Reynolds Number: Definition & Equation

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  • 0:03 Studying Flows
  • 0:37 The Reynolds Number
  • 2:15 Water Flow Through…
  • 4:43 Other Applications
  • 5:20 Lesson Summary
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Lesson Transcript
Instructor
Matthew Bergstresser

Matthew has a Master of Arts degree in Physics Education. He has taught high school chemistry and physics for 14 years.

Expert Contributor
Christianlly Cena

Christianlly has taught college physics and facilitated laboratory courses. He has a master's degree in Physics and is pursuing his doctorate study.

Fluids can be air or liquids. Air and liquids move, which is called flow. In this lesson, we will investigate a property of fluid flow called the Reynolds number. This value determines whether the fluid is in laminar flow or turbulent flow.

Studying Flows

Let's start this lesson by taking a second to imagine that we're doing a research project on water flowing through pipes. We want to compare the diameters of two pipes exhibiting laminar flow, which is when water flows smoothly in a predictable fashion. Turbulent flow is when water flows chaotically, making predictions involving its flow difficult. Turbulent flow can cause vibrations, which can cause premature wear in a system leading to failure. Let's focus now on the details surrounding the Reynolds number, which tells us if flow is laminar or turbulent.

The Reynolds Number

The Reynolds number is the ratio of a fluid's inertial force to its viscous force. Inertial force involves force due to the momentum of the mass of flowing fluid. Think of it as a measure of how difficult it would be to change the velocity of a flowing fluid. Viscous forces deal with the friction of a flowing fluid. Think of pouring a cup of tea versus pouring cooking oil. The cooking oil has a higher viscosity because it's more resistant to flow.

If you are thinking inertial force and viscous force are very similar, you are correct. In fact, they are so similar that they have the same units! This means the Reynolds number is unitless. We can determine whether fluid flow is laminar or turbulent based on the Reynolds number.

If the Reynolds number is less than 2300, the flow is laminar. Any Reynolds number over 4000 indicates turbulent flow. In between these values indicates transient flow, which means the fluid flow is transitioning between laminar and turbulent flow. This usually happens only for a short period of time at the beginning or end of fluid flow when a valve or faucet is turned on or off. Let's look at the equation for the Reynolds number.


Reynolds number equation
Reynolds


To explain the variables:

  • R is the Reynolds number, which is unitless
  • ρ is the fluid density in kilograms-per-cubic-meter (kg/m3)
  • v is the velocity in meters-per-second (m/s)
  • D is the diameter of the pipe in meters (m)
  • μ is the viscosity of the fluid in pascal-seconds (Pa⋅s)

Now, let's look at the calculations for the flow through two different pipes.

Water Flow Through Closed Pipes

When designing something, it's smart to make a model first to test it before you go into full-scale production of it. In the real world, if you put in a pipe system and it doesn't work effectively, it has to be fixed, which means time and money are lost. The model has to be comparable to the real thing. Dynamic similitude is when the model and full-scale system have the same proportional conditions. The geometric shapes have to be identical, the velocities of fluids through the systems have to be proportional along with the pressure, viscosity of the fluid, and shear stresses (the non-perpendicular forces on the fluid). An example of how a model and actual system are not in dynamic similitude would be having one pipe horizontal and one pipe not horizontal.

For our project, we want a Reynolds number of 2200 for the water flowing through the two separate pipes. The first pipe has a diameter of 2.75 cm (0.0275 m). The density of water is 1000 kg/m3, and the viscosity of water is 0.0013 kg/(m⋅ s). What velocity does the water have to be through the pipe to fit these parameters? Let's calculate:


calc1


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Additional Activities

Simple Experiment on Laminar and Turbulent Flow

Materials:

  1. Any soft transparent hose of considerable length (i.e., flex pipe of 1 meter long)
  2. Syringe
  3. Dye or Food Coloring (Blue or Red)
  4. Bucket

Experimental Set-up

In this experiment, the dependency of laminar and turbulent flow on the velocity of water in a pipe is tested.

  1. Find a variable water source such as a faucet or a water dispenser for this experiment.
  2. Straighten the hose and attach one of its end to the water source. Make sure that the connection between the hose to the water source offers no leak.
  3. Place a bucket at the other end of the hose. This will serve as the outlet to where the liquid are deposited.
  4. Take the syringe and fill it with dye or food coloring.
  5. Puncture the hose using this syringe filled with dye. It should be punctured in the part of the hose near the water source. Make sure that the needle of the syringe pierces through the hose and that there are no leaks.
  6. Set the flow of the water source to low and while water is running, press the syringe to pump the dye.
  7. Record your observations regarding the flow of the dye.
  8. Repeat the procedure number 6 but with the flow of the water source set to high.
  9. Again, record your observations regarding the flow of the dye.

Follow-Up Questions

  1. Is there a difference in the motion of the dye when the flow of the water is varied?
  2. Which part did you observe the dye creeping its way through the water?
  3. Which part did you see eddies occurring within the dye?
  4. What type of flow is exhibited in each of the scenario above?
  5. How do you distinguish laminar from turbulent flow?
  6. What can you infer about the relationship between the Reynolds number to the velocity of water in the hose?

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