A layer that is an integral of everything upstream

How much uphill a layer can take is this collection’s account of separation, and it works with a shape parameter — a number combining the local pressure gradient with the layer’s own thickness, which reaches a critical value where the flow lets go.

The pressure gradient in that parameter is local. The thickness is not, and this essay is about how badly non-local it is.

The integral #

Thwaites’ method gives the momentum thickness of a laminar boundary layer as

$$\theta^2(x) = \frac{0.45\,\nu}{U(x)^6}\int_0^x U(\xi)^5\,d\xi,$$

which is an integral over the whole surface upstream, weighted by the fifth power of the external velocity there.

That is a memory in the sense this collection uses the word. What the layer is doing at a station is a functional of the entire velocity history along the surface, not a function of the conditions at the station — and the weighting is severe, because a fifth power concentrates almost all of the integral wherever the flow was fastest.

Two histories, one station #

The demonstration is a pair of external velocity distributions constructed to agree exactly at a station half-way along, in value and in gradient, and to differ everywhere upstream of it.

One accelerates steadily from the leading edge. The other does most of its accelerating in the first tenth and then coasts. At the station both are at 1.59994 in units of the free stream — the same number to five figures, with a difference of exactly 0.000000000 — and the two velocity distributions are identical from the station onwards, both falling at a rate of 1.2 free-stream velocities per chord, so from there on the two layers are given exactly the same adverse gradient. Their momentum thicknesses there are 0.000937 and 0.000678, a ratio of 1.382, and they separate at 0.5283 and 0.5842 of the surface — 5.6 per cent of the chord apart, which is 10.6 per cent of the earlier one’s own run from the leading edge, from a common condition.

Their momentum thicknesses there differ by a factor of 1.38. The one that reached its speed early has been running fast for longer, its fifth-power weighting has accumulated more, and it arrives at the station with a thicker layer.

And two different separations #

A thicker layer at the same gradient is a layer nearer to separating, because the shape parameter is the thickness squared times the gradient over the viscosity.

So the two layers reach the critical value at different places: 0.528 and 0.584 of the surface, five and a half per cent of the length apart. That is a large difference for an aerodynamic surface — it is the difference between a section that separates at the trailing edge and one that separates before it.

How large the family of matched histories is #

Two distributions are a demonstration; the question a designer cares about is how much freedom there really is, and the answer is that there is a great deal.

The constraint imposed here is stronger than a station’s worth. The two external velocities agree at the station to 0.000000000 and then coincide for the whole of the rest of the surface, so the only thing the two layers can differ in is what happened before — and they still separate 5.6 per cent of a chord apart. Weaken the constraint to what a designer actually matches, which is a velocity and a gradient at one station, and the admissible set is a whole function space with two conditions on it; within it the accumulated integral can be made almost anything, bounded only by how fast the flow is allowed to go.

Concretely: a distribution that reaches a high speed early and coasts accumulates a large integral, and one that arrives at the station having only just got there accumulates a small one. The ratio between the extremes is bounded by the fifth power of the ratio of speeds available, which for a realistic range is a factor of several rather than the 1.38 measured between the two tame cases used here.

That is the same shape of statement as exact in the total, free in the profile makes in general — a finite number of conditions leaves a function’s worth of freedom — and it is why matching local conditions between two designs is so weak a comparison.

Why the weighting is a fifth power #

The exponent is not arbitrary and it is worth knowing where it comes from, because it says how severe the memory is.

Thwaites’ method rests on an empirical observation: the quantity $\theta^2 U'/\nu$ evolves along the surface in a way that is nearly independent of the pressure gradient’s history, so a single universal function relates its growth to itself. Integrating that observation gives the $U^5$ weighting, with the 0.45 being the constant that fits the exact solutions.

So the fifth power is a measured exponent rather than a derived one, and it is what makes the method work: an integral method needs the profile family to be summarised by one parameter, and this is the combination for which that is nearly true.

Its consequence is that the memory is dominated by the fastest part of the run. Halving the external velocity over a stretch reduces that stretch’s contribution by a factor of 32, so a layer that has been through a slow region has almost forgotten it — while a fast region contributes far out of proportion to its length.

What the solver computed, and how it was checked #

Thwaites’ integral is evaluated at two thousand stations along each surface, and the shape parameter and the separation point follow from it — the momentum thicknesses at the common station read 0.000937 and 0.000678, which is the whole of the difference the local conditions cannot see. There is no boundary-layer solution here: the method is an integral one, which is the point — it summarises the layer with a single number that is explicitly an integral of the past.

Three checks. That the two external velocities really do agree at the station, to better than two per cent — they agree to zero, both reading 1.59994 — which is the whole basis of the comparison and would be an easy thing to get wrong by construction. That the two momentum thicknesses differ by at least thirty per cent there; they differ by 38.2 per cent, so the effect is present. And that both layers separate somewhere on the surface, since a comparison in which one of them never separates would be a comparison of two different situations.

The one thing the layer does forget #

It would be wrong to leave the impression that a boundary layer remembers everything, because there is a specific and important thing it does not.

It forgets the profile’s shape. Thwaites’ method works because the family of profiles a laminar layer passes through is nearly one-parameter: given the thickness and the pressure gradient, the shape is determined. Whatever shape the layer had further upstream has been erased by diffusion across the layer, which happens on the time it takes momentum to cross the layer — much shorter than the running time.

So the layer’s memory is one number rather than a function, which is precisely why an integral method exists at all. A layer that remembered its whole profile could not be summarised by a thickness, and every integral method in the subject would be wrong.

That is exactly the structure of a duct that forgets everything but one number: the fast modes are erased and the slow one survives, so the state that has to be carried is low-dimensional. The two essays are the same statement about the same mechanism, in an external flow and an internal one.

Where that fails is where integral methods fail, and it is near separation: the profile family stops being one-parameter as the flow reverses, the layer starts remembering more than a thickness, and the method’s accuracy collapses exactly where the answer is wanted.

What this does to a design method #

The engineering consequence is direct and is the reason inverse design exists.

A local criterion cannot predict separation. Anything computed from the pressure and its gradient at a station — a local shape factor, a local Reynolds number, a rule of thumb about how much adverse gradient is allowable — is missing the term that carries most of the information. Two designs with identical local conditions separate in different places.

The history is the design variable. What a designer controls is the whole pressure distribution, and the thing to control is not the peak adverse gradient but where the acceleration was spent. That is the content of every laminar-flow aerofoil: a long favourable region keeps the layer thin, and the thin layer then tolerates a steeper recovery.

And “the same pressure distribution” has to mean all of it. Two sections matched in their pressure peaks and differing in their forward regions are not equivalent, which is a common way of comparing sections and is a misleading one.

Where the memory goes in a turbulent layer #

A turbulent boundary layer has the same structure and a shorter memory, and the difference is worth having.

The turbulent momentum-integral methods carry the same kind of history — the momentum thickness at a station is an integral of the skin friction and the pressure gradient over everything upstream — but the layer’s own response time is shorter, because turbulent mixing redistributes momentum across the layer in a fraction of the time molecular diffusion needs.

So a turbulent layer forgets an upstream disturbance in a few boundary-layer thicknesses of running length, and a laminar one does not forget at all within the length available. That is one of the practical reasons turbulent layers are more forgiving to design with, and it is the same statement as the one about relaxation lengths in the cost of going turbulent: mixing is what gives a flow a finite memory.

What a measurement at a station is measuring #

There is a warning here for anybody instrumenting a surface, and it is the practical form of the whole essay.

A single traverse says what the layer is, not why. A hot-wire traverse at a station returns a profile, and from it a thickness and a shape factor. Those are the accumulated result of everything upstream, and nothing in the traverse says which part of the history produced them.

Two stations are worth far more than one. The growth between them is a local statement — it is the momentum-integral equation evaluated over that interval — and it is what a comparison with a computation should be made against. A single station compares an accumulated quantity, so a disagreement there could have originated anywhere upstream.

And the leading edge matters more than it looks. The fifth-power weighting means the region where the flow is fastest dominates, and on most aerodynamic surfaces that is close to the leading edge, where the pressure distribution is hardest to measure and hardest to compute. A per cent of error in the external velocity there is worth five per cent in the accumulated integral.

The general form of that is the one the instrument in the answer keeps making: an integrated quantity carries a weighted sum of everything that has happened to it, and interpreting one requires knowing the weight.

The same integral, seen from the wall #

There is a second way of reading the result which connects it to the rest of this collection.

The wall the fluid is listening to computes the kernel by which a wall’s motion reaches the fluid above it — a diffusion kernel in time. A boundary layer is the same physics with the time replaced by a distance, because the fluid is being convected: what has happened to a parcel by the time it reaches a station is what happened over the time it took to get there.

So the two essays are one kernel in two coordinates, and the fifth power in Thwaites’ integral is what the diffusion kernel becomes once the convection has been folded in and the profile family has been summarised by one parameter. The severity of the weighting is the convection: fast fluid spends less time at each station, so the accumulation is dominated by where the fluid was fastest rather than by where it lingered.

Where else the same construction decides an answer #

The structure — a local criterion applied to a quantity that is an integral of a history — turns up wherever a threshold is being tested, and three instances in this collection are worth putting together.

Transition. Whether a layer goes turbulent is decided by an amplification integrated along the surface, not by a local Reynolds number: the e-to-the-N methods integrate a growth rate over the whole unstable region, and two surfaces with the same local conditions transition in different places.

Cavitation. Whether a bubble grows is decided by the pressure history a nucleus has been through, not by the pressure where it is now, because the nucleus needs time to respond.

And fatigue. Whether a blade cracks is decided by an accumulated cycle count rather than by the stress it is under, which is a row that meets the row before it’s eventual subject.

In every case the local criterion is the one that gets used, because it is the one that can be evaluated from a snapshot — and in every case the integral is the one that decides. The transferable rule is to ask, of any threshold criterion, whether the quantity being compared with the threshold is local or accumulated, and to expect scatter in the second case that is not scatter at all.

The same structure on a particle rather than a wall is two essays back, on the same machinery.

What the picture cannot show #

The two external velocity distributions are drawn and the two layers are drawn, and the thing that distinguishes them — the accumulated integral — is drawn only as its weighting. The integral itself is a running total, and a figure of it would show two curves separating steadily and would say less than the weighting does about why.

Nothing here shows a velocity profile. Thwaites’ method does not compute one: it computes a thickness and a shape parameter, and the profile is recovered afterwards from a family. That is the method’s strength and it means the figures cannot show the layer, only its summary.

What this makes of a surface-pressure measurement #

The result changes what can be inferred from a pressure distribution, which is the measurement most often available on a real body.

A pressure distribution is a boundary condition, not a state of the layer. Two bodies with the same pressure at a station have different layers there if their upstream distributions differed, so the station’s separation margin cannot be read off the local pressure.

The integral is what has to be carried forward, and the good news is that it is cheap: one quadrature along the surface gives the momentum thickness, and the shape factor follows from it. That is why an integral method survives in design practice long after it stopped being the most accurate available.

And the failure case is the one to remember. A body redesigned to fix a separation by relieving the pressure gradient at the separation point will usually fail, because the layer arriving there was set further upstream and is unchanged.

Who found it, and when #

Thwaites’ method is from 1949 and is a correlation rather than a theory: he collected the known exact solutions, found the combination in which they nearly collapse, and fitted the relation. It remains the standard hand method for a laminar layer and its accuracy — a few per cent in thickness, rather more in separation position — has not been improved on by anything as simple.

The momentum-integral equation it rests on is von Kármán’s, from 1921, and is exact: four profiles, one drag is this collection’s account of how much it determines and how much it leaves open. Thwaites’ contribution is the closure that turns one exact equation into a usable method.

Limits recorded rather than smoothed over #

Laminar, incompressible, two-dimensional. All three matter. A turbulent layer needs a different closure, a compressible one needs a temperature history as well, and a three-dimensional one has a cross-flow that carries its own memory.

The separation criterion is approximate. Thwaites’ separation value is a fitted number and the method is known to be least accurate exactly there, where the profile family it assumes is being stretched furthest. The five and a half per cent difference between the two cases is more reliable than either absolute position.

The two distributions are constructions. They are written to agree at a station and differ before it, which is what the argument needs and is not what any body produces. A real pair of sections differing only in their forward pressure distribution would show the same effect and would show it mixed with several others.

One Reynolds number, one fluid. The numbers here are for air at a metre of surface and a metre a second. Thwaites’ integral carries the viscosity as a simple factor, so the thicknesses scale with the square root of it and the ratio between the two histories does not move at all — which is the part of the result that transfers.

And the fifth power is empirical. It is the exponent that makes the correlation collapse, not a consequence of the equations. A different closure gives a different weighting and the same qualitative conclusion: the layer is an integral of its past, and the only question is how it is weighted.