A layer released from a gradient overshoots the shape it left

A layer whose stress lags the gradient separates sooner took a flat-plate turbulent layer at thirty metres a second and put a sudden adverse pressure gradient in front of it. The layer’s shear stress is carried by large eddies that were made upstream, in no gradient at all, and they take several thicknesses to turn over and become eddies of the new conditions. Green’s lag-entrainment method writes that history as an equation for the entrainment coefficient, which trails its equilibrium value; set it at equilibrium everywhere and the layer has no memory. The lagged layer entrained about half what the gradient asked, grew its shape factor faster, and separated in 35 thicknesses where the memoryless one took 60.

It ended on the opposite step. A layer that has grown an overlarge wake in an adverse gradient — at the end of a diffuser, approaching a wing’s trailing edge, behind a ridge — is often suddenly released into a flat or accelerating flow, and its outer eddies, made in the gradient, keep their large shear stress for a while. The question was whether the lag then acts the other way: a layer that recovers faster than its shape alone would predict.

It does, and by enough to overtake. The lag that made the layer separate sooner makes it recover sooner, and past where a memoryless layer would stop. The lagged layer reaches the release with the worse shape and, a dozen thicknesses later, has the better one.

Two layers through one gradient and its release #

The calculation is the earlier essay’s, with the outer flow given an end. The layer develops on a flat plate at thirty metres a second to a thickness of about 23 millimetres at x = 1 metre. From there the outer speed falls as $(1 + (x - 1)/0.4)^m$ with $m = -0.3$ — a strong deceleration, one that takes the lagged layer’s shape factor from 1.35 to 1.66 — until x = 1.6 metres, where the deceleration stops. After the release the outer flow is either flat or accelerating as a power of the distance past it.

Both layers are Green, Weeks and Brooman’s integral layer, with its momentum thickness, shape factor and entrainment coefficient and the same skin-friction and shape relations. In one, the entrainment obeys the lag equation, Bradshaw’s shear-stress transport rewritten for it. In the other, the entrainment is set at every station to the equilibrium value for the layer’s own shape — the shape-only layer, which has no history. Distances after the release are measured in the layer’s thickness there, 55 millimetres for the lagged layer.

The reference for “recovered” is a flat-plate layer in equilibrium at the same momentum-thickness Reynolds number: the shape factor and friction a layer would have if it had never seen the gradient but had grown to the same thickness. Neither layer starts there, and the question is how each approaches it.

Arrives worse, leaves better #

The first figure draws both layers’ shape factors over the whole run. In the gradient they part exactly as the earlier essay found: the lagged layer entrains too little, so its shape factor climbs faster, and at the release it is 1.66 against the shape-only layer’s 1.57. The lagged layer is nearer separation and has the thicker, slower wake.

After the release both fall. The lagged layer falls faster, and 11.8 thicknesses past the release it crosses below the shape-only layer. By the end of the plate, at 4.5 metres, it is at 1.282 against 1.301. It reached the release with the worse shape and left it with the better one.

The same mechanism did both. In the gradient the entrainment trailed a rising equilibrium, so it was too low, and a layer that entrains too little for its gradient thickens its wake. After the release the entrainment trails a falling equilibrium, so it is too high, and a layer that entrains too much for its conditions fills its wake in faster. The lag does not favour separation or recovery; it exaggerates whatever the gradient is doing to the shape, in either direction, for as long as the eddies remember the old conditions.

Past the flat plate #

The second figure zooms in on the release. Five thicknesses after it the lagged layer is at 1.48, the shape-only layer at 1.44, and a flat-plate layer of the same Reynolds number would be at 1.30: both are still far from recovered, the lagged one further. Forty thicknesses after it the lagged layer is at 1.287, the flat plate’s equilibrium is 1.293, and the shape-only layer is at 1.307.

So the lagged layer does not merely catch up with the memoryless one. It overshoots the flat plate itself, reaching a shape factor slightly below the one a layer that never saw the gradient would have, because it is still entraining at the rate its gradient-grown eddies set. The shape-only layer, which can only approach its equilibrium from the side it starts on, never gets there: it approaches the flat plate’s value from above, slowly, as its shape relaxes.

The overshoot is small — half a per cent in the shape factor — but its sign is the point. A designer estimating how a layer recovers after a diffuser by a shape-only method will put the recovered layer a little thicker and closer to separation than it is, and will put the distance to recovery in the wrong place.

The eddies keep entraining #

The third figure shows the mechanism directly. Before the release the lagged layer’s entrainment is below its equilibrium, as it was in the earlier essay. At the release the equilibrium value falls — a layer with no adverse gradient needs less entrainment to hold its shape — but the lagged entrainment, which can only relax towards its equilibrium at the rate the eddies turn over, stays where it was. Ten thicknesses past the release the lagged layer entrains 0.0267 against an equilibrium of 0.0218; thirty thicknesses past, 0.0164 against 0.0117. For a long distance it is entraining a fifth to two fifths more than its shape would ask.

That excess is the history the earlier essay described, seen from the other side. A gradient-grown layer has a deep, energetic outer region — the wake an adverse gradient grows — whose eddies were set up to carry a large shear stress against the deceleration. Released, they keep carrying it, and a large shear stress in the outer layer means a layer that drags fresh fluid in and mixes momentum down towards the wall faster than its current shape would require. The layer’s shape relaxes at the rate the eddies allow, and the eddies relax more slowly than the shape.

The same structure, a quantity relaxing towards a target that has moved, is a dissipation that lags its production in a turbulence closure: when the production falls, the dissipation stays high for a while, and the turbulence decays faster than an equilibrium estimate says.

The friction overshoots too #

The fourth figure follows the wall. A layer with a smaller shape factor has a fuller profile and a larger friction at the same Reynolds number, so the lagged layer’s friction rises faster after the release than the shape-only layer’s, and keeps rising after the shape-only layer has levelled. At its largest, 34.9 thicknesses past a flat release, it is 4.1 per cent above the shape-only layer’s and 1.005 times the flat plate’s at the same Reynolds number: recovered fully, where the memoryless layer is still four per cent short.

The dial accelerates the release. With the outer flow speeding up as the power 0.1 of the distance past the release, the excess over the shape-only layer reaches 5.6 per cent, at 26.5 thicknesses; at the power 0.2, 7.7 per cent at 24.7. An accelerating flow drives the friction above the flat plate’s anyway — to 1.12 and 1.22 times it at those stations — and the lagged layer, carrying the gradient’s eddies into it, takes more of that gain sooner.

The consequence for heat transfer and for the wall’s own drag follows the friction, through the Reynolds analogy, and it runs the same way: a released layer heats or cools the wall more than a shape-only method predicts, over a stretch of twenty to forty thicknesses after the release.

Twelve thicknesses, whatever the gradient was #

The fifth figure repeats the calculation for decelerations from a power of −0.1, which takes the layer’s shape factor only to 1.39, to −0.35, which takes it to 1.81, near separation. The friction’s excess grows with the gradient’s strength, from 1.3 per cent to 4.7 after a flat release and from 4.9 to 8.4 after an accelerating one: the stronger the gradient, the more shear stress the eddies carry out of it.

The crossing does not move. After a flat release the lagged layer overtakes the shape-only one between 11.4 and 12.0 thicknesses past the release, whatever gradient it came out of; after an accelerating release, between 4.1 and 6.2. The crossing distance is set by the lag equation’s own relaxation length — the distance over which the entrainment approaches its equilibrium, a few thicknesses times a factor of order one — rather than by how far from equilibrium the layer started. The earlier essay found the same kind of number going the other way: a layer meeting a sudden gradient reached its new shape in about twenty thicknesses whatever the step. Both are the eddies’ turnover distance, measured in the layer’s own thickness.

Where layers are released #

The released layer is not a laboratory curiosity; it is most of the places a designer meets an adverse gradient. A wind-tunnel or engine diffuser decelerates its flow over a few metres and then hands the layer to a settling chamber, a combustor or a constant-area duct. A wing’s upper surface carries an adverse gradient from its suction peak to its trailing edge, and a slotted flap releases the main element’s layer into the flap’s favourable gradient. A layer over a ridge or a hill sees the adverse gradient of the lee slope and is released at the foot; how far downwind a surface is remembered measured the same kind of memory in the atmosphere, for a change of roughness rather than of pressure.

In each the designer needs two numbers from a layer that has just been released: how near separation it still is, and what friction or heat transfer it now delivers to the wall. A shape-only method answers both a little wrongly, in the directions the figures show, and over a distance — a dozen to forty thicknesses — that is short in metres but long against the gaps between a diffuser’s exit and the next component. In a compressor stage the next blade row may arrive within that distance, and the layer it receives is still the gradient’s.

The method’s two layers also bracket what a cruder estimate does. A guess with a constant in it, a mixing length that adjusts instantly to the local profile, has no history and behaves like the shape-only layer; a layer that is an integral of everything upstream is the opposite extreme, remembering forever. The lag equation sits between them, with a memory a few thicknesses long, and the crossing at twelve thicknesses is that memory’s length in the measurement that matters.

The direction of the error is also the direction of the safety margin. A designer who uses a shape-only method to decide whether a second gradient downstream of the release will separate the layer will think the layer starts the second gradient with a fuller wake than it has — the conservative error — once more than a dozen thicknesses have passed, and the optimistic one before. How much uphill a layer can take measured the margin a laminar layer has; for a turbulent one just released, the margin depends on how long ago the release was.

What the release costs downstream #

The lagged layer arrives at the release thicker, and some of that stays. At the end of the plate its momentum thickness is one per cent larger than the shape-only layer’s: the extra momentum deficit it built up in the gradient is not undone by its faster recovery afterwards, only its shape is. So a shape-only method is wrong in two directions at once after a release. It makes the layer slightly too thin, which is optimistic for the drag the wall has already paid, and its shape too full of wake and its friction too low, which is pessimistic about how close the layer is to separating again and about the friction the wall downstream still has to pay.

The convention: distances in thicknesses at the release #

The shape factor is the displacement thickness over the momentum thickness; the entrainment coefficient is the rate at which the layer’s volume flux grows per unit distance over the outer speed; the skin friction is the wall shear over the outer dynamic pressure, given in thousandths. “Shape-only” means the entrainment set at every station to its equilibrium value for the layer’s own shape; “flat plate” means a layer in equilibrium in no gradient at the same momentum-thickness Reynolds number. Distances after the release are in the layer’s thickness there, estimated as the momentum thickness times the sum of the two shape factors, as the earlier essay estimated it.

What the calculation leaves out #

The integral method’s own closures. Green’s equations, their friction law and their shape relations are fits to measured layers, and the lag constant in particular was fitted to layers in adverse gradients. How well it describes relaxation after a release is a question for measurements, not for this calculation, and a closure with no memory at all describes what a one-equation account cannot hold.

Curvature and a real trailing edge. At a wing’s trailing edge the release coincides with the wall ending and the layer becoming a wake, which is a different calculation; at a diffuser’s exit the flow often becomes a jet. The release here is onto a continuing flat wall.

Three-dimensional and unsteady effects. A diffuser’s layer is often not two-dimensional and not steady, and a layer near separation at the release may have intermittent backflow that the integral method does not represent.

The near-wall layer’s own memory. The method lets the outer shear stress lag; the inner layer is assumed to adjust at once, through the friction law, which is good for the inner layer’s short time scale and is the reason the friction follows the shape.

Who worked it out #

Relaxing turbulent layers were measured after sudden changes in pressure gradient by Bradshaw and his colleagues from the 1960s, and by Smits and others for layers released after curvature and gradients; Green, Weeks and Brooman’s lag-entrainment method of 1973 was built from Head’s entrainment method and Bradshaw’s shear-stress transport, and its ability to follow relaxing layers was one of the tests it was judged on. The finding here that a released layer overtakes the memoryless one at about twelve thicknesses whatever the gradient, and overshoots the flat plate’s shape, is this calculation’s, from the method above.

How it was checked #

A layer run with no gradient at all ends within 0.4 per cent of the flat plate’s equilibrium shape factor, which is the reference’s own consistency. Halving the march’s step moves the crossing distance by three parts in a million. And the layers before the release are the earlier essay’s, which checked the method against the flat plate’s friction and against its own equilibrium layers.

Still open: a layer that is released twice #

A real diffuser’s exit is rarely the end. Behind a compressor’s stage the flow decelerates, is released, and decelerates again in the next stage, and a wing’s upper surface passes from a favourable gradient into an adverse one and, behind a flap, into a favourable one again. If the eddies take about twelve thicknesses to forget a gradient, a second gradient that arrives sooner than that meets a layer still carrying the first one’s shear stress. The next calculation would march a layer through a deceleration, a release of chosen length and a second deceleration, and ask how short the release can be before the second gradient’s separation distance is set by the first gradient’s history rather than by the layer’s shape at its start.