Reynolds Number and Turbulence

Transcript

0:00Introduction

Hello everyone! Okay, picture this. Water flows gently through a pipe, in smooth, neat layers. Now push it a little faster, and suddenly the flow breaks up into swirls and eddies. So where exactly is the line between smooth flow and messy flow? Is it decided by the speed alone? Keep that question in mind. Today we meet one single number that settles it: the Reynolds number. Quick questions on it do come up in JEE, so let's begin.

0:31The Reynolds number

So, here it is. Capital R with a small subscript e: we read it R sub e, the Reynolds number. R sub e equals rho times v times capital D, over eta. Four quantities, one number. Okay? Now, what are they? Rho, the Greek letter, is the density of the fluid. Small v is the speed of the flow. Capital D is the diameter of the pipe. Diameter, not radius! Remember that. And eta, another Greek letter, is the coefficient of viscosity of the fluid. So, our opening question: it is not the speed alone. Density, diameter and viscosity all matter too. Right? And what does this number measure? The ratio of inertial forces to viscous forces: inertial forces on top, viscous forces at the bottom. Viscous forces are the internal friction between the layers of the fluid. Clear? Now tell me. If R sub e is very large, which force is winning? The inertial forces! Yes, exactly! Large R sub e: inertia wins, and the flow turns turbulent. Small R sub e: viscosity wins, and the flow stays streamline. Got it?

1:48Why the Reynolds number has no units

Next, a quick check. Square brackets around a quantity mean its dimensions; capital M, L and T stand for mass, length and time. Density is mass per volume: M L to the power minus three. Speed: L T to the power minus one. And the diameter is just a length: L, okay? And eta? From Newton's law of viscosity, its dimensions are M L to the power minus one, T to the power minus one. Right? Now multiply the three on top, and divide by the bottom. What is left? Nothing at all! Correct! On top, M once; L to the power minus three, times L, times L, gives L to the power minus one; and T to the power minus one. That is exactly the bottom. Everything cancels, and we get one. With me? So the Reynolds number is dimensionless: a pure number, with no units at all. But careful: every quantity must be in one system of units. Use S I units: kilograms, metres and seconds. Clear?

2:59Streamline, unsteady or turbulent?

Now, the main use of this number. When R sub e is less than one thousand, the flow is usually streamline. The sign that opens to the right means less than. Notice the word usually: a guide, not a sharp wall. Okay? When R sub e is greater than two thousand, the flow is turbulent: eddies and irregular motion. This sign opens to the left, and means greater than. Right? So tell me. What about a value in between: more than one thousand, but less than two thousand? Unsteady! Very good! Between one thousand and two thousand, the flow is unsteady. This is the transition zone between the two kinds of flow. So: below one thousand, streamline; above two thousand, turbulent; in between, unsteady. Got it?

3:50The critical velocity

Now, a natural question. For a given fluid and pipe, at what speed does the flow change its kind? For that, we need v from the formula. Make v the subject. Try it. Easy, right? Multiply both sides by eta, and divide by rho times capital D, okay? So, v equals R sub e times eta, over rho D, just as you see. The double arrow means implies. With me? Now put R sub e at the transition value. Then this speed gets a special name: the critical velocity, v with a small subscript c, read v sub c. v sub c equals R sub e times eta, over rho D, with R sub e at the transition. Clear? Quick question. Take a more viscous liquid, a larger eta, and keep everything else the same. Is the critical velocity higher, or lower? Higher! Perfect! Eta sits on top, so a larger eta gives a larger v sub c. More friction between the layers, so the flow keeps its calm up to a higher speed. Right?

4:58Example 1: Re of water in a 2 cm pipe, diameter not radius

Now, Example one. Water, with eta equal to ten to the power minus three pascal second, flows at zero point one metres per second, in a pipe of diameter two centimetres. On screen, P a s means pascal second, m s to the power minus one means metres per second, and c m means centimetres. Find R sub e, and describe the flow. Pause the video, and try it first. Done? Okay, let's check. First, the trap. Some students take the radius, one centimetre. Which value goes into the formula? Two centimetres, the diameter! Correct! In S I units, that is zero point zero two metres. The density of water, rho, is one thousand kilograms per cubic metre; k g m to the power minus three on screen. And the speed is zero point one metres per second. Okay? Now substitute. One thousand times zero point one is one hundred; times zero point zero two gives two. Two over ten to the power minus three: two thousand. So R sub e is two thousand. Right? Now describe the flow. Streamline, or turbulent? Careful! Neither, really. It sits exactly at two thousand, right where the turbulent range begins. So the flow is at the edge of turbulence. See, that answers our opening puzzle: R sub e sets the line, and this slow water is already on it. With the radius, you would get one thousand, half the true value. That is the trap! Clear?

6:37Example 2: max streamline speed of water in a 1 cm pipe

Now a slightly harder one. Find the largest speed for streamline flow of water in a one centimetre pipe. Take R sub e equal to one thousand. Hint: you need the density and the viscosity of water, the same values as in Example one. Pause the video, and try it. Done? Let's see. The largest speed for streamline flow: which formula gives that? The critical velocity, exactly! v sub c equals R sub e times eta, over rho times capital D, okay? And R sub e is one thousand here, the upper limit for streamline flow. Got it? Now the data, all in S I units. For water, rho is one thousand kilograms per cubic metre, and eta is ten to the power minus three pascal second. The pipe's size is its diameter: one centimetre, that is zero point zero one metres. Right? Substitute. On top, one thousand times ten to the power minus three. At the bottom, one thousand times zero point zero one. With me? Now look closely before you multiply. Can you spot something that cancels? The thousands! Very good! One thousand for R sub e on top, and one thousand for rho at the bottom: they cancel. So v sub c is ten to the power minus three, over zero point zero one. That is zero point one metres per second, the largest speed for streamline flow in this pipe. That's it!

8:14Quick revision: Re formula, flow ranges, critical velocity

So, quick revision. The Reynolds number, R sub e, equals rho v D over eta. It is dimensionless, capital D is the diameter, and every quantity goes in S I units. Below one thousand, the flow is usually streamline; above two thousand, turbulent; in between, unsteady. And the critical velocity, v sub c, equals R sub e times eta, over rho D, with R sub e at the transition. Now go and try the practice problems on this page. You can do them. All the best!