3.3. Using the Continuity Equation

Now, we also apply the continuity equation and this will give us a second equation. So, V1 y1 = V2 y2.

The continuity equation is a fundamental principle in fluid mechanics that states that the mass flow rate must remain constant from one cross-section of a channel to another, provided there are no leaks. In this equation, 'V1' represents the velocity of the flow at the first section, 'y1' is the depth of the flow at that section, and 'V2' and 'y2' represent the corresponding velocity and depth at a second section. This equation helps us understand how fluid behavior changes as it moves through different areas of a channel.

Imagine water flowing through a garden hose. If you squeeze the hose, the water moves faster at the point of constriction (similar to V2), but the total amount of water flowing through must stay the same as it enters and exits the hose. This is akin to how the continuity equation operates in a fluid system.

So, if y1 V1 is known, then we can express y2 in terms of known quantities: V1y1 = V2y2, hence V2 = (V1 y1) / y2.

This part of the equation shows how the depth 'y2' can be deduced from the known values of 'V1' and 'y1'. It emphasizes that changes in the flow speed (V1 vs. V2) are inversely related to the depth changes of the water in the channel. As water depth decreases, the velocity increases, maintaining the continuity of flow.

Think about a river flowing into a narrow canyon. As the river enters the canyon, the width of the river decreases, causing the water to flow faster as it gets deeper ('y2'). This is a practical illustration of the relationship between depth and velocity in various landscapes.

Now, equation 1 and 2 can be combined to give a cubic equation, if you do in terms of y you can find this equation 1 y2 whole cube - 1.90 y2 square + 0.513 = 0.

Once we have our continuity equation and energy equation established, we can manipulate them algebraically to form a cubic equation. This is essential to solve for 'y2', which represents the water depth downstream. Cubic equations can have multiple solutions, referred to as roots, which can be positive or negative, but only the positive roots make physical sense in this context.

Consider a puzzle where different pieces can fit together in various configurations. The cubic equation acts similarly – it provides multiple solutions, but only the ones that fit the physical context (like depth being positive) are relevant. You're piecing together data to understand how water behaves under specific conditions.

Now, if we solve this, we will get 3 solutions, y2 is equal to 1.72 feet, y2 is going to be another, another value is going to be 0.638 feet and this is a negative value. So, of course, we are going to neglect the negative values.

After solving the cubic equation, we find three potential solutions for 'y2', which correspond to different possible depths downstream, including a negative value which has no physical significance. The positive solutions (1.72 feet and 0.638 feet) represent plausible water depths that can be utilized for further analysis or design.

Imagine shooting an arrow and landing near the target. Just like you assess which hit is closest to the bullseye (valid solutions) versus those that miss entirely (invalid solutions), in mathematics, we focus on realistic results that fit our physical expectations.

The corresponding elevations of the free surfaces are either y2 + z2 is going to be one, I mean, if we take 1.72 as y2 2.22 feet or if we take 0.638 feet then it will be 1.14 feet.

This chunk explains how to find the elevations of the water surface based on the values obtained for 'y2'. The total water surface elevation combines the depth 'y2' and any additional height (z2) due to ramps or other features in the channel. This elevation is critical for understanding how high the water will flow at different locations.

If you have a glass of water filled to a certain level and then you add a few ice cubes, the water's surface elevation changes. Similarly, here, by knowing the depth and additional raised surface due to conditions, we can visualize how water behaves in channels, like filling a container.

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