Similitude (model)

Resemblance or similarity theory physics is a technical term in physics and refers to a theory in the dimensionless ratios using a physical process ( original ) to a process model (model) is returned. This theory is widely applied both in the light of theoretical considerations and experiments. Typical uses are the fluid mechanics and heat transfer.

The precursor of the similarity theory, the Reynolds similarity law, which was set up in 1883 by Osborne Reynolds and states that the flows at the original and the model run mechanically similar if the Reynolds numbers are the same ( Re).

Introduction

The similarity theory is to form combinations of a known and accessible (model ) system to draw conclusions about a planned and experimentally inaccessible (real ) system, which, for example, larger or smaller, faster or slower or in other dimensions only quantitatively differs from the known system. Where appropriate further boundary conditions must be adhered to so that conclusions are possible. The theory is applied for example in the following cases:

• If the aerodynamics of a new aircraft type is to be investigated and optimized in the wind tunnel, but no wind channel is available, which is large enough to hold the plane in original size. We experimented instead with a smaller model. The similarity theory is concerned with what should be taken so that it is possible to transfer the measurement results from the model test on the scheduled aircraft and its size. As well as what factors those calculated in the model measurements must be converted.
• If a larger aircraft is to be constructed, but are already measured and empirical values ​​from a smaller aircraft model. Based on the known performance data of existing aircraft type allows the similarity theory to extrapolate a design for a larger aircraft.
• When a series of engines will be developed with a gradation of different performance. The similarity theory makes it possible to make a basic design where certain parameters are then systematically varied to achieve the different services. It saves the hassle of having to go through the complete design process for the various power levels each time.

Today's meaning

Since nowadays very powerful computers are available, and many have very complex relationships can be calculated directly. The similarity theory has thus lost some of its significance. About 50 years ago, when such calculations on this scale had not yet possible to put the theory of similarity in many cases the only way, for example, to develop a commercial airliner. This was done by identifying a model empirical data and these were then extrapolated using the theory of similarity to the target system.

Even today the similarity theory has meaning to without complex calculations to estimate trends and limitations, or to develop a sense of dimensions and quantities. Also, the concept of equivalent speed in flying the similarity theory was an important component.

Method

If all dimensionless parameters that describe a physical system between the original model and the same, so it is ensured that the two systems are physically similar in running processes. Results from the model can then be transferred without restriction to the original. From the equality of the dimensionless parameters to requirements arising in the model, always including the geometrical similarity between original and model.

Difficulties start appearing in the selection of appropriate indicators. In addition, can be kept constant often not all dimensionless parameters. In this case, the transferability of results is limited. Nevertheless, the similarity theory can be an important tool to simplify experiments and the derivation of physical relationships.

Example of use

For an action film a scene should be rotated in the derailed a train and crashes off a bridge. Because of limited budgets, the scene is to be simulated with the help of a model train, scale H0, ie 1:87.

Would the scene be easily rotated only in macro setting to simulate a real size, then the train would be unrealistic quickly fall from the bridge. So the scene has to be rotated in slow motion to convey a realistic impression.

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

The physical formula that applies to the case law, is:

And, as is usual, s for distances, ie spatial extent is, and t is the time and g is the acceleration due to gravity.

If we now substitute this formula for reality and model related. The index r is intended to represent reality, the index m for the model. Then we have:

According to the scale of 1:87.

A transformation gives:

Or

One second in the model thus corresponds to 9.3 seconds in reality. Thus, the scene must be around 9 - to 10 -fold are included slow motion to convey a realistic impression of the falling speed of the train.

In this modeling we have assumed that it depends only on the rate of fall, and, for example, influences can be neglected through the air friction while falling. For this example, a valid assumption. In another scenario, such as a parachute jump, however, the air friction would play a central role. Why then more formulas would have to be taken into account. The modeling using the similarity theory is not a procedure that can be executed on a fixed schedule, but requires a basic understanding of the processes and their physical modeling, as well as experience with the expected quantities.

Difficulties and limitations

The theory of similarity, there is no method that can be used in Scheme F. The modeling depends heavily on the question from, and requires knowledge and experience in assessing which variables can be neglected and which must be modeled. In detail, for example, the following problems may occur:

• With appropriate complexity of the compliance of conditions may be necessary, which conflict with each other.
• The similarity rules can specify the material properties, for which it is difficult or impossible to find real materials that have these properties.

• It can be difficult to distinguish be different phenomena against each other. For example, a flow is affected by the shape of an object on the one hand, on the other, of the surface properties, such as roughness and adhesion forces.

Analogies

In physics, there are quite different phenomena that can be described with the same mathematical means, for example,

• Electrical current flow <- > magnetic flux <- > heat conduction.
• Voltage <- > magnetic excitation
• Electric field theory ( eg antennas) <- > fluid mechanics (eg, injection molding).
• Spring -mass system <- > coil - capacitor system.

One speaks in this case of analogies. Thus, as in philosophy, one can also view in physics the physical similarity as a special case of physical analogy.