Four profiles, one drag

The momentum integral is an exact statement. Integrate the boundary-layer momentum equation across the layer and every term but two disappears: on a flat plate,

$$\frac{d\theta}{dx} = \frac{\tau_w}{\rho U^2},$$

with $\theta$ the momentum thickness. Nothing has been assumed about the shape of the velocity profile and nothing in that equation asks for it. It is a conservation law integrated across a region, which is the same kind of statement as the drag a wake survey measures.

The Kármán–Pohlhausen method then does the obvious thing: guess a profile, evaluate both sides, and solve the ordinary differential equation that results.

The whole profile enters through two numbers #

Write the profile as $u/U = f(\eta)$ with $\eta = y/\delta$. Then $\theta = \delta I$ with $I = \int_0^1 f(1-f)\,d\eta$, and $\tau_w = \mu U f'(0)/\delta$. Substituting both into the momentum integral and solving gives

$$C_f\sqrt{\mathrm{Re}_x} = \sqrt{2\,f'(0)\,I}.$$

Two numbers. The wall slope of the guess, and one integral of it. Every other feature of the assumed profile — where it is steepest, how it approaches the edge, whether it has an inflection — is invisible to the drag.

Each guess is a point in a plane, the contours of constant drag are hyperbolae, and everything about a profile except where it sits in that plane does not reach the answer.

The two numbers are not arbitrary either, and it is worth seeing where each comes from. The wall slope is the stress, so it is the right-hand side of the momentum integral. The integral $I$ is the momentum thickness in units of the layer’s own height, so it converts the equation’s unknown — the momentum thickness — into the height the profile is written in terms of. Between them they close the equation, and nothing else is asked for.

That also settles a question a reader might have about the displacement thickness. It does not appear in the flat-plate answer at all. It appears in the pressure-gradient case, through the shape factor, which is why the flat plate is the clean case for this argument and the retarded stream is not.

Which means the freedom can be measured #

Two bumps are added to Pohlhausen’s cubic with their amplitudes chosen so that the wall slope is untouched and the momentum integral comes back to exactly where it was. Both are genuine velocity profiles: they rise from zero at the wall to one at the edge and stay between the two throughout, at the largest amplitude for which that is true.

They are twenty-three per cent of the free stream apart inside the layer, and their drags agree to 1.7 parts in 10¹⁶.

What they do not share is the shape factor: 2.69 against 3.52. That is the difference between a layer comfortably attached and one on the point of separating, and it is precisely the quantity the drag cannot see.

The first attempt at this construction used a single bump with no wall slope, on the reasoning that preserving one number was the hard part. It moved the momentum integral by four per cent and the drag with it — a demonstration of the opposite of the claim. Two conditions need two amplitudes, which is obvious after the fact and was not before.

There is a way of stating the whole result that makes it sound less like an accident of the algebra. The momentum integral is one equation. A profile is a function. Solving one equation for a function is under-determined by an infinite amount, and the method closes the gap by assuming the function up to one parameter — its thickness. So the answer is a projection: whatever the assumed family can represent, in the direction the constraint can see.

What the two-number result adds is that the direction the constraint can see is only two-dimensional, so most of what the assumed family offers is never used. That is why the method is robust, and it is also why making the assumed profile more elaborate — a fifth-order polynomial, say — buys nothing on a flat plate: the extra freedom lands in directions the constraint is orthogonal to.

What four honest guesses actually give #

Now the guesses somebody would actually make. A straight line gives 0.577 against Blasius’ 0.664 — thirteen per cent low. A parabola gives 0.730 — ten per cent high. Pohlhausen’s cubic gives 0.646, 2.7 per cent low, and a quarter sine gives 0.655, 1.4 per cent low.

The span across the four is twenty-three per cent. That is the honest number, and it is not three.

The four sort themselves cleanly, and not by how sophisticated they look. The two within three per cent are the two that satisfy the conditions the true profile satisfies: no curvature at the wall, because the momentum equation at $y = 0$ with no pressure gradient says $u'' = 0$ there; and no slope at the edge, because the profile has to meet the free stream smoothly. The two that are ten and thirteen per cent out satisfy one or neither.

So the celebrated insensitivity of the momentum integral is real and belongs to a smaller family than the one it is usually claimed for. The constraint is the same exact statement in all four cases. What changed is which profiles were being swept — which is the general form this collection has already put a number on: the residual freedom is a property of the constraint and the admissible set together, and quoting an insensitivity without saying which family was swept quotes half of an answer.

The wall-curvature condition is worth dwelling on because it is the one that is easy to omit and does most of the work. At the wall the fluid is at rest, so every inertial term in the momentum equation vanishes there and what is left is $\mu\,\partial^2 u/\partial y^2 = dp/dx$. On a flat plate the pressure gradient is zero, so the profile has exactly no curvature at the wall — it is locally straight, to second order.

The parabola violates that by construction: a parabola has constant curvature everywhere, so it has the wrong curvature at the wall by the largest amount available. That single defect is most of its ten per cent, and it is a defect a reader can see in the drawing rather than one that emerges from the arithmetic.

The same condition is what makes the criterion useful in a pressure gradient. There the wall curvature is not zero but $\rho^{-1}dp/dx$, and in an adverse gradient it is positive — the profile is concave at the wall, which is the geometric statement that an adverse gradient is bending the profile towards separation long before the wall slope reaches zero. A profile family that cannot represent that curvature cannot represent the approach to separation, whatever it does at the wall.

The quantity the same machinery cannot reach #

Drag is an integral of the profile. Separation is the wall slope going to zero, which is a point value of a derivative — the one class of quantity that no number of integral constraints reaches.

Pohlhausen’s quartic family is indexed by a pressure-gradient parameter, and its wall slope is $2 + \Lambda/6$, so it reaches zero at $\Lambda = -12$ by construction. The separation criterion is therefore an assumption of the family rather than a result of the calculation — worth noticing before the number it produces is quoted.

Notice what the momentum integral is still doing at separation. The wall slope has gone to zero and the momentum integral is a healthy tenth, so the drag stays finite while the layer lets go. The two quantities are not merely different; one is heading for a singularity and the other is not moving much, on the same profile.

On Howarth’s linearly retarded stream — the standard test — the exact answer is 0.1199 of the deceleration length. Thwaites’ correlation, which is a fit to exact solutions rather than a profile family, gives 0.1231. The quartic family gives 0.1565: thirty per cent late, from the same momentum integral that got the flat plate’s drag right to two per cent.

Why the contrast is the point #

It would be easy to read this as a comparison of methods and conclude that Thwaites is better than Pohlhausen, which is true and is not what the measurement is about.

The same exact constraint, applied through the same machinery to the same flow, is worth two per cent on one quantity and thirty per cent on another. The difference is not in the constraint and not in the effort; it is in what class of quantity is being asked for. An integral of the profile is reached; a local value of its derivative is not.

That is also why Thwaites does better. His method does not assume a profile at all: it correlates the quantity that decides separation directly against exact solutions of the boundary-layer equations, so it is supplying the local information from outside rather than deducing it from an integral. The improvement is not a better approximation; it is a different kind of input.

What the shape factor is for #

The quantity the two matched profiles disagree about deserves its own paragraph, because it is the one that carries all the information the drag does not.

The shape factor is the displacement thickness over the momentum thickness, and it is the standard diagnostic for how close a layer is to separating: about 2.6 for a laminar layer on a flat plate, about 4 at laminar separation, about 1.4 for a turbulent one and about 2.4 where a turbulent layer lets go. Every integral method that predicts separation predicts it by tracking that number.

So the situation is not that the momentum integral is blind and nothing can be done. It is that the momentum integral supplies one equation and the shape factor is a second unknown, so a second equation is needed — an energy integral, an entrainment relation, or a correlation like Thwaites’ — and that equation is where the separation prediction actually comes from. A method quoted as “the momentum integral method” is usually two statements, one exact and one fitted, and the fitted one is doing the part this page is about.

Reading the two matched profiles again with that in mind: they have the same momentum thickness and different displacement thicknesses, so they differ in exactly the variable the second equation would have to supply. The construction is not a curiosity; it is a picture of what the second equation is for.

Why Blasius has the value it has #

It is worth asking what the exact answer is doing in this picture, since it is not a member of the family.

Blasius’ profile is the similarity solution of the boundary-layer equations, so it satisfies the momentum integral automatically — it satisfies the equation the integral came from, at every station, pointwise. Its two numbers are a wall slope of 0.332 in similarity variables and a momentum integral that puts $\theta$ at 0.664 times the square root of $\nu x/U$, and the product under the square root gives 0.664.

So the exact solution is one point in the same plane as the four guesses, and the question the comparison asks is how close a guess lands to it. The quarter sine lands 1.4 per cent away not because it resembles Blasius closely — it does not, being visibly fuller near the wall — but because its two numbers happen to sit near the same hyperbola.

That is the last thing worth extracting from the plane picture. Proximity in the plane is not resemblance of the profiles. Two profiles can be near in drag and far apart in shape, which is the matched pair; and two can look alike and give different drags, which is the cubic and the parabola, whose curves are hard to tell apart in a drawing and which are five per cent apart in the answer.

The habit this suggests #

Three things follow, and they apply well beyond boundary layers.

Say which family. “The momentum integral is insensitive to the profile” is a statement about the profiles somebody had in mind, and the honest form names them. Over four ordinary guesses the answer is twenty-three per cent; over the two that satisfy the wall conditions it is three; over all profiles it is unbounded.

Ask whether the wanted quantity is an integral or a point value. The first will be reached by an integral method and the second will not, and no amount of care in the integration changes which one it is.

And treat a criterion built into a family as an assumption. $\Lambda = -12$ is where the quartic’s wall slope vanishes because of how the quartic was written. It is not a discovery about separation, and the thirty per cent is how much that matters.

The same shape appears wherever a method is asked to deliver something more local than the statement it rests on. It is why the point at which a layer lets go is the hardest number in this subject to predict, why the maximum lift is harder than the lift, and why a scheme’s conserved quantities are the ones worth checking: what an exact statement determines is what it is a statement about.

Where the same two numbers turn up again #

The flat-plate result has a compact form that is worth carrying, because it recurs.

$C_f\sqrt{\mathrm{Re}_x} = \sqrt{2 f'(0) I}$ is a product of a wall quantity and a bulk quantity, under a square root. A profile that is steep at the wall has a small $I$, because a steep profile reaches the free stream early and leaves little deficit; a profile that is gentle at the wall has a large $I$. The two move in opposite directions, and the square root of their product moves much less than either.

That cancellation is the real reason the method is robust, and it is not confined to boundary layers. The same structure appears in the friction a wall model predicts, where the profile near the wall and the profile far from it trade off in an integral across the pipe; and in a flowmeter’s calibration, where the kinetic-energy and momentum coefficients of an approach profile move together and their combination moves less.

In each case a total is being formed from a profile by an integral with a weight, and the weight is smooth. Smooth weights average out shape. That is what makes integral methods work, and it is the same fact as their inability to reach anything local — one property, read twice.

The company this result keeps #

Three other results in this collection have the same two-number structure, and seeing them together makes the pattern predictive rather than anecdotal.

Thin-aerofoil theory reduces a whole camber line to three coefficients, so two sections differing by half their camber have identical lift and moment. There the constraint is two weighted integrals of a load and the freedom is every harmonic above the second.

A flowmeter reduces an approach profile to two coefficients, so two profiles differing by half the mean velocity read identically. There the constraint is an energy balance and a momentum balance and the freedom is everything else about the shape.

A wall model’s friction depends on an additive constant far more than on the buffer layer it was written about, because the integral weights the outer region and the disagreement is in a sliver.

In each case a function is being reduced to a small number of weighted integrals, and the question worth asking of any such reduction is the same one: how many integrals, with what weights, and is the wanted answer among them? That question has an arithmetic answer, and this collection has computed what the answer is worth in general.

Why the family is the thing to state, not the constraint #

The habit this essay recommends is one sentence long and worth repeating in a form that transfers.

When an insensitivity is quoted, ask what was varied. Not what was held fixed — that is the constraint, and it is usually stated — but what was allowed to move. An answer insensitive to a two-parameter family is a much weaker statement than one insensitive to every profile, and the two are routinely reported in the same words.

The same question exposes the reverse error. A method reported as sensitive to an assumption may simply have been swept over an implausibly wide family: a boundary-layer profile with reverse flow in it, a pore-size distribution spanning six decades, a wall model without a viscous sublayer. A sensitivity to inadmissible members is not a sensitivity.

So the useful form of a robustness claim has three parts: the constraint, the family, and the number. “The drag from the momentum integral varies by three per cent across profiles satisfying both wall conditions” is a statement somebody can check and use. “The momentum integral is insensitive to the profile” is not, and it is what nearly every account of the method says.

What is not claimed #

Blasius is the reference and is not a measurement. The comparison is between an approximate method and an exact solution of the same equations, both of which are boundary-layer approximations to the Navier–Stokes equations. Real flat-plate friction differs from Blasius by an amount that has nothing to do with anything on this page.

The two matched profiles are constructed. They are built to share both numbers, and one of them has a dip in it that no boundary layer would have. What they establish is that the drag cannot distinguish them, not that anybody would offer the second.

Thirty per cent is Howarth’s number. A different pressure-gradient history gives a different error, and the quartic family does better on gentler decelerations. What is general is the direction — profile families predict separation late, because the family has to reach $\Lambda = -12$ before its wall slope vanishes and the real profile’s does so sooner.

And none of this is an argument against integral methods. They are fast, they are robust, they carry the exact statement, and they get the quantity they are about right to a few per cent from a guess. The argument is only about which quantity that is.