Osborne Reynolds’ Experiment: The Transition from Laminar to Turbulent Flow Abstract: The objective of this experiment was to determine the interval of the Reynold’s number where flow through an idealized straight, smooth pipe with constant pressure transitioned from laminar flow to turbulent flow. This experiment was carried out utilizing a model similar to Reynold’s own design. Dye was injected into a stream of water flowing through a clear pipe and out into a reservoir to demonstrate if there were any disturbances within the flow. Water exiting the bottom valve was collected in a one liter graduated cylinder for a period of about thirty seconds and the accompanying volume and exact time was recorded. The Reynold’s number was calculated for three different flow rates around the transition which was noted by the movement of the dye in the flow and whether it was a straight or windy and broken up line. The collection of many volume and time datum allowed for an average volume flow rate that was more representative of the actual volume flow rate than one set of data could have produced, leading to more accurate results. The accepted Reynold’s number of 2300 was within the experimental interval of calculated Reynold’s numbers for the transition from laminar to turbulent flow.
CEE 171L Lab Report Group 3 Lab Section 1A 1 | Page
Monday Lab, 9-11 am
Table of Contents: 1. Introduction & Hypothesis………………………………………………………………………..…… Page 3 2. Methods………………………………………………………………………………… ………………..…….Page 4 3. Analysis of Results………………………………………………………………………………….. ……..Page 5 4. Interpretation of Results & Conclusions…. …………………………………………….…...….Page 7 5. Appendix………………………………………………………………………………… ……………………..Page 8 6. Works Cited……………………………………………………………………………………… …………...Page 9
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Introduction and Hypothesis: Liquids do not always flow in a perfect, smooth stream. Once velocity increases past a certain threshold (determined by the diameter of the pipe, density and viscosity of the liquid, and velocity), puffs begin to develop in the liquid and produce turbulence. A smooth flow of liquid is called laminar while an uneven flow is called turbulent. The first person to mention that there may be in fact two types of viscous flows was an engineer named G.H.L Lagen in 1839. He deduced an equation that described the difference in pressure, taking into account the entrance effects and some unknown constant. ∆ p= ( const )
LQ +entrance effects(¿ 351) 4 R
He observed that after increasing the volume flow rate in a pipe past a certain point, his equation no longer held true. What he did not know is that the constant in his equation related to the viscosity of the liquid. In 1883, an engineering professor by the name of Osborne Reynolds discovered that the constant in Lagen’s equation was in fact: ℜ=
ρVd μ
This is the equation for what we now call the Reynold’s number. The experiment he used to determine this is what was duplicated to generate the results presented in this report. The significance of the Reynolds number is that it relates given values (density, velocity, length/diameter, and viscosity) to the type of flow that the liquid will experience. The larger the Reynolds number, the more turbulent the flow. The accepted Reynolds number for transition from laminar to turbulent flow in a smooth, circular pipe is 2300 (White 352).
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Methods: In order to reproduce Reynold’s experiment, we were provided with an apparatus that was comprised of a reservoir of room temperature water, a pump, a large container over a thin, smooth, cylindrical pipe, and two exits for the liquid (one spillover tube at the top of the container and one at the bottom of the pipe), as well as necessary tools such as a timer and a one liter graduated cylinder. There were two valves to control the flow of the water over the duration of the experiment; one was located at the bottom to control the flow of water from the reservoir to the container and the other was located at the junction of the pipe exit and one of the tubes. The pump was turned on to transfer the water in the reservoir at the bottom of the apparatus to the container at the top and adjusted the valves to let in as much water to where there was a small amount of spillover from the top tube while maintaining a constant water level in the container to sustain a constant pressure, and therefore, constant flow rate, during data collection. We gathered Set Up of Reynold’s Experiment water exiting from the tube connected to the pipe exit for a time of about thirty seconds in order to calculate the volume flow rate. Data was collected over a spectrum of flows ranging from laminar to turbulent. 4 | Page
In the data shown in the Analysis of Results section, it should be noted that discrepancies of the collection time exist because it was impossible for human reflexes to stop the timer at exactly thirty seconds each time. There is also human error in the amount of liquid collected in the graduated cylinder since one person was operating the timer and another person was observing the amount of water exiting into the graduated cylinder and removing the cylinder at the thirty second mark.
Analysis of Results: Constants:
Temperature of Water: 15°C Diameter of Pipe: 0.0045m Density of Water [1]: 1001 kg/m3 Viscosity of Water [2]: 0.00115 kg/m·s
Table of Collected Data & Calculations: Time (s)
Volume of Liquid (L)
30.44
0.288
30.33
0.280
30.19
0.280
30.12
0.280
30.29
0.280
30.52
0.369
30.23
0.358
30.33
0.360
30.25
0.368
30.22
0.367
Volume Flow Rate, Q (m3/s)
Velocity, V [4] (m/s) [3]
Reynold’s Descripti Number, on of [5] Re Flow
9.4612E-06
0.595
2328
9.2318E-06
0.580
2271
9.2746E-06
0.583
2282
9.2961E-06
0.585
2287
9.244E-06
0.581
2274
1.209E-05
0.760
2975
1.1843E-05
0.745
2914
1.1869E-05
0.746
2920
1.2165E-05
0.765
2993
1.2144E-05
0.764
2988
LaminarTransitional LaminarTransitional LaminarTransitional LaminarTransitional LaminarTransitional TransitionalTurbulent TransitionalTurbulent TransitionalTurbulent TransitionalTurbulent TransitionalTurbulent 5 | Page
30.03 30.42 30.03 30.08 30.13
0.463 0.464 0.455 0.461 0.460
1.5418E-05 1.5253E-05 1.5152E-05 1.5326E-05 1.5267E-05
0.969 0.959 0.953 0.964 0.960
3793 3753 3728 3771 3756
Turbulent Turbulent Turbulent Turbulent Turbulent
[1], [2], [3], [4], [5]
: Refer to Appendix for Calculations
Statistical Analysis: In order to calculate the following values, only the values for the LaminarTransitional description were implemented.
Mean Critical Reynolds Number: 2289 Standard Deviation of the Critical Reynolds Number: 20.42
The Mean Critical Reynolds Number gives us the average of our calculated values. The standard deviation gives us how much, on average, the data set deviates from the mean. Based upon these calculations, I conclude that 2300 is a valid value for the Reynold’s number for the transition from laminar to turbulent flow in a smooth, cylindrical pipe (otherwise known as the critical Reynolds number).
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Interpretation of Results & Conclusions: Discussion of Concepts: Through this experiment, we determined that the Reynold’s number is truly dependent on velocity. While we were not able to fully determine its dependency on the other factors such as density of the water, diameter of the pipe, or viscosity, we observed that the Reynold’s number generally increased with increasing velocity. This makes sense because the Reynold’s number maintains a linear relationship with velocity in the equation. Also, the faster the liquid is moving, the higher tendency it will have to develop turbulence. Validity of Initial Hypothesis: The experimental data determined that the mean critical Reynold’s number was 2289 with a standard deviation of 20.42. This means that values falling anywhere within 20.44 above (for a total value of 2309.42) or below 2289 (for a total value of 2268.58) is a valid estimate for the critical Reynolds number. Since the widely accepted critical Reynold’s Number of 2300 falls within the standard deviation range, it is indeed a transitional Reynolds number. Error Assessment: There is some error in the experimentally calculated mean critical Reynold’s number due to a few parameters. One of these is solely human error, which had the largest impact on our calculations. Our team found it was difficult to carry out the data collection effectively with two people since the they would have had to have impeccable synchronization in order to produce the most accurate data. The problem was that the person operating the timer was starting either a small amount of time before or after the liquid collection actually began and ended. Another parameter of our errors was that we were not constant with who the operators were throughout the experiment. The inconsistency in reflexes and synchronization between operators adds another variable to our experiment along with the time and volume of the liquid. As with all correctly conducted experiments, all variables other than the ones desired to be measured should be kept constant as to isolate any changes occurring to the variables. This was one factor that should have been considered prior to beginning the experiment. One other factor to our error to consider is the fact that the volume data collected was being read off of the bottom of the meniscus in a large, 1 liter graduated cylinder with markers that were very spread out in terms of significant figures. This error could have been avoided if the volume readings 7 | Page
were obtained by weighing the dry graduated cylinder and taring the scale, performing the liquid collection, weighing the graduated cylinder with the liquid and obtaining the mass of the collected liquid, and then dividing the mass of the collected liquid by the density of the liquid (taking into account the temperature of the liquid) to obtain the true volume of the collected liquid with more significant figures. This method would have taken longer than the original, but yielded more accurate results.
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Appendix:
[1]
ρ ≈ 1000−0.0178 ( T ° C−4 ° C )1.7 ∓0.2
[2]
μ 273 K 273 K ln =1.704−5.306 + 7.003 μ0 TK TK
[3]
π Q= AV = d 2 V 4
[4]
V=
4Q π d2
[5]
ℜ=
ρVd μ
(
)
(
2
)
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Works Cited: 1. White, Frank M. Fluid Mechanics. Seventh ed. New York: McGraw-Hill, 2011. Print.
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