PART II

INVISCID INCOMPRESSIBLE FLOW

In part II,we deal with the flow of a fluid which has

constant density– incompressible flow,This applies to the

flow of liquids,such as water flow,and to low-speed flow

of gases,The material covered here is applicable to low-

speed flight through the atmosphere—flight at a Mach

number of about 0.3

Chapter 3

Fundamentals of Inviscid,

incompressible Flow

Theoretical fluid dynamics,being a difficult subject,is for

convenience,commonly divided into two branches,one

treating frictionless or perfect fluids,the other treating of

viscous or imperfect fluids,The frictionless fluid has no

existence in nature,but is hypothesized by mathematicians in

order to facilitate the investigation of important laws and

principles that may be approximately true of viscous or natural

fluids,

3.1 Introduction and Road Map

From an aerodynamic point of view,at

air velocities between 0 to 360km/h the

air density remains essentially constant,

varying only a few percent,

Purpose of this chapter is to establish

some fundamental relations applicable to

inviscid,incompressible flows and to

discuss some simple but important flow

fields and applications,

3.2 Bernoulli’s Equation

c o n s tVp ?? 2

2

1 ?

Derivation of Bernoulli’s equation

Step 1,x component momentum equation

without viscous effect and body forces

x

p

Dt

Du

?

????

or

x

p

z

uw

y

uv

x

uu

t

u

?

???

?

??

?

??

?

??

?

? ????

For steady flow,,then 0??? tu

x

p

z

uw

y

uv

x

uu

?

???

?

??

?

??

?

?

?

1

Multiply both sides with dx

dx

x

pdx

z

uwdx

y

uvdx

x

uu

?

???

?

??

?

??

?

? ???

Along a streamline in 3D space,there are

0

0

??

??

udyv d x

w d xudz

Substituting the differential equations of the

streamline into x component momentum equation

dx

x

pdz

z

uudy

y

uudx

x

uu

?

???

?

??

?

??

?

?

?

1

or

dx

x

p

dz

z

u

dy

y

u

dx

x

u

u

?

?

????

?

?

??

?

?

?

?

?

?

?

?

?

?

?

1

as

dz

z

udy

y

udx

x

udu

?

??

?

??

?

??

We have

dx

x

pudu

?

???

?

1

or

dx

x

pdu

?

???

?

1

2

1 2

In the same way,we can get

dy

y

pdv

?

???

?

1

2

1 2 dz

z

pdw

?

???

?

1

2

1 2

??

?

?

??

?

?

?

?

?

?

?

?

?

?

???? dz

z

p

dy

y

p

dx

x

p

wvud

?

1

)(

2

1 222

2222 Vwvu ???

as

and

dpdz

z

pdy

y

pdx

x

p ?

?

??

?

??

?

?

Then,we have

dpdV

?

1

2

1 2 ?? or V d Vdp ???

V d Vdp ???

The above equation is called Euler’s Equation,

precondition,inviscid,without body force,along a

streamline,

usage,setup the relation between dpa n ddV

Step 2,integration of Euler’s equation

For incompressible flows,

The integration from point 1 to point 2 along a

streamline is

c o n s t??

?? ??

2

1

2

1

V

V

p

p

V d Vdp ?

or

??

?

?

??

?

?

????

22

2

1

2

2

12

VV

pp ?

or

2

11

2

22 2

1

2

1 VpVp ?? ???

The above equation is called Bernoulli’s Equation,

precondition,steady,inviscid,incompressible,without

body force,along a streamline,

usage,setup the relation between at point

1 on a streamline to at another point 2 on

the same streamline,

11 pa n dV

22 pa n dV

As point 1 and point 2 are arbitrarily chosen on a

streamline,so

c o n s tVp ?? 2

2

1 ?

Along a streamline

1,As there are no stipulation being made to whether

the flow is rotational or irrotational,it satisfies for

both irrotational flows and rotational flows as well,

2,The value of the constant will change from one

streamline to another,

3,For irrotational flows

c o n s tVp ?? 2

2

1 ? Through out the flow

Bernoulli’s equation is also a relation for mechanical

energy in an incompressible flow

c o n s tVp ?? 2

2

1 ?

where is the kinetic energy per unit volume 22V?

As Bernoulli’s equation can also be derived from the

energy equation,the energy equation is redundant

for the analysis of inviscid,incompressible flow,

The way for solving inviscid,incompressible flows,

1,Obtain the velocity field from the governing

equations,

2,Obtain the pressure field from Bernoulli’s equation

3.3 Incompressible flow in a duct,the

Venturi tube and low-speed wind tunnel

Assumption of quasi-one-dimensional flows

1,Flow-field properties are uniform across any cross

section,

2,All the flow-field properties are assumed to be

functions of x only,

Respect to continuity equation,and for steady case

0222111 ??? VAVA ??

or

222111 VAVA ?? ?

For incompressible flow

2211 VAVA ?

Applications of venturi tube

Measurements of flow speed in a duct,

2

212

2

1 )(

2 VppV ???

?

From Bernoulli’s equation

From continuity equation,we have

1

2

1

2 VA

AV ?

Then

2

1

2

2

1

12

2

1 )(

2

V

A

A

ppV ??

?

?

??

?

?

???

?

? ?1)(

)(2

2

21

12

1 ?

?

?

AA

pp

V

?

3.4 Pitot tube,measurement of airspeed

Definition of static pressure,the measurement of

the static pressure,

Definition of stagnation pressure,the measurement

of the stagnation pressure,

The usage of Pitot tube,

The pressure at point C is the stagnation pressure

0p

The pressure at point A is the static pressure

1p

The speed at point C is equal to zero

The speed at point A is equal to free stream velocity

1V

The speed and pressure and at point A and B are the

same,apply Bernoulli’s equation,we have

22

2

1

2

1

BBAA VpVp ?? ???

0

2

1

0

2

11 ??? pVp ?

or

?

)(2 10

1

pp

V

?

?

Combined instrument for the measurement of both

static and total(stagnation) pressure,

Definition of dynamic pressure,the relationship

between of the static pressure,total pressure and

dynamic pressure,

2

2

1 Vq ??

For incompressible flows

0

2

11 2

1 pVp ?? ?

static

pressure

dynamic

pressure

total

pressure

Keep in mind !!!!!

Any result derived from Bernoulli’s equation holds for

incompressible flow only

3.5 Pressure coefficient

Definition of the pressure coefficient

?

???

q

ppC

p

where

2

2

1

??? ? Vq ?

The pressure coefficient is a nondimensional value,

it can be used throughout aerodynamics,for both

compressible flow and incompressible flow,

For incompressible flow,the pressure coefficient

can be expressed in terms of velocity only

If and are defined as the freestream velocity

and pressure,And and are the velocity and

pressure at an arbitrary point in the flow,Then,with

Bernoulli’s equation,

?V ?

p

V p

22

2

1

2

1 VpVp ?? ???

??

or

)(

2

1 22 VVpp ???

?? ?

?

???

q

ppC

p

? )(

2

1 22 VVpp ???

?? ?

2

22

2

1

)(

2

1

?

?

?

?

?

?

?

?

V

VV

q

pp

C

p

?

?

or

2

1 ??

?

?

??

?

?

??

?

?

??

?

V

V

q

pp

C p

Valid for

incompressible

flow only

??

??

?

?

?

??

??

??

??

??

??

ppVV

Cqpp

CVV

CVV

CVV

CV

p

p

p

p

p

0

0

0

10

3.6 Condition on velocity for

incompressible flow

Contents in the center branch

1,Laplace’s equation

2,General philosophy and use in solving problems

3,Synthesis of complex flows from a superposition of

elementary flows

Basic condition on velocity in an incompressible

flow,

From the definition for the divergence of velocity

? ?

Dt

VD

V

V ?

?

1??? ?

If the density of the fluid is equal to a constant,then

? ? 0?

Dt

VD ?

That means,for incompressible flow,the velocity

must satisfies

0??? V?

If we consider the continuity equation

? ? 0????

?

? V

t

?

??

And apply it for the incompressible flows,then,the

continuity equation becomes

00 ???? V??

?

0??? V?

3.7 Governing equation for irrotational,

incompressible flow,Laplace’s Equation

The continuity equation for incompressible flow is

0??? V?

???V?

Therefore,for incompressible and irrotational flows

0)( 2 ?????? ?? Laplace’s equation

If the flow is irrotational,then,the velocity can be

expressed with velocity potential

Solutions of Laplace’s Equation are called harmonic

functions,

Solutions of Laplace’s Equation

in Cartesian coordinates,

02

2

2

2

2

2

2 ?

?

?

?

?

?

?

?

?

??

zyx

???

?

in cylindrical coordinates,

0

11

2

2

2

2

2

2 ?

?

?

?

?

?

??

?

?

?

?

?

?

?

?

?

??

zrr

r

rr

?

?

??

?

For 2D incompressible flow,a stream function can

be defined,and the relation between velocity and

stream function is

x

v

y

u

?

???

?

?? ??,

The continuity equation,expressed in Cartesian

coordinates is,

0?

?

??

?

????

y

v

x

uV?

0

22

?

??

?

?

??

?

??

?

?

?

?

?

?

?

?

?

?

???

?

?

??

?

?

?

?

?

?

xyyxxyyx

????

The very definition and use of stream function is a

statement of the continuity equation of mass,and

therefore Equations bellow

x

v

y

u

?

???

?

?? ??,

Can be used in place of the continuity equation itself,

If the flow is irrotational,then

0?

?

??

?

?

y

u

x

v

? xvyu ?

???

?

?? ??,

0???

?

?

??

?

?

?

?

?

?

??

?

?

?

?

?

?

?

?

?

?

yyxx

??

?

02

2

2

2

?

?

?

?

?

?

yx

??

Laplace’s equation

Important conclusions

1,Any irrotational,incompressible flow has a velocity

potential and stream function(for 2D flow) that

both satisfy Laplace’s Equation

2,Conversely,any solution of Laplace’s Equation

represents the velocity potential or stream

function(2D) for an irrotational and incompresible

flow,

Superposition of the solutions of Laplace’s Equation

The Laplace’s Equation is a second-order linear

partial differential equation,The sum of any

particular solutions of a linear differential equation is

also a solution of the equation(Laplace’s Equation),

For example,if represent n separate

solutions of the Laplace’s equation,then the sum

n????,,,,321 ??

n????? ????? ??321

is also a solution of the Laplace’s equation

Conclusion respects to the superposition of the

solutions of Laplace’s equation

A complicated flow pattern for an irrotational,

incompressible flow can be synthesized by adding

together a number of elementary flows which are

also irrotational and incompressible,

How to make the difference between irrotational

and incompressible flows around different

aerodynamic shapes,--- Boundary conditions

3.7.1 Infinity boundary conditions

0,?

?

???

?

???

?

??

?

??

? xyvVyxu

????

At infinity in all directions

3.7.2 Wall boundary conditions

0?? nV ??

On the surface of a solid body,respecting to

or

0?

?

?

n

?

?

0?

?

?

s

?

or

c o n s tbyys u r f a c e ?? ???

On the surface of a solid body,respecting to ?

On the surface of a solid body,respecting to

streamline

s u r f a c e

b

u

v

dx

dy

?

?

?

?

?

??

3.8 Interim summary

General approach to solve the incompressible and

irrotational flows

1,Solve Laplace’s equation for or along with

the proper boundary conditions,These solutions

are usually in the form of a sum of elementary

solutions,

2,Obtain the flow velocity from or

3,Obtain the pressure from Bernoulli’s equation

? ?

???V?

xvyu ??????? ??,

22

2

1

2

1

?? ??? VpVp ??

3.9 Uniform flow,our first elementary flow

There are a series of elementary flows which can be

superimposed to synthesize more complex

incompressible flows,Right now,we are starting to

introduce them one by one,

Uniform flow,if the velocity is a constant at every

point in the flow field,then,such a flow can be

called as uniform flow,

It is easy to see,a uniform flow is a physical possible

incompressible flow and it is irrotational

0??? V? 0??? V?

Velocity vector is in x direction,and it is a constant

As this is a irrotational flow,then,the velocity

components can be expressed by

y

v

x

u

?

??

?

?? ??,

0,??

?

???

?

?

? vyVux

??

or

After integration being carried out

c o n s txV ?? ??

or

xV ???

Reason?

As this is a divergence free flow,then,the velocity

components can be expressed by

0,?

?

????

?

??

? xvVyu

??

0,???

?

???

?

?

? vxVuy

??

or

After integration being carried out

c o n s tyV ?? ??

yV ???

or

Characteristics of Uniform flow

0???? ?

C

sdV ?

?

1,Circulation around any closed curve in a uniform

flow is zero,

2,The velocity potential and stream function satisfies

Laplace’s equation,

3.10 Source flow,

our second elementary flow

Definition of source flow,

1,All the streamline are straight line emanating from

a center point O,

2,Velocity along the streamline varies inversely with

the distance from point O,

3,Source flow is a physical possible flow(that means

the divergence of velocity is zero at every point

except the origen.where it is infinite,Thus,the

origin is a singular point,

4,Source flow is irrotational at every point,

The origin,where the streamlines emanating from,

can be interpreted as a discrete source or sink,

The radial flow surrounding the origin can be

looked as the flow induced by the discrete source

or sink,

Superposition of distributions of singularities over

arbitrary bodies can be used to synthesize the

irrotational,incompressible flows around the bodies,

Definition of the strength of a discrete source or sink,

r

cV

r ?

0??V

Line source in three-dimensional space

Mass flow across the surface of the cylinder

rrr Vrldr lVlrdVm ??????

??

?? ???

2

0

2

0

2)(?

Volume flow across the surface of the cylinder

rr lV

mv ?

?

2??

??

The rate of volume flow per unit length along the

cylinder,

rrVl

v ?2??? ?

r

V r

?2

???

Velocity potential for a source

r

V

r r ?

?

2

???

?

?

01 ??

?

?

??

? V

r

)(ln

2

?

?

? fr ???

)( rgc o n s t ???

?

c o n s tr ??? ln

2 ?

?

?

rln

2 ?

? ??

Stream function for a source

r

V

r r ??

?

2

1 ???

?

?

0??

?

??

?

? V

r

)(

2

rf??? ?

?

?

)(?? gc o n s t ??

?

c o n s t??? ?

?

?

2

?

?

?

?

2

??

Description of stream function for a source

c o n s t?? ? c o n s t??

?

a straight line from the origin

Description of velocity potential for a source

c o n s t?? ? c o n s tr ?

?

a circle with its center at the origin

The velocity potential and stream function of a

source satisfy Laplace’s equation,

Conclusion,

source flow is a viable elementary flow for use in

building more complex flows

3.11 Combination of a uniform flow with a

source and sink

In a polar coordinate system,the stream functions of

a uniform flow and a source or sink can be written as,

?

?

?

2

???? s i nrV ?? and

Combination of two simple elementary flows,

Combination of a uniform flow and a source flow,

The stream function of the resulting flow is,

?

?

??

2

s i n ??? ? rV

as it satisfies Laplace’s equation,then,the stream

function above represents a viable irrotational,

incompressible flow,

Streamlines of the combined flow,

c o n s trV ???? ? ?

?

??

2

s i n

Velocity field of the combined flow,

r

V

r

V r

?

?

?

?

2

c o s1 ???

?

??

?

??? s i n???

?

??? V

r

V

Velocity field of a uniform flow,

?c o s?? VV r ?? s i n??? VV

Velocity field of a source flow,

r

V r

?2

?? 0??V

Conclusion,

The combined velocity filed of a uniform flow and

a source flow is simply direct sum of the two velocity

fields,Therefore,not only can we add the values of

velocity potential or stream function to obtain more

complex flows,we can add their derivatives,i.e.,the

velocities as well,

Analysis of the shape of the streamlines in the

combined flow field,

Equation of streamlines of the resulting flow,

c o n s trV ???? ? ?

?

??

2

s i n

Every streamline can be denoted as a solid wall

Among all the streamlines,which one we are

interested most?

A special point in the flow field --- stagnation point

At the stagnation point

0

2

c o s1 ????

?

??

? rVrV r ???

?

0s i n ???

?

???

? ?

?

? VrV

Solving for and, We find one stagnation point

located at

r ?

),2(),( ??? ??? Vr

The stagnation point is located directly upstream of

the source for a distance

)2( ?? V?

)2( ?? V??V increase ? decrease

? increase ? )2( ?? V?

increase

The value of stream function at the stagnation point is

c o n s t

V

V ?????

?

? ????? 2s in2

c o n s t???

2

?

Hence,the streamline that passing through the

stagnation point is described by,and it is

the curve ABC shown in the figure,

2???

Conclusion,(for inviscid flow)

1,Any streamline of the combined flow could be

replaced by a solid surface,

2,Curve ABC is a dividing streamline,

3,The entire region inside ABC could be replaced

with a solid body of the same shape,

4,The streamline ABC extends downstream to

infinity,and will not intersect with each other,that

means,the body is not closed,

Superposition of a uniform flow,a source flow,and

a sink flow,

The stream function of the resulting flow is,

21 22s in ??????

?????

? rV

or

)(

2

s in 21 ??

?

?? ???? ? rV

are functions of

21,?? ba n dr ?,

Setting,we can get two stagnation points at A

and B,And these two stagnation points are located

such that,

0?V

?

?

???

V

b

bOBOA

?

2

The equation of the streamline is

c o n s trV ????? ? )(

2

s i n 21 ??

?

??

At point A,

???? ??? 21

At point B,

021 ??? ???

So,the values of stream function at two stagnation

points are zero

The two stagnation points are on the same streamline,

and the equation the stagnation streamline is

0)(

2

s i n 21 ????? ??

?

?rV

It is the equation of an oval,

It is also the dividing streamline,

The region inside the oval can be replaced by a solid

body with the shape given by the equation above,

The region outside the oval can be interpreted as the

inviscid,potential,incompressible flow over the oval,

3.12 Doublet flow,our third elementary

flow

※ Source-sink pair degenerate a doublet

Stream function of a induced flow field by a source-

sink pair

?

?

??

?

? ???????

2

)(

2 21

Let distance,but remains constant,

then,the limit of the stream function denotes a

special flow called doublet

0?l ???l

?

?

?

?

?

? ???

??

?

?

?

?

d

c o n s tl 2

l im

,0

For an infinitesimal,the geometry of fig.3.19 yields ?d

badlrbla ???? ???,co s,s i n

Hence,

?

??

c o s

s i n

lr

l

b

ad

?

??

?

?

?

?

?

? ???

??

?

?

?

?

d

c o n s tl 2

l im

,0

?

??

c o s

s i n

lr

l

b

ad

????

?

?

?

?

?

?

?

???

?? ?

?

?

?

? c o s

s i n

2

lim

,0 lr

l

c o n s tl

or

?

?

?

?

?

?

?

??

?? ?

?

?

??

? c o s

s i n

2

lim

,0 lrc o n s tl

r

?

?

?? s i n

2

??

or Stream function of a doublet

In a similar way we can get

r

?

?

?? c o s

2

?

Velocity potential of a

doublet

Streamlines of a doublet flow

cc o n s t

r

???? ?

?

?? s i n

2

or

?

?

? s i n

2 c

r ??

Let,then we have an equation in polar

coordinate system

cd ?? 2??

?s i ndr ?

It is a circle with diameter d,on the vertical axis and

with the center located directly above the origin,

yddryxr ???? ?s i n222

?

222 )2()2( ddyx ???

Direction of a doublet flow,

the direction of a doublet is pointed from the sink to

the source

Detailed explanation for the limit as,

0?l

Doublet is a singularity that induces about it the

double-lobed circular flow,

3.13 Nonlifting flow over a circular cylinder

Some thing should be mentioned first

Uniform flow + a source

flow over a semi-infinite body

Uniform flow + a source-sink pair

flow over an oval-shaped body

Uniform flow + a doublet

Stream function combined with a uniform flow and

a doublet

r

rV ?

?

??? s i n

2

s i n ?? ?

or

??

?

?

??

?

?

??

?

? 221s in rVrV ?

?

??

Let

?? VR ?? 22

??

?

?

??

?

?

?? ?

2

2

1s in

r

R

rV ??

Stream function for a uniform flow + a doublet flow

Stream function for a flow over a circular cylinder of

radius R,

Prove,

Step 1,Velocity field

??

?

?

c o s11)c o s(

11

2

2

2

2

?? ??

?

?

??

?

?

????

?

?

??

?

?

??

?

?

? V

r

R

r

R

rV

rr

V r

?

?

?

?

?

?

??

?

?

??

?

?

????

?

?

?? ?? )s in(1

2

)s in( 2

2

3

2

??

?

? Vr

R

r

R

rV

r

V

?

?

? s in1 2

2

???

?

?

??

?

?

???

?

?

?? V

r

R

r

V

Step 2,Stagnation points

set

0c o s1

2

2

???

?

?

??

?

?

? ? ?V

r

R 0s i n1

2

2

???

?

?

??

?

?

? ? ?V

r

R

There are two stagnation points,located at

)0,(),( Rr ?? ),(),( ?? Rr ?

and

Point A Point B

Step 3,The value of stram function at two stagnation

points

01s in

,02

2 Rr

r

R

rV

?

?

? ???

?

?

??

?

?

??

??

??

The streamline with passing through both

stagnation points,the equation of this streamline is

0??

01s in

2

2

???

?

?

??

?

?

?? ?

r

R

rV ??

The equation above is satisfied with for all Rr ? ?

The equation above is satisfied with for all ??,0? r

The streamline is a dividing streamline,

therefore,we can replace the flow inside the circle

by a solid body,and the external flow will not know

the difference,

The inviscid irrotational,incompressible flow over a

circular cylinder of radius R can be synthesized by a

uniform flow with velocity and a doublet of

strength,where R is related to and through

0??

?V

?

?V

?

?

?

V

R

?

?

2

The entire flow field is symmetrical about both the

horizontal and vertical axes through the center of

the circular cylinder

There is no net lift and net drag over the cylinder

d’Alembert paradox,drag due to the viscous effect

Velocity distribution over the circular cylinder

pressure distribution over the circular cylinder

Example 3.9

3.14 Vortex flow,

Our fourth elementary flow

The flows we have introduced with superposition

of the elementary flows,such as uniform flow,

source or sink flow,and doublet flow,can be used to

present several flows around semi-infinite body,oval,

and circular cylinder,

Now,one thing we have to keep in mind,

there are no net lift exerted on the bodies

mentioned above,

What will happen if a vortex flow joins into the

superposition

Description of a vortex flow

Streamlines,circles

0?rV

r

CV ?

?

Definition of vortex flow,

1,All the streamline are circles with its center located

at point O,

2,Velocity along the streamline varies inversely with

the distance from point O,

3,Vortex flow is a physical possible flow(that means

the divergence of velocity is zero at every

point).where it is infinite.,

4,Vortex flow is irrotational at every point except the

origin,Thus,the origin is a singular point

Evaluation of C,

r

C

r

c o n s tV ??

?

Circulation around a given circular streamline

)2( rVsdV

C

??? ?????? ?

?

or

r

V

?? 2

??? ?

?2

???C

For vortex flow,the circulation along any streamline

is the same,that’s to say,,And is

called as the strength of the vortex,

C?2??? ?

What is the value of vorticity at r=0

CSdV

S

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

?? Stokes theorem

dSVSdVC

SS ????

??????

???

)(2 ?

The vorticity is perpendicular to the paper

Choose a circle with its center at the origin,and its

radius r approached to zero,Not matter how close

the circle approaches to the origin,the circulation

about the circle remains C?2???

0?r

dSVdSV

S

??

???????

That is

dSVC ?????2

or

dS

CV ?2??? ?

??

dS

C?2

as 0?r

or

???? V?

as 0?r

Vortex flow is irrotational everywhere except at the

point r = 0,where the vorticity is infinite,

The origin,where the vortex is located,is a singular

point in the flow field,

Velocity potential and stream function of the vortex

flow

?

?

?

2

???

rln

2 ?

? ??

3.15 Lifting flow over a circular cylinder

??

?

?

??

?

?

?? ?

2

2

1 1s in r

R

rV ??

Nonlifting over a circular cylinder

c o n s tr ??? ln

22 ?

?

Vortex flow

with

Rc o n s t ln

2 ?

???

R

rln

22 ?

? ??

R

r

r

R

rV ln

2

1s in

2

2

21 ?????

?

???

?

?

??

?

?

???? ?

After superimpose the two flows

?? ofv a l u e sa l lf o rRr 0???

Therefore,r=R is part of a streamline of the flow,

Influence of the value of on the value of the

stream function along the circle r =R

?

is a special case for the flows around a

circular cylinder,

0??

Determined value of is another condition for

the flow around any cylinder being defined

uniquely,

?

the streamlines about the horizontal axis is no longer

symmetrical,There must be a normal force exerted

on the circular cylinder

the streamlines about the vertical axis is still

symmetrical,There would be no horizontal force

exerted on the circular cylinder

Velocity field

?c o s1 2

2

???

?

?

??

?

?

?? V

r

R

V r

r

V

r

R

V

?

??

2

s in1 2

2 ?

???

?

?

??

?

?

??? ?

Stagnation point

0c o s1 2

2

???

?

?

??

?

?

?? ? ?V

r

R

V r

0

2

s in1 2

2

?

?

???

?

?

??

?

?

??? ?

r

V

r

R

V

?

??

With r=R

??

?

?

??

?

? ?

??

? RV?

?

4

a r c s in

Note,

1

4

??

? RV?

1

4

??

? RV?

1

4

??

? RV?

1

4

??

? RV?

For case

14 ??

? RV?

0c o s1 2

2

???

?

?

??

?

?

?? ? ?V

r

R

V r

? 22 ??? or??

r

V

r

R

V

?

??

2

s in1 2

2 ?

???

?

?

??

?

?

??? ?

?

2

2

44

R

RVRV

r ???

?

?

??

?

? ?

?

?

?

?? ??

Velocity on the surface of the cylinder

R

VVV

?

??

2

s i n2 ????? ?

Pressure coefficient along the surface of the cylinder

?

?

?

?

?

?

?

?

??

?

?

??

?

? ?

?

?

????

??

2

2

2

s in2

s in41

RVRV

VC p

??

?

??

The lift per unit span of the circular cylinder

??? ?VL ?

Kutta-Joukowski theorem,

the lift per unit span is directly proportional to the

circulation,

Comparison between the theoretical result and

real case,

Creation of lift on a spinning cylinder

3.16 The Kutta-Joukowski theorem and

the generation of lift

The lift per unit span of the circular cylinder

??? ?VL ?

It also valid in general for cylinder bodies of arbitrary

cross section

If the airfoil is producing lift,the velocity field around

the airfoil will be such that the line integral of velocity

around A will be finite

? ??? A sdV ??

Based on Kutta-Joukowski theorem,the lift per unit

span on the airfoil is given as

??? ?VL ?

Kutta-Joukowski theorem is simply an alternate way

of expressing the consequence of the surface

pressure distribution,

It is not quite proper to say that circulation,causes”

lift,

In the theory of incompressible,potential flow,it is

generally much easier to determine the circulation

around the body rather than calculate the detailed

surface pressure distribution,

How to calculate the circulation for a given body in a

given incompressible,inviscid flow?

3.17 Nonlifting flows over arbitrary bodies,

the numerical source panel method

Construction of a source panel

1,Fig.(a),m = 5

2,Fig.(b),m=11,total strength is the same as case 1

3,Fig.(c),m=101,total strength is the same as case 1

4,Fig.(d),m=101,but the source strength is reduced

5,Fig.(e),

6,Fig.(f),Boundary conditions at inclined surface

???? Vm 2,?

Investigation of nonlifting flows with indirect method

1,Superposition of known elementary flows

2,Find the stagnation points

3,Find dividing streamline

4,Replace the inner region of the dividing stramline

with a solid body

Flows over semi-infinite body,oval,circular cylinder

Find a correct combination of elementary flows to

synthesize the flow over a given body,

------- direct method,

Purpose of this section,present a direct method for

nonlifting flows with source panel method,

------- numerical solution

Under which condition,this numerical method is

valid,

------- incompressible,potential flows

Concept of the source sheet,

infinite number of line sources distributed side by

side,the strength of each line source if infinitesimal

small,

Definition for the strength of source sheet,

source strength per unit length along s

Discontinuity of the normal velocity component

across the source sheet,

?

?

Velocity potential at point P contributed by a small

section of the source sheet,

rdsd ln

2 ?

?? ?

Velocity potential at point P contributed by the

whole source sheet,

rdsyx

b

a

ln

2

),( ??

?

??

A given body of arbitrary shape in free stream flow

A source sheet is used to cover the surface of the

body,Condition for the distribution of )(s?

The source sheet is approximated by a series of

source panels,

The source strength per unit length is a constant

over a given panel,but varies from one panel to the

next,

?

If there are n source panels distributed along the

body surface,then,there would be n unknowns,

that is the source strength per unit length

j?

The velocity potential induced at point P due to the

jth panel is

???

b

a

jpj

j

j dsrln2 ?

?

?

The velocity potential induced at point P due to all

the panels

? ??

??

???

n

j

b

a

jpj

j

n

j

j dsrP

11

ln

2

)(

?

?

??

where

22 )()(

jjpj yyxxr ????

Let point P is located at the control point on panel i

? ?

?

?

n

j

j

jij

j

ii dsryx

1

ln

2

),(

?

?

?

where

22 )()(

jijiij yyxxr ????

Component of normal to the ith panel is

?V

iin VnVV ?c o s,??? ???

??

Normal component of velocity induced at the

control point of the ith panel by the source panels

? ?),( ii

i

n yxnV ??

??

? ?? ?

?

?

?

?

??

n

ij

j

j

jij

i

ji

n dsr

n

V

)(

1

ln

22 ?

??

Boundary condition

0,??? nn VV

? ? 0c o sln

22

)(

1

??

?

?

? ?

?

?

? ? i

n

ij

j

j

jij

i

ji

Vdsr

n

?

?

??

Let

? ? j

j

ij

i

ji dsrnI ? ?

?? ln

,

0c o s

22

)(

1

,??? ?

?

?

? i

n

ij

j

ji

ji

VI ?

?

??

It is a linear algebraic equations with n unknowns

After the n unknowns been solved,we can

calculate the tangential velocity component on the

surface of the body

is VV ?s i n,?? ?

? ?? ?

?

?

?

?

?

?

?

n

j

j

jij

j

s dsr

ss

V

1

ln

2 ?

??

? ?? ?

?

??

?

?

????

n

j

j

jij

j

issi dsr

s

VVVV

1

,ln

2

s in

?

?

?

With Bernoulli’s equation

2

1 ??

?

?

??

?

?

??

?V

V

C ip

INVISCID INCOMPRESSIBLE FLOW

In part II,we deal with the flow of a fluid which has

constant density– incompressible flow,This applies to the

flow of liquids,such as water flow,and to low-speed flow

of gases,The material covered here is applicable to low-

speed flight through the atmosphere—flight at a Mach

number of about 0.3

Chapter 3

Fundamentals of Inviscid,

incompressible Flow

Theoretical fluid dynamics,being a difficult subject,is for

convenience,commonly divided into two branches,one

treating frictionless or perfect fluids,the other treating of

viscous or imperfect fluids,The frictionless fluid has no

existence in nature,but is hypothesized by mathematicians in

order to facilitate the investigation of important laws and

principles that may be approximately true of viscous or natural

fluids,

3.1 Introduction and Road Map

From an aerodynamic point of view,at

air velocities between 0 to 360km/h the

air density remains essentially constant,

varying only a few percent,

Purpose of this chapter is to establish

some fundamental relations applicable to

inviscid,incompressible flows and to

discuss some simple but important flow

fields and applications,

3.2 Bernoulli’s Equation

c o n s tVp ?? 2

2

1 ?

Derivation of Bernoulli’s equation

Step 1,x component momentum equation

without viscous effect and body forces

x

p

Dt

Du

?

????

or

x

p

z

uw

y

uv

x

uu

t

u

?

???

?

??

?

??

?

??

?

? ????

For steady flow,,then 0??? tu

x

p

z

uw

y

uv

x

uu

?

???

?

??

?

??

?

?

?

1

Multiply both sides with dx

dx

x

pdx

z

uwdx

y

uvdx

x

uu

?

???

?

??

?

??

?

? ???

Along a streamline in 3D space,there are

0

0

??

??

udyv d x

w d xudz

Substituting the differential equations of the

streamline into x component momentum equation

dx

x

pdz

z

uudy

y

uudx

x

uu

?

???

?

??

?

??

?

?

?

1

or

dx

x

p

dz

z

u

dy

y

u

dx

x

u

u

?

?

????

?

?

??

?

?

?

?

?

?

?

?

?

?

?

1

as

dz

z

udy

y

udx

x

udu

?

??

?

??

?

??

We have

dx

x

pudu

?

???

?

1

or

dx

x

pdu

?

???

?

1

2

1 2

In the same way,we can get

dy

y

pdv

?

???

?

1

2

1 2 dz

z

pdw

?

???

?

1

2

1 2

??

?

?

??

?

?

?

?

?

?

?

?

?

?

???? dz

z

p

dy

y

p

dx

x

p

wvud

?

1

)(

2

1 222

2222 Vwvu ???

as

and

dpdz

z

pdy

y

pdx

x

p ?

?

??

?

??

?

?

Then,we have

dpdV

?

1

2

1 2 ?? or V d Vdp ???

V d Vdp ???

The above equation is called Euler’s Equation,

precondition,inviscid,without body force,along a

streamline,

usage,setup the relation between dpa n ddV

Step 2,integration of Euler’s equation

For incompressible flows,

The integration from point 1 to point 2 along a

streamline is

c o n s t??

?? ??

2

1

2

1

V

V

p

p

V d Vdp ?

or

??

?

?

??

?

?

????

22

2

1

2

2

12

VV

pp ?

or

2

11

2

22 2

1

2

1 VpVp ?? ???

The above equation is called Bernoulli’s Equation,

precondition,steady,inviscid,incompressible,without

body force,along a streamline,

usage,setup the relation between at point

1 on a streamline to at another point 2 on

the same streamline,

11 pa n dV

22 pa n dV

As point 1 and point 2 are arbitrarily chosen on a

streamline,so

c o n s tVp ?? 2

2

1 ?

Along a streamline

1,As there are no stipulation being made to whether

the flow is rotational or irrotational,it satisfies for

both irrotational flows and rotational flows as well,

2,The value of the constant will change from one

streamline to another,

3,For irrotational flows

c o n s tVp ?? 2

2

1 ? Through out the flow

Bernoulli’s equation is also a relation for mechanical

energy in an incompressible flow

c o n s tVp ?? 2

2

1 ?

where is the kinetic energy per unit volume 22V?

As Bernoulli’s equation can also be derived from the

energy equation,the energy equation is redundant

for the analysis of inviscid,incompressible flow,

The way for solving inviscid,incompressible flows,

1,Obtain the velocity field from the governing

equations,

2,Obtain the pressure field from Bernoulli’s equation

3.3 Incompressible flow in a duct,the

Venturi tube and low-speed wind tunnel

Assumption of quasi-one-dimensional flows

1,Flow-field properties are uniform across any cross

section,

2,All the flow-field properties are assumed to be

functions of x only,

Respect to continuity equation,and for steady case

0222111 ??? VAVA ??

or

222111 VAVA ?? ?

For incompressible flow

2211 VAVA ?

Applications of venturi tube

Measurements of flow speed in a duct,

2

212

2

1 )(

2 VppV ???

?

From Bernoulli’s equation

From continuity equation,we have

1

2

1

2 VA

AV ?

Then

2

1

2

2

1

12

2

1 )(

2

V

A

A

ppV ??

?

?

??

?

?

???

?

? ?1)(

)(2

2

21

12

1 ?

?

?

AA

pp

V

?

3.4 Pitot tube,measurement of airspeed

Definition of static pressure,the measurement of

the static pressure,

Definition of stagnation pressure,the measurement

of the stagnation pressure,

The usage of Pitot tube,

The pressure at point C is the stagnation pressure

0p

The pressure at point A is the static pressure

1p

The speed at point C is equal to zero

The speed at point A is equal to free stream velocity

1V

The speed and pressure and at point A and B are the

same,apply Bernoulli’s equation,we have

22

2

1

2

1

BBAA VpVp ?? ???

0

2

1

0

2

11 ??? pVp ?

or

?

)(2 10

1

pp

V

?

?

Combined instrument for the measurement of both

static and total(stagnation) pressure,

Definition of dynamic pressure,the relationship

between of the static pressure,total pressure and

dynamic pressure,

2

2

1 Vq ??

For incompressible flows

0

2

11 2

1 pVp ?? ?

static

pressure

dynamic

pressure

total

pressure

Keep in mind !!!!!

Any result derived from Bernoulli’s equation holds for

incompressible flow only

3.5 Pressure coefficient

Definition of the pressure coefficient

?

???

q

ppC

p

where

2

2

1

??? ? Vq ?

The pressure coefficient is a nondimensional value,

it can be used throughout aerodynamics,for both

compressible flow and incompressible flow,

For incompressible flow,the pressure coefficient

can be expressed in terms of velocity only

If and are defined as the freestream velocity

and pressure,And and are the velocity and

pressure at an arbitrary point in the flow,Then,with

Bernoulli’s equation,

?V ?

p

V p

22

2

1

2

1 VpVp ?? ???

??

or

)(

2

1 22 VVpp ???

?? ?

?

???

q

ppC

p

? )(

2

1 22 VVpp ???

?? ?

2

22

2

1

)(

2

1

?

?

?

?

?

?

?

?

V

VV

q

pp

C

p

?

?

or

2

1 ??

?

?

??

?

?

??

?

?

??

?

V

V

q

pp

C p

Valid for

incompressible

flow only

??

??

?

?

?

??

??

??

??

??

??

ppVV

Cqpp

CVV

CVV

CVV

CV

p

p

p

p

p

0

0

0

10

3.6 Condition on velocity for

incompressible flow

Contents in the center branch

1,Laplace’s equation

2,General philosophy and use in solving problems

3,Synthesis of complex flows from a superposition of

elementary flows

Basic condition on velocity in an incompressible

flow,

From the definition for the divergence of velocity

? ?

Dt

VD

V

V ?

?

1??? ?

If the density of the fluid is equal to a constant,then

? ? 0?

Dt

VD ?

That means,for incompressible flow,the velocity

must satisfies

0??? V?

If we consider the continuity equation

? ? 0????

?

? V

t

?

??

And apply it for the incompressible flows,then,the

continuity equation becomes

00 ???? V??

?

0??? V?

3.7 Governing equation for irrotational,

incompressible flow,Laplace’s Equation

The continuity equation for incompressible flow is

0??? V?

???V?

Therefore,for incompressible and irrotational flows

0)( 2 ?????? ?? Laplace’s equation

If the flow is irrotational,then,the velocity can be

expressed with velocity potential

Solutions of Laplace’s Equation are called harmonic

functions,

Solutions of Laplace’s Equation

in Cartesian coordinates,

02

2

2

2

2

2

2 ?

?

?

?

?

?

?

?

?

??

zyx

???

?

in cylindrical coordinates,

0

11

2

2

2

2

2

2 ?

?

?

?

?

?

??

?

?

?

?

?

?

?

?

?

??

zrr

r

rr

?

?

??

?

For 2D incompressible flow,a stream function can

be defined,and the relation between velocity and

stream function is

x

v

y

u

?

???

?

?? ??,

The continuity equation,expressed in Cartesian

coordinates is,

0?

?

??

?

????

y

v

x

uV?

0

22

?

??

?

?

??

?

??

?

?

?

?

?

?

?

?

?

?

???

?

?

??

?

?

?

?

?

?

xyyxxyyx

????

The very definition and use of stream function is a

statement of the continuity equation of mass,and

therefore Equations bellow

x

v

y

u

?

???

?

?? ??,

Can be used in place of the continuity equation itself,

If the flow is irrotational,then

0?

?

??

?

?

y

u

x

v

? xvyu ?

???

?

?? ??,

0???

?

?

??

?

?

?

?

?

?

??

?

?

?

?

?

?

?

?

?

?

yyxx

??

?

02

2

2

2

?

?

?

?

?

?

yx

??

Laplace’s equation

Important conclusions

1,Any irrotational,incompressible flow has a velocity

potential and stream function(for 2D flow) that

both satisfy Laplace’s Equation

2,Conversely,any solution of Laplace’s Equation

represents the velocity potential or stream

function(2D) for an irrotational and incompresible

flow,

Superposition of the solutions of Laplace’s Equation

The Laplace’s Equation is a second-order linear

partial differential equation,The sum of any

particular solutions of a linear differential equation is

also a solution of the equation(Laplace’s Equation),

For example,if represent n separate

solutions of the Laplace’s equation,then the sum

n????,,,,321 ??

n????? ????? ??321

is also a solution of the Laplace’s equation

Conclusion respects to the superposition of the

solutions of Laplace’s equation

A complicated flow pattern for an irrotational,

incompressible flow can be synthesized by adding

together a number of elementary flows which are

also irrotational and incompressible,

How to make the difference between irrotational

and incompressible flows around different

aerodynamic shapes,--- Boundary conditions

3.7.1 Infinity boundary conditions

0,?

?

???

?

???

?

??

?

??

? xyvVyxu

????

At infinity in all directions

3.7.2 Wall boundary conditions

0?? nV ??

On the surface of a solid body,respecting to

or

0?

?

?

n

?

?

0?

?

?

s

?

or

c o n s tbyys u r f a c e ?? ???

On the surface of a solid body,respecting to ?

On the surface of a solid body,respecting to

streamline

s u r f a c e

b

u

v

dx

dy

?

?

?

?

?

??

3.8 Interim summary

General approach to solve the incompressible and

irrotational flows

1,Solve Laplace’s equation for or along with

the proper boundary conditions,These solutions

are usually in the form of a sum of elementary

solutions,

2,Obtain the flow velocity from or

3,Obtain the pressure from Bernoulli’s equation

? ?

???V?

xvyu ??????? ??,

22

2

1

2

1

?? ??? VpVp ??

3.9 Uniform flow,our first elementary flow

There are a series of elementary flows which can be

superimposed to synthesize more complex

incompressible flows,Right now,we are starting to

introduce them one by one,

Uniform flow,if the velocity is a constant at every

point in the flow field,then,such a flow can be

called as uniform flow,

It is easy to see,a uniform flow is a physical possible

incompressible flow and it is irrotational

0??? V? 0??? V?

Velocity vector is in x direction,and it is a constant

As this is a irrotational flow,then,the velocity

components can be expressed by

y

v

x

u

?

??

?

?? ??,

0,??

?

???

?

?

? vyVux

??

or

After integration being carried out

c o n s txV ?? ??

or

xV ???

Reason?

As this is a divergence free flow,then,the velocity

components can be expressed by

0,?

?

????

?

??

? xvVyu

??

0,???

?

???

?

?

? vxVuy

??

or

After integration being carried out

c o n s tyV ?? ??

yV ???

or

Characteristics of Uniform flow

0???? ?

C

sdV ?

?

1,Circulation around any closed curve in a uniform

flow is zero,

2,The velocity potential and stream function satisfies

Laplace’s equation,

3.10 Source flow,

our second elementary flow

Definition of source flow,

1,All the streamline are straight line emanating from

a center point O,

2,Velocity along the streamline varies inversely with

the distance from point O,

3,Source flow is a physical possible flow(that means

the divergence of velocity is zero at every point

except the origen.where it is infinite,Thus,the

origin is a singular point,

4,Source flow is irrotational at every point,

The origin,where the streamlines emanating from,

can be interpreted as a discrete source or sink,

The radial flow surrounding the origin can be

looked as the flow induced by the discrete source

or sink,

Superposition of distributions of singularities over

arbitrary bodies can be used to synthesize the

irrotational,incompressible flows around the bodies,

Definition of the strength of a discrete source or sink,

r

cV

r ?

0??V

Line source in three-dimensional space

Mass flow across the surface of the cylinder

rrr Vrldr lVlrdVm ??????

??

?? ???

2

0

2

0

2)(?

Volume flow across the surface of the cylinder

rr lV

mv ?

?

2??

??

The rate of volume flow per unit length along the

cylinder,

rrVl

v ?2??? ?

r

V r

?2

???

Velocity potential for a source

r

V

r r ?

?

2

???

?

?

01 ??

?

?

??

? V

r

)(ln

2

?

?

? fr ???

)( rgc o n s t ???

?

c o n s tr ??? ln

2 ?

?

?

rln

2 ?

? ??

Stream function for a source

r

V

r r ??

?

2

1 ???

?

?

0??

?

??

?

? V

r

)(

2

rf??? ?

?

?

)(?? gc o n s t ??

?

c o n s t??? ?

?

?

2

?

?

?

?

2

??

Description of stream function for a source

c o n s t?? ? c o n s t??

?

a straight line from the origin

Description of velocity potential for a source

c o n s t?? ? c o n s tr ?

?

a circle with its center at the origin

The velocity potential and stream function of a

source satisfy Laplace’s equation,

Conclusion,

source flow is a viable elementary flow for use in

building more complex flows

3.11 Combination of a uniform flow with a

source and sink

In a polar coordinate system,the stream functions of

a uniform flow and a source or sink can be written as,

?

?

?

2

???? s i nrV ?? and

Combination of two simple elementary flows,

Combination of a uniform flow and a source flow,

The stream function of the resulting flow is,

?

?

??

2

s i n ??? ? rV

as it satisfies Laplace’s equation,then,the stream

function above represents a viable irrotational,

incompressible flow,

Streamlines of the combined flow,

c o n s trV ???? ? ?

?

??

2

s i n

Velocity field of the combined flow,

r

V

r

V r

?

?

?

?

2

c o s1 ???

?

??

?

??? s i n???

?

??? V

r

V

Velocity field of a uniform flow,

?c o s?? VV r ?? s i n??? VV

Velocity field of a source flow,

r

V r

?2

?? 0??V

Conclusion,

The combined velocity filed of a uniform flow and

a source flow is simply direct sum of the two velocity

fields,Therefore,not only can we add the values of

velocity potential or stream function to obtain more

complex flows,we can add their derivatives,i.e.,the

velocities as well,

Analysis of the shape of the streamlines in the

combined flow field,

Equation of streamlines of the resulting flow,

c o n s trV ???? ? ?

?

??

2

s i n

Every streamline can be denoted as a solid wall

Among all the streamlines,which one we are

interested most?

A special point in the flow field --- stagnation point

At the stagnation point

0

2

c o s1 ????

?

??

? rVrV r ???

?

0s i n ???

?

???

? ?

?

? VrV

Solving for and, We find one stagnation point

located at

r ?

),2(),( ??? ??? Vr

The stagnation point is located directly upstream of

the source for a distance

)2( ?? V?

)2( ?? V??V increase ? decrease

? increase ? )2( ?? V?

increase

The value of stream function at the stagnation point is

c o n s t

V

V ?????

?

? ????? 2s in2

c o n s t???

2

?

Hence,the streamline that passing through the

stagnation point is described by,and it is

the curve ABC shown in the figure,

2???

Conclusion,(for inviscid flow)

1,Any streamline of the combined flow could be

replaced by a solid surface,

2,Curve ABC is a dividing streamline,

3,The entire region inside ABC could be replaced

with a solid body of the same shape,

4,The streamline ABC extends downstream to

infinity,and will not intersect with each other,that

means,the body is not closed,

Superposition of a uniform flow,a source flow,and

a sink flow,

The stream function of the resulting flow is,

21 22s in ??????

?????

? rV

or

)(

2

s in 21 ??

?

?? ???? ? rV

are functions of

21,?? ba n dr ?,

Setting,we can get two stagnation points at A

and B,And these two stagnation points are located

such that,

0?V

?

?

???

V

b

bOBOA

?

2

The equation of the streamline is

c o n s trV ????? ? )(

2

s i n 21 ??

?

??

At point A,

???? ??? 21

At point B,

021 ??? ???

So,the values of stream function at two stagnation

points are zero

The two stagnation points are on the same streamline,

and the equation the stagnation streamline is

0)(

2

s i n 21 ????? ??

?

?rV

It is the equation of an oval,

It is also the dividing streamline,

The region inside the oval can be replaced by a solid

body with the shape given by the equation above,

The region outside the oval can be interpreted as the

inviscid,potential,incompressible flow over the oval,

3.12 Doublet flow,our third elementary

flow

※ Source-sink pair degenerate a doublet

Stream function of a induced flow field by a source-

sink pair

?

?

??

?

? ???????

2

)(

2 21

Let distance,but remains constant,

then,the limit of the stream function denotes a

special flow called doublet

0?l ???l

?

?

?

?

?

? ???

??

?

?

?

?

d

c o n s tl 2

l im

,0

For an infinitesimal,the geometry of fig.3.19 yields ?d

badlrbla ???? ???,co s,s i n

Hence,

?

??

c o s

s i n

lr

l

b

ad

?

??

?

?

?

?

?

? ???

??

?

?

?

?

d

c o n s tl 2

l im

,0

?

??

c o s

s i n

lr

l

b

ad

????

?

?

?

?

?

?

?

???

?? ?

?

?

?

? c o s

s i n

2

lim

,0 lr

l

c o n s tl

or

?

?

?

?

?

?

?

??

?? ?

?

?

??

? c o s

s i n

2

lim

,0 lrc o n s tl

r

?

?

?? s i n

2

??

or Stream function of a doublet

In a similar way we can get

r

?

?

?? c o s

2

?

Velocity potential of a

doublet

Streamlines of a doublet flow

cc o n s t

r

???? ?

?

?? s i n

2

or

?

?

? s i n

2 c

r ??

Let,then we have an equation in polar

coordinate system

cd ?? 2??

?s i ndr ?

It is a circle with diameter d,on the vertical axis and

with the center located directly above the origin,

yddryxr ???? ?s i n222

?

222 )2()2( ddyx ???

Direction of a doublet flow,

the direction of a doublet is pointed from the sink to

the source

Detailed explanation for the limit as,

0?l

Doublet is a singularity that induces about it the

double-lobed circular flow,

3.13 Nonlifting flow over a circular cylinder

Some thing should be mentioned first

Uniform flow + a source

flow over a semi-infinite body

Uniform flow + a source-sink pair

flow over an oval-shaped body

Uniform flow + a doublet

Stream function combined with a uniform flow and

a doublet

r

rV ?

?

??? s i n

2

s i n ?? ?

or

??

?

?

??

?

?

??

?

? 221s in rVrV ?

?

??

Let

?? VR ?? 22

??

?

?

??

?

?

?? ?

2

2

1s in

r

R

rV ??

Stream function for a uniform flow + a doublet flow

Stream function for a flow over a circular cylinder of

radius R,

Prove,

Step 1,Velocity field

??

?

?

c o s11)c o s(

11

2

2

2

2

?? ??

?

?

??

?

?

????

?

?

??

?

?

??

?

?

? V

r

R

r

R

rV

rr

V r

?

?

?

?

?

?

??

?

?

??

?

?

????

?

?

?? ?? )s in(1

2

)s in( 2

2

3

2

??

?

? Vr

R

r

R

rV

r

V

?

?

? s in1 2

2

???

?

?

??

?

?

???

?

?

?? V

r

R

r

V

Step 2,Stagnation points

set

0c o s1

2

2

???

?

?

??

?

?

? ? ?V

r

R 0s i n1

2

2

???

?

?

??

?

?

? ? ?V

r

R

There are two stagnation points,located at

)0,(),( Rr ?? ),(),( ?? Rr ?

and

Point A Point B

Step 3,The value of stram function at two stagnation

points

01s in

,02

2 Rr

r

R

rV

?

?

? ???

?

?

??

?

?

??

??

??

The streamline with passing through both

stagnation points,the equation of this streamline is

0??

01s in

2

2

???

?

?

??

?

?

?? ?

r

R

rV ??

The equation above is satisfied with for all Rr ? ?

The equation above is satisfied with for all ??,0? r

The streamline is a dividing streamline,

therefore,we can replace the flow inside the circle

by a solid body,and the external flow will not know

the difference,

The inviscid irrotational,incompressible flow over a

circular cylinder of radius R can be synthesized by a

uniform flow with velocity and a doublet of

strength,where R is related to and through

0??

?V

?

?V

?

?

?

V

R

?

?

2

The entire flow field is symmetrical about both the

horizontal and vertical axes through the center of

the circular cylinder

There is no net lift and net drag over the cylinder

d’Alembert paradox,drag due to the viscous effect

Velocity distribution over the circular cylinder

pressure distribution over the circular cylinder

Example 3.9

3.14 Vortex flow,

Our fourth elementary flow

The flows we have introduced with superposition

of the elementary flows,such as uniform flow,

source or sink flow,and doublet flow,can be used to

present several flows around semi-infinite body,oval,

and circular cylinder,

Now,one thing we have to keep in mind,

there are no net lift exerted on the bodies

mentioned above,

What will happen if a vortex flow joins into the

superposition

Description of a vortex flow

Streamlines,circles

0?rV

r

CV ?

?

Definition of vortex flow,

1,All the streamline are circles with its center located

at point O,

2,Velocity along the streamline varies inversely with

the distance from point O,

3,Vortex flow is a physical possible flow(that means

the divergence of velocity is zero at every

point).where it is infinite.,

4,Vortex flow is irrotational at every point except the

origin,Thus,the origin is a singular point

Evaluation of C,

r

C

r

c o n s tV ??

?

Circulation around a given circular streamline

)2( rVsdV

C

??? ?????? ?

?

or

r

V

?? 2

??? ?

?2

???C

For vortex flow,the circulation along any streamline

is the same,that’s to say,,And is

called as the strength of the vortex,

C?2??? ?

What is the value of vorticity at r=0

CSdV

S

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

?? Stokes theorem

dSVSdVC

SS ????

??????

???

)(2 ?

The vorticity is perpendicular to the paper

Choose a circle with its center at the origin,and its

radius r approached to zero,Not matter how close

the circle approaches to the origin,the circulation

about the circle remains C?2???

0?r

dSVdSV

S

??

???????

That is

dSVC ?????2

or

dS

CV ?2??? ?

??

dS

C?2

as 0?r

or

???? V?

as 0?r

Vortex flow is irrotational everywhere except at the

point r = 0,where the vorticity is infinite,

The origin,where the vortex is located,is a singular

point in the flow field,

Velocity potential and stream function of the vortex

flow

?

?

?

2

???

rln

2 ?

? ??

3.15 Lifting flow over a circular cylinder

??

?

?

??

?

?

?? ?

2

2

1 1s in r

R

rV ??

Nonlifting over a circular cylinder

c o n s tr ??? ln

22 ?

?

Vortex flow

with

Rc o n s t ln

2 ?

???

R

rln

22 ?

? ??

R

r

r

R

rV ln

2

1s in

2

2

21 ?????

?

???

?

?

??

?

?

???? ?

After superimpose the two flows

?? ofv a l u e sa l lf o rRr 0???

Therefore,r=R is part of a streamline of the flow,

Influence of the value of on the value of the

stream function along the circle r =R

?

is a special case for the flows around a

circular cylinder,

0??

Determined value of is another condition for

the flow around any cylinder being defined

uniquely,

?

the streamlines about the horizontal axis is no longer

symmetrical,There must be a normal force exerted

on the circular cylinder

the streamlines about the vertical axis is still

symmetrical,There would be no horizontal force

exerted on the circular cylinder

Velocity field

?c o s1 2

2

???

?

?

??

?

?

?? V

r

R

V r

r

V

r

R

V

?

??

2

s in1 2

2 ?

???

?

?

??

?

?

??? ?

Stagnation point

0c o s1 2

2

???

?

?

??

?

?

?? ? ?V

r

R

V r

0

2

s in1 2

2

?

?

???

?

?

??

?

?

??? ?

r

V

r

R

V

?

??

With r=R

??

?

?

??

?

? ?

??

? RV?

?

4

a r c s in

Note,

1

4

??

? RV?

1

4

??

? RV?

1

4

??

? RV?

1

4

??

? RV?

For case

14 ??

? RV?

0c o s1 2

2

???

?

?

??

?

?

?? ? ?V

r

R

V r

? 22 ??? or??

r

V

r

R

V

?

??

2

s in1 2

2 ?

???

?

?

??

?

?

??? ?

?

2

2

44

R

RVRV

r ???

?

?

??

?

? ?

?

?

?

?? ??

Velocity on the surface of the cylinder

R

VVV

?

??

2

s i n2 ????? ?

Pressure coefficient along the surface of the cylinder

?

?

?

?

?

?

?

?

??

?

?

??

?

? ?

?

?

????

??

2

2

2

s in2

s in41

RVRV

VC p

??

?

??

The lift per unit span of the circular cylinder

??? ?VL ?

Kutta-Joukowski theorem,

the lift per unit span is directly proportional to the

circulation,

Comparison between the theoretical result and

real case,

Creation of lift on a spinning cylinder

3.16 The Kutta-Joukowski theorem and

the generation of lift

The lift per unit span of the circular cylinder

??? ?VL ?

It also valid in general for cylinder bodies of arbitrary

cross section

If the airfoil is producing lift,the velocity field around

the airfoil will be such that the line integral of velocity

around A will be finite

? ??? A sdV ??

Based on Kutta-Joukowski theorem,the lift per unit

span on the airfoil is given as

??? ?VL ?

Kutta-Joukowski theorem is simply an alternate way

of expressing the consequence of the surface

pressure distribution,

It is not quite proper to say that circulation,causes”

lift,

In the theory of incompressible,potential flow,it is

generally much easier to determine the circulation

around the body rather than calculate the detailed

surface pressure distribution,

How to calculate the circulation for a given body in a

given incompressible,inviscid flow?

3.17 Nonlifting flows over arbitrary bodies,

the numerical source panel method

Construction of a source panel

1,Fig.(a),m = 5

2,Fig.(b),m=11,total strength is the same as case 1

3,Fig.(c),m=101,total strength is the same as case 1

4,Fig.(d),m=101,but the source strength is reduced

5,Fig.(e),

6,Fig.(f),Boundary conditions at inclined surface

???? Vm 2,?

Investigation of nonlifting flows with indirect method

1,Superposition of known elementary flows

2,Find the stagnation points

3,Find dividing streamline

4,Replace the inner region of the dividing stramline

with a solid body

Flows over semi-infinite body,oval,circular cylinder

Find a correct combination of elementary flows to

synthesize the flow over a given body,

------- direct method,

Purpose of this section,present a direct method for

nonlifting flows with source panel method,

------- numerical solution

Under which condition,this numerical method is

valid,

------- incompressible,potential flows

Concept of the source sheet,

infinite number of line sources distributed side by

side,the strength of each line source if infinitesimal

small,

Definition for the strength of source sheet,

source strength per unit length along s

Discontinuity of the normal velocity component

across the source sheet,

?

?

Velocity potential at point P contributed by a small

section of the source sheet,

rdsd ln

2 ?

?? ?

Velocity potential at point P contributed by the

whole source sheet,

rdsyx

b

a

ln

2

),( ??

?

??

A given body of arbitrary shape in free stream flow

A source sheet is used to cover the surface of the

body,Condition for the distribution of )(s?

The source sheet is approximated by a series of

source panels,

The source strength per unit length is a constant

over a given panel,but varies from one panel to the

next,

?

If there are n source panels distributed along the

body surface,then,there would be n unknowns,

that is the source strength per unit length

j?

The velocity potential induced at point P due to the

jth panel is

???

b

a

jpj

j

j dsrln2 ?

?

?

The velocity potential induced at point P due to all

the panels

? ??

??

???

n

j

b

a

jpj

j

n

j

j dsrP

11

ln

2

)(

?

?

??

where

22 )()(

jjpj yyxxr ????

Let point P is located at the control point on panel i

? ?

?

?

n

j

j

jij

j

ii dsryx

1

ln

2

),(

?

?

?

where

22 )()(

jijiij yyxxr ????

Component of normal to the ith panel is

?V

iin VnVV ?c o s,??? ???

??

Normal component of velocity induced at the

control point of the ith panel by the source panels

? ?),( ii

i

n yxnV ??

??

? ?? ?

?

?

?

?

??

n

ij

j

j

jij

i

ji

n dsr

n

V

)(

1

ln

22 ?

??

Boundary condition

0,??? nn VV

? ? 0c o sln

22

)(

1

??

?

?

? ?

?

?

? ? i

n

ij

j

j

jij

i

ji

Vdsr

n

?

?

??

Let

? ? j

j

ij

i

ji dsrnI ? ?

?? ln

,

0c o s

22

)(

1

,??? ?

?

?

? i

n

ij

j

ji

ji

VI ?

?

??

It is a linear algebraic equations with n unknowns

After the n unknowns been solved,we can

calculate the tangential velocity component on the

surface of the body

is VV ?s i n,?? ?

? ?? ?

?

?

?

?

?

?

?

n

j

j

jij

j

s dsr

ss

V

1

ln

2 ?

??

? ?? ?

?

??

?

?

????

n

j

j

jij

j

issi dsr

s

VVVV

1

,ln

2

s in

?

?

?

With Bernoulli’s equation

2

1 ??

?

?

??

?

?

??

?V

V

C ip