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**Model Analysis in Fluid Mechanics: Application, Examples, Selection, Similarity, Prototype and Laws. [with solved examples, solutions, formula and equations]**

**Introduction to Model Analysis: **

In recent years, hydraulic model studies are being made in the study and analysis of many problems in fluid mechanics. A hydraulic problem may of course be analysed by analytical methods, but these analytical methods involve a number of approximations and assumptions and hence their applications often become restricted. In many cases, the analytical methods involve highly complicated equations which cannot be solved.

In spite of the vast progress made in the field of fluid mechanics, the solutions to various complex flow patterns cannot be obtained by analytical methods alone. The available analytical methods need many simplification so much so that their applications become not only restricted but also theoretical.

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There are many cases where it is impossible or impracticable to make a satisfactory and complete mathematical analysis of the problems. A hydraulic model may sometimes provide the only means of ascertaining and eliminating certain undesirable conditions.

It is only through model experiments and research that improvements in the existing works, safe and economic design and construction of new works and furtherance of knowledge on many aspects of hydraulic engineering can be effected. These days, model studies have a wide field of applications – dams, rivers and harbours, hydraulic machines, structures, ships, aircraft, seepage problems etc.

In short, model experimentation has become an indispensable tool to a designing engineer.

Model studies are made for two purposes, namely –

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(i) To obtain information about the likely performance of the prototype, and

(ii) To help in the design and to avoid costly mistakes and to evolve an economic solution of a hydraulic problem.

Since, comparatively, it is inexpensive to modify the construction of a model, there is ample scope to try several alternative designs in the model before adopting a final one. Obviously such trials would be excessively costly if they were undertaken with the full scale system. A model study is intended to provide not only qualitative but also quantitative indications of the characteristics of the prototype.

A hydraulic model offers itself as a powerful design tool which establishes a fluid system from the observations on which the performance of the prototype could be inferred. Models are the only recourse to the nearest approach to the solution of many hydraulic problems.

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They are very useful in studying the relative merits of alternative designs. They clearly indicate the weak points of a particular design. Today, the importance of hydraulic models, especially in large projects, has been acknowledged in all quarters.

The idea of using models to study the characteristics of hydraulic structures was conceived by Sir Isaac Newton who stated the principle of similitude in his work Principia. The idea however remained dormant until Froude adopted small scale models to study and obtain satisfactory designs for a ship’s hull.

Today the small scale model is the most powerful instrument for research in all laboratories for studying the hydraulic structures with the principles of hydraulic similitude. Obviously the choice of model scales and the proper interpretation of model test results will depend to a great extent on the experience of the hydraulic engineer, even though the laws that govern the hydraulic models are well established.

**Fields of Application of Model Analysis****: **

It should not be presumed that model studies provide the ready answers to every hydraulic problem. It will not be possible to devise a suitable model test to interpret the results of the model tests, unless the basic theory of the phenomenon under study is understood. It is likely that in some cases it may be impracticable to build a model study in situations where the results can be predicted correctly by theory.

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In spite of these limitations, model tests have proved to be very valuable in a number of cases and hence the use of models in hydraulic engineering is increasing steadily.

**The following are some examples where model studies have been of great value: **

**i. Dams: **

The design of every major dam is checked by model tests before its construction. Besides the model of the dam, all its connected works like spillways, penstocks and gates are studied in order to get detailed information on the flow of water and its effect on the structure. A model study can be helpful in deciding what type of dam may be best suited for any locality and what site can be the best choice for placing a dam.

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Studies of the models of sluice openings in dams to ascertain the best opening for maximum discharge, studies of various profiles of inlet bell mouths and outlets and studies of the behaviours of gates under operating conditions are of great value. In a penstock where a change of cross-section from a rectangle to a circle for the intake from a dam structure, necessary curves are better decided by model studies.

**ii. Rivers and Harbours: **

Hydraulic engineers all over the world are involved in considerable amount of work devoted to the dredging of rivers, straightening of channels, protection of banks and bottoms from erosion, various forms of river control and improvement. Harbours and estuaries pose special problems of model design, as it is necessary to duplicate the natural tidal cycles in the model and this may naturally need elaborate machinery.

Model studies can help in studying the wave action of harbours concerned mainly with the effectiveness of the proposed breakwaters for providing protection from waves, and the extent of damage the waves may inflict on the breakwaters. Model studies of breakwater are very essential as there are a number of instances where the breakwaters have failed to provide the adequate protection anticipated of them.

**iii. Hydraulic Machines: **

Small scale model tests are very useful to obtain performance data for hydraulic turbines and other turbo machines and centrifugal pumps. Model study is a very valuable guide to designers of turbo machines.

**iv. Structures: **

Where prototype testing is difficult or costly, models provide the proper solution. Many structural tests like deflection tests and destructive and non-destructive testing of structures can be satisfactorily performed on models. Photo elastic and analogy methods of testing of models of dams are very common.

**v. Ships: **

Towing models in a canal are helpful in estimating Drag force and wave patterns of naval vessels.

**vi. Seepage Problems: **

It is possible to make studies of seepage flow on geometrically similar sectional models in glass flumes, to find out the uplift pressures on hydraulic structures.

The above are only a few examples of applications of models. It should not be taken for granted that model investigations alone can solve our hydraulic problems. Unless we are able to diagnose the fundamental nature of the problems, the results obtained from the model may be incomplete and may be unreliable. Thus the correct applicability of model technique is a well-balanced, well studied and well done combination of model experimentation and analysis.

Analysis involves factors unsolved due to the ignorance of the experiments, which can be solved with the aid of models. Though a well-designed model enables the hydraulic engineer to solve specific problems like those of river control etc., yet the performance is not always a cent percent miniature of the prototypes, because of what can be called model limitations.

It is for this reason that the results obtained from some model experiments are qualitative only. The design, operation and correct interpretation of results, therefore, is a specialised job, requiring quite an amount of experience and good visualisation of the various assumptions made.

Today model analysis has provided the field engineers with a novel tool for studying the various complicated problems and for fixing and regulating the forces of nature to suit his need by properly designed courses.

**Selection of Scale for a Model:**

Selecting a proper scale is a very important part in planning a model.

**The following are the factors which influence the selection of scale for the models: **

(a) The space available to accommodate the model.

(b) Quantum of water supply available at the model station.

(c) Type of results desired, i.e. qualitative or quantitative.

(d) Expenditure involved.

It is very important to note that the scale of the model be such that the value of the Reynold number in the model within the range of discharge in which it is experimented, is not below the value corresponding to turbulent flow. If, in any case, with a scale assumed, it is found that the flow in the model is not turbulent while the flow in the prototype is turbulent, it will be necessary to make adjustment in the scale.

In such cases we have to adopt distorted models, i.e., models in which the vertical and horizontal scales are not the same. Selection of a proper scale is the primary and essential consideration in order to achieve the desired results, from a study on a hydraulic model, and it is here, mainly, the wide experience of the hydraulic engineer comes into play.

**The scale ratios generally adopted are as follows: **

(i) Dams and spillways — 1/30 to 1/400

(ii) Head works, gates, canals — 1/5 to 1/25

(iii) Rivers, harbours and estuaries — 1/100 to 1/1000

**Similarity of Behaviour of Model and Prototype****: **

When the components of a model have the same shape as the corresponding components of the prototype, then the two systems are geometrically similar. Usually, geometric similarity is maintained in most of the models of fabricated structures.

There is a point to point correspondence between a model and its prototype. By examining a particular property at a particular point in the model, we can determine the corresponding property at the corresponding point in the prototype.

It is for such determination, models are made and experimented. It should be noted that the concept of similarity extends too many characteristics besides geometry. It is not enough if the model looks like the prototype, but it should function like the prototype. In fact geometric similarity is not really the requirement but similarity of performance is the requirement.

From this point of view, in the case of models of rivers, estuaries and harbours it will not be feasible to provide geometric similarities, since the depth of water in such models may be at the order of about 5 mm. When such small depths are involved, the flow is dependent on Weber’s number i.e., the flow is strongly influenced by surface tension. Hence models of rivers and harbours are generally distorted i.e., the depths of water are made relatively greater.

A model is intended to provide quantitative indications of the characteristics of the prototype. The similarity and the test conditions of the model are generally determined by dimensional analysis taking into account the various independent variables influencing the problems. If all the independent dimensionless parameters have the same values for the model and prototype then the two are said to be absolutely similar.

It is of course obvious that absolute similarity is impossible without geometric similarity. But it may be noted that it is not feasible to impose absolute similarity in a model test and hence some of the independent dimensionless parameters which are anticipated to have only secondary influences or which influence the phenomenon in a predictable manner are allowed to deviate from their correct values.

When such departures are made from absolute similarity, the hydraulic engineer must be able to justify such departures. For instance, the influence of viscosity on drag on a ship may be estimated by theory and consequently it will not be necessary to preserve the correct value of the Reynold’s number in a towing test of a model.

There is, however, an important precaution to be observed while ignoring certain non-dimensional products. There are situations where forces of negligible importance on the behaviour of the prototype may affect significantly the behaviour of the model. Ocean waves, for instance, are not influenced by surface tension.

But, if in a model, the waves are less than 25 mm length, their nature is dominated by surface tension. This means the Weber’s number is a very important parameter of the model while its importance is negligible in the prototype. Such disturbing influences are called scale effects and should be guarded against by building the models as large as feasible.

**Similitude****: **

In order the observations made on the performance of the model are useful to predict the performance of the prototype, it is very necessary that the model should represent the prototype in every respect i.e., the model and prototype should have similar properties.

This similarity between a prototype and its model is called similitude. For absolute similitude between a model and the prototype the following types of similarities should exist.

**(i) Geometric Similarity: **

This is similarity of form. This similarity is said to exist between a model and the prototype if the ratio of corresponding linear dimensions of the model and prototype is the same.

if l_{m}, b_{m}, d_{m}, h_{m} etc. refer to certain linear dimensions of the model and if l_{p}, b_{p}, d_{p}, h_{p} etc. refer to corresponding linear dimensions of the prototype, then the condition for geometric similarity is –

**(ii) Kinematic Similarity: **

This is similarity of motion. This similarity is said to exist when the ratios of corresponding kinematic quantities at corresponding points of the model and prototype are the same.

If v_{m1}, v_{m2}, v_{m3} etc. be the velocities at certain points in the model, and if v_{p1}, vp_{2} vp_{3} etc. be the velocities at corresponding points in the prototype, then the condition to be satisfied is –

**(iii) Dynamic Similarity: **

This is similarity of masses and forces. This similarity is said to exist between the model and the prototype when (a) the ratio of masses of corresponding fluid particles in motion are the same, and (b) the ratio of forces on corresponding fluid particles are the same. Dynamic similarity also includes geometric and kinematic similarities.

**A fluid mass, in general, may be subjected to the following forces: **

For absolute dynamic similarity all the above equations (i) to (v) should be satisfied. But in nature it is not possible to satisfy all these equations simultaneously. For instance, if we satisfy the condition [(F_{i}/F_{g})_{m}] = [(F_{i}/F_{g})_{p}] we cannot, at the same time, satisfy the condition [(F_{i}/F_{v})_{m}] = [(F_{i}/F_{v})_{p}].

Hence complete dynamic similarity cannot be practically achieved.

But in the practical cases we may handle, in any one phenomenon only one of the above mentioned forces is predominant and the other forces are insignificant. For instance, in a study of flow over a weir, gravitational force is the predominant force. In a case of flow of a fluid through a conduit at very low velocities, the viscous force is predominant.

Thus we find in a phenomenon, a particular force alone is usually predominant. Hence, for practical purposes, a model may be taken to be dynamically similar to the prototype if the ratio of the inertial to the predominant force is the same in the model and the prototype.

**Froud’s Law****: **

When gravitational forces alone are predominant, a model may be taken to be dynamically similar to the prototype if the ratio of the inertial to the gravitational forces is the same in the model and the prototype.

For this case the condition for dynamic similarity is –

The table given above shows the scale factors for various quantities when Froude’s Law is applicable.

Froude’s law guarantees dynamic similarity only if the gravitational force is the predominating force.

**This law has application in the following cases: **

(i) Flow over spillways.

(ii) Flow through sluices, orifices.

(iii) Surface waves.

(iv) Flow through channels.

**Reynolds Law****: **

If viscous forces alone are predominant, a model, may be taken to be dynamically similar to the prototype if the ratio of the inertial to the viscous forces is the same in the model and the prototype.

For this case the condition for dynamic similarity is –

Similarly, we can determine the scale factors for other quantities.

Reynolds law has applications, only in such problems where the viscous forces alone are predominant.

**The following are some of the examples where Reynolds law has applications:**

(i) Incompressible fluid flow in closed pipes.

(ii) Motion of a body fully submerged in a fluid.

Example – Motion of submarines

(iii) Motion of air planes.

(iv) Flow of a fluid very close to bodies fully submerged.

**Euler’s Law****: **

When pressure forces alone are predominant, a model may be taken to be dynamically similar to the prototype when the ratio of the inertial to the pressure forces is the same in the model and the prototype.

For this case –

The reciprocal of Euler number is called Newton number.

**The fields of application of Euler’s law are: **

(i) Pressure rise due to sudden closure of valves.

(ii) Discharge through orifices, mouthpieces, sluices etc.

**Mach Law****: **

When elastic forces alone are predominant a model may be taken to be dynamically similar to the prototype if the ratio of the inertial to the elastic forces is the same in the model and the prototype.

For this case, the condition for dynamic similarity is –

Mach’s Law has application in –

(i) Aerodynamic testing.

(ii) Flow of gases exceeding the velocity of sound √(E/ρ).

(iii) Water hammer problems.

**Weber Law****: **

When surface tensile forces alone are predominant a model may be taken to be dynamically similar to the prototype when the ratio of inertial to the surface tensile forces is the same in the model and prototype.

For this case, the condition for dynamic similarity is –

**The fields of application of Weber law are the following: **

(i) Capillary movement of water in soils.

(ii) Flow of a liquid at a very small depth over a surface.

(iii) Flow over weirs at very small heads.

(iv) Spray of liquid from the exit of a discharging tube resulting in the formation of drops of liquids.

**Fixed Bed Models****: **

Models of streams whose beds and banks cannot be eroded are called fixed bed models. Such models are useful in the investigation of problems covering long regions of a river in which variations in bed configuration are not to be considered.

**Such models are used in the following studies: **

(i) To study the changes caused by placing obstacles to flow like dams, piers etc.

(ii) Effect on navigation conditions and back water profiles during floods.

In these models it is necessary to distort the slope in order to overcome the high resistance and also to provide a sufficiently high value of the Reynolds number to ensure turbulent flow conditions.

The distortion of slope required can be estimated as follows. We know that the velocity of a stream is given by Manning’s formula, v = (1/N)m^{2/3}i^{1/2}, where m is the hydraulic mean depth and N is Manning’s roughness coefficient.

i = v^{2}N^{2}/m^{4/3}

Hence if the distortion and also the Manning’s coefficient for the prototype are known, we can determine the Manning’s coefficient for the model.

**Movable Bed Model****: **

These are models of channels and streams meant for investigating problems involving erosion, transportation and deposition of the material of the channel bad. These models are very difficult to design and are found to be unreliable in a number of situations.

We know the bed load of the moving streams can be moved due to the tractive force exerted by the stream. When geometrically similar models are made it is seen that the tractive force available is insufficient to produce corresponding bed material movement when sand is adopted as the bed material.

Hence in order to provide adequate tractive force, movable bed models are generally made distorted so as to provided steeper, slopes. Such a distortion of the model is made by exaggerating the vertical scale, to provide adequate slope.

To achieve this a number of models may be experimented with different vertical scales using a given bed material and finally a satisfactory vertical scale which provides a general movement of the bed material at the discharge corresponding to the prototype discharge producing the bed movement is selected.

It is equally important that the scale selected ultimately, is such that it provides a sufficiently high value of the Reynolds number to ensure turbulent flow, even at minimum discharge condition. In these cases the value of another dimensionless constant called Karman number is determined. Karman number is given by –

The above dimensionless constant is also called Roughness Reynolds number.

R_{K} should be greater than 50 to provide roughness of sand grain and greater than 100 for any type of roughness in order to maintain turbulent flow.

We may however bring down the value of the tractive force if lighter materials of specific gravity slightly greater than unity are adopted. Some of the materials adopted are bakelite powder pumice, coal dust, stone dust, wax balls, glass balls, mixture of grains of plastics of different specific gravity etc. By using such lighter bed material the extent of distortion can be minimized.

**Distorted Models****: **

Models of rivers and harbours are usually constructed following different scales for horizontal and vertical dimensions. It is not feasible to provide geometrical similarity in these problems. Because, if geometric similarity is strictly followed the depths of flow will become too small to be measured accurately.

Moreover, due to very small depths of flow, surface tensile effects may alter the flow condition. Further it is necessary to maintain a turbulent flow condition as it exists in the prototype.

Hence the vertical scale is exaggerated resulting in a distorted model. The spirit behind the design of a model is that is not enough if it looks like the prototype but should behave like the prototype.

In other words, it is hydraulic similitude and not geometric similitude which is the main governing factor in model designs. Hence in order to achieve hydraulic similitude it may become necessary to adopt distorted models.

**In general, the distortion introduced in a model may be of the following types: **

(i) Geometric distortion –This is a distortion introduced by adopting different scales for horizontal and vertical dimensions.

(ii) Configuration distortion – In this case the bed slope of the model is increased, otherwise the model is geometrically similar. This case amounts to placing a geometrically similar model in a tilted position compared to the positions of the prototype.

(iii) Hydraulic distortion – This is a type of distortion such that some hydraulic quality say velocity or time of discharge may be changed.

(iv) Material distortion – This type of distortion involves the use of materials different from those in the prototype. Surface materials, surface roughness or the medium in which the model works may be changed.

**Merits of Distorted Models: **

(i) The necessary hydraulic similitude is obtained.

(ii) Depth of flow is increased affording precise measurements.

(iii) Height of waves is increased affording precise measurements.

(iv) Viscous effects which are practically absent in the prototype can be practically eliminated in the model. For instance, by increasing the bed slope in an otherwise geometrically similar model, the velocity can be increased, thus decreasing viscous effects.

(v) Movement of silt and sand can be satisfactorily brought about to match the corresponding behaviour of the prototype.

(vi) By adopting a distorted model the size of the model can be reduced thus simplifying the operation of the model.

(i) Due to unequal horizontal and vertical scales the pressure and velocity distribution are not truly reproduced in the model.

(ii) The wave pattern in the model will be different from that in the prototype due to depth distortion.

(iii) Slopes, bends and earth cuts are not truly reproduced.

**Scale Ratios of Comparison of Quantities in Geometrically Distorted Models****: **

**Scale Effect****: **

This is a defect which occurs in certain models due to which the computed properties of the prototype from model experiments deviate much from the actual properties of the prototype.

We know in the prototype usually large depths and high velocities occur and surface tension for instance has little effect on the motion of the fluid. But in the model if constructed to a small scale, the depth of flow may be very low and surface tensile force will also affect the flow the fluid.

Further, the flow may not be turbulent in the model while in the prototype turbulent flow prevails. In such cases, unless the model is made sufficiently large, the properties of the model will not be similar to those of the prototype.

There are also situations where more than one type of force governs the flow rendering difficulty in making the models dynamically similar to the prototype. For instance, we know in the case of ships, the resistance to the motion is a function of Froude’s and Reynolds numbers. Obviously equality of Froude numbers in the model and prototype, and equality of Reynolds numbers in the model and prototype cannot be achieved simultaneously.

In all these cases where the computed properties of the prototype computed from those of the models, deviate from the actual properties of the prototype, we say there is scale effect in the model. The nature of the scale effect may be such that it can be considerably minimized by using a bigger model. But there may also be a type of scale effect which cannot be avoided.

In such cases we may allow the scale effect and estimate its value. In the case of the problem of determining the resistance to the motion of ships, the model is designed satisfying condition of equality of Froude numbers.

The wave-making resistance for the prototype can be calculated by similitude principles. But the frictional resistance (which depends on the Reynolds number) is calculated by an empirical formula.