A layer whose stress lags the gradient separates sooner

An adverse gradient grows the wake, not the log law took a turbulent boundary layer into a pressure gradient that slows its outer flow and found where the change goes. The logarithmic layer near the wall is left alone; the wake, the outer two thirds of the layer, grows, and with it the shape factor, until the layer separates. Every layer in that essay was an equilibrium layer in Clauser’s sense, its wake strength a function of one number, Clauser’s β, the pressure gradient measured in the layer’s own units of wall stress and displacement thickness. Such layers exist — Clauser built them in a wind tunnel by tuning the gradient along the wall — but they exist because the gradient has been the same, in those units, for as long as the layer can remember.

That essay ended on the question a real diffuser asks. A layer that has grown on a flat plate meets a diffuser’s adverse gradient all at once, at the throat. It cannot be in equilibrium there: the large eddies that carry its shear stress were made upstream, in no gradient at all, and they take time to become the eddies an adverse gradient makes. How many layer thicknesses does the adjustment take, how far does the friction stray on the way, and is a layer hit hard separated before it has had time to adjust?

A closure with no memory at all gave the general reason to expect an answer of some tens of thicknesses: an eddy-viscosity model, which sets the stress from the local strain, is a frequency response evaluated at zero, and after a step in the strain it is wrong by more than half for about a turnover of the large eddies. A layer travels many of its own thicknesses in a turnover. This essay puts numbers on that for the layer a diffuser sees.

Two ways to let a layer entrain #

The calculation is an integral one: the layer is described by its momentum thickness, its shape factor and how fast it draws in the outer, irrotational fluid — its entrainment. A layer that is an integral of everything upstream found that the momentum thickness is already a history; the question here is whether the entrainment is too.

Head’s method of 1958 answers that it is not. Entrainment is fixed by the layer’s shape: a layer of a given shape factor draws in fluid at a given rate, whatever happened to it upstream. Green, Weeks and Brooman’s lag-entrainment method of 1973 answers that it is. They gave the entrainment an equation of its own, derived from Bradshaw’s transport equation for the shear stress, so that it relaxes towards the value an equilibrium layer of that shape would have, over a distance set by the large eddies. In a slowly changing gradient the two agree. After a step they do not, and the difference is what the eddies’ history is worth.

The layer here grows on a flat plate in air at 30 metres a second until it is 23 millimetres thick, and then, at a station called the step, meets an outer flow that decelerates as a power of distance downstream. The power is the dial: −0.1 is gentle, −0.4 separates the layer within a metre of the step, −0.8 within two tenths of a metre. Distances downstream are measured in the layer’s thickness at the step.

The entrainment trails the gradient #

The first figure is the entrainment itself at a power of −0.2: the lagged layer’s, the value its own shape would have in equilibrium, and the value that would hold its shape steady against the local gradient.

At the step the gradient appears at once and the entrainment does not. Five thicknesses downstream the lagged layer is drawing in fluid at 16.7 thousandths of the outer speed, where holding its shape against the gradient would need 30.8 — barely more than half. It is still entraining as the flat plate’s eddies did. The deficit closes over the following twenty or so thicknesses, the entrainment rising as the outer eddies grow, and by forty thicknesses it has overshot slightly, 26.7 against 24.5, before settling.

An adverse gradient slows the outer flow, and a slower outer flow, in a layer that keeps entraining at its old rate, leaves the slowed fluid inside the layer. The layer’s momentum deficit grows, and its profile empties out from the middle. That is the opposite of protection.

It reaches the new shape quickly, and overshoots #

The second figure shows the consequence for the shape factor, with Clauser’s equilibrium value at the lagged layer’s own local β beside it — computed independently, through Coles’s profile with the wake strength that an adverse gradient grows the wake used, Das’s fit to the measured equilibrium layers.

At the step the layer’s shape factor is the flat plate’s, 1.34, while the equilibrium value for the gradient it now meets is 1.47. The lagged layer climbs to it in about seventeen thicknesses, overshoots slightly, to 1.482 against 1.470 at twenty-five, and from there tracks the equilibrium within a hundredth as both drift down with the slowly relaxing β. The shape-only layer is slower: twenty thicknesses on its shape factor is 1.445 against the lagged layer’s 1.479, and it approaches the equilibrium value only from below, after eighty.

It is worth being exact about which layer has the memory here. Both carry their history in their momentum thickness. The lagged layer also carries it in its shear stress, which is the physical fact the experiments show; and because the stress lags, the mean profile adjusts faster, not slower. The shape-only layer’s entrainment follows its shape, which follows its entrainment, a loop that makes the mean profile sluggish. A calculation that lets the eddies keep up with the gradient thereby lets the profile fall behind it.

Why a slow answer is a worse one #

The momentum budget makes the mechanism plain. The momentum thickness grows at a rate that is half the skin friction plus a term proportional to the deceleration, the shape factor plus two, and the momentum thickness itself. On a flat plate only the friction term acts. In an adverse gradient the second term dominates within a few thicknesses, and it is indifferent to the eddies: the pressure gradient acts on all of the layer’s fluid at once, slowing what is already slow in proportion. What the eddies decide is how that loss is shared out across the layer. A layer entraining vigorously carries fresh outer fluid down towards the wall and keeps its profile full; a layer entraining at the flat plate’s rate does not, and the slowed fluid accumulates in the lower half of the wake, which is exactly a rising shape factor.

So the shear stress lagging behind the gradient means the layer’s eddies are too weak for the job they have just been given. In the equilibrium layers of the previous essay, the eddies and the gradient grew up together, and the wake grew in step with β. Here the gradient arrives first, and for the first twenty thicknesses the layer is run by eddies made for a different flow. A closure with no memory at all found an eddy-viscosity model wrong by sixty per cent immediately after a step in the strain; Green’s method is, in effect, the integral-method version of that essay’s lagged stress, and the shape-only layer is the eddy-viscosity version, answering the gradient at once.

The friction overshoots, then falls through #

The third figure is the skin friction. At the step the layer still has the flat plate’s friction, 19 per cent above the equilibrium value for the gradient it now meets: the near-wall flow has not yet been slowed. That is the overshoot the question asked about, and it is not small. It decays as the gradient slows the inner layer, and the lagged layer’s friction falls through the equilibrium value between ten and seventeen thicknesses on and settles about five per cent below it, a gap that is mostly the two calibrations of the friction law disagreeing rather than any lag; the shape-only layer’s falls more slowly and stays above the lagged one throughout, by 5.7 per cent twenty thicknesses on at this power.

The dial moves the power from −0.1 to −0.4. The shape of the response does not change; its size does. At −0.4 the friction falls to zero: the layer separates.

Twenty thicknesses, whatever the step #

The fourth figure measures the two closures’ disagreement for six powers from −0.05 to −0.3: the largest excess of the lagged layer’s shape factor over the shape-only layer’s, and how far downstream it falls.

The excess grows faster than the gradient: 0.006 at −0.05, 0.034 at −0.2, 0.086 at −0.3. Where it falls hardly moves: 19 thicknesses downstream at the gentlest step and 31 at the hardest, most of the growth coming as the layer itself thickens under the steeper gradients. That is the relaxation length the question asked for, and it is a property of the eddies rather than of the gradient: a few large-eddy turnovers, each of which carries the layer several of its own thicknesses downstream. A dissipation that lags its production found the cascade’s delivery time to be a fraction of a turnover; the shear stress, carried by the largest eddies, takes several.

It separates sooner #

The fifth figure is where each layer separates. At a power of −0.35 the lagged layer separates 103 thicknesses downstream of the step and the shape-only one 166; at −0.4, 35 and 60; at −0.5, 16 and 24; at −0.8, 7 and 8. The lagged layer always separates first. The relative gap is largest in the gentlest gradients that separate the layer at all — at −0.35 and −0.4 the shape-only calculation puts separation more than half as far again downstream — and closes in the steepest, where both layers are swept to separation within a few thicknesses, faster than either closure’s history matters.

This is the answer to the diffuser’s question, and it is the reason lag-entrainment methods came to be preferred to Head’s for adverse gradients in the design codes for wings and intakes. A calculation that lets the eddies keep up is optimistic about separation, and optimistic by most exactly where a designer would trust it, in gradients just steep enough to separate the layer over a long diffuser. Some of a diffuser’s stall margin is in the shape of its gradient, and some is in how suddenly the gradient arrives.

The steep end of the figure has its own reading. At a power of −0.8 the outer flow loses a third of its speed within the first ten thicknesses, and both layers separate inside the relaxation length, before the lagged layer’s eddies have done much catching up and before the shape-only layer’s entrainment has had much to follow. There the two calculations agree within a thickness or two, and neither is a statement about eddies; it is the momentum budget’s deceleration term, acting on a layer that has no time to respond in either closure. Between the gentle and the steep there is a band of gradients, roughly the powers from −0.35 to −0.5 in this configuration, where the gradient is steep enough to separate the layer and slow enough for the eddies’ history to decide where. That band is where most real diffusers are designed to work, close to the edge of stall, because that is where they recover the most pressure in the least length.

Read the other way, the figure says what a shape-only calculation is worth there. It is not wrong about whether the layer separates; both closures separate it at every power from −0.35 down. It is wrong about where, by a margin of the same order as the distance itself.

How long twenty thicknesses is #

The relaxation length is easiest to judge in real distances. The layer here is 23 millimetres thick at the step, so twenty thicknesses is nearly half a metre and the lagged layer’s separation at a power of −0.4, 35 thicknesses on, is eight tenths of a metre downstream. A wind-tunnel diffuser, a ventilation duct or an engine intake is a metre or two long, and over most of it the layer in it is not in equilibrium at all. On a wing of one metre chord at the same speed the layer is about a centimetre thick at mid-chord, and twenty thicknesses is a fifth of the chord: the recovery region behind the suction peak, where the gradient arrives most suddenly, is shorter than the time its layer needs to adjust. Equilibrium layers are the exception in practice; they are the rule only in the laboratories that built them on purpose.

What was checked #

On a flat plate the two closures are the same layer to a part in a thousand, with a shape factor of 1.31 and a friction within four per cent of the one-seventh-power law at the same momentum-thickness Reynolds number. In a steady gentle gradient carried far downstream, at a power of −0.12, the two layers settle together, their shape factors within 0.003 and their friction within half a per cent, which checks that the lag is a transient and not a bias. And the lagged layer’s shape factor, computed by Green’s equations, was compared with Clauser’s equilibrium value computed by an independent route — Coles’s composite profile with Das’s wake strength at the lagged layer’s own β — and the two agree within a hundredth once the transient is over. The calculation refuses an accelerating power and one steeper than −1.

What the calculation leaves out #

Integral methods are calibrated, not derived. Every constant in Green’s equations, and the shape-only entrainment law, was fitted to measured layers. The relaxation length is the closure’s, consistent with what measured layers show, and not a prediction from first principles. A guess with a constant in it made the same point about the mixing length.

Separation is a criterion. An integral method cannot follow a layer through separation; the calculation stops where the shape factor passes 2.4 or the friction reaches zero, which is where real layers detach to within the scatter of the measurements.

The step is sharp. A real diffuser’s gradient rises over a short distance rather than at a point. A rise shorter than the relaxation length, a few tens of thicknesses, is a step to the layer; a longer one is not, and the lag’s effect fades as the rise lengthens.

Two dimensions. Diffusers stall first in their corners, where secondary flows feed the layers on two walls at once.

Who worked it out #

Clauser’s equilibrium layers are of 1954 and 1956, Coles’s law of the wake of 1956, and Head’s entrainment method of 1958. Bradshaw, Ferriss and Atwell’s shear-stress transport equation of 1967 showed that the stress in a boundary layer is carried, not set locally, and Green, Weeks and Brooman built the lag-entrainment method on it in 1973. How much uphill a layer can take and Stratford’s limiting layers of 1959 are the older half of the same question; how far downwind a surface is remembered is its atmospheric cousin, a layer adjusting to a change of the surface rather than of the gradient.

Still open: a layer released from a gradient #

The step here goes one way, from no gradient into an adverse one. The opposite step, a layer leaving an adverse gradient for a flat or favourable one, is what a diffuser’s exit and a wing’s trailing edge give it, and the lag should act the other way: a layer with an overgrown wake and outer eddies made for deceleration keeps entraining at its old, high rate when the gradient relaxes. The next calculation would release a layer from an equilibrium adverse gradient, follow how long its wake and its friction take to recover, and ask whether a layer that came close to separating is left more resistant to the next adverse gradient, or less, for how many thicknesses afterwards.