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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)