Why Wind Tunnel Models Work: Dimensional Analysis of Aerodynamic Drag

A full size aircraft costs a fortune to build, and more to fly. So before anybody builds one, a small copy of it goes into a wind tunnel, and somebody measures the force on the copy. The strange thing is that this works, and today I want to show you exactly why. Here is the aircraft you actually care about. Air comes at it, and the air pushes back. The part of that push which lies along the flow is what we call drag, and it is what the engines have to fight. It sets the fuel burn, the range, and the top speed. Getting it wrong by ten percent is an expensive mistake. One measurement of the wing matters throughout. Call the distance from the front of it to the back L. And you cannot put this aircraft in a tunnel, because no tunnel is that big. So you build a model instead. It sits in the working section of a tunnel, with the walls a little way above it and below it, and the whole thing is eight times smaller. Run the tunnel, and air blows over the model exactly as it blows over the aircraft. You put a balance underneath it and you read a force, in newtons, the same way you would if the aircraft were up there. And here is the question. This thing is eight times smaller than that thing, sitting in different air at a different speed. Why should the reading on the balance tell you anything at all about the aircraft? The answer is two numbers, and this whole lecture is about where they come from. The first is called the Reynolds number. It is built from the density of the air, and from the speed of the flow, and from the size of the body, all divided in the end by the viscosity of that same air. The second is the drag coefficient. That is the drag force itself, divided by a quantity built out of the density, the speed and an area, so that every unit cancels and a pure number is left behind. And the claim, which we are going to earn rather than assume, is this. Match the Reynolds number between the model and the aircraft, and the drag coefficient comes out the same. Get that one number right, and the model is telling you the truth.

Before any algebra at all, let us just ask what the drag on a body could possibly depend on. Here is a body, and here is air moving past it from the left. Four candidates suggest themselves. The first is how big the body is. One length will do to stand for that, and I will call it L. Double the size, and you have doubled everything the flow has to get around. The second is how fast the air is going, which is V. Not the speed of the aircraft and the speed of the air separately, notice. Only the relative speed between them, because a flow does not know whether the body or the air is doing the moving. Then the air itself, which brings two numbers of its own. Air has mass, so a given volume of it weighs something. That is the density, rho. Push through denser air and there is simply more mass to shove out of the way. And air is very slightly sticky. Layers of it drag on one another, and on the skin of the body. That stickiness is the viscosity, mu, and it is the reason a wing has friction to fight as well as pressure. You might reasonably ask why gravity is not on that list, or the temperature. Gravity does not push a wing sideways, and temperature acts only through rho and through mu, both of which are already on it. Choosing this list is the one creative step in the whole method, and it is the one place you can go badly wrong. So: four quantities in, and one quantity out. The drag, D. Now the trick. We are going to work out the shape of the relation between them without solving a single equation of motion. Every one of these five is built out of three basic dimensions: mass, length and time. I am going to write those upright, so that a dimension never gets mistaken for a symbol. A size is a length, and nothing else. A speed is a length divided by a time. Density is mass per unit volume, so mass divided by a length cubed. Viscosity, if you chase its definition through, works out as mass, divided by a length, divided by a time. Take that last one on trust for the moment. And drag is a force. Force is mass times acceleration, so its dimensions are mass, times length, divided by time squared. Now count. There are five quantities. Look down the last column, and between them they use three independent dimensions and no more. Mass appears, length appears, time appears, and that is the lot. Five take away three is two. And that little subtraction is a theorem, one of the most useful in all of engineering. A relation among n quantities built from k independent dimensions can always be rewritten as a relation among exactly n minus k dimensionless groups. Two groups. Not three, not five. Whatever the true law of drag turns out to be, however ugly the equations of motion sitting behind it, it can be written as one dimensionless number depending on one other dimensionless number. So let us go and build them.

Two groups, then. Let us build them by hand, so that you can see for yourself there is nothing hidden in the machinery. Start with the drag, and multiply it by the other three quantities, each raised to some power we do not know yet. Call the whole product Pi one, and demand that it come out dimensionless. Write out the dimensions of everything on the right hand side. Drag brings mass, length, and time to the minus two. Density brings mass over length cubed, all raised to the power a. The speed and the size follow on the next line. For Pi one to be a pure number, every dimension has to cancel completely. Take mass first. The drag brings one power of it, density brings a of them, and nothing else on that line carries any mass at all. So one plus a must be zero, and a is minus one. Time next. Drag carries time to the minus two, and the speed to the power b contributes minus b. So minus two minus b is zero, and b is minus two. And length. One from the drag, minus three a from the density, b from the speed, and c from the size. Put in the a and the b we already have, and c comes out at minus two as well. So there is the first group. Drag, divided by density times speed squared times size squared. Every dimension has cancelled, and what is left is a bare number with no units on it anywhere. In practice nobody writes it quite like that. There is a one half out in front, borrowed from the kinetic energy in the flow, and a reference area A standing in for L squared. That is the drag coefficient, and it is what wind tunnels actually report. Now the second group, by exactly the same procedure. This time we start from the viscosity, because the drag has already been used up. Viscosity is mass, over a length, over a time. The other three are as before. Mass again. Viscosity brings one power, density brings a of them, so a is minus one, exactly as it was last time. Time. Viscosity has time to the minus one, and the speed contributes minus b. So b is minus one. And length. Minus one from the viscosity, plus three from that minus three a, minus one from the speed, and c. That comes to one plus c, so c is minus one. Which gives viscosity over density times speed times size. That is a perfectly good dimensionless group, and it is conventional to turn the whole thing upside down. That is the Reynolds number, and it has a meaning worth carrying around. Rho V squared is roughly the pressure the flow's own momentum can push with. Mu V over L is roughly the stress that the stickiness can manage. Their ratio is Re, so a large Reynolds number means inertia is winning. So here is what the theorem promised, now delivered. Five quantities, three dimensions, two groups, and only two. Whatever the drag law is, it must be expressible as the drag coefficient equal to some function of the Reynolds number, and of nothing else at all. Notice carefully what has not happened here. Dimensional analysis has not told you what that function is. It cannot. Finding f still takes a wind tunnel, or a very large computer. What it has told you is that one number is enough to look the answer up. And that, precisely, is the wind tunnel's licence to exist.

Enough theory. Here are two experiments, and I want you to notice first how little they have in common. Sizes first. The aircraft's wing measures two metres from front to back. The model's measures a quarter of a metre. That is a factor of eight. Speeds. The aircraft flies at forty five metres a second. The tunnel runs at ninety. Twice as fast. And it is not even the same air. The tunnel is pressurised, so the air inside it is four times as dense as the air out there. Viscosity, oddly, hardly notices pressure at all, so that one number really is identical in both columns. Eight times smaller. Twice as fast. Four times denser. Nothing about these two flows looks remotely alike, and the two drag forces will not be alike either. So feed those numbers into the Reynolds number. Density, times speed, times size, all over the viscosity. For the aircraft, the top line is one and a quarter, times forty five, times two. One and a quarter times forty five is fifty six and a quarter, and times two is a hundred and twelve point five. Divide that by the viscosity and you get six million. Now the model. Five, times ninety, times a quarter. Five times ninety is four hundred and fifty, and a quarter of that is a hundred and twelve point five. The very same top line. And of course it is. One eighth of the size, times twice the speed, times four times the density. An eighth, times two, times four, is one. So the product is unchanged, and the Reynolds number is unchanged. Six million again. Which is the whole of it, really. Similarity is not a coincidence you hope for. It is something you engineer, by choosing the tunnel's speed and pressure so that one product comes out right. So what does that buy you? Here is the drag coefficient of this shape, measured, and plotted against Reynolds number. The axis is in millions, so six on it means six million. Dimensional analysis promised us that a curve like this exists, and that it needs only one input. It said nothing whatever about the shape of it. Somebody had to go out and measure that. Our two experiments both sit here. Not near one another. At the same point, reading the same drag coefficient of three hundredths. That is what dynamic similarity means. Two flows at the same Reynolds number have the same drag coefficient, the same separation, the same wake, the same everything, once you take the units back out. So here is the procedure an engineer actually runs. Measure the force on the model. Suppose the balance reads thirty eight newtons. Divide it by a half rho V squared times the model's area, using the model's own density, speed and area. Out comes a drag coefficient of nought point zero three zero. Now turn it around. Multiply that same coefficient by a half rho V squared times the aircraft's area, using the aircraft's numbers this time. A hundred and fifty two newtons. That is a quantitative prediction about a machine nobody has built, made from a force measured on a small object in a box.

Now the honest part. Everything so far rested on a list of four quantities, and that list had something missing from it. Air is springy. Squeeze it and it pushes back, and the speed at which that push travels through it is the speed of sound. Call it a. It is a length over a time, like any other speed. Here is why it matters. A body moving through air sends pressure signals out ahead of itself, at the speed of sound, warning the air to get out of the way. Those circles are the signals. At a quarter of the speed of sound the signals run far out in front, and the air has plenty of notice. Push the speed up towards the speed of sound, and they bunch against the nose. The air gets almost no warning at all. So put a on the list. Six quantities now, still three dimensions, and six take away three is three. There is a third group, and it is the easiest one in the lecture: a speed, divided by a speed. That is the Mach number. Our result should really have read like this: the drag coefficient is a function of the Reynolds number, and of the Mach number as well, and we simply left the second one out. Which raises an awkward question. Why did ignoring it work? Here is the drag coefficient of a wing plotted against Mach number, at one fixed Reynolds number. Below about a third, that curve is flat. Compressibility is there, of course, but it is doing nothing you could measure on a balance. Our aircraft at forty five metres a second is at Mach nought point one three. The sound speed is about three hundred and forty. The model at ninety is at nought point two six. Both of them are down here in the flat part, so both may pretend the air is incompressible, and the two of them agree. But look what happens further along. Past about Mach nought point seven, patches of the flow over the wing go supersonic, shock waves form, and the coefficient does this. That is drag divergence, and it is not a small correction. The coefficient can triple. Two flows matched on Reynolds number but sitting on opposite sides of that rise do not resemble each other in the slightest. So test an airliner cruising at Mach nought point eight five, and you must match both numbers at once. And now you have a real problem. Matching the Reynolds number with a model eight times smaller means running it eight times faster. Matching the Mach number, in the same gas at the same temperature, means running it at exactly the same speed. You cannot do both. Which is why the expensive tunnels exist. Pressurise the air and rho goes up, so Reynolds goes up without touching the speed. Chill the gas to a hundred kelvin and the viscosity falls, and the speed of sound falls with it. Cryogenic nitrogen, at four atmospheres, in a tunnel that costs more than the aircraft. All of it to buy back one dimensionless number. And the same trap has other names. Put a seaplane hull down on water and gravity joins the list, because gravity is what makes the waves. Out comes the Froude number, speed over the root of g times length. Match Reynolds and you get the waves wrong. Match Froude and you get the friction wrong. The model builder splits the difference and corrects the rest by calculation. So the lesson is not that dimensional analysis is unreliable. It is exact. The lesson is that it is only ever as good as the list you started from. Write down every quantity that could matter. Count your dimensions. Then check that every group you produced is actually matched. Miss one, and your beautiful small model is quietly telling you about a flow that does not exist.