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 2

Aerodynamics,Some Fundamental

Principles and Equations

There is so great a difference between a fluid and a collection

of solid particles that the laws of pressure and of equilibrium

of fluids are very different from the laws of the pressure and

equilibrium of solids,

Jean Le Rond d’Alembert,1768

2.1 Introduction and Road Map

Preparation of tools for the analysis of

aerodynamics

Every aerodynamic tool we developed in

this and subsequent chapters is

important for the analysis and

understanding of practical problems

Orientation offered by the road map

2.2 Review of Vector relations

2.2.1 to 2.2.10 Skipped over

2.2.11 Relations between line,surface,and

volume integrals

The line integral of A over C is related to the surface integral

of A(curl of A) over S by Stokes’ theorem,

? ? SAsA dd

SC

????? ???

Where aera S is bounded by the closed curve C,

The surface integral of A over S is related to the volume

integral of A(divergence of A) over V by divergence’ theorem,

? ?dVd

VS

????? ???? ASA

Where volume V is bounded by the closed surface S,

If p represents a scalar field,a vector relationship analogous

to divergence theorem is given by gradient theorem,

dVppd

VS

????? ??S

2.3 Models of the fluid,control

volumes and fluid particles

Importance to create physical feeling from

physical observation,

How to make reasonable judgments on difficult

problems,

In this chapter,basic equations of aerodynamics

will be derived,

Philosophical procedure involved with the

development of these equations

1,Invoke three fundamental physical principles which are

deeply entrenched in our macroscopic observations of

nature,namely,

a,Mass is conserved,that’s to say,mass can be neither

created nor destroyed,

b,Newton’s second law,force=mass? acceleration

c,Energy is conserved; it can only change from one form to

another

2,Determine a suitable model of the fluid,

3,Apply the fundamental physical principles listed in item 1

to the model of the fluid determined in item2 in order to

obtain mathematical equations which properly describe

the physics of the flow,

Emphasis of this section,

1,What is a suitable model of the fluid?

2,How do we visualize this squishy substance in

order to apply the three fundamental principles?

3,Three different models mostly used to deal with

aerodynamics,

finite control volume （ 有限控制体）

infinitesimal fluid element （ 无限小流体微团）

molecular （ 自由分子）

2.3.1 Finite control volume approach

Definition of finite control volume,

a closed volume sculptured within a finite region of

the flow,The volume is called control volume V,

and the curved surface which envelops this region

is defined as control surface S,

Fixed control volume and moving control volume,

Focus of our investigation for fluid flow,

2.3.2 Infinitesimal fluid element approach

Definition of infinitesimal fluid element,

an infinitesimally small fluid element in the flow,

with a differential volume,

It contains huge large amount of molecules

Fixed and moving infinitesimal fluid element,

Focus of our investigation for fluid flow,

The fluid element may be fixed in space with fluid moving

through it,or it may be moving along a streamline with velocity

V equal to the flow velocity at each point as well,

2.3.3 Molecule approach

Definition of molecule approach,

The fluid properties are defined with the use of

suitable statistical averaging in the microscope

wherein the fundamental laws of nature are

applied directly to atoms and molecules,

In summary,although many variations on the theme

can be found in different texts for the derivation of

the general equations of the fluid flow,the flow

model can be usually be categorized under one of the

approach described above,

2.3.4 Physical meaning of the

divergence of velocity

Definition of,

is physically the time rate of change of

the volume of a moving fluid element of fixed

mass per unit volume of that element,

V???

V???

Analysis of the above definition,

Step 1,Select a suitable model to give a frame

under which the flow field is being described,

a moving control volume is selected,

Step 2,Select a suitable model to give a frame

under which the flow field is being described,

a moving control volume is selected,

Step 3,How about the characteristics for this

moving control volume?

volume,control surface and density will be

changing as it moves to different region of the

flow,

Step 4,Chang in volume due to the movement of

an infinitesimal element of the surface dS over,

? ?? ? ? ? SdtVdSntVV ???? ???????

t?

The total change in volume of the whole control

volume over the time increment is obviously

given as bellow t?

? ??? ??

S

SdtV

??

Step 5,If the integral above is divided by,the

result is physically the time rate change of the

control volume

t?

? ? ???? ????

?

?

SS

SdVSdtV

tDt

DV ????1

Step 6,Applying Gauss theorem,we have

??? ???

V

dVV

Dt

DV ?

Step 7,As the moving control volume approaches

to a infinitesimal volume,,Then the above

equation can be rewritten as

V?

? ?

??? ???

V

dVV

Dt

VD

?

? ?

Assume that is small enough such that is the

same through out, Then,the integral can be

approximated as,we have

V? V???

V?

? ? VV ????

? ? VV

Dt

VD ?? ???? ? ?

Dt

VD

V

V ?

?

1??? ?or

Definition of,

is physically the time rate of change of

the volume of a moving fluid element of fixed

mass per unit volume of that element,

V???

V???

Another description of and,

?? ?

S

SdV ?? V

???

Assume is a control surface corresponding to control

volume,which is selected in the space at time,

At time the fluid particles enclosed by at time will

have moved to the region enclosed by the surface,

The volume of the group of particles with fixed identity

enclosed by at time is the sum of the volume in region

A and B,And at time,this volume will be the sum of the

volume in region B and C,

As time interval approaches to zero,coincides with,

If is considered as a fixed control volume,then,the

region in A can be imagined as the volume enter into the

control surface,C leave out,

V

S

t

1t S t

1S

S

t

1t

1S

S

S

Based on the argument above,the integral of can

be expressed as volume flux through fixed control surface,

Further,can be expressed as the rate at which fluid

volume is leaving a point per unit volume,

?? ?

S

SdV ??

V???

The average value of the velocity component on the right-

hand x face is

)2)(( xxuu ????

The rate of volume flow out of the right-hand x face is

? ? zyxxuu ?????? )2)((

That into the left-hand x face is

? ? zyxxuu ?????? )2)((

The net outflow from the x faces is

zyxxu ????? )( per unit time

The net outflow from all the faces in x,y,z directions per

unit time is

? ? zyxzwyvxu ??????????? )()()(

The flux of volume from a point is

? ?

zyx

zyxzwyvxu

V

flu xin flo wflu xo u tflo w

V ???

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

?

)()()(lim

0

)()()(lim 0 zwyvxuV f l u xi n f l o wf l u xo u t f l o wV ???????????

2.4 Continuity equation

In this section,we will apply fundamental

physical principles to the fluid model,More

attention should be given for the way we

are progressing in the derivation of basic

flow equations,

Derivation of continuity equation

Step 1,Selection of fluid model,A fixed finite

control volume is employed as the frame for the

analysis of the flow,Herein,the control surface and

control volume is fixed in space,

Step 2,Introduction of the concept of mass flow,

Let a given area A is arbitrarily oriented in a flow,

the figure given bellow is an edge view,If A is small

enough,then the velocity V over the area is uniform

across A,The volume across the area A in time

interval dt can be given as

AdtVV o l u m e n )(?

The mass inside the shaded volume is

AdtVM a s s n )(??

The mass flow through is defined as the mass

crossing A per unit second,and denoted as m?

dt

AdtVm n )(???

or

AVm n???

The equation above states that mass flow through A

is given by the product

Area X density X component of flow velocity normal

to the area

mass flux is defined as the mass flow per unit area

nVA

mf l u xM a s s ??? ?

Step 3,Physical principle Mass can be neither

created nor destroyed,

Step 4,Description of the flow field,control volume

and control surface,

),,,(),,,,( tzyxVVtzyx ?? ?? ??

:Sd?

Directional elementary surface area on the control surface

:dV Elementary volume inside the finite control volume

Step 5,Apply the mass conservation law to this

control volume,

Net mass flow out of control

volume through surface S

Time rate decrease of mass

inside control volume

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 2

Aerodynamics,Some Fundamental

Principles and Equations

There is so great a difference between a fluid and a collection

of solid particles that the laws of pressure and of equilibrium

of fluids are very different from the laws of the pressure and

equilibrium of solids,

Jean Le Rond d’Alembert,1768

2.1 Introduction and Road Map

Preparation of tools for the analysis of

aerodynamics

Every aerodynamic tool we developed in

this and subsequent chapters is

important for the analysis and

understanding of practical problems

Orientation offered by the road map

2.2 Review of Vector relations

2.2.1 to 2.2.10 Skipped over

2.2.11 Relations between line,surface,and

volume integrals

The line integral of A over C is related to the surface integral

of A(curl of A) over S by Stokes’ theorem,

? ? SAsA dd

SC

????? ???

Where aera S is bounded by the closed curve C,

The surface integral of A over S is related to the volume

integral of A(divergence of A) over V by divergence’ theorem,

? ?dVd

VS

????? ???? ASA

Where volume V is bounded by the closed surface S,

If p represents a scalar field,a vector relationship analogous

to divergence theorem is given by gradient theorem,

dVppd

VS

????? ??S

2.3 Models of the fluid,control

volumes and fluid particles

Importance to create physical feeling from

physical observation,

How to make reasonable judgments on difficult

problems,

In this chapter,basic equations of aerodynamics

will be derived,

Philosophical procedure involved with the

development of these equations

1,Invoke three fundamental physical principles which are

deeply entrenched in our macroscopic observations of

nature,namely,

a,Mass is conserved,that’s to say,mass can be neither

created nor destroyed,

b,Newton’s second law,force=mass? acceleration

c,Energy is conserved; it can only change from one form to

another

2,Determine a suitable model of the fluid,

3,Apply the fundamental physical principles listed in item 1

to the model of the fluid determined in item2 in order to

obtain mathematical equations which properly describe

the physics of the flow,

Emphasis of this section,

1,What is a suitable model of the fluid?

2,How do we visualize this squishy substance in

order to apply the three fundamental principles?

3,Three different models mostly used to deal with

aerodynamics,

finite control volume （ 有限控制体）

infinitesimal fluid element （ 无限小流体微团）

molecular （ 自由分子）

2.3.1 Finite control volume approach

Definition of finite control volume,

a closed volume sculptured within a finite region of

the flow,The volume is called control volume V,

and the curved surface which envelops this region

is defined as control surface S,

Fixed control volume and moving control volume,

Focus of our investigation for fluid flow,

2.3.2 Infinitesimal fluid element approach

Definition of infinitesimal fluid element,

an infinitesimally small fluid element in the flow,

with a differential volume,

It contains huge large amount of molecules

Fixed and moving infinitesimal fluid element,

Focus of our investigation for fluid flow,

The fluid element may be fixed in space with fluid moving

through it,or it may be moving along a streamline with velocity

V equal to the flow velocity at each point as well,

2.3.3 Molecule approach

Definition of molecule approach,

The fluid properties are defined with the use of

suitable statistical averaging in the microscope

wherein the fundamental laws of nature are

applied directly to atoms and molecules,

In summary,although many variations on the theme

can be found in different texts for the derivation of

the general equations of the fluid flow,the flow

model can be usually be categorized under one of the

approach described above,

2.3.4 Physical meaning of the

divergence of velocity

Definition of,

is physically the time rate of change of

the volume of a moving fluid element of fixed

mass per unit volume of that element,

V???

V???

Analysis of the above definition,

Step 1,Select a suitable model to give a frame

under which the flow field is being described,

a moving control volume is selected,

Step 2,Select a suitable model to give a frame

under which the flow field is being described,

a moving control volume is selected,

Step 3,How about the characteristics for this

moving control volume?

volume,control surface and density will be

changing as it moves to different region of the

flow,

Step 4,Chang in volume due to the movement of

an infinitesimal element of the surface dS over,

? ?? ? ? ? SdtVdSntVV ???? ???????

t?

The total change in volume of the whole control

volume over the time increment is obviously

given as bellow t?

? ??? ??

S

SdtV

??

Step 5,If the integral above is divided by,the

result is physically the time rate change of the

control volume

t?

? ? ???? ????

?

?

SS

SdVSdtV

tDt

DV ????1

Step 6,Applying Gauss theorem,we have

??? ???

V

dVV

Dt

DV ?

Step 7,As the moving control volume approaches

to a infinitesimal volume,,Then the above

equation can be rewritten as

V?

? ?

??? ???

V

dVV

Dt

VD

?

? ?

Assume that is small enough such that is the

same through out, Then,the integral can be

approximated as,we have

V? V???

V?

? ? VV ????

? ? VV

Dt

VD ?? ???? ? ?

Dt

VD

V

V ?

?

1??? ?or

Definition of,

is physically the time rate of change of

the volume of a moving fluid element of fixed

mass per unit volume of that element,

V???

V???

Another description of and,

?? ?

S

SdV ?? V

???

Assume is a control surface corresponding to control

volume,which is selected in the space at time,

At time the fluid particles enclosed by at time will

have moved to the region enclosed by the surface,

The volume of the group of particles with fixed identity

enclosed by at time is the sum of the volume in region

A and B,And at time,this volume will be the sum of the

volume in region B and C,

As time interval approaches to zero,coincides with,

If is considered as a fixed control volume,then,the

region in A can be imagined as the volume enter into the

control surface,C leave out,

V

S

t

1t S t

1S

S

t

1t

1S

S

S

Based on the argument above,the integral of can

be expressed as volume flux through fixed control surface,

Further,can be expressed as the rate at which fluid

volume is leaving a point per unit volume,

?? ?

S

SdV ??

V???

The average value of the velocity component on the right-

hand x face is

)2)(( xxuu ????

The rate of volume flow out of the right-hand x face is

? ? zyxxuu ?????? )2)((

That into the left-hand x face is

? ? zyxxuu ?????? )2)((

The net outflow from the x faces is

zyxxu ????? )( per unit time

The net outflow from all the faces in x,y,z directions per

unit time is

? ? zyxzwyvxu ??????????? )()()(

The flux of volume from a point is

? ?

zyx

zyxzwyvxu

V

flu xin flo wflu xo u tflo w

V ???

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

?

)()()(lim

0

)()()(lim 0 zwyvxuV f l u xi n f l o wf l u xo u t f l o wV ???????????

2.4 Continuity equation

In this section,we will apply fundamental

physical principles to the fluid model,More

attention should be given for the way we

are progressing in the derivation of basic

flow equations,

Derivation of continuity equation

Step 1,Selection of fluid model,A fixed finite

control volume is employed as the frame for the

analysis of the flow,Herein,the control surface and

control volume is fixed in space,

Step 2,Introduction of the concept of mass flow,

Let a given area A is arbitrarily oriented in a flow,

the figure given bellow is an edge view,If A is small

enough,then the velocity V over the area is uniform

across A,The volume across the area A in time

interval dt can be given as

AdtVV o l u m e n )(?

The mass inside the shaded volume is

AdtVM a s s n )(??

The mass flow through is defined as the mass

crossing A per unit second,and denoted as m?

dt

AdtVm n )(???

or

AVm n???

The equation above states that mass flow through A

is given by the product

Area X density X component of flow velocity normal

to the area

mass flux is defined as the mass flow per unit area

nVA

mf l u xM a s s ??? ?

Step 3,Physical principle Mass can be neither

created nor destroyed,

Step 4,Description of the flow field,control volume

and control surface,

),,,(),,,,( tzyxVVtzyx ?? ?? ??

:Sd?

Directional elementary surface area on the control surface

:dV Elementary volume inside the finite control volume

Step 5,Apply the mass conservation law to this

control volume,

Net mass flow out of control

volume through surface S

Time rate decrease of mass

inside control volume