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