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,
