Reynolds Number Regimes

In today’s lesson, we will be discussing the following topics:

1. Intro to Reynolds Number

2. Fluid Flow Graphs

3. Reynolds Number Ranges

4. Reynolds Number Equation

5. Example – Using Reynolds Number

Video coming soon

In this chapter, we will focus on flow in ducts. Since this entire course is related to fluid flow, this chapter will study ducts because piping systems are encountered in many types of engineering designs. There are different pipe geometries (eg. valve, bend, and more – we will get to those), different velocities of the fluid (flow rates), and different properties for each fluid.

Why do we want to know this information? This will be important in designing these pipes when they are encountered in real life.

Intro to Reynolds Numbers

There is no general analysis of fluid flow yet. There are many different solutions, many approximations done by the computer, and a lot of experimental data too – basically, a lot of theory. But there is no ONE general solution, because fluids change behaviour and properties at different variables, such as density, viscosity, velocity and more. There is a number known as the Reynolds number (Re), that depends on some of these variables. This is dimensionless and helps to categorize different types of fluid flow.

There are three categorizations of fluid flow using the Reynolds Number (Re): Laminar flow at low Re (smooth flow), transition to turbulence at intermediate Re (from laminar to turbulent), and turbulent flow at high Re.

Fluid Flow Graphs

Here is a breakdown of what each type of flow looks like, graphing velocity (u) with time (t). Notice that the velocity fluctuations are much higher when the flow becomes turbulent in image (c). This is not a graph of the Reynolds Number – just the different types of flow in terms of velocity.

Reynolds Number Ranges

Here is a breakdown of the Reynolds Number ranges and the corresponding types of fluid flow at each value:

Reynolds Number Equation

Note the following equation, which shows how the Reynolds Number (Re) is a function of the Force Coefficient (Cf). This equation has been used in a previous lesson, but here it is again, with all the variables and units labeled. The pVL/u in the brackets is the equation for the Reynolds Number itself, so it is dependent on the density, length that the fluid travels, velocity and viscosity.

Example – Using Reynolds Number

We are given the Reynolds number in this question, and we have other variables that are used in the Reynolds number equation. We are looking for velocities. Therefore, we can use the Reynolds number equation twice, substitute in the given values (once for water, once for air), and solve for velocity of both air and water.

I had given the Reynolds Number equation above with units and labels for each variable, but here it is again. Note that the L (length) can also be labeled as d in some equations. It is the same thing.

Make sure all units are in kilograms (mass), meters (length) and seconds (time) when you substitute them in – I had to convert the length to meters. Viscosity (u) and density (p) of air and water (at 20C) can be found in textbook tables or online.

Substitute these numbers in and isolate for velocity in each equation.

These are the final answers for velocity!