Aerodynamics Fundamentals: Bernoulli's Equation
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Bernoulli's Equation
06.54
| 00:00 | One of the most fundamental principles in aerodynamics is the relationship between pressure and velocity, and this is expressed through what is known as Bernoulli's equation. |
| 00:08 | This equation will use the principle of conservation of energy to allow us to derive a relationship between the pressure of the fluid at a point and the velocity of the fluid at a point. |
| 00:18 | You might be wondering where you've heard of Bernoulli's equation before. |
| 00:22 | This is the exact same principle that carburettors use to function, so if you even understand the basics of how carbs operate, you might have a head start here. |
| 00:31 | Let's consider fluid flowing uniformly through a horizontal pipe. |
| 00:35 | If the fluid is flowing along, we're not adding or subtracting energy from it, so the energy level of the fluid at each point is constant. |
| 00:43 | Let's now give that fluid a velocity, say one meter per second. |
| 00:47 | At all points in this pipe, the fluid will have the same velocity, as the pipe is constant volume and cross-sectional area at all points. |
| 00:55 | Now, let's modify the pipe. |
| 00:57 | I'm going to shrink the pipe to a point where the fluid is flowing at a constant velocity, and I'm going to shrink the area halfway along to be half the cross -sectional area. |
| 01:02 | So, say our inlet was one meter squared, and this area halfway along the pipe is now half a meter squared. |
| 01:09 | We're going to assume that our fluid doesn't compress or change density. |
| 01:13 | Don't worry, we'll discuss that assumption more in an upcoming module. |
| 01:17 | Our flow is traveling at one meter per second at the inlet over an area of one meter squared. |
| 01:23 | This means that we're flowing one meter cubed of fluid through the inlet. |
| 01:27 | No fluid flow can enter or leave the pipe halfway along. |
| 01:31 | It's bound by the pipe walls. |
| 01:32 | So, we also have one meter cubed per second at the narrower midway section of the pipe. |
| 01:39 | However, the cross-sectional area at this point is half. |
| 01:43 | This means that we have one meter cubed per second flowing through an area of half a meter squared. |
| 01:49 | So, the flow speed will increase to two meters per second. |
| 01:53 | We now have a bit of a conundrum. |
| 01:55 | We haven't added any energy to the to the fluid flow. |
| 01:58 | However, it has increased in speed, increasing its kinetic energy. |
| 02:03 | There has to be a conservation of energy here as a fundamental law of physics. |
| 02:08 | The energy required to increase this velocity must come from somewhere, so where is it? Let's consider the case of a water tank with a pipe outlet at the bottom. |
| 02:19 | This is a situation that's easy to visualize as you've likely encountered it at some point in your life. |
| 02:23 | If we fill the tank to a high level, the water will flow faster out of the pipe. |
| 02:28 | If I fill it to a low level, it will trickle out of the pipe. |
| 02:32 | This is a consequence of the conservation of gravitational potential energy into kinetic energy, not dissimilar to a ball rolling down a hill. |
| 02:40 | The gravitational energy of an object is given by its mass times gravity times height, and for a fluid this is instead given as density times gravity times height, referred to as rho g h. |
| 02:52 | The kinetic energy of a mass is given as a half times the mass times the velocity squared, or in the case of our fluid, we again use density instead of mass, giving the formula half times rho times v squared. |
| 03:08 | In the case of our tank, the gravitational potential energy at the start is equivalent to our kinetic energy at the exit, so rho times g times h equals a half times rho times v squared. |
| 03:19 | Finally, let's consider the case of a static fluid tank of a certain height. |
| 03:24 | If we consider a slice down it, let's look at the force at a particular horizontal location. |
| 03:29 | As we move down through the tank, more and more of the fluid above our chosen location will weigh down on that location. |
| 03:36 | In fact, it's that same mass multiplied by gravity that's applying the force as we discussed before. |
| 03:43 | So, what you end up with is a scenario where force, or in our case pressure, within a fluid is given as p equals rho times g times h squared. |
| 03:52 | So, as we can see, the pressure is linked to the gravitational potential energy of the fluid. |
| 03:58 | When we combine these three key terms that we've discussed, gravitational potential energy, kinetic energy and static pressure, what we get is known as Bernoulli's equation. |
| 04:09 | Essentially, this is an equation where what is known as total pressure equals the sum of these terms. |
| 04:16 | So, pt, the total pressure, equals a half times rho times v squared plus ps, the static pressure, plus rho times g times h. |
| 04:26 | If we haven't added or removed energy from our fluid system, the total pressure remains constant throughout. |
| 04:33 | So, if we were to speed up our flow and increase our kinetic energy, our pressure will have to drop or our height will have to change. |
| 04:41 | Now, that we have this formula, let's go back and look at our pipe example from before. |
| 04:45 | It's a horizontal pipe, so the gravitational potential isn't changing significantly and we can ignore it. |
| 04:51 | Our speed started at one meter per second and is now sped up to two meters per second. |
| 04:56 | To simplify the calculations, I'm going to say the density of our fluid is five kilograms per meter cubed. |
| 05:03 | Comparing our total pressure at two points with Bernoulli's principle, we can see that our inlet has a velocity-based term, known as the dynamic pressure, of a half times five times one squared equals two and a half pascals. |
| 05:20 | While our narrowed-min section has a dynamic pressure of a half times five times two squared equals ten pascals. |
| 05:30 | If we look at the difference between these two values, it gives a static pressure differential of seven point five pascals. |
| 05:37 | The pressure of the fluid at the narrowed section is seven point five pascals lower than the pressure at the inlet. |
| 05:43 | Through this, we can see how we're able to use the velocity difference between one point and another in the fluid difference between the two points. |
| 05:52 | The key thing we really need to take away from all this is that as the velocity increases, the pressure will decrease, as this is a key concept of car aerodynamics. |
| 06:01 | It's worth noting too that we generally tend to ignore gravity-based effects on air around race cars, as the heights are so small with respect to the density of air that the numbers become quite irrelevant compared to the much larger deltas in other quantities like velocity and pressure. |
| 06:18 | Let's summarise what we've covered in this module by looking at the different aspects before we move on. |
| 06:21 | Bernoulli's equation, based on the conservation of energy, describes the relationship between pressure and velocity in fluid dynamics. |
| 06:29 | When fluid flows through a pipe and its cross-sectional area narrows, velocity increases while pressure decreases. |
| 06:35 | This principle is key to understanding car aerodynamics, where increased airflow speeds reduce pressure, affecting performance. |
