Dimensional Analysis And Similitude

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.