6.5. Velocity Calculation After the Jump

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.

The hydraulic jump is a phenomenon that occurs when water transitions from a supercritical state to a subcritical state. This transition dramatically changes the water's behavior, particularly in terms of its velocity and depth. To assess this change, we start by calculating the Froude number (Fr1) before the jump using the formula Fr1 = V1 / √(g * y1), where V1 is the velocity of water before the jump, g is the acceleration due to gravity, and y1 is the depth of the water before the jump.

Think of a person running at high speed (supercritical flow) who suddenly encounters a thick mattress. They can't keep running at the same speed and must slow down to navigate through it (subcritical flow). The transition point they feel represents the hydraulic jump.

So, Fr1 is given by V1 divided by under root gy1, V1 was given 5.5, y1 was given 0.2, we see, 18 feet per second means, 5.5 metres per second and this is 30 meter. So, the Froude number 1 comes out to be 3.92 and this is greater than 1, which means hydraulic jump will occur.

To determine Fr1 numerically, we substitute the values we know into the Froude number formula. With V1 = 5.5 m/s and y1 = 0.2 m, we calculate Fr1. Since the resulting Fr1 value (3.92) is greater than 1, it confirms that the flow is supercritical, and hence, a hydraulic jump is expected to occur.

Like a speeding car hitting a bump (the hydraulic jump), which causes it to lift and slow down; in fluid dynamics, an increase in the gravitational effect or height can make the water behave differently.

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. Fr1 whole square which already we found out in the previous slide was 3.92, plug these values here, so what comes out is y2 by y1 is 5.07.

After determining the Froude number, we use it to calculate the depth ratio (y2/y1) using the provided formula. Substituting Fr1 = 3.92 into the formula yields a depth ratio of 5.07. This means that the depth of water after the jump (y2) can be found by multiplying this depth ratio with the initial depth y1.

Imagine filling a cup with water—if you suddenly tilt the cup (the jump), the water level rises dramatically as you transition from a static to dynamic state, just like how water depth increases after a hydraulic jump.

Therefore, y2 is going to be 5.07 into 0.2 and that is 1.01 metre, so this is going to be 1.01 metre. 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.

From our depth ratio, we compute y2, which is 1.01 meters. Next, to find the velocity after the hydraulic jump (V2), we apply the principle of conservation of mass, expressed through the flow rate equation: A1V1 = A2V2. Given the dimensions and liquid properties, we can rearrange this to get V2 = (V1 * y1) / y2, eventually yielding a value for V2.

Imagine a funnel: as the liquid narrows at the bottom, it speeds up. Similarly, in hydraulic jumps, the flow velocity changes based on depth differences—narrow sections increase speed just like the funnel.

Froude number at location 2 will be V2 by under root gy2. V2 we have calculated here, this was the reason we were calculating V2, for calculating the Froude number.

Once we find V2, we can calculate the Froude number after the jump (Fr2) using the same formula as before: Fr2 = V2 / √(g * y2). By inserting our calculated values for V2 and y2, we obtain Fr2. A key outcome is observing that Fr2 is less than 1, indicating subcritical flow after the jump.

Think of coming off a slippery slide into a shallow pool: when you land, you slow down significantly. This transition represents how the flow switches from supercritical (fast) to subcritical (slow) after the 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.

To find the energy loss during the hydraulic jump, we utilize the energy conservation principle. The energy loss is the difference between the total energy at the upstream section (before the jump) and the downstream section (after the jump). This involves using the known velocities and depths to calculate energy forms (potential and kinetic) at both sections and their difference gives us the head loss.

Imagine going down a slide—the gravitational potential energy converts to kinetic energy as you descend, but some energy gets lost as heat due to friction on the surface. Similarly, in hydraulic jumps, energy is lost as water dynamics change.

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