Post on 19-Jan-2018
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Elementary Mechanics of Fluids
CE 319 FDaene McKinney
DimensionalAnalysis
Dimensional Analysis• Want to study pressure drop as function of
velocity (V1) and diameter (do)• Carry out numerous experiments with
different values of V1 and do and plot the data
0
1
4
0
12
1
4
0
121
21
20
20
0
21
1
1
2
12
2
22
ddfC
dd
V
p
ddVp
VVp
VpVp
p
5 parameters:p, , V1, d1, do
2 dimensionless parameters:p/(V2/2), (d1/do)
p1
p0
V1
A1
V0
A0
Much easier to establish functional relations with 2 parameters, than 5
Exponent Method),,,( VLfF
VLorLV
bbTacbaL
ccMMLLTLTMLTLM
VLcba
cba
11
31110001
110:1310:
110:)()())((
)(
Force F on a body immersed in a flowing fluid depends on: L, V, , and
n = 5 No. of dimensional parametersj = 3 No. of dimensionsk = n - j = 2 No. of dimensionless parameters
Select “repeating” variables: L, V, and Combine these with the rest of the variables: F &
Reynolds number
F L V
MLT-2 L LT-1 ML-3 ML-1T-1
Exponent Method
)(
)(
220:2310:110:
)()())((
)(
22
12222
3120002
fLV
F
fandVLF
bbTacbaLccM
MLLTLMLTTLM
VLFcba
cba
Dimensionless force is a function of the Reynolds number
F L V
MLT-2 L LT-1 ML-3 ML-1T-1
Exponent Method1. List all n variables involved in the problem
– Typically: all variables required to describe the problem geometry (D) or define fluid properties (, ) and to indicate external effects (dp/dx)
2. Express each variables in terms of MLT dimensions (j)3. Determine the required number of dimensionless parameters (n – j)4. Select a number of repeating variables = number of dimensions
– All reference dimensions must be included in this set and each must be dimensionalls independent of the others
5. Form a dimensionless parameter by multiplying one of the nonrepeating variables by the product of the repeating variables, each raised to an unknown exponent
6. Repeat for each nonrepeating variable7. Express result as a relationship among the dimensionless parameters
Example (8.7)• Find: Drag force on rough sphere is function
of D, , , V and k. Express in form:
),( 213 f
FD D V k
MLT-2 L ML-3 ML-1T-1 LT-1 L
VDorDV
bbTacbaL
ccMMLLTLTMLTLM
VDcba
cba
11
31110001
110:1310:
110:)()())((
)(
n = 6 No. of dimensional parametersj = 3 No. of dimensionsk = n - j = 3 No. of dimensionless parameters
Select “repeating” variables: D, V, and Combine these with nonrepeating variables: F, & k
Example (8.7)FD D V k
MLT-2 L ML-3 ML-1T-1 LT-1 L
Dk
bbTacbaL
ccMMLLTLLTLM
VDkcba
cba
2
310002
00:1310:
00:)()())((
)(
Select “repeating” variables: D, V, and Combine these with nonrepeating variables: F, & k
223
3120003
220:2310:110:
)()())((
)(
DVF
bbTacbaLccM
MLLTLMLTTLM
VDF
D
cba
cbaD
),(22 DkVDf
DVFD
Common Dimensionless No’s.• Reynolds Number (inertial to viscous forces)
– Important in all fluid flow problems
• Froude Number (inertial to gravitational forces)– Important in problems with a free surface
• Euler Number (pressure to inertial forces)– Important in problems with pressure differences
• Mach Number (inertial to elastic forces)– Important in problems with compressibility effects
• Weber Number (inertial to surface tension forces)– Important in problems with surface tension effects
Vd
ghVF
cV
EVM
/
2LVW
2VpC p
Similitude• Similitude
– Predict prototype behavior from model results
– Models resemble prototype, but are • Different size (usually smaller) and may
operate in • Different fluid and under• Different conditions
– Problem described in terms of dimensionless parameters which may apply to the model or the prototype
– Suppose it describes the prototype
– A similar relationship can be written for a model of the prototype
)321 ,...,,( nf
)321 ,...,,( npppp f
)321 ,...,,( nmmmm f
Similitude• If the model is designed
& operated under conditions that
• then
npnm
pm
pm
...33
22
pm 11
Similarity requirements or modeling laws
Dependent variable for prototype will be the same as in the model
Example• Consider predicting the drag on a
thin rectangular plate (w*h) placed normal to the flow.
• Drag is a function of: w, h, , , V
),,,,( VhwfFD
),(
),(
22
321
Vw
hwf
VwF
f
D
),(
),(
22
321
m
mmm
m
m
mmm
Dm
mmmwV
hwf
VwF
f
),(
),(
22
321
p
ppp
p
p
ppp
Dp
ppp
wVhw
fVw
F
f
• Dimensional analysis shows:• And this applies BOTH to a model
and a prototype
• We can design a model to predict the drag on a prototype.
• Model will have:
• And the prototype will have:
Example
pm
p
m
p
p
mm
p
ppp
m
mmmpm V
ww
VwVwV
33
Dmm
p
m
p
m
pDp
ppp
Dp
mmm
Dmpm F
V
V
w
wF
Vw
F
VwF
22
222211
•Similarity conditionsGeometric similarity
Dynamic similarity
Then
pp
mm
p
p
m
mpm w
hhw
hw
hw
22 Gives us the size of the model
Gives us the velocity in the model
Example (8.28)• Given: Submarine moving below surface in
sea water (=1015 kg/m3, =1.4x10-6 m2/s). Model is 1/20-th scale in fresh water (20oC).
• Find: Speed of water in the testdynamic similarity and the ratio of drag force on model to that on prototype.
• Solution: Reynolds number is significant parameter.
smV
sm
VLL
V
LVLV
m
pp
m
m
pm
p
pp
m
mm
pm
/6.28
/24.1
1120
ReRe
504.0
201
26.28
10151000 22
22
22
2222
p
m
ppp
mmm
p
m
ppp
p
mmm
m
FF
lVlV
FF
lV
F
lVF