FLOW NETS - Universiti Teknologi Malaysia · 2008-08-11 · FLOW NETS For any two-dimensional...

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IDEAL FLOW THEORY FLOW NETS For any two-dimensional irrotational flow of a ideal fluid, two series of lines may be drawn : (1) lines along which ψ is constant (2) lines along which φ is constant SECTION B 1

Transcript of FLOW NETS - Universiti Teknologi Malaysia · 2008-08-11 · FLOW NETS For any two-dimensional...

Page 1: FLOW NETS - Universiti Teknologi Malaysia · 2008-08-11 · FLOW NETS For any two-dimensional irrotational flow of a ideal fluid, two series of ... for two-dimensional flow, the flow

IDEAL FLOW THEORY

FLOW NETS For any two-dimensional irrotational flow of a ideal fluid, two series of lines may be drawn : (1) lines along which ψ is constant (2) lines along which φ is constant

SECTION B 1

Page 2: FLOW NETS - Universiti Teknologi Malaysia · 2008-08-11 · FLOW NETS For any two-dimensional irrotational flow of a ideal fluid, two series of ... for two-dimensional flow, the flow

IDEAL FLOW THEORY

stream line ψ perpendicular to the velocity potential φ These lines together form a grid of quadrilaterals having 90º corners. This grid is known as a flow net. It is provides a simple yet valuable indication of the flow pattern.

SECTION B 2

Page 3: FLOW NETS - Universiti Teknologi Malaysia · 2008-08-11 · FLOW NETS For any two-dimensional irrotational flow of a ideal fluid, two series of ... for two-dimensional flow, the flow

IDEAL FLOW THEORY

COMBINING FLOW

PATTERNS 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

Page 4: FLOW NETS - Universiti Teknologi Malaysia · 2008-08-11 · FLOW NETS For any two-dimensional irrotational flow of a ideal fluid, two series of ... for two-dimensional flow, the flow

IDEAL FLOW THEORY

21 ψψψψ +∆+=AP

ψψψψ ∆++= 21AQ

The resultant flow pattern may therefore be constructed graphically simply by joining the points for which the 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 +=∂∂

+∂∂

=+∂∂

=∂∂

=ψψψψψ

21 vvx

v +=∂∂

−=ψ

Net velocity potential ;

.......321 +++= φφφφnet

SECTION B 5

Page 6: FLOW NETS - Universiti Teknologi Malaysia · 2008-08-11 · FLOW NETS For any two-dimensional irrotational flow of a ideal fluid, two series of ... for two-dimensional flow, the flow

IDEAL FLOW THEORY

BASIC PATTERNS OF FLOW Uniform Flow ;

velocity components ;

αα

sincos⋅=⋅=

qvqu

stream function ;

vxuy −=ψ velocity potential ;

vyux +=φ

SECTION B 6

Page 7: FLOW NETS - Universiti Teknologi Malaysia · 2008-08-11 · FLOW NETS For any two-dimensional irrotational flow of a ideal fluid, two series of ... for two-dimensional flow, the flow

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 streamlines uniformly 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

Page 8: FLOW NETS - Universiti Teknologi Malaysia · 2008-08-11 · FLOW NETS For any two-dimensional irrotational flow of a ideal fluid, two series of ... for two-dimensional flow, the flow

IDEAL FLOW THEORY

The strength m of a source is the total volume rate of flow from it. The velocity q at radius r is given by;

rmqπ2 velocitylar toperpendicu area

flow of rate volume==

velocity components ;

0

2

=∂∂

−=′

=∂⋅

∂=′

rv

rm

ru

ψθ

ψ

stream function;

πθψ

2m

source =

SECTION B 8

Page 9: FLOW NETS - Universiti Teknologi Malaysia · 2008-08-11 · FLOW NETS For any two-dimensional irrotational flow of a ideal fluid, two series of ... for two-dimensional flow, the flow

IDEAL FLOW THEORY

velocity potential ;

Crmsource += ln

2πφ ( I ) at 00,0 =⇒== Crφ

rmsource ln

2πφ =

( II ) at AmCAr ln2

,0π

φ −=⇒==

⎟⎠⎞

⎜⎝⎛=

Arm

source ln2π

φ

SECTION B 9

Page 10: FLOW NETS - Universiti Teknologi Malaysia · 2008-08-11 · FLOW NETS For any two-dimensional irrotational flow of a ideal fluid, two series of ... for two-dimensional flow, the flow

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, ψ , φ are therefore the same as those for a source but with the signs reversed.

SECTION B 10

Page 11: FLOW NETS - Universiti Teknologi Malaysia · 2008-08-11 · FLOW NETS For any two-dimensional irrotational flow of a ideal fluid, two series of ... for two-dimensional flow, the flow

IDEAL FLOW THEORY

stream function;

πθψ

2sinkm

−= velocity potential ;

Crm+−= ln

2sink πφ ( I ) at 0,0 == rφ

rm ln2sink π

φ −= ( II ) at Ar == ,0φ

⎟⎠⎞

⎜⎝⎛−=

Arm ln

2sink πφ

SECTION B 11

Page 12: FLOW NETS - Universiti Teknologi Malaysia · 2008-08-11 · FLOW NETS For any two-dimensional irrotational flow of a ideal fluid, two series of ... for two-dimensional flow, the flow

IDEAL FLOW THEORY

Vortex ;

2 types ; 1. Irrotational vortex 2. Forced vortex

SECTION B 12

Page 13: FLOW NETS - Universiti Teknologi Malaysia · 2008-08-11 · FLOW NETS For any two-dimensional irrotational flow of a ideal fluid, two series of ... for two-dimensional flow, the flow

IDEAL FLOW THEORY

Irrotational vortex ; Circulation ;

δθδδ )(vortex rvvr ⋅′+′⋅=Γ

rv π2vortex ⋅′=Γ vorticity ;

0vortex =′

+′

=rv

rv

δδζ

stream function ;

rr

ln2vortex π

ψ Γ−=

velocity potential ;

θπ

φ2vortexΓ

=

SECTION B 13

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IDEAL FLOW THEORY

Forced vortex ;

rv ⋅=′ ω vorticity ;

ωςζ 20 =⇒≠

rv

rv

δδωζ′

+′

==∴ 2

SECTION B 14

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IDEAL FLOW THEORY

COMBINATION OF

BASIC FLOW PATTERNS Linear and Source ;

Stream function ;

sourcelinearncombinatio ψψψ +=

θπ

θψ ⋅+⋅−=2

sin mrUncombinatio

SECTION C 1

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IDEAL FLOW THEORY

stagnation point S is the point where the resultant velocity is zero.

UmBOSπ2

== stream function at 0=θ ;

02

sin0 =⋅+⋅−== θπ

θψψmU

It is called ‘stagnation line’. The body whose contour is formed by the combination of uniform rectilinear flow and a source is known as a half body, since it has a nose but no tail, or Rankine body.

SECTION C 2

Page 17: FLOW NETS - Universiti Teknologi Malaysia · 2008-08-11 · FLOW NETS For any two-dimensional irrotational flow of a ideal fluid, two series of ... for two-dimensional flow, the flow

IDEAL FLOW THEORY

Distance from origin to 0=ψ ;

θπθsin2 ⋅

=Umr

Asymptote y ;

⎟⎠⎞

⎜⎝⎛−=⋅=

Um

Umry

2 and

2sinθ

Velocity components ;

θπ

cos2

⋅−=′ Ur

mu

θsin⋅=′ Uv

SECTION C 3

Page 18: FLOW NETS - Universiti Teknologi Malaysia · 2008-08-11 · FLOW NETS For any two-dimensional irrotational flow of a ideal fluid, two series of ... for two-dimensional flow, the flow

IDEAL FLOW THEORY

If rectilinear flow comes from the other side ;

2m

ncombinatio =ψ ( )

θπθπ

sin2 ⋅−

=U

mr

θπ

cos2

⋅+=′ Ur

mu θsin⋅−=′ Uv

SECTION C 4

Page 19: FLOW NETS - Universiti Teknologi Malaysia · 2008-08-11 · FLOW NETS For any two-dimensional irrotational flow of a ideal fluid, two series of ... for two-dimensional flow, the flow

IDEAL FLOW THEORY

Source and Sink ;

In this situation, the assumption again being made that the fluid extends to infinity in all directions.

SECTION C 5

Page 20: FLOW NETS - Universiti Teknologi Malaysia · 2008-08-11 · FLOW NETS For any two-dimensional irrotational flow of a ideal fluid, two series of ... for two-dimensional flow, the flow

IDEAL FLOW THEORY

Combination of stream function ; sinksource ψψψ +=ncombinatio

( )212θθ

πψ −=

mncombinatio

⎟⎟⎠

⎞⎜⎜⎝

⎛+−

= −222

1 2tan2 yAx

Aymncombinatio π

ψ Component velocity ;

( ) ( ) ⎥⎦

⎤⎢⎣

+++

−+−

−= 22222 yAx

AxyAx

Axmuπ

( ) ( ) ⎥⎦

⎤⎢⎣

++−

+−= 22222 yAx

yyAx

ymvπ

velocity potential ;

⎟⎟⎠

⎞⎜⎜⎝

⎛⋅=

2

1ln2 r

rmncombinatio π

φ

SECTION C 6

Page 21: FLOW NETS - Universiti Teknologi Malaysia · 2008-08-11 · FLOW NETS For any two-dimensional irrotational flow of a ideal fluid, two series of ... for two-dimensional flow, the flow

IDEAL FLOW THEORY

Source, Sink and Linear ;

Combination of stream function ;

linearncombinatio ψψψψ ++= sinksource

UyyAx

Aymncombinatio −⎥

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+−

= −222

1 2tan2π

ψ

SECTION C 7

Page 22: FLOW NETS - Universiti Teknologi Malaysia · 2008-08-11 · FLOW NETS For any two-dimensional irrotational flow of a ideal fluid, two series of ... for two-dimensional flow, the flow

IDEAL FLOW THEORY

Component velocity ;

( ) ( )U

yAxAx

yAxAxmu −⎥

⎤⎢⎣

+++

−+−

−= 22222π

( ) ( ) ⎥⎦

⎤⎢⎣

++−

+−−

= 22222 yAxy

yAxymv

π

value of x ;

1+=UAmAx

π value of ymax ;

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛= −

Ay

Umy max1

max tanπ

SECTION C 8

Page 23: FLOW NETS - Universiti Teknologi Malaysia · 2008-08-11 · FLOW NETS For any two-dimensional irrotational flow of a ideal fluid, two series of ... for two-dimensional flow, the flow

IDEAL FLOW THEORY

COMBINATION OF

BASIC FLOW PATTERNS Doublet ;

SECTION D 1

Page 24: FLOW NETS - Universiti Teknologi Malaysia · 2008-08-11 · FLOW NETS For any two-dimensional irrotational flow of a ideal fluid, two series of ... for two-dimensional flow, the flow

IDEAL FLOW THEORY

Stream function ;

( ) θπµθθ

πψ sin

22 21 rm

ncombinatio =−= velocity components ;

θπµ cos

2 2ru =′

θπµ sin

2 2rv =′

22 rq

πµ

= velocity potential ;

θπµφ cos

2 r−=

SECTION D 2

Page 25: FLOW NETS - Universiti Teknologi Malaysia · 2008-08-11 · FLOW NETS For any two-dimensional irrotational flow of a ideal fluid, two series of ... for two-dimensional flow, the flow

IDEAL FLOW THEORY

Doublet and Uniform ;

Stream function ;

θπµψ sin

2⎟⎠⎞

⎜⎝⎛ −= Ur

rncombinatio

SECTION D 3

Page 26: FLOW NETS - Universiti Teknologi Malaysia · 2008-08-11 · FLOW NETS For any two-dimensional irrotational flow of a ideal fluid, two series of ... for two-dimensional flow, the flow

IDEAL FLOW THEORY

velocity potential ;

θπµφ cos

2⎟⎠⎞

⎜⎝⎛ +−= Ur

rncombinatio

0=ncombinatioψ , 0=θ , πθ =

Ur

πµ

2=

SECTION D 4

Page 27: FLOW NETS - Universiti Teknologi Malaysia · 2008-08-11 · FLOW NETS For any two-dimensional irrotational flow of a ideal fluid, two series of ... for two-dimensional flow, the flow

IDEAL FLOW THEORY

UAr

πµ

222 ==

stream function ;

⎟⎟⎠

⎞⎜⎜⎝

⎛−−= 2

2

1sinrAUrncombinatio θψ

velocity potential ;

⎟⎟⎠

⎞⎜⎜⎝

⎛−−= 2

2

1cosrAUrncombinatio θφ

SECTION D 5

Page 28: FLOW NETS - Universiti Teknologi Malaysia · 2008-08-11 · FLOW NETS For any two-dimensional irrotational flow of a ideal fluid, two series of ... for two-dimensional flow, the flow

IDEAL FLOW THEORY

velocity components ;

⎟⎟⎠

⎞⎜⎜⎝

⎛−−=′ 2

2

1cosrAUu θ

⎟⎟⎠

⎞⎜⎜⎝

⎛+=′ 2

2

1sinrAUv θ

velocity at cylinder surface ;

Ar = , °= 90θ 0=′u Uv 2=′

Pressure coefficient CP ;

θρ

22

21

12 sin41−=−

=U

PPCP

SECTION D 6

Page 29: FLOW NETS - Universiti Teknologi Malaysia · 2008-08-11 · FLOW NETS For any two-dimensional irrotational flow of a ideal fluid, two series of ... for two-dimensional flow, the flow

IDEAL FLOW THEORY

Pressure at cylinder surface ; ( )θρ 22

21

12 sin41−+= UPP Drag force FD ;

∫ =⋅= 0cosθdFFD Lift force FL ;

∫ =⋅−= 0sinθdFFL In real situation, both of these force are exist.

SECTION D 7

Page 30: FLOW NETS - Universiti Teknologi Malaysia · 2008-08-11 · FLOW NETS For any two-dimensional irrotational flow of a ideal fluid, two series of ... for two-dimensional flow, the flow

IDEAL FLOW THEORY

Doublet, vortex and Uniform ;

stream function ;

⎟⎠⎞

⎜⎝⎛Γ

−⎟⎟⎠

⎞⎜⎜⎝

⎛−−=

Ar

rAUr C

ncombinatio ln2

1sin 2

2

πθψ

velocity components ;

⎟⎟⎠

⎞⎜⎜⎝

⎛−−=′ 2

2

1cosrAUu θ

rrAUv C

πθ

21sin 2

2 Γ+⎟⎟

⎞⎜⎜⎝

⎛+=′

SECTION D 8

Page 31: FLOW NETS - Universiti Teknologi Malaysia · 2008-08-11 · FLOW NETS For any two-dimensional irrotational flow of a ideal fluid, two series of ... for two-dimensional flow, the flow

IDEAL FLOW THEORY

velocity at cylinder surface ;

Ar = , 0=′u

AUv C

πθ

2sin2 Γ

+=′ stagnation point S ;

Ar =

UAC

πθ

4sin Γ

−=

0sin0=

=Γθ

C

SECTION D 9

Page 32: FLOW NETS - Universiti Teknologi Malaysia · 2008-08-11 · FLOW NETS For any two-dimensional irrotational flow of a ideal fluid, two series of ... for two-dimensional flow, the flow

IDEAL FLOW THEORY

0sin0=

=Γθ

C

0.1sin4−<

<Γθ

πUAC

0.1sin4−=

=Γθ

πUAC

0.1sin4−>

>Γθ

πUAC

(Impossible)

SECTION D 10