# Fluid Dynamics Basics

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1 Fluid Dynamics Basics Bernoulli s Equation A very important equation in fluid dynamics is the Bernoulli equation. This equation has four variables: velocity ( ), elevation ( ), pressure ( ), and density ( ). It also has a constant ( ), which is the acceleration due to gravity. Here is Bernoulli s equation: To understand and use this equation, we must know about streamlines. Streamlines are curves that are tangent to the velocity vector of the flow. In other words, they show the direction a fluid element will travel in at any point in time. A streamline is best illustrated by examples: If you throw a leaf in a stream of water, a streamline is the path of the leaf as it floats downstream. Of course the leaf can take any number of paths depending on where it lands in the stream after it was thrown. Streamlines exist underwater as well. Imagine a submerged particle that is neutrally buoyant (meaning it neither sinks nor floats). The particle, such as a waterlogged twig, also follows a streamline down the river. Streamlines can also be visualized with dye in water or smoke in air: See Figure 1. Figure 1. Streamlines of smoke around an airplane wing in a wind tunnel. (Image source: NASA, When values for velocity, pressure, etc. are plugged into Bernoulli s equation, the result of the equation will be the same (constant) at every point along the streamline. Take, for example, a water reservoir at the top of a hill. Imagine a streamline from the reservoir to a residence, where the water is used. At the reservoir, the velocity of the water is very small as it slowly moves toward the spillway. For a water particle near the surface of the reservoir, the pressure is also very small. The elevation above the residence, however, is quite large. Let s plug some of these values into Bernoulli s equation: velocity, = 0.01 elevation, = 1000 m pressure, = 1 Pa density of water, = acceleration due to gravity, = What is the value of Bernoulli s equation with these values? 2. What are the units of this value? Tippy Tap Plus Piping Activity Fluid Dynamics Basics Handout 1

3 tee, branched flow Flow 1.0 elbow, regular 90 o 0.3 elbow, long radius 90 o 0.2 elbow, long radius 45 o 0.2 Tippy Tap Plus Piping Activity Fluid Dynamics Basics Handout 3

4 return bend, 180 o 0.2 globe valve, fully open 10 gate valve, fully open 0.15 Tippy Tap Plus Piping Activity Fluid Dynamics Basics Handout 4

6 Note that the summation symbol ( ) means to add up the losses from all the different sources. A less compact-way to write this equation is: Combining Bernoulli s Equation With Head Loss We can now combine the concepts of Bernoulli s equation and head loss to understand the Fluidic Energy Equation. We divide each term in Bernoulli s equation by the specific weight of the fluid ( ). The specific weight of a fluid is simply the density of the fluid multiplied by the acceleration of gravity:.thus, the effects of head loss result in: Recall our discussion of streamlines from the first section. The left side of this equation represents the state of a fluid particle at the start of our streamline, for example sitting in the water reservoir at the top of a hill. The right side of this equation is the state of our fluid particle at the end of our streamline, for example at the water tap of the residence at the bottom of a hill. The energy of the fluid particle in the reservoir is equal to the energy of the fluid particle at the tap plus the head losses. In other words, because of head losses, some of the energy is sapped from the fluid particle during its journey from the reservoir to the tap. Again, consider our pipeline from the reservoir to the residential tap. Suppose you measure the pressure and velocity at two points in the piping system (along the same streamline) and discover that they are the same for both points. In other words, and. 10. If we now consider head losses along our streamline, what has to be true about the change in elevation between points one and two? 11. If there are no change in elevation or pressure between points one and two, what would you expect the velocity to be at point two ( ), relative to point one? Tippy Tap Plus Piping Activity Fluid Dynamics Basics Handout 6

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