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

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