PART I

FUNDAMENTAL PRINCIPLES

（基本原理）

In part I,we cover some of the basic principles

that apply to aerodynamics in general,These are

the pillars on which all of aerodynamics is based

Chapter 1

Aerodynamics,

Some Introductory Thoughts

The term,aerodynamics” is generally used for problems

arising from flight and other topics involving the flow of air,

Ludwig Prandtl,1949

Aerodynamics:The dynamics of gases,especially of

atmospheric interactions with moving objects,

The American Heritage

Dictionary of English Language,1969

1.1 Importance of Aerodynamics,

Historical Examples

Sea battle between English fleet and

Spanish fleet,English channel,8-8-1588

(英国与西班牙海战，英吉利海峡）

First flight of Wright brothers,12-27-1903

(怀特兄弟首次飞行）

Minimizing of aerodynamic heating of

ICBMs

(洲际弹道导弹气动热降低问题）

Impetus to the study of fluidmechnics

（ 流体力学研究的推动力）

1,Newton’s sine-square law

2,Experiments carried out by D’Alembert

3,Euler’s description of the flow model

1,Newton’s sine-square law

a) Newton considered a fluid flow as a uniform,

rectilinear stream of particles,much like a cloud

of pellets from a shotgun blast,

b) Newton assumed that upon striking a surface

inclined at a angle to the stream,the particles

would transfer their normal momentum to the

surface but their tangential momentum would

be preserved,Hence,after collision with the

surface,the particles would then move along the

surface,This led to an expression for the

hydrodynamics force on the surface which varies

as

?

?2sin

2,D’Alembert

The experiment results show,the rule that for

oblique resistance varies with the sine square of

the angle of the incidence holds good only for

angle between 50 and 90 deg and must be

abandoned for lesser angles

3,Euler noted

The fluid moving toward a body,before reaching

the latter,bends its direction and its velocity so

that when it reaches the body it flows pass it

along the surface,and exercise no other force

on the body except the pressure corresponding

to the single points of the contact.”

4,Real case for fluid approaching a body

All the fluid particles are in random motion,and

has a average velocity,During their motion,they

collide with each other,The molecules strike on

to the solid surface will be rebounded,and these

rebounded molecules will make collision to other

molecules,This process transfers the message

of the existence of the body,and most of the

particles will go other round,

After the collision between fluid particles and solid

surface,the momentum change of the particles

is in the perpendicular direction of the surface,

First flight of Wright brothers

Dec,17,1903

Wilbur and Orville Wright's Wright Flyer

was the first successful airplane,On

December 17,1903,at Kitty Hawk,North

Carolina,Orville Wright flew the first

heavier-than-air machine in a powered,

controlled,and sustained flight,The Flyer,

constructed of wood,wire,and muslin,went

a distance of 120 feet in 12 seconds,It was a

tremendous success,coming from a long

series of aeronautics experiments that the

Wright Brothers started in 1899 with a kite,

At the rear of the 1903 Wright Flyer one finds a pair of

pusher propellers,The propellers are long,thin,twisted

pieces of wood which are spun at high speed,

Control of roll,WING WARP

Overview of Wright Brothers Discoveries

Aerodynamic heating of the reentry vehicle

ICBMs reentry the atmosphere at the speeds of

from 6 to 6.7km/s,

The aerodynamic heating of the reentry vehicles

becomes severe,the cover of the war head will

be heated up to 10,000K,

Blunt reentry body design can minimize the

aerodynamic heating problem,

1.2 Aerodynamics:Classification and

Practical Objectives

(空气动力学：分类和应用目标）

Distinction of solids,liquids,and gases

Practical applications in engineering

Solids,liquids,and gases in a container

The solid object will not change,its shape and boundaries

will remind the same,

The liquid will change its shape to conform to that of the

container and will take take on the same boundaries as the

container up to the maximum depth of the liquid,

The gas will completely fill the container,taking on the same

boundaries as the container,

Solid and,fluid”(a liquid or a gas) under

a tangential force ==? deformation

固体和流体在受到剪应力时，各自形状所发生的变化

方式截然不同 。

Under a force applied tangentially to the surface of a solid

body,the solid body will undergo a finite deformation,and the

tangential force per unit area—the shear stress—will usually be

proportional to the amount of deformation,

If the case happens for a fluid,then,the fluid will experience a

continuously increasing deformation and the shear stress will

usually be proportional to the rate of the deformation,

?? ?Solid,

fluid,?? ??

:

:

:

?

?

?

?

Shear stress 剪应力

Deformation 变形

Rate of deformation 变形率

Mechanics distinction of solids,liquids,and gases

Distinction of solids,liquids,and gases

respects to the intermolecular forces

Fluid dynamics is subdivided into three areas,

Hydrodynamics --- flow of liquids

Gas dynamics --- flow of gases

Aerodynamics --- flow of air

Practical objectives of Aerodynamics

1,The prediction of forces and moments on and

heat transfer to,bodies moving through a fluid,

2,Determination of flows moving internally

through ducts

3,External aerodynamics

4,Internal aerodynamics

1.3 Road Map of this chapter

What’s the usage of the road map

1,At the beginning of each chapter,road

map give you the sense for you get to

know where you are,where you are

going,and how can you get there

2,Show the interrelationship of the

materials in the chapter

3,At the end of the chapter,after you look

back over the road map,you will see

where you started,where you are now,

and what you learned in between,

1.4 Some fundamental

Aerodynamic Variables

1,Aerodynamic variables are something like

technical vocabulary for the physical science

and engineering understanding

2,First introduced aerodynamic variables,

pressure,density,temperature,and flow velocity

The velocity description of a fluid is quite

different to that of a solid body,

Velocity of a flowing gas at any fixed point B in space is the

velocity of a small fluid element as it sweeps through B,

1.5 Aerodynamic forces and

moments

Aerodynamic forces and moments on a

moving body are due to only two basic

sources,

1,Pressure distribution over the body surface

2,Shear stress distribution over the body

surface

Both pressure and shear stress have

dimensions of force per unit area,

pressure acts normal to the body surface,

shear stress acts tangential to the surface,

The net effect of the pressure and shear stress

distribution results in a aerodynamic force R

and moment M on the body,

The resultant force R can be split into components

L = lift, component of R perpendicular to

D = drag, components of R parallel to

(wind system)

?V

?V

N = normal force,

component of R perpendicular to c

A = axial force,

components of R parallel to c

(body system)

After the pressure and shear stress distributions

being defined,and the geometry shape of the body

being known,the resultant aerodynamic force can be

obtained by the integration of the pressure and shear

stress distributions along the surface of the body,

From Eqs,(1.7),(1.8) and (1.11),we can see clearly,

that the sources of the aerodynamic lift,drag,and

moments on a body are the pressure and shear

stress distribution integrated over the body,

The basic task of theoretical aerodynamics is to

calculate p(s) and τ(s) for a given body shape and

freestream conditions,and then obtain the

aerodynamic forces and moments with the use of

Eqs,(1.7),(1.8) and (1.11)

Dimensionless aerodynamic force and moment

coefficients are even more important than the

aerodynamic forces and moments,

Definition of and

density and velocity in the freestream,which is

far ahead of the body,

?? ?V

Definition of dynamic pressure

The dynamic pressure has the unit of pressure

2

2

1

??? ? Vq ?

Definition of dimensionless force and moment

coefficients

Lift coefficient,

Sq

LC

L

?

?

Drag coefficient,

Sq

DC

D

?

?

Normal force coefficient,

Sq

NC

N

?

?

Axial force coefficient,

Sq

NC

N

?

?

Moment coefficient,

Slq

MC

M

?

?

, reference area

, reference length

S

l

Definition of and may be different for different

shapes of the body being concerned,

S l

The symbols in capital letters,such as

represents the force and moment coefficients for a

three-dimensional body,

The symbols in lowercase letters

denote the force and moment coefficients for a two-

dimensional body

ANMDL CandCCCC,,,

mdl candcc,

2

'''

,,

cq

Mc

cq

Dc

cq

Lc

mdl

???

???

are force and moments per unit span

''',,MDL

)1(cS ?

Two additional dimensionless quantities of

immediate use are

?

???

q

ppC

p

Pressure coefficient

?

?

q

c f ?

Skin friction coefficient

Where is the free stream pressure

?p

1.6 Center of pressure（ 压力中心）

The center of the pressure is a point on the body

about which the aerodynamic moment contributed

by the pressure and shear stress distributions is

equal to zero,

If is defined as the moment generated by the

distributed loads,and is the component of the

resultant force,then the pressure center must be

located downstream of the leading edge

'LEM

'N

cpx

'

'

N

Mx LE

cp ??

If the angle of attack is small,,thus '' NL ?

'

'

L

Mx LE

cp ??

It is clear to see that as lift approaches to zero,the

center of pressure moves to infinity,So,the center

of pressure is not always a convenient concept in

aerodynamics,There are other ways to define the

force-and-moment system on an airfoil

''

4

'' 4 LxMcLM

cpcLE ?????

1.7 Dimensional analysis,The

Buchingham PI theorem（ 量纲分

析,PI定理）

※ What physical quantities determine the variation

of the aerodynamic forces and moments? On a

physical,intuitive basis,we expect R is depend on,

1,Freestream velocity

2,Freestream density

3,Viscosity of the fluid

4,The size of the body

5,The compressibility of the fluid

)23.1(),,,,( ????? acVfR ??

※ How to find a precise functional relation for the

equation above? Execute huge amount of wind

tunnel experiment might be one way,

Is there any other way can do more effectively?

Method of dimensional analysis

※ An obvious fact for the dimensional analysis

???? ???

All the terms in this physical relation must have the

same dimensions

※ Buckingham PI theorem

1,Let K to be the number of fundamental dimensions

required to describe the physical variables

2,Let represent N physical variables in the

physical relation N

PPP,,,21 ?

0),,( 211 ?NPPPf ?

3,Then the physical relation can be reexpressed as a

relation of (N-K) dimensionless products,

0),,( 212 ???? ? KNf ?

4,Every product is a dimensionless product of a set

of K physical variables plus one other physical

variable,

),,,( 12131 ??? KK PPPPf ?

),,,( 22142 ??? KK PPPPf ?

),,,( 215 NKKN PPPPf ??? ?

????????????

5,is called repeating variables,These

variables should include all the K dimensions used in

the problem,

KPPP ?,,21

※ Aerodynamic force on a given body at a given

angle of attack,

1,Eq,(1.23)

)23.1(),,,,( ????? acVfR ??

can be expressed as

)27.1(0),,,,,( ????? acVRg ??

2,Following Buckingham theorem and our physical

intuition,the fundamental dimensions are m,l and t,

Hence,K=3

3,The physical variables and their dimensions are

111

132

][,][,][

,][,][,][

?

?

??

?

?

?

?

?

?

???

???

ltatmllc

ltVmlm ltR

?

?

and N=6

4,As explained by Buckingham theorem,Eq.(1.27)

can be reexpressed in terms of N-K=3 dimensionless

products,that is ?

)28.1(0),,( 3212 ????f

5,Now,we chose as repeating variables,

from Eq.(1.26),these products are

cV,,???

?

),,,(31 RcVf ???? ?

),,,(42 ????? ?? cVf

),,,(53 ????? acVf ?

5,Assume

RcV ebd ???? ?1

in dimensional form

? ? )()()()( 2131 ????? m l tlltml ebd

6,As is dimensionless,then 1?

02:

013:

01:

???

?????

??

btfo r

ebdlfo r

dmfo r

7,The above Equations give d=-1,b=-2,and e=-2,

then we have

22

221

1 cV

RcVR

??

??

?

?

? ??? ??

or

Sq

R

SV

R

?

??

???

2

1

2

1

?

where S is defined as reference area

8,In the same way,we can obtain the remaining

products as follows

?

????

?

? cV

2

Reynolds Number 雷诺数

1? RC

is a force coefficient,defined as

?

???

a

V

3

Mach Number 马赫数

9,Inserting all the products into Eq,(1.28) ?

0),,

5.0

( 22 ?

?

?

?

??

?? a

VcV

SV

Rf

?

?

?

or

0)R e,,(2 ??MCf R

or

)( R e,6 ?? MfC R

10,Important conclusion,

In the general function form,R is expressed with

five independent physical variables

After our dimensional analysis,R can be expressed

with only two independent variables

? R can be expressed in terms of a dimensionless

force coefficient

? is a function of only Re and

RC ?M

11,Important applications of Re and,

similarity parameters ?

M

12,As lift and drag are components of the resultant

force,then the lift and drag coefficients are also

functions of only Re and,

?M

)( R e,7 ?? MfC L

)( R e,8 ?? MfC D

Moreover,a relation similar to aerodynamic forces

holds for aerodynamic moments,and dimension

analysis yields

)( R e,9 ?? MfC M

13,If the angle of attack is allowed to vary,then,

the lift,drag and moment coefficients will in general

depend on the value of, ?

),( R e,10 ??? MfC L

),( R e,11 ??? MfC D

),( R e,12 ??? MfC M

14,Other similarity parameters associated with

thermodynamics and heat transfer,

Physical variables should be added

temperature,specific heat,thermal conductivity,

temperature of the body surface

Fundamental dimension should be added

unit of the temperature(K)

Similarity parameters created

vp cc ?TTw ??

? kc p?Pr

1.8 Flow similarity（ 流动相似）

※ Definition of flow similarity

Different flows are dynamically similar if,

1,The streamline patterns are geometrically similar

2,The distributions of etc.,

throughout the flow field are the same when

plotted against common nondimensional

coordinates,

3,The force coefficients are the same

,,,??? TTppVV

※ Criteria to ensure flow similarity

1,The bodies and any other solid boundaries are

geometrically similar for both flows,

2,The similarity parameters are identical for both

flows,

3,Reynolds and Mach number are the most

dominant similarity parameters for many

aerodynamic problems,

※ Examples 1.4 and 1.5

1.9 Fluid Statics,Buoyancy Force

(流体静力学：浮力）

Skipped over

1.10 Types of Flow （ 流动类型）

1,The purpose for categorizing different types of

flow,

2,The strategy to simplify the flow problems,

3,Itemization and comparison of different types of

flow,and brief description of their most important

physical phenomena,

1.10.1 Continuum versus free molecule

flow

1,Definition of mean-free path,

2,Continuum flow,

3,Free molecule flow

4,In most aerodynamic problems,we will always

treat the fluid as continuum flow,

?

d???

d??

1.10.2 Inviscid versus viscous flow

1,The random motion of the molecule will transport

their mass,momentum,and energy from one

location to another in the fluid,This transport on

a molecule scale gives rise to the phenomena of

mass diffusion,viscosity,and thermal conduction,

All real flows exhibit the effect of these transport

phenomena; such flows are call viscous flows,

2,A flow that is assumed free with all these

phenomena above is called inviscid flow,

3,Inviscid flow is approached in the limit as the

Reynolds number goes to infinity,

4,The flow with high Reynolds number,can be

assumed to be inviscid,And the influence of

friction,thermal conduction,and diffusion is

limited in the boundary layer,

5,The inviscid theory can be used to predicts the

pressure distribution and lift,However,it cannot

predicts total drag,

6,Flows dominated by viscous effects,

Flow around airfoil

at high angle of

attack

Flow around blunt

body

7,No inviscid theory can independently predict the

aerodynamics of such flows,

1.10.3 Incompressible versus

compressible Flows

1,A flow in which the density is constant is called

incompressible,In contrast,a flow where the

density is variable is called compressible,

2,All the flows are compressible,more or less

3,There are a number of aerodynamic problems

that can be modeled as being incompressible

without any detrimental loss of accuracy,

4,In many cases,whether the compressibility

should be considered or not,is manly based on

?

the Mach number of the flow,

1.10.4 Mach number regimes

1,Local definition

1?M

1?M

1?M

Subsonic if

Sonic if

Supersonic if

Where is the local Mach number at an arbitrary point

in a flow field,

M

2,Definition for whole flow field

3,Block diagram categorizing the types of

aerodynamic flows

1.11 Applied aerodynamics,The

aerodynamic coefficients — Their

magnitude and variations

1,Difference between the fundamentals and

applications of aerodynamics,

2,Aerodynamic coefficients,such as lift,drag,and

moment coefficients,are the primary language of

application external aerodynamics,

3,Typical values for the aerodynamic coefficients for

some common aerodynamic shapes and it’s

variation with Mach number and Reynolds

number,

4,Some typical drag coefficients for various aerodynamic

configurations in low speed flows,

)1(' dSSqDC D ?? ?

Comparison through case a to c,

the Reynolds numbers for all these three cases are

the same based on d (diameter),

the wakes are getting smaller in size from a to c

also becomes smaller from case a to c

DC

Comparison between case b and d,

the Reynolds number in case b,

the Reynolds number in case d,

is the same for case b to d

for a circular cylinder is relatively independent

of Reynolds number between Re= and

510

410

DC

DC

410 510

Comparison between case b to e,

the Reynolds number in case b,

the Reynolds number in case e,

in case e is 0.6

smaller wake behind the cylinder in case e

compared to that in case b,

510

710

DC

Note,With based on the frontal projected area

(S=d(1) per unit span),the value of range from

a maximum 2 to numbers as low as 0.12,

Magnitude of Reynolds number of a flow around a

circular cylinder at standard sea level,where,

smVsmkgmkgmd /45,/10789.1,/23.1,1 53 ?????? ??? ??

Then the Reynolds number is,

6

5 1009.3107 8 9.1

)1)(45)(23.1(Re ??

??? ?

??

?

? dV

for practical applications in aerodynamics,the

values of Re are in millions,

Pressure drag and skin friction drag,

The total drag exerted on the bodies are combined

with pressure drag and skin friction drag,

the drag of the vertical flat plate and the circular

cylinder is dominated by pressure drag,whereas,

in contrast,most of the drag of the streamlined

body is due to skin friction,

Drag on a flat plate at zero angle of attack,

Here,the drag is completely due to shear stress,

there is no pressure force in the drag direction,

)1('' cqDSqDC f ?? ??

The reference area is the planform area

From the above figure,we can conclude that

1,is a strong function of Re

2,The value of depends on whether the flow

over the plate surface is laminar or turbulent,

3,The magnitudes of range typically from 0.001

to 0.01 over a large range of Re,

DC

DC

DC

Drag coefficient of a complete low-speed aircraft,

Drag coefficient of a complete high-speed aircraft,

Lift coefficient of an airfoil,

Lift coefficient increases linearly with angle of

attack until reaches near 14 degrees,And

beyond this angle of attack,lift coefficient

decreases precipitously,

The ratio of lift to drag is a very important

characteristic for flight performance,

The L/D ratio for NACA 63-210 at is 130,

This is much lager than that of a complete aircraft,

Application of flap

02??

?

Application of flap (High-lift device),

In the take-off and landing phases,the flight

speed is very slow compared with cruise phases,

And,as we know,the lift is proportional to the

square of the flight speed,So,with the same

shape and angle of attack,the lift at take-off and

landing phases will be much smaller than that of

the cruise phase,

Flaps mounted at the trailing edge of the wing

are used to increase the lift or lift coefficient

during the take-off and landing of an aircraft,

SqCL L ??

Moment coefficient,

FUNDAMENTAL PRINCIPLES

（基本原理）

In part I,we cover some of the basic principles

that apply to aerodynamics in general,These are

the pillars on which all of aerodynamics is based

Chapter 1

Aerodynamics,

Some Introductory Thoughts

The term,aerodynamics” is generally used for problems

arising from flight and other topics involving the flow of air,

Ludwig Prandtl,1949

Aerodynamics:The dynamics of gases,especially of

atmospheric interactions with moving objects,

The American Heritage

Dictionary of English Language,1969

1.1 Importance of Aerodynamics,

Historical Examples

Sea battle between English fleet and

Spanish fleet,English channel,8-8-1588

(英国与西班牙海战，英吉利海峡）

First flight of Wright brothers,12-27-1903

(怀特兄弟首次飞行）

Minimizing of aerodynamic heating of

ICBMs

(洲际弹道导弹气动热降低问题）

Impetus to the study of fluidmechnics

（ 流体力学研究的推动力）

1,Newton’s sine-square law

2,Experiments carried out by D’Alembert

3,Euler’s description of the flow model

1,Newton’s sine-square law

a) Newton considered a fluid flow as a uniform,

rectilinear stream of particles,much like a cloud

of pellets from a shotgun blast,

b) Newton assumed that upon striking a surface

inclined at a angle to the stream,the particles

would transfer their normal momentum to the

surface but their tangential momentum would

be preserved,Hence,after collision with the

surface,the particles would then move along the

surface,This led to an expression for the

hydrodynamics force on the surface which varies

as

?

?2sin

2,D’Alembert

The experiment results show,the rule that for

oblique resistance varies with the sine square of

the angle of the incidence holds good only for

angle between 50 and 90 deg and must be

abandoned for lesser angles

3,Euler noted

The fluid moving toward a body,before reaching

the latter,bends its direction and its velocity so

that when it reaches the body it flows pass it

along the surface,and exercise no other force

on the body except the pressure corresponding

to the single points of the contact.”

4,Real case for fluid approaching a body

All the fluid particles are in random motion,and

has a average velocity,During their motion,they

collide with each other,The molecules strike on

to the solid surface will be rebounded,and these

rebounded molecules will make collision to other

molecules,This process transfers the message

of the existence of the body,and most of the

particles will go other round,

After the collision between fluid particles and solid

surface,the momentum change of the particles

is in the perpendicular direction of the surface,

First flight of Wright brothers

Dec,17,1903

Wilbur and Orville Wright's Wright Flyer

was the first successful airplane,On

December 17,1903,at Kitty Hawk,North

Carolina,Orville Wright flew the first

heavier-than-air machine in a powered,

controlled,and sustained flight,The Flyer,

constructed of wood,wire,and muslin,went

a distance of 120 feet in 12 seconds,It was a

tremendous success,coming from a long

series of aeronautics experiments that the

Wright Brothers started in 1899 with a kite,

At the rear of the 1903 Wright Flyer one finds a pair of

pusher propellers,The propellers are long,thin,twisted

pieces of wood which are spun at high speed,

Control of roll,WING WARP

Overview of Wright Brothers Discoveries

Aerodynamic heating of the reentry vehicle

ICBMs reentry the atmosphere at the speeds of

from 6 to 6.7km/s,

The aerodynamic heating of the reentry vehicles

becomes severe,the cover of the war head will

be heated up to 10,000K,

Blunt reentry body design can minimize the

aerodynamic heating problem,

1.2 Aerodynamics:Classification and

Practical Objectives

(空气动力学：分类和应用目标）

Distinction of solids,liquids,and gases

Practical applications in engineering

Solids,liquids,and gases in a container

The solid object will not change,its shape and boundaries

will remind the same,

The liquid will change its shape to conform to that of the

container and will take take on the same boundaries as the

container up to the maximum depth of the liquid,

The gas will completely fill the container,taking on the same

boundaries as the container,

Solid and,fluid”(a liquid or a gas) under

a tangential force ==? deformation

固体和流体在受到剪应力时，各自形状所发生的变化

方式截然不同 。

Under a force applied tangentially to the surface of a solid

body,the solid body will undergo a finite deformation,and the

tangential force per unit area—the shear stress—will usually be

proportional to the amount of deformation,

If the case happens for a fluid,then,the fluid will experience a

continuously increasing deformation and the shear stress will

usually be proportional to the rate of the deformation,

?? ?Solid,

fluid,?? ??

:

:

:

?

?

?

?

Shear stress 剪应力

Deformation 变形

Rate of deformation 变形率

Mechanics distinction of solids,liquids,and gases

Distinction of solids,liquids,and gases

respects to the intermolecular forces

Fluid dynamics is subdivided into three areas,

Hydrodynamics --- flow of liquids

Gas dynamics --- flow of gases

Aerodynamics --- flow of air

Practical objectives of Aerodynamics

1,The prediction of forces and moments on and

heat transfer to,bodies moving through a fluid,

2,Determination of flows moving internally

through ducts

3,External aerodynamics

4,Internal aerodynamics

1.3 Road Map of this chapter

What’s the usage of the road map

1,At the beginning of each chapter,road

map give you the sense for you get to

know where you are,where you are

going,and how can you get there

2,Show the interrelationship of the

materials in the chapter

3,At the end of the chapter,after you look

back over the road map,you will see

where you started,where you are now,

and what you learned in between,

1.4 Some fundamental

Aerodynamic Variables

1,Aerodynamic variables are something like

technical vocabulary for the physical science

and engineering understanding

2,First introduced aerodynamic variables,

pressure,density,temperature,and flow velocity

The velocity description of a fluid is quite

different to that of a solid body,

Velocity of a flowing gas at any fixed point B in space is the

velocity of a small fluid element as it sweeps through B,

1.5 Aerodynamic forces and

moments

Aerodynamic forces and moments on a

moving body are due to only two basic

sources,

1,Pressure distribution over the body surface

2,Shear stress distribution over the body

surface

Both pressure and shear stress have

dimensions of force per unit area,

pressure acts normal to the body surface,

shear stress acts tangential to the surface,

The net effect of the pressure and shear stress

distribution results in a aerodynamic force R

and moment M on the body,

The resultant force R can be split into components

L = lift, component of R perpendicular to

D = drag, components of R parallel to

(wind system)

?V

?V

N = normal force,

component of R perpendicular to c

A = axial force,

components of R parallel to c

(body system)

After the pressure and shear stress distributions

being defined,and the geometry shape of the body

being known,the resultant aerodynamic force can be

obtained by the integration of the pressure and shear

stress distributions along the surface of the body,

From Eqs,(1.7),(1.8) and (1.11),we can see clearly,

that the sources of the aerodynamic lift,drag,and

moments on a body are the pressure and shear

stress distribution integrated over the body,

The basic task of theoretical aerodynamics is to

calculate p(s) and τ(s) for a given body shape and

freestream conditions,and then obtain the

aerodynamic forces and moments with the use of

Eqs,(1.7),(1.8) and (1.11)

Dimensionless aerodynamic force and moment

coefficients are even more important than the

aerodynamic forces and moments,

Definition of and

density and velocity in the freestream,which is

far ahead of the body,

?? ?V

Definition of dynamic pressure

The dynamic pressure has the unit of pressure

2

2

1

??? ? Vq ?

Definition of dimensionless force and moment

coefficients

Lift coefficient,

Sq

LC

L

?

?

Drag coefficient,

Sq

DC

D

?

?

Normal force coefficient,

Sq

NC

N

?

?

Axial force coefficient,

Sq

NC

N

?

?

Moment coefficient,

Slq

MC

M

?

?

, reference area

, reference length

S

l

Definition of and may be different for different

shapes of the body being concerned,

S l

The symbols in capital letters,such as

represents the force and moment coefficients for a

three-dimensional body,

The symbols in lowercase letters

denote the force and moment coefficients for a two-

dimensional body

ANMDL CandCCCC,,,

mdl candcc,

2

'''

,,

cq

Mc

cq

Dc

cq

Lc

mdl

???

???

are force and moments per unit span

''',,MDL

)1(cS ?

Two additional dimensionless quantities of

immediate use are

?

???

q

ppC

p

Pressure coefficient

?

?

q

c f ?

Skin friction coefficient

Where is the free stream pressure

?p

1.6 Center of pressure（ 压力中心）

The center of the pressure is a point on the body

about which the aerodynamic moment contributed

by the pressure and shear stress distributions is

equal to zero,

If is defined as the moment generated by the

distributed loads,and is the component of the

resultant force,then the pressure center must be

located downstream of the leading edge

'LEM

'N

cpx

'

'

N

Mx LE

cp ??

If the angle of attack is small,,thus '' NL ?

'

'

L

Mx LE

cp ??

It is clear to see that as lift approaches to zero,the

center of pressure moves to infinity,So,the center

of pressure is not always a convenient concept in

aerodynamics,There are other ways to define the

force-and-moment system on an airfoil

''

4

'' 4 LxMcLM

cpcLE ?????

1.7 Dimensional analysis,The

Buchingham PI theorem（ 量纲分

析,PI定理）

※ What physical quantities determine the variation

of the aerodynamic forces and moments? On a

physical,intuitive basis,we expect R is depend on,

1,Freestream velocity

2,Freestream density

3,Viscosity of the fluid

4,The size of the body

5,The compressibility of the fluid

)23.1(),,,,( ????? acVfR ??

※ How to find a precise functional relation for the

equation above? Execute huge amount of wind

tunnel experiment might be one way,

Is there any other way can do more effectively?

Method of dimensional analysis

※ An obvious fact for the dimensional analysis

???? ???

All the terms in this physical relation must have the

same dimensions

※ Buckingham PI theorem

1,Let K to be the number of fundamental dimensions

required to describe the physical variables

2,Let represent N physical variables in the

physical relation N

PPP,,,21 ?

0),,( 211 ?NPPPf ?

3,Then the physical relation can be reexpressed as a

relation of (N-K) dimensionless products,

0),,( 212 ???? ? KNf ?

4,Every product is a dimensionless product of a set

of K physical variables plus one other physical

variable,

),,,( 12131 ??? KK PPPPf ?

),,,( 22142 ??? KK PPPPf ?

),,,( 215 NKKN PPPPf ??? ?

????????????

5,is called repeating variables,These

variables should include all the K dimensions used in

the problem,

KPPP ?,,21

※ Aerodynamic force on a given body at a given

angle of attack,

1,Eq,(1.23)

)23.1(),,,,( ????? acVfR ??

can be expressed as

)27.1(0),,,,,( ????? acVRg ??

2,Following Buckingham theorem and our physical

intuition,the fundamental dimensions are m,l and t,

Hence,K=3

3,The physical variables and their dimensions are

111

132

][,][,][

,][,][,][

?

?

??

?

?

?

?

?

?

???

???

ltatmllc

ltVmlm ltR

?

?

and N=6

4,As explained by Buckingham theorem,Eq.(1.27)

can be reexpressed in terms of N-K=3 dimensionless

products,that is ?

)28.1(0),,( 3212 ????f

5,Now,we chose as repeating variables,

from Eq.(1.26),these products are

cV,,???

?

),,,(31 RcVf ???? ?

),,,(42 ????? ?? cVf

),,,(53 ????? acVf ?

5,Assume

RcV ebd ???? ?1

in dimensional form

? ? )()()()( 2131 ????? m l tlltml ebd

6,As is dimensionless,then 1?

02:

013:

01:

???

?????

??

btfo r

ebdlfo r

dmfo r

7,The above Equations give d=-1,b=-2,and e=-2,

then we have

22

221

1 cV

RcVR

??

??

?

?

? ??? ??

or

Sq

R

SV

R

?

??

???

2

1

2

1

?

where S is defined as reference area

8,In the same way,we can obtain the remaining

products as follows

?

????

?

? cV

2

Reynolds Number 雷诺数

1? RC

is a force coefficient,defined as

?

???

a

V

3

Mach Number 马赫数

9,Inserting all the products into Eq,(1.28) ?

0),,

5.0

( 22 ?

?

?

?

??

?? a

VcV

SV

Rf

?

?

?

or

0)R e,,(2 ??MCf R

or

)( R e,6 ?? MfC R

10,Important conclusion,

In the general function form,R is expressed with

five independent physical variables

After our dimensional analysis,R can be expressed

with only two independent variables

? R can be expressed in terms of a dimensionless

force coefficient

? is a function of only Re and

RC ?M

11,Important applications of Re and,

similarity parameters ?

M

12,As lift and drag are components of the resultant

force,then the lift and drag coefficients are also

functions of only Re and,

?M

)( R e,7 ?? MfC L

)( R e,8 ?? MfC D

Moreover,a relation similar to aerodynamic forces

holds for aerodynamic moments,and dimension

analysis yields

)( R e,9 ?? MfC M

13,If the angle of attack is allowed to vary,then,

the lift,drag and moment coefficients will in general

depend on the value of, ?

),( R e,10 ??? MfC L

),( R e,11 ??? MfC D

),( R e,12 ??? MfC M

14,Other similarity parameters associated with

thermodynamics and heat transfer,

Physical variables should be added

temperature,specific heat,thermal conductivity,

temperature of the body surface

Fundamental dimension should be added

unit of the temperature(K)

Similarity parameters created

vp cc ?TTw ??

? kc p?Pr

1.8 Flow similarity（ 流动相似）

※ Definition of flow similarity

Different flows are dynamically similar if,

1,The streamline patterns are geometrically similar

2,The distributions of etc.,

throughout the flow field are the same when

plotted against common nondimensional

coordinates,

3,The force coefficients are the same

,,,??? TTppVV

※ Criteria to ensure flow similarity

1,The bodies and any other solid boundaries are

geometrically similar for both flows,

2,The similarity parameters are identical for both

flows,

3,Reynolds and Mach number are the most

dominant similarity parameters for many

aerodynamic problems,

※ Examples 1.4 and 1.5

1.9 Fluid Statics,Buoyancy Force

(流体静力学：浮力）

Skipped over

1.10 Types of Flow （ 流动类型）

1,The purpose for categorizing different types of

flow,

2,The strategy to simplify the flow problems,

3,Itemization and comparison of different types of

flow,and brief description of their most important

physical phenomena,

1.10.1 Continuum versus free molecule

flow

1,Definition of mean-free path,

2,Continuum flow,

3,Free molecule flow

4,In most aerodynamic problems,we will always

treat the fluid as continuum flow,

?

d???

d??

1.10.2 Inviscid versus viscous flow

1,The random motion of the molecule will transport

their mass,momentum,and energy from one

location to another in the fluid,This transport on

a molecule scale gives rise to the phenomena of

mass diffusion,viscosity,and thermal conduction,

All real flows exhibit the effect of these transport

phenomena; such flows are call viscous flows,

2,A flow that is assumed free with all these

phenomena above is called inviscid flow,

3,Inviscid flow is approached in the limit as the

Reynolds number goes to infinity,

4,The flow with high Reynolds number,can be

assumed to be inviscid,And the influence of

friction,thermal conduction,and diffusion is

limited in the boundary layer,

5,The inviscid theory can be used to predicts the

pressure distribution and lift,However,it cannot

predicts total drag,

6,Flows dominated by viscous effects,

Flow around airfoil

at high angle of

attack

Flow around blunt

body

7,No inviscid theory can independently predict the

aerodynamics of such flows,

1.10.3 Incompressible versus

compressible Flows

1,A flow in which the density is constant is called

incompressible,In contrast,a flow where the

density is variable is called compressible,

2,All the flows are compressible,more or less

3,There are a number of aerodynamic problems

that can be modeled as being incompressible

without any detrimental loss of accuracy,

4,In many cases,whether the compressibility

should be considered or not,is manly based on

?

the Mach number of the flow,

1.10.4 Mach number regimes

1,Local definition

1?M

1?M

1?M

Subsonic if

Sonic if

Supersonic if

Where is the local Mach number at an arbitrary point

in a flow field,

M

2,Definition for whole flow field

3,Block diagram categorizing the types of

aerodynamic flows

1.11 Applied aerodynamics,The

aerodynamic coefficients — Their

magnitude and variations

1,Difference between the fundamentals and

applications of aerodynamics,

2,Aerodynamic coefficients,such as lift,drag,and

moment coefficients,are the primary language of

application external aerodynamics,

3,Typical values for the aerodynamic coefficients for

some common aerodynamic shapes and it’s

variation with Mach number and Reynolds

number,

4,Some typical drag coefficients for various aerodynamic

configurations in low speed flows,

)1(' dSSqDC D ?? ?

Comparison through case a to c,

the Reynolds numbers for all these three cases are

the same based on d (diameter),

the wakes are getting smaller in size from a to c

also becomes smaller from case a to c

DC

Comparison between case b and d,

the Reynolds number in case b,

the Reynolds number in case d,

is the same for case b to d

for a circular cylinder is relatively independent

of Reynolds number between Re= and

510

410

DC

DC

410 510

Comparison between case b to e,

the Reynolds number in case b,

the Reynolds number in case e,

in case e is 0.6

smaller wake behind the cylinder in case e

compared to that in case b,

510

710

DC

Note,With based on the frontal projected area

(S=d(1) per unit span),the value of range from

a maximum 2 to numbers as low as 0.12,

Magnitude of Reynolds number of a flow around a

circular cylinder at standard sea level,where,

smVsmkgmkgmd /45,/10789.1,/23.1,1 53 ?????? ??? ??

Then the Reynolds number is,

6

5 1009.3107 8 9.1

)1)(45)(23.1(Re ??

??? ?

??

?

? dV

for practical applications in aerodynamics,the

values of Re are in millions,

Pressure drag and skin friction drag,

The total drag exerted on the bodies are combined

with pressure drag and skin friction drag,

the drag of the vertical flat plate and the circular

cylinder is dominated by pressure drag,whereas,

in contrast,most of the drag of the streamlined

body is due to skin friction,

Drag on a flat plate at zero angle of attack,

Here,the drag is completely due to shear stress,

there is no pressure force in the drag direction,

)1('' cqDSqDC f ?? ??

The reference area is the planform area

From the above figure,we can conclude that

1,is a strong function of Re

2,The value of depends on whether the flow

over the plate surface is laminar or turbulent,

3,The magnitudes of range typically from 0.001

to 0.01 over a large range of Re,

DC

DC

DC

Drag coefficient of a complete low-speed aircraft,

Drag coefficient of a complete high-speed aircraft,

Lift coefficient of an airfoil,

Lift coefficient increases linearly with angle of

attack until reaches near 14 degrees,And

beyond this angle of attack,lift coefficient

decreases precipitously,

The ratio of lift to drag is a very important

characteristic for flight performance,

The L/D ratio for NACA 63-210 at is 130,

This is much lager than that of a complete aircraft,

Application of flap

02??

?

Application of flap (High-lift device),

In the take-off and landing phases,the flight

speed is very slow compared with cruise phases,

And,as we know,the lift is proportional to the

square of the flight speed,So,with the same

shape and angle of attack,the lift at take-off and

landing phases will be much smaller than that of

the cruise phase,

Flaps mounted at the trailing edge of the wing

are used to increase the lift or lift coefficient

during the take-off and landing of an aircraft,

SqCL L ??

Moment coefficient,