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Transcript of FLOW NETS - Universiti Teknologi 2008-08-11آ  FLOW NETS For any two-dimensional irrotational 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

• 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

• 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

• 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

• IDEAL FLOW THEORY

Velocity components ;

( ) 212121 uuyyyyu +=∂ ∂

+ ∂ ∂

=+ ∂ ∂

= ∂ ∂

= ψψψψψ

21 vvx v +=

∂ ∂

−= ψ

Net velocity potential ;

.......321 +++= φφφφnet

SECTION B 5

• IDEAL FLOW THEORY

BASIC PATTERNS OF FLOW Uniform Flow ;

velocity components ;

α α

sin cos ⋅= ⋅=

qv qu

stream function ;

vxuy −=ψ velocity potential ;

vyux +=φ

SECTION B 6

• 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

• 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;

r mq π2 velocitylar toperpendicu area

flow of rate volume ==

velocity components ;

0

2

= ∂ ∂

−=′

= ∂⋅

∂ =′

r v

r m

r u

ψ θ

ψ

stream function;

π θψ

2 m

source =

SECTION B 8

• IDEAL FLOW THEORY

velocity potential ;

Crmsource += ln2π φ ( I ) at 00,0 =⇒== Crφ

rmsource ln2π φ =

( II ) at A mCAr ln 2

,0 π

φ −=⇒==

⎟ ⎠ ⎞

⎜ ⎝ ⎛=

A rm

source ln2π φ

SECTION B 9

• 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

• IDEAL FLOW THEORY

stream function;

π θψ

2sink m

−= velocity potential ;

Crm +−= ln 2sink π

φ ( I ) at 0,0 == rφ

rm ln 2sink π

φ −= ( II ) at Ar == ,0φ

⎟ ⎠ ⎞

⎜ ⎝ ⎛−=

A rm ln

2sink π φ

SECTION B 11

• IDEAL FLOW THEORY

Vortex ;

2 types ; 1. Irrotational vortex 2. Forced vortex

SECTION B 12

• IDEAL FLOW THEORY

Irrotational vortex ; Circulation ;

δθδδ )(vortex rvvr ⋅′+′⋅=Γ

rv π2vortex ⋅′=Γ vorticity ;

0vortex = ′

+ ′

= r v

r v

δ δζ

stream function ;

r r

ln 2vortex π

ψ Γ−= velocity potential ;

θ π

φ 2vortex Γ

=

SECTION B 13

• IDEAL FLOW THEORY

Forced vortex ;

rv ⋅=′ ω vorticity ;

ωςζ 20 =⇒≠

r v

r v

δ δωζ ′

+ ′

==∴ 2

SECTION B 14

• IDEAL FLOW THEORY

COMBINATION OF BASIC FLOW PATTERNS Linear and Source ;

Stream function ;

sourcelinearncombinatio ψψψ +=

θ π

θψ ⋅+⋅−= 2

sin mrUncombinatio

SECTION C 1

• IDEAL FLOW THEORY

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

U mBOS π2

== stream function at 0=θ ;

0 2

sin0 =⋅+⋅−== θπ θψψ

mU 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

• IDEAL FLOW THEORY

Distance from origin to 0=ψ ;

θπ θ sin2 ⋅

= U mr

Asymptote y ;

⎟ ⎠ ⎞

⎜ ⎝ ⎛−=⋅=

U m

U mry

2 and

2 sinθ

Velocity components ;

θ π

cos 2

⋅−=′ U r

mu

θsin⋅=′ Uv

SECTION C 3

• IDEAL FLOW THEORY

If rectilinear flow comes from the other side ;

2 m

ncombinatio =ψ ( )

θπ θπ

sin2 ⋅ −

= U

mr

θ π

cos 2

⋅+=′ U r

mu θsin⋅−=′ Uv

SECTION C 4

• 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

• IDEAL FLOW THEORY

Combination of stream function ; sinksource ψψψ +=ncombinatio

( )212 θθπψ −= m

ncombinatio

⎟⎟ ⎠

⎞ ⎜⎜ ⎝

⎛ +−

= − 222 1 2tan

2 yAx Aym

ncombinatio π ψ Component velocity ;

( ) ( ) ⎥⎦ ⎤

⎢ ⎣

++ +

− +−

− = 22222 yAx

Ax yAx

Axmu π

( ) ( ) ⎥⎦ ⎤

⎢ ⎣

++ −

+− = 22222 yAx

y yAx

ymv π

velocity potential ;

⎟⎟ ⎠

⎞ ⎜⎜ ⎝

⎛ ⋅=

2

1ln 2 r

rm ncombinatio π

φ

SECTION C 6

• IDEAL FLOW THEORY

Source, Sink and Linear ;

Combination of stream function ;

linearncombinatio ψψψψ ++= sinksource

Uy yAx

Aym ncombinatio −⎥

⎤ ⎢ ⎣

⎡ ⎟⎟ ⎠

⎞ ⎜⎜ ⎝

⎛ +−

= − 222 1 2tan

2π ψ

SECTION C 7

• IDEAL FLOW THEORY

Component velocity ;

( ) ( ) U

yAx Ax

yAx Axmu −⎥

⎤ ⎢ ⎣

++ +

− +−

− = 22222π

( ) ( ) ⎥⎦ ⎤

⎢ ⎣

++ −

+− −

= 22222 yAx y

yAx ymv

π

value of x ;

1+= UA mAx

π value of ymax ;

⎥ ⎦

⎤ ⎢ ⎣

⎡ ⎟ ⎠ ⎞

⎜ ⎝ ⎛= −

A y

U my max1max tanπ

SECTION C 8

• IDEAL FLOW THEORY

COMBINATION OF BASIC FLOW PATTERNS Doublet ;

SECTION D 1

• IDEAL FLOW THEORY

Stream function ;

( ) θ π µθθ

π ψ sin

22 21 r m

ncombinatio =−= velocity components ;

θ π µ cos

2 2r u =′

θ π µ sin

2 2r v =′

22 r q

π µ

= velocity potential ;

θ π µφ cos

2 r −=

SECTION D 2

• IDEAL FLOW THEORY

Doublet and Uniform ;

Stream function ;

θ π µψ sin

2 ⎟ ⎠ ⎞

⎜ ⎝ ⎛ −= Ur

rncombinatio

SECTION D 3

• IDEAL FLOW THEORY

velocity potential ;

θ π µφ cos

2 ⎟ ⎠ ⎞

⎜ ⎝ ⎛ +−= Ur

rncombinatio

0=ncombinatioψ , 0=θ , πθ =

U r

π µ

2 =

SECTION D 4

• IDEAL FLOW THEORY

U Ar

π µ

2 22 ==

stream function ;

⎟⎟ ⎠

⎞ ⎜⎜ ⎝

⎛ −−= 2

2

1sin r AUrncombinatio θψ

velocity potential ;

⎟⎟ ⎠

⎞ ⎜⎜ ⎝

⎛ −−= 2

2

1cos r AUrncombinatio θφ

SECTION D 5

• IDEAL FLOW THEORY

velocity components ;

⎟⎟ ⎠

⎞ ⎜⎜ ⎝

⎛ −−=′ 2

2

1cos r AUu θ

⎟⎟ ⎠

⎞ ⎜⎜ ⎝

⎛ +=′ 2