22 Basic Flow

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IDEAL FLOW THEORY

FLOWNETSFor any two-dimensional irrotational

flow of a ideal fluid, two series of

lines may be drawn :

(1) lines along which is constant(2) lines along which is constant

SECTION B 1

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IDEAL FLOW THEORY

stream line perpendicular to the

velocity potential

These lines together form a grid of

quadrilaterals having 90corners.

This grid is known as aflow net.

It is provides a simple yet valuable

indication of the flow pattern.

SECTION B 2

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IDEAL FLOW THEORY

COMBININGFLOWPATTERNS

If two or more flow patterns are

combined, the resultant flow pattern

is described by a stream function that

at any point is the algebraic sum of

the stream functions of the

constituent flow at that point.

By this principle complicated motions

may be regarded as combinations of

simpler ones.

SECTION B 3

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IDEAL FLOW THEORY

21 ++=AP

++= 21AQ

The resultant flow pattern may

therefore be constructed graphically

simply by joining the points for whichthe total stream function has the

same value.

This method was first described by

W.J.M.Rankine(1820-1872)

SECTION B 4

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IDEAL FLOW THEORY

Velocity components ;

( ) 2121

21 uuyyyy

u +=

+

=+

=

=

21 vvx

v +=

=

Net velocity potential ;

.......321 +++= net

SECTION B 5

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IDEAL FLOW THEORY

BASIC PATTERNS OF FLOWUniform Flow ;

velocity components ;

sin

cos

=

=

qv

qu

stream function ;vxuy=

velocity potential ;vyux+=

SECTION B 6

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IDEAL FLOW THEORY

Source Flow ;

A source is a point from which fluid

issues uniformly in all directions. If

for two-dimensional flow, the flow

pattern consists of streamlinesuniformly spaced and directed

radially outward from one point in

the reference plane, the flow is said to

emerge from a line source.

SECTION B 7

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IDEAL FLOW THEORY

The strength mof a source is the total

volume rate of flow from it.

The velocity qat radius ris given by;

r

mq

2velocitylar toperpendicuarea

flowofratevolume==


0

2

=

=

=

=

rv

r

m

ru

stream function;

2

msource

=

SECTION B 8

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IDEAL FLOW THEORY

velocity potential ;

Crm

source += ln2

( I ) at 00,0 === Cr

rm

source ln2

=

( II ) atA

mCAr ln

2,0

===

=

A

rmsource ln

2

SECTION B 9

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IDEAL FLOW THEORY

Sink ;

A sink, the exact opposite of a

source, is a point to which the fluid

converges uniformly and from which

fluid is continuously removed.

The strength of a sink is considered

negative, and the velocities, ,

are therefore the same as those for a

source but with the signs reversed.

SECTION B 10

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IDEAL FLOW THEORY

stream function;

2sink

m=


Crm

+= ln2

sink

( I ) at 0,0 == r

rm

ln2

sink

=

( II ) at Ar== ,0

=

A

rmln

2sink

SECTION B 11

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IDEAL FLOW THEORY

Vortex ;

2 types ;

1.Irrotational vortex2.Forced vortex

SECTION B 12

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IDEAL FLOW THEORY

Irrotational vortex ;

Circulation ;

)(vortex rvvr +=

rv 2vortex =

vorticity ;

0vortex =

+

=r

v

r

v

stream function ;

r

r

ln

2vortex

=


2

vortex

=

SECTION B 13

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IDEAL FLOW THEORY

Forced vortex ;

rv =

vorticity ;

20 =

r

v

r

v

+

== 2

SECTION B 14

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IDEAL FLOW THEORY

COMBINATIONOFBASICFLOWPATTERNS

Linear and Source ;

Stream function ;

sourcelinearncombinatio +=

+=2

sin m

rUncombinatio

SECTION C 1

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IDEAL FLOW THEORY

stagnation point Sis the point where

the resultant velocity is zero.

U

mBOS

2==

stream function at 0= ;

02

sin0

=+==

m

U

It is called stagnation line.

The body whose contour is formed bythe combination of uniform

rectilinear flow and a source is

known as a half body, since it has a

nose but no tail, orRankinebody.

SECTION C 2

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IDEAL FLOW THEORY

Distance from origin to 0= ;

sin2 =

U

mr

Asymptotey;

==

U

m

U

mry

2and

2sin

Velocity components ;

cos2

= Ur

mu

sin= Uv

SECTION C 3

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IDEAL FLOW THEORY

If rectilinear flow comes from the

other side ;

2

mncombinatio =

( )

sin2

= U

m

r

cos2

+= Ur

mu

sin= Uv

SECTION C 4

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IDEAL FLOW THEORY

Source and Sink ;

In this situation, the assumptionagain being made that the fluid

extends to infinity in all directions.

SECTION C 5

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IDEAL FLOW THEORY

Combination of stream function ;

sinksource +=ncombinatio

( )21

2

=

mncombinatio

+=

222

1 2tan

2 yAx

Aym

ncombinatio

Component velocity ;

( ) ( )

++

+

+

=22222 yAx

Ax

yAx

Axm

u

( ) ( )

++

+=

22222 yAx

y

yAx

ymv


=

2

1ln

2 r

rmncombinatio

SECTION C 6

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IDEAL FLOW THEORY

Source, Sink and Linear ;

Combination of stream function ;

linearncombinatio ++= sinksource

UyyAx

Aymncombinatio

+

= 222

1 2tan

2

SECTION C 7

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IDEAL FLOW THEORY

Component velocity ;

( ) ( ) U

yAx

Ax

yAx

Axmu

++

+

+

=

22222

( ) ( )

++

+

=

2222

2 yAx

y

yAx

ymv

value of x ;

1+=UA

mAx

value of ymax;

=

A

y

U

my max

1

max tan

SECTION C 8

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IDEAL FLOW THEORY

COMBINATIONOFBASICFLOWPATTERNSDoublet ;

SECTION D 1

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IDEAL FLOW THEORY

Stream function ;

( )

sin

22 21

r

mncombinatio ==


cos

2 2r

u =

sin

2 2r

v =

22 r

q

=


cos2 r

=

SECTION D 2

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IDEAL FLOW THEORY

Doublet and Uniform ;

Stream function ;

sin

2

= Ur

rncombinatio

SECTION D 3

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IDEAL FLOW THEORY


cos

2

+= Urr

ncombinatio

0=ncombinatio , 0= , =

Ur

2=

SECTION D 4

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IDEAL FLOW THEORY

UAr

2

22 ==

stream function ;

=

2

2

1sinr

AUrncombinatio


=

2

2

1cos

r

AUrncombinatio

SECTION D 5

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IDEAL FLOW THEORY


=

2

2

1cosr

AUu

+=

2

2

1sin

r

AUv

velocity at cylinder surface ;

r= , =90

0=u Uv 2=

Pressure coefficient CP;

2

2

2

1

12sin41=

=

U

PPCP

SECTION D 6

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IDEAL FLOW THEORY

Pressure at cylinder surface ;

222

1

12 sin41+= UPP

Drag force FD ;

== 0cosdFFD

Lift force FL;

== 0sindFFL

In real situation, both of these force

are exist.

SECTION D 7

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IDEAL FLOW THEORY

Doublet, vortex and Uniform ;

stream function ;

= A

r

r

A

Ur

C

ncombinatio ln

21sin

2

2


= 2

2

1cosrAUu

rr

AUv C

21sin

2

2 +

+=

SECTION D 8

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IDEAL FLOW THEORY

velocity at cylinder surface ;r= ,

0=u

AUv C

2

sin2

+=

stagnation point S ;

r=

UA

C

4sin =

0sin

0

==

C

SECTION D 9

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IDEAL FLOW THEORY

0sin

0

=

=

C

0.1sin

4

UAC

(Impossible)

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22 Basic Flow