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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