Dimensional Analysis and Similitude

Dimensional Analysis and Similitude

CEE 331

Summer 2000

CEE 331

Summer 2000

Why?Why?

“One does not want to have to show and relate the results for all possible velocities, for all possible geometries, for all possible roughnesses, and for all possible fluids...”

“One does not want to have to show and relate the results for all possible velocities, for all possible geometries, for all possible roughnesses, and for all possible fluids...”

Wilfried Brutsaert in “Horton, Pipe Hydraulics, and the Atmospheric Boundary Layer.” in Bulletin of the American Meteorological Society. 1993.

On Scaling...On Scaling...

“...the writers feel that they would well deserve the flood of criticism which is ever threatening those venturous persons who presume to affirm that the same laws of Nature control the flow of water in the smallest pipes in the laboratory and in the largest supply mains running over hill and dale. In this paper it is aimed to present a few additional arguments which may serve to make such an affirmation appear a little less ridiculous than heretofore.”

“...the writers feel that they would well deserve the flood of criticism which is ever threatening those venturous persons who presume to affirm that the same laws of Nature control the flow of water in the smallest pipes in the laboratory and in the largest supply mains running over hill and dale. In this paper it is aimed to present a few additional arguments which may serve to make such an affirmation appear a little less ridiculous than heretofore.”

Saph and Schoder, 1903Saph and Schoder, 1903

Why?Why?

Suppose I want to build an irrigation canal, one that is bigger than anyone has ever built. How can I determine how big I have to make the canal to get the desired flow rate? Do I have to build a section of the canal and test it?

Suppose I build pumps. Do I have to test the performance of every pump for all speed, flow, fluid, and pressure combinations?

Suppose I want to build an irrigation canal, one that is bigger than anyone has ever built. How can I determine how big I have to make the canal to get the desired flow rate? Do I have to build a section of the canal and test it?

Suppose I build pumps. Do I have to test the performance of every pump for all speed, flow, fluid, and pressure combinations?

Dimensional AnalysisDimensional Analysis

The case of Frictional Losses in Pipes (NYC)

Dimensions and Units Theorem Assemblage of Dimensionless Parameters Dimensionless Parameters in Fluids Model Studies and Similitude

The case of Frictional Losses in Pipes (NYC)

Dimensions and Units Theorem Assemblage of Dimensionless Parameters Dimensionless Parameters in Fluids Model Studies and Similitude

Frictional Losses in Pipescirca 1900

Frictional Losses in Pipescirca 1900

Water distribution systems were being built and enlarged as cities grew rapidly

Design of the distribution systems required knowledge of the head loss in the pipes (The head loss would determine the maximum capacity of the system)

It was a simple observation that head loss in a straight pipe increased as the velocity increased (but head loss wasn’t proportional to velocity).

Water distribution systems were being built and enlarged as cities grew rapidly

Design of the distribution systems required knowledge of the head loss in the pipes (The head loss would determine the maximum capacity of the system)

It was a simple observation that head loss in a straight pipe increased as the velocity increased (but head loss wasn’t proportional to velocity).

Two Opposing TheoriesTwo Opposing Theories

agrees with the “law of a falling body”

f varies with velocity and is different for different pipes

agrees with the “law of a falling body”

f varies with velocity and is different for different pipes

Fits the data well for any particular pipe

Every pipe has a different m and n.

What does g have to do with this anyway?

Fits the data well for any particular pipe

Every pipe has a different m and n.

What does g have to do with this anyway?

“In fact, some engineers have been led to question whether or not water flows in a pipe according to any definite determinable laws whatsoever.”

Saph and Schoder, 1903

hl is mechanical energy lost to thermal energy expressed as p.e.

h fld

Vgl 2

2h mVl

n

h mVln

Research at Cornell!Research at Cornell!

Augustus Saph and Ernest Schoder under the direction of Professor Gardner Williams

Saph and Schoder had concluded that “there is practically no difference between a 2-in. and a 30-in. pipe.”

Conducted comprehensive experiments on a series of small pipes located in the basement of Lincoln Hall, (the principle building of the College of Civil Engineering)

Chose to analyze their data using ________

Augustus Saph and Ernest Schoder under the direction of Professor Gardner Williams

Saph and Schoder had concluded that “there is practically no difference between a 2-in. and a 30-in. pipe.”

Conducted comprehensive experiments on a series of small pipes located in the basement of Lincoln Hall, (the principle building of the College of Civil Engineering)

Chose to analyze their data using ________

Saph and Schoder ConclusionsSaph and Schoder Conclusions

Oh, and by the way, there is a “critical velocity” below which this equation doesn’t work. The “critical velocity” varies with pipe diameter and with temperature.

Check units...

hl is in ft/1000ft

V is in ft/s

d is in ft

hd

Vl 0.296 to 0.469

1.251.74 to 2.00

Oops!!Oops!!

The Buckingham TheoremThe Buckingham Theorem

“in a physical problem including n quantities in which there are m dimensions, the quantities can be arranged into n-m independent dimensionless parameters”

We reduce the number of parameters we need to vary to characterize the problem!

“in a physical problem including n quantities in which there are m dimensions, the quantities can be arranged into n-m independent dimensionless parameters”

We reduce the number of parameters we need to vary to characterize the problem!

Assemblage of Dimensionless Parameters

Assemblage of Dimensionless Parameters

Several forces potentially act on a fluid Sum of the forces = ma (the inertial force) Inertial force is always present in fluids

problems (all fluids have mass) Nondimensionalize by creating a ratio with

the inertial force The magnitudes of the force ratios for a

given problem indicate which forces govern

Several forces potentially act on a fluid Sum of the forces = ma (the inertial force) Inertial force is always present in fluids

problems (all fluids have mass) Nondimensionalize by creating a ratio with

the inertial force The magnitudes of the force ratios for a

given problem indicate which forces govern

Force parameter dimensionless Mass (inertia) ______ Viscosity ______ ______ Gravitational ______ ______ Pressure ______ ______ Surface Tension ______ ______ Elastic ______ ______

Force parameter dimensionless Mass (inertia) ______ Viscosity ______ ______ Gravitational ______ ______ Pressure ______ ______ Surface Tension ______ ______ Elastic ______ ______

Forces on FluidsForces on Fluids

R

F

p Cp W

K M

Dependent variableDependent variable

Inertia as our Reference ForceInertia as our Reference Force

F=ma Fluids problems always (except for statics)

include a velocity (V), a dimension of flow (l), and a density ()

F=ma Fluids problems always (except for statics)

include a velocity (V), a dimension of flow (l), and a density ()

F a F

a

f

f M

L T2 2f

ML T2 2

L lL l T T M M

fi fi

lV

lV

l3 l3

Vl

2

Vl

2

Viscous ForceViscous Force

What do I need to multiply viscosity by to obtain dimensions of force/volume?

What do I need to multiply viscosity by to obtain dimensions of force/volume?

Cf Cf

fC

fC

LTMTL

M

C22

LTMTL

M

C22

LTC

1 LTC

1

μ

i

ff μ

i

ff

2l

VC 2l

VC

Vlμ

i

ff

Vl

μ

i

ff

VlR

VlR

Reynolds number

L l Tl

V M l 3

fi Vl

2

2lV

lV 2

Gravitational ForceGravitational Force

gC g

g

f

gC g

g

f

2

22

TLTL

M

Cg

2

22

TLTL

M

Cg

3LM

Cg 3LM

Cg

g

i

ff g

i

ff

gC gC

glV 2

g

i

ff

glV 2

g

i

ff

glVFgl

VF

Froude number

L l Tl

V M l 3

fi Vl

2

lV 2

g

Pressure ForcePressure Force

pC p

p

f

pC p

p

f

2

22

LTMTL

M

C p

2

22

LTMTL

M

C p

LC p

1L

C p

1

p

i

ff p

i

ff

lC p

1l

C p

1

pV 2

p

i

ff

pV 2

p

i

ff 2

2C

Vp

p 2

2C

Vp

p

Pressure Coefficient

L l Tl

V M l 3

fi Vl

2

lV 2

lp

Dimensionless parametersDimensionless parameters

Reynolds Number

Froude Number

Weber Number

Mach Number

Pressure Coefficient

(the dependent variable that we measure experimentally)

Reynolds Number

Froude Number

Weber Number

Mach Number

Pressure Coefficient

(the dependent variable that we measure experimentally)

VlR

VlR

glVFgl

VF

2

2C

Vp

p 2

2C

Vp

p

lV

W2

lVW

2

cV

M cV

M

AVd

2

Drag2C

AVd

2

Drag2C

Application of Dimensionless Parameters

Application of Dimensionless Parameters

Pipe Flow Pump characterization Model Studies and Similitude

dams: spillways, turbines, tunnels harbors rivers ships ...

Pipe Flow Pump characterization Model Studies and Similitude

dams: spillways, turbines, tunnels harbors rivers ships ...

Example: Pipe FlowExample: Pipe Flow

fpC

fpC

Inertial

diameter, length, roughness height

Reynolds

l/D

viscous

/D

DDl

,,R

What are the important forces?______, ______. Therefore _________ number.

What are the important geometric parameters? _________________________ Create dimensionless geometric groups

______, ______ Write the functional relationship

What are the important forces?______, ______. Therefore _________ number.

What are the important geometric parameters? _________________________ Create dimensionless geometric groups

______, ______ Write the functional relationship

Example: Pipe FlowExample: Pipe Flow

R,

Df

lD

C p

R,

Df

lD

C p

R,f

Df

lD

C p

R,f

Df

lD

C p

2

2C

Vp

p 2

2C

Vp

p Cp proportional to l

f is friction factor

C flD Dp FH IK, ,

R

How will the results of dimensional analysis guide our experiments to determine the relationships that govern pipe flow?

If we hold the other two dimensionless parameters constant and increase the length to diameter ratio, how will Cp change?

How will the results of dimensional analysis guide our experiments to determine the relationships that govern pipe flow?

If we hold the other two dimensionless parameters constant and increase the length to diameter ratio, how will Cp change?

0.01

0.1

1E+03 1E+04 1E+05 1E+06 1E+07 1E+08R

fric

tion

fact

or

laminar

0.050.04

0.03

0.020.015

0.010.0080.006

0.004

0.002

0.0010.0008

0.0004

0.0002

0.0001

0.00005

smooth

lD

C pf

lD

C pf

D

D

Each curve one geometryCapillary tube or 24 ft diameter tunnelWhere is temperature?Compare with real data!Where is “critical velocity”?Where do you specify the fluid?At high Reynolds number curves are flat.Frictional Losses in Straight PipesFrictional Losses in Straight Pipes

What did we gain by using Dimensional Analysis?

What did we gain by using Dimensional Analysis?

Any consistent set of units will work We don’t have to conduct an experiment on

every single size and type of pipe at every velocity

Our results will even work for different fluids

Our results are universally applicable We understand the influence of temperature

Any consistent set of units will work We don’t have to conduct an experiment on

every single size and type of pipe at every velocity

Our results will even work for different fluids

Our results are universally applicable We understand the influence of temperature

Model Studies and Similitude:Scaling Requirements

Model Studies and Similitude:Scaling Requirements

Mach Reynolds Froude Weber

C fp M, R, F,W,geometrya fC fp M, R, F,W,geometrya f

dynamic similitude geometric similitude

all linear dimensions must be scaled identically roughness must scale

kinematic similitude constant ratio of dynamic pressures at corresponding

points streamlines must be geometrically similar _______, __________, _________, and _________

numbers must be the same

dynamic similitude geometric similitude

all linear dimensions must be scaled identically roughness must scale

kinematic similitude constant ratio of dynamic pressures at corresponding

points streamlines must be geometrically similar _______, __________, _________, and _________

numbers must be the same

Relaxed Similitude RequirementsRelaxed Similitude Requirements

same sizesame size

Impossible to have all force ratios the same unless the model is the _____ ____ as the prototype

Need to determine which forces are important and attempt to keep those force ratios the same

Impossible to have all force ratios the same unless the model is the _____ ____ as the prototype

Need to determine which forces are important and attempt to keep those force ratios the same

Similitude ExamplesSimilitude Examples

Open hydraulic structures Ship’s resistance Closed conduit Hydraulic machinery

Open hydraulic structures Ship’s resistance Closed conduit Hydraulic machinery

Scaling in Open Hydraulic Structures

Examples spillways channel transitions weirs

Important Forces inertial forces gravity: from changes in water surface elevation viscous forces (often small relative to gravity forces)

Minimum similitude requirements geometric Froude number

VlR

VlR

glVFgl

VF

Froude similarityglVFgl

VFpm FF pm FF

pp

2p

mm

2m

Lg

V

LgV

pp

2p

mm

2m

Lg

V

LgV

p

2p

m

2m

L

V

LV

p

2p

m

2m

L

V

LV

m

pr L

LL

m

pr L

LL rr LV rr LV

rr

rr L

VL

t rr

rr L

VL

t

2/5rrr LLL rrrr LAVQ 2/5rrr LLL rrrr LAVQ

3r2

r

r3rrrrr L

tL

LaMF 3r2

r

r3rrrrr L

tL

LaMF

difficult to change g

Froude number the same in model and prototype

________________________

define length ratio (usually larger than 1)

velocity ratio

time ratio

discharge ratio

force ratio

Example: Spillway ModelExample: Spillway Model

A 50 cm tall scale model of a proposed 50 m spillway is used to predict prototype flow conditions. If the design flood discharge over the spillway is 20,000 m3/s, what water flow rate should be tested in the model?

A 50 cm tall scale model of a proposed 50 m spillway is used to predict prototype flow conditions. If the design flood discharge over the spillway is 20,000 m3/s, what water flow rate should be tested in the model?

000,1002/5 rr LQ 000,1002/5 rr LQ

pm FF pm FF 100rL 100rL

smsm

Qm3

3

2.0000,100

000,20 smsm

Qm3

3

2.0000,100

000,20

Ship’s ResistanceShip’s Resistance

FR,,C

Drag22 l

fAV d

FR,,C

Drag22 l

fAV d

Viscosity, roughnessViscosity, roughness

gravitygravity

ReynoldsReynolds FroudeFroude

Skin friction ______________ Wave drag (free surface effect) ________ Therefore we need ________ and ______

similarity

Skin friction ______________ Wave drag (free surface effect) ________ Therefore we need ________ and ______

similarity

Water is the only practical fluidWater is the only practical fluid

Reynolds and Froude Similarity?Reynolds and Froude Similarity?

VlR

VlR

ppmm lVlV ppmm lVlV

p

m

m

p

l

l

V

V

p

m

m

p

l

l

V

V

r

r

LV

1r

r

LV

1

Reynolds

rL1

rLrL

1 rL

glVFgl

VF

rr LV rr LV

Froude

Lr = 1Lr = 1

p

ppp

m

mmmlVlV

p

ppp

m

mmmlVlV

Ship’s ResistanceShip’s Resistance

Can’t have both Reynolds and Froude similarity

Froude hypothesis: the two forms of drag are independent

Measure total drag on Ship Use analytical methods to

calculate the skin friction Remainder is wave drag

Can’t have both Reynolds and Froude similarity

Froude hypothesis: the two forms of drag are independent

Measure total drag on Ship Use analytical methods to

calculate the skin friction Remainder is wave drag

FR,,C

D22

total

Df

AVd

FR,,C

D22

total

Df

AVd

FfAV

2D

2

w

FfAV

2D

2

w

R,

2D

2

f

Df

AV

R,

2D

2

f

Df

AV

totalD totalD wf DD

empiricalempirical

analyticalanalytical

Closed Conduit Incompressible Flow

Closed Conduit Incompressible Flow

viscosityviscosityinertiainertia

velocityvelocity

Forces __________ __________

If same fluid is used for model and prototype VD must be the same Results in high _________ in the model

High Reynolds number (R) Often results are independent of R for very

high R

Forces __________ __________

If same fluid is used for model and prototype VD must be the same Results in high _________ in the model

High Reynolds number (R) Often results are independent of R for very

high R

Example: Valve CoefficientExample: Valve Coefficient

The pressure coefficient, , for a 600-mm-diameter valve is to be determined for 5 ºC water at a maximum velocity of 2.5 m/s. The model is a 60-mm-diameter valve operating with water at 5 ºC. What water velocity is needed?

The pressure coefficient, , for a 600-mm-diameter valve is to be determined for 5 ºC water at a maximum velocity of 2.5 m/s. The model is a 60-mm-diameter valve operating with water at 5 ºC. What water velocity is needed?

2

2C

Vp

p 2

2C

Vp

p

Example: Valve CoefficientExample: Valve Coefficient

Note: roughness height should scale! Reynolds similarity

Note: roughness height should scale! Reynolds similarity

VlR

VlR

p

pp

m

mmDVDV

p

pp

m

mmDVDV

VDR

VDR

m

ppm D

DVV

m

ppm D

DVV

m

msmVm 06.0

6.0)/5.2(

mmsm

Vm 06.06.0)/5.2(

ν = 1.52 x 10-6 m2/s ν = 1.52 x 10-6 m2/s

Vm = 25 m/s Vm = 25 m/s

Use water at a higher temperatureUse water at a higher temperature

Example: Valve Coefficient(Reduce Vm?)

Example: Valve Coefficient(Reduce Vm?)

What could we do to reduce the velocity in the model and still get the same high Reynolds number?

What could we do to reduce the velocity in the model and still get the same high Reynolds number?

VlR

VlR

VDR

VDRDecrease kinematic viscosityDecrease kinematic viscosityUse a different fluidUse a different fluid

Example: Valve CoefficientExample: Valve Coefficient

Change model fluid to water at 80 ºC Change model fluid to water at 80 ºC

p

pp

m

mmDVDV

p

pp

m

mmDVDV

VDR

VDR

mp

ppmm D

DVV

mp

ppmm D

DVV

msmx

msmsmxVm 06.0/1052.1

6.0)/5.2(/10367.026

26

msmxmsmsmx

Vm 06.0/1052.16.0)/5.2(/10367.0

26

26

νm = ______________νm = ______________

νp = ______________νp = ______________

Vm = 6 m/s Vm = 6 m/s

0.367 x 10-6 m2/s

1.52 x 10-6 m2/s

Approximate Similitude at High Reynolds Numbers

Approximate Similitude at High Reynolds Numbers

High Reynolds number means ______ forces are much greater than _______ forces

Pressure coefficient becomes independent of R for high R

High Reynolds number means ______ forces are much greater than _______ forces

Pressure coefficient becomes independent of R for high R

inertialinertialviscousviscous

Pressure Coefficient for a Venturi Meter

Pressure Coefficient for a Venturi Meter

1

10

1E+00 1E+01 1E+02 1E+03 1E+04 1E+05 1E+06

R

Cp

VlR

VlR

2

2C

Vp

p 2

2C

Vp

p

Similar to rough pipes in Moody diagram!Similar to rough pipes in Moody diagram!

Hydraulic Machinery: PumpsHydraulic Machinery: Pumps

rr l

V1r

r lV

1

streamlines must be geometrically similar streamlines must be geometrically similar

rr lV rr lV

Rotational speed of pump or turbine is an additional parameter additional dimensionless parameter is the ratio

of the rotational speed to the velocity of the water _________________________________

homologous units: velocity vectors scale _____ Now we can’t get same Reynolds Number!

Reynolds similarity requires Scale effects

Rotational speed of pump or turbine is an additional parameter additional dimensionless parameter is the ratio

of the rotational speed to the velocity of the water _________________________________

homologous units: velocity vectors scale _____ Now we can’t get same Reynolds Number!

Reynolds similarity requires Scale effects

Dimensional Analysis SummaryDimensional Analysis Summary

enables us to identify the important parameters in a problem

simplifies our experimental protocol (remember Saph and Schoder!)

does not tell us the coefficients or powers of the dimensionless groups (need to be determined from theory or experiments)

guides experimental work using small models to study large prototypes

enables us to identify the important parameters in a problem

simplifies our experimental protocol (remember Saph and Schoder!)

does not tell us the coefficients or powers of the dimensionless groups (need to be determined from theory or experiments)

guides experimental work using small models to study large prototypes

Dimensional analysis:

endend

100,000

1,000,000

10,000,000

1800 1850 1900 1950 2000

year

popu

latio

n

NYC populationNYC population

Cro

ton

Cat

skil

l

Del

awar

e

New

Cro

ton

Supply Aqueducts and Tunnels

Catskill AqueductCatskill Aqueduct

Delaware TunnelDelaware Tunnel

East Delaware tunnelEast Delaware tunnel

West Delaware tunnelWest Delaware tunnel

Shandaken TunnelShandaken Tunnel

Neversink TunnelNeversink Tunnel

Delaware AqueductDelaware Aqueduct

10 km

Rondout Reservoir

West Branch

Reservoir

Flow Profile for Delaware Aqueduct

Flow Profile for Delaware Aqueduct

Rondout Reservoir(EL. 256 m)

West Branch Reservoir(EL. 153.4 m)

70.5 km

Hudson River crossing El. -183 m)

Sea Level

(Designed for 39 m3/s)

p Vg

z Hp V

gz H hp t l

11

12

12

222

22 2

p Vg

z Hp V

gz H hp t l

11

12

12

222

22 2

Ship’s Resistance: We aren’t done learning yet!

Ship’s Resistance: We aren’t done learning yet!

FASTSHIPS may well ferry cargo between the U.S. and Europe as soon as the year 2003. Thanks to an innovative hull design and high-powered propulsion system, FastShips can sail twice as fast as traditional freighters. As a result, valuable cargo should be able to cross the Atlantic Ocean in 4 days.

FASTSHIPS may well ferry cargo between the U.S. and Europe as soon as the year 2003. Thanks to an innovative hull design and high-powered propulsion system, FastShips can sail twice as fast as traditional freighters. As a result, valuable cargo should be able to cross the Atlantic Ocean in 4 days.

Port ModelPort Model A working scale model was used to eliminated danger to boaters from

the "keeper roller" downstream from the diversion structure

http://ogee.hydlab.do.usbr.gov/hs/hs.html

Hoover Dam SpillwayHoover Dam Spillway

A 1:60 scale hydraulic model of the tunnel spillway at Hoover Dam for investigation of cavitation damage preventing air slots.

A 1:60 scale hydraulic model of the tunnel spillway at Hoover Dam for investigation of cavitation damage preventing air slots.

http://ogee.hydlab.do.usbr.gov/hs/hs.html

Irrigation Canal ControlsIrrigation Canal Controlshttp://elib.cs.berkeley.edu/cypress.html

SpillwaysSpillways

Frenchman Dam and spillway (in use).Lahontan Region (6)

DamsDams

Dec 01, 1974Cedar Springs Dam, spillway & ReservoirSanta Ana Region (8)

Dec 01, 1974Cedar Springs Dam, spillway & ReservoirSanta Ana Region (8)

SpillwaySpillway

Mar 01, 1971Cedar Springs Spillway construction.Santa Ana Region (8)

Mar 01, 1971Cedar Springs Spillway construction.Santa Ana Region (8)

Kinematic ViscosityKinematic Viscosity

1.00E-07

1.00E-06

1.00E-05

1.00E-04

1.00E-03

mercur

y

carb

on te

trach

loride

water

ethyl

alcoh

ol

kero

sene air

sae 1

0W

SAE 10W

-30

SAE 30

glyce

rine

kine

mat

ic v

isco

sity

20C

(m

2 /s)

Kinematic Viscosity of WaterKinematic Viscosity of Water

0.0E+00

5.0E-07

1.0E-06

1.5E-06

2.0E-06

0 20 40 60 80 100

Temperature (C)

Kin

emat

ic V

isco

sity

(m

2 /s)