Physical Similarity Principles
Introduction
Laboratory tests are usually carried out under altered conditions of the operating variables from the actual ones in practice. These variables, in case of experiments relating to fluid flow, are pressure, velocity, geometry of the working systems and the physical properties of the working fluid.
The pertinent questions arising out of this situation are:
1. How to apply the test results from laboratory experiments to the actual problems?
2. Is it possible, to reduce the large number of experiments to a lesser one in achieving the same objective?
Answer of the above two questions lies in the principle of physical similarity. This principle is useful for the following cases:
1. To apply the results taken from tests under one set of conditions to another set of conditions
2. To predict the influences of a large number of independent operating variables on the performance of a system from an experiment with a limited number of operating variables.
Concept and Types of Physical Similarity
The primary and fundamental requirement for the physical similarity between two problems is that the physics of the problems must be the same.
For an example, two flows: one governed by viscous and pressure forces while the other by gravity force cannot be made physically similar. Therefore, the laws of similarity have to be sought between problems described by the same physics.
Definition of physical similarity as a general proposition.
Two systems, described by the same physics, operating under different sets of conditions are said to be physically similar in respect of certain specified physical quantities; when the ratio of corresponding magnitudes of these quantities between the two systems is the same everywhere.
In the field of mechanics, there are three types of similarities which constitute the complete similarity between problems of same kind.
Geometric Similarity: If the specified physical quantities are geometrical dimensions, the similarity is called Geometric Similarity,
Kinematic Similarity: If the quantities are related to motions, the similarity is called Kinematic Similarity
Dynamic Similarity: If the quantities refer to forces, then the similarity is termed as Dynamic Similarity.
Geometric Similarity
Geometric Similarity implies the similarity of shape such that, the ratio of any length in one system to the corresponding length in other system is the same everywhere.
This ratio is usually known as scale factor.
Therefore, geometrically similar objects are similar in their shapes, i.e., proportionate in their physical dimensions, but differ in size.
In investigations of physical similarity,
(i) the full size or actual scale systems are known as prototypes
(ii) the laboratory scale systems are referred to as models
(iii) use of the same fluid with both the prototype and the model is not necessary
(iv) Model need not be necessarily smaller than the prototype. The flow of fluid through an injection nozzle or a carburettor, for example, would be more easily studied by using a model much larger than the prototype.
(v) The model and prototype may be of identical size, although the two may then differ in regard to other factors such as velocity, and properties of the fluid.
If l1 and l2 are the two characteristic physical dimensions of any object, then the requirement of geometrical similarity is
(The second suffices m and p refer to model and prototype respectively) where lr is the scale factor or sometimes known as the model ratio. Figure 1 shows three pairs of geometrically similar objects, namely, a right circular cylinder, a parallelepiped, and a triangular prism.
In the entire above cases model ratio is 1/2
Geometric similarity is perhaps the most obvious requirement in a model system designed to correspond to a given prototype system.
A perfect geometric similarity is not always easy to attain. Problems in achieving perfect geometric similarity are:
• For a small model, the surface roughness might not be reduced according to the scale factor (unless the model surfaces can be made very much smoother than those of the prototype). If for any reason the scale factor is not the same throughout, a distorted model results.
• Sometimes it may so happen that to have a perfect geometric similarity within the available laboratory space, physics of the problem changes. For example, in case of large prototypes, such as rivers, the size of the model is limited by the available floor space of the laboratory; but if a very low scale factor is used in reducing both the horizontal and vertical lengths, this may result in a stream so shallow that surface tension has a considerable effect and, moreover, the flow may be laminar instead of turbulent. In this situation, a distorted model may be unavoidable (a lower scale factor”for horizontal lengths while a relatively higher scale factor for vertical lengths. The extent to which perfect geometric similarity should be sought therefore depends on the problem being investigated, and the accuracy required from the solution.
Kinematic Similarity
Kinematic similarity refers to similarity of motion.
Since motions are described by distance and time, it implies similarity of lengths (i.e., geometrical similarity) and, in addition, similarity of time intervals.
If the corresponding lengths in the two systems are in a fixed ratio, the velocities of corresponding particles must be in a fixed ratio of magnitude of corresponding time intervals.
If the ratio of corresponding lengths, known as the scale factor, is lr and the ratio of corresponding time intervals is tr, then the magnitudes of corresponding velocities are in the ratio lr/tr and the magnitudes of corresponding accelerations are in the ratio lr/t2 r.
A well-known example of kinematic similarity is found in a planetarium. Here the galaxies of stars and planets in space are reproduced in accordance with a certain length scale and in simulating the motions of the planets, a fixed ratio of time intervals (and hence velocities and accelerations) is used.
When fluid motions are Kinematiclly similar, the patterns formed by streamlines are geometrically similar at corresponding times.
Since the impermeable boundaries also represent streamlines, Kinematiclly similar flows are possible only past geometrically similar boundaries.
Therefore, geometric similarity is a necessary condition for the kinematic similarity to be achieved, but not the sufficient one.
For example, geometrically similar boundaries may ensure geometrically similar streamlines in the near vicinity of the boundary but not at a distance from the boundary.
Dynamic Similarity
Dynamic similarity is the similarity of forces.
In dynamically similar systems, the magnitudes of forces at correspondingly similar points in each system are in a fixed ratio.
In a system involving flow of fluid, different forces due to different causes may act on a fluid element. These forces are as follows:
According to Newton’s law, the resultant FR of all these forces, will cause the acceleration of a fluid element. Hence
Moreover, the inertia force 
is defined as equal and opposite to the resultant accelerating force 
Therefore Eq. 17.1 can be expressed as
For dynamic similarity, the magnitude ratios of these forces have to be same for both the prototype and the model.
The inertia force is usually taken as the common one to describe the ratios as (or putting in other form we equate the non dimensionalised forces in the two systems)
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