# Dimensional Analysis And Similitude

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| By Trneelakantan
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Trneelakantan
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Questions: 14 | Attempts: 765

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This is prepared for SASTRA University students by Dr. T. R. Neelakantan, Prof., School of Civil Engineering, SASTRA University, Thanjavur, Tamilnadu, India.

• 1.

### 1. Match List I with List II and select the correct answer using the codes given below the lists:List I(Fluid problems)Flow over a spillway damFlow through a butterfly valveRise of moisture in the steam of a plantWater hammer in a penstockList II(Model laws)Euler NumberFroude NumberMach NumberReynolds NumberWeber Number

• A.

A-2 B-4 C-5 D-3

• B.

A-4 B-2 C-3 D-1

• C.

A-2 B-3 C-1 D-4

• D.

A-5 B-1 C-4 D-2

A. A-2 B-4 C-5 D-3
Explanation
The correct answer is A-2 B-4 C-5 D-3. This means that the fluid problem "Flow over a spillway dam" can be modeled using the Euler Number, "Flow through a butterfly valve" can be modeled using the Reynolds Number, "Rise of moisture in the steam of a plant" can be modeled using the Weber Number, and "Water hammer in a penstock" can be modeled using the Froude Number.

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• 2.

### Match List I with List II and select the correct answer using the codes given below the lists: List I (Flow problem under study) Rise of gas bubbles in liquid Flow of gas in a pipe Flow over a spillway dam Flight of supersonic jet List II (Model laws) Euler Number Froude Number Mach Number Reynolds Number Weber Number

• A.

A-3 B-5 C-4 D-1

• B.

A-3 B-4 C-2 D-1

• C.

A-5 B-2 C-4 D-3

• D.

A-5 B-4 C-2 D-3

D. A-5 B-4 C-2 D-3
Explanation
The rise of gas bubbles in liquid corresponds to the Weber Number, as it relates to the balance between the inertia and surface tension forces. The flow of gas in a pipe corresponds to the Reynolds Number, as it characterizes the flow regime and determines whether it is laminar or turbulent. The flow over a spillway dam corresponds to the Froude Number, as it relates to the ratio of inertial forces to gravitational forces. The flight of a supersonic jet corresponds to the Mach Number, as it relates to the ratio of the speed of the object to the speed of sound in the surrounding medium.

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• 3.

### Both Reynolds and Froude numbers assume significance in one of the following examples:

• A.

Motion of a submarine at large depths

• B.

Motion of ship in rough seas

• C.

Cruising of a missile in air

• D.

Flow over spillways

B. Motion of ship in rough seas
Explanation
The Froude number is a dimensionless number that represents the ratio of the inertia forces to the gravitational forces in a fluid flow. It is commonly used in the study of ship hydrodynamics, particularly in rough seas. The Reynolds number, on the other hand, represents the ratio of inertial forces to viscous forces in a fluid flow. While it is also relevant in ship hydrodynamics, it is not specifically associated with rough seas. Therefore, the correct answer is the motion of a ship in rough seas.

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• 4.

### Dynamic similarity is said to exist between the flow over two geometrically similar bodies when

• A.

Geometrical similarity exists between the two flows

• B.

Velocities at corresponding points are the same

• C.

The flow fields have geometrical similarity and force field similarity

• D.

The forces acting on the bodies are equal.

C. The flow fields have geometrical similarity and force field similarity
Explanation
Dynamic similarity is a concept used in fluid mechanics to compare the behavior of flows over different geometrically similar bodies. When dynamic similarity exists, it means that the flow fields of the two bodies have both geometrical similarity and force field similarity. Geometrical similarity refers to the similarity in the shape and dimensions of the bodies, while force field similarity refers to the similarity in the forces acting on the bodies. Therefore, the correct answer is that dynamic similarity exists when the flow fields have both geometrical similarity and force field similarity.

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• 5.

### If two geometrically similar models having a scale ratio of Lr are operated in a given laboratory at the same Froude number, then all the corresponding accelerations will be in the ratio of

• A.

Lr^1.5

• B.

Lr

• C.

Lr^0.5

• D.

1

D. 1
Explanation
If two geometrically similar models are operated in a laboratory at the same Froude number, it means that they are experiencing the same flow conditions. In this case, the Froude number is a dimensionless quantity that represents the ratio of inertial forces to gravitational forces. Since the models are geometrically similar, their shapes and sizes are proportional to each other with a scale ratio of Lr.

When the Froude number is the same, it implies that the ratio of inertial forces to gravitational forces is the same for both models. As acceleration is related to inertial forces, it can be concluded that the corresponding accelerations of the two models will also be in the same ratio of 1. Therefore, the correct answer is 1.

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• 6.

### Match List I with List II and select the correct answer using the codes given below the lists: List I (Parameters) Inertial forces and gravitational force Inertial forces and elastic forces Inertial forces and surface tension Inertial forces and viscous force List II (Number) Reynolds Number Weber Number Froude Number Mach Number

• A.

A-2 B-3 C-4 D-1

• B.

A-1 B-4 C-2 D-3

• C.

A-3 B-4 C-2 D-1

• D.

A-3 B-2 C-4 D-1

C. A-3 B-4 C-2 D-1
Explanation
The correct answer is A-3 B-4 C-2 D-1. Inertial forces and gravitational force are related to the Froude Number, inertial forces and elastic forces are related to the Reynolds Number, inertial forces and surface tension are related to the Weber Number, and inertial forces and viscous force are related to the Mach Number.

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• 7.

### A river model is constructed to a horizontal scale of 1:1000 and a vertical scale of 1:100.  If the model discharge were 0.1m3/s, then the discharge in the river (in m3/s) would be

• A.

1,000

• B.

10,000

• C.

100,000

• D.

1000,000

D. 1000,000
Explanation
The river model is constructed to a horizontal scale of 1:1000 and a vertical scale of 1:100. This means that every unit of measurement in the model represents 1000 units in the actual river horizontally and 100 units vertically. Therefore, if the model discharge is 0.1m3/s, the actual discharge in the river would be 1000 times greater, resulting in a discharge of 1000m3/s.

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• 8.

### Match List I with List II and select the correct answer using the codes given below the lists: List I (Non-dimensional numbers) Mach number Thoma number Reynolds number Weber number List II (Application) 1. Waves in an ocean2. Launching of rockets3. Cavitation phenomenon4. Capillary flow in soil 5. Motion of a submarine

• A.

A-1 B-3 C-5 D-2

• B.

A-2 B-3 C-5 D-4

• C.

A-3 B-2 C-4 D-1

• D.

A-2 B-1 C-3 D-4

B. A-2 B-3 C-5 D-4
Explanation
The Mach number is a non-dimensional number used to describe the speed of an object relative to the speed of sound in the surrounding medium. It is commonly used in the application of launching rockets, which is represented by option B-3.

The Thoma number is a non-dimensional number used to describe the stability of a fluid flow. It is not directly related to any of the given applications, so option A-2 is incorrect.

The Reynolds number is a non-dimensional number used to describe the flow characteristics of a fluid. It is commonly used in the application of cavitation phenomenon, which is represented by option C-5.

The Weber number is a non-dimensional number used to describe the ratio of inertial forces to surface tension forces in a fluid flow. It is commonly used in the application of capillary flow in soil, which is represented by option D-4.

Therefore, the correct answer is A-2 B-3 C-5 D-4.

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• 9.

### A spillway model constructed to a scale of 1:4 gives a discharge of 5 m3/s.   The discharge from the prototype would be

• A.

160 m3/s

• B.

640 m3/s

• C.

10 m3/s

• D.

20 m3/s

A. 160 m3/s
Explanation
The scale of 1:4 means that the dimensions of the model are 1/4th of the dimensions of the prototype. Since the discharge is directly proportional to the dimensions of the spillway, the discharge from the prototype can be calculated by multiplying the discharge from the model (5 m3/s) by the scale factor (4). Therefore, the discharge from the prototype would be 5 m3/s * 4 = 20 m3/s.

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• 10.

### A river model is made to a length scale ratio of 1/100 and depth scale ratio of 1/16.  A peak discharge of 25,600 m3/s in the river will be simulated in the model with a discharge of

• A.

10 cumec

• B.

4 cumec

• C.

1.6 cumec

• D.

0.64 cumec

B. 4 cumec
Explanation
In a river model, the length scale ratio is 1/100 and the depth scale ratio is 1/16. This means that every 1 unit of length in the model represents 100 units in the actual river, and every 1 unit of depth in the model represents 16 units in the actual river.

To simulate a peak discharge of 25,600 m3/s in the river, we need to find the corresponding discharge in the model. Since the scale ratio for length is 1/100, the discharge in the model will also be scaled down by a factor of 1/100. Therefore, the discharge in the model will be 25,600 m3/s * (1/100) = 256 m3/s.

Among the given options, the closest value to 256 m3/s is 4 cumec (cubic meters per second). Therefore, the correct answer is 4 cumec.

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• 11.

### If the physical quantities involved in a fluid flow phenomenon are discharge Q, diameter D, acceleration due to gravity g, dynamic viscosity μ and density ρ then the number of π parameters needed to express the function F (Q, D, g, m, ρ) = 0 are

• A.

2

• B.

3

• C.

4

• D.

5

A. 2
Explanation
The function F(Q, D, g, μ, ρ) = 0 involves the physical quantities discharge Q, diameter D, acceleration due to gravity g, dynamic viscosity μ, and density ρ. The number of π parameters needed to express this function is 2. This is because the Buckingham Pi theorem states that the number of π parameters needed is equal to the number of variables (n) minus the number of fundamental dimensions (m). In this case, there are 5 variables (Q, D, g, μ, ρ) and 3 fundamental dimensions (mass, length, time), so the number of π parameters is 5 - 3 = 2.

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• 12.

### In a 1: 100 scale model of a harbour, the time which will correspond to the prototype tidal period of 12 hours will be

• A.

0.12 hour

• B.

1.2 hours

• C.

12 hours

• D.

120 hours

B. 1.2 hours
Explanation
In a 1:100 scale model, the ratio of the model's time to the prototype's time is 1:100. Therefore, if the prototype tidal period is 12 hours, the corresponding time in the model will be 1/100th of 12 hours, which is 0.12 hours.

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• 13.

### Apart form inertia force, which of the following forces is most important in motion of submarines under water?

• A.

Viscous force

• B.

Gravity force

• C.

Compressive force

• D.

Surface tension force

A. Viscous force
Explanation
The most important force in the motion of submarines under water, apart from inertia force, is viscous force. Viscous force refers to the resistance that a fluid (in this case, water) exerts on an object moving through it. In the case of submarines, the shape and design of the hull are crucial in minimizing the drag caused by viscous force. This force plays a significant role in determining the speed and efficiency of the submarine's movement through the water.

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• 14.

### A model of reservoir is emptied in 10 minutes.  If the model scale is 1:25, the time taken by the prototype to empty itself, would be

• A.

250 minutes

• B.

50 minutes

• C.

6250 minutes

• D.

2 minutes

B. 50 minutes
Explanation
The model of the reservoir is scaled down by a factor of 25. This means that the time taken to empty the model is also scaled down by the same factor. Therefore, if the model takes 10 minutes to empty, the prototype (actual reservoir) will take 25 times longer, which is 250 minutes.

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• Current Version
• Mar 22, 2023
Quiz Edited by
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• Jan 19, 2009
Quiz Created by
Trneelakantan

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