Introduction

Often model size is specified so as to make testing possible leaving the questions -
What experimental variables should be included in the experimental plan?
What should be the values of the the experimental variables so that the full-size situation is accurately represented?

A useful concept to start answering such questions is -
Starting with the most general belief that the model-size and full-size situations are different, quantitative system comparisons have to be based on quantities that are qualitatively similar in both systems. Pure numbers are such a quantity. An underlying concept for model and experiment scaling is to compare the values of pure numbers or unit-less or dimensionless quantities for the model-size and full size objects or processes.

Calling the pure number/unit-less quantity a similarity parameter there are two more general questions that arise naturally -
Given the many physical quantities common to most systems, what are the similarity parameters?
The expectation is that there are many similarity parameters, and so which should be used to describe the correspondence between model-size and full-size behavior?

Many terms have been used and precise definitions of some of them and some others will be useful below.
Physical quantity
Dimension
Primary quantity
Unit
Unit-less
Dimensionless
Similarity
Similarity parameter - the dimensionless quantity selected to describe the correspondence between the behaviors of model-size and fill-size objects.

Buckingham Pi Theorem

For n physical quantities q_i that describe a system,
and if the equation f ( q_i) = 0 is the only relation between the q's

and if m is the minimum numbers of primary quantities needed to define the dimensions of the q's

then the n dimensional physical quantities can be combined into k independent unit-less products, p, of the q's such that the equation above

The immediately useful result is

Defining n physical quantities of interest that take m primary dimensional quantities to define, there are k = n - m independent unit-less or dimensionless products of the p's and these are similarity parameters that can be used for model scaling and experiments.

In testing of a model beam for studying the behavior of a full-size beam
the physical quantities of interest could be q₁ = beam length = L, q₂ = beam width = b, q₃ = beam height = h, q₄ = beam deflection = d, q₅ = applied load = P
the primary quantities are the length dimension, [L], the time dimension [t], the mass dimension [m]

the number of independent similarity parameters = k = n - m = 5 - 2 = 2

In practice the similarity parameters are formulated based on
- the intent of the modeling and experiments,
- insight into system behavior
- the equation describing system behavior, if they are known

If in modeling beam bending interest is with beam deflection
the intuitive modeling parameters d / L, d / b, h / L might be of interest
as well as use of the deflection-load equation d = P L³ / C E I

rearranging the load-deflection relation to d / L = P L² / C E I
enables comparison of model and full-size beam behavior

( d / L )_model = ( d / L )
or
( P L² / C E I )_m = P L² / C E I

Scaling - Size Scaling, Behavior Scaling

Not to bring up the old joke of mathematical arguments starting with "Lets consider (some completely unrealistic mathematical model of a physical situation)", but let's consider a spherical elephant that we want to model in our small laboratory. A handy size model of the spherical elephant is a spherical mouse.

The behavior of interest has to be clearly defined. Let's cast our consideration in terms of temperature. That is, we want to know how to scale the elephant so that the system behavior in terms of temperature is the same for the full-size and model-size systems. And we'll say that the temperature of the creature is proportional to the ratio of the heat generation rate to heat dissipation rate,i.e.,

T = Q_generation / Q_dissipation

for our spherical elephant the volume V is (4/3) Pi r³
the heat generation rate is q / unit time
and the heat generated in unit time is Q_g = q V = (4/3) Pi q r³

and for the model size elephant Q_gm = (4/3) Pi q r_m³

for the full size elephant the surface area is A = 4 Pi r²
for the model size elephant the surface area is A_m = 4 Pi r_m²

with heat dissipation rate of k per unit area per unit time
heat dissipated from the elephant is Q_d = k A = k 4 Pi r²
and from the model is Q_dm = k A = k 4 Pi r_m²

The temperatures of the elephant and model (ratios of the heat generated to heat dissipated) are

T = q r / 3 k
T_m = q r_m / 3 k

Now to the real point of all this. For the same metabolic rate and surface heat transfer characteristics, the ratio of model size system temperature to full size system temperature is

T_m / T = r_m / r

T_m = T ( r_m / r )

The point is that
- system behavior can differ with scaling in a different way than linear size scaling.

Or, for our particular example,
- geometric scaling is linear and preserves appearance,
- system behavior depends on volume that is described with three dimensions and surface area that exists in two dimensions.

Finally, and repeating,
- if we want to model system behavior (thermal effects) using simple linear size scaling may not be accurate.

We can look to how to bring the temperatures into correspondence by considering the heat generation rate as metabolic rate does vary with animal size - but not linearly. Named after the biologist Max Kleiber, Kleiber's Law says metabolic rate scales as (body mass)^3/4.

Scaling - Geometric Scaling, Dynamic Scaling

Atraditional way to introduce similitude is by describing the modeling of ships. It's traditional since much of the early development of modeling theory was in ship modeling applications and it's also traditional in that it's used so often. It's clear that running model tests in a towing tank is easier than running force measurement tests on full size ships in full size bodies of water.

Dynamically similar points are points at corresponding locations and at corresponding times that have proportional velocities and proportional accelerations.

For our purposes the total force acting on a ship will be the sum of the friction force and the force associated with the energy dissipated when waves are created.

F_total = F_friction + F_wave

Experimental observations of full size and model size ships showed that the number of waves along the hull lengths of different size ships are the same when the values of { Velocity / (Length) }^1/2 are the same. That is for model and full size similitude

{ V / ( L ) }^1/2 ]_m = { V / ( L ) }^1/2 ]

Using V / ( g L )^1/2 as our similarity parameter (sample units are m / { ( m/sec² ) m }^1/2 ).

Similitude, in this case, is described by.

[ V / ( g L )^1/2 ]_m = [ V / ( g L )^1/2 ]

L² / V = constant

By way of contrast to the inertial force we can consider the viscous force effects. The Reynold's Number is the ratio of viscous force to inertial force.

Re = V L / µ
µ is dynamic viscosity
using kinematic viscosity nu = µ / density = µ / rho
Re = rho V L / nu

First, making waves - the model velocity is

Fn_model = Fn
( V / {g L}^1/2 )_m = V / {g L}^1/2
V_m = V { L_m / L}^1/2

( V L / nu )_m = V L / nu
running the full size and model ships in the same fluid
V_m = V ( L / L_m )

Can these two model velocities be equal?

V ( L / L_m ) = V { L_m / L}^1/2
only when
L / L_m = 1
The length of the real and model ships have to be equal to include both types of energy dissipation in the model.

Scaling - Beam Bending

The behavior of interest has to be specified and, say, the concern is =-
for similar deflection behavior for a model that is a factor of n smaller than the full size beam, what load should be applied to the model?

The characteristic length of the model beam is L / n. A dimensionless quantity including the deflection, d, is d / l.

d / l = d_m / l_m

d = P l³ / C E I

P l² / C E I = P_m l_m² / C E_m I_m
for the same loading situations and elastic modulus
P l² / I = P_m l_m² / I_m

l_m = l / n
I_m = b_m h_m³ / 12
I_m = ( b / n ) ( h / n )³ / 12
I_m = ( 1 / 12 )( 1 / n⁴ )( b h³ / 12 )
I_m = I / n⁴

P = P_m n²
P_m = P / n²