A layer's memory of one gradient helps it through the next, unless the next comes too soon

A layer released from a gradient overshoots the shape it left followed a turbulent boundary layer out of an adverse pressure gradient and into a flat stretch. Green’s lag-entrainment model, in which the layer’s shear stress — carried by its large eddies — answers a change in the outer flow only over several thicknesses, made the released layer arrive with a worse shape than a layer whose entrainment followed its own shape at once, and then, twelve thicknesses later, have a better one: the eddies made in the gradient kept entraining after it had gone, and the layer’s shape factor fell past the flat plate’s.

A real flow is rarely that simple. Behind a compressor stage the flow decelerates, is released, and decelerates again in the next stage; a wing’s upper surface passes from a favourable gradient to an adverse one and, behind a flap, into a favourable one and then adverse again. The essay ended by asking what happens if a second gradient arrives before the layer has forgotten the first — how short the release can be before the second gradient’s separation is set by the first gradient’s history rather than by the layer’s shape when it begins.

Two gradients and a gap #

The run is the earlier essay’s, extended. A layer on a flat plate in air at thirty metres a second meets, at one metre, an outer flow that slows as the power −0.3 of the distance, until 1.6 metres. There it is released: the outer flow holds its speed for a stated length. Then it slows again, as the power −0.4, until the layer separates — until its shape factor reaches 2.4, which in this kind of model is where the skin friction has fallen to zero. The release is measured in the layer’s thickness at the first release, 55 millimetres; the separation distance in the thickness at the second gradient’s start.

Each case is run three ways. Green’s model with its lag is the layer with a history. The shape-only model, in which the entrainment is always the equilibrium value for the layer’s current shape, is a layer with no lag at all, in either gradient. And the third is the comparison that answers the question cleanly: the lagged layer, stopped at the second gradient’s start, restarted with exactly the same thickness and shape but with its entrainment reset to the equilibrium value for that shape, and marched on with the lag as before. That layer meets the second gradient with the same geometry and the same physics and no memory of the first gradient. The difference between its separation and the real layer’s is what the history is worth.

Too soon, and memory costs #

The entrainment shows why the answer has two signs. At the end of the first gradient the lagged layer has not caught up with what the gradient was asking: its eddies were made upstream, where the gradient was weaker, and it is entraining 0.033 where its shape — by then a shape factor of 1.66, a layer well into its adverse-gradient wake — would entrain 0.040 in equilibrium. If the second gradient arrives at once, it meets a layer that is already behind, and the second gradient’s own lag compounds the first’s. The layer separates 2.91 thicknesses into the second gradient, against 3.16 for the same shape with no memory: the history costs eight per cent.

Released, the deficit reverses within a few thicknesses. The outer flow has stopped decelerating, so the shape’s equilibrium entrainment falls quickly towards the flat plate’s — a relaxed shape asks for less — while the eddies, which answer only over several thicknesses, keep entraining at nearly the rate the first gradient set. By twelve thicknesses the layer entrains 0.025 where its shape would entrain 0.020: a quarter more than it needs. If the second gradient arrives now, it meets a layer whose large eddies are already working harder than its shape suggests, and the layer resists separation better than its shape factor alone would predict.

Later, and memory helps #

The figure is the answer to the earlier essay’s question in one curve. A release shorter than 3.26 of the first-release thicknesses leaves the history a cost; a longer one makes it a help, which peaks at five and a half per cent of the separation distance for releases of a dozen to twenty thicknesses and fades slowly after — still one per cent at sixty. The history has a useful life. The first few thicknesses after a release are spent letting the entrainment catch up with a shape that is improving faster than it is; the next forty are spent with eddies that remember a stronger gradient than the layer now needs to resist; after that the memory has worn away and the layer is what its shape says it is.

That is the reverse of the comfortable reading of the earlier essay. The overshoot found there — the released layer’s shape factor falling below the flat plate’s — looked like a layer becoming robust. It is, but only once the overshoot has had time to happen; a second gradient that arrives during the release’s first three thicknesses finds the layer at its most vulnerable, still paying for the lag it accumulated in the first.

How much release to give #

The separation distance itself, history and all, is longest after a release of about fifteen thicknesses. Shorter releases leave the layer’s shape unrecovered: it enters the second gradient with a shape factor of 1.66 after no release, 1.50 after four thicknesses, and separates within three to four of its own thicknesses. Longer releases let the layer recover fully, but also let it grow: a layer that has run sixty thicknesses on a flat stretch is thicker and its shape relaxed to the flat plate’s, and a fixed second gradient, whose length scale does not grow with it, is relatively steeper for it. The best release in this model is where the shape has recovered and the layer has not yet grown much — about a dozen to twenty thicknesses — and that is also where the history helps most.

For a compressor, where the release is the gap between one blade row and the next and the layer’s thickness at a blade’s trailing edge is a few per cent of its chord, a dozen thicknesses is a third of a chord or so. Real axial gaps are of that order, chosen mostly for the blade rows’ unsteady interaction; the model says the gap is also near where a layer passing from one row to the next is most robust, and that a gap much shorter would hand the next row a layer that is worse than its shape factor admits.

Almost whatever the gradients #

The crossing hardly moves. With the first gradient held at −0.3, the second can be −0.35, −0.4 or −0.5 and the crossing stays at 3.26 thicknesses to three figures: the crossing is a property of the release, not of what follows it, because it is where the entrainment’s lag behind the improving shape turns into a lead, and that happens before the second gradient arrives. With the first gradient varied it moves a little — 3.08 thicknesses after −0.2, 3.38 after −0.35 — because a stronger first gradient leaves a larger deficit for the release to make up.

So the rule is simple enough to carry: a second adverse gradient should not begin within about three thicknesses of the end of the first. Beyond that, the first gradient’s eddies are an asset; within it, they are a debt. The rule needs no knowledge of the second gradient’s strength, which is what makes it usable at the design stage, before the downstream loading is settled.

Why the shape-only layer is the wrong comparison #

The obvious comparison is the shape-only layer, and the figure draws it: at every release length it separates later than the lagged layer, by between seven and thirty per cent. That is not the history’s doing. The shape-only layer has no lag in either gradient, and the lag in the second gradient alone — the effect the essay before last found, a layer that cannot entrain as fast as a sudden gradient asks — costs any lagged layer separation distance, history or none. A fresh flat-plate layer meeting the second gradient with no first gradient at all separates seventeen of its thicknesses in with the lag and twenty-seven without. Comparing the lagged and shape-only layers after a release mixes the two lags, and the first gradient’s history is only a part of the difference.

The restart separates them. It gives the second gradient a layer that is identical to the real one in every quantity the shape-only theory knows about — thickness, shape factor, Reynolds number — and in its physics, the lag included, and differs only in the one quantity that carries the history. Whatever separates the two runs is the history and nothing else. That is the experiment a wind tunnel cannot do, since no tunnel can reset a layer’s eddies while keeping its profile, and it is why the integral model, for all its approximations, can answer the question cleanly.

In metres, and against a fresh layer #

The thickness units hide how much the first gradient costs in the first place. In metres, the layer that meets the second gradient with no release separates sixteen centimetres into it; after a release of twelve thicknesses, sixty-six centimetres long, it separates thirty-seven centimetres in; after sixty thicknesses, three and a third metres of release, forty-seven. A layer that never met the first gradient, arriving at the second as a flat-plate layer three centimetres thick, separates fifty-six centimetres in. The first gradient, in other words, costs the second most of its room whatever the release, by thickening the layer and fattening its wake; the history effect this essay is about is a correction of several per cent to that, of one sign or the other depending on the release.

That ordering is worth keeping in view when reading the crossing. The decision a designer can make — whether a second gradient should begin two thicknesses or five after a release — moves the separation point by a few per cent. The decision whether there should be a first gradient at all moves it by a factor of three. The history matters most where everything else has been fixed: in a row of blades whose spacing and loading are set, the gap between rows is what is left to choose, and the crossing says to choose it longer than three thicknesses.

Why the shape factor is not the whole state #

The refuted claim — that a layer’s resistance to a gradient is set by its shape factor where the gradient begins — is the assumption behind every one-equation integral method, Head’s among them, and behind the common practice of quoting a separation criterion as a shape factor. Green’s model has a second state variable, the entrainment, which carries the shear stress of the large eddies and changes on its own timescale. Two layers with the same thickness and shape but different entrainment are different layers, and the second gradient can tell them apart.

An adverse gradient grows the wake, not the inner law of the wall, and the wake is where the large eddies live; the history this essay measures is the wake’s memory. The inner layer near the wall adjusts within a fraction of a thickness and has no memory to speak of, so the effect is entirely an outer-layer one, and it is of the size the outer layer’s slowness suggests: several per cent, over tens of thicknesses.

Where else a layer meets two gradients #

The same arithmetic applies wherever a turbulent layer passes from one adverse gradient into another with a pause between. A wing’s high-lift system is the classic case: the flow over the main element decelerates towards its trailing edge, is released into the slot’s favourable gradient, and decelerates again over the flap. A slot so short that the flap’s gradient begins within three thicknesses of the main element’s trailing edge hands the flap a layer that is worse than its shape factor admits; a longer one hands it a layer whose eddies are still working, which is part of why a well-designed slot does more than reset the layer, though what a slot does is mostly a matter of the outer flow, and the layer’s history is a small part of it. A diffuser with a step or a short parallel section in its wall, used to break a long deceleration into two, is the same case again, and the model says the parallel section should be at least three of the layer’s thicknesses long to do more good than harm.

The model’s reach #

The layer is Green’s lag-entrainment integral model, as in the earlier essays: momentum thickness, shape factor and entrainment coefficient, marched with Head’s shape relations, Green’s lag equation and a skin-friction law in the shape factor and the momentum-thickness Reynolds number. Separation is taken where the shape factor reaches 2.4. The outer flow is prescribed and does not respond to the layer’s growth; near separation a real outer flow would be changed by the displacement, and a coupled calculation would move every separation point. The release is an exactly constant outer speed, and its length is in the thickness at the first release; the separation distance is in the thickness at the second gradient’s start, which grows with the release. Integral models of this kind are calibrated on equilibrium and mildly non-equilibrium layers; the double gradient here is within the range they were built for, but the numbers are the model’s, and the per-cent-level effects are of the size integral models’ disagreements with each other can reach.

Who measured layers that remember #

Clauser’s equilibrium layers of the 1950s showed that a layer in a constant pressure-gradient parameter keeps a constant shape; Bradshaw in the 1960s measured how slowly a layer’s turbulence answers a change in gradient, and Green, Weeks and Brooman built the lag-entrainment method on that measurement in 1973. Layers that pass through successive gradients — in diffusers with interrupted walls, and on cascades of blades — were measured in the decades after, and history effects of a few per cent in the separation point are the order those experiments report. The crossing at three thicknesses, and the restart that isolates the history, are this calculation’s.

How it was checked #

The march is the earlier essays’ and was checked there against a flat plate’s equilibrium and against step halving. The restart reproduces the marched layer’s thickness and shape at the second gradient exactly, by construction, and differs only in the entrainment, whose reset value is the equilibrium for that shape. A fresh flat-plate layer meeting the second gradient with no first gradient separates seventeen of its thicknesses in, which is the baseline the once-decelerated layers fall short of. The crossing, found by bisection on the release length, is the same to three figures for three second gradients. The model refuses a negative release, a second gradient that does not decelerate and a first that accelerates.

Still open: a gradient that oscillates #

Every gradient here is steady. Behind a compressor’s blade rows the layer on the next row’s blades sees the previous row’s wakes passing, a gradient that rises and falls at the blade-passing frequency, many times within the layer’s memory. The next calculation would give the outer flow a periodic deceleration of stated amplitude and period superposed on a mean gradient and ask whether a layer whose memory is longer than the period averages the oscillation — separating where the mean gradient alone would put it — or rectifies it, entraining on the decelerating half more than it loses on the other, and so separating later than the mean would allow.