15.1.3. Control Volume Concept

No doubt we already discussed about these mass conservation equations, linear momentum equations, energy conservation equations. These equations will apply for a control volume. So it is a control volume concept what we discussed more with RTT, Reynolds transport theorems. So that the thing concepts are used for the open channel flow to solve the flow depth, energy losses, the discharge flow depth, energy losses, energy loss.

In fluid mechanics, a control volume is a defined region in space through which fluid can flow in and out. It is crucial for applying conservation laws, such as mass, momentum, and energy. When we discuss control volume concepts, it often involves using the Reynolds Transport Theorem (RTT), which helps convert the changes in quantity within a control volume to flow across its boundaries. By applying these principles, we can analyze and predict various aspects of flow, such as the depth of the flow, the energy losses encountered, and how the discharge (the amount of fluid flowing per unit time) behaves within both natural and manmade channels.

Imagine a water tank where water is being filled and drained at the same time. The tank's boundaries form a control volume. Using the concepts of mass and energy conservation, engineers can determine the water level at any time by considering how much water enters and exits the tank.

One of the manmade channels we India is a country we are very much leader in constructing the canals if you look at the history of the Ganga canals you always can google or look in the wikipedia's what is a Ganga canals. The canal was constructed way back 1842 to 1854 so it is about 12 years the constructions what had happened the Ganga Canal.

The Ganga canals are a prime example of early engineering in fluid mechanics and open channel flow management in India. Constructed between 1842 and 1854, these canals were designed to manage water flow efficiently for irrigation and navigation. Their successful operation for nearly 170 years demonstrates the timeless nature of the underlying principles of fluid mechanics used in their design.

Think of a vital roadway or a railway system. Just like these transportation infrastructures are built to facilitate smooth travel for vehicles, the Ganga canals were designed to channel and direct water flow efficiently, ensuring agricultural lands and cities had enough water supply.

If you look at that, when you have let me have the river is flowing like this okay river is flowing and this is what you have the bed level or the canals this is the bed level. this is what you have the free surface. That is what you have the free surface okay having the flow depth of H. If you have these things and if you create any disturbance okay let me there is a one big stone is there. okay or just dump a stones here. It create a disturbance to these flow systems okay or you throw a stone to a river.

When water flows over a surface, it maintains a certain pattern or depth, which we refer to as flow depth (H). If something disrupts this flow, such as throwing a stone into the river, it creates a disturbance that affects the flow patterns. This disturbance causes waves to form on the water's surface, which propagate upstream and downstream from the point of disturbance, illustrating how changes in flow conditions can travel through the water.

Consider a calm pond where you toss a pebble. The ripples that radiate outwards represent the waves propagating from the point of disturbance. Just like those ripples, disturbances in a river flow can cause waves to travel in various directions, changing the characteristics of the flow.

If you look at that the flow proud numbers what we define it as an inertia force by the gravity forces and when you do this inertia forces gravity forces we are getting the flow proud numbers is a functions of the speed of water okay.

Flow regimes can be categorized based on the Froude number, which is a dimensionless number defined as the ratio of inertial forces to gravitational forces in a flow. If the Froude number is less than 1, the flow is considered subcritical, meaning gravity forces dominate inertia forces. If it is equal to 1, it’s critical flow, where both forces are balanced. If it is greater than 1, supercritical flow occurs, where inertia forces overpower gravity. Understanding these regimes is vital for predicting how water will behave under different conditions.

Imagine a river flowing gently (subcritical flow) versus a river in flood (supercritical flow). In the former case, the water moves slowly, while in the latter, it rushes rapidly, and understanding these behaviors is crucial for effective river management and flood control.

Here you will have a supercritical flow. then you will have a subcritical flow. So if you look at that same thing is a subcritical flow supercritical channel. In between whenever a convergence of flow happens from the supercritical to subcritical flow there will be a formations of the eddies and there are a lot of energy dissipations will happen it their energy loss will happen it.

A hydraulic jump occurs when there is a transition from supercritical to subcritical flow, often resulting in the formation of turbulent eddies and significant energy dissipation. This phenomenon typically occurs in channels where flow conditions change abruptly, like at the base of a dam. Hydraulic jumps are important because they can effectively dissipate energy, which helps to prevent erosion downstream.

Think of a steep water slide at a theme park. As water rushes down the slide quickly (supercritical), it suddenly hits a flat pool (subcritical), creating splashes and ripples. This change in flow results in a mix of turbulent energy, similar to what happens during a hydraulic jump.

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