Geometric Similarity

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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,
·          the full size or actual scale systems are known as prototypes
·          the laboratory scale systems are referred to as models
·          use of the same fluid with both the prototype and the model is not necessary
·           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.
·          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 land l2 are the two characteristic physical dimensions of any object, then the requirement of geometrical similarity is
     (model ratio)
(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 5.1 shows three pairs of geometrically similar objects, namely, a right circular cylinder, a parallelopiped, and a triangular prism.
Fig 17.1   Geometrically Similar Objects
In all the 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

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