058:0160
Jianming Yang
Fall 2012
Chapter 5
1
Chapter 5: Dimensional Analysis and Modeling
1 The Need for Dimensional Analysis
Dimensional analysis is a procedure by which the functional relationship can be
expressed in terms of 𝑘 nondimensional parameters, in which 𝑘 < 𝑛 with 𝑛 the number
of variables. Such a reduction is significant since in an experimental or numerical
investigation a reduced number of experiments or calculations is extremely beneficial
1. Reduction in Variables:
If
𝐹 (𝐴1 , 𝐴2 , ⋯ , 𝐴𝑛 ) = 0,
dimensional form
𝐹 = functional form, 𝐴𝑖 = dimensional variables
Then 𝑓 (𝛱1 , 𝛱2 , ⋯ , 𝛱𝑘<𝑛 ) = 0 nondimensional form (number of variables reduced)
or 𝛱1 = 𝛱1 (𝛱2 , ⋯ , 𝛱𝑘 )
𝛱𝑟 = nondimensional parameters= 𝛱𝑟 (𝐴𝑖 ),
i.e., 𝛱𝑟 consists of nondimensional groupings of 𝐴𝑖 ’s
Thereby reduces number of experiments/simulations required to determine 𝑓 vs. 𝐹
2. Helps in understanding physics
3. Useful in data analysis and modeling
4. Fundamental to concept of similarity and model testing
Enables scaling for different physical dimensions and fluid properties
058:0160
Jianming Yang
Fall 2012
Chapter 5
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2 The 𝚷 Theorem
In a physical problem including 𝑛 dimensional variables in which there are 𝑚
dimensions, the variables can be arranged into 𝑘 = 𝑛– 𝑗 independent nondimensional
parameters 𝛱𝑘 (where usually 𝑗 ≤ 𝑚).
𝐹 (𝐴1 , 𝐴2 , ⋯ , 𝐴𝑛 ) = 0
𝑓(𝛱1 , 𝛱2 , ⋯ , 𝛱𝑘 ) = 0
𝐴𝑖 ’s = dimensional variables required to formulate problem (𝑖 = 1, ⋯ , 𝑛)
𝛱𝑟 ’s = nondimensional parameters consisting of groupings of 𝐴𝑖 ’s (𝑟 = 1, ⋯ , 𝑘)
𝐹, 𝑓 represents functional relationships between 𝐴𝑛 ’s and 𝛱𝑘 ’s, respectively
𝑗 is the reduction of variables, 𝑗 ≤ 𝑚 (𝑚 is the number of dimensions)
The reduction 𝑗 is the number of scaling variables that do not form a pi among
themselves.
Each desired pi group will be a power product of these 𝑗 variables plus one additional
variable with a nonzero exponent.
Dimensions and Equations
Basic dimensions: F, L, and t or M, L, and t
F and M related by F = Ma = MLT-2
058:0160
Jianming Yang
Fall 2012
Chapter 5
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Dimensional Analysis
Methods for determining 𝛱𝑟 ’s
1. Functional Relationship Method
Identify functional relationships 𝐹 (𝐴𝑖 ) and 𝑓 (𝛱𝑟 ) by first determining 𝐴𝑖 ’s and then
evaluating 𝛱𝑟 ’s
a. The Pi theorem method
b. Ipsen’s step-by-step method
2. Nondimensionalize governing differential equations and initial/boundary conditions
Select appropriate quantities for nondimensionalizing the GDE, IC, and BC, e.g. for
M, L, and t
Put GDE, IC, and BC in nondimensional form
Identify 𝛱𝑟 ’s
058:0160
Jianming Yang
Fall 2012
Chapter 5
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058:0160
Jianming Yang
Fall 2012
Chapter 5
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Suppose one knew that the force 𝐹 on a particular body shape immersed in a stream of
fluid depended only on the body length 𝐿, stream velocity 𝑉, fluid density 𝜌, and fluid
viscosity 𝜇; that is,
𝐹 = 𝑓 (𝐿, 𝑈, 𝜌, 𝜇)
Use the pi theorem to derive
058:0160
Jianming Yang
Fall 2012
Chapter 5
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Jianming Yang
Fall 2012
Chapter 5
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Jianming Yang
Fall 2012
Chapter 5
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Jianming Yang
Fall 2012
Chapter 5
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Jianming Yang
Chapter 5
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Fall 2012
3 Common Dimensionless Parameters for Fluid Flow Problems
Most common physical quantities of importance in fluid flows are: (without heat transfer)
1
2
3
4
5
6
7
8
V,
,
g,
,
,
K,
p,
L
velocity density gravity viscosity surface tension compressibility pressure change length
n=8
m=3
 5 dimensionless parameters
VL inertia forces V 2 / L

1) Reynolds number =

viscous forces V / L2
R
e
Rcrit distinguishes among flow regions: laminar or turbulent value varies depending
upon flow situation
V
inertia forces
2) Froude number =

gL gravity force
V 2 / L

Fr
important parameter in free-surface flows
V 2 L
inertia force
V 2 / L
3) Weber number =


surface tension force  / L2 We
important parameter at gas-liquid or liquid-liquid interfaces and when these surfaces
are in contact with a boundary
058:0160
Jianming Yang
Chapter 5
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Fall 2012
4) Mach number =
V
V
inertia force
 
compressib ility force
k / a
Ma
where a is the speed of sound in fluid
Paramount importance in high speed flow (V > c)
5) Pressure Coefficient =
p
V 2

pressure force p / L
inertia force V 2 / L
Cp
(Euler Number)
4 Nondimensionalization of the Basic Equation
It is very useful and instructive to nondimensionalize the basic equations and boundary
conditions. Consider the situation for  and  constant and for flow with a free surface
Continuity:
V  0
Momentum:

DV
  p  gz    2V
Dt
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Jianming Yang
Chapter 5
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Boundary Conditions:
1) fixed solid surface: V  0
2) inlet or outlet: V = Vo p = po
3) free surface:
w

(z = )
t
p  pa   Rx1  Ry1 
All variables are now nondimensionalized in terms of  and
U = reference velocity
L = reference length
V 
*
V
U
t* 
tU
L
x 
*
x
L
p* 
p  gz
U 2
All equations can be put in nondimensional form by making the substitution
VV U
*

 t * U 


t t * t L t *
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Jianming Yang

Chapter 5
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Fall 2012



î  ĵ  k̂
x y z
 x *
 y*
 z *
 *
î  *
ĵ  *
k̂
x x
y y
z z
1
 *
L
u 1 
U u *
*

Uu 
and
etc.
x L x *
L x *


Result: *  V*  0
DV

*
  * p* 
*2 V
Dt
VL
*
1)
2)
3)
V 0
V
*
V  o
U
*
*
w  *
t
*
Re-1
p* 
p* 
po
V 2
po
gL *

*1
*1



z

R

R
x
y
2
2
2
V U
V L
pressure coefficient
Fr-2
We-1
058:0160
Jianming Yang
Fall 2012
Chapter 5
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058:0160
Jianming Yang
Fall 2012
Chapter 5
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5 Similarity and Model Testing
Flow conditions for a model test are completely similar if all relevant dimensionless
parameters have the same corresponding values for model and prototype
Π𝑖,model = Π𝑖,prototype 𝑖 = 1, ⋯ , 𝑘 (𝑘 = 𝑛 − 𝑗)
Enables extrapolation from model to full scale
Complete similarity usually is not possible. Often it is necessary to use 𝑅𝑒, or 𝐹𝑟, or 𝑀𝑎
scaling, i.e., select the most important Π and accommodate others as best possible.
Types of Similarity:
1) Geometric Similarity (similar length scales):
A model and prototype are geometrically similar if and only if all body dimensions
in all three coordinates have the same linear-scale ratios
𝛼 = 𝐿model /𝐿prototype (𝛼 < 1, i. e. , 1⁄10 , or 1⁄50)
2) Kinematic Similarity (similar length and time scales):
The motions of two systems are kinematically similar if homologous (same relative
position) particles lie at homologous points at homologous times
3) Dynamic Similarity (similar length, time and force (or mass) scales):
In addition to the requirements for kinematic similarity the model and prototype
forces must be in a constant ratio
058:0160
Jianming Yang
Fall 2012
5.1 Model Testing in Water with a Free Surface
𝐹 = 𝐹(𝐹𝑟, 𝑅𝑒)
𝐹𝑟 similarity
𝐹𝑟model = 𝐹𝑟prototype
𝑉prototype
𝑉model
=
√𝑔𝐿model √𝑔𝐿prototype
Froude scaling:
𝑉model /𝑉prototype = √𝛼
𝑅𝑒 similarity
𝑅𝑒model = 𝑅𝑒prototype
𝑉model 𝐿model 𝑉prototype 𝐿prototype
=
𝜐model
𝜐prototype
Reynolds scaling:
𝜐model
𝑉model 𝐿model
=
= 𝛼 3/2
𝜐prototype 𝑉prototype 𝐿prototype
Impossible to achieve both Fr and Re similarity at the same time, e.g, if prototype is
water flow, and 𝛼 = 1/10, (see Table 1.4 in textbook)
𝜐model = 𝛼 3/2 𝜐water = 3.1 × 10−8 m2 ⁄𝑠 < 𝜐mercury = 1.2 × 10−7 m2 ⁄𝑠
Chapter 5
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058:0160
Jianming Yang
Chapter 5
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Fall 2012
Alternatively one could maintain 𝑅𝑒 similarity and obtain
𝑉model = 𝑉prototype /𝛼
But if 𝛼 = 1/10,
𝑉model = 10𝑉prototype
High speed testing is difficult and expensive.
Different body forces?
2
2
𝑉prototype
𝑉model
=
𝑔model 𝐿model 𝑔prototype 𝐿prototype
2
𝐿prototype
𝑔model
𝑉model
1 1
1
= 2
= 2 = 3
𝑔prototype 𝑉prototype
𝐿model
𝛼 𝛼 𝛼
𝑔prototype
𝑔model =
𝛼3
But if 𝛼 = 1/10,
𝑔model = 1000𝑔prototype
Impossible to achieve
In hydraulics model studies, 𝐹𝑟 scaling used, but lack of 𝑊𝑒 similarity can cause
problems. Therefore, often models are distorted, i.e. vertical scale is increased by 10 or
more compared to horizontal scale
058:0160
Jianming Yang
Chapter 5
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Fall 2012
5.2 Model Testing in Air
𝐹 = 𝐹(𝑅𝑒, 𝑀𝑎)
𝑅𝑒 similarity
𝑅𝑒model = 𝑅𝑒prototype
𝑉model 𝐿model 𝑉prototype 𝐿prototype
=
𝜐model
𝜐prototype
𝑀𝑎 similarity
𝑀𝑎model = 𝑀𝑎prototype
𝑉model 𝑉prototype
=
𝑎model 𝑎prototype
i.e.,
𝜐model
𝜐prototype
=
𝐿model
𝑎model
𝐿prototype 𝑎prototype
=
𝐿model
𝐿prototype
=𝛼
again not possible
Therefore, in wind tunnel testing Re scaling is also violated
(
𝑎model
𝑎prototype
= 1)