PHYS426 - Laminar Flow

Robert Payne & Julius Wang

1. Introduction

Flow can be characterized as laminar, turbulent, or transitional, depending on the relative importance of the inertial and viscous forces acting on the fluid. This can be quantified with a dimensionless ratio called the Reynold's number,

\begin{equation} \label{Re} Re = \frac{UL}{\nu} \end{equation}

where $U$ is the mean flow speed, $L$ is the hydraulic diameter of the pipe or channel, and $\nu$ is the kinematic viscosity of the fluid. If $\nu$ is large relative to $UL$, then $Re$ is small and viscous forces are dominant over inertial forces. If instead $UL$ is large compared to $\nu$, then $Re$ is large and inertial forces dominate over viscous forces.

Consider the flow in which viscous forces dominate over inertial forces. Such a flow is called laminar flow, wherein fluid particles tend to flow along smooth, continuous layers, with no cross-currents and mixing of mass or energy. Laminar flow can be easily visualized when dye is inserted into such a flow, as it will form a thin ribbon, tracing out and following a continuous streamline. Quantitatively, a flow is considered laminar when its Reynold's number is less than $2000$. Typical examples include flow over an aircraft wing and through through pipes.

Now consider the flow in which inertial forces are dominant. Such flows are turbulent, wherein fluid particles no longer flow in parallel layers, but instead whose motion is chaotic and often subject to rapid change in flow velocity and pressure. There is substantial exchange of energy and momentum between layers, and the formation of eddies of varying sizes. Dyes inserted into the flow will rapidly diffuse and spread throughout the fluid. Quantitatively, turbulent flow occurs for flows whose Reynold's number is greater than $4000$. Examples of turbulence include flow around a golf ball, smoke rising from a lit match, and general atmospheric circulation.

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Finally, as its name suggests, transitional flow occurs in the transition from laminar flow to turbulent flow. Such flow occurs for Reynold's numbers in between $2000$ and $4000$. Qualitatively, such a flow would appear similar to laminar but with some mixing between layers and instability in the streamlines.

In this experiment, laminar and turbulent flow (and the transition between them) is investigated with two parts. The first parts uses an Osborne-Reynold's apparatus allowing for ample control of the flow regime. Measured parameters allow for the calculation of the flow's Reynold's number in order to determine consistency between theory and observation. The second part of this experiment concerns Taylor-Couette flow, wherein dye is inserted into a viscous fluid contained between two cylindrical walls. The inner wall is allowed to rotate, exerting a stress on the fluid and mixing the dye. When the wall is rotated in the opposite direction, the dye should ideally return to its original position, demonstrating that laminar flow is reversible.

2. Procedure

Osborne-Reynolds Apparatus

The head tank of the Osborne Reynold’s apparatus (fig. 2) was filled with water until the hypodermic tube was submerged. The dye reservoir was filled with a solution of water and blue dye, and the dye flow control valve was released so that the dye solution began to drain into the head tank and bellmouth. The inflow control valve was carefully adjusted such that the flow through the test section was laminar. The flow rate out was measured by timing how long it took to fill a large beaker to a given volume. The temperature of the water was also noted, as the viscosity of water is temperature-dependent. This was repeated for transitional and turbulent flow.

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Taylor-Couette Flow

A container consisting of an outer cylindrical wall and an inner cylindrical wall was filled with corn syrup (fig. 3). The outer wall was fixed in place while the inner wall could be rotated in order to exert a stress on the fluid. Using a pipette, several drops of dye (mixed with corn syrup as to create a neutrally- buoyant solution) were carefully inserted into the corn syrup. The inner cylinder was slowly rotated, exerting stress on the syrup and dye. After the dye had sufficiently spread and mixed, the inner wall was then slowly rotated back the opposite direction until the dye returned as close to its original location as possible.

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3. Results

Osborne-Reynolds Apparatus

Three Reynold's numbers were calculated for each flow regime, based on the measured parameters below.

$\ $

T = Water temperature = 12.5°C

D = Pipe diameter = $0.015$m

$\nu$ = Kinematic viscosity = $1.218\times 10^{-6}$m$^2$/s

A = Cross-sectional area of the pipe = $1.77\times 10^{-4}$m$^2$

$\ $

Let $U$ denote the speed of the flow inside the pipe. It was found that for speeds $U < 0.19$m/s, the flow was laminar. For speeds $U>0.23$m/s, the flow was instead turbulent. Moreover, for the three laminar flows, the average Reynold's number was found to be approximately $387$. For the three transitional flows, the average Reynold's number was found to be $2539$. Finally, for the three turbulent flows, the average Reynold's number was found to be $7576$. These numbers and their respective observed flow regimes are consistent with theory:

$\ $

Laminar Flow : $Re<2000$

Transitional Flow: $2000<Re<4000$

Turbulent Flow: $Re>4000$

$\ $

Sample Calculations:

$u=\frac{\mathrm{volume}}{\mathrm{time}}$=$\frac{(0.6)(0.001\mathrm{m}^3)}{105\mathrm{s}}$=$5.714\times 10^{-6}\mathrm{m}^3/\mathrm{s}$

$U=\frac{u}{A}$=$\frac{5.714\times10^{-6}\mathrm{m}^3/\mathrm{s}}{1.77\times10^{-4}\mathrm{m}^2}$=$0.03228\mathrm{m}/\mathrm{s}$

$Re=\frac{UD}{\nu}$=$\frac{(0.03228\mathrm{m}/\mathrm{s})(0.015\mathrm{m})}{1.218\times10^{-6}\mathrm{m}^2/\mathrm{s}}=397.73$

Because $Re<2000$, the flow should be laminar.

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$\ $ $\ $ $\ $ $\ $ $\ $ $\ $ $\ $ $\ $ $\ $ $\ $ $\ $ $\ $ $\ $ $\ $ $\ $ $\ $ $\ $ $\ $ $\ $ $\ $ $\ $ $\ $ $\ $ $\ $ $\ $ $\ $ $\ $ Figure 4: From left to right: Laminar flow, transitional flow, turbulent flow.

Taylor Couette Flow

We rotated the cylinder with slow speed, mixing the two dyes. Afterwards, the inner cylinder was reversed. As anticipated, the two dyes separated and returned to their original positions with minor differences, demonstrating that the flow was laminar.

4. Discussion

Osborne-Reynolds Apparatus

The experiment producing various flow regimes with the Osborne-Reynolds (OR) apparatus was found to produce results consistent with theory. For laminar flow, Reynold's numbers of $398$, $379$, and $383$ were found, which are indeed less than the theoretical threshold of $Re<2000$. This flow was indeed observed to resemble "ribbons" of dye, closely following constant streamlines. For transitional flow, Reynold's numbers of $2399$, $2899$, $2319$ were found, again falling in between the theoretical threshold $2000<Re<4000$. This flow qualitatively resembled laminar flow with slightly more instability and curviness in the flow. Finally, Reynold's numbers of $8117$, $6494$, $8117$ were observed for turbulent flow, consistent with $Re>4000$. The flow in this regime quickly spread the dye out and did not follow constant streamlines, but instead mixed the dye quickly throughout the flow.

It's important to note that even the best laminar flow achieved exhibited small fluctuations uncharacteristic of its flow regime. Moreover, transitional flow was sometimes observed to suddenly transform into laminar flow or turbulent flow for periods of time. These indicate the presence of systematic errors with a substantial effect on the observed flow regime. Firstly, the water pump, while running, would cause the hydraulics table and OR apparatus to vibrate. To remedy this, the OR apparatus was placed firmly on the concrete floor. However, it could not be distanced very far from the Hydraulics table due to the lengths of tubes available, and hence the vibration may have affected the OR apparatus. This would serve to increase turbulence, as the shaking could breed instabilities that exponentially grow into turbulent flow. These kinds of instabilities may have also been induced through the handling of the apparatus, particularly through turning the flow control valve in order to achieve a certain regime. Air bubbles were also an issue throughout the course of the procedure. These typically occurred upon turning on the pump, and would take a few minutes to disappear. Future renditions of this experiment may wish to further distance the apparatus from the pump, and take special care in applying changes to the flow. After turning the pump on, the flow should be given several minutes to rid the system of air bubbles as much as possible.

One might expect temperature to have an effect on the viscosity of water, its Reynold's number, and hence the observed flow regime. However, fluctuations from room temperature most likely did not have a substantial effect on what was observed in our experiment. For instance, even if the temperature were to have suddenly increased from $12.5$°C to $15$°C, the kinematic viscosity of water would only change from around $1.218\times10^{-6}$m$^2$/s to $1.139\times10^{-6}$m$^2$/s, a percent difference of approximately 6%. Since the relative change in the Reynold's number equals the relative change in viscosity, $\frac{\delta Re}{Re} = -\frac{\delta \nu}{\nu} \approx -0.06$, $Re$ would experience a decrease of around 6%. Unless the flow is near one of the flow regime thresholds, which was not the case for the flows measured in this experiment, we would not expect this to significantly change the flow.

Taylor-Couette Flow

Upon rotating the inner cylinder counterclockwise at a sufficiently slow rate, two dyes of different colours suspended in the oil were gradually mixed. Upon rotating the cylinder back, laminar flow was observed and the two dyes "unmixed", returning for the most part to their original locations. This demonstrates that, as expected, laminar flow is reversible.

This experiment was, however, subject to substantial error. The inner cylinder was not perfectly fixed at the center of the outer cylinder, meaning that the setup did not have perfect cylindrical symmetry. The inner cylinder was also observed to move around a bit when rotating, also affecting the distance between the inner and outer cylinders. This would ideally be fixing the inner cylinder about an axis at the center of the larger cylinder, such that it can still be rotated. Friction between the inner cylinder and the floor of the container, in addition to it being rotated by hand, resulted in a non-steady rotation rate, and hence non-steady application of stress on the fluid. One could perhaps use a motor to rotate the cylinder at a steady rate, and apply grease to the boundary between the inner cylinder and floor to decrease friction. Finally, the density difference between the oil and the dye caused unwanted vertical motion. This was managed by creating a dye-oil solution that was ideally neutrally buoyant, but the dye was still observed to sink slightly over the course of the experiment.

5. Hints & Suggestions

Osborne-Reynolds Apparatus

Taylor-Couette Flow