# Similarity Theory

Similarity theory or similarity physics is a technical term in physics and describes a theory in which a physical process (original) is traced back to a model process (model) with the help of dimensionless key figures . This theory is often applied both against the background of theoretical considerations and experiments. Classic areas of application are fluid dynamics and heat transfer .

A well-known example from the similarity theory is Reynolds' Law of Similarity , which was established by Osborne Reynolds in 1883 and states that the flows on the original and on the model are mechanically similar if the Reynolds numbers ( Re ) match.

## history

Aristotle already considered the behavior of geometrically similar wooden rods against bending. Galileo Galilei investigated the breaking strength of geometrically similar cylinders and speculated about the possible size of animals. Isaac Newton used the concept of mechanical similarity in the movement of bodies in liquids, but it was not until 1847 that J. Bertrand pronounced the principle of mechanical similarity in full rigor and generality.

## introduction

The similarity theory is concerned with drawing conclusions about a planned and experimentally inaccessible (real) system from a known and accessible (model) system. B. larger or smaller, faster or slower or only differs quantitatively from the known system in other dimensions. Additional boundary conditions may have to be observed so that conclusions can be drawn. The theory is applied z. B. in the following cases:

• If the aerodynamics of a new type of aircraft are to be examined and optimized in the wind tunnel, but no wind tunnel is available that is large enough to accommodate the aircraft in its original size. Instead, you experiment with a smaller model. The similarity theory deals with what needs to be taken into account so that it is possible to transfer the measurement results from the model experiment to the planned aircraft and its size. As well as the factors with which the measured variables determined on the model must be converted.
• If a larger aircraft is to be constructed, but measurements and empirical values ​​from a smaller aircraft model are already available. Based on the known performance data of the existing aircraft type, the similarity theory enables a design for a larger aircraft to be extrapolated.
• If a series of motors is to be developed, with a gradation of different performance data. The similarity theory makes it possible to make a basic design in which certain parameters are then systematically varied in order to achieve the various services. This saves the effort of going through the entire construction process for the various performance levels.

## Today's meaning

Since very powerful computers are available nowadays, many very complex relationships can be calculated directly. The similarity theory has lost some of its importance. About 50 years ago, when such calculations were not yet possible to this extent, the similarity theory was often the only possibility, e.g. # 8239; B. to develop an airliner. This was done by determining empirical data on a model and then extrapolating them to the target system with the help of the similarity theory.

Even today, the similarity theory is still important in order to be able to estimate tendencies and limits or to develop a feeling for dimensions and quantities without time-consuming calculations. The similarity theory was also an important part of the equivalent speed in aviation.

## Procedure

If all dimensionless key figures that describe a physical system remain the same between the original and the model, this ensures that the two systems are physically similar in the processes that take place. Results from the model can then be transferred to the original without restriction. The equality of the dimensionless key figures results in requirements for the model, which always include the geometric similarity between the original and the model.

Difficulties arise initially in the selection of suitable key figures. In addition, often not all dimensionless key figures can be kept constant. In this case the transferability of the results is limited. Nevertheless, the similarity theory can be an important tool for simplifying experiments and deriving physical relationships.

## Application example

For an action film, a scene is to be shot in which a train derails and falls from a bridge. Due to the limited budget, the scene is to be recreated with the help of a model railway of nominal size H0 on a scale of 1:87.

If the scene were simply rotated in the macro setting to simulate a real size, then the train would fall unrealistically quickly from the bridge because the gravitational acceleration does not scale with the model - it is constant. The scene must also be shot in slow motion to give a more realistic impression.

The similarity theory provides an answer to the question of how strong the slow motion must be.

The physical formula that applies to the laws of case is:

${\ displaystyle s = {\ frac {1} {2}} gt ^ {2}}$ Where, as is common practice , s stands for distance , i.e. spatial expansion, and t for time and g for acceleration due to gravity .

Let us now relate this formula for reality and model.
The index r should stand for reality, the index m for the model. Then we get:

${\ displaystyle {\ frac {s_ {r}} {s_ {m}}} = {\ frac {{\ frac {1} {2}} gt_ {r} ^ {2}} {{\ frac {1} {2}} gt_ {m} ^ {2}}} = {\ frac {t_ {r} ^ {2}} {t_ {m} ^ {2}}} = {\ frac {87} {1}} }$ corresponding to the scale of 1:87.

A transformation gives:

${\ displaystyle t_ {m} ^ {2} = {\ frac {t_ {r} ^ {2}} {87}}}$ or.

${\ displaystyle t_ {m} = {\ frac {t_ {r}} {\ sqrt {87}}} = {\ frac {t_ {r}} {9.3}}}$ One second in the model corresponds to 9.3 seconds in reality. The scene has to be recorded in around 9 to 10 times slow motion in order to give a realistic impression of the falling speed of the train.

In this modeling, we have assumed that only the speed of fall is important, and B. Influences from air friction when falling can be neglected. A valid assumption for this example. In another scenario, e.g. B. a parachute jump on the other hand, the air friction would play a central role. Therefore further formulas would have to be considered. Modeling with the help of the similarity theory is therefore not a procedure that can be processed according to a fixed scheme, but requires a basic understanding of the processes and their physical modeling, as well as experience with the quantities to be expected.

## Difficulties and limits

The similarity theory is not a procedure that can be applied according to scheme F. The modeling depends strongly on the question and requires knowledge and experience in assessing which quantities can be neglected and which have to be modeled. In detail z. B. the following problems arise:

• Given the corresponding complexity, it may be necessary to comply with conditions that conflict with one another.
• The similarity rules can specify material properties for which it is difficult or impossible to find real materials that have these properties.
Example: If currents play a role, the viscosity of the flowing medium must be increased at the same time as the size is reduced in order to achieve similar conditions. This can e.g. B. can be achieved in ship models by replacing water with an oil of suitable viscosity. However, this changes the buoyancy, so that the weight of the ship models must then also be suitably adapted.
In an experiment in the wind tunnel, air is the flow medium. To increase their viscosity, either the temperature is usually lowered or the pressure is increased. Lowering the temperature can lead to undesirable effects such as icing. Even the pressure in a wind tunnel cannot be increased at will and only at high cost (pressure resistance).
• It can be difficult to differentiate between different phenomena.
For example, a flow is influenced on the one hand by the shape of an object and on the other hand by surface properties such as roughness and adhesive forces.
One must therefore be able to ensure that the influence of one factor is negligible compared to the influence of the other factor; and this both for the model and for the system to be modeled. Otherwise one could observe a certain vortex formation on the model, but one would have to be able to determine which proportions of the observed vortices can be attributed to the shape and which proportion to the surface properties in order to be able to set up the appropriate formulas with which the measurement results on the Target object can be extrapolated.

## Analogies

In physics there are very different phenomena that can be described with the same mathematical means, e.g. B.

• electric current flow <-> magnetic flow <-> heat conduction <-> diffusion.
• electrical voltage <-> magnetic excitation
• electric field theory (e.g. antennas) <-> fluid mechanics (e.g. injection molding).
• Spring-mass system <-> coil-capacitor system.

In this case one speaks of analogies . Just as in philosophy , physical similarity in physics can also be viewed as a special case of physical analogy .