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I have calculated the following short table concerning the stationary flow of water in a 1 meter long pipe with a diameter of 16 mm and a completely smooth inner surface. The flow is driven by the pressure difference (in kilograms force per square cm) at the ends as shown in the first column. The second and third columns have been jointly calculated using Poseuilles law based on the assumption that the flow is laminar. The fourth and fifth collumns have been jointly calculated using Darcys formula and the Colebrooke equation for the friction factor and assuming the flow is turbulent.
As an example of the ambiguity we can take the first entry where the same pressure difference yields 2 different velocities, namely 19.6 cm/s versus 12.6 cm/s, the first having a Reynolds number 3139 indicating a possibly turbulent flow contrary to the laminar assumption. However the lower second velocity is associated with a Reynolds number 2019 that indicates a laminar flow, which is also contrary to the turbulent assumption! So how can one determine (without a measurement) which calculation gives the correct answer or is this just not possible when the Reynolds number lies somewhere in the transition region?

Pressure    ASSUMED LAMINAR             ASSUMED TURBULENT
difference  Velocity    Reynolds        Velocity  Reynolds
kp/cm2      cm/sec      Number          cm/sec    Number
2.5         19.6        3139            12.6      2019
2.6         20.4        3265            12.9      2066
2.7         21.2        3390            13.2      2113
2.8         22.0        3516            13.5      2160
2.9         22.8        3641            13.8      2205
3.0         23.5        3767            14.1      2250
3.1         24.3        3893            14.3      2295
3.2         25.1        4018            14.6      2338
3.3         25.9        4144            14.9      2424
3.5         27.5        4395            15.4      2466
3.6         28.3        4520            15.7      2508
3.7         29.0        4646            15.9      2549
3.8         29.8        4772            16.2      2589
3.9         30.6        4897            16.4      2629
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  • $\begingroup$ There are no completely smooth inner surfaces. Also, pressure differences should be in Newtons/square-meter or Newtons/square cm. $\endgroup$ – David White Jul 9 '19 at 15:43
  • $\begingroup$ "Smooth" compared to thickness of boundary layer. 1 Kilopont/cm2 = grsvitstional force of 1 kilo acting on 1 cm2. More helpful comments, please. $\endgroup$ – Jens Jul 9 '19 at 19:06
  • $\begingroup$ A "kilo" sounds a LOT like a kilogram, so there is much room for confusion in using such an abbreviation. $\endgroup$ – David White Jul 9 '19 at 19:48

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