6.9. Solving Another Example Problem

The question is determining the depth after the jump, the Froude numbers before and after the jump. So, this is simple application of the formula that we have.

In this section, we are introduced to a problem related to hydraulic jumps. The objective is to determine the depth of water after the jump, as well as the Froude numbers before and after this hydraulic jump. A hydraulic jump occurs when the flow changes from supercritical (fast and shallow) to subcritical (slower and deeper). We will use certain formulae to solve this problem.

Imagine a river flowing fast down a hill. When it reaches a flat area, the water suddenly slows down and deepens. This change in flow is similar to what happens in a hydraulic jump.

So, it is pretty simple to calculate Froude number 1. We will go step by step, the way you should be solving to. Fr1 is given by V1 divided by under root gy1.

To find the Froude number before the jump (Fr1), we need to use the formula: Fr1 = V1 / √(g * y1), where V1 is the velocity of water before the jump, g is the acceleration due to gravity (approximately 9.81 m/s²), and y1 is the depth of water before the jump. Substituting the known values, we find Fr1 is equal to 3.92.

Think of a race car speeding down a straight track. Its speed (V1) and the area (depth) it covers on the track (y1) are critical for determining how fast it moves compared to gravity (g).

Since Fr1 comes out to be 3.92 and this is greater than 1, which means hydraulic jump will occur.

Since the calculated Froude number (Fr1) is greater than 1, it indicates that the flow is supercritical, meaning it has a high velocity and low depth. This condition will lead to the formation of a hydraulic jump, where the flow will transition to subcritical flow, slower and deeper.

Imagine water rushing down a steep slide; it flows rapidly (supercritical). Once it reaches a flat surface, it suddenly slows down (subcritical) - that's similar to a hydraulic jump.

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.

To find the depth after the jump (y2), we use the ratio formula: y2 / y1 = 1/2 * (–1 + √(1 + 8 * Fr1²)). After substituting our Fr1 value (3.92), we calculate y2. This calculation shows how the depth increases after the hydraulic jump.

Think of a sponge soaking up water as it drops into a larger pool. Just as the water from the sponge spreads out and creates a depth change in the pool, the water flows more deeply after the jump.

Now, the way we obtain V2 is by equating the flow rate. So, A1 V1 is equal to A2 V2.

To find the velocity after the jump (V2), we use the principle of conservation of flow rate, which states that the flow entering must equal the flow exiting. We set up the equation A1 * V1 = A2 * V2. Knowing the areas and previous velocity, we can calculate V2.

Imagine a hose spraying water; if you block part of the hose, the water must speed up to maintain the same flow rate. Similarly, as the water flows into a narrower section after the jump, it speeds up.

Froude number at location 2 will be V2 by under root gy2. So, Froude number 2 comes out to be 0.343.

To calculate the Froude number after the jump (Fr2), we use the formula: Fr2 = V2 / √(g * y2). Here, we substitute our previously determined V2 and y2 values. The resulting Fr2 indicates the flow is now subcritical, as Fr2 is less than 1.

Think about a car slowing down on a flat road after going downhill. The balance of speed, depth (road level), and gravity plays a similar role in determining how fluid behaves after 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 energy loss during the hydraulic jump can be calculated by taking the total energy at the first section and subtracting the total energy at the second section. The equation includes potential and kinetic energy terms. After calculation, we determine that the head loss is 0.671 meters.

It's like when water splashes out of a tub when you jump in; some energy is lost during the transition, which in our case translates to head loss in the hydraulic jump.

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