PHYS 426 - Laminar Flow¶
Joel Nordine and Cora Tangermann
1. Abstract¶
In general, the flow of a fluid can be characterized as either laminar or turbulent. Laminar flow occurs in slow-moving, viscous fluids, where viscous forces dominate. Turbulent flows occur in fast-moving, low viscosity fluids, where inertial forces dominate. These states are classified by the Reynolds number, a dimensionless quantity. This lab seeks to measure the laminar, transitional and turbulent flows of water using an Osborne-Reynolds apparatus, as well as observe the reversible behaviour of the Taylor-Couette flow. Overall, the Osborne-Reynold's apparatus was able to adequately show the different flow regimes, and the Taylor-Couette flow was able to be observed.
2. Theory¶
Reynold's Number and the Osborne-Reynolds Apparatus¶
The Reynold's Number characterizes the behaviour of a flow, and takes the following form
$ Re = \frac{UL}{\nu} $
where U is the speed of the flow, L is the characteristic length, and $\nu$ is the viscosity of the fluid. For $Re<2000$, viscous forces will dominate the flow, and will therefore be laminar. In this case, the fluid will experience no mixing, as the energy exchange is small between particles. In constrast, turbulent flows occur when $Re>4000$. During these flows, significantly more energy is being exchanged between particles, causing the flow to have more mixing and random movements. For $2000<Re<4000$, the flow will be in a transitional state, where some mixing is occurring, but the flow is not quite turbulent or laminar.
To visualize this effect, the Osborne-Reynold's Apparatus is set up such that a stream of dye can be funneled into a pipe, from which the outflow rate can be changed to control the fluid's velocity. The apparatus consists of a large cylindrical tank with a bell-mouthed funnel in the middle connected to the outflow pipe. The tank is half-filled with marbles to negate mixing effects. A dye tank is mounted on the roof of the tank, with a needle pointing into the tank, allowing for dye to be injected directly into the fluid. The tank can be filled using the inflow pipe, and the water level is bounded by an overflow pipe at the top of the tank.
To measure Reynold's Number with the Osborne-Reynold's apparatus, the characteristic length is given by the diameter of the outflow pipe, and the flow speed can be determined by measuring the volume of water released per unit time, and dividing it by the cross-sectional area of the pipe. The expression for the Reynold's Number can then be expanded to,
$ Re = \frac{VL}{tA \nu} $
where V is the volume of water released by the tank over a time period t, and A is the cross sectional area of the pipe. By marking certain volume levels on the tank and releasing the same amount of water for a each calculation, the Reynold's Number becomes only dependant on time.
Taylor-Couette Flow¶
Taylor-Couette flow is characterized by the flow of a fluid between two cylinders, with the inner cylinder being rotated to exert a frictional force on the fluid. For low Reynold's Numbers, the behaviour of the flow is reversible, as the viscous forces dominate and slide the 'layers' of the fluid. The apparatus consists of two concentric cylinders, with the space between them filled with a viscous fluid such as canola oil. Streaks of neutrally buoyant dye are made in the fluid, and the middle cylinder is slowly rotated, allowing the fluid to apparently mix. If after, the middle cylinder is rotated the opposite direction back to the initial state, the dyes will un-mix, and the fluid will return to it's initial conditions.
3. Prodedure¶
Osborne-Reynold's Apparatus¶
The Osborne Reynolds apparatus was set up on top of the water tank, and filled with water. The diameter of the outflow pipe was measured to be $L= 0.015m$ ($A = 0.00018m^2$), and the water temperature was 293K, which implies a viscosity of $\nu=1.002\times 10^-6$. Tape was placed at the top of the water column, and then 2.0L ($0.002m^3$) of water was removed from the tank through the outlet pipe. Another piece of tape was placed at the new water level. This would allow for $V = 0.002m^3$ of water to be released for every trial. A solution of dark blue dye and water was prepared and placed in the dye resevoir and the needle was centered over the bellmouth entry.
The outflow pipe was then opened using the control valve with the water pump still going, and the system was given time to stablize and flush out any air bubbles. The water pump was then shut off, and the dye was allowed to flow into the tank. The time taken for the volume of water to drop past the bottom of each strip of tape was measured, and used to calculate the Reynold's Number for that flow.
Taylor-Couette Flow¶
First, relevant repairs to the apparatus were made using electrical tape. Then the main reservoir was filled with vegetable oil, and two colours of dyed oil were prepared. About a tablespoon of each colour were deposited in the clear oil, using a dropper to form streaks. The inner cylinder was slowly turned a full rotation, and then slowly rotated back to the starting position. This was repeated until a visually significant result was achieved and recorded.
4. Results¶
Osborne-Reynold's Apparatus¶
Sample Calculation of Re:
$ Re = \frac{VL}{tA \nu} = \frac{(0.002m^3)(0.015m)}{(21s)(0.00018m^2)(1.002\times 10^-6)} = 7920 $
| Time (s) | Reynold's Number | Flow Type |
|---|---|---|
| 21 | 7920 | Turbulant |
| 30 | 5540 | Turbulant |
| 35 | 4750 | Turbulant |
| 47 | 3540 | Transitional |
| 53 | 3140 | Transitional |
| 80 | 2080 | Transitional |
| 135 | 1230 | Laminar |
| 388 | 429 | Laminar |
| 6300 | 26.4 | Laminar |
Videos of Laminar Flows : https://youtu.be/7I7immfn8FE
Videos of Transitional Flows : https://youtu.be/FRUA1zUscu0
Videos of Turbulent Flows : https://youtu.be/yS2318AjSV0
Taylor-Couette Flow¶
Upon rotating the innner cylinder back to its original location, the fluid was found to unmix as expected.
Video of Taylor-Couette Flow : https://youtube.com/shorts/XWjk0B7KN54?feature=share
5. Discussion¶
Overall, the flow types observed with the Osborne-Reynold's apparatus were consistent with the Reynold's Numbers obtained. The trials with $Re<2000$ exhibited laminar flow, as a clearly defined stream was observed going down the outflow pipe. The trials with $Re>4000$ exhibited turbulent flow, and no clear structure could be seen in the pipe due to mixing. The fluctuation between laminar and turbulent flow for trials with Reynold's Numbers of $2000<Re<4000$ is consistent with the expected behaviour of the fluid. In these conditions, a structure could be made out in the outflow pipe, however mixing was clearly apparent.
In calculation of the Reynold's Number for each trial, the main source of uncertainty can be associated with the measuring of the volume of water released from the tank. During the experimental procedure, some water could be observed backflowing into the pump, which would influence the time taken to reach the desired volume. The change in water level throughout the process would also influence the flow speed of the water. Additionally, the methods used failed to account for the additional volume from the dye. The consistent results suggest that these sources of uncertainty were negligible relative to the scale of the other variables.
The temperature of the water was measured throughout the experiment, and did not vary, implying very little uncertainty.
The biggest difficulty involved in conducting this procedure was finding a way to measure the flow rate, while simultaneously managing the tanks water level and monitoring the flow regime. Additionally, the dye was observed to have a noticable curve while the pump was running, implying that the marbles had were not doing enough to negate the mixing from the inflowing water. By marking out a 2.0L volume on the tank, the flow rate could be measured passively with a stopwatch, without having the tank constantly filling. Future versions of this experiment could more precisely mark the water levels to their corresponding volumes in the tank, or provide a more consistent way to manage the water height. An device to automatically measure the flow rate would also greatly improve the accuracy of the results.
Despite estimating a low Reynolds’s number for this setup, the oil and apparatus did not provide a demonstration as striking as desired for a couple of reasons.
First, the dye and the oil were mostly immiscible, and the coloured oil tended to sink to the bottom of the reservoir. Upon contact with the bottom, the dye itself would separate from the oil and stick to the surface. In an attempt to solve this issue, a water bath was prepared and each coloured oil mixture was placed in the bath to slightly raise its temperature, with mixed results.
The apparatus itself was not ideal; the width of the annular reservoir was too large, causing some of the dyed oil to not be affected by the rotation of the inner cylinder. The method of securing the inner cylinder lacked stability, which made uniform rotation challenging to achieve. The inner cylinder would skip along the bottom of the reservoir, causing impulses of fast flow, causing irreversible mixing at some points.
To solve these issues, clear syrup could be used for its higher viscosity and mixing ability, and the inner cylinder could be replaced by one with a larger radius. It would also be helpful if the inner cylinder was secured to the bottom of the apparatus in a way that restricts its motion to rotation about a fixed axis, with a suitable handle to facilitate gentle motions.
Hints and Suggestions¶
By pre-marking out a certain volume on the tank, the flow speed can be measured passively, while the behaviour in the outflow tube can be observed.
Allowing the pump to run before releasing the dye allows the system some time to flush out any air bubbles.
Ensure that the dye is not allowed to linger in the tank without the outflow tube open, otherwise the water will slowly be dyed, making it difficult to view the flow regime.
Higher viscosity fluids such as corn syrup should be used, rather than vegetable oil.
The dye must be inserted quickly to create a streak in the liquid.
6. References¶
$^1$Czubai, Andrew, et al. “Experiment #7: Osborne Reynolds’ Demonstration.” Applied Fluid Mechanics Lab Manual, Mavs Open Press, 14 Aug. 2019, uta.pressbooks.pub/appliedfluidmechanics/chapter/experiment-7/.
from sympy import *
import numpy as np
import matplotlib.pyplot as plt
from sympy import Sum, factorial, oo, IndexedBase, Function
import scipy.integrate as integrate
import scipy.special as special
ts = np.array([21, 30, 35, 47, 53, 80, 388, 6300])
vol = 0.002 #m^3
d = 0.015 #m
A = 0.00018
nu = 1.002 * 10**-6 #m2/2 at water temperature at 20C
def Re (t):
q = vol/t
u = q/A
return (u*d)/nu