PHAROS UNIVERSITY ME 253 FLUID MECHANICS II
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Transcript of PHAROS UNIVERSITY ME 253 FLUID MECHANICS II
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PHAROS UNIVERSITYME 253 FLUID MECHANICS II
Boundary Layer (Two Lectures)
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Flow Past Flat Plate• Dimensionless numbers involved Re Ma Fr
Ul U U
c gl
• for external flow: Re>100 dominated by inertia, Re<1 – by viscosity
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Boundary Layer
Flows over bodies. Examples include the flows over airfoils, ship hulls, etc.
Boundary layer flow over a flat plate with no external pressure variation.
laminar turbulenttransition
Dye streakU U U
U
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Boundary layer characteristicsfor large Reynolds number flow can be dividedinto boundary region where viscous effect are important and outside region where liquid can be treated as inviscid Rex
Ux
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The Boundary-Layer Concept
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The Boundary-Layer Concept
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Boundary Layer Thicknesses
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Boundary Layer Thicknesses
• Disturbance Thickness,
Displacement Thickness, *
Momentum Thickness,
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Boundary Layer Thickness
Boundary layer thickness is defined as the height above the surface where the velocity reaches 99% of the free stream velocity.
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Boundary Layer(BL)
Three Thicknesses of a Boundary Layer
d*
1
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Displacement Thickness *
Volume flux:0
udy Q
Ideal flux: 1Q U 10
udy U
1 0
udy
U
*1 0
(1 )u
dyU
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Velocity Distribution
U
SolidBoundary
Equivalent Flow Rate
U
Velocity Defect
VelocityDefect
*
Ideal FluidFlow
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Eqn. for Displacement Thickness
• By equating the flow rate for velocity defect to flow rate for ideal fluid
–
• If density is constant, this simplifies to
–
* would always be smaller than
0
* dyuUU
0
* 1 dyU
u
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Displacement Thickness Laminar B.L.
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Eqn. for Momentum Thickness
• By equating the momentum flux rate for velocity defect to that for ideal fluid
–
• If density is constant, this simplifies to
–
would always be smaller than * and
0
2 uUudyU
01 dyU
u
U
u
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Momentum ThicknessThe rate of mass flow across an element of the boundary layer is ( u dy) and the mass has a momentum ( u2 dy ) The same mass outside the boundary layer has the momentum ( u ue dy)
is a measure of the reduction in momentum transport in the B. Layer
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Empirical Equations of Laminar B. Layer Parameters
• Boundary Layer Thickness
• Momentum Thickness
• Displacement Thickness
• Skin Friction Coefficient
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Skin Friction Coefficient
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Boundary layer characteristics• Boundary layer thickness
• Boundary layer displacement thickness: *
0
1u
dyU
• Boundary layer momentum thickness (defined in terms of momentum flux):
2
0 0
1u u
bU b u U u dy bU U
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Drag on a Flat Plate
• Drag on a flat plate is related to the momentum deficit within the boundary layer
2
0
2
1
w
u ubU b
U U
db bU
dx x
D
D
• Drag and shear stress can be calculated by assuming velocity profile in the boundary layer
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Boundary Layer Definition Boundary layer thickness (d): Where the local velocity reaches to 99% of the free-stream velocity. u(y=d)=0.99U Displacement thickness (δ*): Certain amount of the mass has been displaced (ejected) by the presence of the boundary layer ( mass conservation). Displace the uniform flow away from the solid surface by an amount δ*
0 0* ( ) , or * (1 )
uU w U u wdy dy
U
Amount of fluid being displaced outward
*
U-u
equals
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Laminar Flat-PlateBoundary Layer: Exact Solution
• Governing Equations
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Laminar Flat-PlateBoundary Layer: Exact Solution
• Boundary Conditions
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Laminar Flat-PlateBoundary Layer: Exact Solution
• Results of Numerical Analysis
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MOMENTUM INTEGRAL EQN
• BOUNDARY LAYER EQUATIONS
• BOUNDARY CONDITIONS y=0, u=v=0 & y = δ , u= U
• Integrating the momentum equation w.r.t y in the interval [0,δ]
0
y
v
x
u
2
2
)(y
u
dx
dUU
y
uv
x
uu
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MOMENTUM INTEGRAL EQN
dyy
udy
dx
dUUdy
y
uvdy
x
uu
0
2
2
000
dyy
udy
dx
dUUdy
y
vuUvdy
x
uu y
0
2
2
00
0
0
)(
000
0
0
)(
y
y y
udy
dx
dUUdy
x
uuUvdy
x
uu
0000
2
y
y
udy
dx
dUUdy
x
uUdy
x
uu
dyy
udy
dx
dUUdy
y
uvdy
x
uu
0
2
2
000
dyy
udy
dx
dUUdy
y
vuUvdy
x
uu y
0
2
2
00
0
0
)(
dyy
udy
dx
dUUdy
y
uvdy
x
uu
0
2
2
000
dyy
udy
dx
dUUdy
y
vuUvdy
x
uu y
0
2
2
00
0
0
)(
dyy
udy
dx
dUUdy
y
uvdy
x
uu
0
2
2
000
dyy
udy
dx
dUUdy
y
vuUvdy
x
uu y
0
2
2
00
0
0
)(
dyy
udy
dx
dUUdy
y
uvdy
x
uu
0
2
2
000
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MOMENTUM INTEGRAL EQN
dyU
u
dx
dUUdy
U
u
dx
dUdy
U
uU
dx
dUudy
dx
dU
dyU
u
dx
dUUdy
U
u
dx
dUdy
U
uU
dx
ddyu
dx
d
dydx
dUUdyu
dx
dUdyu
dx
d
00
2
00
0
22
0
2
0
22
0
2
0
000
2
2
)( 00 )(
yy
u
dyUuUu )/1(/0
UdyuU /)(*0
DISPLACEMENT THICKNESS
MOMENTUM THICKNESS
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MOMENTUM INTEGRAL EQN
0
00
2
0
0000
2
0
000
2
0
000
2
0
22
0
2
0
000
2
112
12
121
2
)(
dyU
u
dx
dUUdy
U
u
U
u
dx
dUU
dx
dU
dyU
u
dx
dUUdy
U
u
dx
dUUdy
dx
dUUdy
U
u
U
u
dx
dUU
dx
dU
dydx
dUUdy
U
u
U
u
dx
dUUdy
U
u
U
u
dx
dU
dydx
dUUdy
U
u
dx
dUUdy
U
u
dx
dUdy
U
u
dx
dUUdy
U
u
dx
dU
dydx
dUUdyu
dx
dUdyu
dx
d
/)2( 0
*2 dx
dUU
dx
dU
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MOMENTUM INTEGRAL EQN
• VON KARMAN MOMENTUM INTEGRAL EQUATION
/)2( 0
*2 dx
dUU
dx
dU
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KARMAN POHLHAUSEN METHOD FOR FLOW OVER FLAT PLATE
/)2(
0
0
2
1
0*2
2
dx
dUU
dx
dU
dx
dUU
dx
dpdx
dUU
dx
dp
constUp BERNOULLI’S EQUATION
xU
2
21 2
0REVISED KARMAN EQUATION
FOR NO EXTERNAL PRESSURE
NO IMPOSED PRESSURE
VON KARMAN EQUATION
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KARMAN POHLHAUSEN METHOD FOR FLOW OVER FLATE PLATE
• Dimensionless velocity u/U can be expressed at any location x as a function of the dimensionless distance from the wall y/δ.
• Boundary (at wall) conditions y=0, u=0 ; d2u/dy2 = 0• At outer edge of boundary layer (y=δ) u=U, du/dy=0• Applying boundary conditions a=0,c=0, b=3/2, d= -1/2
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y
dy
cy
bay
U
u
3
2
1
2
3
yy
U
uVelocity profile
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KARMAN POHLHAUSEN METHOD FOR FLOW OVER FLATE PLATE
3
2
1
2
3
yy
U
u
xU
2
21 2
0
INSERT INTO THIS EQUATION
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KARMAN POHLHAUSEN METHOD FOR FLOW OVER FLATE PLATE
xU
dyyyyyy
xU
dyyyyy
xU
dyU
u
U
u
xU
xU
2
0
64322
0
32
0
220
280
39
4
1
2
3
2
1
4
9
2
3
2
1
2
3*
2
1
2
31
1
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KARMAN POHLHAUSEN METHOD FOR FLOW OVER FLATE PLATE
• At the solid surface, Newton’s Law of Viscosity gives:
xU
U
therefore
U
yU
yU
yy
u
yy
2
0
3
0
0
280
39
2
3
2
3
2
1
2
3
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KARMAN POHLHAUSEN METHOD FOR FLOW OVER FLATE PLATE
Re
64.4
*13
2*140
0,0,0
13
140
2
13
140
2
x
xUx
Cx
CU
x
xU
eRx
0.5
KARMAN POHLHAUSEN SOLUTION
BLASIUS SOLUTION
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Boundary Layer Parameters• BOUNDARY LAYER THICKNESS INCREASES AS THE
SQUARE ROOT OF THE DISTANCE x FROM THE LEADING EDGE AND INVERSELY AS SQUARE ROOT OF FREE STREAM VELOCITY.
• WALL SHEAR STRESS IS INVERSELY PROPORTIONAL TO THE SQUARE ROOT OF x AND DIRECTLY PROPORTIONAL TO 3/2 POWER OF U
• LOCAL & AVERAGE SKIN FRICTION VARY INVERSELY AS SQUARE ROOT OF BOTH x & U
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Use of the Momentum Equation for Flow with Zero Pressure Gradient
• Simplify Momentum Integral Equation(Item 1)
The Momentum Integral Equation becomes
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Use of the Momentum Equation for Flow with Zero Pressure Gradient
• Laminar Flow– Example: Assume a Polynomial Velocity Profile
(Item 2)
• The wall shear stress w is then (Item 3)
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Use of the Momentum Equation for Flow with Zero Pressure Gradient
• Laminar Flow Results(Polynomial Velocity Profile)
Compare to Exact (Blasius) results!
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Use of the Momentum Equation for Flow with Zero Pressure Gradient
• Turbulent Flow– Example: 1/7-Power Law Profile (Item 2)
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Use of the Momentum Equation for Flow with Zero Pressure Gradient
• Turbulent Flow Results(1/7-Power Law Profile)
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Example
Assume a laminar boundary layer has a velocity profile as u(y)=U(y/ for 0y and u=U for y>, as shown. Determine the shear stress and the boundary layer growth as a function of the distance x measured from the leading edge of the flat plate.
u(y)=U(y/
u=U y
x
2
w 0
0 0
U
For a laminar flow ( ) from the profile.
Substitute into the definition of the momentum thickness:
U y(1 ) (1 ) , since u
.6
w
y
d
dxUu
y
u u y ydy dy
U U
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2 2
2 2
x
3 2
1U , U
66 12
Separation of variables: , integrate 12( ) ,U U U
13.46 3.46 , where Re
Re
3.46 ,
0.289 10.289 ,
Re
w
x
w w
x
Ud d
dx dx
d x xx
U x
x U x
xx
U
U U U
x x
Note: In general, the velocity distribution is not a straight line. A laminar flat-plate boundary layer assumes a Blasius profile. The boundary layer thickness and the wall shear stress w behave as:
(9.14). ,Re
332.0.(9.13) ,
Re
0.50.5 2
x
w
x
Ux
xU
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Laminar Boundary Layer Development
x( )
x0 0.5 1
0
0.5
1• Boundary layer growth: x• Initial growth is fast• Growth rate d/dx 1/x, decreasing downstream.
w x( )
x0 0.5 1
0
5
10
• Wall shear stress: w 1/x• As the boundary layer grows, the wall shear stress decreases as the velocity gradient at the wall becomes less steep.
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Momentum Integral Relation for Flat-plate BL
xx
y
hU
CV
Stream line
P
d
Free stream
U=const P=const z b
Steady & incompressible
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Momentum Integral Relation for Flat-plate BL
:X Outlet2
0b u dy
Inlet2b u h
.2 2
0M bU h b u dy
2 2
0D bU h b u dy
Continuity0
hU udy
0
uh dy
U
0( )D b u U u dy
2bU 2dD d
bUdx dx
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Meantime
0( ) ( )
x
wD x b x dx
w
dDb
dx
2w
dU
dx
For flat plate boundary layer
U const 0dU
dx
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2wd
dx U
' '
K a m a n
0(1 )
u udy
U U
2u a by cy
0 0y u y u U 0u
y
2
20, , -
U Ua b c
2
2
2u y y
U 0 ( )y x
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2 2
2 20
2 2( )(1 )
y y y ydy
y
令 =
1 2 2
0(2 )(1 2 )dy
2
15
2
15
d d
dx dx
2d
dx U
2|w y o
u U
y
15d x
U
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0, 0x
21 15
2x
U
5.5 5.5x
xU Ux
5.5
Rex x
*( ) 1.83
Re
x
x x
( ) 0.74
Re
x
x x
*3 7.5 *
H
Shape factor
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2
0.73
1 2w
f
x
CU Re
Skin-friction coefficient
0
l
f wX bdx
21 2f
D
XC
U bl Drag coefficient
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Boundary Layer Equation
Inviscid
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3 Boundary Layer Equation
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Boundary Layer Equation
2-D,steady,incompressible,neglect body force
0u v
x y
2 2
2 2
1( )
u u p u uu v
x y x x y
2 2
2 2
1( )
v v p v vu v
x y y x y
1[ ]Re
[ ]L
[ ]L
[ ]L
[ ]L
[1][1] [1] [1]
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For BL, 1L
2
2
1( )
u u p uu v
x y x y
0u v
x y
0p
y
( )P P x
dp
dx
External flow U U 0V (Inviscid Flow)
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Euler Equation1dU dP
Udx dX
2
2
u u dU uu v U
x y dx y
2
2( )
u u
y y y
1( )
u
y y
1
y
u
y
____
' 'uu v
y
0, 0y u v , ( )y u U x
Laminar flow
Turbulent flow
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Blasius 1908
u
U ( )
y y
x x U
'( ) ( )u
fU
( ) ( )f d '' '1
( ( ) ) 02
d
''' ''1( ) 0
2f f f
0, 0y '0, ( ) 0u f
, by ', 1u U f
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5.0
xx Re
* 1.721
xx Re
0.664
xx Re
|w o
u U
y x U
''| (0)o
f Uf
x U
2
0.66412
wf
x
CReU
1.328
ReD
L
C
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U =10m/s, =17x10-6 m2/s
DISPLACEMENT AND MOMENTUN THICKNESS
• Typical distribution of , * and
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Influence of Adverse Pressure GradientAdverse Pressure Gradient dp/dx>0 can cause flow separation
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Pressure Gradients in Boundary-Layer Flow
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Boundary Layer and separation
gradient pressure
favorable ,0
x
P
gradient no ,0
x
P
0, adverse
pressure gradient
P
x
Flow accelerates Flow decelerates
Constant flow
Flow reversalfree shear layerhighly unstable
Separation point
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Flow Separation
SeparationBoundary layer
Wake
Stagnation point
21/ 2
P PCp
U
Inviscid curve 21 4sinCp
Turbulent
Laminar
1.0
0
-1.0
-2.0
-3.0
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Drag Coefficient: CD
Supercritical flowturbulent B.L.
Stokes’ Flow, Re<1
Relatively constant CD
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Drag
• Drag Coefficient
with
or
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Influence of Adverse Pressure GradientAdverse Pressure Gradient dp/dx>0 can cause flow separation
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WHY DOES BOUNDARY LAYER SEPARATE?• Adverse pressure gradient interacting with velocity profile through B.L.• High speed flow near upper edge of B.L. has enough speed to keep moving
through adverse pressure gradient• Lower speed fluid (which has been retarded by friction) is exposed to same
adverse pressure gradient is stopped and direction of flow can be reversed• This reversal of flow direction causes flow to separate
– Turbulent B.L. more resistance to flow separation than laminar B.L. because of fuller velocity profile
– To help prevent flow separation we desire a turbulent B.L.
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EXAMPLE OF FLOW SEPARATION
• Velocity profiles in a boundary layer subjected to a pressure rise– (a) start of pressure rise– (b) after a small pressure rise– (c) after separation
• Flow separation from a surface– (a) smooth body– (b) salient edge
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BOUNDARY LAYER SEPARATION
• Separation takes place due to excessive momentum loss near the wall in a boundary layer trying to move downstream against increasing pressure, i.e., which is called adverse pressure gradient.
dx
dp
y
u
dx
dp
dx
dUU
y
u
y
ydx
dUU
y
uv
x
uu
wall
wallwall
1
1)(
2
2
2
2
MOMENTUM EQUATION
AT WALL v=u=0
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BOUNDARY LAYER SEPARATION
• In adverse gradient, the second derivative of velocity is positive at the wall, yet it must be negative at the outer layer (y=δ) to merge smoothly with the mainstream flow U(x).
• It follows that the second derivative must pass through zero somewhere in between, at the point of inflexion, and any boundary layer profile in an adverse must exhibit a characteristic S –shape.
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BOUNDARY LAYER SEPARATION
• In FAVOURABLE GRADIENT, profile is rounded, no point of inflexion, no separation.
• In ZERO PRESSURE GRADIENT, point of inflexion is at the wall itself. No separation.
• In ADVERSE GRADIENT, point of inflexion (PI) occurs in the boundary layer, its distance from the wall increasing with the strength of the adverse gradient
• CRITICAL CONDITION is reached where the wall shear is exactly zero (∂u/∂y =0). This is defined as separation point
0w SHEAR STRESS AT WALL IS ZERO
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BOUNDARY LAYER SEPARATION
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BOUNDARY LAYER SEPARATION• The mathematical explanation of flow-separation :
– The point of separation may be defined as the limit between forward and reverse flow in the layer very close to the wall, i.e., at the point of separation
– This means that the shear stress at the wall, .But at this point, the adverse pressure continues to exist and at the downstream of this point the flow acts in a reverse direction resulting in a back flow.
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EXAMPLE: SLATS AND FLAPS