WEBVTT 00:00:00.035 --> 00:00:04.378 A full size aircraft costs a fortune to build, and more to fly. 00:00:04.481 --> 00:00:08.462 So before anybody builds one, a small copy of it goes into a wind tunnel, 00:00:08.545 --> 00:00:11.123 and somebody measures the force on the copy. 00:00:11.320 --> 00:00:16.162 The strange thing is that this works, and today I want to show you exactly why. 00:00:17.465 --> 00:00:20.090 Here is the aircraft you actually care about. 00:00:20.704 --> 00:00:23.480 Air comes at it, and the air pushes back. 00:00:24.277 --> 00:00:27.932 The part of that push which lies along the flow is what we call drag, 00:00:28.015 --> 00:00:30.419 and it is what the engines have to fight. 00:00:30.615 --> 00:00:34.262 It sets the fuel burn, the range, and the top speed. 00:00:34.458 --> 00:00:37.954 Getting it wrong by ten percent is an expensive mistake. 00:00:38.750 --> 00:00:41.421 One measurement of the wing matters throughout. 00:00:41.525 --> 00:00:44.159 Call the distance from the front of it to the back L. 00:00:44.242 --> 00:00:48.515 And you cannot put this aircraft in a tunnel, because no tunnel is that big. 00:00:49.312 --> 00:00:51.379 So you build a model instead. 00:00:51.575 --> 00:00:54.768 It sits in the working section of a tunnel, with the walls a little 00:00:54.768 --> 00:00:58.601 way above it and below it, and the whole thing is eight times smaller. 00:00:59.397 --> 00:01:02.764 Run the tunnel, and air blows over the model exactly as it blows 00:01:02.764 --> 00:01:06.654 over the aircraft. You put a balance underneath it and you read a force, 00:01:06.654 --> 00:01:10.207 in newtons, the same way you would if the aircraft were up there. 00:01:11.004 --> 00:01:12.676 And here is the question. 00:01:12.780 --> 00:01:15.321 This thing is eight times smaller than that thing, 00:01:15.404 --> 00:01:17.935 sitting in different air at a different speed. 00:01:18.132 --> 00:01:22.185 Why should the reading on the balance tell you anything at all about the aircraft? 00:01:22.795 --> 00:01:27.347 The answer is two numbers, and this whole lecture is about where they come from. 00:01:27.543 --> 00:01:29.936 The first is called the Reynolds number. 00:01:30.040 --> 00:01:31.988 It is built from the density of the air, 00:01:32.071 --> 00:01:35.506 and from the speed of the flow, and from the size of the body, 00:01:35.589 --> 00:01:39.456 all divided in the end by the viscosity of that same air. 00:01:40.369 --> 00:01:42.819 The second is the drag coefficient. 00:01:43.016 --> 00:01:47.438 That is the drag force itself, divided by a quantity built out of the density, 00:01:47.521 --> 00:01:53.338 the speed and an area, so that every unit cancels and a pure number is left behind. 00:01:53.949 --> 00:01:58.024 And the claim, which we are going to earn rather than assume, is this. 00:01:58.221 --> 00:01:59.266 Match the Reynolds number 00:01:59.266 --> 00:02:03.644 between the model and the aircraft, and the drag coefficient comes out the same. 00:02:03.840 --> 00:02:07.707 Get that one number right, and the model is telling you the truth. 00:02:08.819 --> 00:02:10.454 Before any algebra at all, 00:02:10.537 --> 00:02:14.088 let us just ask what the drag on a body could possibly depend on. 00:02:14.171 --> 00:02:18.108 Here is a body, and here is air moving past it from the left. 00:02:18.212 --> 00:02:20.697 Four candidates suggest themselves. 00:02:21.493 --> 00:02:23.897 The first is how big the body is. 00:02:24.001 --> 00:02:27.645 One length will do to stand for that, and I will call it L. 00:02:27.728 --> 00:02:31.943 Double the size, and you have doubled everything the flow has to get around. 00:02:32.740 --> 00:02:36.593 The second is how fast the air is going, which is V. 00:02:36.676 --> 00:02:40.310 Not the speed of the aircraft and the speed of the air separately, notice. 00:02:40.506 --> 00:02:42.225 Only the relative speed between them, 00:02:42.225 --> 00:02:46.486 because a flow does not know whether the body or the air is doing the moving. 00:02:47.283 --> 00:02:50.883 Then the air itself, which brings two numbers of its own. 00:02:50.987 --> 00:02:54.865 Air has mass, so a given volume of it weighs something. 00:02:55.317 --> 00:02:57.161 That is the density, rho. 00:02:57.244 --> 00:03:01.517 Push through denser air and there is simply more mass to shove out of the way. 00:03:01.942 --> 00:03:04.683 And air is very slightly sticky. 00:03:04.879 --> 00:03:08.688 Layers of it drag on one another, and on the skin of the body. 00:03:09.001 --> 00:03:10.859 That stickiness is the viscosity, 00:03:10.859 --> 00:03:15.352 mu, and it is the reason a wing has friction to fight as well as pressure. 00:03:16.056 --> 00:03:20.771 You might reasonably ask why gravity is not on that list, or the temperature. 00:03:21.223 --> 00:03:25.356 Gravity does not push a wing sideways, and temperature acts only through rho 00:03:25.356 --> 00:03:28.642 and through mu, both of which are already on it. 00:03:28.839 --> 00:03:32.285 Choosing this list is the one creative step in the whole method, 00:03:32.368 --> 00:03:35.295 and it is the one place you can go badly wrong. 00:03:36.091 --> 00:03:39.702 So: four quantities in, and one quantity out. 00:03:39.899 --> 00:03:42.001 The drag, D. Now the trick. 00:03:42.198 --> 00:03:43.533 We are going to work out the shape 00:03:43.533 --> 00:03:47.446 of the relation between them without solving a single equation of motion. 00:03:48.843 --> 00:03:52.382 Every one of these five is built out of three basic dimensions: mass, 00:03:52.465 --> 00:03:53.964 length and time. 00:03:54.160 --> 00:03:55.784 I am going to write those upright, 00:03:55.867 --> 00:03:58.956 so that a dimension never gets mistaken for a symbol. 00:03:59.567 --> 00:04:02.316 A size is a length, and nothing else. 00:04:02.399 --> 00:04:05.233 A speed is a length divided by a time. 00:04:05.937 --> 00:04:10.918 Density is mass per unit volume, so mass divided by a length cubed. 00:04:11.370 --> 00:04:14.236 Viscosity, if you chase its definition through, 00:04:14.319 --> 00:04:18.836 works out as mass, divided by a length, divided by a time. 00:04:19.149 --> 00:04:21.808 Take that last one on trust for the moment. 00:04:22.512 --> 00:04:24.451 And drag is a force. 00:04:24.555 --> 00:04:30.115 Force is mass times acceleration, so its dimensions are mass, times length, 00:04:30.198 --> 00:04:32.080 divided by time squared. 00:04:32.876 --> 00:04:34.119 Now count. 00:04:34.316 --> 00:04:36.488 There are five quantities. 00:04:36.800 --> 00:04:38.250 Look down the last column, 00:04:38.333 --> 00:04:42.444 and between them they use three independent dimensions and no more. 00:04:42.640 --> 00:04:47.854 Mass appears, length appears, time appears, and that is the lot. 00:04:48.651 --> 00:04:51.334 Five take away three is two. 00:04:52.246 --> 00:04:54.439 And that little subtraction is a theorem, 00:04:54.522 --> 00:04:57.274 one of the most useful in all of engineering. 00:04:57.587 --> 00:05:01.894 A relation among n quantities built from k independent dimensions can always 00:05:01.894 --> 00:05:06.457 be rewritten as a relation among exactly n minus k dimensionless groups. 00:05:07.254 --> 00:05:10.297 Two groups. Not three, not five. 00:05:10.609 --> 00:05:13.127 Whatever the true law of drag turns out to be, 00:05:13.210 --> 00:05:17.030 however ugly the equations of motion sitting behind it, it can be written 00:05:17.030 --> 00:05:20.989 as one dimensionless number depending on one other dimensionless number. 00:05:21.186 --> 00:05:22.998 So let us go and build them. 00:05:24.098 --> 00:05:25.539 Two groups, then. 00:05:25.735 --> 00:05:26.455 Let us build them 00:05:26.455 --> 00:05:30.775 by hand, so that you can see for yourself there is nothing hidden in the machinery. 00:05:31.572 --> 00:05:35.424 Start with the drag, and multiply it by the other three quantities, 00:05:35.507 --> 00:05:38.399 each raised to some power we do not know yet. 00:05:38.503 --> 00:05:43.020 Call the whole product Pi one, and demand that it come out dimensionless. 00:05:43.817 --> 00:05:47.324 Write out the dimensions of everything on the right hand side. 00:05:47.427 --> 00:05:51.794 Drag brings mass, length, and time to the minus two. 00:05:51.990 --> 00:05:56.807 Density brings mass over length cubed, all raised to the power a. 00:05:56.890 --> 00:05:59.944 The speed and the size follow on the next line. 00:06:00.741 --> 00:06:05.699 For Pi one to be a pure number, every dimension has to cancel completely. 00:06:05.895 --> 00:06:07.264 Take mass first. 00:06:07.347 --> 00:06:09.342 The drag brings one power of it, 00:06:09.425 --> 00:06:14.265 density brings a of them, and nothing else on that line carries any mass at all. 00:06:14.348 --> 00:06:18.737 So one plus a must be zero, and a is minus one. 00:06:19.650 --> 00:06:21.972 Time next. Drag carries time 00:06:21.972 --> 00:06:26.709 to the minus two, and the speed to the power b contributes minus b. 00:06:26.894 --> 00:06:31.168 So minus two minus b is zero, and b is minus two. 00:06:31.953 --> 00:06:33.196 And length. 00:06:33.381 --> 00:06:36.595 One from the drag, minus three a from the density, 00:06:36.678 --> 00:06:39.709 b from the speed, and c from the size. 00:06:39.882 --> 00:06:44.922 Put in the a and the b we already have, and c comes out at minus two as well. 00:06:45.695 --> 00:06:47.516 So there is the first group. 00:06:47.599 --> 00:06:52.778 Drag, divided by density times speed squared times size squared. 00:06:52.975 --> 00:06:54.773 Every dimension has cancelled, 00:06:54.856 --> 00:06:58.258 and what is left is a bare number with no units on it anywhere. 00:06:58.962 --> 00:07:01.746 In practice nobody writes it quite like that. 00:07:01.829 --> 00:07:06.310 There is a one half out in front, borrowed from the kinetic energy in the flow, 00:07:06.393 --> 00:07:09.526 and a reference area A standing in for L squared. 00:07:09.609 --> 00:07:14.428 That is the drag coefficient, and it is what wind tunnels actually report. 00:07:16.080 --> 00:07:19.563 Now the second group, by exactly the same procedure. 00:07:19.667 --> 00:07:24.521 This time we start from the viscosity, because the drag has already been used up. 00:07:25.062 --> 00:07:29.057 Viscosity is mass, over a length, over a time. 00:07:29.160 --> 00:07:31.332 The other three are as before. 00:07:31.606 --> 00:07:32.907 Mass again. 00:07:33.104 --> 00:07:38.791 Viscosity brings one power, density brings a of them, so a is minus one, 00:07:38.874 --> 00:07:40.872 exactly as it was last time. 00:07:41.181 --> 00:07:46.891 Time. Viscosity has time to the minus one, and the speed contributes minus b. 00:07:46.974 --> 00:07:49.100 So b is minus one. 00:07:49.478 --> 00:07:52.460 And length. Minus one from the viscosity, 00:07:52.543 --> 00:07:57.583 plus three from that minus three a, minus one from the speed, and c. 00:07:57.687 --> 00:08:01.228 That comes to one plus c, so c is minus one. 00:08:01.770 --> 00:08:06.217 Which gives viscosity over density times speed times size. 00:08:06.321 --> 00:08:08.711 That is a perfectly good dimensionless group, 00:08:08.794 --> 00:08:12.161 and it is conventional to turn the whole thing upside down. 00:08:12.958 --> 00:08:17.324 That is the Reynolds number, and it has a meaning worth carrying around. 00:08:17.428 --> 00:08:22.281 Rho V squared is roughly the pressure the flow's own momentum can push with. 00:08:22.478 --> 00:08:27.181 Mu V over L is roughly the stress that the stickiness can manage. 00:08:27.377 --> 00:08:32.579 Their ratio is Re, so a large Reynolds number means inertia is winning. 00:08:33.976 --> 00:08:37.146 So here is what the theorem promised, now delivered. 00:08:37.343 --> 00:08:42.011 Five quantities, three dimensions, two groups, and only two. 00:08:42.323 --> 00:08:46.991 Whatever the drag law is, it must be expressible as the drag coefficient equal 00:08:46.991 --> 00:08:50.858 to some function of the Reynolds number, and of nothing else at all. 00:08:51.561 --> 00:08:54.279 Notice carefully what has not happened here. 00:08:54.383 --> 00:08:58.781 Dimensional analysis has not told you what that function is. It cannot. 00:08:58.864 --> 00:09:03.148 Finding f still takes a wind tunnel, or a very large computer. 00:09:03.852 --> 00:09:07.577 What it has told you is that one number is enough to look the answer up. 00:09:07.660 --> 00:09:12.212 And that, precisely, is the wind tunnel's licence to exist. 00:09:13.278 --> 00:09:15.656 Enough theory. Here are two experiments, 00:09:15.740 --> 00:09:18.992 and I want you to notice first how little they have in common. 00:09:19.603 --> 00:09:23.837 Sizes first. The aircraft's wing measures two metres from front to back. 00:09:23.921 --> 00:09:28.218 The model's measures a quarter of a metre. That is a factor of eight. 00:09:29.270 --> 00:09:33.298 Speeds. The aircraft flies at forty five metres a second. 00:09:33.402 --> 00:09:36.434 The tunnel runs at ninety. Twice as fast. 00:09:37.137 --> 00:09:39.494 And it is not even the same air. 00:09:39.692 --> 00:09:41.062 The tunnel is pressurised, 00:09:41.062 --> 00:09:45.068 so the air inside it is four times as dense as the air out there. 00:09:45.520 --> 00:09:49.047 Viscosity, oddly, hardly notices pressure at all, 00:09:49.130 --> 00:09:52.684 so that one number really is identical in both columns. 00:09:53.480 --> 00:09:54.964 Eight times smaller. 00:09:55.048 --> 00:09:57.904 Twice as fast. Four times denser. 00:09:58.520 --> 00:10:01.257 Nothing about these two flows looks remotely alike, 00:10:01.340 --> 00:10:04.302 and the two drag forces will not be alike either. 00:10:05.699 --> 00:10:08.181 So feed those numbers into the Reynolds number. 00:10:08.264 --> 00:10:12.709 Density, times speed, times size, all over the viscosity. 00:10:12.792 --> 00:10:17.565 For the aircraft, the top line is one and a quarter, times forty five, times two. 00:10:18.268 --> 00:10:21.424 One and a quarter times forty five is fifty six and a quarter, 00:10:21.508 --> 00:10:24.315 and times two is a hundred and twelve point five. 00:10:24.398 --> 00:10:27.940 Divide that by the viscosity and you get six million. 00:10:28.992 --> 00:10:30.258 Now the model. 00:10:30.873 --> 00:10:33.463 Five, times ninety, times a quarter. 00:10:33.660 --> 00:10:35.216 Five times ninety is four hundred 00:10:35.216 --> 00:10:39.302 and fifty, and a quarter of that is a hundred and twelve point five. 00:10:39.406 --> 00:10:41.172 The very same top line. 00:10:41.598 --> 00:10:43.200 And of course it is. 00:10:43.304 --> 00:10:47.880 One eighth of the size, times twice the speed, times four times the density. 00:10:48.076 --> 00:10:51.826 An eighth, times two, times four, is one. 00:10:52.024 --> 00:10:56.003 So the product is unchanged, and the Reynolds number is unchanged. 00:10:56.086 --> 00:10:57.620 Six million again. 00:10:58.532 --> 00:11:00.473 Which is the whole of it, really. 00:11:00.668 --> 00:11:03.197 Similarity is not a coincidence you hope for. 00:11:03.281 --> 00:11:07.194 It is something you engineer, by choosing the tunnel's speed and pressure 00:11:07.194 --> 00:11:09.504 so that one product comes out right. 00:11:10.901 --> 00:11:12.628 So what does that buy you? 00:11:12.712 --> 00:11:15.868 Here is the drag coefficient of this shape, measured, 00:11:15.952 --> 00:11:18.240 and plotted against Reynolds number. 00:11:18.436 --> 00:11:22.779 The axis is in millions, so six on it means six million. 00:11:23.576 --> 00:11:27.138 Dimensional analysis promised us that a curve like this exists, 00:11:27.222 --> 00:11:29.463 and that it needs only one input. 00:11:29.660 --> 00:11:32.190 It said nothing whatever about the shape of it. 00:11:32.806 --> 00:11:34.988 Somebody had to go out and measure that. 00:11:35.786 --> 00:11:38.132 Our two experiments both sit here. 00:11:38.236 --> 00:11:39.862 Not near one another. 00:11:40.174 --> 00:11:44.459 At the same point, reading the same drag coefficient of three hundredths. 00:11:45.511 --> 00:11:48.182 That is what dynamic similarity means. 00:11:48.634 --> 00:11:52.509 Two flows at the same Reynolds number have the same drag coefficient, 00:11:52.592 --> 00:11:53.821 the same separation, 00:11:53.905 --> 00:11:58.758 the same wake, the same everything, once you take the units back out. 00:11:59.462 --> 00:12:03.178 So here is the procedure an engineer actually runs. 00:12:03.282 --> 00:12:05.326 Measure the force on the model. 00:12:05.430 --> 00:12:08.449 Suppose the balance reads thirty eight newtons. 00:12:09.060 --> 00:12:12.462 Divide it by a half rho V squared 00:12:12.658 --> 00:12:17.848 times the model's area, using the model's own density, speed and area. 00:12:17.953 --> 00:12:22.122 Out comes a drag coefficient of nought point zero three zero. 00:12:22.824 --> 00:12:24.286 Now turn it around. 00:12:24.370 --> 00:12:29.579 Multiply that same coefficient by a half rho V squared times the aircraft's area, 00:12:29.662 --> 00:12:32.032 using the aircraft's numbers this time. 00:12:32.829 --> 00:12:35.032 A hundred and fifty two newtons. 00:12:35.116 --> 00:12:38.759 That is a quantitative prediction about a machine nobody has built, 00:12:38.843 --> 00:12:42.350 made from a force measured on a small object in a box. 00:12:43.401 --> 00:12:44.969 Now the honest part. 00:12:45.166 --> 00:12:46.466 Everything so far rested 00:12:46.466 --> 00:12:50.600 on a list of four quantities, and that list had something missing from it. 00:12:51.652 --> 00:12:52.926 Air is springy. 00:12:53.010 --> 00:12:54.854 Squeeze it and it pushes back, 00:12:54.938 --> 00:12:59.060 and the speed at which that push travels through it is the speed of sound. 00:12:59.257 --> 00:13:03.948 Call it a. It is a length over a time, like any other speed. 00:13:04.652 --> 00:13:06.406 Here is why it matters. 00:13:06.509 --> 00:13:10.222 A body moving through air sends pressure signals out ahead of itself, 00:13:10.306 --> 00:13:14.103 at the speed of sound, warning the air to get out of the way. 00:13:14.207 --> 00:13:16.367 Those circles are the signals. 00:13:17.164 --> 00:13:20.818 At a quarter of the speed of sound the signals run far out in front, 00:13:20.902 --> 00:13:23.329 and the air has plenty of notice. 00:13:23.526 --> 00:13:27.973 Push the speed up towards the speed of sound, and they bunch against the nose. 00:13:28.077 --> 00:13:30.492 The air gets almost no warning at all. 00:13:31.289 --> 00:13:32.973 So put a on the list. 00:13:33.077 --> 00:13:35.826 Six quantities now, still three dimensions, 00:13:35.910 --> 00:13:38.732 and six take away three is three. 00:13:38.835 --> 00:13:43.012 There is a third group, and it is the easiest one in the lecture: a speed, 00:13:43.096 --> 00:13:44.769 divided by a speed. 00:13:45.194 --> 00:13:47.053 That is the Mach number. 00:13:47.249 --> 00:13:50.001 Our result should really have read like this: 00:13:50.198 --> 00:13:53.551 the drag coefficient is a function of the Reynolds number, 00:13:53.635 --> 00:13:57.769 and of the Mach number as well, and we simply left the second one out. 00:13:59.072 --> 00:14:01.488 Which raises an awkward question. 00:14:01.685 --> 00:14:03.787 Why did ignoring it work? 00:14:04.100 --> 00:14:07.882 Here is the drag coefficient of a wing plotted against Mach number, 00:14:07.966 --> 00:14:10.533 at one fixed Reynolds number. 00:14:11.329 --> 00:14:13.872 Below about a third, that curve is flat. 00:14:13.976 --> 00:14:15.915 Compressibility is there, of course, 00:14:15.915 --> 00:14:19.109 but it is doing nothing you could measure on a balance. 00:14:20.022 --> 00:14:24.864 Our aircraft at forty five metres a second is at Mach nought point one three. 00:14:25.316 --> 00:14:28.045 The sound speed is about three hundred and forty. 00:14:28.749 --> 00:14:31.695 The model at ninety is at nought point two six. 00:14:31.779 --> 00:14:33.509 Both of them are down here in the flat part, 00:14:33.509 --> 00:14:37.968 so both may pretend the air is incompressible, and the two of them agree. 00:14:38.880 --> 00:14:41.319 But look what happens further along. 00:14:41.516 --> 00:14:44.662 Past about Mach nought point seven, patches of the flow 00:14:44.662 --> 00:14:50.120 over the wing go supersonic, shock waves form, and the coefficient does this. 00:14:50.916 --> 00:14:54.679 That is drag divergence, and it is not a small correction. 00:14:54.783 --> 00:14:56.943 The coefficient can triple. 00:14:57.139 --> 00:15:00.715 Two flows matched on Reynolds number but sitting on opposite sides 00:15:00.715 --> 00:15:03.886 of that rise do not resemble each other in the slightest. 00:15:05.398 --> 00:15:08.751 So test an airliner cruising at Mach nought point eight five, 00:15:08.835 --> 00:15:11.506 and you must match both numbers at once. 00:15:12.121 --> 00:15:14.339 And now you have a real problem. 00:15:14.764 --> 00:15:15.902 Matching the Reynolds number 00:15:15.902 --> 00:15:20.128 with a model eight times smaller means running it eight times faster. 00:15:20.580 --> 00:15:25.306 Matching the Mach number, in the same gas at the same temperature, means running it 00:15:25.306 --> 00:15:27.361 at exactly the same speed. 00:15:27.976 --> 00:15:29.602 You cannot do both. 00:15:30.654 --> 00:15:33.430 Which is why the expensive tunnels exist. 00:15:33.533 --> 00:15:36.143 Pressurise the air and rho goes up, 00:15:36.227 --> 00:15:38.813 so Reynolds goes up without touching the speed. 00:15:38.897 --> 00:15:42.679 Chill the gas to a hundred kelvin and the viscosity falls, 00:15:42.763 --> 00:15:45.109 and the speed of sound falls with it. 00:15:45.813 --> 00:15:47.308 Cryogenic nitrogen, 00:15:47.392 --> 00:15:51.654 at four atmospheres, in a tunnel that costs more than the aircraft. 00:15:52.106 --> 00:15:55.032 All of it to buy back one dimensionless number. 00:15:56.429 --> 00:15:58.833 And the same trap has other names. 00:15:59.146 --> 00:16:02.649 Put a seaplane hull down on water and gravity joins the list, 00:16:02.733 --> 00:16:05.253 because gravity is what makes the waves. 00:16:05.450 --> 00:16:09.746 Out comes the Froude number, speed over the root of g times length. 00:16:10.659 --> 00:16:13.233 Match Reynolds and you get the waves wrong. 00:16:13.317 --> 00:16:15.892 Match Froude and you get the friction wrong. 00:16:15.976 --> 00:16:20.029 The model builder splits the difference and corrects the rest by calculation. 00:16:20.407 --> 00:16:25.350 So the lesson is not that dimensional analysis is unreliable. It is exact. 00:16:25.434 --> 00:16:29.986 The lesson is that it is only ever as good as the list you started from. 00:16:30.783 --> 00:16:33.454 Write down every quantity that could matter. 00:16:33.651 --> 00:16:35.358 Count your dimensions. 00:16:35.555 --> 00:16:38.884 Then check that every group you produced is actually matched. 00:16:38.968 --> 00:16:43.319 Miss one, and your beautiful small model is quietly telling you about a flow 00:16:43.403 --> 00:16:45.191 that does not exist.