3. Department of Civil Engineering

Welcome back students. So, we start with solving some questions that are related to hydraulic jumps, so rapidly varied flow. So, this entire lecture will be dedicated to solving some of the basics and a little more complex problems, those type of problems which you encountered in exams like GATE or IES.

The introduction outlines the focus of the lecture, which is centered on hydraulic jumps, a specific kind of fluid behavior in hydraulic engineering. Hydraulic jumps occur when water transitions from a supercritical flow (fast and shallow) to a subcritical flow (slow and deep), often seen in spillways or free-surface flows. Understanding this phenomenon is crucial for engineers working with fluid systems.

Think of a water slide at a water park. At first, the water flows quickly over the slide's edge (supercritical flow). As it splashes down into the pool, it suddenly slows down and spreads out (subcritical flow), demonstrating a hydraulic jump.

So, we start with one problem as below, it says that water on the horizontal apron of 30 metre wide spillway. So, it is 100 feet or, sorry, that is 30 meter has a depth of 0 point, so 0.06 feet is equivalent to 0.20 metre and a velocity of 18 feet per second that means, 5.5 metres per second.

In this section, a problem setup is introduced that involves water flow over a spillway. It specifies critical values such as the width of the spillway (30 meters), the depth of water (0.20 meters), and the velocity of flow (5.5 meters per second). These details are essential for calculating the characteristics of the hydraulic jump that will occur.

Imagine you're at a swimming pool with a slide. The slide is wide (the spillway), and the water flows at a certain speed (velocity) while being a specific depth. These characteristics determine how the water flows from the slide into the deeper pool below.

So, we know the conditions here, at number 1, so it is pretty simple to calculate Froude number 1. We will go step by step, the way you should be solving to. ... Froude number 1 comes out to be 3.92 and this is greater than 1, which means hydraulic jump will occur.

The Froude number is a dimensionless value that helps assess the state of flow, calculated as the ratio of the flow's inertial forces to gravitational forces. A Froude number greater than 1 indicates supercritical flow conditions, which mean hydraulic jumps can occur when the flow meets resistance, such as flowing over a spillway.

You can liken this to driving a car: at a certain speed (analogous to supercritical flow), the car can easily pass over bumps (like obstacles in flow). However, if you were to decelerate (subcritical flow), going over these bumps would be much smoother and controlled, indicating a different flow state.

So, in the second step, we obtain depth ratio, we had the formula which said y2 by y1 is equal to 1/2 into multiplied by – 1 + 1 + 8 Fr1 square.

This step focuses on calculating the depth of the water after the hydraulic jump using the derived formula. The depth before the jump (y1) and the calculated Froude number (Fr1) help define the relationship between the initial and final depths (y2 and y1). By substituting these values, engineers can understand how the depth varies after the jump occurs.

Consider a water balloon being dropped from a height. Just before it bursts at impact (the jump), it retains a certain shape and volume. After bursting, the water splays out, creating a new 'depth' as it spreads out on the ground (analogous to the changed depth after the jump).

Now, the way we obtain V2 is by equating the flow rate. So, A1 V1 is equal to A2 V2, V gets cancel out, so V2 will be V1 y1 divided by y2.

This calculation involves using the principle of conservation of mass – the flow rate must remain constant before and after the jump. By equating the product of the area and velocity of flow before and after the jump, engineers can solve for the new velocity (V2) after the jump, ensuring the flow continuity is maintained.

Imagine squeezing a bottle of ketchup. If you apply pressure at the top (initial flow), the ketchup comes out faster (initial velocity). However, if you let it flow freely into a wider plate, it slows down (final velocity) because it has to fill a larger area, akin to flow transitioning in a hydraulic jump.

Now, we have to also obtain the head loss, that the energy loss. We simply use this equation, total energy at section 1 minus total energy minus section 2...

The head loss represents the energy loss in the system due to the hydraulic jump. This loss is calculated by subtracting the total energy before the jump from the total energy after. It’s essential in hydraulic engineering as it quantifies the energy transformation and loss during the hydraulic jump, which affects system design and efficiency.

Think of a roller coaster: as it climbs, it gains potential energy. However, when it races down and undergoes curves (like hydraulic jumps), some energy is lost to friction and turbulence. The difference in energy from the top to the bottom of the ride shows the energy loss due to conditions along the path.

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