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LITTORAL CÔTE D’OPALE
Fluid MechanicsChapter 1 : Similitude
Mathieu Bardoux
IUT du Littoral Côte d’OpaleDépartement Génie Thermique et Énergie
2nd year
Objectives of similitude models
Summary
1 Objectives of similitude models
2 Dimensional analysis
3 Similitude conditions
4 Dimensional formula
5 Complete and partial similarities
Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 2nd year 2 / 32
Objectives of similitude models
Similitude modela powerful tool for engineering
I Testing of a design prior to building⇒ small scale modelI Validation of theoretical models
I Fluid mechanics⇒ complex fluid dynamics problems⇒ lack ofreliable calculations or computer simulations
Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 2nd year 3 / 32
Objectives of similitude models
Similitude modela powerful tool for engineering
I Testing of a design prior to building⇒ small scale modelI Validation of theoretical modelsI Fluid mechanics⇒ complex fluid dynamics problems⇒ lack of
reliable calculations or computer simulations
Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 2nd year 3 / 32
Dimensional analysis
Summary
1 Objectives of similitude models
2 Dimensional analysis
3 Similitude conditions
4 Dimensional formula
5 Complete and partial similarities
Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 2nd year 4 / 32
Dimensional analysis
Dimensional analysis : for what purpose ?
Never do any calculation unless you already know the result.
Find relations between measurable quantities in variousphysical phenomena
⇒ search for homogeneous relations
Compare experiences conducted under different conditions
⇒ group variables into dimensionless numbers
Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 2nd year 5 / 32
Dimensional analysis
Dimensional analysis : for what purpose ?
Never do any calculation unless you already know the result.
Find relations between measurable quantities in variousphysical phenomena
⇒ search for homogeneous relationsCompare experiences conducted under different conditions
⇒ group variables into dimensionless numbers
Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 2nd year 5 / 32
Dimensional analysis
Dimensional analysis : for what purpose ?
Never do any calculation unless you already know the result.
Find relations between measurable quantities in variousphysical phenomena⇒ search for homogeneous relations
Compare experiences conducted under different conditions
⇒ group variables into dimensionless numbers
Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 2nd year 5 / 32
Dimensional analysis
Dimensional analysis : for what purpose ?
Never do any calculation unless you already know the result.
Find relations between measurable quantities in variousphysical phenomena⇒ search for homogeneous relations
Compare experiences conducted under different conditions
⇒ group variables into dimensionless numbers
Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 2nd year 5 / 32
Dimensional analysis
Dimensional analysis : for what purpose ?
Never do any calculation unless you already know the result.
Find relations between measurable quantities in variousphysical phenomena⇒ search for homogeneous relations
Compare experiences conducted under different conditions⇒ group variables into dimensionless numbers
Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 2nd year 5 / 32
Dimensional analysis
What is a dimension ?
Common language :Dimension ≈ SizeMy car has huge dimensions
Physicist language :Dimension , Size
Dimension ≈ Nature of a physical quantity
Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 2nd year 6 / 32
Dimensional analysis
What is a dimension ?
Common language :Dimension ≈ SizeMy car has huge dimensions
Physicist language :Dimension , Size
Dimension ≈ Nature of a physical quantity
Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 2nd year 6 / 32
Dimensional analysis
What is a dimension ?
Common language :Dimension ≈ SizeMy car has huge dimensions
Physicist language :Dimension , SizeDimension ≈ Nature of a physical quantity
Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 2nd year 6 / 32
Dimensional analysis
What is a dimension ?
Some dimensions : length, volume, mass, power, speed. . .
Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 2nd year 7 / 32
Dimensional analysis
What is a dimension ?
900kg
200hp
500L
1,70m80km/h
Some dimensions : length, volume, mass, power, speed. . .
Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 2nd year 7 / 32
Dimensional analysis
Dimension , unit
6ft
5yd
20in
Length units: meter, inch, yard, parsec, toise, miles, kilometers. . .
Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 2nd year 8 / 32
Dimensional analysis
Dimensional analysis : basics
I In physics, each measurable quantity is associated to adimension (which is distinct from the unit in which it is given)
I Every equation must respect dimensional homogeneityI Every physical quantity derive from a couple of base quantities
Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 2nd year 9 / 32
Dimensional analysis
Dimensional analysis : basics
I In physics, each measurable quantity is associated to adimension (which is distinct from the unit in which it is given)
I Every equation must respect dimensional homogeneity
I Every physical quantity derive from a couple of base quantities
Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 2nd year 9 / 32
Dimensional analysis
Dimensional analysis : basics
I In physics, each measurable quantity is associated to adimension (which is distinct from the unit in which it is given)
I Every equation must respect dimensional homogeneityI Every physical quantity derive from a couple of base quantities
Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 2nd year 9 / 32
Dimensional analysis
Rayleigh’s method of dimensional analysisPrinciple
Let’s say a physical phenomenon: what is the law that governs it?
I draw up an inventory of all the independent variables involved inthe phenomenon
I write the (algebraic) law as a product of these variables to acertain power
I write the dimensions of all these variables as a combination ofbase quantities
I combine them in a dimensionaly homogeneous equation
Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 2nd year 10 / 32
Dimensional analysis
Rayleigh’s method of dimensional analysisPrinciple
Let’s say a physical phenomenon: what is the law that governs it?
I draw up an inventory of all the independent variables involved inthe phenomenon
I write the (algebraic) law as a product of these variables to acertain power
I write the dimensions of all these variables as a combination ofbase quantities
I combine them in a dimensionaly homogeneous equation
Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 2nd year 10 / 32
Dimensional analysis
Rayleigh’s method of dimensional analysisPrinciple
Let’s say a physical phenomenon: what is the law that governs it?
I draw up an inventory of all the independent variables involved inthe phenomenon
I write the (algebraic) law as a product of these variables to acertain power
I write the dimensions of all these variables as a combination ofbase quantities
I combine them in a dimensionaly homogeneous equation
Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 2nd year 10 / 32
Dimensional analysis
Rayleigh’s method of dimensional analysisPrinciple
Let’s say a physical phenomenon: what is the law that governs it?
I draw up an inventory of all the independent variables involved inthe phenomenon
I write the (algebraic) law as a product of these variables to acertain power
I write the dimensions of all these variables as a combination ofbase quantities
I combine them in a dimensionaly homogeneous equation
Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 2nd year 10 / 32
Dimensional analysis
Rayleigh’s method of dimensional analysisPrinciple
Let’s say a physical phenomenon: what is the law that governs it?
I draw up an inventory of all the independent variables involved inthe phenomenon
I write the (algebraic) law as a product of these variables to acertain power
I write the dimensions of all these variables as a combination ofbase quantities
I combine them in a dimensionaly homogeneous equation
Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 2nd year 10 / 32
Dimensional analysis
Rayleigh’s method of dimensional analysisSimple gravity pendulum
#»g
m
l
What is the value of T ?
I draw up an inventory of all the independent variables involved inthe phenomenon
Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 2nd year 11 / 32
Dimensional analysis
Rayleigh’s method of dimensional analysisSimple gravity pendulum
#»g
m
l
What is the value of T ?
I draw up an inventory of all the independent variables involved inthe phenomenon
Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 2nd year 11 / 32
Dimensional analysis
Rayleigh’s method of dimensional analysisSimple gravity pendulum
#»g
m
l
What is the value of T ?
I draw up an inventory of all the independent variables involved inthe phenomenon
Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 2nd year 11 / 32
Dimensional analysis
Rayleigh’s method of dimensional analysisSimple gravity pendulum
I draw up an inventory of all the independent variables involved inthe phenomenon
m , l ,g
I write the (algebraic) law as a product of these variables to acertain power
I write the dimensions of all these variables as a combination ofbase quantities
I combine them in a dimensionaly homogeneous equation
Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 2nd year 12 / 32
Dimensional analysis
Rayleigh’s method of dimensional analysisSimple gravity pendulum
I draw up an inventory of all the independent variables involved inthe phenomenon
I write the (algebraic) law as a product of these variables to acertain power
T = mαlβgγ
I write the dimensions of all these variables as a combination ofbase quantities
I combine them in a dimensionaly homogeneous equation
Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 2nd year 12 / 32
Dimensional analysis
Rayleigh’s method of dimensional analysisSimple gravity pendulum
I draw up an inventory of all the independent variables involved inthe phenomenon
I write the (algebraic) law as a product of these variables to acertain power
I write the dimensions of all these variables as a combination ofbase quantities
[T ] = T [m] = M [l ] = L [g] = L ·T−2
I combine them in a dimensionaly homogeneous equation
Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 2nd year 12 / 32
Dimensional analysis
Rayleigh’s method of dimensional analysisSimple gravity pendulum
I draw up an inventory of all the independent variables involved inthe phenomenon
I write the (algebraic) law as a product of these variables to acertain power
I write the dimensions of all these variables as a combination ofbase quantities
I combine them in a dimensionaly homogeneous equation
T = MαLβ(LT−2)γ
Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 2nd year 12 / 32
Dimensional analysis
Rayleigh’s method of dimensional analysisSimple gravity pendulum
I draw up an inventory of all the independent variables involved inthe phenomenon
I write the (algebraic) law as a product of these variables to acertain power
I write the dimensions of all these variables as a combination ofbase quantities
I combine them in a dimensionaly homogeneous equation
T = MαLβ(LT−2)γ
α = 0,γ = −12,β =
12
Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 2nd year 12 / 32
Dimensional analysis
Rayleigh’s method of dimensional analysisSimple gravity pendulum
I draw up an inventory of all the independent variables involved inthe phenomenon
I write the (algebraic) law as a product of these variables to acertain power
I write the dimensions of all these variables as a combination ofbase quantities
I combine them in a dimensionaly homogeneous equation
T =
√lg
Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 2nd year 12 / 32
Dimensional analysis
Dimensionless quantitiesConcept of invariance
These rectangles are of different sizes
I Lenght (dimensional) changeI Proportions (dimensionless) remain the same
Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 2nd year 13 / 32
Dimensional analysis
Dimensionless quantitiesConcept of invariance
These rectangles are identical
I Lenght (dimensional) changeI Proportions (dimensionless) remain the same
Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 2nd year 13 / 32
Dimensional analysis
Dimensionless quantitiesConcept of invariance
These rectangles are of different sizes, yet identical
I Lenght (dimensional) changeI Proportions (dimensionless) remain the same
Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 2nd year 13 / 32
Dimensional analysis
Buckingham π theoremStatement
Theorem
I If there is a physically meaningful equation involving a certainnumber n of physical variables, depending on k base quantities,then the original equation can be rewritten in terms of a set ofp = n − k dimensionless parameters π1, π2, . . . , πp constructedfrom the original variables.
I f(x1,x2, . . .xn) = 0 becomes :
φ(π1,π2, . . .πp) = 0
where π1,π2, . . . ,πp are dimensionless independant parameters,functions of the variables (x1,x2, . . .xn).
The number of variables has therefore decreased by k ,simplifying the study of the phenomenon.
Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 2nd year 14 / 32
Dimensional analysis
Buckingham π theoremApplication : drag force
A sphere in movement through a viscous fluid :
I Variables : D , v , ρ, µ, FI 5-variable equation : f(D ,v ,ρ,µ,F) = 0I n = 5 variables dépending on k = 3 base quantitiesI There is a relation of p = n − k = 2 variables :
φ(π1,π2) = 0
Find π1 and π2
Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 2nd year 15 / 32
Dimensional analysis
Buckingham π theoremApplication : drag force
Dimensional variables in the initial equation :
D = L
v = L ·T−1
ρ = M · L−3
µ = M · L−1 ·T−1
F = M · L ·T−2
π1 =F
ρD 2v2π2 =
ρvDµ
π1 is called drag coefficient (Cd ), π2 is calledReynolds number (Re).
Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 2nd year 16 / 32
Dimensional analysis
Buckingham π theoremApplication : drag force
Dimensional variables in the initial equation :
D = L
v = L ·T−1
ρ = M · L−3
µ = M · L−1 ·T−1
F = M · L ·T−2
π1 =F
ρD 2v2π2 =
ρvDµ
π1 is called drag coefficient (Cd ), π2 is calledReynolds number (Re).
Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 2nd year 16 / 32
Dimensional analysis
Buckingham π theoremApplication : drag force
Dimensional variables in the initial equation :
D = L
v = L ·T−1
ρ = M · L−3
µ = M · L−1 ·T−1
F = M · L ·T−2
π1 =F
ρD 2v2π2 =
ρvDµ
π1 is called drag coefficient (Cd ), π2 is calledReynolds number (Re).
Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 2nd year 16 / 32
Dimensional analysis
Buckingham π theoremApplication : drag force
Every cases relating to the drag of a sphere in a viscous fluide arereduced to a single curve Cd = f(Re) !
102 104 106103 105 107
0.1
0.5
1.0
1.5
Re
Cd
Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 2nd year 17 / 32
Similitude conditions
Summary
1 Objectives of similitude models
2 Dimensional analysis
3 Similitude conditions
4 Dimensional formula
5 Complete and partial similarities
Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 2nd year 18 / 32
Similitude conditions
Similitude conditions
Are the measurements on scale models transferable to the prototype?
I Yes, if and only if both configurations check the similarityconditions.
I Three conditions :Geometric similarityKinematic similarityDynamic similarity
Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 2nd year 19 / 32
Similitude conditions
Geometric similarity
I Two geometrical objects are called similar if they both have thesame shape. Thus one can be obtained from the other byuniformly scaling.
A2A1
B2
B1
C2
C1
I The ratio of two homologous distances, connecting twohomologous points, has a constant value : this is the geometric
similarity scale L ∗ =L1L2
.
I Base quantity : L.
Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 2nd year 20 / 32
Similitude conditions
Geometric similarity
I Two geometrical objects are called similar if they both have thesame shape. Thus one can be obtained from the other byuniformly scaling.
A2A1
B2
B1
C2
C1
I The ratio of two homologous distances, connecting twohomologous points, has a constant value : this is the geometric
similarity scale L ∗ =L1L2
.
I Base quantity : L.
Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 2nd year 20 / 32
Similitude conditions
Geometric similarityIs it enough ?
Photo L. Shyamal, licence cc-by-sa-2.5
Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 2nd year 21 / 32
Similitude conditions
Geometric similarityIs it enough ?
Photo L. Shyamal, licence cc-by-sa-2.5
Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 2nd year 21 / 32
Similitude conditions
Kinematic similarity
I Two geometrically similar systems are also cinematically similarif two homologous particles occupy homologous positions athomologous times.
I In this case, velocity vectors and acceleration vectors athomologous points will also have homologous modules anddirections in homologous times.
I Fluide streamlines are then similar.I Base quantities : L et T .
L ∗ et T ∗ are different in general case !
Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 2nd year 22 / 32
Similitude conditions
Dynamic similarity
I Two systems, geometrically and cinematically similar, are alsodynamically similar if their homologous parts are subjected tohomologous force systems, at homologous times.
I The similarity is complete if the ratios between all1 homologousforces are equal.
I Ratio of inertia forces : If there is geometric and kinematicsimilarity, there is also dynamic similarity for inertial forces, if themass distribution is similar (ρ∗ = ρ1ρ2 = cste).
I Base quantities : L , M & T .
1viscosity, pressure, gravity, surface tension, elasticity. . .Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 2nd year 23 / 32
Similitude conditions
Dynamic similarity
Ratio of inertial forces to pressure forces (Euler number):
mapS
=(ρL 3)(L /T 2)
pL 2=
(ρL 2)(L 2/T 2)pL 2
=ρL 2v2
pL 2=ρv2
p= Eu
Ratio of inertial forces to viscosity forces:
ma
µdvdz L2=
ρL 3LT−2
µLT−1L−1L 2=ρvLµ
= Re
Viscous forces dominate at low Reynolds numbers, inertial forcesdominate at high Reynolds numbers.
Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 2nd year 24 / 32
Similitude conditions
Dynamic similarity
Ratio of inertial forces to gravity forces (Froude number):
mamg
=ρL 2v2
ρL 3g=
v2
Lg= Fr2
Ratio of inertial forces to elasticity forces (Mach number):
maES
=ρL 2v2
EL 2=ρv2
E=Ma2
This has a major role in the flow of compressible fluids.Ratio of inertial forces to surface tension forces (Weber number):
maσL
=ρL 2v2
σL=ρLv2
σ=We
Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 2nd year 25 / 32
Dimensional formula
Summary
1 Objectives of similitude models
2 Dimensional analysis
3 Similitude conditions
4 Dimensional formula
5 Complete and partial similarities
Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 2nd year 26 / 32
Dimensional formula
Dimensional homogeneity
I Let’s be G , derived from base quantities L , M & T ; its scale ratiois :
G ∗ =G1G2
= f(L ∗,M ∗,T ∗)
I f(L ∗,M ∗,T ∗) is G ’s dimensional formula.I If scaling factor G ∗ is 1, then G is dimensionless, i.e independant
of any unit system.
Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 2nd year 27 / 32
Dimensional formula
Similitude of physical variables
I Let’s be u , v , unknown measurable values, functions of threemeasureable values a , b , c : u = f1(a ,b ,c) et v = f2(a ,b ,c)
I f1 et f2 do not need to be know, as long as they existI There is similarity, if one can find a∗, b ∗, c∗, scaling factors
different from 0 and from∞ so that : uu∗ = f1(aa∗,bb ∗,cc∗) etvv∗ = f2(aa∗,bb ∗,cc∗)
I Scaling factors a∗, b ∗, c∗ must satisfy some relations, whichcontitute similarity conditions.
I Relations between unknown factors u∗, v∗ and a∗, b ∗, c∗ are theresults of similitude.
I This can be obtained without knowledge of f1 nor f2.
Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 2nd year 28 / 32
Complete and partial similarities
Summary
1 Objectives of similitude models
2 Dimensional analysis
3 Similitude conditions
4 Dimensional formula
5 Complete and partial similarities
Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 2nd year 29 / 32
Complete and partial similarities
Complete similarity
I Dimensional ananlyse of independant variables lead to a law ofthe following form :
f(π1,π2, . . .πp) = 0
I If p −1 dimensionless products are identical for two flows, thenthe last one is identical too.
I Equality of p −1 dimensionless products constitutes completesimilitude condition between two flows.
Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 2nd year 30 / 32
Complete and partial similarities
Partial similarity
I In practice, it is rarely possible to achieve the complete similaritycondition. It is rare to obtain the equality of every πcharacterizing the phenomenon.
I One then tries to achieve a limited similarity by neglecting thenumbers π whose influence on the phenomenon studied is theweakest. This is an approximation that leads to a certain degreeof uncertainty.
Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 2nd year 31 / 32
Complete and partial similarities
Conclusion
The advantages of dimensional analysis:I Predict the form of an equationI Check the validity of a resultI Simplify the relationships between many variablesI Study complex phenomena with model systems
Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 2nd year 32 / 32
Objectives of similitude modelsDimensional analysisSimilitude conditionsDimensional formulaComplete and partial similarities