22 Basic Flow
Transcript of 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==
velocity components ;
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=
velocity potential ;
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
=
velocity potential ;
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
velocity potential ;
=
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 ==
velocity components ;
cos
2 2r
u =
sin
2 2r
v =
22 r
q
=
velocity potential ;
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
velocity potential ;
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
velocity potential ;
=
2
2
1cos
r
AUrncombinatio
SECTION D 5
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IDEAL FLOW THEORY
velocity components ;
=
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
velocity components ;
= 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)