flow regimes and drag force
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7/31/2019 Flow Regimes and Drag Force
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18 January 2007 1
Drag on a Sphere and Settling Velocity
DragOne of basic concepts in Hydraulicsand Fluid Mechanics
Drag on a sphereA simple caseNo general theoretical solutionsStill a challenging topic
18 January 2007 2
Drag on a Sphere
Sediment transport
Sludge transport
Air pollution
Food processing
Water purification
L Applications
A
B
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18 January 2007 3
What is Drag?
DragLift
Drag is a hydro-/aero-dynamicforce that opposes an object's
motion through a fluid
18 January 2007 4
Two kind of Drags
P
= dscosPDP
= dssinDf Pressure Drag
Friction Drag
V
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18 January 2007 5
Two kind of Drags
Pressure Drag depends on the shape of the object,
and also called Form Drag.
Friction Drag is related to the surface roughnessand the boundary layer. It is also called SurfaceDrag.
18 January 2007 6
Drag Coefficient
Consider a flow with velocity V approaching an object
V
a b
p
2
fp C*A*2
V
D =A
Stagnation point
V
Pressure head b = Velocity head a(Bernoulli equation)
g
V
g
P a
f
b
2
2
=
Projected area normal to the flow
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18 January 2007 7
V
D
2
fD C*A*2
VF =
Drag Coefficient
Drag
Drag Coefficient
A
18 January 2007 8
Force Balance (Simplified BBO Equation)
FW
FBFD
FD + FB = FW
3
6DgFB
=
24
22 w
DCF DD
=
3
6DgF pW
=
32
2
6)(
24Dg
wDC pD
=
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18 January 2007 9
Settling Velocity & Drag Coefficient
32
2
6)(
24Dg
wDC pD
=
DC
gDw
=
3
4
23
4
w
gDCD
= Here = (p - )/
18 January 2007 10
Re = wD/v
CD
1 2*105
Laminar BL Turbulent BL
General Relationship: CD ~ Re
No inertial
effect
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18 January 2007 11
Re
CD
1
Theoretical Solution
No inertial
effect
For creeping flow (Re = wD/v 2*105, Turbulent boundary layer,Wake size reduces,
Drag sharply decreases
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18 January 2007 13
Flow over a Circular Cylinder
Creeping Flow, Steady, Symmetric Streamlines
(Van Dyke, 1982)
18 January 2007 14
Laminar Flow, Steady, Separated Flow
(Van Dyke, 1982)
Flow over a Circular Cylinder
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18 January 2007 15
Laminar Flow, Steady, Separated Flow
(Van Dyke, 1982)
Flow over a Circular Cylinder
18 January 2007 16
Unsteady Transitional Flow, Karman Vortex Street
(Van Dyke, 1982)
Flow over a Circular Cylinder
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18 January 2007 17
Unsteady, Turbulent Flow
(Van Dyke, 1982)
Flow over a Circular Cylinder
18 January 2007 18
Unsteady, Turbulent Flow
(Van Dyke, 1982)
Flow over a Circular Cylinder
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18 January 2007 19
Settling Velocity of a Sphere
=
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18 January 2007 21
)(RefCD =
)(3
42
v
wDf
w
gD=
(Iteration neededfor computing w)
*
3/1
2
3/1
2
4
3DD
gRe
CD
=
)( *DfRe =
)(Dfw =
18 January 2007 22
( )
=
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18 January 2007 23
Settling Velocity of Naturally Worn Particles
ReCD
32=
2/32/3
1+32
=Re
CD
For Stokes range, it reduces to
18 January 2007 24
1
10
100
1000
10000
100000
1000000
10000000
100000000
1E-07 1E-06 1E-05 0.0001 0.001 0.01 0.1 1 10 100 1000
Re
CD
Experiment
Computation
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18 January 2007 25
0.0000001
0.000001
0.00001
0.0001
0.001
0.01
0.1
1
10
100
1000
10000
0.01 0.1 1 10 100 1000
D*
Re
Measurement
Cheng (1997)
5.12
* )52.125( += DD
w
18 January 2007 26
Km cw
w)1( =
Particle Concentration Effect on Settling Velocity
75.0
75.0
175.01
42.065.4
+
+
=
wD
wD
K
Wm = w
if c = 0
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