5.2. Fluid experiencing change in elevation

If we talk about another simple case, where pressure is 0 or constant. What is an example of fluid experiencing a change in elevation but remaining at a constant pressure? An example is a free jet. What is a free jet? This can be taken as a free jet, you fill a bottle with water or any fluid and make many holes...

The Bernoulli equation along 1 streamline will be p1 / gamma + z1 + V1^2 / 2g = p2 / gamma + z2 + V2^2 / 2g.

In this chunk, we discuss a scenario where the fluid moves upwards through a free jet while maintaining constant pressure. A free jet can be visualized as water flowing out of a bottle through holes, similar to fountains. In this scenario, when applying Bernoulli's equation, we note that both points at the top (p1) and the point below (p2) are exposed to atmospheric pressure, leading to a simplification where the gauge pressure is zero. Therefore, the equation helps us derive how elevation changes (z1 to z2) relate to the velocities at these points (V1 and V2)...

Think about a garden hose with water. When you block part of the end of the hose with your thumb, the water shoots out farther. Here, the hose itself represents a pressure zone while the shooting water illustrates a free jet. Even though you've increased the velocity as the pressure remains constant (by controlling flow with your thumb), the height to which the water can reach showcases the change in elevation due to pressure.

Now talking about hydraulic and energy grade lines, the question is the 2-centimeter diameter jet is 5 meters lower than the surface of the reservoir, what is the flow rate Q?

The jet solution, so, the question again coming back to the question, the 2-centimeter diameter jet is 5 meters lower than the surface of the reservoir...

In this chunk, we tackle a practical problem where a water jet from a reservoir drops a certain height before reaching an external point. Starting from the example's conditions, where the height (z2) is known to be 5 meters, and that the pressure at both points in the jet is atmospheric (setting p1 and p2 to zero), we use Bernoulli's equation to derive the velocities involved. By expressing V2 in terms of the known height difference and applying the diameter of the jet, we can calculate the flow rate (Q) as a function of velocity and area...

Imagine a waterfall—when water flows from a high place to a lower elevation, its velocity increases due to gravity. Just like measuring how fast water falls lets you understand how much water flows, using Bernoulli’s equation we can estimate the quantity flowing through the jet. It’s akin to how we measure the speed of a car descending a hill; the steeper the hill, the faster the car goes.

One of the other applications of Bernoulli’s equation is the stagnation tube. What happens when the water starts flowing in the channel? For example, this and so, the equation along the streamline by Bernoulli equation will be can be given as p / gamma + z + V^2 / 2g = constant.

This chunk emphasizes how Bernoulli's equation applies to different flow scenarios, including stagnation tubes where the water velocity is affected by elevation changes. In essence, as the water ascends to a designated height within the tube, its kinetic energy converts to potential energy, illustrating how flow behavior can be assessed using the same foundational principles of fluid mechanics...

Think of a water slide at a theme park. As you slide down, you pick up speed (kinetic energy). If you reach an upward slope, you notice a slowdown (kinetic energy converting to potential energy). Here, just as in a stagnation tube, the principles of fluid dynamics dictate how far and fast fluid travels based on its elevation and velocity.

There is another thing we should mention—the Pitot tube, which is used to measure the flow velocities in an airplane as well. So, this is how a pitot tube looks like. It is used to measure air speed on airplanes...

In discussing the Pitot tube, we focus on its design and operational principles. This tube measures fluid velocity by comparing the static pressure (static tap) and stagnation pressure (stagnation tap), yielding airflow speed metrics by using Bernoulli’s principle. Notably, the velocities can be obtained through experimentally derived coefficients that help provide accurate results...

Consider a speedometer in your car. Just like it measures how fast you're going by sensing the air pressure pushing against the car's body, the Pitot tube functions similarly. While you're driving, the car's speed changes the pressure around it, which the Pitot uses to determine the vehicle's velocity. This application extends to airplanes, ensuring they fly safely at the right speeds.

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