{"version":1,"lectureId":"01M14TY5WHGF2FPAP90WKFQNRA","attempt":0,"publication":{"slug":"dimensional-analysis-drag","title":"Why Wind Tunnel Models Work: Dimensional Analysis of Aerodynamic Drag","subject":"engineering","summary":"A small model in a wind tunnel is eight times smaller than the aircraft it stands for, and yet the number it hands you is the number you want. This lecture explains why, by listing every quantity that could set the drag on a body in a flow, counting the independent dimensions among them, and letting the Reynolds number and the drag coefficient fall out as the only two groups that can matter. Both are built by hand, exponent by exponent, with no equation of motion solved anywhere. Two experiments at wildly different sizes, speeds and densities are then shown landing on the same point of the same measured curve, and the drag on a full size wing is predicted from a force read off a balance under a model. The lecture closes honestly: at high speed the speed of sound joins the list, the Mach number appears as a third group, and matching Reynolds alone becomes a quiet lie.","metaDescription":"Why a scale model predicts the drag on a real aircraft: count the dimensions, derive Reynolds and the drag coefficient, and see where Mach breaks similarity.","transcript":"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.","watch":{"version":1,"scenes":[{"title":"The Wind Tunnel Question","start":0,"end":131.03370833333332,"objects":{"card":"a Title that says \"Introduction to Fluid Mechanics — Why Wind Tunnel Models Work: Dimensional Analysis of Aerodynamic Drag\"","cd_def":"a Math [text] that says \"$C_D = frac(D, frac(1,2) rho V^2 A)$\"","head_pair":"a Heading that says \"A Model, and the Thing It Stands For\"","head_two":"a Heading that says \"Two Numbers Do All the Work\"","label_model":"a Tex [text] that says \"The model\" (underline=True)","label_real":"a Tex [text] that says \"The aircraft\" (underline=True)","law":"a Math [text] that says \"$C_D = f(upright(\"Re\"))$\"","model_drag":"a Vector [red] labelled \"D\" drawn in tunnel (start=(5.9, 3.3), end=(7.1, 3.3))","model_flow_high":"a Vector [green] labelled \"V\" drawn in tunnel (start=(0.9, 4.3), end=(2.3, 4.3))","model_flow_low":"a Vector [green] drawn in tunnel (start=(0.9, 2.0), end=(2.3, 2.0))","model_span":"a Line [yellow] labelled \"L\" drawn in tunnel (start=(4.2, 2.5), end=(5.7, 2.5))","model_wing":"a Polygon [blue] drawn in tunnel (vertices=((4.2, 3.3), (4.53, 3.51), (5.025, 3.4785), (5.7, 3.3), (5.025,…, fill_opacity=0.35)","promise":"a Text [text] that says \"Match the Reynolds number, and the drag coefficient matches with it.\"","re_def":"a Math [text] that says \"$upright(\"Re\") = frac(rho V L, mu)$\"","real_drag":"a Vector [red] labelled \"D\" drawn in sky (start=(8.3, 3.4), end=(9.6, 3.4))","real_flow_high":"a Vector [green] labelled \"V\" drawn in sky (start=(0.3, 5.2), end=(1.7, 5.2))","real_flow_low":"a Vector [green] drawn in sky (start=(0.3, 1.3), end=(1.7, 1.3))","real_span":"a Line [yellow] labelled \"L\" drawn in sky (start=(2.0, 1.9), end=(8.0, 1.9))","real_wing":"a Polygon [blue] drawn in sky (vertices=((2.0, 3.4), (3.3200000000000003, 4.24), (5.300000000000001, 4.…, fill_opacity=0.35)","sky":"a Figure (x_range=(0.0, 10.0), y_range=(0.0, 6.0), aspect=(10.0, 6.0))","tunnel":"a Figure (x_range=(0.0, 10.0), y_range=(0.0, 6.0), aspect=(10.0, 6.0))","wall_bottom":"a Line [gray] drawn in tunnel (start=(0.5, 1.6), end=(9.5, 1.6))","wall_top":"a Line [gray] drawn in tunnel (start=(0.5, 5.0), end=(9.5, 5.0))"},"beats":[{"start":0,"say":"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.","live":[],"does":[[0,"card is shown on the screen, written out."],[1.5,"card: enter:write-left-to-right."],[16.0445,"card is hidden from the screen — left the board."]]},{"start":17.2445,"say":"Here is the aircraft you actually care about. Air comes at it, and the air pushes back.","live":null,"does":[[17.2445,"sky is shown on the screen, written out."],[17.837,"real_wing is shown on the screen, written out."],[20.217,"real_flow_low is shown on the screen, written out."],[20.217,"real_flow_high is shown on the screen, written out."]]},{"start":23.545499999999997,"say":"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. 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You put a balance underneath it and you read a force, in newtons, the same way you would if the aircraft were up there.","live":["label_real","sky","label_model","tunnel","real_wing","real_flow_low","real_flow_high","real_drag","real_span","model_wing","wall_top","wall_bottom","model_span"],"does":[[60.315,"model_flow_low is shown on the screen, written out."],[60.315,"model_flow_high is shown on the screen, written out."],[66.22399999999999,"model_drag is shown on the screen, written out."]]},{"start":71.46799999999999,"say":"And here is the question. This thing is eight times smaller than that thing, sitting in different air at a different speed. 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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.","live":[],"does":[[83.86399999999999,"head_two is shown on the screen, written out."],[85.153,"re_def is shown on the screen, written out."],[92.47900000000001,"re_def (the \"rho\" part) is emphasized."],[94.162,"re_def (the \"V\" part) is emphasized."],[94.162,"re_def (the \"rho\" part) is no longer emphasized."],[96.287,"re_def (the \"L\" part) is emphasized."],[96.287,"re_def (the \"V\" part) is no longer emphasized."],[98.992,"re_def (the \"L\" part) is no longer emphasized."],[98.992,"re_def (the \"mu\" part) is emphasized."],[101.12799999999999,"re_def (the \"mu\" part) is no longer emphasized."]]},{"start":101.728,"say":"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.","live":["re_def","head_two"],"does":[[103.272,"cd_def is shown on the screen, written out."],[107.301,"cd_def (the \"frac(1,2) rho V^2 A\" part) is emphasized."],[112.525,"cd_def (the \"frac(1,2) rho V^2 A\" part) is no longer emphasized."]]},{"start":115.72649999999999,"say":"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. 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Double the size, and you have doubled everything the flow has to get around.","live":["items","stream","head_what","body","flow_low","flow_high"],"does":[[145.58870833333333,"items (the \"The size of the body\" part) is emphasized."],[149.84970833333333,"size_line is shown on the screen, written out."]]},{"start":156.32470833333332,"say":"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. 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Push through denser air and there is simply more mass to shove out of the way.","live":null,"does":[[178.2627083333333,"items (the \"The density of the air\" part) is emphasized."],[178.2627083333333,"items (the \"The speed of the flow\" part) is no longer emphasized."],[178.7737083333333,"fluid is shown on the screen, written out."]]},{"start":184.56920833333334,"say":"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.","live":["items","stream","head_what","body","flow_low","flow_high","size_line","fluid"],"does":[[191.5997083333333,"items (the \"The density of the air\" part) is no longer emphasized."],[191.5997083333333,"items (the \"The viscosity of the air\" part) is emphasized."]]},{"start":198.9792083333333,"say":"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.","live":null,"does":[[208.4527083333333,"items (the \"The density of the air\" part) is emphasized."],[208.4527083333333,"items (the \"The viscosity of the air\" part) is no longer emphasized."],[209.3937083333333,"items (the \"The density of the air\" part) is no longer emphasized."],[209.3937083333333,"items (the \"The viscosity of the air\" part) is emphasized."]]},{"start":220.45420833333333,"say":"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.","live":null,"does":[[220.45420833333333,"items (the \"The viscosity of the air\" part) is no longer emphasized."],[225.00470833333333,"drag_arrow is shown on the screen, written out."],[233.68920833333334,"head_what is hidden from the screen — left the board."],[233.68920833333334,"items is hidden from the screen — left the board."],[233.68920833333334,"stream is hidden from the screen — left the board."],[233.68920833333334,"body is hidden from the screen — stream left the board."],[233.68920833333334,"flow_low is hidden from the screen — stream left the board."],[233.68920833333334,"flow_high is hidden from the screen — stream left the board."],[233.68920833333334,"size_line is hidden from the screen — stream left the board."],[233.68920833333334,"fluid is hidden from the screen — stream left the board."],[233.68920833333334,"drag_arrow is hidden from the screen — stream left the board."]]},{"start":234.88920833333333,"say":"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.","live":[],"does":[[234.88920833333333,"head_dims is shown on the screen, written out."],[237.44270833333331,"table is shown on the screen, written out."]]},{"start":246.55320833333332,"say":"A size is a length, and nothing else. A speed is a length divided by a time.","live":["head_dims"],"does":[[247.09870833333332,"table is shown on the screen, written out."],[250.05970833333333,"table is shown on the screen, written out."]]},{"start":253.08620833333333,"say":"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.","live":null,"does":[[253.43470833333333,"table is shown on the screen, written out."],[258.8207083333333,"table is shown on the screen, written out."]]},{"start":269.54470833333335,"say":"And drag is a force. Force is mass times acceleration, so its dimensions are mass, times length, divided by time squared.","live":null,"does":[[270.09070833333334,"table is shown on the screen, written out."],[279.62170833333334,"table moves to a new place on the board."],[279.62170833333334,"head_dims is hidden from the screen — left the board."]]},{"start":280.22170833333337,"say":"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.","live":[],"does":[[280.22170833333337,"head_count is shown on the screen, written out."],[285.70170833333333,"table (the \"column=3\" part) is emphasized."],[287.74570833333337,"count is shown on the screen, written out."]]},{"start":296.06620833333335,"say":"Five take away three is two.","live":["count","head_count"],"does":[[296.06620833333335,"table (the \"column=3\" part) is no longer emphasized."],[298.3417083333334,"groups is shown on the screen, written out."]]},{"start":299.77770833333335,"say":"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.","live":["count","groups","head_count"],"does":[[301.9377083333334,"theorem is shown on the screen, written out."]]},{"start":316.16720833333335,"say":"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.","live":["count","groups","theorem","head_count"],"does":[[316.51570833333335,"groups is indicated — a transient flash."],[326.1757083333333,"theorem (the \"dimensionless groups\" part) is emphasized."],[331.73670833333335,"theorem (the \"dimensionless groups\" part) is no longer emphasized."],[332.78004166666665,"count is hidden from the screen — left the board."],[332.78004166666665,"groups is hidden from the screen — left the board."],[332.78004166666665,"head_count is hidden from the screen — left the board."],[332.78004166666665,"table is hidden from the screen — left the board."],[332.78004166666665,"theorem is hidden from the screen — left the board."]]}]},{"title":"Building the Two Groups","start":333.82170833333333,"end":561.8929583333334,"objects":{"cd_def":"a Math [text] that says \"$C_D = frac(D, frac(1,2) rho V^2 A)$\"","group_two":"a Math [text] that says \"$Pi_2 = frac(mu, rho V L)$\"","head_first":"a Heading that says \"Building the First Group\"","head_law":"a Heading that says \"The Whole of It\"","head_second":"a Heading that says \"Building the Second Group\"","law":"a Math [text] that says \"$C_D = f(upright(\"Re\"))$\"","note":"a Text [text] that says \"Dimensional analysis never tells you the function $f$. It tells you that one number is enough to look the answer up.\"","pi_one":"a Math [text] that says \"$Pi_1 = frac(D, rho V^2 L^2)$\"","work_one":"a Derivation [text] that says \"$Pi_1 &= D thin rho^a thin V^b thin L^c \\ &= (upright(\"M\") upright(\"L\") upright(\"T\")^(-2)) (upright(\"M\") upright(\"L\")^(-3))^a \\ & quad quad times (upright(\"L\") upright(\"T\")^(-1))^b (upright(\"L\"))^c \\ upright(\"M\"): quad 1 + a &= 0 quad arrow…$\"","work_two":"a Derivation [text] that says \"$Pi_2 &= mu thin rho^a thin V^b thin L^c \\ &= (upright(\"M\") upright(\"L\")^(-1) upright(\"T\")^(-1)) (upright(\"M\") upright(\"L\")^(-3))^a \\ & quad quad times (upright(\"L\") upright(\"T\")^(-1))^b (upright(\"L\"))^c \\ upright(\"M\"): quad 1 + a &= 0 quad…$\""},"beats":[{"start":333.82170833333333,"say":"Two groups, then. Let us build them by hand, so that you can see for yourself there is nothing hidden in the machinery.","live":[],"does":[[333.82170833333333,"head_first is shown on the screen, written out."]]},{"start":340.93470833333333,"say":"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.","live":["head_first"],"does":[[341.23670833333335,"work_one is shown on the screen, written out."]]},{"start":352.58820833333334,"say":"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.","live":null,"does":[[352.99470833333334,"work_one is shown on the screen, written out."],[367.1467083333333,"work_one is shown on the screen, written out."]]},{"start":368.57120833333335,"say":"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.","live":null,"does":[[373.72570833333333,"work_one is shown on the screen, written out."],[382.2357083333333,"work_one (the \"1 + a\" part) is emphasized."],[386.02070833333335,"work_one (the \"1 + a\" part) is no longer emphasized."]]},{"start":386.6207083333333,"say":"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.","live":null,"does":[[386.9687083333333,"work_one is shown on the screen, written out."]]},{"start":398.6157083333333,"say":"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.","live":null,"does":[[399.0047083333333,"work_one is shown on the screen, written out."]]},{"start":411.93420833333334,"say":"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.","live":null,"does":[[414.31470833333333,"pi_one is shown on the screen, written out."],[423.05670833333335,"pi_one is indicated — a transient flash."]]},{"start":426.29220833333335,"say":"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.","live":["pi_one","head_first"],"does":[[430.6107083333333,"cd_def is shown on the screen, written out."],[442.2097083333333,"A box is drawn around cd_def."],[443.0802083333333,"cd_def moves to a new place on the board."],[443.0802083333333,"head_first is hidden from the screen — left the board."],[443.0802083333333,"pi_one is hidden from the screen — left the board."],[443.0802083333333,"work_one is hidden from the screen — left the board."]]},{"start":444.28020833333335,"say":"Now the second group, by exactly the same procedure. This time we start from the viscosity, because the drag has already been used up.","live":["cd_def"],"does":[[444.28020833333335,"head_second is shown on the screen, written out."],[444.28020833333335,"The box around cd_def is lifted."],[449.0987083333333,"work_two is shown on the screen, written out."]]},{"start":453.2047083333333,"say":"Viscosity is mass, over a length, over a time. The other three are as before.","live":["cd_def","head_second"],"does":[[453.6807083333333,"work_two is shown on the screen, written out."],[458.83570833333334,"work_two is shown on the screen, written out."]]},{"start":460.15520833333335,"say":"Mass again. Viscosity brings one power, density brings a of them, so a is minus one, exactly as it was last time.","live":null,"does":[[460.3177083333333,"work_two is shown on the screen, written out."]]},{"start":469.50920833333333,"say":"Time. Viscosity has time to the minus one, and the speed contributes minus b. So b is minus one.","live":null,"does":[[469.85770833333333,"work_two is shown on the screen, written out."]]},{"start":477.9282083333334,"say":"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.","live":null,"does":[[478.37570833333336,"work_two is shown on the screen, written out."]]},{"start":490.7482083333333,"say":"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.","live":null,"does":[[491.37470833333333,"group_two is shown on the screen, written out."],[500.65070833333334,"group_two becomes \"$upright(\"Re\") = frac(rho V L, mu)$\"."]]},{"start":502.40020833333335,"say":"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.","live":["cd_def","group_two","head_second"],"does":[[517.4117083333333,"group_two (the \"frac(rho V L, mu)\" part) is emphasized."],[521.0927083333333,"group_two (the \"frac(rho V L, mu)\" part) is no longer emphasized."],[521.8817083333333,"cd_def moves to a new place on the board."],[521.8817083333333,"group_two moves to a new place on the board."],[521.8817083333333,"head_second is hidden from the screen — left the board."],[521.8817083333333,"work_two is hidden from the screen — left the board."]]},{"start":523.0817083333334,"say":"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.","live":["cd_def","group_two"],"does":[[523.0817083333334,"head_law is shown on the screen, written out."],[524.6607083333333,"law is shown on the screen, written out."],[537.9767083333332,"A box is drawn around law."]]},{"start":540.0397083333334,"say":"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.","live":["cd_def","group_two","law","head_law"],"does":[[544.6487083333333,"note is shown on the screen, written out."]]},{"start":552.7842083333333,"say":"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.","live":["cd_def","group_two","law","note","head_law"],"does":[[554.5957083333333,"law is indicated — a transient flash."],[560.8512916666666,"cd_def is hidden from the screen — left the board."],[560.8512916666666,"group_two is hidden from the screen — left the board."],[560.8512916666666,"head_law is hidden from the screen — left the board."],[560.8512916666666,"law is hidden from the screen — left the board."],[560.8512916666666,"note is hidden from the screen — left the board."]]}]},{"title":"Two Flows, One Number","start":561.8929583333334,"end":773.6638125000001,"objects":{"curve":"a FunctionPlot [blue] labelled \"C_D\" drawn in curve_axes (function=<function>, x_range=(0.3, 12.0))","curve_axes":"an Axes (x_range=(0.0, 12.0), y_range=(0.0, 0.1), x_ticks_every=2.0)","head_read":"a Heading that says \"Reading the Aircraft Off the Model\"","head_runs":"a Heading that says \"Two Experiments\"","head_same":"a Heading that says \"The Same Number\"","line":"a Line [yellow] drawn in curve_axes (start=(6.0, 0.030000000000000002), end=(0.0, 0.030000000000000002), dashed=True)","line_2":"a Line [yellow] drawn in curve_axes (start=(6.0, 0.030000000000000002), end=(6.0, 0.0), dashed=True)","point":"a Point [yellow] drawn in curve_axes (location=(6.0, 0.030000000000000002))","predict":"a Derivation [text] that says \"$C_D &= frac(D_m, frac(1,2) rho_m V_m^2 A_m) \\ &= frac(38.0, frac(1,2) dot.op 5.0 dot.op 90^2 dot.op 0.0625) = 0.030 \\ D_a &= C_D dot.op frac(1,2) rho_a V_a^2 A_a \\ &= 0.030 dot.op frac(1,2) dot.op 1.25 dot.op 45^2 dot.op 4.0 = 152 thin upr…$\"","re_work":"a Derivation [text] that says \"$upright(\"Re\")_upright(\"a\") &= frac(1.25 dot.op 45 dot.op 2.0, 1.875 times 10^(-5)) \\ &= frac(112.5, 1.875 times 10^(-5)) = 6.0 times 10^6 \\ upright(\"Re\")_upright(\"m\") &= frac(5.0 dot.op 90 dot.op 0.25, 1.875 times 10^(-5)) \\ &= frac(112.5,…$\"","runs":"a Table [text] that says \"Quantity Aircraft Model Size $L$, m $2.0$ $0.25$ Speed $V$, m/s $45$ $90$ Density $rho$ $1.25$ $5.0$ Viscosity $mu$ $1.875 times 10^(-5)$ $1.875 times 10^(-5)$ Reynolds $upright(\"Re\")$ $6.0 times 10^6$ $6.0 times 10^6$\" (rows=(('Quantity', 'Aircraft', 'Model'), ('Size $L$, m', '$2.0$', '$…, header=True)","spot":"a PlotPoint [red] drawn in curve_axes (target='curve', x=6.0)"},"beats":[{"start":561.8929583333334,"say":"Enough theory. Here are two experiments, and I want you to notice first how little they have in common.","live":[],"does":[[561.8929583333334,"head_runs is shown on the screen, written out."],[563.5179583333334,"runs is shown on the screen, written out."]]},{"start":568.4834583333334,"say":"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.","live":["head_runs"],"does":[[569.0409583333334,"runs is shown on the screen, written out."],[575.9609583333333,"runs (the \"row=2\" part) is emphasized."],[577.2839583333333,"runs (the \"row=2\" part) is no longer emphasized."]]},{"start":577.8839583333333,"say":"Speeds. The aircraft flies at forty five metres a second. The tunnel runs at ninety. Twice as fast.","live":null,"does":[[578.3599583333333,"runs is shown on the screen, written out."],[585.4659583333333,"runs (the \"row=3\" part) is emphasized."],[586.5979583333334,"runs (the \"row=3\" part) is no longer emphasized."]]},{"start":587.1979583333333,"say":"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.","live":null,"does":[[590.4539583333334,"runs is shown on the screen, written out."],[595.7019583333333,"runs is shown on the screen, written out."],[601.3909583333334,"runs (the \"row=5\" part) is emphasized."],[603.2254583333333,"runs (the \"row=5\" part) is no longer emphasized."]]},{"start":603.8254583333334,"say":"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.","live":null,"does":[[614.6114583333333,"runs moves to a new place on the board."],[614.6114583333333,"head_runs is hidden from the screen — left the board."]]},{"start":615.8114583333333,"say":"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.","live":[],"does":[[615.8114583333333,"head_same is shown on the screen, written out."],[624.4259583333334,"re_work is shown on the screen, written out."]]},{"start":629.1584583333333,"say":"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.","live":["head_same"],"does":[[636.2409583333333,"re_work is shown on the screen, written out."]]},{"start":639.9174583333333,"say":"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.","live":null,"does":[[640.5909583333333,"re_work is shown on the screen, written out."],[651.3299583333334,"re_work (the \"frac(5.0 dot.op 90 dot.op 0.25, 1.875 times 10^(-5))\" part) is emphasized."]]},{"start":652.9109583333334,"say":"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.","live":null,"does":[[652.9109583333334,"re_work (the \"frac(5.0 dot.op 90 dot.op 0.25, 1.875 times 10^(-5))\" part) is no longer emphasized."],[664.0499583333334,"re_work is shown on the screen, written out."],[667.4519583333333,"runs is shown on the screen, written out."]]},{"start":669.3754583333333,"say":"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.","live":null,"does":[[675.4009583333334,"runs (the \"row=6\" part) is emphasized."],[679.7429583333334,"runs (the \"row=6\" part) is no longer emphasized."],[680.5789583333334,"head_same is hidden from the screen — left the board."],[680.5789583333334,"re_work is hidden from the screen — left the board."],[680.5789583333334,"runs is hidden from the screen — left the board."]]},{"start":681.7789583333333,"say":"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.","live":[],"does":[[681.7789583333333,"head_read is shown on the screen, written out."],[681.7789583333333,"curve_axes is shown on the screen, written out."],[684.8899583333333,"curve is shown on the screen, drawn."]]},{"start":695.2194583333334,"say":"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.","live":["curve_axes","head_read","curve"],"does":[[698.5399583333334,"curve is indicated — a transient flash."]]},{"start":707.2664583333334,"say":"Our two experiments both sit here. Not near one another. At the same point, reading the same drag coefficient of three hundredths.","live":null,"does":[[709.1359583333333,"spot is shown on the screen, written out."],[712.3629583333334,"point is shown on the screen, grown."],[714.0579583333333,"line is shown on the screen, drawn."],[714.9989583333333,"line_2 is shown on the screen, drawn."]]},{"start":716.7134583333334,"say":"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.","live":["curve_axes","head_read","curve","spot","point","line","line_2"],"does":[[716.9989583333333,"point is hidden from the screen."],[716.9989583333333,"line is hidden from the screen."],[716.9989583333333,"line_2 is hidden from the screen."],[718.4079583333333,"spot is indicated — a transient flash."]]},{"start":730.1649583333334,"say":"So here is the procedure an engineer actually runs. Measure the force on the model. Suppose the balance reads thirty eight newtons.","live":["curve_axes","head_read","curve","spot"],"does":[[733.7869583333334,"curve_axes moves to a new place on the board."],[733.7869583333334,"predict is shown on the screen, written out."]]},{"start":739.0314583333334,"say":"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.","live":null,"does":[[739.2869583333334,"predict is shown on the screen, written out."]]},{"start":751.8104583333334,"say":"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.","live":null,"does":[[753.8889583333333,"predict is shown on the screen, written out."]]},{"start":762.5689583333334,"say":"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.","live":null,"does":[[763.1379583333334,"predict is shown on the screen, written out."],[766.6669583333334,"A box is drawn around predict."],[772.6221458333334,"curve_axes is hidden from the screen — left the board."],[772.6221458333334,"curve is hidden from the screen — curve_axes left the board."],[772.6221458333334,"spot is hidden from the screen — curve_axes left the board."],[772.6221458333334,"head_read is hidden from the screen — left the board."],[772.6221458333334,"predict is hidden from the screen — left the board."]]}]},{"title":"Where Similarity Fails","start":773.6638125000001,"end":1020.0183333333334,"objects":{"air_mach":"a Math [text] that says \"$upright(\"Ma\")_upright(\"a\") = frac(45, 340) = 0.13$\"","count_six":"a Math [text] that says \"$n = 6, quad k = 3, quad n - k = 3$\"","dart":"a Polygon [blue] drawn in sound (vertices=((0.34, 0.0), (-0.3, 0.17), (-0.3, -0.17)), fill_opacity=0.9)","dart_speed":"a Vector [green] labelled \"V\" drawn in sound (start=(0.5, 0.0), end=(1.4, 0.0))","divergence":"a Line [gray] drawn in mach_axes (start=(0.72, 0.0), end=(0.72, 0.095), dashed=True)","fixes":"a Block [text] that says \"Pressurise the tunnel: $rho$ rises, so $upright(\"Re\")$ rises at fixed speed. Chill the gas towards $100$ K: $mu$ falls, and so does the speed of sound. Or accept the mismatch, and correct for it afterwards.\"","flat_note":"a Text [text] that says \"Below about $0.3$ the curve is flat, so both of our flows may pretend that air is incompressible.\"","front_1":"a Circle [gray] drawn in sound (center=((-0.72 * mach), 0.0), radius=0.72)","front_2":"a Circle [gray] drawn in sound (center=((-1.44 * mach), 0.0), radius=1.44)","front_3":"a Circle [gray] drawn in sound (center=((-2.16 * mach), 0.0), radius=2.16)","froude":"a Math [text] that says \"$upright(\"Fr\") = frac(V, sqrt(g L))$\"","head_both":"a Heading that says \"You Cannot Match Both\"","head_diverge":"a Heading that says \"Where the Flat Part Ends\"","head_lesson":"a Heading that says \"What the Method Actually Promises\"","head_missing":"a Heading that says \"The Quantity We Left Out\"","law_two":"a Math [text] that says \"$C_D = f(upright(\"Re\"), upright(\"Ma\"))$\"","lesson":"a Text [text] that says \"Dimensional analysis is exact. It is only ever as good as the list of quantities you started from.\"","mach":"a VariableNumber (initial_value=0.25, format_spec='.2f')","mach_axes":"an Axes (x_range=(0.0, 0.98), y_range=(0.0, 0.1), x_ticks_every=0.2)","mach_def":"a Math [text] that says \"$upright(\"Ma\") = frac(V, a)$\"","mach_now":"a VariableNumber (initial_value=0.13, format_spec='.2f')","mach_plot":"a FunctionPlot [blue] drawn in mach_axes (function=<function>, x_range=(0.0, 0.95))","mach_read":"a Point [yellow] labelled \"upright(\"Ma\") = 0.25\" drawn in sound (location=(0.7, 2.0), show_marker=False)","match_ma":"a Math [text] that says \"$upright(\"Match Ma:\") quad V_m = V_a$\"","match_re":"a Math [text] that says \"$upright(\"Match Re:\") quad V_m = 8 V_a$\"","model_mach":"a Math [text] that says \"$upright(\"Ma\")_upright(\"m\") = frac(90, 340) = 0.26$\"","rider":"a PlotPoint [yellow] labelled \"0.13\" drawn in mach_axes (target='mach_plot', x=<VariableNumber mach_now = 0.86>)","sound":"a Figure (x_range=(-5.0, 1.7), y_range=(-2.6, 2.6), aspect=(6.7, 5.2))","sound_dim":"a Math [text] that says \"$[a] = upright(\"L\") upright(\"T\")^(-1)$\""},"beats":[{"start":773.6638125000001,"say":"Now the honest part. Everything so far rested on a list of four quantities, and that list had something missing from it.","live":[],"does":[[773.6638125000001,"head_missing is shown on the screen, written out."],[777.0538125,"sound is shown on the screen, written out."],[777.7268125,"dart is shown on the screen, written out."],[777.7268125,"dart_speed is shown on the screen, written out."]]},{"start":781.7523125,"say":"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.","live":["sound","head_missing","dart","dart_speed"],"does":[[783.4818125,"front_1 is shown on the screen, written out."],[783.7818125000001,"front_2 is shown on the screen, written out."],[784.3818125,"front_3 is shown on the screen, written out."],[792.2708125,"sound moves to a new place on the board."],[792.2708125,"sound_dim is shown on the screen, written out."]]},{"start":795.9708125000001,"say":"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.","live":["sound_dim","sound","head_missing","dart","dart_speed","front_1","front_2","front_3"],"does":[[800.1498125,"mach_read is shown on the screen, written out."],[805.8978125000001,"front_3 is indicated — a transient flash."]]},{"start":808.1113125,"say":"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.","live":["sound_dim","sound","head_missing","dart","dart_speed","front_1","front_2","front_3","mach_read"],"does":[[814.7638125000001,"front_1 is redrawn as the numbers it depends on change."],[814.7638125000001,"front_2 is redrawn as the numbers it depends on change."],[814.7638125000001,"front_3 is redrawn as the numbers it depends on change."],[814.7638125000001,"mach_read is redrawn as the numbers it depends on change."],[814.7638125000001,"mach ticks to 0.92."]]},{"start":822.1553125,"say":"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.","live":null,"does":[[824.4308125000001,"count_six is shown on the screen, written out."],[834.8098125,"mach_def is shown on the screen, written out."]]},{"start":836.1768125000001,"say":"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.","live":["sound_dim","count_six","mach_def","sound","head_missing","dart","dart_speed","front_1","front_2","front_3","mach_read"],"does":[[838.7428125000001,"law_two is shown on the screen, written out."],[844.9768125,"law_two (the \"upright(\"Ma\")\" part) is emphasized."],[848.6803125000001,"count_six is hidden from the screen — left the board."],[848.6803125000001,"head_missing is hidden from the screen — left the board."],[848.6803125000001,"law_two is hidden from the screen — left the board."],[848.6803125000001,"mach_def is hidden from the screen — left the board."],[848.6803125000001,"sound is hidden from the screen — left the board."],[848.6803125000001,"dart is hidden from the screen — sound left the board."],[848.6803125000001,"dart_speed is hidden from the screen — sound left the board."],[848.6803125000001,"front_1 is hidden from the screen — sound left the board."],[848.6803125000001,"front_2 is hidden from the screen — sound left the board."],[848.6803125000001,"front_3 is hidden from the screen — sound left the board."],[848.6803125000001,"mach_read is hidden from the screen — sound left the board."],[848.6803125000001,"sound_dim is hidden from the screen — left the board."],[848.6803125000001,"law_two (the \"upright(\"Ma\")\" part) is no longer emphasized."]]},{"start":849.8803125000001,"say":"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.","live":[],"does":[[849.8803125000001,"head_diverge is shown on the screen, written out."],[849.8803125000001,"mach_axes is shown on the screen, written out."],[857.4148125,"mach_plot is shown on the screen, drawn."]]},{"start":862.1363125,"say":"Below about a third, that curve is flat. Compressibility is there, of course, but it is doing nothing you could measure on a balance.","live":["mach_axes","head_diverge","mach_plot"],"does":[[864.6678125000001,"mach_axes moves to a new place on the board."],[864.6678125000001,"flat_note is shown on the screen, written out."]]},{"start":871.3628125,"say":"Our aircraft at forty five metres a second is at Mach nought point one three. The sound speed is about three hundred and forty.","live":["flat_note","mach_axes","head_diverge","mach_plot"],"does":[[871.9778125,"air_mach is shown on the screen, written out."],[875.2638125000001,"rider is shown on the screen, written out."]]},{"start":879.5093125000001,"say":"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.","live":["air_mach","flat_note","mach_axes","head_diverge","mach_plot","rider"],"does":[[879.9848125000001,"model_mach is shown on the screen, written out."],[882.1098125000001,"rider is redrawn as the numbers it depends on change."],[882.1098125000001,"mach_now ticks to 0.26."]]},{"start":890.5348125,"say":"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.","live":["air_mach","model_mach","flat_note","mach_axes","head_diverge","mach_plot","rider"],"does":[[894.6798125,"divergence is shown on the screen, written out."],[898.3018125000001,"rider is redrawn as the numbers it depends on change."],[898.3018125000001,"mach_now ticks to 0.86."]]},{"start":902.3383125,"say":"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.","live":["air_mach","model_mach","flat_note","mach_axes","head_diverge","mach_plot","rider","divergence"],"does":[[907.6788125,"mach_plot is indicated — a transient flash."],[912.3928125000001,"rider is indicated — a transient flash."],[915.8053125000001,"air_mach is hidden from the screen — left the board."],[915.8053125000001,"flat_note is hidden from the screen — left the board."],[915.8053125000001,"head_diverge is hidden from the screen — left the board."],[915.8053125000001,"mach_axes is hidden from the screen — left the board."],[915.8053125000001,"mach_plot is hidden from the screen — mach_axes left the board."],[915.8053125000001,"rider is hidden from the screen — mach_axes left the board."],[915.8053125000001,"divergence is hidden from the screen — mach_axes left the board."],[915.8053125000001,"model_mach is hidden from the screen — left the board."]]},{"start":917.0053125000001,"say":"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.","live":[],"does":[[917.0053125000001,"head_both is shown on the screen, written out."]]},{"start":926.9168125000001,"say":"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.","live":["head_both"],"does":[[927.7178125,"match_re is shown on the screen, written out."],[933.3838125000001,"match_ma is shown on the screen, written out."],[940.1408125,"match_ma is slashed through — it cancels."]]},{"start":941.9823125,"say":"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.","live":["match_re","match_ma","head_both"],"does":[[945.1178125000001,"fixes is shown on the screen, written out."],[946.5808125000001,"fixes (the \"Pressurise the tunnel\" part) is emphasized."],[951.5728125,"fixes (the \"Chill the gas\" part) is emphasized."],[951.5728125,"fixes (the \"Pressurise the tunnel\" part) is no longer emphasized."]]},{"start":958.2683125000001,"say":"Cryogenic nitrogen, at four atmospheres, in a tunnel that costs more than the aircraft. All of it to buy back one dimensionless number.","live":["match_re","match_ma","fixes","head_both"],"does":[[964.8858125000002,"fixes (the \"Chill the gas\" part) is no longer emphasized."],[966.8883125000001,"fixes is hidden from the screen — left the board."],[966.8883125000001,"head_both is hidden from the screen — left the board."],[966.8883125000001,"match_ma is hidden from the screen — left the board."],[966.8883125000001,"match_re is hidden from the screen — left the board."]]},{"start":968.0883125,"say":"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.","live":[],"does":[[968.0883125,"head_lesson is shown on the screen, written out."],[977.9968125000001,"froude is shown on the screen, written out."]]},{"start":982.1033125000001,"say":"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.","live":["froude","head_lesson"],"does":[[986.4338125,"froude (the \"frac(V, sqrt(g L))\" part) is emphasized."],[990.4158125000001,"froude (the \"frac(V, sqrt(g L))\" part) is no longer emphasized."]]},{"start":993.7093125000001,"say":"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.","live":null,"does":[[998.4588125,"lesson is shown on the screen, written out."]]},{"start":1004.4103125,"say":"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.","live":["froude","lesson","head_lesson"],"does":[[1005.6758125000001,"lesson (the \"list of quantities\" part) is emphasized."],[1013.0948125000001,"lesson (the \"list of quantities\" part) is no longer emphasized."],[1018.9766666666667,"froude is hidden from the screen — left the board."],[1018.9766666666667,"head_lesson is hidden from the screen — left the board."],[1018.9766666666667,"lesson is hidden from the screen — left the board."]]}]}]},"durationSeconds":1020,"chapters":[{"title":"The Wind Tunnel Question","startSeconds":0,"narration":"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."},{"title":"What Drag Can Depend On","startSeconds":131.03370833333332,"narration":"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."},{"title":"Building the Two Groups","startSeconds":333.82170833333333,"narration":"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."},{"title":"Two Flows, One Number","startSeconds":561.8929583333334,"narration":"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."},{"title":"Where Similarity Fails","startSeconds":773.6638125000001,"narration":"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."}]}}