Transcript Introduction to Fluid Mechanics - Pharos University in Alexandria

Pharos University
ME 259 Fluid Mechanics
Lecture # 9
Dimensional Analysis and
Similitude
Main Topics
• Nature of Dimensional Analysis
• Buckingham Pi Theorem
• Significant Dimensionless Groups in
Fluid Mechanics
• Flow Similarity and Model Studies
Objectives
1. Understand dimensions, units, and
dimensional homogeneity
2. Understand benefits of dimensional analysis
3. Know how to use the method of repeating
variables
4. Understand the concept of similarity and
how to apply it to experimental modeling
Dimensions and Units
•
Review
– Dimension: Measure of a physical quantity, e.g., length,
time, mass
– Units: Assignment of a number to a dimension, e.g., (m),
(sec), (kg)
– 7 Primary Dimensions:
1.
2.
3.
4.
5.
6.
7.
Mass
Length
Time
Temperature
Current
Amount of Light
Amount of matter
m
L
t
T
I
C
N
(kg)
(m)
(sec)
(K)
(A)
(cd)
(mol)
Dimensions and Units
– All non-primary dimensions can be formed
by a combination of the 7 primary
dimensions
– Examples
• {Velocity} m/sec = {Length/Time} = {L/t}
• {Force} N = {Mass Length/Time} = {mL/t2}
Dimensional Homogeneity
• Every additive term in an equation must have the same
dimensions
• Example: Bernoulli equation
– {p} = {force/area}={mass x length/time x 1/length2} = {m/(t2L)}
– {1/2V2} = {mass/length3 x (length/time)2} = {m/(t2L)}
– {gz} = {mass/length3 x length/time2 x length} ={m/(t2L)}
Nondimensionalization of Equations
• To nondimensionalize, for example, the Bernoulli
equation, the first step is to list primary dimensions
of all dimensional variables and constants
{p} = {m/(t2L)}
{g} = {L/t2}
{} = {m/L3}
{z} = {L}
{V} = {L/t}
– Next, we need to select Scaling Parameters. For this
example, select L, U0, 0
Nature of Dimensional Analysis
Example: Drag on a Sphere
 Drag depends on FOUR parameters:
sphere size (D); speed (V); fluid density (); fluid
viscosity (m)
 Difficult to know how to set up experiments to
determine dependencies
 Difficult to know how to present results (four
graphs?)
Nature of Dimensional Analysis
Example: Drag on a Sphere
 Only one dependent and one independent
variable
 Easy to set up experiments to determine
dependency
 Easy to present results (one graph)
Nature of Dimensional Analysis
Buckingham Pi Theorem
• Step 1:
List all the parameters involved
Let n be the number of parameters
Example: For drag on a sphere, F, V, D, , m,
& n=5
• Step 2:
Select a set of primary dimensions
For example M (kg), L (m), t (sec).
Example: For drag on a sphere choose MLt
Buckingham Pi Theorem
• Step 3
List the dimensions of all parameters
Let r be the number of primary dimensions
Example: For drag on a sphere r = 3
Buckingham Pi Theorem
• Step 4
Select a set of r dimensional parameters that
includes all the primary dimensions
Example: For drag on a sphere (m = r = 3)
select ϱ, V, D
Buckingham Pi Theorem
• Step 5
Set up dimensionless groups πs
There will be n – m equations
Example: For drag on a sphere
Buckingham Pi Theorem
• Step 6
Check to see that each group obtained is dimensionless
Example: For drag on a sphere
Π2 =
Π2
Re = ϱVD / μ
Significant Dimensionless Groups in
Fluid Mechanics
• Reynolds Number
Mach Number
Significant Dimensionless Groups in
Fluid Mechanics
• Froude Number
Weber Number
Significant Dimensionless Groups in
Fluid Mechanics
• Euler Number
Cavitation Number
Dimensional analysis
• Definition : Dimensional analysis is a process of formulating fluid mechanics problems in
in terms of non-dimensional variables and parameters.
• Why is it used :
• Reduction in variables ( If F(A1, A2, … , An) = 0, then f(P1, P2, … Pr < n) = 0,
where, F = functional form, Ai = dimensional variables, Pj = non-dimensional
parameters, m = number of important dimensions, n = number of dimensional variables, r
= n – m ). Thereby the number of experiments required to determine f vs. F is reduced.
• Helps in understanding physics
• Useful in data analysis and modeling
• Enables scaling of different physical dimensions and fluid properties
Example
Drag = f(V, L, r, m, c, t, e, T, etc.)
From dimensional analysis,
Vortex shedding behind cylinder
Examples of dimensionless quantities : Reynolds number, Froude
Number, Strouhal number, Euler number, etc.
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Similarity and model testing
• Definition : Flow conditions for a model test are completely similar if all relevant
dimensionless parameters have the same corresponding values for model and prototype.
• Pi model = Pi prototype i = 1
• Enables extrapolation from model to full scale
• However, complete similarity usually not possible. Therefore, often it is necessary to
use Re, or Fr, or Ma scaling, i.e., select most important P and accommodate others
as best possible.
• Types of similarity:
• Geometric Similarity : all body dimensions in all three coordinates have the same
linear-scale ratios.
• Kinematic Similarity : homologous (same relative position) particles lie at homologous
points at homologous times.
• Dynamic Similarity : in addition to the requirements for kinematic similarity the model
and prototype forces must be in a constant ratio.
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Dimensional Analysis and Similarity
• Geometric Similarity - the model must be the
same shape as the prototype. Each dimension
must be scaled by the same factor.
• Kinematic Similarity - velocity as any point
in the model must be proportional
• Dynamic Similarity - all forces in the model
flow scale by a constant factor to
corresponding forces in the prototype flow.
• Complete Similarity is achieved only if all 3
conditions are met.
Dimensional Analysis and Similarity
• Complete similarity is ensured if all independent P
groups are the same between model and prototype.
• What is P?
– We let uppercase Greek letter P denote a nondimensional
parameter, e.g.,Reynolds number Re, Froude number Fr,
Drag coefficient, CD, etc.
• Consider automobile experiment
• Drag force is F = f(V,  m, L)
• Through dimensional analysis, we can
reduce the problem to
Flow Similarity and Model Studies
• Example: Drag on a Sphere
Flow Similarity and Model Studies
• Example: Drag on a Sphere
For dynamic similarity …
… then …
Flow Similarity and Model Studies
• Scaling with Multiple Dependent Parameters
Example: Centrifugal Pump
Pump Head
Pump Power
Similitude-Type of Similarities
• Geometric Similarity: is the similarity of shape.
Lp
Lm



Bp
Bm

Dp
Dm
 Lr
Where: Lp, Bp and Dp are Length, Breadth, and diameter of
prototype and Lm, Bm, Dm are Length, Breadth, and
diameter of model.
Lr= Scale ratio
Similitude-Type of Similarities
• Kinematic Similarity: is the similarity of motion.
Vp1
Vm1



Vp 2
Vm 2
 Vr ;
a p1
am1

ap2
am 2
 ar
Where: Vp1& Vp2 and ap1 & ap2 are velocity and
accelerations at point 1 & 2 in prototype and Vm1& Vm2 and
am1 & am2 are velocity and accelerations at point 1 & 2 in
model.
Vr and ar are the velocity ratio and acceleration ratio
Similitude-Type of Similarities
• Dynamic Similarity: is the similarity of forces.
 Fi  p  Fv  p  Fg  p


 Fi m  Fv m  Fg m


 Fr
Where: (Fi)p, (Fv)p and (Fg)p are inertia, viscous and
gravitational forces in prototype and (Fi)m, (Fv)m and (Fg)m
are inertia, viscous and gravitational forces in model.
Fr is the Force ratio
Flow Similarity and Model Studies
• Scaling with Multiple Dependent Parameters
Example: Centrifugal Pump
Head Coefficient
Power Coefficient
Flow Similarity and Model Studies
• Scaling with Multiple Dependent Parameters
Example: Centrifugal Pump
(Negligible Viscous Effects)
If …
… then …
Flow Similarity and Model Studies
• Scaling with Multiple Dependent Parameters
Example: Centrifugal Pump
Specific Speed
Types of forces encountered in fluid Phenomenon
• Inertia Force, Fi: = mass X acceleration in the flowing fluid.
• Viscous Force, Fv: = shear stress due to viscosity X surface
area of flow.
• Gravity Force, Fg: = mass X acceleration due to gravity.
• Pressure Force, Fp: = pressure intensity X C.S. area of flowing
fluid.
Dimensionless Numbers
• These are numbers which are obtained by dividing the
inertia force by viscous force or gravity force or pressure
force or surface tension force or elastic force.
• As this is ratio of once force to other, it will be a
dimensionless number. These are also called nondimensional parameters.
• The following are most important dimensionless numbers.
–
–
–
–
Reynold’s Number
Froude’s Number
Euler’s Number
Mach’s Number
Dimensionless Numbers
• Reynold’s Number, Re: It is the ratio of inertia force to the viscous force of flowing
fluid.
Velocity
Volume

. Velocity
Fi
Time
Time
Re 


Fv Shear Stress. Area Shear Stress. Area
 QV
.
 AV .V  AV .V VL VL





du
V
 .A
m

m .A m .A
dy
L
Mass.

Froude’s Number, Fe: It is the ratio of inertia force to the gravity
force of flowing fluid.
Fe 
Velocity
Volume

. Velocity
Fi
Time
Time


Fg
Mass. Gavitational Acceleraion
Mass. Gavitational Acceleraion
Mass.
 QV
.
 AV .V
V2
V




Volume.g
 AL.g
gL
gL
Dimensionless Numbers
• Eulers’s Number, Re: It is the ratio of inertia force to the pressure force
of flowing fluid.
Velocity
Fi
Time 

Fp
Pr essure. Area
Eu 


Mass.
 QV
.
P. A

 AV .V
P. A

V2

P/
Volume
. Velocity
Time
Pr essure. Area
V
P/
•Mach’s Number, Re: It is the ratio of inertia force to the
elastic force of flowing fluid.
M

Fi

Fe
 Q.V
K .A

Velocity
Time

Elastic Stress. Area
Mass.
 AV .V
K .A
Where : C  K / 

 L2V 2
KL2

Volume
. Velocity
Time
Elastic Stress. Area

V
V

K/ C