Boundary Layers and Drag: From No-Slip to the Drag Crisis

Bernoulli's equation is one of the first genuinely powerful results you meet in fluid mechanics, and it is also the source of one of the first genuine crises of confidence. Apply it carefully to a body sitting in a steady stream, and it tells you, quite calmly, that the body feels no drag whatsoever. Not a little drag. None. Let me set that calculation up, because the reasoning inside it is good. Here is a cylinder seen end on, sitting in a stream that arrives from the left at speed U. The pale lines are streamlines, the paths the fluid follows as it goes past. The fluid in this calculation has no viscosity at all. Nothing sticks and nothing rubs. And with no viscosity the streamlines close up behind the body exactly as tidily as they opened in front of it. Cover half the picture, and you genuinely cannot tell which half you covered. So put Bernoulli to work along one of them. Pressure plus one half rho U squared is a constant, so where the fluid speeds up the pressure falls, and where it slows down the pressure rises. Right at the nose the fluid is brought completely to rest, so the pressure there is as high as it gets. Over the shoulders it has to hurry to get around, so the pressure there drops. And then at the tail the fluid slows to a stop a second time, and the pressure climbs back to exactly the value it had at the front. Which means every push on the front is matched by an equal push on the back. Add the pressure up over the whole surface, and the total along the flow is zero. That is d'Alembert's paradox, and it is not an algebra slip. It is what the ideal theory honestly says. Here is the same body in real air, put beside the prediction. Same shape, same speed. And the flow gets around the front perfectly happily. Then, somewhere near the widest part of the body, it lets go of the surface completely. Those two red lines mark where it leaves. Behind them sits a broad, churning region of slow, disorganised fluid, and that region is called the wake. And now the symmetry is gone. At the nose the pressure is still high, because the fluid is still being stopped there. But at the back it never recovers, because nothing back there slows down in the orderly way the ideal picture assumed. It stays low. You can read that straight off the two pictures. On the left there are two high marks, one at each end, and they cancel each other exactly. On the right there is a high mark at the front and a low one at the back, and nothing cancels anything. High pressure on the front, low pressure on the back, and the difference is a net backward force. That is drag. And almost all of it, on a body shaped like this, is decided by one thing: the point at which the flow chose to let go. What the ideal calculation left out is viscosity, and in particular what viscosity does inside an extremely thin sheet of fluid pressed against the surface. That sheet is called the boundary layer, and it is the subject of the next twenty minutes. And here is the destination. This is the drag coefficient of a sphere, plotted against Reynolds number, which is the one number that says how fast a flow is in the sense that matters. Notice that somewhere out to the right, the curve falls off a cliff. The drop is roughly a factor of four, and it happens because the boundary layer changes its character. So the plan is this. First the wall itself. Then the two quite different ways a body can feel drag. Then separation, the wake, and that cliff, which has a name of its own: the drag crisis. And by the end of it, you will know exactly why a golf ball is covered in dimples.

Everything that follows comes out of one experimental fact, so let me put it on the board first. When a real fluid flows past a solid surface, the fluid actually touching that surface does not slide along it. It sticks. That is the no-slip condition. At the wall, the fluid velocity equals the wall velocity, and for a body sitting still that means zero. It holds for every ordinary fluid, however slippery it looks. Water, air, honey: at the wall, nothing moves relative to the wall. Far from the wall, of course, the fluid is doing whatever it likes. Out here it is travelling at the free stream speed U. So somewhere in between the velocity has to climb from nothing up to U, and it does that over some finite distance. There is the profile. Zero at the wall, rising very steeply just above it, and then flattening out as it joins the outer flow. A velocity that changes like that across a layer of fluid means the fluid is being sheared. And the shear stress at the wall is the viscosity, mu, multiplied by the slope of that profile at the wall. Steep profile, large stress. Gentle profile, small stress. That one quantity is the entire origin of friction drag, and we will come back to it. Now the real question. How far out into the fluid does the wall's influence reach? Put a thin flat plate edge on into a uniform stream, so that the plate barely disturbs the outer flow at all, and follow it from the leading edge. At the leading edge the layer has no thickness whatsoever: the fluid has only just met the plate. A short way downstream, here, the profile looks like this, and the affected fluid reaches up to about here. The convention is to call the edge of the layer the height at which the fluid has recovered ninety nine percent of the free stream speed. That height is delta, and delta is a function of how far along the plate you have travelled. Now watch what happens as I move that station downstream. The layer grows. The profile keeps essentially the same shape, it simply stretches upward, and the plate's influence reaches further and further out into the stream. The growth follows a square root. The thickness goes like the square root of the distance along the plate, and that is exactly what diffusion does. Viscosity carries news of the wall outward, and diffusion always spreads like the square root of time. Whether that layer is thin at all is decided by one dimensionless group, and here it is. Reynolds number: density, times speed, times distance, divided by viscosity. It compares the inertia of the fluid with its stickiness. High Reynolds number means a thin layer. Take air at ten metres per second over a plate one metre long. That is a Reynolds number of about seven hundred thousand, and at the far end of the plate the boundary layer is roughly five millimetres deep. And there is the reconciliation with Bernoulli. Outside those five millimetres, viscosity genuinely is negligible and Bernoulli genuinely does work. Inside them, viscosity is everything. The ideal theory was not wrong. It was blind to a thin strip that decides the answer. One more thing about this layer, and it is the thing the rest of the lecture turns on. Boundary layers come in two kinds. Near the leading edge the layer is laminar: smooth, orderly, sheets of fluid sliding over one another. Further downstream it goes unstable and becomes turbulent: eddies, mixing, fluid tumbling violently between one height and another. On a flat plate the switch happens at a Reynolds number of roughly five hundred thousand. Here the two profiles are drawn against the same axes: speed across the bottom as a fraction of the free stream, height up the side as a fraction of the layer thickness. And the turbulent one is visibly fuller. It holds a much higher speed much closer to the wall, because the eddies are continually dragging fast fluid down out of the outer flow and throwing slow fluid back up. That has two consequences, and they point in opposite directions. Look at the two slopes right at the wall. The turbulent one is far steeper, so the friction is higher. But the fluid down near the wall is now carrying a great deal more momentum. Hold on to that sentence. A turbulent boundary layer costs you more friction, and buys you more momentum in exactly the place the fluid is about to need it.

Before we go anywhere near a sphere, let us be precise about what drag actually is. A body in a stream feels a force on every square millimetre of its surface, and that force always has exactly two parts. There is a pressure, pushing straight into the surface at right angles to it. And there is a shear stress, dragging along the surface in the direction the fluid is sliding. Take the component of each of those along the flow direction, and add them up over the whole body. The tangential parts give you friction drag. The normal parts give you pressure drag, which for historical reasons is also called form drag. Those two are not just different terms in a sum. They behave completely differently, and the cleanest way to see that is to take one object and turn it round. Here is a thin flat plate lying edge on, along the flow. The streamlines barely notice it is there. The pressure on the front edge is essentially the pressure on the back edge, so the pressure contribution is almost exactly nothing. But there is a boundary layer growing along both faces, and at every point of both faces the fluid is rubbing. Add up all of that shear and you have the entire drag. This plate is a pure friction body, and its drag coefficient is about five thousandths. Now turn the same plate through ninety degrees, so it faces the flow squarely. Nothing about the plate has changed. Only its orientation has. The fluid piles up against the front face and is brought to rest, so the pressure there is high. And it cannot get round those sharp edges at all, so it separates immediately, and the whole of the back face sits inside a wake at low pressure. Meanwhile the shear stress on those two faces points up and down, at right angles to the flow, so it contributes precisely nothing to the drag. Every last bit of the force is pressure, and the coefficient has gone from five thousandths to about one point two. Two hundred times the drag, from the same piece of metal. Which tells you that if you want to understand the drag of a real body, the first question is always which of those two bills you are paying. So here is the split for a few bodies. The columns are the fraction of the total drag paid as friction, and the fraction paid as pressure. The plate edge on, as we said, is a hundred percent friction. A well made airfoil is close behind it, around ninety percent friction, because it is shaped precisely so that the flow stays attached all the way to the trailing edge. And now look at the sphere. Roughly five percent friction, ninety five percent pressure. A sphere is much closer to a plate held face on than it is to an airfoil, which is not what most people expect from something so smooth and rounded. So for a sphere, and for most of the blunt objects you actually care about, friction is a rounding error. The drag is pressure drag. Pressure drag is set by the wake. And the wake is set by one thing only: where the boundary layer lets go of the surface. Which means that separation, that one decision the fluid makes, is the whole game. So let us go and watch it happen.

Let us look at the pressure a sphere feels, all the way around its surface. Along the bottom is the angle measured from the front stagnation point, in degrees, so zero is the nose, ninety is the equator, and a hundred and eighty is the very back. Up the side is the pressure coefficient, which is just the pressure measured against the pressure far away and scaled by one half rho U squared. One means fully stopped fluid. Zero means the pressure of the undisturbed stream. The pale curve is what Bernoulli predicts. Coefficient one at the nose, falling as the fluid accelerates round the shoulder, a minimum of minus three at the equator, and then climbing all the way back up to one at the tail. Perfectly symmetric, and it integrates to nothing. Now split it at the equator. From the nose out to ninety degrees the pressure is falling as you move downstream. That is called a favourable gradient: the fluid is being pushed along, running downhill. But from the equator to the tail, the pressure rises as you move downstream. That is an adverse gradient, and now the fluid is being asked to climb. Out in the free stream the climb is easy. That fluid arrives at the equator moving fast, with plenty of kinetic energy to trade back for pressure. But down in the boundary layer, right against the surface, the fluid has already given most of its speed away to friction. It arrives at the hill with almost nothing in the tank. And here is what is actually measured on a real sphere. It follows the ideal curve round the front, and then, at about eighty degrees, it simply stops following it and goes flat. So let me draw what a rising pressure does to a boundary layer. Here is a wall, with the pressure increasing to the right, and three velocity profiles at three places along it. The first one is healthy. It has a good steep slope at the wall, the fluid down there is still moving forward, and the layer is firmly attached. A little further along, the rising pressure has been pushing back on that slow near-wall fluid the whole way, and the slope at the wall has been squeezed down to exactly zero. That point, where the wall slope vanishes, is the separation point. And just past it, the near-wall fluid has actually been driven backwards. The flow reverses, that reversed fluid wedges underneath the layer and lifts it clean off the surface, and the outer flow can no longer follow the body at all. So separation is not the fluid failing to turn a corner. It is the fluid near the wall running out of momentum while it is being asked to climb a pressure hill. Which tells you immediately what would help: more momentum near the wall. On a real sphere at ordinary speeds, with a laminar boundary layer, that happens about eighty degrees from the nose. Just before the widest point, and well short of the back. Downstream of that point the flow leaves the surface along these two lines, and the whole region they enclose is the wake. Broad, slow, and sitting at roughly the low pressure the fluid happened to have at the moment it let go. That is why the measured pressure curve went flat. Over the whole of the back of the sphere the pressure never recovers, because after separation there is no orderly deceleration left to recover it with. High at the nose, low across the entire rear, and the difference is the drag. So here is the sentence this whole lecture has been walking towards. The wider the wake, the bigger the drag. Anything that pushes the separation point further round towards the back narrows that wake, and cuts the drag. So put the two cases next to each other. On the left is the one we have just drawn: the boundary layer is still laminar when it meets the adverse gradient, its near-wall fluid is slow, and it gives up at eighty degrees. On the right, everything about the sphere is identical, but the boundary layer has already gone turbulent before it arrives. Remember what that does to the profile: the eddies are constantly dragging fast fluid down out of the free stream and into the slow region at the wall. So this layer arrives at the pressure hill with far more momentum in it, and it survives much further round. Separation moves back to about a hundred and twenty degrees, well past the widest point of the body. And look at the wake it leaves. Dramatically narrower than the one on the left, because the flow stayed attached long enough to be turned back inward before it let go. Compare the two shaded regions and you are looking at the entire drag crisis. A turbulent boundary layer does cost you more skin friction, and we said so earlier. But it separates later, and the pressure drag it saves is worth vastly more than the friction it costs. One picture, two wakes, and a factor of several in the force. All that remains is to find out when nature makes that switch for you, and how you can make it happen on purpose.

So, the curve I promised you at the start. The drag coefficient of a smooth sphere up the side, and along the bottom the logarithm, base ten, of the Reynolds number. Four on this axis means a Reynolds number of ten thousand, and six means a million. The drag coefficient is simply the drag made dimensionless: the force, divided by one half rho U squared, divided by the frontal area of the body. If nothing interesting ever happened in a fluid, this would be a constant, and the plot would be a horizontal line. Start over on the left, at a Reynolds number of a thousand. The pair beside the marker reads the horizontal coordinate first and the drag coefficient second, and there the coefficient is about zero point four seven. Now walk it up the axis. Ten thousand. A hundred thousand. Three hundred thousand. Across those three whole decades the curve barely moves at all. It sits between about zero point four and zero point five, a long, flat, boring plateau. And that plateau is exactly the picture we drew a moment ago. Laminar separation at eighty degrees, a wide wake, and a pressure drag that hardly cares how fast you go. At a hundred thousand the coefficient is still zero point four seven. And then this happens. In less than half a decade of Reynolds number, the drag coefficient falls from about zero point five to about zero point one. The drag on the sphere collapses to something like a fifth of what it was, while the sphere is being pushed through the fluid harder than before. That is the drag crisis, and the cause is precisely what we watched on the two spheres. At that Reynolds number the boundary layer turns turbulent before it reaches the adverse gradient. Separation jumps from eighty degrees back to a hundred and twenty, the wake narrows, and the pressure on the back comes up. Afterwards the curve creeps upward again, partly because the wake is now narrow enough that friction is starting to matter once more. But it never climbs back anywhere near the plateau. Now, a golf ball. A golf ball is about forty three millimetres across, and off the club face it is travelling at something like seventy metres per second. Put those two numbers into the Reynolds number and you get roughly two hundred thousand. Look where that lands on the curve. Two hundred thousand is a logarithm of five point three, which is right in the middle of the plateau, and comfortably on the wrong side of the crisis. A smooth ball of that size, hit that hard, has a drag coefficient of about zero point five. It is stuck on the expensive side of the cliff, and no golfer alive can swing hard enough to get over it. Unless, of course, you move the cliff. Which is what the dimples are for. They trip the boundary layer into turbulence deliberately, using roughness instead of speed. Here is the same measurement for a dimpled ball. The crisis has moved left, down to a Reynolds number of about fifty thousand, which is far below anything a golf ball will ever experience. So at two hundred thousand the dimpled ball is already past its own crisis, and its drag coefficient is about zero point two five. That is half the drag, near enough, and it is worth something like twice the carry. Which is why every golf ball made in the last century has been covered in dimples, and why a perfectly smooth one, hit properly, is faintly embarrassing to watch. And notice the price you pay for it. The dimpled ball after its crisis sits at about zero point two five, which is well above the zero point one the smooth ball reaches after its own. Turbulence is never free. It is simply far cheaper than a wide wake. So here is the whole chain, in order. No slip at the wall, so there is a thin layer where viscosity rules everything. That layer thickens along the surface, and the fluid inside it, near the wall, is slow. Push that slow fluid into a rising pressure and it stalls, reverses, and lifts off, and the wake it leaves behind is what sets the pressure drag. Make the layer turbulent and it survives the rising pressure further, so the wake is narrower and the drag is lower. Bernoulli was never wrong about any of this. It simply could not see the five millimetres of fluid pressed against the wall. And that, it turns out, is where the drag lives.