A closure with no memory at all

A guess with a constant in it is this collection’s account of the eddy-viscosity hypothesis: the Reynolds stress is taken to be proportional to the mean strain rate, with a coefficient that has to be modelled, and the whole difficulty of turbulence modelling is in that coefficient.

This essay is about a different objection, which does not concern the coefficient at all. The hypothesis is a statement that the stress responds instantaneously, and it does not.

What the stress actually does #

The Reynolds stress is a correlation between two velocity fluctuations, and the fluctuations are eddies. A change in the mean strain rate does not change the eddies; it changes the rate at which they are being distorted, and they take time to be distorted. Step the strain and the equilibrium anisotropy moves from −0.300 to a new value at once while the actual anisotropy is still at −0.300, and the gap between them takes a turnover to close.

The relevant time is the eddy turnover — the same one a dissipation that lags its production measures — and the resulting behaviour is a relaxation: the stress moves towards the value the present strain rate would eventually give it, at a rate set by the turnover.

Step the strain rate and the stress therefore lags. The eddy viscosity, which is the equilibrium value at every instant, jumps immediately.

Immediately after a step the closure is wrong by 60 per cent of the stress it is predicting, and the error decays over about a turnover.

Asked as a frequency #

The sharper way to see it is to ask what the closure is, rather than what it gets wrong.

Drive the strain rate sinusoidally and measure the stress. The response has a magnitude and a phase, both functions of the frequency:

At low frequency the magnitude is one and the phase is zero, and at high frequency the magnitude falls as the reciprocal of the frequency and the phase approaches ninety degrees. An eddy viscosity is already ten per cent wrong when the strain varies at half a turnover, and eighty per cent wrong at five.

An eddy viscosity is that response evaluated at zero frequency. It is not an approximation to the curve; it is one point on it — the point where the magnitude is 1.0000 and the phase is 0.000°, and using it everywhere is asserting that the flow’s strain rate varies infinitely slowly compared with a turnover.

Below a twentieth of a turnover the assertion costs under a tenth of a per cent. At one turnover it costs 29 per cent of the stress and 45 degrees of phase. At five turnovers it costs 98 per cent, and the closure is predicting a stress that is essentially not there.

The same curve, in a different subject #

The shape is one this collection has already met, in aerodynamics rather than turbulence. The lag that makes flutter possible computes Theodorsen’s lift deficiency: the factor by which an aerofoil’s circulatory lift falls short of its quasi-steady value, as a function of how fast the incidence is changing.

That function is one at zero frequency, falls monotonically, and approaches a half — it reads 0.726 at a reduced frequency of 0.254, with a phase of −15.5°. This one is one at zero frequency, falls monotonically, and approaches zero, reading 0.707 at a frequency of one with a phase of −45.0°. The limits differ because the mechanisms differ — a wake removes at most half of the lift, and a turbulence that cannot respond at all carries no stress — and the structure is identical, for the same reason in both cases.

A quantity produced by a process with a time in it cannot follow a forcing that changes faster than that time. The wake’s convection time is one such process and the eddy turnover is another, and setting either response to its zero-frequency value is the same mistake made in two fields.

The parallel is worth pressing because of what it says about the fix. In aeroelasticity, setting $C(k)$ to one was found to move a flutter speed by sixty per cent and was corrected by carrying the frequency dependence. In turbulence modelling the same correction exists — the stress-transport models carry an equation for the stress rather than an algebraic relation, which is exactly the time-domain form of the same repair — and it is used far less than it should be, because it costs six equations rather than two.

What the solver computed, and how it was checked #

A first-order relaxation of the stress anisotropy towards the value the present strain rate would give it, with a relaxation time of one turnover, and the frequency response of the same equation.

This is a model of a model, and it is a deliberately minimal one. What it captures is the structure — that the response is first order with a time constant of a turnover — and it captures nothing else. The real response has a spectrum of times rather than one, and its high-frequency limit is set by rapid distortion theory rather than by an exponential.

Three checks. That the response at zero frequency is one, which is the identification of the eddy viscosity with the limit and is the essay’s central claim. That it is small at high frequency, below five per cent at a hundred turnovers — it reads 1.0 per cent. And that the error after a step exceeds thirty per cent; at a frequency of one it reads 29.3, so there is something to report.

What the phase does, which is worse than what the magnitude does #

The response has two parts and the discussion so far has been about one of them. The other is the more consequential.

A magnitude error means the model predicts too much stress or too little, which shows up as a friction or a spreading rate that is wrong by a factor. That is bad and it is the kind of error a calibration can partly absorb.

A phase error means the stress is predicted to be aligned with the strain when in fact it is not. At one turnover the lag is 45 degrees, which is not a small misalignment: the principal axes of the predicted stress and the real one differ by half of that, and in a flow with curvature or rotation the misalignment changes the sign of what the stress does to the mean flow.

That is why eddy-viscosity models fail characteristically rather than randomly. They are insensitive to streamline curvature, they get rotating flows wrong in a known direction, and they cannot represent the lag between strain and stress in a three-dimensional boundary layer. All three are the same defect: an algebraic relation between two tensors forces them to be aligned, and a lag is exactly a misalignment.

No coefficient can produce a misalignment, which is the sharpest statement of why the repair has to be structural.

Where a flow’s frequencies actually are #

The whole question is whether a flow’s strain rate varies slowly compared with a turnover, so it is worth listing where real flows sit.

A long straight duct or a flat plate. The strain a parcel meets changes over a distance of many boundary-layer thicknesses, and its turnover is a fraction of one. The ratio is well below a tenth, where the magnitude error is under 0.5 per cent and the phase lag under 5.7 degrees, and the closure is fine.

A curved duct, a diffuser, an aerofoil near the trailing edge. The strain changes over a distance comparable with the layer’s own thickness, so the ratio is of order one, the magnitude is 0.707 and the phase is 45 degrees behind — an error of 29.3 per cent in amplitude and a full eighth of a cycle in timing — which is the regime in which eddy-viscosity models are known to perform poorly and are used anyway.

A shock, a rapid contraction, a blade passage. The strain changes over a distance short compared with a turnover, and the ratio is well above one. There the closure is not approximately wrong; it is predicting a quantity that has not had time to exist.

And any unsteady flow. A flow forced at a frequency comparable with the turnover — a rotor, a pulsating flow, a bluff-body wake — has the response evaluated at that frequency, and using the zero-frequency value is the error this essay is about at its full size.

Why the hypothesis works as well as it does #

The essay has been unkind and the hypothesis is nonetheless the basis of nearly every industrial computation ever performed, which needs explaining rather than dismissing.

Most of the flow, most of the time, is slow. In a boundary layer on an aircraft wing or a duct wall, the strain a parcel meets changes over a distance of many layer thicknesses while its turnover is a fraction of one. The ratio is genuinely small over the great majority of the volume, and where it is small the closure is not merely adequate but nearly exact.

And the quantity usually wanted is an integral. A drag, a pressure drop, a heat transfer coefficient is an integral over a surface, and the regions where the closure fails are a small fraction of it. An error of tens of per cent over a few per cent of the area is a small error in the total — unless the regions where it fails are the ones that decide the flow topology, which near separation they are.

So the hypothesis is a good approximation with a known failure set, which is a far more useful thing than either its defenders or its critics usually claim. The practical skill is knowing where the failure set is, and the ratio above is what locates it.

The constant that makes a variance negative is the neighbouring complaint — a different structural defect of the same hypothesis, visible without any unsteadiness at all — and the two together are why the model’s failures are systematic rather than random.

Why the constant cannot fix it #

It is worth closing off the obvious repair, because it is the one that is usually attempted.

A model whose coefficient is tuned reproduces the zero-frequency response exactly and leaves the shape of the curve untouched, because a constant has no frequency in it. Tuning to a flow at one strain rate therefore produces a model that is right at that flow’s characteristic frequency and wrong at every other — which is why coefficients calibrated on one class of flow transfer so badly to another, and why the transfer failures are correlated with how fast the strain changes rather than with anything about the geometry.

That diagnosis is testable and is the useful part of the essay. A model’s calibration should be suspected first in flows whose strain changes on a turnover, and a model that has been recalibrated for a new class of flow has usually absorbed a frequency response into a constant.

The response the closure is a limit of is computed two essays away, on the same machinery.

What a stress-transport model does about it #

The repair exists and its structure is exactly the time-domain version of the frequency response above.

Instead of an algebraic relation between stress and strain, a stress-transport model carries a differential equation for each component of the stress, with production, redistribution and dissipation terms. The redistribution term is the one that relaxes the stress towards isotropy, and its coefficient is the turnover-based time this essay’s model uses.

So a stress-transport model has the memory: solving its equation is convolving the strain history with a kernel, in the same sense that the fluid that has not finished its last deformation solves a constitutive equation with one. The price is six transport equations instead of an algebraic relation, plus a model for the redistribution, which is itself the hardest thing in the subject.

That trade — carry a state and get the memory, or carry a constant and lose it — is the same one the wall the fluid is listening to describes for a diffusion problem, and it comes out the same way: the state is cheaper than the history whenever the kernel is an exponential.

What to compute before choosing a model #

The essay reduces to one measurement, and it is cheap enough to make routinely.

Form the ratio of the turnover time to the strain time. The turnover is the turbulent energy over the dissipation; the strain time is the reciprocal of the mean strain rate. Both come out of any computation and either can be estimated from an experiment.

Read the response off the curve. At a ratio of 0.05 the closure is missing a tenth of a per cent; at 0.5 it is missing eleven per cent and 27 degrees; at 2 it is missing 55 per cent and 63 degrees.

And plot it as a field. The interesting output is a map of that ratio over the flow, which shows where the model is being used inside its assumption and where it is not. Those regions are usually small, and they are usually the ones the answer depends on — a separation, a shock interaction, a leading edge.

That procedure is exactly the one what of order one is worth recommends in general: form the dimensionless group the approximation rests on, and look at where it is not small.

What the picture cannot show #

The frequency response is drawn as a magnitude and a phase and both are the same complex number. A figure of the complex response — a Nyquist plot, a semicircle — would say that more directly and would be unreadable to anybody who does not already know what one is.

Nothing here shows an eddy. The whole essay is about a correlation between velocity fluctuations responding to a strain, and the fluctuations are exactly what a one-point model has averaged away.

What would have to change to fix it #

The obvious repair is to give the closure a memory, and it is worth saying what that costs, because the cost is why it is rarely done.

A relaxation form replaces the algebraic stress-strain relation with an equation that carries the stress as its own variable, relaxing towards the eddy-viscosity value over a turnover time. That is a second-moment closure with a lag in it, and it reproduces the low-frequency limit exactly while following the high-frequency one correctly. It costs six more transported quantities.

A full convolution would be better and is not available, because the kernel is not a property of the fluid — it depends on the strain history that produced the turbulence, so the object being modelled would have to be measured for each flow.

And the honest middle is what most practice does: use the algebraic closure, and know the frequency above which it stops being a model. That number is computable from the turnover time and it is not usually computed, which is the gap this essay is about.

Who found it, and when #

Boussinesq proposed the eddy viscosity in 1877 and the objection is nearly as old: Prandtl’s own writing is careful about the conditions under which it holds. The systematic frequency-domain statement is much more recent and belongs to rapid distortion theory, which computes what turbulence does when strained faster than it can respond — Batchelor and Proudman’s from 1954, and Hunt’s from the 1970s.

Stress-transport modelling is Launder, Reece and Rodi’s in its standard form, from 1975, and its motivation is precisely the failure this essay computes: flows in which the stress and the strain are not aligned, which is what a lag produces.

Limits recorded rather than smoothed over #

A model of a model, again. The relaxation used here is a schematic of what a stress-transport model does, not a solution of anything. Its purpose is to identify the eddy viscosity with a limit, which is a structural statement and does not depend on the relaxation being first order.

A single relaxation time. Real turbulence has a spectrum of them, so the response is a sum rather than one first-order lag, and its high-frequency behaviour is a power law rather than a reciprocal.

Anisotropy as one scalar. The Reynolds stress is a tensor with five independent components, and representing it by one number throws away the alignment question — which is the other half of what an eddy viscosity gets wrong and is not computed here.

No rapid distortion. At very high frequency the correct behaviour is not zero response but the rapid distortion limit, in which the eddies are deformed without interacting. This model gives zero, which is wrong in the same direction and for a different reason.

Homogeneous, and nothing is transported. The model relaxes a stress at a point. In a real flow the stress is also convected and diffused, so a parcel arrives carrying a stress produced somewhere else — which is a second memory, in space rather than in time, and is a layer that is an integral of everything upstream’s subject.

And the analogy with Theodorsen’s function is structural, not quantitative. Both are the response of a lagged process, and the curve drawn beside the turbulent one has the aerofoil’s shape rather than its values. The two are not the same function and are not claimed to be.