# Similitude

Similitude is a concept that is applied when testing a model and comparing it to a real application or prototype. By definition a model “is a representation of a physical system that may be used to predict the behavior of the system in some desired respect.”   In turn, for similitude to occur the model will have to share geometric similarities with the real application.

### Model Theory

In order to develop a model the theory of dimensional analysis will need to be used.  This is done by defining a set of pi terms.

(Eq 1) $Π_1=Φ(Π_2,Π_3,…,Π_n$

In turn, to define the Pi terms you will need to have general knowledge of the physical phenomenon.  In addition, you will need know what variables are required to create the Pi terms.

Next, in order to relate a prototype to a model a similar mathematical function to equation 1 will need to be written for the model.

(Eq 2) $Π_{1m} = Φ(Π_{2m},Π_{3m},…,Π_{nm})$

These two functions will relate to one another as long as the phenomenon occurs for both the prototype and the model.  Hence, similitude will occur.  When similitude occurs the models pi terms, represented by subscript $m$, will be the same as the prototypes pi terms.

$$Π_{2m}=Π_2$$

$$Π_{3m}=Π_3$$

$$:$$

$$Π_{nm}=Π_n$$

As a result, since $Φ$ is the same between the model and prototype than it can be assumed that the following is also true.

(Eq 3) $Π_{1m}=Π_1$

Equation 3 is called the prediction equation.  It states the the measured value $Π_{1m}$ is equal to $Π_1$ creating similitude.  However, in order for this to occur the other pi terms must be equal as shown above.  These additional pi terms define the model design conditions which is also known as similarity requirements or modeling laws.

### Scales

Sometimes a model and a prototype may not be the in the same scale.  As a result, a ratio of the like quantities will need to be taken in order for similitude to occur.  For example if there were two different length variables between the model and the prototype the following ratio will need to be taken.

$\frac{l_1}{l_2}=\frac{l_{1m}}{l_{2m}}$

In turn, this can rewritten as the following.

$\frac{l_{1m}}{l_1}=\frac{l_{2m}}{l_2}$

This ratio will result in a length scale.  This type of scale can be defined for all variables within a problem such as velocity, density, and viscosity.

### Validation

When a model is created different types of assumptions are made to simplify the problem.  Due to these assumptions there will be some uncertainty in the results obtained from the model design.  As a result, it is always a good idea when possible to check the model experimentally.  This way you will be able to confirm that the model has reached similitude. In turn, if the model at first does not agree with the experimental data, this data can than be used to improve the model for future analysis.

### Distorted Models

Even though the requirement to create a model is straight forward.  There is still the possibility that not all known requirements will be satisfied.  As a result, say that $Π_{2m}≠Π_2$ than it can stand to reason that the prediction equation $Π_1=$Π_{1m} is not correct.  Hence, a distorted model will result since not all of the similarity requirements are satisfied.

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