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
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
????? ???
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
????? ???? 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,
????? ??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
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
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
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
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,
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
Step 4,Chang in volume due to the movement of
an infinitesimal element of the surface dS over,
? ?? ? ? ? SdtVdSntVV ???? ???????
The total change in volume of the whole control
volume over the time increment is obviously
given as bellow t?
? ??? ??
Step 5,If the integral above is divided by,the
result is physically the time rate change of the
control volume
? ? ???? ????
DV ????1
Step 6,Applying Gauss theorem,we have
??? ???
DV ?
Step 7,As the moving control volume approaches
to a infinitesimal volume,,Then the above
equation can be rewritten as
? ?
??? ???
? ?
Assume that is small enough such that is the
same through out, Then,the integral can be
approximated as,we have
V? V???
? ? VV ????
? ? VV
VD ?? ???? ? ?
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,
Another description of and,
?? ?
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,
1t S t
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,
?? ?
SdV ??
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
? ?
flu xin flo wflu xo u tflo w
V ???
)()()(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?
AdtVm n )(???
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
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 ?? ?? ??
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