Reynolds Number - Blayne Sarazin

Reynolds Number

Figure 1 The plume from this candle flame goes from laminar to turbulent. If the correct fluid properties are known, the Reynolds number can be calculated to predict where the Laminar-to-Turbulent transition occurs⁹.

The Reynolds number is a dimensionless quantity in fluid mechanics that is used to help predict flow patterns in different fluid flow situations. The Reynolds Number serves as a guide to the laminar-turbulent transition in a particular flow situation,¹ and for the scaling of similar but different-sized flow situations. The Reynolds number is often used to predict the velocity at which a certain fluid flow turns turbulent, while it can also be used to determine what state of flow the fluid in question is currently under. Calculation of the Reynolds number depends heavily on what type of fluid is being utilized, as well as through what type of channel (i.e. pipe flow, duct, open channel, etc.) this fluid is travelling. Figure 1 is a good example of a fluid experiencing all three types of flows: laminar at the bottom, transitional near the middle of the stream (if only very briefly), and turbulent flow towards the top.

The concept was first introduced in 1851 by George Gabriel Stokes,² however it was named by Arnold Sommerfield in 1908³ after Osborne Reynolds, who popularized its use in 1883.⁴

Definition

The Reynolds Number is defined as the ratio of inertial forces to viscous forces in a flowing fluid. It is used in many fluid flow correlations and is used to describe the boundaries of fluid flow regimes (laminar, transitional and turbulent).¹ Viscous force is what tends to keep the layers moving smoothly. When these forces are sufficiently high, this removes any disturbances from the flow and we see what we call laminar flow. However, as velocity increases, inertia forces increase and particles are pushed out of the smoother path. This causes disturbances within the flow, and will eventually lead to what we call turbulent flow.¹¹ Determining whether a flow is experiencing laminar or turbulent flow is quite simple. Reynold's number tells us everything we need to know about the behavior of a given flow scenario.

Reynolds number is determined via the following equation:

[math]\displaystyle{ \mathrm{Re} = \frac{\rho v L}{\mu} }[/math]

where,

• [math]\displaystyle{ \rho }[/math] is the density of the fluid (SI Units: kg/m^3)
• v is the velocity of the fluid (SI Units: m/s)
• L is the characteristic length of the fluid. This varies depending on through what the fluid is flowing (SI Units: m)
• [math]\displaystyle{ \mu }[/math] is the dynamic viscosity of the fluid (SI Units: Ns/m²)

You will sometimes see Reynolds number in a simplified version such as this:

[math]\displaystyle{ \mathrm{Re} = \frac{v L}{\nu} }[/math]

where [math]\displaystyle{ \nu }[/math] is simply the kinematic viscosity of the acting fluid (SI Units: m²/s). ([math]\displaystyle{ \nu = \mu/\rho) }[/math]

The Reynolds Number can be used to determine if flow is laminar, transient or turbulent.⁵ The flow is

• laminar - when Re < 2300
• transient - when 2300 < Re < 4000
• turbulent - when Re > 4000

Flow Types

Reynolds number can tell us the behavior of the flow we are analyzing. Each flow phase corresponds with a specific range of Reynolds numbers. The flow types can be broken down into the following three branches:

Laminar

Laminar flow is the flow that corresponds with low velocities and Reynolds numbers less than 2300.⁵ In this type of flow, the fluid flows in parallel layers, with no disruption between the layers.⁷ At low enough velocities, the fluid will tend to flow without lateral mixing, while adjacent layers simply slide past one another. This can be especially important in microfluidics when you do not want lateral mixing. These phenomenon are displayed well by the streamlines depicted in the laminar flow case of Figure 2. Note that the flow is very clean and without disturbance. Furthermore, laminar flow is entirely reversible, allowing us to return to almost exactly where we began. The following video expertly displays this phenomenon (skip ahead to 20 seconds):

<html>

<iframe width="400" height="250" src="//www.youtube.com/embed/IFR1ETk0Ssc" frameborder="0" allowfullscreen></iframe> </center></html>

Figure 2 Spheres in both Laminar (bottom) and Turbulent (top) flow. The sphere subjected to laminar flow does not display flow separation, mixing, or eddies. The sphere in turbulent flow experiences flow separation and eddy formation behind the sphere.⁸

Transitional (transient)

Transitional or transient flow is the phase of flow that occurs between laminar and turbulent flow, and corresponds to Reynolds numbers that land between 2300 and 4000⁵. In this type of flow, there is a mixture of laminar and turbulent flows present. As Reynolds number increase from 2300 to ~4000, there are an increasing amount of disturbances appearing within the flow.

Turbulent

Turbulent flow is the most common form of flow in nature, and corresponds to the Reynolds numbers higher than a value of 4000⁵. Turbulent flow is ultimately described as chaotic and unpredictable, and is often seen with fluids at high velocities. The flow undergoes irregular fluctuations, or mixing, and continuously changes magnitude and direction⁶. As can be seen from figure 2, the sphere in the upper portion of the figure has steady streamlines in front of it, but severe eddy (vortex) formation behind it. Since turbulent flow is extremely hard to measure compared to laminar flow, experimental tools such as a hot wire probe¹⁰ must be used in order to obtain good results. A hot wire probe is a device that has a very fast response time, with a probe that can respond to temperature changes within 1 millisecond, which makes this tool a good candidate for measuring flows (such as air) that experience constant, rapid change as we see in turbulent flow.

Flow Scenarios

Calculation of the Reynolds number for a given fluid depends on several things such as speed, density, and viscosity. Two of these are properties of the fluid itself and are usually pretty easy to look up. Speed is often a given, or is one of the things you are trying to solve for if initially given a Reynolds number. However, the characteristic length is something that changes based upon what the fluid is flowing through. The following sections will describe how calculation of characteristic length changes, and will show the equations to use in each of these situations.

Pipe Flow

Let's say you have water flowing through a pipe with a diameter of 25 mm at a speed of 3 meters per second. If we take a look at the equation for Reynolds number defined above, we simply need to replace to the characteristic length term with a D, which denotes to diameter of the pipe. The equation then looks as such:

[math]\displaystyle{ \mathrm{Re} = \frac{\rho v D}{\mu} }[/math]

• where D is the diameter of the pipe (SI units: m)

For this specific scenario, the equation with the aforementioned values plugged in becomes:

[math]\displaystyle{ \mathrm{Re} = \frac{(1000kg/m^3)(5 m/s)(0.025 m)}{1.002 Ns/m^2} = 124.75 }[/math]

This is laminar flow based on the previously defined boundaries. To create turbulent flow, the velocity or diameter of the pipe would have to be increased. Increasing either would easily force the flow to approach turbulence.

Duct

Coming soon

Open Channel

Coming Soon

Reynolds Number in Microscopic Flow

Coming soon

In Physiology (Hemodynamics)

Coming Soon

References

1. neutrium.net/fluid_flow/reynolds-number/
2. Stokes, George (1851). "On the Effect of the Internal Friction of Fluids on the Motion of Pendulums". Transactions of the Cambridge Philosophical Society. 9: 8–106.
3. Sommerfeld, Arnold (1908). "Ein Beitrag zur hydrodynamischen Erkläerung der turbulenten Flüssigkeitsbewegüngen (A Contribution to Hydrodynamic Explanation of Turbulent Fluid Motions)". International Congress of Mathematicians . 3: 116–124
4. Reynolds, Osborne (1883). "An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous, and of the law of resistance in parallel channels". Philosophical Transactions of the Royal Society. 174 (0): 935–982
5. www.engineeringtoolbox.com/reynolds-number-d_237.html
6. abyss.uoregon.edu/~js/glossary/turbulent_flow.html
7. Batchelor, G. (2000). Introduction to Fluid Mechanics
8. www.nuclear-power.net/nuclear-engineering/fluid-dynamics/laminar-flow-viscous/
9. en.wikipedia.org/wiki/Reynolds_number
10. web.mst.edu/~cottrell/ME240/Resources/Fluid_Flow/Fluid_flow.pdf
11. http://www.uobabylon.edu.iq/eprints/paper_2_2117_1369.pdf