reynolds number experiment

Engineering Cheat Sheet

The Reynolds Experiment

Background and theory.

This laboratory aims to recreate the Moody diagram.  Moody plotted the Moody Diagram in 1944, and it is now most famous and useful tool in fluid mechanics. The accuracy of the Moody chart is +/- 15 percent [1]. “The Moody Chart gives a good visual summary of laminar and turbulent pipe friction including roughness effects” [1].

The objective of this experiment is to experimentally recreate the Moody Chart, and to observe the transition from laminar to turbulent flow.

The experimental friction factor values are then going to be compared to the Haaland

Equation, which has a margin of error less than 2 percent [1].

                             (Eq. 1)

Where Re is the Reynolds Number, f is the friction factor, and ε/d is relative pipe roughness to diameter of pipe ratio .

Experimental Setup and Procedure

See Figure 1: Moody Diagram below. To begin this experiment you selected any size pipe. It was assumed that all pipes were smooth and there for had a relative pipe roughness of 0. The desired valve was then opened and the mass flow rate was calculated. Afterwards the pressure drop was then measured for that specific mass flow rate. Using the mass flow rate, the velocity of the fluid could be calculated using Eq.2.

Velocity ( V ):

                                              (Eq. 2)

Where V was the velocity (m/s).  M_dot  is the mass flow rate (kg/s). ρ is the density of the fluid (kg/ m 3 ). In this case the fluid used was water so: ρ water   at 20 o C : 998 kg/ m 3 [1]. Finally A is the cross sectional area of the pipe tested (m 2 ).

After the velocity of the water had been calculated the Reynolds Number was then found using Eq 3.

                           (Eq. 3)

Where Re #   is the Reynolds Number. ρ is the density of water (kg/ m 3 ). V is the average velocity of the fluid (m/s). d is the diameter of the pipe used (m).  μ is the dynamic viscosity of the fluid (N s/m 2 ). The fluid used was water so: μ water at 20 o Cis 1.002 x 10 -3 N s/m 2 [1].

Using the Reynolds number the theoretical friction factor could be calculated using Eq. 1 Haaland Equation from above. This friction factor was compared to the actual friction factor, which was calculated using Eq. 4.

                                 (Eq. 4)

where f is the experimental friction factor, d is the diameter of the pipe used (m), ρ is the density of water (kg/ m 3 ), V is the velocity (m/s) and L is distance between measured points (m). 

The experimental friction factor is then plotted against the Reynolds Number to recreate the Mood Diagram.

Experimental Results

All experimental results are presented in the appendix. When comparing the experimental result to the theoretical result the trends are that the % error is much lower with the turbulent regime then the laminar regime. See Table 1: Turbulent Flow below and Table 2: Laminar Flow at the top of the following page.

When this data was graph based off the Moody Diagram (Figure 1), the expectation was to see the laminar flow to linear resemblance line, meanwhile for turbulent flow the expectations were to see more of an exponential style graph. Even though the laminar flow had much higher % errors from theoretical the graph did resemble a linear line. One the other hand the turbulent flows had a much lower % error but due to the lack of range it was difficult to really see any shape to the curve. See Figure 2: Experimental Moody Chart below.

Examining the data of Figure 2 one can see that the experimental Moody Chart is missing a large chunk. The lack of data points in the turbulent regime was due to the fact that when conducting this experiment you could only open the value so much while still being able to get pressure readings. If you exceeded the capacity of the system to read the pressure difference then you were not able to calculate the experimental friction factor. Even though that a complete moody diagram was not completed the turbulent did have much lower % errors ranging from 5.18% – 18.70% error. Where as the laminar flow had a much higher % error ranging from 53.93% – 59.70% . One of the reasons that this could have happened is due to the fact that with laminar flow the measured mass flow rate has a large impact in the accuracy of the results. When you are dealing with turbulent flows and larger mass flow rates some error in the technique calculating mass flow might have a small effect on your over all calculating. Where as when you are dealing with smaller flow rates the same error accumulates and results in a much larger error.  The reason that the technique used to calculate mass flow rate is the most probably cause is because the errors in the laminar flow are very consistence. So even though the experimental friction factors were incorrect or had low precision they were very accurate. Another place this is low precision high accuracy of the laminar regime is seen, is when a linear regression was done the R 2 value was 0.96 which indicates the values are represented well with a linear line. Some of the random error could be reduced by taking more than three repeats at each flow rate setting.

After conducting this experiment it is interesting to see how the friction factor has more of an effect in laminar regime instead of turbulent. This is believed to be cause because laminar flows tend to have a lower velocity there for they are more susceptible to the interior friction of the pipe. 

Some things that could be changed to improve the results of this experiment is improve how the flow rates are calculated. Also if multiple trials were ran without changing the mass flow rate, then it could be easier to catch errors which are due to technique. If the range that pressure measurement could be taken also increased then it would have been easier to reproduce the moody diagram because the data points would be more spread out.

• F. M. White, Fluid Mechanics , 5th edition, , McGraw Hill, 2003
• J.P. Holman, Experimental Methods for Engineers , 7th Edition, McGraw Hill, New York, 2001

Sample Calculations:

Area ( A ):

Reynolds Number (Re #):

Friction Factor ( f ):

Haaland Eq ( f ):

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Reynolds Experiment

For a particular fluid flow, depending on the velocity, the flow shows the laminar, transition and turbulent patterns. In laminar flow, the fluid flows in a layer without disturbing the other layer, whereas in turbulent flow the fluid does not follow any regular layer and the flow is highly chaotic. The flow shows the randomness of turbulent flow due to elevated dissipation. The flow exhibits intermittent behaviour in the transition regime, sometimes laminar and sometimes turbulent. You can compare the essence of the flow with a non-dimensional number called the Reynolds number.

Reynolds number is defined as the ratio of inertia to viscous force in a flow. For a particular fluid i.e. for constant viscosity, the flow transits from laminar to turbulent as the velocity of the flow is increased. The Reynolds number at which the flow starts to transit from laminar is called the critical Reynolds number.

The Reynolds number for a flow in a pipe is obtained using following equation: \[R_e = {\rho V D \over \mu}\] here, ρ is the density, V is the average flow velocity in the pipe, D is the pipe diameter and μ is the dynamic viscosity.

The flow transits from laminar to turbulent for a specific fluid, i.e. for continuous viscosity, as the flow velocity is increased. The critical number of Reynolds is the Reynolds number at which the flow begins to transition from laminar to turbulent. The essential Reynolds number for a flow through a pipe is generally calculated to be 2300. On the basis of the experimental data, Reynolds classified the flow regimes as follows:

Laminar < 2300

2300 < Transition < 4000

Turbulent > 4000

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The Reynolds Experiment

Nicole sharp - february 6, 2014 august 12, 2019.

One of the most famous and enduring of all fluid dynamics experiments is Osborne Reynolds ’ pipe flow experiment, first published in 1883 and recreated in the video above. At the time, it was understood that flows could be laminar or turbulent , though Reynolds’ terminology of direct or  sinuous is somewhat more poetic:

Again, the internal motion of water assumes one or other of two broadly distinguishable forms-either the elements of the fluid follow one another along lines of motion which lead in the most direct manner to their destination, or they eddy about in sinuous paths the most indirect possible. #

There had, however, been no direct evidence of these eddies in a pipe. Reynolds built an apparatus  that allowed him to control the velocity of flow through a clear pipe and simultaneously introduce a line of dye into the flow. He carefully varied the velocity and temperature (and thus viscosity) in his apparatus and not only documented both laminar and turbulent flow but found that the transition from one to another could be described by a dimensionless number he derived from the Navier-Stokes equation . This number was dependent on the fluid’s velocity and kinematic viscosity as well as the diameter of the pipe. This was the birth of the Reynolds number , one of the most important parameters in all of fluid dynamics. (Video credit: S. dos Santos; research credit: O. Reynolds )

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The Reynolds Number

The dimensionless Reynolds number is the ratio of inertial forces to viscous forces and is used to distinguish laminar and turbulent fluid systems.

Introduction

The intricate motion of fluids is incredibly difficult to predict due to its nonlinear nature. As engineers, physicists or biologists, we can luckily predict whether the flow in a given system should be laminar or turbulent, thanks to an Irish genius and physicist named Osborne Reynolds.

Reynolds was a pioneer in the study of fluid dynamics, performing a simple but elegant experiment to demonstrate that the transition point between the two types of flow could be predicted by one simple number, that we now know as the important and convenient Reynolds number, commonly referred to as Re .

Although the concept was introduced by George Stokes in 1851, Arnold Sommerfeld named the Reynolds number in 1908 after Osborne Reynolds!

In general, the Reynolds Number is defined as the ratio of inertial force to viscous forces and quantifies the relative importance of these two types of forces for given flow conditions - an important concept that we will also cover in this blog post.

To this day, the Reynolds number remains the standard mathematical framework to study laminar & turbulent fluid systems.

Formula & Derivation 💦

The Reynolds number is defined as a characteristic length multiplied by a characteristic velocity and divided by the kinematic viscosity.

\[ Re = \frac{\rho VD}{\mu} = \frac{VD}{\nu} \]

• V is the flow velocity
• D is the characteristic dimension of the object (traveled length of the fluid such as the hydraulic diameter, chord length of an airfoil etc.)
• ρ fluid density (kg/m 3 )
• μ dynamic viscosity (Pa•s)
• ν kinematic viscosity (m 2 /s) 👉 ν = μ / ρ

Whenever viscous forces are dominant (slow flow, low Re), they are sufficient enough to keep all the fluid particles in line, then the flow is called laminar . Very low Reynolds Numbers indicate viscous creeping motion, where inertial effects are negligible. When the inertial forces dominate over viscous forces (when the fluid flows faster and Re is larger), the flow is said to be turbulent .

Laminar Flow 💧

Laminar flow is the movement of fluid particles along well-defined paths or streamlines, where all the streamlines are straight and parallel. Hence, the particles move in laminar or layers gliding smoothly over the adjacent layer.

Laminar flow occurs in small diameter pipes in which fluid flows at lower velocities and high viscosity. This type of flow is also called streamline flow or viscous flow.

Characteristics of laminar flow :

• The exception and not the rule for engineering applications
• Fluid fluid travels smoothly or in regular, straight lines
• Layers of water flow over one another at different speeds with virtually no mixing between layers
• The flow velocity profile for laminar flow in circular pipes is parabolic in shape, with a maximum flow at the centre of the pipe and a minimum flow at the pipe walls

The velocity distribution at a cross section of laminar flow will be parabolic in shape with the maximum velocity at the centre being about twice the average velocity in the pipe. In turbulent flow, a fairly flat velocity distribution exists across the section of pipe due to increased momentum transfer to the walls.

Turbulent Flow 🌀

Turbulent flow is defined as the flow in which the fluid particles move in a zigzag way. Due to the movement of fluid particles in a zigzag way, the formation of eddies takes place, which is responsible for high energy loss.

In turbulent flow, the speed of the fluid at a point continuously changes in both magnitude and direction. Turbulent flow tends to occur in large diameter pipes in which fluid flows with high velocity.

The main tool available for the analysis of turbulent flow is CFD analysis. CFD is a branch of fluid mechanics that uses algorithms and numerical analysis to analyse and solve problems that involve turbulent fluid flows.

It is widely accepted that the Navier-Stokes equation or simplified Reynolds-averaged Navier-Stokes equations are the basis for essentially all CFD codes.

The type of flow is determined by a non-dimensional number called the Reynolds number for a pipe flow.

Characteristics of turbulent flow :

• The rule and not an exception common type of flow
• The large diffusive nature of turbulence causes rapid mixing & increased rates of mass, momentum and energy transfer
• The flow velocity profile for turbulent flow is fairly flat across the centre section of a pipe and drops rapidly extremely close to the walls
• Turbulence is rotational, 3-dimensional, and characterised by high levels of fluctuating vorticity
• The randomness or irregular nature of turbulent flows makes a solely deterministic approach impossible.

One must therefore rely on statistical methods !

In principle, it is possible to simulate any turbulent flow by solving the Navier-Stokes Equations exactly using appropriate boundary conditions and suitable numerical procedures such as Direct Numerical Simulation (DNS) which is however too computationally expensive and rarely done in practice.

This opens up the area of turbulence modelling using statistical approaches which will be discussed in future blog posts - so stay tuned and subscribe !

Non-Dimensionalisation of the Navier-Stokes Equations 🔴

One way of finding the Reynolds Number is to non-dimensionalise the momentum equation for an incompressible flow by choosing some appropriate scaling values.

We start with

\[ \rho \mathbf{u} \cdot \nabla \mathbf{u} = -\nabla p + \mu \nabla^2 \mathbf{u} \]

and now choose the appropriate scaling values

\[ \tilde{x} = \frac{x}{L} \]

\[ \mathbf{\tilde{u}} = \frac{\mathbf{u}}{U} \]

\[ \mathbf{\tilde{p}} = \frac{p}{P} \]

where U and P are the characteristic velocity and pressure respectively. This results in:

\[ \rho \frac{U^2}{L} \mathbf{\tilde{u}} \cdot \tilde{\nabla} \tilde{\mathbf{u} } = - \frac{P}{L} \tilde{\nabla} \tilde{p} + \mu \frac{U}{L^2} \tilde{\nabla}^2 \tilde{\mathbf{u}} \]

The next step is to divide through the last big term that appears in the equation, namely

\[ \mu \frac{U}{L^2} \]

and define the characteristic pressure P as

\[ P = \mu \frac{U}{L} \]

which finally results in the equation we need to find the Reynolds Number

\[ Re \, \mathbf{\tilde{u}} \cdot \tilde{\nabla} \tilde{\mathbf{u} } = -\tilde{\nabla} \tilde{p} + \tilde{\nabla}^2 \tilde{\mathbf{u}} \]

For Reynolds Numbers way smaller than 1 (Re<< 1), where viscous effects dominate, we see that the convective term on the left becomes negligible compared to the pressure gradient and viscous stress tensor on the right.

The scenario for Re>>1, we need to divide by

\[ \rho \frac{U^2}{L} \]

and the characteristic pressure P defined as

\[ P = \rho U^2 \]

which yields

\[ \mathbf{\tilde{u}} \cdot \tilde{\nabla} \tilde{\mathbf{u} } = -\tilde{\nabla} \tilde{p} + \frac{1}{Re} \tilde{\nabla}^2 \tilde{\mathbf{u}} \]

The viscous stress tensor on the right-hand side becomes negligible compared to the pressure gradient and the nonlinear convection term on the left.

The Buckingham π-Theorem 🤔

The Buckingham π-theorem (also known as Pi theorem) is used to determine the number of dimensional groups required to describe a phenomena and an alternative way to derive the Reynolds Number. Using this method, we collect all of the units of a problem which is length , velocity , density and viscosity and then combine these quantities into a dimensionless number.

To run you through how that is actually performed, let's recap on the used quantities for a second.

\[ Length = L \]

\[ Velocity = L/T \]

\[ Density = M/L^3 \]

\[ Viscosity = M/LT \]

We can eliminate T by taking the ratio of the velocity and viscosity:

\[ Velocity/Viscosity = \frac{LLT}{TM} = \frac{L^2}{M} \]

The mass can now be eliminated by multiplying with the density

\[ \frac{M}{L^3} \cdot \frac{L^2}{M} = \frac{1}{L} \]

Now we can simply have to multiply with the length and we receive

\[ \frac{1}{L} \cdot L = 1 \]

Summarising all the steps results in

\[ \frac{Velocity \cdot Density \cdot Length}{Viscosity} = Re \]

which can be simplified by formulating the Reynolds Number using the kinematic viscosity that we used in the introduction of this article

\[ \nu = \frac{\mu}{\rho} \]

A more thorough explanation of the π-theorem will be delivered in a separate blog post!

Unit Check ✅

Let’s take each term of the Reynold's Number one-by-one.

• The velocity is Length per Time, so [L/T]
• The primary dimension of ρ is Mass per Length cubed, so [M/L 3 ]
• The dynamic viscosity μ is [M/LT]

Putting all these values in the formula yields

\[ Re = \frac{L}{T} \cdot \frac{M}{L^3} \cdot \frac{LT}{M} \cdot L \]

\[ Re = \frac{L^3 MT}{TL^3M} = 1 \]

Which means the Reynolds Number is dimensionless !

The Reference Length 📏

The reference length or characteristic length in the Reynolds number can be anything convenient, as long as it is consistent, especially when comparing different geometries.

When calculating the Reynolds number for airfoils and wings for example, the chord length is often chosen over the span because the former is a representative function of the lift. For pipes, the diameter is the characteristic length. For rectangular pipes a suitable choice is the hydraulic diameter defined as

\[ D_h = \frac{4A}{P} \]

where A is the cross-section area (in square metre) and P is the wetted perimeter (in metre). The wetted perimeter defines the total perimeter of walls in contact with the flow.

If the characteristic length is unknown, it helps to think about the growth of the boundary layer.

• For flow of fluid over a flat plate, the boundary layer grows along the length of the plate which is the characteristic dimension.  
• For flow in a tube, the boundary grows in the radial direction which is known as the developing length.  
• In the fully developed region, the boundary layer thickness is equal to the diameter of the tube.

Importance of Reynolds Number❗

When viscous forces dominates over the inertial forces, the flow is smooth and if someone would put dye or ink into the fluid, you would see a regular pattern and no mixing whatsoever. The Reynolds Number for this flow would comparatively be low and is known as laminar flow.

On the other hand, when inertial forces are dominant, the value of Reynolds number is comparatively higher and is called turbulent flow and ink would start to show chaotic and mixing behaviour. At low Reynolds Number values of smaller than ≈2,300 for pipe flow, the viscous force is sufficient enough to keep the fluid at a smooth and constant fluid motion. At large Reynolds Number values of around 4,000, the flow tends to produce chaotic eddies, vortices, and other flow instabilities making the flow turbulent .

Note that in the whole Reynolds Number range, there is no sudden jump from laminar to turbulent flow. What we have is a transitional region in between, also called the laminar-turbulent transition.

In between the regime of Re=2,300 & 4,000, this transition occurs and it crinkles up into complicated, random turbulent flow. It is important to know that turbulence distinguishes different type of flows and obstacles where flow becomes turbulent under specific conditions. The transition from laminar to turbulent flow depends on the surface geometry, surface roughness, free-stream velocity, surface temperature, and type of fluid, and much more.

Reynolds Experiment 💧

Reynolds' investigations of the flow of fluids through tubes was published in 1883 (Reynolds, 1883). He showed most effectively that the characteristics of the flow vary with the flow velocity, and demonstrated the features of laminar and turbulent flow.

Reynolds passed water through a pipe at different flow velocities and introduced dye into the tubes to visualise fluid flow behaviour. His apparatus was very simple as can be seen in the illustration below and consisted of a glass-sided tank, 6 feet long, 18 inches deep and 18 inches wide

Inside the tank was a glass tube with 'a trumpet mouth of varnished wood, great care being taken to make the surface of the wood continuous with that of the glass' so that water could be injected with minimal disturbances.

On one side, the tube was connected to an iron pipe equipped with a valve which could be controlled by means of a long lever. On the other was the device for introducing a streak of dye into the trumpet. A float connected to a pulley and dial arrangement was used to register the water-level in the tank and hence the volume being discharged through the glass tube

Experiment 1️⃣

When the velocities were sufficiently low, the streak of colour extended in a beautiful straight line through the tube.

Experiment 2️⃣

At sufficiently low velocities, the streak would shift about the tube, but there was no appearance of sinuosity.

Experiment 3️⃣

As the velocity was increased by small stages, at some point in the tube, always at a considerable distance from the trumpet or intake, the colour band would all at once mix up with the surrounding water, and fill the rest of the tube with a mass of coloured water - this is turbulent flow, showing its diffusive behaviour and causing the dye to spread out.

Read more about the experiment in the original publication by Osborne Reynolds named "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."

Critical Reynolds Number ⚡

After this experiment, Reynolds tried to find the critical condition for an eddying flow to revert into a non-turbulent one as the flow rate is reduced, referring to this as the ' inferior limit '. To do this, he used equipment which allowed water to flow in a disturbed state from the mains supply through a length of pipe and measured the pressure-drop over a five-foot length near the outlet

The Reynolds number at which the flow becomes turbulent is called the critical Reynolds number . The value of the critical Reynolds number is different for different geometries and is a highly complicated process, which is not yet fully understood.

• For flow over a flat plate, the generally accepted value of the critical Reynolds number is Rex ≈ 500000 .
• For flow in a pipe of diameter D, experimental observations show that for “fully developed” flow, laminar flow occurs when ReD < 2300, and turbulent flow occurs when ReD > 3500 .
• For a sphere in a fluid, the characteristic length-scale is the sphere’s diameter, and the characteristic velocity is that of the sphere relative to the fluid. Purely laminar flow only exists up to Re = 10 under this definition.

It should be noted that the value of about 2,300 often quoted for the critical Reynolds Number is only applicable to flow in pipes, and has no relevance to other flow phenomena that depend on a Reynolds number! For instance, the Reynolds Number for water in open channels is about 6,000, and for the flow of air across a plane wing it may be from ≈50,000 upwards. As mentioned earlier, for flow in pipes and tubes engineers take the diameter as the "characteristic length measurement" (reference length).

Important Note ⚠️

I have seen a ton of students making the mistake to assume that the critical Reynolds Number is the same for every type of flow and every obstacle, which is incorrect!

The critical Reynolds Number for a pipe flow for example are not fix and can be delayed up to a Re of 50,000 or even up to 100,000 depending on the conditions without the flow becoming turbulent. The initial critical Reynolds Number found by Reynolds in his experiment for example cannot be reproduced at the same facility today as cars and trucks cause vibration in the ground that trigger a transition into a turbulent state even before the magic threshold of Re = 2,300.

Note that you will often hear that turbulent flows are special and laminar flow is the exception where it is the other way around.

Turbulence is the law, laminar is the special case.

Reynolds Number Examples👇

• A large whale swimming at 10 m/s 👉 Re = 300,000,000
• A duck flying at 20 m/s 👉 Re = 300,000
• Blood flow in the aorta 👉 Re = 1,000
• Blood flow in your brain 👉 Re = 100
• Flapping wings of the smallest flying insect 👉 Re = 30
• A bacterium swimming at 0,01 mm/s 👉 Re = 0,00001

High values of the Reynolds Number on the order of ≈10 million indicate that viscous forces are small (not negligible) and the flow is essentially inviscid. In this case, one can invoke the Euler equations to model the flow. On the other hand, low Reynolds Numbers indicate that viscous forces play an important role and must be considered.

Reynolds' Law of Similarity ✅

If you are investigating two geometrically similar flows, they are both equal to each other as long as they embrace the same Reynolds Number according to Reynolds' Law of Similarity. Example: If you're in the wind tunnel and investigate the flow around a vehicle, you would need to have twice the speed inside the wind tunnel in order to make up for the half-scale model. Mathematically, this would look as follows:

\[ Re_{full} = \frac{\rho VD}{\mu} \]

\[ Re_{half} = \frac{\rho \mathbf{2V} \mathbf{\frac{1}{2}D}}{\mu} \]

Applications of the Reynolds Number👇

The Reynolds number plays an important part in fluid dynamics and heat transfer problems calculations.

• For the Darcy-Weisbach equation for example, it is essential to calculate the friction factor in a few of the equations of fluid mechanics. It relates the loss of pressure or head loss due to friction along the given length of pipe to the average velocity of the fluid flow for an incompressible fluid
• It is used to predict the transition from laminar to turbulent flow and in the scaling of similar but different-sized flow situations, such as between an aircraft model in a wind tunnel and the full-size version.
• The predictions of the onset of turbulence and the ability to calculate scaling effects can be used to help predict fluid behavior on a larger scale, such as in local or global air or water movement, and thereby the associated meteorological and climatological effects.

The Moody Diagram 📊

The Moody Chart (or Moody Diagram) is a diagram used in the calculation of pressure drop or head loss due to friction in pipe flow and is used to find the friction factor for flow in a pipe.

The plot shows the friction factor plotted over the Reynolds Number and relative roughness . Relative roughness is defined as the pipe roughness height, ε (epsilon), divided by the inside diameter, D. Once the Reynolds Number and relative roughness are known, the friction factor can be found from the chart and used in the Darcy-Weisbach equation for calculating the pressures loss caused by friction.

That's all folks! If you’d like to see more Fluid Mechanics blogs like this one, consider subscribing to my latest blogs, tutorials and course updates - and please feel free to leave a comment down below! 🙂

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A simple and affordable experiment to determine Reynolds number

Lewis A Baker 1 and Alison M Taylor 1

Published 24 September 2019 • © 2019 IOP Publishing Ltd Physics Education , Volume 54 , Number 6 Citation Lewis A Baker and Alison M Taylor 2019 Phys. Educ. 54 063004 DOI 10.1088/1361-6552/ab430a

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1 Faculty of Engineering and Physical Sciences, University of Surrey, 388 Stag Hill, Guildford, GU2 7XH, United Kingdom

Lewis A Baker https://orcid.org/0000-0002-0097-4011

Alison M Taylor https://orcid.org/0000-0002-3599-0038

• Received 31 May 2019
• Accepted 10 September 2019
• Published 24 September 2019

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Method : Single-anonymous Revisions: 2 Screened for originality? Yes

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1. Introduction

Students studying a whole range of physics and engineering related disciplines need a solid grasp of the concepts of fluid mechanics: e.g. chemical engineers studying fluid flow through pipelines, civil engineers involved in the construction of structures including dams and tidal barrages need to understand the behaviour of water under both static and dynamic conditions, physicists modelling complex weather patterns—all rely on an understanding of how fluids flow. The Reynolds number (Re) is a useful dimensionless quantity in fluid dynamics which characterises the stability of a fluid. It is defined by the ratio of inertial forces to frictional forces acting on the bulk fluid [ 1 ].

Laminar flow (also described as steady, streamlined, orderly or uniform flow), is characterised by low Reynolds numbers, and describes layers of fluids flowing parallel over one another smoothly. Turbulent flow (also described as disorderly flow), describes fluid flow which is chaotic and disordered since fluid particles move with a large distribution of velocities. It is characterised by high Reynolds numbers [ 2 , 3 ]. The terms 'high' and 'low' here are relative since the Reynolds number depends on the exact shape of fluid flow (the geometry of the pipe or surface flow occurs through) as well as the physical properties of the fluid itself (density and viscosity) given by the equation:

To this end, we describe a simple experiment which is accurate enough to give quantitative insight into the Reynolds numbers for laminar and turbulent flow, but simple enough that students could construct the apparatus as part of the experiment using only basic laboratory equipment. We identify key challenges in performing the experiment as well as opportunities for additional investigation meaning this experimental set-up can be differentiated for many levels of study.

2. Methodology

A small (6 mm) hole is drilled into the bottom of a 1 litre screw-top bottle, see figure 1(a) . The bottle is filled close to the top with tap water whilst using a finger or suitable alternative (such as Blu-Tack TM ) to keep the water in and the screw-cap replaced. Opening the screw-cap a small amount allows the water to exit under laminar flow (an example is given in figure 1(b) ), whilst opening much further allows water to exit under turbulent flow (figure 1(c) ). This screw-cap therefore operates as the flow rate control, by limiting the rate at which atmospheric air is able to replace the volume of water lost during fluid flow. In the case of the screw-cap being fully tightened, fluid flow would cease, due to a drop in pressure above the surface of the water created by an increase in the available volume for the air to expand into due to the loss of fluid. This balances the atmospheric pressure outside the hole in the base of the bottle.

Figure 1.  (a) Image of the experimental set-up. (b) An example of laminar flow and (c) turbulent flow in this experiment.

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A small vertical pen mark can be drawn across the screw-cap and bottle to mark the closed position. Once laminar (or turbulent) flow has been established, two additional vertical pen marks can be drawn on the bottle, marked 'L' for laminar and 'T' for turbulent. This ensures repeat measurements can be taken more reliably, by lining up the mark on the screw-cap with the mark on the bottle, see figure 2 .

Figure 2.  A schematic of the experimental set-up used in this work. A 1 litre plastic screw-top bottle has a 6 mm hole drilled into the bottom. When filled, the screw-cap can be loosened to increase the flow rate of water leaving through the hole. A 500 cm 3 measuring cylinder is used to collect 300 cm 3 of water. A basin is used to contain the experiment and any water spillage. Labels can be used on the bottle and screw-cap to identify the tightness of the cap used for laminar flow (L) and turbulent flow (T) making repeat measurements more reliable.

After the bottle has been marked for the positions of the screw-cap for laminar and turbulent flow, measurements are then taken. Once the flow has been established and appears consistent, the bottle is placed on top of a 500 cm 3 measuring cylinder at which point a stopwatch is started. After 300 cm 3 of water has been collected the stopwatch is stopped. A schematic of this experimental set-up is given in figure 2 . The bottle can then be refilled and the measuring cylinder emptied so that the experiment can be repeated.

Reynolds numbers below this transition can be considered more laminar-like and above this value more turbulent-like. From this we can confirm we are indeed observing laminar flow.

4. Discussion and conclusions

4.1. notes on the methodology.

We found that when establishing laminar flow it helps to open the screw-cap more, before slowly tightening it back up to the desired flow rate. We also found that the initial flow rate (particularly for very low Reynolds numbers) can be unstable, so we suggest that the flow is established and collected over the basin. Once the flow appears consistent, it is then lifted onto the measuring cylinder, where timing may begin.

A 500 cm 3 measuring cylinder is used as a suitable size to place the bottle upon, whilst 300 cm 3 is a reasonable volume to collect with this size of bottle. As the volume of the water inside the bottle decreases the flow rate of water leaving the bottle also decreases. This is because the pressured exerted on the base of the bottle depends on the height of the column of water above it. For this reason, it is a good idea to refill the bottle after each measurement. Over the course of collecting 300 cm 3 we measured a height difference of around 55 mm from an initial height of 180 mm, which corresponds to a pressure difference of approximately 0.54 kPa. This manifests itself as a systematic error in our reported measurements. However, it is worth noting that this drop in pressure does not cause any obvious observable differences in the flow rate over the volume collected, and in all our ensuing calculations, we assume the flow rate is indeed constant through the bottle aperture. Both laminar and turbulent flow regimes can still be clearly identified, suggesting the size of this error is acceptable for determining Reynolds number. This systematic error could be reduced by collecting a smaller volume of water, however, at the expense of an increased relative error in the measured mean time taken to collect this volume of water.

For measurements concerning laminar flow, we found that if the flow was too slow, instabilities were large enough to cause an inconsistent flow rate, for the given diameter of the hole. This can be avoided by increasing the bulk flow speed (increasing the Reynolds number, but still within the laminar region).

Finally, the bottle should be handled by the thick plastic ring below the bottle cap. The reason for this is that disruptions to the screw-cap fixing or to the shape of the bottle will increase or decrease the pressure on the water exiting the bottle, altering the flow rate. Once the bottle has been placed onto the measuring cylinder it can be left alone whilst the measurement is taken.

4.2. Suggestions for differentiated activities

There are a number of options to differentiate these activities:

• (i)   The number of repeats in this work was 30. This could be decreased to perhaps 5 to provide an estimate of the uncertainty in the measurements.
• (ii)   The error analysis in the time taken was propagated through to the calculated Reynolds number. This was done using standard equations derived from normally-distributed errors. This could be made more involved by also propagating estimated errors on the volume of water collected, the characteristic length and the given values of density and viscosity. The analysis could be less involved by the addition of uncertainties (such as those found in A-level syllabi). Finally, the analysis could be left to simply calculating the Reynolds number, without propagating uncertainties, instead leaving a qualitative exercise to identify sources of uncertainties.
• (iii)   Bottles with different characteristic lengths could be constructed and investigated.
• (iv)   Different fluids could be investigated in a similar manner. We note, however, that the quantities involved might need to be altered to suit the availability of the fluid in question.

To conclude, we have described a simple experiment to determine the Reynolds number using tap water and basic laboratory equipment. We have shown that the calculated Reynolds numbers for laminar and turbulent flow are consistent with accepted values. This experiment can be differentiated to suit the level of study with some suggestions given.

Acknowledgments

LAB and AMT thank the University of Surrey Faculty of Engineering and Physical Sciences for support and our students for trying out the experiment.

Biographies

Lewis A. Baker completed his MPhys and PhD in Mathematical Biology and Biophysical Chemistry at the University of Warwick in 2017. He then trained as a secondary-school teacher before combining these experiences in his current position as Teaching Fellow on the Faculty of Engineering and Physical Sciences Foundation Year at the University of Surrey.

Alison M. Taylor completed a B.Eng (Hons) in Materials Technology and then a PhD in Surface Analysis at the University of Surrey. She spent 23 years teaching the physical sciences at a large high performing state secondary school. She currently holds the post of Teaching Fellow on the Faculty of Engineering and Physical Sciences Foundation Year, having returned to the University of Surrey.

Reynolds Number

Article summary & faqs, what is the reynolds number.

Application of Reynolds Number

History of reynolds number, critical reynolds number, reynolds’ law of similarity, laminar vs. turbulent flow, reynolds number regimes, reynolds number and internal flow, power-law velocity profile – turbulent velocity profile, hydraulic diameter, example: reynolds number for primary piping and a fuel bundle, reynolds number and external flow.

• Properties Of Fluids

Reynolds Number

What is a reynolds number.

Reynolds number is a dimensionless quantity that is used to determine the type of flow pattern as laminar or turbulent while flowing through a pipe. Reynolds number is defined by the ratio of inertial forces to that of viscous forces.

Reynolds Number Formula

It is given by the following relation:

\(\begin{array}{l}Reynolds\;Number = \frac{Inertial\;Force}{Viscous\;Force}\end{array} \)

• R e is the Reynolds number
• ρ is the density of the fluid
• V is the velocity of flow
• D is the pipe diameter
• μ is the viscosity of the fluid

If the Reynolds number calculated is high (greater than 2000), then the flow through the pipe is said to be turbulent. If Reynolds number is low (less than 2000), the flow is said to be laminar. Numerically, these are acceptable values, although in general the laminar and turbulent flows are classified according to a range. Laminar flow falls below Reynolds number of 1100 and turbulent falls in a range greater than 2200.

Laminar flow is the type of flow in which the fluid travels smoothly in regular paths. Conversely, turbulent flow isn’t smooth and follows an irregular math with lots of mixing.

An illustration depicting laminar and turbulent flow is given below.

The Reynolds number is named after the British physicist Osborne Reynolds. He discovered this while observing different fluid flow characteristics like flow a liquid through a pipe and motion of an airplane wing through the air. He also observed that the type of flow can transition from laminar to turbulent quite suddenly.

Reynolds Number Example Problems

Problem 1- Calculate Reynolds number, if a fluid having viscosity of 0.4 Ns/m 2  and relative density of 900 Kg/m 3  through a pipe of 20 mm with a velocity of 2.5 m.

Solution 1 – Given that,

 Viscosity of fluid μ

\(\begin{array}{l}\mu =\frac{0.4Ns}{m^{2}}\end{array} \)

Density of fluid ρ

\(\begin{array}{l}\rho=900Kg/m^{2}\end{array} \)

Diameter of the fluid

\(\begin{array}{l}L=20\times 10^{-3}m\end{array} \)

\(\begin{array}{l}R_{e}=\frac{\rho VL}{\mu }\end{array} \)

\(\begin{array}{l}=\frac{900\times 2.5\times 20\times10^{-3}}{0.4}\end{array} \)

\(\begin{array}{l}=112.5\end{array} \)

From the above answer, we observe that the Reynolds number value is less than 2000. Therefore, the flow of liquid is laminar.

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Osbourne Reynolds Apparatus Experiment

INTRODUCTION

        The experiment is conducted mainly to study the criterion of laminar, transition and turbulent flow. In fluid mechanics, internal flow is defined as a flow for which the fluid is confined by a surface. The flow may be laminar or turbulent. Osborne Reynolds (23 August 1832 – 21 February 1912) was a prominent innovator in the understanding of fluid dynamics and mechanics.

        Osborne Reynolds Apparatus consists of water resource for the system supply, fix-head water input to big and small transparent pipes, dye input by injection unit, and water output unit to determine water flow rate. The laminar, transition and turbulent flows can be obtained by varying the water flow rate using the water outlet control valve. Water flow rate and hence the flow velocity is measured by the volumetric measuring tank. The supply tank consists of glass beads to reduce flow disturbances. Flow patterns are visualized using dye injection through a needle valve. The dye injection rate can be controlled and adjusted to improve the quality of flow patterns.

AIMS / OBJECTIVES

• To observe the characteristics of laminar, transition and turbulent flow.
• To prove that the Reynolds number is dimensionless by using the formula;

        In fluid mechanics, Reynolds Number (R e ) is a dimensionless number that is expressed as the ratio of inertial forces (pV 2 /L) to viscous forces (µV/L 2 ). Thus, the Reynolds number can be simplified as followings;

R e  = (pV 2 /L) / (µV/L 2 )

= pVL/µ

Where p is the density of the fluid, V is the mean fluid velocity, L is a characteristic linear dimension, and µ is the dynamic viscosity of the fluid.

When a fluid flows through a pipe the internal roughness (e) of the pipe wall can create local eddy currents within the fluid adding a resistance to flow of the fluid. Pipes with smooth walls such as glass, copper, brass and polyethylene have only a small effect on the frictional resistance. Pipes with less smooth walls such as concrete, cast iron and steel will create larger eddy currents which will sometimes have a significant effect on the frictional resistance. The velocity profile in a pipe will show that the fluid at the centre of the stream will move more quickly than the fluid towards the edge of the stream. Therefore friction will occur between layers within the fluid. Fluids with a high viscosity will flow more slowly and will generally not support eddy currents and therefore the internal roughness of the pipe will have no effect on the frictional resistance. This condition is known as laminar flow.

Reynolds number basically determines the transition of fluid flow form laminar flow to turbulent flow. When the value of Reynolds number is less than 2300, laminar flow will occur and the resistance to flow will be independent of the pipe wall roughness (℮). Meanwhile, turbulent flow occurs when the value of Reynolds number is exceeding 4000.

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For large viscous force, whereby R e  value is less than 2300, viscous effects are great enough to damp any disturbance in the flow and the flow remains laminar. The flow is called laminar because the flow takes place in layers. Any combination of low velocity, small diameter, or high kinematic viscosity which results in R e  value of less than 2300 will produce laminar flow. As Re increases, the viscous damping of flow disturbances or perturbations decreases relative to the inertial effects. Because of a lack of viscous damping, disturbances are amplified until the entire flow breaks down into in irregular motion. There is still a definite flow direction, but there is an irregular motion superimposed on the average motion. Thus, for turbulent flow in a pipe, the fluid is flowing in the downstream direction, but fluid particles have an irregular motion in addition to the average motion. The turbulent fluctuations are inherently unsteady and three dimensional. As a result, particles which pass though a given point in the flow do not follow the same path in turbulent flow even though they all are flowing generally downstream. Flows with 2000 < Re < 4000 are called transitional. The flow can be unstable and the flow switch back and forth between turbulent and laminar conditions.

* A re-entrant bell mouthed glass experimental tube of 16 mm bore and approximately 790 mm long mounted horizontally in a 103 mm bore Perspex tube.

* Dye injector with needle valve control.

* Rotometer flow meter.

* Water supply from a tank with clear test section tube and “bell mouth” entrance.

EXPERIMENTAL PROCEDURES

This experiment demonstrates visually laminar (or streamline) flow and its transition to turbulent flow at a particular velocity.

• Firstly, the apparatus is set up and insert the red dye into the dye reservoir with a steady flow of water.
• The dye is allowed to flow from the nozzle at the entrance of the channel until a colored stream is visible along the passage. The velocity of water flow should be increased if the dye accumulates around the nozzle.
• Adjust the water flow until a laminar flow pattern which is a straight thin line or streamline of dye is able to be seen along the whole passage.
• Collect the volume of water that flows for 10 seconds then measure the amount of water in the volumetric measuring tank. Repeat this step 3 times to get the average and more accurate volume of water. The volume flow rate is calculated from the volume and a known time.
• The water flow rate is increased by opening the pipe vessel and the flow pattern of the fluid is observed. Repeat step 2-4 for transition and turbulent flow.
• Clean all the apparatus after the experiment is done.

SAMPLE CALCULATIONS

Data Given :

Times                        = 3 sec

Density of water, ρ        = 1000 kg/m³

Viscosity, μ                = 10.00 x 10 -4 Ns/m²

Diameter of tube, d        = 16 x 10ˉ³ m

Length,                 = 0.103 m

Area of cross passage, a        = πd²/4

                                                     = π (16 x 10ˉ 0 ³) / 4

                                                     = 2.0106 x 10ˉ 4  m²

From   experiment :

Laminar Flow:

Volume flow rate        = volume/ time

= 8.4 x 10 -5 m 3  / 3s

                        = 2.8 x 10 -5 m 3 /s

Velocity, v = (m / ρa)        = volume flow rate /  area

                        = 2.8 x 10 -5  m 3 /s ÷ 2.0106x 10 -4 m 2

                        = 0.1393 m/s

Reynolds number, Re        = ρvd / μ

= (1000 kgm -3  x 0.1393 m/s x 16 x 10 -3 m) ÷ 10.00 x 10 -4 Ns/m 2

* For laminar flow Re should be less than 2300.

Transition Flow:

Volume flow rate         = volume/ time

                        = 9.6 x 10 -5 m 3  ÷ 3s

                        = 3.2 x 10 -5 m 3 /s

Velocity, v = (m / ρa)        = volume flow rate / x area

                        = 3.2 x 10 -5  m 3 /s ÷ 2.0106 x 10 -4 m

                        = 0.1592 m/s

= (1000 kgm -3 x 0.1592 m/s x 16x 10 -3 m) ÷ 10.00 ˉ 4 Ns/m²

*For transition flow Re should be in between 2300 and 4000

Turbulent Flow:

                        = 16.8 x 10 -5 m 3  ÷ 10s

                        = 5.60 x 10 -5 m 3 /s

                        = 5.60 x 10 -5  m 3 /s ÷ 2.0106 x 10 -4 m 2

                        = 0.2785 m/s

= (1000kgm -3 x 0.2785 m/s x 0.016m) ÷ 10.00 x 10 ˉ4  Ns/m²

*For turbulent flow Re should be more than 4000

        It is necessary to know the differences between laminar, turbulent and transition flow before one is about to conduct this experiment. As for laminar flow, it is defined as a highly ordered fluid motion with smooth streamlines. Turbulent flow is much different with laminar, as it is a highly disordered fluid motion characterized by velocity and fluctuations and eddies, whereas transition flow is known as a flow that contains both laminar and turbulent regions.

        Based on Reynolds apparatus experiment, laminar flow is obtained when a single ordered line is seen after a thin filament of dye is injected into the transparent glass tube. There is not much dispersion of dye can be observed throughout the flowing fluid. Nevertheless, the case is not the same with turbulent flow, as there is obvious dispersion of dye along the glass tube, whereby the lines of dye breaks into myriad entangled threads of dye.

        Throughout the experiment, we observed that the red dye line starts flowing in a straight ordered line through the glass tube, and as the velocity increases after some time, the ordered streamlines is seen to begin to disperse at about the middle of the streamlines, but still remain the straight line at the earlier part. Next, the dispersion started to increase, indicating the turbulent flow. These observations are concluded as the streamlines is undergoing a change of type of flow, which is from laminar flow, transition flow to turbulent flow.

        There are a few careless mistakes that have been done during this experiment. Most of all, the accuracy of collecting the fluid flowing out of the tube within 3 seconds is a bit inaccurate. The one who collect the fluid might not begin right when the person monitoring the stopwatch started ticking on it, and he/she might also not stop collecting exactly after the third second. Therefore, the values calculated in results section might not be exactly 100% correct.

         As a conclusion, as water flow rate is increasing, the Reynolds number will automatically increase as well, and the red dye line change from straight line to swirling streamlines. Likewise, it is proven that Reynolds number is dimensionless, since no unit is representing the value of Reynolds number.  Laminar flow is obtained if the Reynolds number is less than 2300; meanwhile the Reynolds number for turbulent flow is more than 4000. The Reynolds number for transition flow is in between 2300 until 4000.

RECOMMENDATIONS

There are some recommendations to make sure this experiment would attain more accurate and precise results in the future:

• Check whether the water in the tube flows in a correct way and we should also make sure that the flow is stable before measuring the flow rate by monitoring the time taken for collecting an amount of water in the volumetric measuring tank.
• Before injecting the dye into the fluid, we should make sure the dye is not too much and not too insufficient. It will be hard to stable the fluid to get a laminar flow.
• The experiment should be repeated twice to get better result.
• The person collecting the water should synchronize well with the time keeper.
• Fluid Mechanics by Dr. Andrew Sleigh (J. Franzini/E. Finnemore), McGraw Hill.
• F. M. White, Fluid Mechanics (Mc-Graw Hill, Inc., New York, 1994).
• J. Baggett and L. Trefethen, “Low-dimensional models of subcritical transition to turbulence,”Phys. Fluids 9 , 1043 (1997).

Document Details

• Word Count 1914
• Page Count 11
• Level University Degree
• Subject Engineering

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