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Module # 4
Flow Past of Immersed Bodies
1 1
Bodies
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Incompressible Flow
2 2
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Flow Around Objects
3 3
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FLUID FLOW ABOUT IMMERSED BODIES
pp7 p8
p
Drag due to surface stresses composed of normal (pressure) and tangential
(viscous) stresses.
Up4
p1
p2
p3
p6
p5
p7 p8p9
p10
p11
p13p…
p1210
9
8
7
65
4
3
2
1 ……
DRAG
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The transmission of energy from an object passing through afluid to the fluid is known as fluid resistance.
The resistance of an object passing through a fluid increasesas the speed of the object increases and as the viscosity of thefluid increases.
Fluid Resistance
DragDrag◦ Is the
resistance anairplaneexperiencesin movingforwardthrough theair
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At any point on surface:Fn PA P : pressure
Ft A : shear stress
Integrate pressure and shear stress distributions around bodyIntegrate pressure and shear stress distributions around bodysurface
Drag FD - component of resultant force in direction of flowLift FL - component of resultant force perpendicular todirection of flow
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Concept of Drag
Drag is the retarding force exerted on a moving bodyin a fluid medium
It does not attempt to turn the object, simply to slowit down
It is a function of the speed of the body, the size (andIt is a function of the speed of the body, the size (andshape) of the body, and the fluid through which it ismoving
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Drag Force Due to Air
The drag force due to wind (air) acting on an object can be found by:
FD = ½ ρ CDV2A
where: FD = drag force (N)D
CD = drag coefficient (no units)
V = velocity of object (m/s)
A = projected area (m2)
ρ = density of air (kg/m3) {1.2 kg/m3}
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Surface drag is a result of the friction between thesurface and the fluid.
The fluid closest to the object (boundary layer) rubsagainst the object creating friction.
Surface and Form Drag
Form drag occurs when air is driven past anForm drag occurs when air is driven past anobject and is diverted outward creating a lowpressure region behind the object.
high pressure
low pressure
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The orientation of theobject will affect thefrontal area and willplay an important rolein the amount of form
Low form drag
Form Drag
in the amount of formdrag.
High form drag
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Lift and Drag
shear stress and pressure integrated over the surface of abody create force
drag: force component in the direction of upstreamvelocity
lift: force normal to upstream velocity (might have 2components in general case)
D21
2
212
cos sin
sin cos
x w D
y w L
dF p dA dA CU A
dF p dA dA CU A
DD
LL
CD = FD/(1/2 U2A) = CD,pressure + CD,friction
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Projected Area
The projected area used in the FD is the area “seen” by the fluid.
Spherical Particle
A2
2 DR
A
4R
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Projected Area
Cylinder
For objects having shapes other than spherical, it isnecessary to specify the size, geometry and orientationrelative to the direction of flow.
Axis perpendicular to flow Rectangle LDA Axis perpendicular to flow Rectangle LDA
Axis parallel to flow Circle4
2DA
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Lift FL
Resultant FR
C
BA
drag FD
D
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low pressure region
Drag force dueto pressure difference
high pressure region
motion of air
motion of object
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Drag force dueto pressure difference
v
flow speed (high) vair + v reduced pressure
vair (vball)
low pressure region
high pressure region
v flow speed (low) vair - v increased pressure
Boundary layer – air sticks to ball (viscosity) – air dragged around with ball
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lift
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Drag Coefficient
For slow flow around a sphere and Re <10
0
2424
DuReCd
2uAC Recall:
2
20uAC
F dD
03 DuFD
Stokes’ Law for Creeping Flow Around Sphere
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Flow past an object
Character of thesteady, viscous flowpast a circularcylinder: (a) lowReynolds numberReynolds numberflow, (b) moderateReynolds numberflow, (c) largeReynolds numberflow.
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Effect of pressure gradient
inviscid flow viscous flow
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Examples
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• b/h =1 square, CD = 1.18; (disk; CD = 1.17)• CD independent of Re for Re > 1000
Question: CD = FD/(1/2 U2A)What happens to CD if double area (b/h 2b/2h)?What happens to FD if double area (b/h 2b/2h)?
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Drag dependence
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Drag Coefficient
ReCRe d 2410 44.01000 dCRe
for external flow: Re > 100 dominated by inertia, Re < 1 – by viscosity
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Why Different Regions?
As the flow rate increases wake drag becomes an important factor.The streamline pattern becomes mixed at the rear of the particle thuscausing a greater pressure at the front of the particle and thus an extraforce term due to pressure difference. At very high Reynolds numberscompletely separate in the wake.
Streamline separation
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Example
A cylindrical bridge pier 1 meter in diameter is submerged to a depthof 10m in a river at 20°C. Water is flowing past at a velocity of 1.2m/s. Calculate the force in Newtons on the pier.
smkgx
mkg
water
water
3
3
10005.1
2.998
smu 2.10
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2
20uAC
F dk
6
3
30 10192.1
10005.1
12.12.998
smkg
msmmkgDuRe
From figure Cd ≈ 0.35From figure Cd ≈ 0.35
Projected Area = DL = 10 m2
Ns
m
m
kgmFk 515,22.12.99810
2
35.02
22
3
2
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28
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29
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STREAMLINING
Streamlining is the attempt to reduce the drag on a body
CD ~ 0.06CD ~ 2 for flat plate
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StreamliningThe less drag you have…◦ Flying a glider: the further you
can fly
◦ Flying an airplane: the less fuelyou use
Therefore streamlining is importantTherefore streamlining is important◦ A design device by which a body
is shaped to minimize drag
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In general, the importance ofstreamlining to reduce drag.
2-D rectangular cylinder
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STREAMLINING
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STREAMLINING
2 cm
~ same drag AND wake
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Non-circular Channels
Equivalent diameter defined as 4 times the hydraulic radius(rH).
p
H
L
Ar
Where, A = cross-sectional area of channelLp = perimeter of channel in contact with fluid
Hydraulic radius of circular tube,
44
2
D
D
Dr
H
The equivalent diameter is 4 rH.
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For a rectangular duct with width W and height H, the hydraulic diameter is
4 4 2
2( )h
A WH WHD
P W H W H
H
W
Annulus between two circular pipes
DD
444 0
0
22
0
i
i
i
H
DD
DD
DD
r
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Sphericity
Surface area of sphere, Sp = 4 r 2 = Dp2
Volume of sphere, Vp = (4/3) r 3 = (1/6) Dp3
Sphericity (s) : The surface-volume ratio for a sphere ofdiameter Dp divided by the surface-volume ratio for the
37
diameter Dp divided by the surface-volume ratio for theparticle whose Nominal size is Dp.
p
p
p
s
vS
D6
psp
p
DV
S
6
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38
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39
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40
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41
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42
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Response to Superficial Velocities
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44
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45
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46
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ProblemAir ( = 1.22 Kg/m3, = 1.9 X 10 –-5 pa.s) is flowingin a fixed bed of a diameter 0.5 m and height 2.5 m. Thebed is packed with spherical particles of diameter 10 mm.The void fraction is 0.38. The air mass flow rate is 0.5kg/s. Calculate the pressure drop across the bed ofparticles.
47 1
particles.
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Solution Q = volumetric flow rate = = 0.41 m3/s
A = D2 = (0.5)2 = 0.1963 m2
u = = = 2.1 m/s
Rep = =
48 1
p
Rep = 2174
fp = + 1.75 = 1.819 =
0.276 x 105 pa.