) ( ) ( )[ ]( ) ( )[ ] ( ) ( ) 0

22

22222222222

2222222222222222

≈−++=+−+=

−−−+=Γ+−−−+

∞∞

∞∞∞

RBABAUBARBAU

URBBARBAUBABBARBAUπ

Let check

For this value of Γ, we have

This value of Γ satisfies the Kutta Condition0=

=azzd

wd

Joukovsky Airfoils


197

Joukovsky Airfoils Design1. Move the Circle to ĉ and choose Radius R so that the Circle

passes through z = a.


β

xcyc

∞U

R

αx

y

( )0,a

for Center at z = 0.( ) zi

z

RzUzW ln

2

2

πΓ+

+= ∞

2. Change z-ĉ → z

( ) ( )czi

cz

RczUzW ˆln

2ˆˆ

2

−Γ+

−

+−= ∞ π3. Change z → z e-iα

( ) ( )cezi

cez

RcezUzW i

ii ˆln

2ˆˆ

2

−Γ+

−

+−= −−

−∞

αα

α

π4. Compute Γ from Kutta Condition

azazd

Wd

d

Wd

==

==2

0ςς ( )βαπ +=Γ ∞

<<

sin4ˆ

RUac



198

Joukovsky Airfoils Design (continue – 1)

5. Use the Transformation and computez

az

2

+=ζ

22 /1

//

za

zdWd

zd

d

zd

Wd

d

Wd

−== ς

ς6. To Compute Lift use either:

( )βαρπρ +=Γ= ∞∞ sin4 2RUUL6.1 Kutta-Joukovsky

6.2 Blasius( )

=−=− ∫ ς

ςραα d

d

WdieFiFeLi i

yxi

2

2''

6.3 Bernoulli( )

2

2/1

2/ ∞∞

∞ −=−=U

zdWd

U

ppC p ρ

−=

−= ∫∫∫∫

−−

∞

−−

a

a

p

a

a

p

a

a

Upp

a

a

Low xdCxdCU

xdpxdpLUL

2

2

2

2

22

2

2

2

''cos

2/''

cos

1

αρ

α

( ) ( )βαπβαπρ

+≈+==≈

≈∞

sin2sin82/ 42

cR

acL c

R

Uc

LC

'yF

'xF 'xF

∞U 'x

L

α

plane−ς

'y

α


199


7. To compute Pitching Moment about Origin use either:

7.2 Blasius

= ∫ ςς

ςρ

dd

WdiM p

2

20Re

7.1 Bernoulli

+−=

+−=

∫∫

∫∫

−−

∞

−−

a

a

p

a

a

p

a

a

Upp

a

a

Low

SpanUnitper

p

xdxCxdxCU

xdxpxdxpM

UL

2

2

2

2

2

2

2

2

2

''''2

''''0

ρ

'yF

'xF 'xF

∞U 'x

L

α

plane−ς

'y

α

0pM

απρ2sin4

222

0aUM p

= ∞

22

20

a

R

a

L

Mx p

p ≈==

( )βαρπ += ∞ sin4 2RUL


200


8. To Pitching Moment about Any Point x0 is given by:

+=+= ∞ Lmpp C

c

xCcULxMM

x

0220 000 2

ρ 'yF

'xF 'xF

∞U 'x

L

α

plane−ς

'y

α

0pM0xαπ 2sin4 22

0aCc m =

( )βαπ += sin2LC

( )

( )

++≈

++=

∞

<<+

≈

∞

βαπαπρ

βαπαπρ

βα

a

xaU

c

x

c

acUM

ac

px

0221

4

02

222

882

sin22sin420

+

+

≈ ∞

<<+

≈βαπρβα

a

x

a

xaUM

acpx

00221

418

20


201Generation of Joukowsky Profiles through Conformal Mapping

Symmetric Joukowsky Profile

Circular Joukowsky Profile

Cambered Joukowsky Profile


202

Profile Theory Using Conformal Mapping

AERODYNAMICSSOLO

203




204


205


206


207



208

Theodorsen Airfoil Design MethodTheodore Theodorsen working at NACA applied the Joukovsky inReverse and developed the following Design Method:

1. Given an Airfoil in ζ = ξ+i η Plane, arrange it with the Trailing Edge at ξ = 2a and Leading Edge at ξ=-2a

2. Transform from ζ = ξ+i η to z’ =a eψ eiθ through

''

2

z

az +=ς( )1

( )( )

=←==←=

θψηηθψηθψξξθψξ,sinsinh2

,coscosh2

a

a( )( )32

( ) ( )( ) ( )θψηηηψ

θψξξηθ

,/sinh2

,/sin2

222

222

=←++−=

=←++=

app

app( )( )54

Theodore Theodorsen (1897 – 1978)

planez''y

'xθ

ψea

x

yplanez −

φ0

0ψaeR =

ξplane−ς

η

a2a2−

Given ξ, η find ψ, θ using

22

221:

−

−=

aap

ηξwhere

T. Theodorsen, “Theory of Wing Sections with Arbitrary Shapes”, NACA Rept. 411, 1931 T. Theodorsen, I.E. Garrick, “General Potential Theory of Arbitrary Wing Sections”, NACA Rept. 452, 1933

Profile Theory Using Conformal MappingAERODYNAMICS

1931

SOLO

smb://upload.wikimedia.org/wikipedia/en/f/f2/Theodore_theodorsen.jpg

209


ξplane−ς

η

a2a2−

planez''y

'xθ

ψea

Theodorsen Airfoil Design Method (continue – 1)

3. Transform from z’ =a eψ eiθ to z = (a eψ0) eiф) through

( )

+=−−= ∑

∞

=10 expexp'

nn

nn

z

BiAzizz εψψ

Equaling Real and Imaginary Parts:

( ) ( )∑∞

=

+−=−=

1 00

sincosn

nn

nn n

R

An

R

B φφθφε( )8where An, Bn can be found by the following Iterative Procedure:

( ) ( )

( ) ( )

( )∫

∫

∫

=

=

=

π

π

π

φφψπ

ψ

φφφψπ

φφφψπ

2

0

0

2

00

2

00

2

1

sin1

cos1

d

dnR

B

dnR

A

nn

nn( )9

( )10

( )11x

yplanez −

φ0

0ψaeR =

Start with ( ) ( )φψθψ ≅

( )7 ( ) ( )∑∞

=

+=−

1 00

0 sincosn

nn

nn n

R

Bn

R

A φφψψ



210


ξplane−ς

η

a2a2−

planez''y

'xθ

ψea

x

yplanez −

φ0

0ψaeR =


4. Given Airfoil, Compute An, Bn, Cp, Γ

( ) ( )[ ] ( )[ ]( ) ( )[ ]222

22200

2

2

/1sinsinh

/1sinsin1

12/

0

θψθψθεεαφα

ρψ

dd

edd

U

q

U

ppC

T

p

++++++−=

−=−=

∞∞

∞

Procedure:

ii φφ −+1

4.2 Take , compute again An, Bn, ψ0 and εi+1 using (9), (10), (11) and (8) until is less then some predefined value .

ii εθφ +=

4.3 Compute Pressure Distribution

θφε −=4.1 Assume ε small and take . Compute An, Bn, ψ0 and using (9), (10), (11) and after that using (8).

θφ =0

where α0 is the Angle of Attack, and εT is the ε of the Trailing Edge



211


ξplane−ς

η

a2a2−

planez''y

'xθ

ψea

x

yplanez −

φ0

0ψaeR =


4. Given Airfoil, Compute An, Bn, Cp, Γ

( )TUea εαπ ψ +=Γ ∞ 0sin4 0

Procedure (continue):

4.4 Compute Γ

where α0 is the Angle of Attack, and εT is the ε of the Trailing Edge

4.5 Compute Lift

Γ= ∞UL ρ

5. Given we can compute for the Airfoil( ) 0,ψφε

5.1 From Compute An, Bn( ) ( ) ( )∑∞

=

+−=

1 00

sincosn

nn

nn n

R

An

R

B φφφε

5.2 Compute ( ) ( )φεφφθ −=

5.4 Compute ξ and η using (2) and (3).

5.3 Compute ( ) ( )∑∞

=

++=

1 00

0 sincosn

nn

nn n

R

Bn

R

A φφψψ




212


The Profile Theory was Initiated by Max Munk a student of Prandtl, who worked with him at the development of “Lifting Line Theory”, at Götingen University in Germany, between 1918-1919. He moved in 1920 to USA and worked at NACA for six years. At NACA, Munk developed an engineering-oriented method for Theoretical Prediction of Airfoil Lift and Moments, a method still in use today.His Theory applies to Thin Airfoils (t/c < 10%) and Small Angles of Attack. He approximate an Infinitely Thin Airfoil with its Main Camber Line. He published his results in a 1922 report, “General Theory of Thin Wings Sections” NACA Report 142.

Michael Max Munk(1890 – 1986)

Hermann Glauert(1892-1934)

Munk derived his results by using the idea of Conformal Mapping, from the Theory of Complex Variables. One year later, W. Birnbaum, in Germany, derived the same results by replacing the Main Camber Line with a Vortex Sheet (Singularities), given a simpler derivation of the Equations of Thin Airfoils. Finally in 1926 Hermann Glauert, in England, applied the solution of Fourier Series to the Solutions of those Equations. Glauert Hermann, “The Elements of Airfoil and Airscrew Theory”, Cambridge University Press, 1926.It is Glauert’s formulation that is still in use today.


213

2-D Inviscid Incompressible FlowProfile Theory by the Method of Singularities

Assumptions:1.Two Dimensional (x, z)2.Low Velocities (Incompressible)3.Irrotational4.Thin Airfoils5.Small Angles of Attack

Use Small Perturbation Theory:

( ) ( ) 0,0, 20

2

222 =Φ∇⇒=

∂Φ∂−Φ∇

≈zx

xMzx

M

( ) ( ) ( ) ( )zxzUxUzx ,sincos, ϕαα ++=Φ ∞∞ ( ) 0,2 =∇ zxϕBoundary Conditions: The Normal Velocity Component on the Airfoil Surface is Zero

nxd

zd

V

Velocity( ) ( ) ( )

zz

UxUzz

Uxx

U

zwUxuUzxV

Thin

∂∂++≈

∂∂++

∂∂+=

+++=Φ∇=

∞∞

<<

<<∂∂∞∞

∞∞

ϕαϕαϕα

ααα

ˆsinˆcos

sinˆcos,1

1:

Normal to Airfoil Surface zxd

zdxn

Surface

ˆˆˆ +−≅

0ˆ =

∂∂++−=⋅ ∞∞

SurfaceSurfacez

Uxd

zdUnV

ϕα

−=

∂∂=

−=

∂∂=

∞

∞

αϕ

αϕ

xd

zdU

zw

xd

zdU

zw

Lower

Lower

Lower

Upper

Upper

Upper

SOLO

214


Profile Theory by the Method of Singularities- Solution:

Solution is a Superposition (Linear Equations) of the Solutions for:• Skeleton (Camber Profile) • Teardrop (Symmetric Airfoil with same Thickness as the Original Airfoil)

Since the Small perturbation Theory leads to a Laplace’s Equationwe may use distribution of solutions (Singularities) to Laplace’s Equation-Sheet of Infinite Line Vortices on the Skeleton (needed for Lift production)-Sheet of Sources, Sinks on the Teardrop

( ) 0,2 =∇ zxϕ

The concept of replacing the Airfoil Surface with a Vortex Sheet is more then justa mathematical device; it also has physical significance. In the real life there is a Thin Boundary Layer on the Surface, due to friction between Flow and Airfoil.

Thickness

tt

Camber

CC

xd

zdU

zxd

zdU

zxd

zdU

zwCB

±=

∂∂+

−=

∂∂=

−=

∂∂= ∞∞∞

ϕαϕαϕ..

SOLO

215


Profile Theory by the Method of Singularities – Solution (continue -1)

Skeleton (Camber Profile)

Assume a Infinite Line (in +y direction) Vortex Sheet γ (x1) (to be defined) distributed on the x axis, between 0 ≤ x ≤ c.The total Circulation Γ is given by

x

z

c0

1x

( ) 11 xdxγ

( ) ( )( )1

111 2

,xx

xdxxxwd

−=

πγ

The contribution of the Vortex Sheet γ (x1) distributed between 0 ≤ x ≤ c must satisfy the Boundary Conditions on the Airfoil Surface

Using Biot-Savart Formula for a Two Dimensional Flow the tangent velocity caused by γ (x1) at x is given by

( )∫=Γc

xdx0

11γ

( ) ( )( )

−=

−= ∞

=

=∫∫

xSurface

cx

x xd

zdU

xx

xdxxxwd α

πγ

1

111

0 2,

1

1

SOLO

216


Profile Theory by the Method of Singularities – Solution (continue – 2)


x

z

c0

1x

( ) 11 xdxγ

( )( )

−=

− ∞∫xSurface xd

zdU

xx

xdx απ

γ1

11

2

Perform a transformation of variables

( ) cxxdcxdcx =→==→==→−= 111111111 &002/sin2/cos1 πθθθθθ( ) 2/cos1 θ−= cx

( )

−=

−⋅

∞∫x

xd

zdUd αθ

θθθγθ

π

π

0

11

11

coscos

sin

2

1

Solution for a Flat Plate dz/dx = 0

( ) αθθθ

θγθπ

π

∞=−⋅

∫ Ud0

11

11

coscos

sin

2

1

The Solution, that must also satisfy the Kutta Condition γ (π) = 0, is

( )1

11 sin

cos12

θθαθγ += ∞U

x

z

c0 1x

( ) 11 xdxγ

SOLO

217




Solution for a Flat Plate dz/dx = 0

To check the solution let substitute it in the Integral

Use Glauert Integral (1926)

Therefore

x

z

c0 1x

( ) 11 xdxγ

θθπθ

θθθπ

sin

sin

coscos

cos

0

11

1 nd

n =−∫

==

=1

00

n

n

π

( )∫∫ −

+=−⋅ ∞

ππ

θθθ

θπαθ

θθθγθ

π 0

11

1

0

11

11

coscos

cos1

coscos

sin

2

1d

Ud

( ) αθθθ

θπαθ

θθθγθ

π

ππ

∞∞ =

−+=

−⋅

∫∫ UdU

d0

11

1

0

11

11

coscos

cos1

coscos

sin

2

1

( )1

11 sin

cos12

θθαθγ += ∞U

SOLO

218




x

z

c0

1x

( ) 11 xdxγ

Solution for a Given Camber Profile z = z (x)

( )

−=

−⋅

∞∫x

xd

zdUd αθ

θθθγθ

π

π

0

11

11

coscos

sin

2

1

To determine the Vorticity Distribution we will write γ (θ1) as a Fourier Series(suggested by Glauert) that has to satisfy the Kutta Condition γ (θ1=π) = 0.

( )

++= ∑∞

=∞

11

1

101 sinsin

cos12

nn

PlateFlat

nAAU θθ

θθγ

To find the parameters An let substitute γ (θ1) in the Integral above

−=

−

+−

+∞

∞

=

∞∞ ∑ ∫∫xn

n xd

zdUd

nA

Ud

AU αθθθθθ

πθ

θθθ

π

π

π

π

1 0

11

11

0

11

10

coscos

sinsin

coscos

cos1

SOLO

219



x

z

c0

1x

( ) 11 xdxγ


−=− ∞

∞

=∞∞ ∑ xd

zdUnAUAU

nn αθ

10 cos

( )[ ] ( )[ ]

( )[ ] ( )[ ] θθ

θθθ

θθ

θθθ

θθπ

θθθθθ

π

ππ

nnnn

dnn

dn

IntegralGlauert

cossin

cossin2

2

1

sin

1sin1sin

2

1

coscos

1cos1cos

2

1

coscos

sinsin1

0

11

11

0

11

11

−=−=+−−=

−+−−=

− ∫∫

Therefore

or ∑∞

=

+−=1

0 cosn

n nAAxd

zd θα

Let compute

( ) ∑ ∫∫∫∞

=

+−=1 00

0

0

coscoscoscosn

n dmnAdmAdmxd

zd πππ

θθθθθαθθ

=≠

=∫ nm

nmdmn

2/

0coscos

0 πθθθ

π


SOLO

220


Profile Theory by the Method of Singularities - Solution (continue – 6)


x

z

c0

1x

( ) 11 xdxγ


∫−=π

θπ

α0

10

1d

xd

zdA

For a Symmetric Airfoil the Skeleton has d z/d x =0 (like for a Flat Plate)A0 = α, An = 0 for n=1,2,…

∑∞

=

+−=1

0 cosn

n nAAxd

zd θα

∫=π

θθπ 0

11cos2

dnxd

zdAn

SOLO

221




x

z

c0

1x

( ) 11 xdxγ


Lift Computation

( )( )

( )

( )

++=

++⋅=

==Γ

∫∑∫

∫ ∑

∫∫

∞

=∞

∞

=∞

−=

ππ

π

πθ

θθθθθ

θθθθ

θ

θθθγγ

0

11110

110

0

111

11

10

0

111

2/cos1

0

11

sinsincos1

sinsinsin

cos12

2

1

sin2

111

dnAdAUc

dnAAUc

dcxdx

nn

nn

cxc

( )1022

AAUc +=Γ ∞π

SOLO

222




x

z

c0

1x

( ) 11 xdxγ


Lift Computation (continue) ( )1022

AAUc +=Γ ∞π

( )10

2

22

AAcU

UL +=Γ= ∞∞ πρρ

( )102

2

21

: AAcU

LCL +==

∞

πρ ∫=

π

θθπ 0

111 cos2

dxd

zdA

πα

2=d

Cd L

The Angle of Attack α0 for which Lift is Zero is given by:

∫∫ +−=+=ππ

θθπ

θπ

α0

11

0

1010 cos22

220 dxd

zdd

xd

zdAA ( )∫ −=

π

θθπ

α0

110 cos12

dxd

zd

∫−=π

θπ

α0

10

1d

xd

zdA

SOLO

223




x

z

c0

1x

( ) 11 xdxγ


Chordwise Load Distribution

The Difference between the Upper and Lower Surface Flow Velocities can be computed in the following way:

( ) ( ) ( ) ( ) ( )[ ] 1111111 :& xdxVxVxdxdxxd LowererUpper −=Γ=Γ γ

Therefore ( ) ( ) ( )111 xVxVx LowererUpper −=γ

Also because the zero thickness of the Camber Surface( ) ( )112 xVxVU LowererUpper +≈∞

We have ( ) ( ) ( )121

212 xVxVUx LowererUpper −=∞γ

Use Bernoulli’s Equality ( ) ( ) ( ) ( )1211

21 2

1

2

1xVxpxVxp LowererLowerUpperUpper ρρ +=+

We get

( ) ( ) ( ) ( )[ ] ( )112

12

11 2

1xUxVxVxpxp LowererUpperUpperLower γρρ ∞=−=−

SOLO

224




x

z

c0

1x

( ) 11 xdxγ


Chordwise Load Distribution (continue)

We get

( ) ( ) ( )( )

++==− ∑

∞

=∞

−=

∞1

11

10

2cos1

2

1

111 sinsin

cos12

11

nn

x

UpperLower nAAUxUxpxp θθ

θργρθ

We can recover the Lift Equation using

( ) ( )[ ]( )

Γ=

++=

−==

∞

∞

=∞

−=

∫ ∑

∫∫

UdnAAcU

xdxpxpLdL

nn

x

c

UpperLower

c

ρθθθθ

θρπθ

0

111

11

10

2cos1

2

1

0

111

0

sinsinsin

cos12

11

( )10

2

22

AAcU

UL +=Γ= ∞∞ πρρ

SOLO

∫=π

θθπ 0

111 cos2

dxd

zdA

∫−=π

θπ

α0

10

1d

xd

zdA

225




x

z

c0

1x

( ) 11 xdxγ


Pitching Moment

Let MLE be the Pitching Moment about the Leading Edge

( ) ( )[ ] ( ) 1111111 xdxxUxdxpxpxMd UpperLowerLE γρ ⋅−=−⋅−= ∞

The Pitching Moment Coefficient: ( )2// 22cUMC LEmLE ∞= ρ

( )

( )∫ ∑ −

++−=

∞

=∞

∞

∞−= πθ

θθθθθ

θ

ρ

ρ

0

1111

11

10

22

cos12

1

sin2

1cos1

2

1sin

sin

cos12

21

11

dccnAAUcU

UC

nn

x

mLE

( )∫ ∑∑

−+−−=

∞

=

∞

=

π

θθθθθθ0

11

111

1112

0 2sinsin2

1sinsincos1 dnAnAA

nn

nn

( )210210 224422

AAAAAA −+−=

−+−= ππππ

( )102 AACL += π( ) ( )21210 44

122

4AACAAAC LmLE

−−−=−+−= ππ

SOLO

∫−=π

θπ

α0

10

1d

xd

zdA ∫=

π

θθπ 0

111 cos2

dxd

zdA

226




Solution for a Given Camber Profile z = z (x)Pitching Moment (continue)

Define the Center of Pressure Position as

( )102 AACL += π( ) ( )21210 44

122

4AACAAAC LmLE

−−−=−+−= ππ

( )

−+=

+

−+=

+−+=−=−=

2110

21

10

210

142

14

2/

2/

4

AAC

c

AA

AAc

AA

AAAc

C

Cc

L

Mx

L

L

mLECP

LE

π

LEML

cx

xM

For any point at a distance x from the Leading Edge we have

( ) ( )10210 2224

AAc

xAAAC

c

xCC Lmm LEx

++−+−=+= ππ

For x = c/4 we have: ( )1244

14/

AACCC Lmm LEc−=+= π

For a Thin Airfoil the Aerodynamic Center of the Section is at the Quarter-Chord Point, x = c/4.

Since A1 and A2 depend on the camber only, the section moment is independent of Angle of Attack. The point about which the section Moment Coefficient is independent of the Angle of Attack is called Aerodynamic Center of the Section.

SOLO

227




Solution for a Given Camber Profile z = z (x)Pitching Moment (continue)

Comparison of the Aerodynamic Coefficients calculated using Thin Airfoil Theory for two Cambered Airfoils:(a)NACA 2412 (b) NACA 2418 Data from Abbott and von Doenhoff (1949)

Comparison of the theoretical and the experimental Section Moment Coefficient (about the Aerodynamic Center) for two Cambered Airfoils:(a)NACA 2412 (b) NACA 2418 Data from Abbott and von Doenhoff (1949)

SOLO

228


Profile Theory by the Method of Singularities - Solution (continue – 14)Teardrop (Symmetric Airfoil)

xtz ( ) xdxσc0 P

Q

The Teardrop (Symmetric Airfoil) Surface is defined by

( ) ( ) ( ) 000 ==≤≤= cffcxxfzt

To find the Velocity Distribution over the Airfoil we use the Teardrop that has the same thickness as the Airfoil. The Flow, at Zero Angle of Attack, is symmetric on the Teardrop, producing Zero Lift (the Lift was computed on the Camber Profile). Therefore we will use a Sheet of Source, Sinks,σ (x1), distributed on x axis, 0 ≤ x1 ≤ c, to compute the Perturbed Velocity Distribution.

We shall make a First Order Approximation, that the Flow Perturbation are small, compared to Free Stream Velocity U∞, and that zt is small. Then the Flux cross any line such as PQ= 2 zt ,located at x, is 2 zt U ∞ . But all the Fluid generated by the Sources between Leading Edge and x must pass the line PQ. Therefore

( ) t

x

zUxdx ∞=∫ 20

11σ ( )xd

zdUx t

∞= 2σxd

d

SOLO

229


Profile Theory by the Method of Singularities - Solution (continue – 15)Teardrop (Symmetric Airfoil) (continue – 1)

xtz ( ) xdxσc0 P

Q

The Algebraic Sum of Sources and Sinks is Zero.

We have ( ) ( ) 020

0

11 == =

=∞∫ cx

xt

c

xzUxdxσ

( )xd

zdUx t

∞= 2σ

At the Leading Edge d zt/ dx > 0 (Sources), at the Trailing Edge d zt/ dx < 0 (Sinks).

To find the Perturbed Velocity Distribution let define first x = (1-cos θ)/2, write the function f as a function of θ, and express the function as a Fourier Series:

( ) ( ) ∑∞

===

11 sin

2

1

nn nBcfxf θθ

where Bn is given by

( )∫=π

θθθπ 0

1 sin4

dnfc

Bn

SOLO

230



For Sources and Sinks we found that exists only a Radial Velocity Component.In our case the Source/Sink σ (x1) dx1 will produce at a point P (x ) a Velocity Perturbation in x direction d uP given by

The Total Perturbation Velocity, due to all Sources/Sinks, is given by

( ) ( )( ) ( )1

11

1

111 2

2

2,

xx

xdxd

zdU

xx

xdxxxud

t

P −=

−=

∞

ππσ

Perform coordinate transformation

c0 P

x1x

Pu( ) 11 xdxσ

( ) ( )∫ −= ∞

c

P xx

xdxdfd

Uxu

0 1

11

π

( ) ( ) ∑∞

=

==1

1111

111

1

1 cos2

1

nn dnBncd

d

fdxd

xd

xfd θθθθθ ( )θθ coscos

2 11 −=− cxx

( ) ( )∫∑

−=

∞

=∞π

θθθ

θ

π 0

11

11

coscos

cosd

nBnU

xu nn

Pto obtain

SOLO

231



c0 P

x1x

Pu( ) 11 xdxσWe get

Use Glauert Integral

( )( )

∑∞

=∞

−==

1

2/cos1

sin

sin

n

nx

P

nBnUxu

θθθ

( )∫=π

θθθπ 0

1 sin4

dnfc

Bn

θθπθ

θθθπ

sin

sin

coscos

cos

0

11

1 nd

n =−∫

( ) ( )∫∑

−=

∞

=∞π

θθθ

θ

π 0

11

11

coscos

cosd

nBnU

xu nn

P

to obtain

SOLO

232

Flow over a Slender Body of Revolution Modeled by Source Distribution

AERODYNAMICS



SOLO


233

Airfoil DesignThe velocities at the Aviation beginning were Low Subsonic, therefore theAirfoils were designed for Subsonic Velocities. The Design was for HighLift to Drag Ratios.

In 1939 Eastman Jacobs at the NACA Langley, designed and tested the first Laminar Flow Airfoil. He create a Family of Airfoils calledNACA Sections.

Eastman Nixon Jacobs (1902 –1987)

Historical Overview of Subsonic Airfoils Shapes.

Examples of airfoils in nature and within various vehicles

AERODYNAMICSSOLO

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

234

Airfoil DesignAERODYNAMICSSOLO

http://www.google.com/url?sa=i&rct=j&q=&esrc=s&frm=1&source=images&cd=&cad=rja&docid=21OTOw5KJvBh0M&tbnid=X_K6LL7jCMZS-M:&ved=0CAUQjRw&url=http%3A%2F%2Fhyperphysics.phy-astr.gsu.edu%2Fhbase%2Ffluids%2Fairfoil.html&ei=Ug_xUe2iG4zFPKWZgNAO&psig=AFQjCNHGSHMebCkbr5pRJuciiAK5kf8VrQ&ust=1374838930159120

235

NACA Airfoils

Airfoil geometry can be characterized by the coordinates of the upper and lower surface. It is often summarized by a few parameters such as: •maximum thickness, •maximum camber, •position of max thickness, •position of max camber, •nose radius.

The Airfoil here is of an Infinite Span, flying in a Incompressible Flow. The Wing Profileis the Cross Section of the Wing.

The NACA 4 digit and 5 digit airfoils were created by superimposing a simple meanline shape with a thickness distribution that was obtained by fitting a couple of popular airfoils of the time:

( ) ( )5325.0 1015.2843.3537.126.2969.2.0/ xxxxxty ⋅−⋅+⋅−⋅−⋅⋅±=The camberline of 4-digit sections was defined as a parabola from the leading edge to the position of maximum camber, then another parabola back to the trailing edge.

NACA 4-Digit Series: 4 4 1 2 max camber position max thickness in % chord of max camber in % of chord in 1/10 of c

AERODYNAMICSSOLO

236

NACA Airfoils

After the 4-digit sections came the 5-digit sections such as the famous NACA 23012. These sections had the same thickness distribution, but used a camberline with more curvature near the nose. A cubic was faired into a straight line for the 5-digit sections. NACA 5-Digit Series: 2 3 0 1 2approx max position max thickness camber of max camber in% of chord in% chord in 2/100 of c

The 6-series of NACA airfoils departed from this simply-defined family. These sections were generated from a more or less prescribed pressure distribution and were meant to achieve some laminar flow.

NACA 6-Digit Series: 6 3 2 - 2 1 2Six- location half width ideal Cl max thickness Series of min Cp of low drag in tenths in% of chord in 1/10 chord bucket in 1/10 of Cl

SOLO

237

AERODYNAMICS

NACA Airfoils

SOLO

238

NACA Airfoils

Geometry of the most important NACA Profiles(a)Four-Digit Profiles(b)Five-Digit Profiles(c)6-Series Profiles

AERODYNAMICSSOLO

239

NACA Airfoils

12.04.002.0

2142

===

↓↓↓↓

c

t

c

xh

c

h

NACA

Lower Surface

Upper Surface

AERODYNAMICSSOLO

240

Effects of the Reynolds Number (Viscosity)µ

ρ cVRe =:

Effects of the Reynolds Number on the Lift and Drag characteristics of NACA 4412

AERODYNAMICS

NACA Airfoils

SOLO


241

Lifting-Line Theory

The Prandtl Lifting-Line Theory, also called the Lanchester–Prandtl Wing Theory is a mathematical model for predicting the Lift Distribution over a Three-Dimensional Wing based on its geometry.

Frederick William Lanchester

(1868 –1946)


The theory was expressed independently by Frederick W. Lanchester in 1907, and by Ludwig Prandtl in 1918–1919 after working with Albert Betz and Max Munk.In this model, the vortex strength reduces along the wingspan, and the loss in vortex strength is shed as a vortex-sheet from the trailing edge, rather than just at the wing-tips.

Albert Betz

(1885 – 1968 ),


1907 1918–1919

The Lifting-Line Theory makes use of the concept of Circulation and of the Kutta–Joukowski Theorem,

so that instead of the lift distribution function, the unknown effectively becomes the distribution of circulation over the span, Γ(y).

The lift distribution over a wing can be modeled with the concept of Circulation

A vortex is shed downstream for every span-wise change in lift

The Upwash and DownwashInduced by the Shed Vortex canBe computed at each NeighborSegment


SOLO

http://en.wikipedia.org/wiki/File:1931_-_Frederick_Lanchester.jpg

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

http://en.wikipedia.org/wiki/File:Lifting_line_theory_illustration_(1).svg



242

Frederick William Lanchester

(1868 –1946)

In 1907 Frederick William Lanchester, an English engineer(automotive and aerodynamics) published a two-volume work, Aerodynamics, dealing with the problems of powered flight. In it, he developed a model for the vortices that occur behind wings during flight, which included the first full description of Lift and Drag. His book was not well received in England, but created interest in Germany where the scientist, Ludwig Prandtl mathematically confirmed the correctness of Lanchester’s vortex theory (Lanchester visited Prandtl and vonKarman in Gotingen in 1908). In his second volume, he turned his attention to aircraft stability, aerodonetics, developing Lanchester's Phugoid Theory which contained a description of oscillations and stalls. During this work he outlined the basic layout almost all aircraft have used since then. Lanchester’s contribution to aeronautical science was not recognised until the end of his life.

1907Lifting-Line Theory

Subsonic Incompressible Flow (ρ∞ = const.) about Wings of Finite Span (AR < ∞)SOLO

http://en.wikipedia.org/wiki/File:1931_-_Frederick_Lanchester.jpg

243

Aerodynamic Load Distribution for a Rectangular Wingin Subsonic Airstream(a)Differential Pressure distribution along the chord for several spanwise stations.(b) Spanwise Lift Distribution

AERODYNAMICS


Consider a Subsonic IncompressibleFlow passing over a Finite Span Wing.

As a consequence of the tendency of thePressure acting on the Top Surface near the Tip of the Wing to equalize with those on the Bottom Surface, the Lift Force perUnit Span decreases toward the Tips.

The difference between the 3-D Flow andthe 2 – D Flow (over an Infinite Span Wing)is the Spanwise variation of Lift.

Lifting-Line Theory

SOLO

244

Generation of the Trailing Vortices due to the SpanwiseLoad Distribution:(a)View from bottom(b)View from Trailing Edge(c)Formation of the Tip Vortex(d)Smoke Flow Pattern showing Tip Vortex

AERODYNAMICS


Where the Flow from Upper Surface and theLower Surface join at the Trailing Edge, the difference in Spanwise Velocity Components will cause the Air to roll up into a number of Streamwise Vortices, distributed along the Span.

Since the Lift Force acting on the Wing SectionAt a given Spanwise location is related to theStrength of the Circulation. Therefor to evaluateThe Spanwise Lift distribution we can use a VortexSystem.

Lifting-Line Theory

SOLO

245

AERODYNAMICS


Vortex System of a Wing of Finite Span

Vortex System

Bound Vortex

Starting Vortex

The Vortex System consists of:•Bound Vortex around the Wing•Trailing (free) Vortices•Starting Vortex

The Bound Vortex around the Wing,the Two Free Vortices and theStarting Vortex form a closed VortexLine in agreement with HelmholtzVortex Theorem.

Lifting-Line Theory

SOLO

246

AERODYNAMICS


Bound + Trailing Vortex= Horseshoe Vortex

A schematic Vortex System for a Unswept Wing

Finite Unswept Wing

We replace the Spanwise Lift Distribution by a Single Bound Vortex System (the axis of which is normal to the plane of symmetry and passes through the Aerodynamic Center of the Lifting Surface). The single Vortex has a Circulation Γ whose strength varies along the Span Γ = Γ (y).

The Vortex System consists of theBound Vortex System and the related System of Trailing Vortices.The strength of the Trailing Vortex is

yyd

d ∆Γ=Γ∆

Lifting-Line Theory

SOLO

247

Start with Biot-Savart Formula for a Semi-infinite 2D Vortex

θπ

θθθπ

π

π

ˆ4

sinˆ4 2/ h

dh

VΓ=Γ= ∫

AERODYNAMICSSubsonic Incompressible Flow (ρ∞ = const.) about Wings of Finite Span (AR < ∞)

ydyd

d Γ

1yy

( )ywy1δ

1yy −

Trailing Vortex

θ

2

πθ =

h

θ

Semi-infinite 2D Vortex

The downstream-drifting free vortices produce a downwash velocity w behind and at the wing. The induced Velocity δ wy1 caused by theSemi-infinite Trailing Vortex located at y1,at a point y is given, using Biot-Savart, by

( ) ( )14

11 yy

yyd

dywy −

∆Γ=∆π

Hence the downwash induced velocity is

( ) ( )∫+

− −

Γ

=2/

2/ 11 4

1 b

b

i ydyy

ydd

ywπ

Lifting-Line Theory

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248

AERODYNAMICS


Di – Induced drag

Lift Effective Lift,acts normal to theEffective Flow Direction

Induced Flow

Because of the downwash velocity theFlow is disturbed. We have: α – Angle of Attack relative to Undisturbed Flow αi - Induced Angle of Attack αe - Effective Angle of Attack

( )

( )∫+

− −

Γ

−=

−=−=

2/

2/ 14

1

:

b

b

iie

ydyy

ydd

V

V

wy

πα

αααα

( ) ( )yVyl Γ= ∞ρBased on Kutta-Joukowsky Theorem, the Lift on an elemental Airfoil Section of theWing is

and the Induced Drag

Integrating over the entire span we get

( ) ( ) ( ) ( )yywylyd ii Γ−=−= ∞ραtan

( )∫+

− ∞ Γ=2/

2/

b

bydyVL ρ

( ) ( )∫+

− ∞ Γ−=2/

2/

b

bi dyyywD ρ

Lifting-Line Theory

SOLO

249

AERODYNAMICS

Subsonic Incompressible Flow (ρ∞ = const.) about Wings of Infinite Span (AR → ∞)

For Wings of Infinite Span (AR →∞) Γ is constant along the Span, i.e.

( ) ( )yVyl Γ= ∞ρBased on Kutta-Joukowsky Theorem, the Lift on an elemental Airfoil Section of theWing is

and the Induced Drag ( ) ( ) ( ) ( ) 0tan =Γ−=−= ∞ yywylyd ii ρα

Lifting-Line Theory

( ) ( ) 04

1

11

=−

∆Γ=∆yy

yyd

dywy π

0=Γ

∞→ARyd

d

( ) ( ) 04

1

11 =

−

Γ

= ∫∞+

∞−

ydyy

ydd

ywi π

For Wings of Infinite Span (AR →∞) the Induced Drag is zero

and

The Downwash Induced Velocity is zero for Infinite Span.

SOLO

250

AERODYNAMICS


Elliptic Circulation Distribution

Elliptic Circulation Distribution

A simple Circulation Distribution, which has also practical applications, is given by the Elliptical relation

( )2

12

0

bs

s

yy =

−Γ=Γ

then

( )2

1

1

20

bs

s

y

ssy

yyd

d =

−

−

Γ=Γ

( ) ( )

( )

( )∫

∫

∫

+

−

+

−

+

−

−−Γ−=

−

−

Γ−=

−

Γ

=

s

s

s

s

s

s

i

ydyyys

y

s

yd

yysy

sy

s

ydyy

ydd

yw

122

0

1

2

0

11

4

14

4

1

π

π

π

Lifting-Line Theory

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251

AERODYNAMICS


Elliptic Circulation Distribution (continue – 1)

Developing the two integrals gives

( )( )

( )( ) ( )

−−+

−−−Γ−=

−−Γ−=

∫∫

∫+

−

+

−

+

−

s

s

s

s

s

s

i

yyys

ydy

yyys

ydyy

s

yyys

ydy

syw

122

1

122

10

122

01

4

4

π

π

πθθ

θθ π

π

π

π

θ==

−=

− ∫∫∫+

−

+

−

⋅=+

−

2/

2/

2/

2/222

sin

22 sin

cosd

ss

d

ys

yd sys

s

( ) ( )Iy

yyys

ydy

yyys

ydy

I

s

s

s

s

1

1221

122

1 =−−

=−− ∫∫

+

−

+

−

( ) [ ]Iys

ywi 10

1 4+Γ−= π

πtherefore

but

( ) ( ) [ ] [ ]( )

044

122

0011 =

−−=⇒+Γ−=−Γ−⇒+==−= ∫

+

−

s

s

iiyyys

ydIsI

ssI

ssywsyw π

ππ

π

and ( )bs

ywbs

i 240

2/0

1

Γ−=Γ−==

Elliptic Circulation Distribution and theResultant Downwash Velocity

Lifting-Line Theory

SOLO

252



The Lift Coefficient of the Wing (Area = S)

alsoElliptic Circulation Distribution and the

Resultant Downwash Velocity

( )

∫

∫∫+

−∞

⋅=

+

−∞

+

− ∞

⋅⋅Γ=

−Γ=Γ=

2/

2/0

sin

2

0

coscos

1

π

π

φφφφρ

ρρ

dsV

yds

yVydyVL

sy

s

s

s

s

0

2/

0 42Γ=Γ= ∞

=

∞ VbsVLbs

ρππρ

SV

b

SV

LCL

0

2 221

Γ==∞

π

ρ

( ) ( ) ss

yds

y

sdyyywD

s

s

s

si 241

4

20

2

00 πρρρ Γ=

−ΓΓ=Γ= ∞

+

−∞

+

− ∞ ∫∫

208

Γ= ∞ρπiD

SVSV

DC i

Di 2

20

2 421

Γ== ∞

∞

ρπ

ρ

Lifting-Line Theory

SOLO

253



We found


SV

bCL

0

2

Γ= πSV

CiD 2

20

4

Γ= ∞ρπ

b

SVCL

π2

0 =Γ2

22

2

21

4 b

SC

b

SVC

SVC LL

Di ππρπ =

= ∞

Since the Wing Aspect Ratio is

S

bAR

2

=

we have

AR

CC L

Di π

2

=

Lifting-Line Theory

For AR → ∞0lim =

∞→ iDAR

C ( ) 0

2

0 1lim Γ=

−Γ=Γ

∞→ s

yy

s

SOLO

254



We found


SV

bCL

0

2

Γ= π

b

SVCL

π2

0 =ΓAR

C

b

SC

b

SVC

VbLLL

i πππα ===

2

2

2

1

Since the Wing Aspect Ratio isS

bAR

2

=

Lifting-Line Theory

SOLO

VbV

w bw

ii

i

20

20

Γ=−=

Γ−=

α

α

π

απ

AR

aa

AR

CL0

0

12

1

2

+=

+=

( )

−=−=

AR

CC L

i

a

L παπααπ 22

0

we have

255


( ) ( ) ( )

−

Γ

−=Γ ∫+

−

2/

2/ 1

1

4

1

2

1 b

b

L ydyy

ydd

VCycVy

παα

We have

and

c (y) is the local chord length

Di – Induced drag


Induced Flow

( ) ( )yVyl Γ= ∞ρ

( ) ( ) ( ) ( )

( )

( ) ( )

−

Γ

−=

∂∂≈=

∫+

−∞

∞∞

2/

2/ 1

12

22

4

1

2

1

2

1

2

1

b

b

L

e

C

LL

ydyy

ydd

VCycV

yC

ycVyCycVyl

L

παρ

αα

ρρ

α

α

From those two relations we get

Use the Transformation

1111 sin2

cos2

,cos2

φφφ byd

by

by =⇒−=−=

( ) ( )( )

−

Γ

−=Γ ∫+π

α φφφφ

φφπ

αφφ0

11

1

11 sin2coscos

2

sin2

4

1

2

1d

bb

bdd

VCcV L

Techniques for General Spanwise Circulation Distribution

Lifting-Line Theory

SOLO

256


We have

Assuming Γ ( =0) = Γ ( =π)=0 let consider theϕ ϕfollowing Fourier development of Γ ( ) ϕ

where the coefficients An have to be determined. Substitute this in the previous equation and use Glauert Integral Formula

Di – Induced drag


Induced Flow

( ) ( ) ( )

−

Γ

−=Γ ∫+π

α φφφ

φπ

αφφ0

11

1

coscos2

1

2

1d

dd

VCcV L

Techniques for General Spanwise Circulation Distribution (continue – 1)

1111 sin2

cos2

,cos2

φφφ byd

by

by =⇒−=−=

( ) ∑∞

==Γ

1sin2

n n nAbV φφ

φφπφ

φφφπ

sin

sin

coscos

cos

0

11

1 nd

n =−∫

( ) ( ) ( )

( )

−=

−−==Γ

∑

∫ ∑∑

∞

=

+ ∞

=∞

=

1

0

11

11

sinsin2

1

coscos

cos2

2

1

2

1sin2

nn

L

n nLn n

nnAb

CcV

dnnAbV

VCcVnAbV

φφ

ππ

αφ

φφφ

φπ

αφφφ

α

π

α

Lifting-Line Theory

SOLO

257


We have


1111 sin2

cos2

,cos2

φφφ byd

by

by =⇒−=−=

( ) ( ) ( )

( )

−=

−−==Γ

∑

∫ ∑∑

∞

=

+ ∞

=∞

=

1

0

11

11

sinsin2

1

coscos

cos

2

1sin2

nn

L

n nLn n

nnA

bCcV

dnnA

bCcVnAbV

φφ

παφ

φφφφ

αφφφ

α

π

α

Rearranging( ) ( ) φαφφφφ

µ

α

µ

α sin44

sinsin1

Ln Ln C

b

cC

b

cnnA =

+∑∞

=

or

[ ] ( )α

φµφαµµφφ Ln n Cb

cnnA

4:sinsinsin

1==+∑∞

= Monoplane Equation

[ ] NinnA i

N

n iin ,,1sinsinsin1

==+∑ =φαµµφφ

Di – Induced drag


Induced Flow

Lifting-Line Theory

SOLO

Assume that we know c (y)=c (-b cos /2), and using Cϕ Lα (2-D) = 2π, and we want to find the first N Fourier coefficients (Ai, i=1,…,N). We can chose N different0 <ϕ i < π (i=1,…,N), and compute the N equations, with N unknowns (Ai, i=1,…,N).

258


We have


( ) ( ) ∑∞

=∞∞ =Γ=1

2 sin42

1n n nAbVVl φρφρφ

( ) ( )

( ) ( )[ ] 122

1 0

22

0 10

cos22/

2/

2

11cos1cos

2

1

sinsin22

1sin

2

AbVdnnAbV

dnAbVbVdb

VydyVL

n n

n n

by

b

b

πρφφφρ

φφφρφφφρρ

π

ππφ

∞∞

=∞

∞

=∞∞

−=+

− ∞

=+−−=

=Γ=Γ=

∑ ∫

∫ ∑∫∫

( ) ∑∞

==Γ

1sin2

n n nAbV φφ

Lift Span

Total Lift

11

2

2

21

AARAS

b

SV

LCL ⋅⋅=⋅⋅==

∞

ππρ

122

2

1AbVL πρ∞=

Di – Induced drag


Induced Flow

Lifting-Line Theory

SOLO

259


We have



( ) ∑∞

==Γ

1sin2

n n nAbV φφ

Induced Velocity

Induced Drag on Span

( ) ( ) ( )

VnnAVdn

AnV

b

db

bnnAVbyd

yy

ydd

yw

in nn

nGlauert

n

yd

yd

dd

d

n n

byb

b

i

αφφφφ

φφπ

φφ

φφ

φφ

ππ

φφπ

π

π

φφ

φ

≈=−

=

−−=

−

Γ

=

∑∑ ∫

∫ ∑∫

∞

=

∞

=

Γ

∞

=

−=+

−

1 11

1

sin

sin

01

0

1

1

cos22/

2/ 11

sinsincoscos

cos

coscos2

sin2

sin2

1cos2

4

1

4

1

1

1

Di – Induced drag


Induced Flow

Lifting-Line Theory

SOLO

260

AERODYNAMICS


We have


( ) ( ) ( )

( ) ( ) ∑∑ ∑ ∫∫

∑ ∑ ∫

∫ ∑∑∫

∞

=∞∞

=

∞

=

⋅

∞

∞

=

∞

=∞

∞

=

∞

=∞

−=+

− ∞

=

++−=

=

=Γ=

1

222

1 1

0

00

22

1 1 0

22

0 11 1

cos22/

2/

2coscos

2

sinsin22

sin2

sin2sinsin

,

n nn m mn

n m mn

m mn n

by

b

bi

AnbV

dnmdnmAAnbV

dmnAAnbV

db

mAbVnnAV

dyyywD

nm

πρφφφφρ

φφφρ

φφφφφ

ρρ

π

δπ

π

π

πφ

( ) ∑∞

==Γ

1sin2

n n nAbV φφ

Induced Velocity

Induced Drag on Span

VnnAVb

yw in ni αφφ

φ ≈=

−= ∑∞

=1 11

11 sinsin

cos2

Total Drag on Wing

∑∞

=∞=

1

222

2 n ni AnbV

D πρ

Di – Induced drag


Induced Flow


Lifting-Line Theory

SOLO

261


We have


∑∞

=∞ ⋅⋅=

1

222

2 n ni AnbV

D πρ

∑∑ ∞

=

∞

=∞

⋅⋅=⋅⋅==1

2

1

22

2

2

n nn ni

D AnARAnS

b

SV

DC

iππ

ρ

Di – Induced drag


Induced Flow

11

2

2

21

AARAS

b

SV

LCL ⋅⋅=⋅⋅==

∞

ππρ

We found

∑∞

=

++++⋅

⋅=

⋅

⋅=

1 21

27

21

25

21

23

22

1

2 7531

nL

onsDistributilSymmetrica

TermsOddOnly

nLD A

A

A

A

A

A

AR

C

A

An

AR

CC

i

ππ

( ) 0753

:12

1

27

21

25

21

23

2

≥+++=+⋅⋅

= A

A

A

A

A

A

AR

CC L

Diδδ

π

CDi is minimum when δ = 0 : A1≠0, A3=A5=A7=…=0. In this case( ) onDistributiEllipticAbV φφ sin2 1=Γ

Lifting-Line Theory

SOLO

262

AERODYNAMICS


Experimental Drag Polar for a Wing with anAspect Ratio AR = 5 compared with theTheoretical Induced Drag

AR

CC D

Dv π

2

=

Effect of the Aspect Ratio on the Drag Polar for Rectangular Wings (AR from 1 to 7) (a) Measured Drag Polars (b) Drag Polar converted to AR = 5

SOLO


263



The Lifting-Line Theory gives good results for Unswept Wings with high AspectRatios. For small Aspect Ratios or Highly Swept Wings or Delta Wings we need an improved method to compute the Lifting Flow Field (for Incompressible Flow and Small Angles of Attack). The Lifting-Surface Theory approximates the continuous distribution of bound vorticity over Wing Surface by a finite number of Horseshoe Vortices.

Sketch of Coordinate System, Elemental Panels, and Horseshoe Vortices for a typical Wing in theVortex Lattice Method

The individual Horseshoe Vortices are placed in Trapezoidal Panels called Finite Elements or Lattices. Hence the procedure of obtaining a Numerical Solution for the Flow is namedVortex Lattice Method (VLM).

The VLM models the lifting surfaces, such as a wing, of an aircraft as an infinitely thin sheet of discrete vortices to compute lift and induced drag. The influence of the thickness, viscosity is neglected.

SOLO

264



Boundary Conditions

The Boundary Conditions require that the Flow is Tangent to the Wing Surface(Non-penetrating Flow Condition). For our simplified Lattice Model we require that this condition is satisfied at one point on the Flat Panel . This Point, where theBoundary Conditions are satisfied is called Control Point. We want that the Numerical Coefficients will not be affected by a change (small) in the Angle of Attack. From the Thin Airfoil Theory we found that Aerodynamic Center of the Section is at the Quarter-Chord Point, x = c/4. Therefore we placethe Bound Vortex on the Quarter-Chord Point of the Flat Panel.

BoundVortex

Lattice

TrailingVortex

TrailingVortex

∞Uαsin∞U

α

rV

π2

Γ=Lattice

4

c

BoundVortex

Boundary Conditions: Vr

UU =Γ=≈ ∞∞ παα

2sin

Γ=Γ== ∞∞

−

∞∞∞∞∞ U

rUcUcUl

JoukowskyKutta

AirfoilSymmetric

ρπ

πρπαρ2

22

1 222

cr =

Control Point Position

SOLO

265



Boundary Conditions

Sketch of the Distributed Horseshoe Vortices representing the Lifting Flow Field over a Swept Wing. It incudes the position of Bound Vortices (at c/4) and of Control Points (at 3c/4). This is known as the “1/4 – ¾ rule”. This placement works well and has become a rule of tumb. It was discovered by Italian Pistolesi.

ENRICO PISTOLESI(1889 - 1968)

SOLO

266



( )34 r

rldrVd n

×Γ=

π( )rVd

The Velocity induced at point C by a Vortex Filament of strength Γn (n is the index of n-Panel) and length dl is given by Biot-Savart Law

From Figure

θsinpr

r =

( )[ ] ( ) ( )( )

( )( ) θ

θθθθ

θθθθ

θθθθθθθθθ

2sinsinsin

sin

sinsin

sincossincoscotcot

dr

d

dr

d

ddrdrld

pp

pp

≈+

=

++−+=+−=

( )21

2121

21

21/ coscos

4sin

4

2

1rr

rr

rrr

rrd

rV

p

n

p

nABC

××−Γ=

××

Γ= ∫ θθπ

θθπ

θ

θ

The Velocity induced at point C by a Vortex Filament A B is given by

Velocity Induced by a General Horseshoe Vortex

SOLO

267



CV

( )21

2121/ coscos

4 rr

rr

rV

p

nABC

××−Γ= θθ

π

The Velocity induced at point C by a Vortex Filament A B is given by

Velocity Induced by a General Horseshoe Vortex

Define ABr =:0

We have

20

202

10

101

0

21 cos,cos,rr

rr

rr

rr

r

rrrp

⋅=⋅=

×= θθ

−⋅

××Γ=

2

2

1

102

21

21/ 4 r

r

r

rr

rr

rrV n

ABC

πTherefore

SOLO

268



Let write

( ) ( ) ( )( ) ( ) ( )( ) ( ) ( ) kzzjyyixxr

kzzjyyixxr

kzzjyyixxr

nnn

nnn

nnnnnn

2222

1111

1212120

−+−+−=

−+−+−=

−+−+−=

ABAB

r

r

r

rr

rr

rrV n

ABC

ΩΨ

−⋅

××Γ=

2

2

1

102

21

21/ 4π

( ) ( ) ( ) ( )[ ]( ) ( ) ( ) ( )[ ]( ) ( ) ( ) ( )[ ]( ) ( ) ( ) ( )[ ]( ) ( ) ( ) ( )[ ]( ) ( ) ( ) ( )[ ]

−−−−−+

−−−−−+

−−−−−

−−−−−+

−−−−−+

−−−−−

=

××=Ψ

21221

22112

21221

1221

2112

1221

2

21

21

nnnn

nnnn

nnnn

nnnn

nnnn

nnnn

AB

yyxxyyxx

zzxxzzxx

zzyyzzyy

kyyxxyyxx

jzzxxzzxx

izzyyzzyy

rr

rr

( ) ( ) ( ) ( ) ( ) ( )[ ]( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )[ ]( ) ( ) ( ) 2

22

22

2

212212212

21

21

21

112112112

2

2

1

10

nnn

nnnnnnnnn

nnn

nnnnnnnnn

AB

zzyyxx

zzzzyyyyxxxx

zzyyxx

zzzzyyyyxxxx

r

r

r

rr

−+−+−

−−+−−+−−−

−+−+−

−−+−−+−−=

−⋅=Ω

Contribution of Bounded Vortex

SOLO

269



Let write ( )( ) ( ) ( )( ) ( ) ( ) kzzjyyixxr

kzzjyyixxr

ixxADr

nnn

nnn

nn

1112

1131

310

−+−+−=

−+−+−=

−==

ADAD

r

r

r

rr

rr

rrV n

ADC

ΩΨ

−⋅

××Γ=

2

2

1

102

21

21/ 4π

( ) ( )( ) ( )[ ] ( )nnnn

nn

AD

xxyyzz

kyyjzz

rr

rr

132

12

1

11

2

21

21

−−+−−−−=

××=Ψ

( ) ( )( ) ( ) ( )

( )( ) ( ) ( )

−+−+−

−+−+−+−

−−=

−⋅=Ω

21

21

21

1

21

21

23

313

2

2

1

10

nnn

n

nnn

nnn

AD

zzyyxx

xx

zzyyxx

xxxx

r

r

r

rr

Contribution of Trailing Vortex A∞

The Velocity induced at point C by a Vortex Filament A D (D→∞)

( ) ( )( ) ( )[ ]

( )( ) ( ) ( )

−+−+−

−+−+−−−−Γ=∞ 2

12

12

1

12

12

1

11/ 1

4nnn

n

nn

nnnAC

zzyyxx

xx

yyzz

kyyjzzV

π

Taking x3n →∞, we obtain

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270



Finally we obtain, for the nth Panel.

( ) ( )( ) ( )[ ]

( )( ) ( ) ( )

−+−+−

−+−+−−−−Γ=∞ 2

12

12

1

12

12

1

11/ 1

4nnn

n

nn

nnnAC

zzyyxx

xx

yyzz

kyyjzzV

π

( ) ( ) ( ) ( )[ ]( ) ( ) ( ) ( )[ ]( ) ( ) ( ) ( )[ ]( ) ( ) ( ) ( )[ ]( ) ( ) ( ) ( )[ ]( ) ( ) ( ) ( )[ ]

( ) ( ) ( ) ( ) ( ) ( )[ ]( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )[ ]( ) ( ) ( )

AB

AB

nnn

nnnnnnnnn

nnn

nnnnnnnnn

nnnn

nnnn

nnnn

nnnn

nnnn

nnnn

nABC

zzyyxx

zzzzyyyyxxxx

zzyyxx

zzzzyyyyxxxx

yyxxyyxx

zzxxzzxx

zzyyzzyy

kyyxxyyxx

jzzxxzzxx

izzyyzzyy

V

Ω

Ψ

−+−+−

−−+−−+−−−

−+−+−

−−+−−+−−

−−−−−+

−−−−−+

−−−−−

−−−−−+

−−−−−+

−−−−−

Γ=

22

22

22

212212212

21

21

21

112112112

21221

22112

21221

1221

2112

1221

/ 4π

( ) ( )( ) ( )[ ]

( )( ) ( ) ( )

−+−+−

−+−+−−−−Γ−=∞ 2

22

22

2

22

22

2

22/ 1

4nnn

n

nn

nnnBC

zzyyxx

xx

yyzz

kyyjzzV

π

∞∞ ++= BCACABCC VVVV ///

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271



We obtained, for the nth Panel.

∞∞ ++= BCACABCC VVVV ///

Let the point (x,y,z) be the Control Point of the mth Panel, which will be designated as (xm, ym, zm).The Velocity induced at the m Control Point by the Vortex representing the nth Panel is designated as .We saw that this can be written as

where the coefficients depend on the geometry of the nth Horseshoe Vortex and its position relative to the mth Control Point.

nmV ,

nnmnm CV Γ= ,,

nmC ,

∑=

Γ=N

nnnmm CV

2

1,

Since the governing equations are Linear, the Velocity induced at mth Panel, by the 2N (Symmetry of the Wing) Vortices is the sum of the influence of all Vortices.

We have 2N equations, one for each Control Point, and 2N unknowns Γn.

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272



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The total induced velocity at the point m on the surface is due to the 2N vortices (N on each side of the planform) is

∑ =Γ=++= N

n nnmindmindmindmindm CkwjviuV2

1 ,,,,,

The solution requires the satisfaction of Boundary Conditions for the Total Velocity, which is the sum of the induced and free stream velocity. The freestream velocity is introduce the possibility of considering vehicles at combined angle of attack and sideslip

kVjViVV

βαββα cossinsincoscos ∞∞∞∞ +−=

so that the total velocity at point m is:

( ) ( ) ( ) kwVjvViuVVVV indmindmindmindmm

,,,, cossinsincoscos +++−++=+= ∞∞∞∞ βαββα

The values of the unknown circulations Γn, are found by satisfying the non-penetration boundary condition at all the control points simultaneously. For steady flow this is

0=⋅ nV

were the surface is given by

( ) 0,, =zyxF

Therefore 00 =∇⋅⇒=∇∇⋅ FV

F

FV

273



SOLO

This can be written as

00 =∇⋅⇒=∇∇⋅ FV

F

FV

( ) ( ) ( )[ ] 0cossinsincoscos ,,, =

∂∂+

∂∂+

∂∂⋅+++−++ ∞∞∞ k

z

Fj

y

Fi

x

FkwVjvViuV indmindmindm

βαββα

( ) ( ) ( )[ ]0

cossinsincoscos2

1 .

2

1 .

2

1 .

=

∂∂+

∂∂+

∂∂

⋅Γ++Γ+−+Γ+ ∑∑∑ =∞=∞=∞

kz

Fj

y

Fi

x

F

kCVjCViCVN

n nnm

N

n nnm

N

n nnm kji

βαββαor

( ) ( ) ( ) 0cossinsincoscos2

1 .

2

1 .

2

1 . =Γ+∂∂+Γ+−

∂∂+Γ+

∂∂ ∑∑∑ =∞=∞=∞

N

n nnm

N

n nnm

N

n nnm kjiCV

z

FCV

y

FCV

x

F βαββα

Performing the dot product we obtain:

Nmz

F

y

F

x

FVC

z

FC

y

FC

x

FN

n nnmnmnm kji2,,1cossinsincoscos

2

1 ... =

∂∂+

∂∂−

∂∂−=Γ

∂∂+

∂∂+

∂∂∑ = ∞ βαββα

Rearranging:

This is the general equation to obtain the circulations coefficients Γn.

274



SOLO

If the surface is in the x – y plane, and the sideslip is zero (β = 0), we obtain a simpler form.In this case the natural description of the surface is

and

( )yxfz ,=

The gradient of F is

( ) ( ) 0,,, =−= yxfzzyxF

1,, =∂∂

∂∂−=

∂∂

∂∂−=

∂∂

z

F

y

f

y

F

x

f

x

F

Nmx

fVCC

y

fC

x

fN

n nnmnmnm kji2,,1sincos

2

1 ... =

−

∂∂=Γ

+

∂∂−

∂∂−∑ = ∞ αα

This equation provides the solution for the circulations coefficients Γn, for this case.

Using this we obtain

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SOLO

Nmx

fVCw

N

n

m

cnnmm k

2,,12

1 . =

−

∂∂=Γ= ∑ = ∞ α

Consider the simple planar surface case, where there is no dihedral. Use the thin airfoil theory were boundary conditions can be applied on the mean surface, and not the actual camber surface. We also use the small angle approximations. Under these assumptions:

We have the following equation which satisfies the boundary conditions and can be used to relate the circulation distribution and the wing camber and angle of attack:

Nmx

f

VC

N

n

m

cnnm k

2,,12

1 . =

−

∂∂=

Γ∑ =∞

α

We have two cases:1.Given camber slopes and α , solve for the circulation strengths, (Γ/V∞) [ a system of 2N simultaneous linear equations].or2.Given (Γ/V∞), which corresponds to a specified surface loading, we want to find the camber and α required to generate this loading (only requires simple algebra, no system of equations must be solved).

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The Classical Vortex Lattice Method

There are many different vortex lattice schemes. In this section we describe the “classical”implementation. Knowing that vortices can represent lift from our airfoil analysis, and that oneapproach is to place the vortex and then satisfy the boundary condition using the “1/4 - 3/4 rule,”we proceed as follows:1. Divide the planform up into a lattice of quadrilateral panels, and put a horseshoevortex on each panel.

2. Place the bound vortex of the horseshoe vortex on the 1/4 chord element line of each panel.3. Place the control point on the 3/4 chord point of each panel at the midpoint in the spanwise direction (sometimes the lateral panel centroid location is used) .4. Assume a flat wake in the usual classical method.5. Determine the strengths of each Γn required to satisfy the boundary conditions by solving a system of linear equations. The implementation is shown schematically in Figure.

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A comparison between Panel Methods and Vortex Lattice Methods are:

Similar to Panel methods:• singularities are placed on a surface• the non-penetration condition is satisfied at a number of control points• a system of linear algebraic equations is solved to determine singularity strengths

Different from Panel methods:• Oriented toward lifting effects, and classical formulations ignore thickness• Boundary conditions (BCs) are applied on a mean surface, not the actual surface (not an exact solution of Laplace’s equation over a body, but embodies some additional approximations, i.e., together with the first item, we find ∆Cp, not Cpupper and Cplower)• Singularities are not distributed over the entire surface• Oriented toward combinations of thin lifting surfaces (recall Panel methods had no limitations on thickness).

278

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The Laplace’s Equation is used for Incompressible Flow. One of the key features of Laplace’s Equation is the property that allows the equation to be converted from a 3D problem to a 2D problem for finding the potential on the surface. The solution is then found using by distributing “singularities” of unknown strength over discretized portion of the surface: panels.The flow is found by representing the surface by a number of panels, and solving a linear set of algebraic equations to determine the unknown strengths of the singularities.

Subsonic Flow: Elliptic PDE, each point influences every other point.Supersonic Flow: Hyperbolic PDE, discontinuities exist, “zone of influence”

solution dependency.


279

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The flow pattern is uniquely defined by using either:

on ɸ ∑ + κ Dirichlet Problem Designor ∂ /∂n on ɸ ∑ + κ Neumann Problem Analysis

We can have also a mixed boundary condition, a + b ∂ /∂n ɸ ɸ on ∑ + κ.

The Dirichlet Problem corresponds to aerodynamic case where a surface pressure distribution is specified and the surface shape must be found. The Neumann Problem is used when the flow over the surface is defined (usually parallel to the surface.


Johann Peter Gustav Lejeune Dirichlet

1805-1859

280

AERODYNAMICS

Wing Configurations

Low wing Mid wingShoulder wing

High wingParasol wing

Monoplane - one wing plane. Since the 1930s most airplanes have been monoplanes. The wing may be mounted at various positions relative to the fuselage.

Biplane - two wing planes of similar size, stacked one above the other. The most common configuration until the 1930s, when the monoplane took over. The Wright Flyer I was a biplane.

Biplane Unequal-span biplane Sesquiplane Inverted sesquiplane

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

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

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

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

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

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

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

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

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

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

281

AERODYNAMICS

Wing ConfigurationsTriplane - three planes stacked one above another. Triplanes such as the Fokker Dr.I (Manfred von Richthofen - Red Baron, WWI As with 80 victories) enjoyed a brief period of popularity during the First World War due to their manoeuvrability, but were soon replaced by improved biplanes

Quadruplane - four planes stacked one above another. A small number of the Armstrong Whitworth F.K.10 were built in the First World War but never saw service.

Multiplane - many planes, sometimes used to mean more than one or more than some arbitrary number. The term is occasionally applied to arrangements stacked in tandem as well as vertically. The 1907 Multiplane of Horatio Frederick Phillips flew successfully with two hundred wing foils, while the nine-wing Caproni Ca.60 flying boat was airborne briefly before crashing.

Triplane Quadruplane Multiplane

Fokker DR1 Triplane2 × 7.92 mm (.312 in)

"Spandau" lMG 08 machine guns


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

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

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

282

AERODYNAMICS

Wing ConfigurationsWing Sweep

• Straight - extends at right angles to the line of flight. The most structurally-efficient wing, it is common for low-speed designs, such as the P-80 Shooting Star and sailplanes.

• Swept back, (aka "swept wing") - The wing sweeps rearwards from the root to the tip. In early tailless examples, such as the Dunne aircraft, this allowed the outer wing section to act like a conventional empennage (tail) to provide aerodynamic stability. At transonic speeds swept wings have lower drag, but can handle badly in or near a stall and require high stiffness to avoid aeroelasticity at high speeds. Common on high-subsonic and early supersonic designs e.g. the Hawker Hunter.

• Forward swept - the wing angles forward from the root. Benefits are similar to backwards sweep, also it avoids the stall problems and has reduced tip losses allowing a smaller wing, but requires even greater stiffness to avoid aeroelastic flutter as on the Sukhoi Su-47. The HFB-320 Hansa Jet used forward sweep to prevent the wing spar passing through the cabin. Small shoulder-wing aircraft may use forward sweep to maintain a correct CoG.

Straight Swept back, Forward swept


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

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

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

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

283

AERODYNAMICS

Wing ConfigurationsWing Sweep

Some types of variable geometry vary the wing sweep during flight:

• Swing-wing - also called "variable sweep wing". The left and right hand wings vary their sweep together, usually backwards. Seen in a few types of military aircraft, such as the General Dynamics F-111.

• Oblique wing - a single full-span wing pivots about its midpoint, so that one side sweeps back and the other side sweeps forward. Flown on the NASA AD-1 research aircraft.

Variable sweep Variable-geometry

NASA AD-1

General Dynamics F-111


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

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

http://upload.wikimedia.org/wikipedia/commons/f/f8/NASA_AD-1_in_flight.jpg

http://en.wikipedia.org/wiki/File:F-111A_Wing_Sweep_Sequence.jpg

284

AERODYNAMICS

Wing ConfigurationsChord Variation Along Span

The wing chord may be varied along the span of the wing, for both structural and aerodynamic reasons.

• Constant chord - parallel leading & trailing edges. Simplest to make, and common where low cost is important, e.g. in the Piper J-3 Cub but inefficient as the outer section generates little lift. Sometimes known as the Hershey Bar wing in North America due to its similarity in shape to a chocolate bar

• Tapered - wing narrows towards the tip, with straight edges. Structurally and aerodynamically more efficient than a constant chord wing, and easier to make than the elliptical type. It is one of the most common wing planforms, as seen on the F4F Wildcat

• Trapezoidal - a low aspect ratio tapered wing, where the leading edge sweeps back and the trailing edge sweeps forwards as on the Lockheed F-22 Raptor.

• Inverse tapered - wing is widest near the tip. Structurally inefficient, leading to high weight. Flown experimentally on the XF-91 Thunderceptor in an attempt to overcome the stall problems of swept wings.

• Compound tapered - taper reverses towards the root. Typically braced to maintain stiffness. Used on the Westland Lysander army cooperation aircraft to increase visibility for the pilot.

• Constant chord with tapered outer section - common variant seen for example on many Cessna types and the English Electric Canberra.

Constant Chord Tapered Trapezoidal Reverse tapered

Compound tapered

Constant chord,tapered outer


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

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

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

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

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

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

285

AERODYNAMICS

Wing ConfigurationsDelta WingsTriangular planform with swept leading edge and straight trailing edge. Offers the advantages of a swept wing, with good structural efficiency and low frontal area. Disadvantages are the low wing loading and high wetted area needed to obtain aerodynamic stability. Variants are

• Tailless Delta - a classic high-speed design, used for example in the widely built Dassault Mirage III series.

• Tailed Delta - adds a conventional tailplane, to improve handling. Popular on Soviet types such as the Mikoyan-Gurevich MiG-21.

• Cropped Delta - tip is cut off. This helps avoid tip drag at high angles of attack. At the extreme, merges into the "tapered swept" configuration.

• Compound Delta or double delta - inner section has a (usually) steeper leading edge sweep e.g. Saab Draken. This improves the lift at high angles of attack and delays or prevents stalling. Seen in tailless form on the Tupolev Tu-144 and the Space Shuttle. The HAL Tejas has an inner section of reduced sweep.

• Ogival Delta - a smoothly blended "wineglass" double-curve encompassing the leading edges and tip of a cropped compound delta. Seen in tailless form on the Concorde supersonic transports.

Tailless Delta Tailed Delta Cropped Delta Compound Delta

Ogival Delta


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

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

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

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

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

286

AERODYNAMICS

Wing ConfigurationsTailplanes and foreplanes

The classic aerofoil section wing is unstable in pitch, and requires some form of horizontal stabilizing surface. Also it cannot provide any significant pitch control, requiring a separate control surface (elevator) mounted elsewhere.

• Conventional - "tailplane" surface at the rear of the aircraft, forming part of the tail or empennage.• Canard - "foreplane" surface at the front of the aircraft. Common in the pioneer years, but from the

outbreak of World War I no production model appeared until the Saab Viggen appeared in 1967.• Tandem - two main wings, one behind the other. Both provide lift; the aft wing provides pitch stability

(as a usual tailplane) . An example is the Rutan Quickie. To provide longitudinal stability, the wings must differ in aerodynamic characteristics : wing loading and aerofoils must be different between the two wings.

• Three surface - used to describe types having both conventional tail and canard auxiliary surfaces. Modern examples include the Sukhoi Su-33 and Piaggio P.180 Avanti. Pioneer examples included the Voisin-Farman I and Curtiss No. 1.

• Tailless - no separate surface, at front or rear. The lifting and stabilizing surfaces may be combined in a single plane, as on the Short SB.4 Sherpa whose whole wing tip sections acted as elevons. Alternatively the aerofoil profile may be modified to provide inherent stability. Aircraft having a tailplane but no vertical tail fin have also been described as "tailless".

Conventional


Canard Tandem Three Surfaces Tailless

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

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

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

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

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

287

AERODYNAMICS

Wing ConfigurationsDihedral and Anhedral


Angling the wings up or down spanwise from root to tip can help to resolve various design issues, such as Stability and Control in Flight.

• Dihedral - the tips are higher than the root as on the Boeing 737, giving a shallow 'V' shape when seen from the front. Adds lateral stability.

• Anhedral - the tips are lower than the root, as on the Ilyushin Il-76; the opposite of dihedral. Used to reduce stability where some other feature results in too much stability

Dihedral Biplane with Dihedralon both wings

Biplane with Dihedralon lower wing

Some biplanes have different degrees of dihedral/anhedral on different wings; e.g. the Sopwith Camel had a flat upper wing and dihedral on the lower wing, while the Hanriot HD-1 had dihedral on the upper wing but none on the lower.

Anhedral

Ilyushin Il-76Boeing 737, Sopwith Camel

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

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

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

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

http://www.google.com/url?sa=i&source=images&cd=&cad=rja&docid=HV0Fha-PT1L5BM&tbnid=za8PLBovGeLATM:&ved=0CAgQjRw&url=http%3A%2F%2Fwww.defenceweb.co.za%2Findex.php%3Foption%3Dcom_content%26view%3Darticle%26id%3D18004&ei=XceiUum3BKen0AW9pIDICA&psig=AFQjCNGR9diWwhDuzXRsjYHcNEm9GMp8HA&ust=1386485981110251

http://en.wikipedia.org/wiki/File:Sopwith_F-1_Camel_2_USAF.jpg

288

AERODYNAMICS

Wing ConfigurationsPolyhedral Wings


In a polyhedral wing the dihedral angle varies along the span.

• Gull wing - sharp dihedral on the wing root section, little or none on the main section, as on the PZL P.11 fighter. Sometimes used to improve visibility forwards and upwards and may be used as the upper wing on a biplane as on the Polikarpov I-153.

• Inverted gull - anhedral on the root section, dihedral on the main section. The opposite of a gull wing. May be used to reduce the length of wing-mounted undercarriage legs or allow a larger propeller. Two well-known examples of the inverted gull wing are World War II's American F4U Corsair , and the German Junkers Ju 87 Stuka dive bomber.

• Cranked - tip section dihedral differs from the main section. The wingtips may crank upwards as on the F-4 Phantom II or downwards as on the Northrop XP-56 Black Bullet. (Note that the term "cranked" varies in usage. Here, it is used to help clarify the relationship between changes of dihedral nearer the wing tip vs. nearer the wing root. See also Cranked arrow planform.)

Gull WingInverted Gull Wing Upward Cranked Tips Downward Cranked Tips

PZL P.11 Junkers Ju 87 Stuka F-4 Phantom

Northrop XP-56 Black Bullet

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

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

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

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

http://www.google.com/url?sa=i&source=images&cd=&cad=rja&docid=x1S8-3YobMtgTM&tbnid=kQ_yt8Y2rMgIRM:&ved=0CAgQjRw4Yg&url=http%3A%2F%2Fwww.nurflugel.com%2FNurflugel%2FNorthrop%2Fxp-56%2Fbody_xp-56.html&ei=69aiUsT-FoTE0QXpzoFI&psig=AFQjCNEWMIK7ZIeMYtHiOPIygTqPXXbz4g&ust=1386489963416898

289

AERODYNAMICSWing Configurations

Variable Planform


• Variable-Sweep Wing or Swing-Wing. The left and right hand wings vary their sweep together, usually backwards. The first successful wing sweep in flight was carried out by the Bell X-5 in the early 1950s. In the Beech Starship, only the canard foreplanes have variable sweep.

• Oblique Wing - a single full-span wing pivots about its midpoint, as used on the NASA AD-1 , so that one side sweeps back and the other side sweeps forward.

• Telescoping Wing - the outer section of wing telescopes over or within the inner section of wing, varying span, aspect ratio and wing area, as used on the FS-29 TF glider. The Makhonine Mak-123 was an early example.[22]

• Extending Wing or Expanding Wing - part of the wing retracts into the main aircraft structure to reduce drag and low-altitude buffet for high-speed flight, and is extended only for takeoff, low-speed cruise and landing. The Gérin Varivol biplane, which flew in 1936, extended the leading and trailing edges to increase wing area.[23]

• Bi-directional Wing - a proposed design in which a low-speed wing and a high-speed wing are laid across each other in the form of a cross. The aircraft would take off and land with the low-speed wing facing the airflow, then rotate a quarter-turn so that the high-speed wing faces the airflow for supersonic flight

Variable sweep(swing-wing)

Variable-geometryoblique wing

Telescoping wing Extending wing Bi-directional flying wing

Bell X-5 NASA AD-1 Makhonine Mak-123

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290

AERODYNAMICSWing ConfigurationsMinor Aerodynamic Surfaces


Aircraft may have additional minor aerodynamic surfaces. Some of these are treated as part of the overall wing configuration:

• Winglet - a small vertical fin at the wingtip, usually turned upwards. Reduces the size of vortices shed by the wingtip, and hence also tip drag.

• Strake - a small surface, typically longer than it is wide and mounted on the fuselage. Strakes may be located at various positions in order to improve aerodynamic behaviour. Leading edge root extensions (LERX) are also sometimes referred to as wing strakes.

• Chine - long, narrow sideways extension to the fuselage, blending into the main wing. As well as improving low speed (high angle of attack) handling, provides extra lift at supersonic speeds for minimal increase in drag. Seen on the Lockheed SR-71 Blackbird.

• Moustache - small high-aspect-ratio canard surface having no movable control surface. Typically is retractable for high speed flight. Deflects air downward onto the wing root, to delay the stall. Seen on the Dassault Milan and Tupolev Tu-144

Moustache, chines, wingletsand nose and ventral strakes

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291


Minor Surface Features


Additional minor features may be applied to an existing aerodynamic surface such as the main wing:

• High-lift devices - some of these are visible aerodynamic features: • Slot - a spanwise gap behind the leading edge section, which forms

a small aerofoil or slat extending along the leading edge of the wing. Air flowing through the slot is deflected by the slat to flow over the wing, allowing the aircraft to fly at lower air speeds. Leading edge slats are moveable extensions which open and close the slot.

• Flap - trailing-edge (or leading-edge) wing section which may be angled downwards for low-speed flight, especially when landing. Some types also extend backwards to increase wing area.

• Wing spanwise flow control devices : • Vortex generator - small triangular protrusion on the upper leading wing surface; usually,

several are spaced along the span of the wing. The vortices re-energise the boundary layer and thereby both reduce the stall speed and improve the effectiveness of control surfaces at low speeds.

• Wing fence - a flat plate extending along the wing chord and for a short distance vertically. Used to control spanwise airflow over the wing.

• Vortilon - a flat plate attached to the underside of the wing near its leading edge, roughly parallel to normal airflow, used to increase lift and reduce stalling at low speeds.

• Notched leading edge.[27]

• Dogtooth leading edge

Vortex generators, root fillet, flap,anti-shock body and wing fence

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292


Minor Surface Features


Vortex generators, root fillet, flap,anti-shock body and wing fence

• Leading edge extensions of various kinds.• Anti-shock body - a streamlined "pod" shaped body added to the

leading or trailing edge of an aerodynamic surface, to delay the onset of shock stall and reduce transonic wave drag. Examples include the Küchemann carrots on the wing trailing edge of the Handley Page Victor B.2, and the tail fairing on the Hawker Sea Hawk.

• Fillet - a small curved infill at the junction of two surfaces, such as a wing and fuselage, blending them smoothly together to reduce drag.

• Fairings of various kinds, such as blisters, pylons and wingtip pods, containing equipment which cannot fit inside the wing, and whose only aerodynamic purpose is to reduce the drag created by the equipment

Handley Page Victor B.2 Hawker Sea Hawk

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293

Wing Parameters

Airfoil: The cross-sectional shape obtained by the intersection of the wing with the perpendicular plane

1. Wing Area, S, is the plan surface of the wing.

2. Wing Span, b, is measured tip to tip.

3. Wing average chord, c, is the geometric average. The product of the span andthe average chord is the wing area (b x c = S).

4. Aspect Ratio, AR, is defined as:

( )∫−

=2/

2/

b

b

dyycS

( )b

Sdyyc

bc

b

b

== ∫−

2/

2/

1

S

bAR

2

=

AERODYNAMICSSOLO

294

Wing Parameters (Continue)

5. The root chord, , is the chord at the wing centerline, and the tip chord, is the chord at the tip.

6. Taper ratio,

7. Sweep Angle, is the angle between the line of 25 percent chord and the perpendicularto root chord.

8. Mean aerodynamic chord,

rc

Λ

r

t

c

c=λ

tc

λ

( )[ ]∫−

=2/

2/

21~b

b

dyycS

c

c~

AERODYNAMICSSOLO

295

Wing Parameters (Continue)

AERODYNAMICS

Illustration of Wing Geometry

Planform, xy plane

Dihedral (V form), yz plane

Profile, twist xz plane

Geometric Designation of Wings of various planform

Swept-backWing

DeltaWing

EllipticWing

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296

Wing Design Parameters

•Planform - Aspect Ratio - Sweep - Taper - Shape at Tip - Shape at Root•Chord Section - Airfoils - Twist•Movable Surfaces - Leading and Trailing-Edge Devices - Ailerons - Spoilers•Interfaces - Fuselage - Powerplants - Dihedral Angle

AERODYNAMICSSOLO

297Stengel, Aircraft Flight Dynamics, Princeton, MAE 331, Lecture 5

AERODYNAMICSSOLO


Airfoil Effects

•Camber increases Zero-α Lift Coefficient•Thickness - Increases α for stall and the stall break - Reduces Subsonic Drag - Increases Transonic Drag - Causes abrupt Pitching Moment variation•Profile Design - Can reduce C.P. (Static Margin) variation with α - Affects Leading-Edge and Trailing-Edge Flow Separation

AERODYNAMICSSOLO


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308

At High Angles of Attack

-Flow Separates

-Wing loses Lift

Flow SeparationProduces Stall

AERODYNAMICS

Stall is a reduction in the lift coefficient generated by a foil as angle of attack increases. This occurs when the critical angle of attack of the foil is exceeded. The critical angle of attack is typically about 15 degrees, but it may vary significantly depending on the fluid, foil, and Reynolds number.

SOLO

309

AERODYNAMICSSOLO


310

AERODYNAMICSSOLO

Continue to Aerodynamics – Part II


311

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 toTheoretical 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

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312

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 theirHistorical Development”,

Prentice-Hall, 1979

L.J. Clancy“Aerodynamics”,

John Wiley & Sons, 1975

References (continue – 1)

AERODYNAMICSSOLO

313

Frank G. Moore“Approximate Methods

for Missile Aerodynamics”, AIAA, 2000

Thomas J. Mueller, Ed.“Fixed and Flapping WingAerodynamics for Micro Air

Vehicle Applications”, AIAA, 2002

Richard S. Shevell“Fundamentals of Flight”, Prentice Hall, 2nd Ed., 1988 Ascher H. Shapiro

“The Dynamics and Thermodynamicsof 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


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314

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


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February 9, 2015 315

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TechnionIsraeli Institute of Technology

1964 – 1968 BSc EE1968 – 1971 MSc EE

Israeli Air Force1970 – 1974

RAFAELIsraeli Armament Development Authority

1974 –2013

Stanford University1983 – 1986 PhD AA

316


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.


317


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)


Sir Geoffrey Ingram Taylor

(1886 – 1975)

ENRICO PISTOLESI(1889 - 1968)

Antonio Ferri(1912 – 1975)

Osborne Reynolds (1842 –1912)

318

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)

319

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)

321

SOLO Complex VariablesConformal Mapping

Transformations or Mappings

x

y

u

v

r

xd

yd

r

ud

vdA B

CD

'A

'B

'C'DThe set of equations ( )

( )

==

yxvv

yxuu

,

,

define a general transformation or mapping between (x,y) plane to (u,v) plane.

If for each point in (x,y) plane there corresponds one and only one point in (u,v)plane, we say that the transformation is one to one.

vdv

rud

u

rvdy

v

yx

v

xudy

u

yx

u

x

yvdv

yud

u

yxvd

v

xud

u

xyydxxdrd

u

r

u

r

∂∂+

∂∂=

∂∂+

∂∂+

∂∂+

∂∂=

∂∂+

∂∂+

∂∂+

∂∂=+=

∂∂

∂∂

1111

1111

If is a vector that defines a point A in (x,y) plane, we have: ( ) ( )vuryxr ,,

=r

The area dx dy of a region A,B,C,D, in (x,y) plane is mapped in the area A’,B’,C’,D’, du dv in the (u,v) plane. We have

zvdudu

y

v

x

v

y

u

xvdudy

v

yx

v

xy

u

yx

u

x

vdudv

r

u

rzydxdydxd

y

r

x

rSd

yx

11111

1

11

∂∂

∂∂−

∂∂

∂∂=

∂∂+

∂∂×

∂∂+

∂∂=

∂∂×

∂∂==

∂∂×

∂∂=

If x and y are differentiable

322


Transformations or Mappings( )( )

==

yxvv

yxuu

,

,

The transformation is one to one if and only if, for distinct points A, B, C, D, in (x,y)we obtain distinct points A’,B’,C’,D’, in (u,v). For this a necessary (but not sufficient)condition:

''''det1det

11

DCBA

ABCD

Sd

v

y

u

y

v

x

u

x

zvdud

v

y

u

y

v

x

u

x

zvdudu

y

v

x

v

y

u

xzydxdSd

∂∂

∂∂

∂∂

∂∂

=

∂∂

∂∂

∂∂

∂∂

=

∂∂

∂∂−

∂∂

∂∂==

Transformation is one to one 00 '''' ≠⇔≠ DCBAABCD SdSd

( )( ) 0det:

,

, ≠

∂∂

∂∂

∂∂

∂∂

=∂∂

v

y

u

y

v

x

u

x

vu

yxJacobian of theTransformation

By symmetry (change x,y to u,v) we obtain:

ABCDDCBA Sd

y

v

x

v

y

u

x

u

Sd

∂∂

∂∂

∂∂

∂∂

=det''''

1detdet =

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

v

y

u

y

v

x

u

x

y

v

x

v

y

u

x

u

one to one

transformation ( )( )

( )( ) 1

,

,

,

, =∂∂

∂∂

vu

yx

yx

vu

x

y

u

v

r

xd

yd

r

ud

vdA B

CD

'A

'B

'C'D

323


Complex Mapping

In the case that the mapping is done by a complex function, i.e.

( ) ( )yixfzfviuw +==+=

we say that f is a complex mapping.If f (z) is analytic, then according to Cauchy-Riemann equation:

( )( )

( ) 2222

det,

,

zd

zfd

y

ui

x

u

y

u

x

u

x

v

y

u

y

v

x

u

y

v

x

v

y

u

x

u

yx

vu =∂∂+

∂∂=

∂∂+

∂∂=

∂∂

∂∂−

∂∂

∂∂=

∂∂

∂∂

∂∂

∂∂

=∂∂

x

v

y

u

y

v

x

u

∂∂−=

∂∂

∂∂=

∂∂

&

If follows that a complex mapping f (z) is one to one in regions where df/dz ≠ 0.

Points where df/dz = 0 are called critical points.

324


Complex Mapping – Riemann’s Mapping Theorem

In the case that the mapping is done by a complex function, i.e.( ) ( )yixfzfviuw +==+=

Georg Friedrich BernhardRiemann1826 - 1866

we have:

x

y

u

vC 'C

1

RR' Let C be the boundary of a region R in the z plane,

and C’ a unit circle, centered at the origin of thew plane, enclosing a region R’.

The Riemann Mapping Theorem states that for each pointin R , there exists a function w = f (z) that performs aone to one transformation to each point in R’.

Riemann’s Mapping Theorem demonstrates the existence of theone to one transformation to region R onto R’, but it not providesthis transformation.

325


Complex Mapping (continue – 1)( )( )

==

yxvv

yxuu

,

,

x

y

u

v

r

2zd

1zd

r

2wd

1wdA

B

C

'A

'B

'C


Consider a point A in (x,y) plane mapped to pointA’ in (u,v) plane

Consider a small displacement from A to Bdefined as dz1, that is mapped to a displacementfrom A’ to B’ defined as dw1

( ) ( ) ( )

+

===1

1

argarg

11

arg

11

zdzd

zfdi

AA

wdi Aezdzd

zfdzd

zd

zfdewdwd

Consider also a small displacement from A to C defined as dz2, that is mapped to a displacement from A’ to C’ defined as dw2

( ) ( ) ( )

+

===2

2

argarg

22

arg

22

zdzd

zfdi

AA

wdi Aezdzd

zfdzd

zd

zfdewdwd

We can see that dw ≠ 0 if dz ≠ 0, i.e. a one-to-one transformation, if and only if

( )0≠

Azd

zfd

326


Complex Mapping (continue – 2)( )( )

==

yxvv

yxuu

,

,

x

y

u

v

r

2zd

1zd

r

2wd

1wdA

B

C

'A

'B

'C


Consider a point A in (x,y) plane mapped to pointA’ in (u,v) plane

( ) ( ) ( )

+

===1

1

argarg

11

arg

11

zdzd

zfdi

AA

wdi Aezdzd

zfdzd

zd

zfdewdwd

( ) ( ) ( )

+

===2

2

argarg

22

arg

22

zdzd

zfdi

AA

wdi Aezdzd

zfdzd

zd

zfdewdwd

We can see that:

( ) ( ) ( ) ( )

12

1212

argarg

argargargargargarg

zdzd

zdzd

zfdzd

zd

zfdwdwd

AA

−=

+−

+=−

Consider two small displacements from A to BAnd from A to C, defined as dz1 and dz2, that are mapped to displacements from A’ to B’ and from A’ to C’, defined as dw1 and dw2

Therefore the angular magnitude and sense between dz1 to dz2 is equal to that between dw1 to dw2. Because of this the transformation or mapping is called aConformal Mapping.Return to Joukovsky Airfoils

327

SOLO

Glauert Integral Formula (1926) Proof

θθπθ

θθθπ

sin

sin

coscos

cos

0

11

1 nd

n =−∫

Consider the Integral

∫ −=

π

θθθθ

θ

0

11

1 sincoscos

cos: d

nI

( ) ( )

( )

( )

( )

( ) ( ) ( ) ( ) ( )11111

1

1

1

111

21

cos21

sin21

sin21

cos

1

21

sin2

21

cos

21

sin2

21

cos

21

sin21

sin2

1

coscos

1

θθθθθθθθθθ

θθ

θθ

θθ

θθθθθθ

−++−+

−

−+

+

+=

−+=

−

But( ) ( ) ( )

( ) ( ) ( )111

111

sinsin2

1

2

1cos

2

1sin

sinsin2

1

2

1sin

2

1cos

θθθθθθ

θθθθθθ

+=−+

−=−+

Therefore( )

( )

( )

( )

−

−+

+

+=

−1

1

1

1

1

21

sin

21

cos

21

sin

21

cos

2

1sin

coscos

1

θθ

θθ

θθ

θθθ

θθ


328

SOLO

Glauert Integral Formula (1926) Proof (continue – 1)

( )

( )

( )

( )

( )

( )∫ ∫∫

− +

+=

−

−+

+

+=

−=

π π

π

π

θθθθ

θθθθ

θθ

θθ

θθ

θθθθ

θθθ

0

11

1

1

11

1

1

1

1

0

11

1 cos

21

sin

21

cos

2

1cos

21

sin

21

cos

21

sin

21

cos

2

1sin

coscos

cos: dndnd

nI

Change variables

Define

11 θθθ dxdx =⇒+=

( ) ∫∫∫+

−

+

−

+

−

+=−=πθ

πθ

πθ

πθ

πθ

πθ

θθθ xdx

xnxn

xdx

xnxn

xdnnxx

x

I

2sin

2cossin

2

sin

2sin

2coscos

2

coscos

2sin

2cos

2

1

∫+

−

=πθ

πθ

xdx

xnx

Yn

2sin

2coscos

: ∫+

−

=πθ

πθ

xdx

xnx

Zn

2sin

2cossin

:

Compute

( )[ ]( ) 01sinsin

2cos

2sin2

2sin

2cos1coscos

00

1 =−+=

−=

−−=− ∫∫∫∫

+

−

+

−

+

−

+

−−

πθ

πθ

πθ

πθ

πθ

πθ

πθ

πθ

xdxnxdxnxdxx

xnxdx

xxnxn

YY nn

( )[ ]( ) 01coscos

2cos

2

1cos2

2sin

2cos1sinsin

00

1 =−+=

−=

−−=− ∫∫∫∫

+

−

+

−

+

−

+

−−

πθ

πθ

πθ

πθ

πθ

πθ

πθ

πθ

xdxnxdxnxdx

xnxdx

xxnxn

ZZ nn

329

SOLO

Glauert Integral Formula (1926) Proof (continue – 2)

nn Zn

Yn

dn

I2

sin

2

cos

coscos

cossin:

0

11

1 θθθθθ

θθπ

+=−

= ∫

Therefore

02

sin

2sin

2sin1

2sin

2coscos 2

11 =

−

===== ∫∫+

−

+

−−

πθ

πθ

πθ

πθ

xd

x

x

xdx

xx

YYY nn

( ) ππθ

πθ

πθ

πθ

πθ

πθ

2cos12

cos2

2sin

2cossin

211 =+====== ∫ ∫∫

+

−

+

−

+

−− xdxxd

xxd

x

xx

ZZZ nn

and

θπθθθ

θθπ

ndn

I sincoscos

cossin:

0

11

1 =−

= ∫

θθπθ

θθθπ

sin

sin

coscos

cos

0

11

1 nd

n =−∫

q.e.d.

330Ray Whitford, “Design for Air Combat”


Flap configurations and (graphs) effect on section lift and drag characteristics of a 25%-chord flap of each type deflected 30°.


Fig 75 Northrop F-5E wing flaps

Fig 76 Lift and drag benefits at various flop settings. The angles given ore for leading and trailing-edge flaps respectively.6

Fig 77 F-18 trimmed drag-due-to-lift. The curve indicated by circles is for leading and trailing-edge flap angles of 0° and 0° respectively; triangles ore for 5°/8°; and squares far 10°/12°.25


Fig 169 Supersonic area-ruling.

334

Aircraft designed to fly at greater angles of attack use a delta wing: a wing shaped like a triangle when viewed from above. Many delta wings additionally feature tapered leading edges, which are used to combat the increased drag that occurs as the angle of attack increases (during a dog fight, for example). These tapered leading edges affect the airflow over the wing in a way that decreases drag and increases lift.

http://www.concept2.com/oars/how-made-and-tested/vortex-edge

Vortex lift is produced by two vortices which separate along the entire length of the side edges and roll up rapidly into two nearly conical, spiral shaped coils above the leeward surface.Such fast spiraling vortices induce large suction (low pressure) over the leeward surface of the foil generating extra lift

http://seagatesail.com/technology/delta-wing-sail/

335http://defence.pk/threads/design-characteristics-of-canard-non-canard-fighters.178592/

aerodynamics part i

Science

atmospheric pressure

pressure units

table of content

atmospheric reradiation

values of density

units of kgm3

units of nm2

mach numberaerodynamics