Fluid Mechanics - Chapter 1 : Similitude · Fluid Mechanics Chapter 1 : Similitude Mathieu Bardoux IUT du Littoral Côte d’Opale Département Génie Thermique et Énergie 2nd year

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


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



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


Dimensional analysis

Summary








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





Find relations between measurable quantities in variousphysical phenomena

⇒ search for homogeneous relationsCompare experiences conducted under different conditions






Find relations between measurable quantities in variousphysical phenomena⇒ search for homogeneous relations

Compare experiences conducted under different conditions






Find relations between measurable quantities in variousphysical phenomena⇒ search for homogeneous relations

Compare experiences conducted under different conditions⇒ group variables into dimensionless numbers



What is a dimension ?

Common language :Dimension ≈ SizeMy car has huge dimensions

Physicist language :Dimension , Size

Dimension ≈ Nature of a physical quantity




Common language :Dimension ≈ SizeMy car has huge dimensions

Physicist language :Dimension , SizeDimension ≈ Nature of a physical quantity




Some dimensions : length, volume, mass, power, speed. . .




900kg

200hp

500L

1,70m80km/h

Some dimensions : length, volume, mass, power, speed. . .



Dimension , unit

6ft

5yd

20in

Length units: meter, inch, yard, parsec, toise, miles, kilometers. . .



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





I Every equation must respect dimensional homogeneity

I Every physical quantity derive from a couple of base quantities





I Every equation must respect dimensional homogeneityI Every physical quantity derive from a couple of base quantities



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



Rayleigh’s method of dimensional analysisSimple gravity pendulum

#»g

m

l

What is the value of T ?






m , l ,g









T = mαlβgγ









[T ] = T [m] = M [l ] = L [g] = L ·T−2









T = MαLβ(LT−2)γ








T = MαLβ(LT−2)γ

α = 0,γ = −12,β =

12








T =

√lg



Dimensionless quantitiesConcept of invariance

These rectangles are of different sizes

I Lenght (dimensional) changeI Proportions (dimensionless) remain the same




These rectangles are identical





These rectangles are of different sizes, yet identical




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.



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




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).




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


Similitude conditions

Summary









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



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.



Geometric similarityIs it enough ?

Photo L. Shyamal, licence cc-by-sa-2.5



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 !



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


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.



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


Dimensional formula

Summary







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.


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.


Complete and partial similarities

Summary








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.



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.



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


Objectives of similitude modelsDimensional analysisSimilitude conditionsDimensional formulaComplete and partial similarities

Fluid Mechanics - Chapter 1 : Similitude · Fluid Mechanics Chapter 1 : Similitude Mathieu Bardoux IUT du Littoral Côte d’Opale Département Génie Thermique et Énergie 2nd year

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