WEBVTT 00:00:00.046 --> 00:00:01.324 Bernoulli's equation is one 00:00:01.324 --> 00:00:04.979 of the first genuinely powerful results you meet in fluid mechanics, 00:00:05.062 --> 00:00:09.324 and it is also the source of one of the first genuine crises of confidence. 00:00:09.520 --> 00:00:13.572 Apply it carefully to a body sitting in a steady stream, and it tells you, 00:00:13.572 --> 00:00:17.195 quite calmly, that the body feels no drag whatsoever. 00:00:17.392 --> 00:00:18.832 Not a little drag. 00:00:18.936 --> 00:00:19.831 None. 00:00:21.483 --> 00:00:25.466 Let me set that calculation up, because the reasoning inside it is good. 00:00:25.662 --> 00:00:29.296 Here is a cylinder seen end on, sitting in a stream that arrives 00:00:29.296 --> 00:00:31.199 from the left at speed U. 00:00:31.282 --> 00:00:36.577 The pale lines are streamlines, the paths the fluid follows as it goes past. 00:00:37.373 --> 00:00:40.750 The fluid in this calculation has no viscosity at all. 00:00:40.833 --> 00:00:43.132 Nothing sticks and nothing rubs. 00:00:43.236 --> 00:00:48.238 And with no viscosity the streamlines close up behind the body exactly as tidily 00:00:48.321 --> 00:00:50.014 as they opened in front of it. 00:00:50.097 --> 00:00:54.603 Cover half the picture, and you genuinely cannot tell which half you covered. 00:00:55.307 --> 00:00:58.059 So put Bernoulli to work along one of them. 00:00:58.256 --> 00:01:03.631 Pressure plus one half rho U squared is a constant, so where the fluid speeds up 00:01:03.631 --> 00:01:07.939 the pressure falls, and where it slows down the pressure rises. 00:01:08.136 --> 00:01:10.932 Right at the nose the fluid is brought completely to rest, 00:01:11.015 --> 00:01:14.035 so the pressure there is as high as it gets. 00:01:14.831 --> 00:01:19.195 Over the shoulders it has to hurry to get around, so the pressure there drops. 00:01:19.278 --> 00:01:22.992 And then at the tail the fluid slows to a stop a second time, 00:01:23.075 --> 00:01:27.011 and the pressure climbs back to exactly the value it had at the front. 00:01:27.808 --> 00:01:31.942 Which means every push on the front is matched by an equal push on the back. 00:01:32.138 --> 00:01:34.400 Add the pressure up over the whole surface, 00:01:34.483 --> 00:01:37.120 and the total along the flow is zero. 00:01:37.316 --> 00:01:41.413 That is d'Alembert's paradox, and it is not an algebra slip. 00:01:41.496 --> 00:01:44.701 It is what the ideal theory honestly says. 00:01:45.842 --> 00:01:49.918 Here is the same body in real air, put beside the prediction. 00:01:50.022 --> 00:01:52.530 Same shape, same speed. 00:01:52.843 --> 00:01:55.978 And the flow gets around the front perfectly happily. 00:01:56.520 --> 00:01:59.397 Then, somewhere near the widest part of the body, 00:01:59.480 --> 00:02:01.803 it lets go of the surface completely. 00:02:02.418 --> 00:02:05.089 Those two red lines mark where it leaves. 00:02:05.401 --> 00:02:10.044 Behind them sits a broad, churning region of slow, disorganised fluid, 00:02:10.127 --> 00:02:12.124 and that region is called the wake. 00:02:12.921 --> 00:02:15.522 And now the symmetry is gone. 00:02:15.835 --> 00:02:17.818 At the nose the pressure is still high, 00:02:17.901 --> 00:02:20.422 because the fluid is still being stopped there. 00:02:21.222 --> 00:02:23.217 But at the back it never recovers, 00:02:23.300 --> 00:02:27.806 because nothing back there slows down in the orderly way the ideal picture assumed. 00:02:28.002 --> 00:02:29.501 It stays low. 00:02:29.926 --> 00:02:32.538 You can read that straight off the two pictures. 00:02:32.735 --> 00:02:35.984 On the left there are two high marks, one at each end, 00:02:36.067 --> 00:02:38.541 and they cancel each other exactly. 00:02:38.853 --> 00:02:42.567 On the right there is a high mark at the front and a low one at the back, 00:02:42.650 --> 00:02:44.833 and nothing cancels anything. 00:02:45.746 --> 00:02:47.172 High pressure on the front, 00:02:47.255 --> 00:02:51.551 low pressure on the back, and the difference is a net backward force. 00:02:51.655 --> 00:02:53.397 That is drag. 00:02:53.501 --> 00:02:58.053 And almost all of it, on a body shaped like this, is decided by one thing: 00:02:58.157 --> 00:03:01.001 the point at which the flow chose to let go. 00:03:02.653 --> 00:03:05.891 What the ideal calculation left out is viscosity, 00:03:05.974 --> 00:03:10.165 and in particular what viscosity does inside an extremely thin sheet 00:03:10.165 --> 00:03:12.360 of fluid pressed against the surface. 00:03:12.673 --> 00:03:14.912 That sheet is called the boundary layer, 00:03:14.995 --> 00:03:17.956 and it is the subject of the next twenty minutes. 00:03:18.753 --> 00:03:20.843 And here is the destination. 00:03:21.040 --> 00:03:25.601 This is the drag coefficient of a sphere, plotted against Reynolds number, 00:03:25.684 --> 00:03:30.212 which is the one number that says how fast a flow is in the sense that matters. 00:03:30.409 --> 00:03:34.404 Notice that somewhere out to the right, the curve falls off a cliff. 00:03:35.107 --> 00:03:36.489 The drop is roughly a factor 00:03:36.489 --> 00:03:40.832 of four, and it happens because the boundary layer changes its character. 00:03:41.029 --> 00:03:42.713 So the plan is this. 00:03:42.816 --> 00:03:44.617 First the wall itself. 00:03:44.813 --> 00:03:48.506 Then the two quite different ways a body can feel drag. 00:03:48.610 --> 00:03:50.187 Then separation, 00:03:50.270 --> 00:03:55.797 the wake, and that cliff, which has a name of its own: the drag crisis. 00:03:56.338 --> 00:03:57.405 And by the end of it, 00:03:57.488 --> 00:04:00.913 you will know exactly why a golf ball is covered in dimples. 00:04:01.953 --> 00:04:05.445 Everything that follows comes out of one experimental fact, 00:04:05.529 --> 00:04:07.639 so let me put it on the board first. 00:04:07.723 --> 00:04:11.903 When a real fluid flows past a solid surface, the fluid actually touching 00:04:11.903 --> 00:04:14.643 that surface does not slide along it. 00:04:14.747 --> 00:04:16.060 It sticks. 00:04:16.856 --> 00:04:19.191 That is the no-slip condition. 00:04:19.503 --> 00:04:23.227 At the wall, the fluid velocity equals the wall velocity, 00:04:23.311 --> 00:04:26.609 and for a body sitting still that means zero. 00:04:26.713 --> 00:04:30.963 It holds for every ordinary fluid, however slippery it looks. 00:04:31.160 --> 00:04:33.633 Water, air, honey: 00:04:33.830 --> 00:04:37.151 at the wall, nothing moves relative to the wall. 00:04:38.366 --> 00:04:42.116 Far from the wall, of course, the fluid is doing whatever it likes. 00:04:42.313 --> 00:04:45.631 Out here it is travelling at the free stream speed U. 00:04:45.715 --> 00:04:50.100 So somewhere in between the velocity has to climb from nothing up to U, 00:04:50.184 --> 00:04:53.041 and it does that over some finite distance. 00:04:53.583 --> 00:04:55.209 There is the profile. 00:04:55.521 --> 00:04:58.920 Zero at the wall, rising very steeply just above it, 00:04:59.004 --> 00:05:02.035 and then flattening out as it joins the outer flow. 00:05:02.232 --> 00:05:03.776 A velocity that changes like 00:05:03.776 --> 00:05:07.632 that across a layer of fluid means the fluid is being sheared. 00:05:08.428 --> 00:05:11.432 And the shear stress at the wall is the viscosity, 00:05:11.516 --> 00:05:15.534 mu, multiplied by the slope of that profile at the wall. 00:05:15.986 --> 00:05:18.599 Steep profile, large stress. 00:05:19.051 --> 00:05:21.606 Gentle profile, small stress. 00:05:22.058 --> 00:05:25.516 That one quantity is the entire origin of friction drag, 00:05:25.600 --> 00:05:27.679 and we will come back to it. 00:05:29.493 --> 00:05:31.154 Now the real question. 00:05:31.258 --> 00:05:35.311 How far out into the fluid does the wall's influence reach? 00:05:35.414 --> 00:05:39.815 Put a thin flat plate edge on into a uniform stream, so that the plate 00:05:39.815 --> 00:05:44.669 barely disturbs the outer flow at all, and follow it from the leading edge. 00:05:45.465 --> 00:05:48.903 At the leading edge the layer has no thickness whatsoever: 00:05:49.006 --> 00:05:51.503 the fluid has only just met the plate. 00:05:52.304 --> 00:05:54.367 A short way downstream, here, 00:05:54.451 --> 00:05:59.074 the profile looks like this, and the affected fluid reaches up to about here. 00:05:59.870 --> 00:06:02.575 The convention is to call the edge of the layer the height 00:06:02.575 --> 00:06:07.115 at which the fluid has recovered ninety nine percent of the free stream speed. 00:06:07.312 --> 00:06:08.703 That height is delta, 00:06:08.787 --> 00:06:12.886 and delta is a function of how far along the plate you have travelled. 00:06:14.100 --> 00:06:17.816 Now watch what happens as I move that station downstream. 00:06:18.268 --> 00:06:19.929 The layer grows. 00:06:20.126 --> 00:06:24.628 The profile keeps essentially the same shape, it simply stretches upward, 00:06:24.712 --> 00:06:28.810 and the plate's influence reaches further and further out into the stream. 00:06:29.607 --> 00:06:32.023 The growth follows a square root. 00:06:32.219 --> 00:06:36.094 The thickness goes like the square root of the distance along the plate, 00:06:36.178 --> 00:06:38.907 and that is exactly what diffusion does. 00:06:39.104 --> 00:06:41.910 Viscosity carries news of the wall outward, 00:06:41.994 --> 00:06:45.536 and diffusion always spreads like the square root of time. 00:06:46.333 --> 00:06:50.022 Whether that layer is thin at all is decided by one dimensionless group, 00:06:50.106 --> 00:06:51.512 and here it is. 00:06:51.708 --> 00:06:58.257 Reynolds number: density, times speed, times distance, divided by viscosity. 00:06:58.454 --> 00:07:02.135 It compares the inertia of the fluid with its stickiness. 00:07:02.932 --> 00:07:05.765 High Reynolds number means a thin layer. 00:07:05.869 --> 00:07:09.956 Take air at ten metres per second over a plate one metre long. 00:07:10.153 --> 00:07:14.588 That is a Reynolds number of about seven hundred thousand, and at the far end 00:07:14.588 --> 00:07:18.257 of the plate the boundary layer is roughly five millimetres deep. 00:07:19.309 --> 00:07:22.073 And there is the reconciliation with Bernoulli. 00:07:22.270 --> 00:07:24.508 Outside those five millimetres, 00:07:24.592 --> 00:07:29.283 viscosity genuinely is negligible and Bernoulli genuinely does work. 00:07:29.735 --> 00:07:32.871 Inside them, viscosity is everything. 00:07:33.323 --> 00:07:35.553 The ideal theory was not wrong. 00:07:35.865 --> 00:07:39.326 It was blind to a thin strip that decides the answer. 00:07:40.978 --> 00:07:42.592 One more thing about this layer, 00:07:42.592 --> 00:07:45.448 and it is the thing the rest of the lecture turns on. 00:07:45.645 --> 00:07:48.432 Boundary layers come in two kinds. 00:07:48.629 --> 00:07:52.910 Near the leading edge the layer is laminar: smooth, orderly, 00:07:52.994 --> 00:07:55.828 sheets of fluid sliding over one another. 00:07:56.624 --> 00:08:00.665 Further downstream it goes unstable and becomes turbulent: 00:08:00.769 --> 00:08:06.343 eddies, mixing, fluid tumbling violently between one height and another. 00:08:06.539 --> 00:08:09.697 On a flat plate the switch happens at a Reynolds number 00:08:09.697 --> 00:08:11.950 of roughly five hundred thousand. 00:08:12.654 --> 00:08:16.533 Here the two profiles are drawn against the same axes: 00:08:16.729 --> 00:08:21.208 speed across the bottom as a fraction of the free stream, height up the side 00:08:21.292 --> 00:08:23.824 as a fraction of the layer thickness. 00:08:24.136 --> 00:08:26.854 And the turbulent one is visibly fuller. 00:08:27.465 --> 00:08:30.713 It holds a much higher speed much closer to the wall, 00:08:30.797 --> 00:08:35.011 because the eddies are continually dragging fast fluid down out of the outer 00:08:35.011 --> 00:08:37.845 flow and throwing slow fluid back up. 00:08:38.897 --> 00:08:43.065 That has two consequences, and they point in opposite directions. 00:08:43.169 --> 00:08:45.477 Look at the two slopes right at the wall. 00:08:45.561 --> 00:08:49.227 The turbulent one is far steeper, so the friction is higher. 00:08:49.311 --> 00:08:53.747 But the fluid down near the wall is now carrying a great deal more momentum. 00:08:54.799 --> 00:08:59.486 Hold on to that sentence. A turbulent boundary layer costs you more friction, 00:08:59.570 --> 00:09:03.902 and buys you more momentum in exactly the place the fluid is about to need it. 00:09:04.980 --> 00:09:10.275 Before we go anywhere near a sphere, let us be precise about what drag actually is. 00:09:10.472 --> 00:09:14.614 A body in a stream feels a force on every square millimetre of its surface, 00:09:14.698 --> 00:09:17.636 and that force always has exactly two parts. 00:09:18.340 --> 00:09:22.764 There is a pressure, pushing straight into the surface at right angles to it. 00:09:22.960 --> 00:09:24.757 And there is a shear stress, 00:09:24.841 --> 00:09:28.383 dragging along the surface in the direction the fluid is sliding. 00:09:29.597 --> 00:09:32.590 Take the component of each of those along the flow direction, 00:09:32.674 --> 00:09:35.125 and add them up over the whole body. 00:09:35.739 --> 00:09:38.418 The tangential parts give you friction drag. 00:09:38.502 --> 00:09:40.566 The normal parts give you pressure drag, 00:09:40.650 --> 00:09:44.053 which for historical reasons is also called form drag. 00:09:44.965 --> 00:09:49.444 Those two are not just different terms in a sum. They behave completely differently, 00:09:49.528 --> 00:09:53.336 and the cleanest way to see that is to take one object and turn it round. 00:09:54.988 --> 00:09:58.774 Here is a thin flat plate lying edge on, along the flow. 00:09:58.971 --> 00:10:01.607 The streamlines barely notice it is there. 00:10:01.711 --> 00:10:05.237 The pressure on the front edge is essentially the pressure on the back edge, 00:10:05.321 --> 00:10:08.689 so the pressure contribution is almost exactly nothing. 00:10:09.300 --> 00:10:12.327 But there is a boundary layer growing along both faces, 00:10:12.411 --> 00:10:15.640 and at every point of both faces the fluid is rubbing. 00:10:15.836 --> 00:10:19.111 Add up all of that shear and you have the entire drag. 00:10:19.215 --> 00:10:21.349 This plate is a pure friction body, 00:10:21.433 --> 00:10:24.626 and its drag coefficient is about five thousandths. 00:10:25.330 --> 00:10:29.963 Now turn the same plate through ninety degrees, so it faces the flow squarely. 00:10:30.067 --> 00:10:32.014 Nothing about the plate has changed. 00:10:32.098 --> 00:10:34.189 Only its orientation has. 00:10:34.892 --> 00:10:38.361 The fluid piles up against the front face and is brought to rest, 00:10:38.445 --> 00:10:40.303 so the pressure there is high. 00:10:40.500 --> 00:10:44.700 And it cannot get round those sharp edges at all, so it separates immediately, 00:10:44.784 --> 00:10:48.836 and the whole of the back face sits inside a wake at low pressure. 00:10:49.633 --> 00:10:53.160 Meanwhile the shear stress on those two faces points up and down, 00:10:53.244 --> 00:10:57.935 at right angles to the flow, so it contributes precisely nothing to the drag. 00:10:58.248 --> 00:11:00.497 Every last bit of the force is pressure, 00:11:00.581 --> 00:11:04.820 and the coefficient has gone from five thousandths to about one point two. 00:11:05.523 --> 00:11:08.891 Two hundred times the drag, from the same piece of metal. 00:11:09.204 --> 00:11:12.730 Which tells you that if you want to understand the drag of a real body, 00:11:12.814 --> 00:11:16.449 the first question is always which of those two bills you are paying. 00:11:17.846 --> 00:11:20.296 So here is the split for a few bodies. 00:11:20.609 --> 00:11:24.054 The columns are the fraction of the total drag paid as friction, 00:11:24.138 --> 00:11:26.206 and the fraction paid as pressure. 00:11:26.909 --> 00:11:30.904 The plate edge on, as we said, is a hundred percent friction. 00:11:31.101 --> 00:11:35.640 A well made airfoil is close behind it, around ninety percent friction, because it 00:11:35.640 --> 00:11:40.274 is shaped precisely so that the flow stays attached all the way to the trailing edge. 00:11:41.070 --> 00:11:43.018 And now look at the sphere. 00:11:43.102 --> 00:11:46.795 Roughly five percent friction, ninety five percent pressure. 00:11:47.410 --> 00:11:51.470 A sphere is much closer to a plate held face on than it is to an airfoil, 00:11:51.554 --> 00:11:55.479 which is not what most people expect from something so smooth and rounded. 00:11:56.392 --> 00:12:00.499 So for a sphere, and for most of the blunt objects you actually care about, 00:12:00.583 --> 00:12:02.508 friction is a rounding error. 00:12:02.592 --> 00:12:04.462 The drag is pressure drag. 00:12:04.566 --> 00:12:06.780 Pressure drag is set by the wake. 00:12:06.864 --> 00:12:09.288 And the wake is set by one thing only: 00:12:09.372 --> 00:12:12.055 where the boundary layer lets go of the surface. 00:12:12.999 --> 00:12:16.040 Which means that separation, that one decision the fluid makes, 00:12:16.040 --> 00:12:17.001 is the whole game. 00:12:17.085 --> 00:12:19.153 So let us go and watch it happen. 00:12:20.269 --> 00:12:24.566 Let us look at the pressure a sphere feels, all the way around its surface. 00:12:24.763 --> 00:12:28.405 Along the bottom is the angle measured from the front stagnation point, 00:12:28.489 --> 00:12:31.029 in degrees, so zero is the nose, 00:12:31.113 --> 00:12:35.387 ninety is the equator, and a hundred and eighty is the very back. 00:12:36.299 --> 00:12:39.736 Up the side is the pressure coefficient, which is just the pressure 00:12:39.736 --> 00:12:45.112 measured against the pressure far away and scaled by one half rho U squared. 00:12:45.564 --> 00:12:48.061 One means fully stopped fluid. 00:12:48.165 --> 00:12:51.532 Zero means the pressure of the undisturbed stream. 00:12:52.329 --> 00:12:55.244 The pale curve is what Bernoulli predicts. 00:12:55.556 --> 00:13:00.453 Coefficient one at the nose, falling as the fluid accelerates round the shoulder, 00:13:00.537 --> 00:13:02.833 a minimum of minus three at the equator, 00:13:02.917 --> 00:13:06.366 and then climbing all the way back up to one at the tail. 00:13:06.563 --> 00:13:10.000 Perfectly symmetric, and it integrates to nothing. 00:13:10.797 --> 00:13:12.922 Now split it at the equator. 00:13:13.119 --> 00:13:14.024 From the nose out 00:13:14.024 --> 00:13:17.857 to ninety degrees the pressure is falling as you move downstream. 00:13:18.053 --> 00:13:20.256 That is called a favourable gradient: 00:13:20.340 --> 00:13:23.859 the fluid is being pushed along, running downhill. 00:13:24.656 --> 00:13:29.684 But from the equator to the tail, the pressure rises as you move downstream. 00:13:29.996 --> 00:13:34.467 That is an adverse gradient, and now the fluid is being asked to climb. 00:13:35.264 --> 00:13:37.935 Out in the free stream the climb is easy. 00:13:38.038 --> 00:13:39.141 That fluid arrives 00:13:39.141 --> 00:13:44.541 at the equator moving fast, with plenty of kinetic energy to trade back for pressure. 00:13:44.853 --> 00:13:48.136 But down in the boundary layer, right against the surface, 00:13:48.220 --> 00:13:51.646 the fluid has already given most of its speed away to friction. 00:13:52.443 --> 00:13:55.729 It arrives at the hill with almost nothing in the tank. 00:13:55.926 --> 00:13:59.503 And here is what is actually measured on a real sphere. 00:13:59.699 --> 00:14:02.985 It follows the ideal curve round the front, and then, 00:14:02.985 --> 00:14:07.385 at about eighty degrees, it simply stops following it and goes flat. 00:14:08.782 --> 00:14:12.684 So let me draw what a rising pressure does to a boundary layer. 00:14:12.880 --> 00:14:16.151 Here is a wall, with the pressure increasing to the right, 00:14:16.235 --> 00:14:19.789 and three velocity profiles at three places along it. 00:14:20.585 --> 00:14:22.394 The first one is healthy. 00:14:22.478 --> 00:14:26.921 It has a good steep slope at the wall, the fluid down there is still moving forward, 00:14:27.005 --> 00:14:29.351 and the layer is firmly attached. 00:14:30.566 --> 00:14:31.864 A little further along, 00:14:31.948 --> 00:14:35.361 the rising pressure has been pushing back on that slow near-wall fluid 00:14:35.361 --> 00:14:40.342 the whole way, and the slope at the wall has been squeezed down to exactly zero. 00:14:40.794 --> 00:14:45.393 That point, where the wall slope vanishes, is the separation point. 00:14:46.305 --> 00:14:50.811 And just past it, the near-wall fluid has actually been driven backwards. 00:14:51.263 --> 00:14:55.802 The flow reverses, that reversed fluid wedges underneath the layer and lifts 00:14:55.802 --> 00:15:00.609 it clean off the surface, and the outer flow can no longer follow the body at all. 00:15:01.406 --> 00:15:05.945 So separation is not the fluid failing to turn a corner. It is the fluid near 00:15:05.945 --> 00:15:10.695 the wall running out of momentum while it is being asked to climb a pressure hill. 00:15:11.147 --> 00:15:13.632 Which tells you immediately what would help: 00:15:13.829 --> 00:15:16.082 more momentum near the wall. 00:15:17.385 --> 00:15:21.191 On a real sphere at ordinary speeds, with a laminar boundary layer, 00:15:21.275 --> 00:15:23.958 that happens about eighty degrees from the nose. 00:15:24.154 --> 00:15:27.731 Just before the widest point, and well short of the back. 00:15:28.528 --> 00:15:32.577 Downstream of that point the flow leaves the surface along these two lines, 00:15:32.661 --> 00:15:35.297 and the whole region they enclose is the wake. 00:15:35.610 --> 00:15:39.859 Broad, slow, and sitting at roughly the low pressure the fluid happened 00:15:39.859 --> 00:15:42.344 to have at the moment it let go. 00:15:43.141 --> 00:15:46.113 That is why the measured pressure curve went flat. 00:15:46.217 --> 00:15:50.234 Over the whole of the back of the sphere the pressure never recovers, because after 00:15:50.234 --> 00:15:54.507 separation there is no orderly deceleration left to recover it with. 00:15:54.704 --> 00:16:00.091 High at the nose, low across the entire rear, and the difference is the drag. 00:16:00.888 --> 00:16:04.534 So here is the sentence this whole lecture has been walking towards. 00:16:04.731 --> 00:16:07.599 The wider the wake, the bigger the drag. 00:16:07.796 --> 00:16:11.987 Anything that pushes the separation point further round towards the back narrows 00:16:11.987 --> 00:16:14.240 that wake, and cuts the drag. 00:16:15.544 --> 00:16:18.122 So put the two cases next to each other. 00:16:18.318 --> 00:16:21.036 On the left is the one we have just drawn: 00:16:21.140 --> 00:16:24.968 the boundary layer is still laminar when it meets the adverse gradient, 00:16:25.052 --> 00:16:29.500 its near-wall fluid is slow, and it gives up at eighty degrees. 00:16:30.714 --> 00:16:33.660 On the right, everything about the sphere is identical, 00:16:33.744 --> 00:16:37.704 but the boundary layer has already gone turbulent before it arrives. 00:16:38.016 --> 00:16:42.556 Remember what that does to the profile: the eddies are constantly dragging fast 00:16:42.556 --> 00:16:47.177 fluid down out of the free stream and into the slow region at the wall. 00:16:47.602 --> 00:16:51.500 So this layer arrives at the pressure hill with far more momentum in it, 00:16:51.584 --> 00:16:54.186 and it survives much further round. 00:16:54.638 --> 00:16:57.990 Separation moves back to about a hundred and twenty degrees, 00:16:58.074 --> 00:17:00.583 well past the widest point of the body. 00:17:01.379 --> 00:17:03.377 And look at the wake it leaves. 00:17:03.574 --> 00:17:07.844 Dramatically narrower than the one on the left, because the flow stayed attached 00:17:07.928 --> 00:17:11.156 long enough to be turned back inward before it let go. 00:17:12.208 --> 00:17:16.841 Compare the two shaded regions and you are looking at the entire drag crisis. 00:17:16.945 --> 00:17:20.332 A turbulent boundary layer does cost you more skin friction, 00:17:20.416 --> 00:17:22.181 and we said so earlier. 00:17:22.494 --> 00:17:23.919 But it separates later, 00:17:24.003 --> 00:17:28.776 and the pressure drag it saves is worth vastly more than the friction it costs. 00:17:29.572 --> 00:17:33.579 One picture, two wakes, and a factor of several in the force. 00:17:33.891 --> 00:17:37.580 All that remains is to find out when nature makes that switch for you, 00:17:37.664 --> 00:17:40.161 and how you can make it happen on purpose. 00:17:41.306 --> 00:17:44.210 So, the curve I promised you at the start. 00:17:44.406 --> 00:17:47.759 The drag coefficient of a smooth sphere up the side, 00:17:47.843 --> 00:17:52.058 and along the bottom the logarithm, base ten, of the Reynolds number. 00:17:52.255 --> 00:17:58.118 Four on this axis means a Reynolds number of ten thousand, and six means a million. 00:17:58.915 --> 00:18:03.405 The drag coefficient is simply the drag made dimensionless: the force, 00:18:03.489 --> 00:18:08.924 divided by one half rho U squared, divided by the frontal area of the body. 00:18:09.120 --> 00:18:13.424 If nothing interesting ever happened in a fluid, this would be a constant, 00:18:13.508 --> 00:18:16.179 and the plot would be a horizontal line. 00:18:17.092 --> 00:18:20.889 Start over on the left, at a Reynolds number of a thousand. 00:18:20.993 --> 00:18:23.939 The pair beside the marker reads the horizontal coordinate first 00:18:24.023 --> 00:18:26.099 and the drag coefficient second, 00:18:26.183 --> 00:18:29.794 and there the coefficient is about zero point four seven. 00:18:30.590 --> 00:18:32.623 Now walk it up the axis. 00:18:32.936 --> 00:18:34.353 Ten thousand. 00:18:34.549 --> 00:18:36.164 A hundred thousand. 00:18:36.361 --> 00:18:38.056 Three hundred thousand. 00:18:38.160 --> 00:18:42.085 Across those three whole decades the curve barely moves at all. 00:18:42.282 --> 00:18:46.484 It sits between about zero point four and zero point five, a long, 00:18:46.484 --> 00:18:48.424 flat, boring plateau. 00:18:49.221 --> 00:18:52.983 And that plateau is exactly the picture we drew a moment ago. 00:18:53.180 --> 00:18:55.360 Laminar separation at eighty degrees, 00:18:55.444 --> 00:19:00.146 a wide wake, and a pressure drag that hardly cares how fast you go. 00:19:00.343 --> 00:19:04.245 At a hundred thousand the coefficient is still zero point four seven. 00:19:05.042 --> 00:19:06.761 And then this happens. 00:19:07.557 --> 00:19:10.074 In less than half a decade of Reynolds number, 00:19:10.158 --> 00:19:15.035 the drag coefficient falls from about zero point five to about zero point one. 00:19:15.232 --> 00:19:19.049 The drag on the sphere collapses to something like a fifth of what it was, 00:19:19.133 --> 00:19:22.698 while the sphere is being pushed through the fluid harder than before. 00:19:23.611 --> 00:19:25.082 That is the drag crisis, 00:19:25.166 --> 00:19:28.650 and the cause is precisely what we watched on the two spheres. 00:19:28.847 --> 00:19:31.842 At that Reynolds number the boundary layer turns turbulent 00:19:31.842 --> 00:19:34.362 before it reaches the adverse gradient. 00:19:34.675 --> 00:19:39.270 Separation jumps from eighty degrees back to a hundred and twenty, the wake narrows, 00:19:39.354 --> 00:19:41.723 and the pressure on the back comes up. 00:19:42.636 --> 00:19:45.376 Afterwards the curve creeps upward again, partly 00:19:45.376 --> 00:19:49.718 because the wake is now narrow enough that friction is starting to matter once more. 00:19:49.915 --> 00:19:52.911 But it never climbs back anywhere near the plateau. 00:19:54.308 --> 00:19:56.247 Now, a golf ball. 00:19:56.444 --> 00:19:59.367 A golf ball is about forty three millimetres across, 00:19:59.451 --> 00:20:03.875 and off the club face it is travelling at something like seventy metres per second. 00:20:03.979 --> 00:20:05.035 Put those two numbers 00:20:05.035 --> 00:20:08.716 into the Reynolds number and you get roughly two hundred thousand. 00:20:09.513 --> 00:20:11.522 Look where that lands on the curve. 00:20:11.834 --> 00:20:13.611 Two hundred thousand is a logarithm 00:20:13.611 --> 00:20:17.860 of five point three, which is right in the middle of the plateau, and comfortably 00:20:17.860 --> 00:20:19.869 on the wrong side of the crisis. 00:20:20.666 --> 00:20:22.419 A smooth ball of that size, 00:20:22.419 --> 00:20:26.925 hit that hard, has a drag coefficient of about zero point five. 00:20:27.121 --> 00:20:28.712 It is stuck on the expensive side 00:20:28.712 --> 00:20:32.881 of the cliff, and no golfer alive can swing hard enough to get over it. 00:20:33.193 --> 00:20:35.377 Unless, of course, you move the cliff. 00:20:36.591 --> 00:20:38.693 Which is what the dimples are for. 00:20:38.890 --> 00:20:41.905 They trip the boundary layer into turbulence deliberately, 00:20:41.989 --> 00:20:44.359 using roughness instead of speed. 00:20:44.811 --> 00:20:47.842 Here is the same measurement for a dimpled ball. 00:20:48.545 --> 00:20:52.838 The crisis has moved left, down to a Reynolds number of about fifty thousand, 00:20:52.922 --> 00:20:56.696 which is far below anything a golf ball will ever experience. 00:20:56.893 --> 00:21:01.314 So at two hundred thousand the dimpled ball is already past its own crisis, 00:21:01.398 --> 00:21:04.881 and its drag coefficient is about zero point two five. 00:21:05.585 --> 00:21:10.586 That is half the drag, near enough, and it is worth something like twice the carry. 00:21:10.670 --> 00:21:14.685 Which is why every golf ball made in the last century has been covered in dimples, 00:21:14.769 --> 00:21:17.378 and why a perfectly smooth one, hit properly, 00:21:17.462 --> 00:21:19.576 is faintly embarrassing to watch. 00:21:20.372 --> 00:21:22.587 And notice the price you pay for it. 00:21:22.671 --> 00:21:26.790 The dimpled ball after its crisis sits at about zero point two five, 00:21:26.874 --> 00:21:30.985 which is well above the zero point one the smooth ball reaches after its own. 00:21:31.181 --> 00:21:33.051 Turbulence is never free. 00:21:33.155 --> 00:21:35.896 It is simply far cheaper than a wide wake. 00:21:37.292 --> 00:21:39.824 So here is the whole chain, in order. 00:21:39.928 --> 00:21:45.025 No slip at the wall, so there is a thin layer where viscosity rules everything. 00:21:45.222 --> 00:21:47.321 That layer thickens along the surface, 00:21:47.405 --> 00:21:50.854 and the fluid inside it, near the wall, is slow. 00:21:51.766 --> 00:21:56.422 Push that slow fluid into a rising pressure and it stalls, reverses, 00:21:56.422 --> 00:22:01.438 and lifts off, and the wake it leaves behind is what sets the pressure drag. 00:22:01.890 --> 00:22:05.672 Make the layer turbulent and it survives the rising pressure further, 00:22:05.756 --> 00:22:09.008 so the wake is narrower and the drag is lower. 00:22:09.805 --> 00:22:13.126 Bernoulli was never wrong about any of this. It simply 00:22:13.126 --> 00:22:16.815 could not see the five millimetres of fluid pressed against the wall. 00:22:16.899 --> 00:22:20.325 And that, it turns out, is where the drag lives.