A dissipation that lags its production

Where the energy goes sets out the cascade: energy enters turbulence at the large scales, passes down through the eddies without being dissipated, and is destroyed by viscosity at the smallest ones. The limit that is not the value adds the result that the rate is set at the top and not at the bottom — the dissipation is a large-scale quantity with no viscosity in it.

Both are statements about a steady cascade. This essay is about the interval after something changes, and the interval is the cascade’s own delivery time.

One quantity follows and three do not #

The experiment is a step: take a patch of turbulence in equilibrium with a steady strain and increase the strain rate by a factor of two and a half.

The production follows immediately, multiplying by 6.28 — which is the strain squared, because production is a stress times a strain and the stress is itself proportional to the strain. Nothing in it is delayed: it is a product of quantities that exist now.

The turbulent energy moves by 0.68 per cent and the dissipation by 0.93. Both are integrals of what has already happened: the energy is what has accumulated, and the dissipation is the rate at which the small scales are receiving, which depends on what the large scales handed over a turnover ago.

So a step in the forcing puts the turbulence badly out of balance, and the imbalance is where its memory is.

How far out of balance, and for how long #

The ratio of production to dissipation settles, in a steady homogeneous shear, at a value this model puts at 2.09 — a number set by two closure coefficients and independent of the strain rate, which is what makes a step in the strain a clean test.

Immediately after the step the ratio is 520 per cent above that. It falls to a tenth of the departure after a third of a turnover and to a hundredth after three quarters of one.

So the recovery is fast in absolute terms and slow compared with the thing that caused it. The strain changed in zero time; the turbulence was still 109 per cent out of equilibrium a third of a turnover later and took 0.769 turnovers to come back within ten per cent; and anything happening on a time scale between those two — which is most of what happens in an engineering flow — is happening while the turbulence is out of equilibrium.

While production exceeds dissipation the energy grows, and the turnover time falls as it does — so the turbulence responds by getting faster as well as stronger, and it is that acceleration which eventually closes the gap.

What it does to the standard scaling #

The most-used relation in turbulence modelling is that the dissipation is the cube of a velocity scale divided by an integral length, with a coefficient of order one. It is the basis of every one-point closure and of most estimates anybody makes by hand.

Through the transient it is badly wrong, and the honest way to see that is to hold the integral length where it was — which is what the large eddies actually do for a turnover, since they are the slow ones. The coefficient then moves by a factor of 530 across the step.

Taking the length from the flow’s own energy and dissipation instead makes the coefficient exactly one by construction and measures nothing at all. That was the first version of this computation, and it is a good example of a check that cannot fail: the quantity being tested had been defined in terms of the quantities it was being compared with.

Where the energy is while it is out of balance #

The book-keeping is worth following, because it says where the missing energy has gone and answers the obvious objection.

At the instant of the step the strain rate is multiplied by 2.5; the production is multiplied by 6.277; the dissipation by 1.0093 and the energy by 1.0068. Energy enters at the rate of the production and leaves at the rate of the dissipation, and the ratio of the two goes from 2.073 — its equilibrium value is 2.091 — to a peak of 12.96, which is 6.2 times where it started. The difference is not lost: it is stored, in the turbulence’s own kinetic energy, which grows.

That is the whole of the memory. The turbulence’s state is one number — how much energy it is carrying — and that number is an integral of the imbalance since the beginning of time. A patch of turbulence arriving at a station with a history of strong straining behind it carries more energy than one that has just been strained, at the same local strain rate.

The parallel with the boundary layer of a layer that is an integral of everything upstream is exact rather than approximate: a thickness there and an energy here, both accumulated, both invisible in the local conditions, and both deciding what happens next.

And in both cases the storage is what makes an integral method possible. A quantity that accumulates can be carried as a state, and carrying it is much cheaper than carrying the history that produced it.

Why this matters more than it looks #

An imbalance that lasts a turnover sounds like a detail, and it is not, for two reasons.

Most flows are never in equilibrium. A turbulence passing through a contraction, round a bend, over a step, through a shock or into an adverse pressure gradient is being strained differently every turnover. The equilibrium the standard scaling assumes is reached only in flows deliberately built to have it — homogeneous shear, decaying grid turbulence, a long straight pipe — and those are the flows the coefficients were measured in.

And the departure is not small. A production-to-dissipation ratio of 12.96 where the model assumes 2.09 is not a correction; it is the model being wrong about which of its two terms dominates, by a factor of 6.2.

That is why the non-equilibrium behaviour of the dissipation has been an active subject since the 2010s, and why the measurements that started it — grid turbulence in which the standard scaling failed by a factor of several in a region where it was supposed to be exact — were surprising rather than merely interesting.

What the solver computed, and how it was checked #

A two-equation model of homogeneous shear, with the energy equation exact given the production and the dissipation and the dissipation equation the standard modelled one. The flow is run to equilibrium at one strain rate, stepped, and run on.

Three checks. That the production jumps by more than a factor of three across the step, so there is something to be out of balance about. That the dissipation moves by less than one per cent at the same instant — it moves by 0.93 per cent against the production’s 528 — which is the essay’s central claim and is what distinguishes a lag from a slow response. And that the departure from equilibrium exceeds ten per cent and lasts between a tenth and twenty turnovers: it peaks at 519 per cent and is back inside ten per cent after 0.769 turnovers, having passed through 1.09 at 0.309.

The equilibrium value itself is the check that made the rest possible. Measuring the departure from one rather than from the model’s own fixed point made the ratio look as though it never settled, because in homogeneous shear it settles at 2.09 and not at unity.

What a turnover actually is #

The time constant deserves a paragraph, because “a turnover” is used loosely and the quantity it stands for is specific.

It is the turbulent kinetic energy divided by the dissipation rate — the time the energy-containing eddies would take to hand over all of their energy at the current rate. Equivalently it is the integral length divided by the velocity fluctuation, which is the time a large eddy takes to turn over once.

Two things follow. It is a large-scale time: the small eddies turn over far faster, in proportion to their own size to the two-thirds power, so the cascade’s lower reaches respond almost immediately and the delay is entirely at the top. And it changes during the transient — from 4.8 before the step to 1.9 after, in this run — because the energy grows and the dissipation grows faster.

That second point is why the recovery is quicker than a fixed-time-constant estimate suggests. The turbulence is not relaxing at a rate; it is relaxing at a rate that is itself accelerating, which is what makes the ratio come back in a third of a turnover rather than in one.

What the model is, and is not #

It is worth being explicit, because a two-equation model is being used to criticise the assumptions behind two-equation models.

The energy equation is exact: the rate of change of turbulent energy is the production minus the dissipation, and nothing is modelled in it. The dissipation equation is entirely a model — it has two fitted coefficients and no derivation — so what is computed here is what that model says about its own transient.

That is a weaker statement than a measurement and it is not vacuous. The model was fitted to equilibrium flows and it nonetheless exhibits a lag, because the lag follows from the structure of the equations rather than from the coefficients: the dissipation cannot respond faster than its own equation allows, and its equation has a time in it.

What the model cannot say is how long the real lag is. Direct simulation and experiment put it at one to a few turnovers, which is the same order, and that agreement is the reason the model is worth quoting here at all.

The scales the energy is crossing while it lags are the subject of the essay after next, and the same machinery draws them.

The same lag, seen from every side #

The structure is one this collection has now met four times, and putting them together is the point of the collection.

An aerofoil’s wake returns a lift deficiency that is one at low frequency and a half at high — the lag that makes flutter possible.

A viscoelastic fluid returns a stress that is a convolution of the strain rate over a relaxation time — the fluid that has not finished its last deformation.

A rotor’s inflow takes a fraction of a wake convection time to arrive — the inflow that takes time to arrive.

And a turbulent cascade takes a turnover to deliver.

Each is a process with a time in it, forced by something that changes faster, and each produces the same two consequences: a response that cannot follow, and a phase lag that a steady theory sets to zero. The next essay makes that comparison exact.

How to tell whether a flow is in equilibrium #

Since almost nothing is, it is worth having a test, and the model supplies one that needs only quantities a measurement can produce.

Compare the strain time with the turnover time. The turnover is the turbulent energy divided by the dissipation; the strain time is the reciprocal of the mean strain rate. Their ratio is the parameter that decides everything, and it is a local, measurable, dimensionless number.

Below about a tenth, equilibrium holds. The turbulence adjusts far faster than the strain changes, so the standard relations apply and an eddy viscosity is reasonable.

Above about one, it does not. The turbulence is being strained faster than it can respond, its state is dominated by where it came from, and every relation calibrated in equilibrium is being used outside its range.

That ratio has a name in this collection’s other fields — it is a Deborah number with a turnover for a relaxation time, or a reduced frequency with a turnover for a chord — and a solid if it is not given time is where the general form of it is set out. Computing it before choosing a model is a minute’s work and is skipped almost universally.

What the picture cannot show #

The cascade is not drawn anywhere here, because a two-equation model does not have one: it has an energy and a dissipation, and the passage of energy between scales is exactly what has been modelled away. A figure of the spectrum during the transient — energy piling up at the large scales and taking a turnover to reach the small ones — is the picture the essay is about, and this machinery cannot produce it.

Nothing here shows the flow either. Homogeneous shear has no boundaries, no geometry and nothing to look at; the whole of it is four numbers evolving in time.

Where the departure shows up in practice #

The model the departure is a departure from is a closure with no memory at all, which is the same quantity written as an instantaneous relation and is what most computations actually carry.

The model the departure is a departure from is a closure with no memory at all, which is the same quantity written as an instantaneous relation and is what most computations actually carry.

Three flows in which the imbalance is not a transient detail but the dominant behaviour, and all three are ordinary.

A contraction. Air entering a wind-tunnel contraction is strained hard over a distance short compared with a turnover, so the turbulence leaving it is far from equilibrium — which is why the turbulence in a working section depends on the contraction’s shape and not only on its ratio.

A shock. Turbulence crossing a shock is compressed in a distance of a few mean free paths, which is instantaneous on any turbulent time scale. What emerges is amplified and profoundly out of equilibrium, and it relaxes over the next several turnovers — during which it is being used.

And a stator row. Turbulence in a turbomachine is strained by every blade it passes, at a rate set by the blade passing frequency, which is comparable with or faster than its own turnover. It is never in equilibrium anywhere in the machine, which is one of the reasons turbomachinery computations need their models calibrated rather than merely selected — a row that meets the row before it is what is doing the straining.

In all three the flow of interest lives entirely inside the interval this essay computes.

What the lag costs a computation, in cores #

The engineering consequence is a scheduling one, and it is worth stating in the units a computation is budgeted in.

A model whose dissipation follows its production instantaneously has one fewer state to carry and one fewer equation to advance, and it can be advanced at the flow’s own time step. A model that carries the lag has to resolve the lag, which means a step short compared with the turnover time — and near a step change in the strain that turnover time is itself changing.

So the cost of the memory is not one extra equation; it is the time step. That is the same trade that appears everywhere in this collection: a quantity that can be eliminated makes a problem cheaper by more than the one variable it removes, because eliminating it removes a timescale as well.

The corollary is the one worth carrying into a modelling decision. If the flow’s strain changes slowly compared with a turnover, the lag can be dropped and the saving is real. If it does not, the saving is a wrong answer that arrives sooner.

Who found it, and when #

The two-equation model is Launder and Spalding’s in its standard form, from 1974, building on Kolmogorov and Prandtl. Its dissipation equation has been criticised since it was written and has not been replaced.

The non-equilibrium dissipation scaling is much more recent — Vassilicos and colleagues from about 2012 — and its claim is that the departure from the standard relation is not confined to transients but persists in whole regions of ordinary flows. That is a stronger statement than this essay makes and it is not settled; what is settled, and is what is computed here, is that the transient exists and lasts a turnover.

Limits recorded rather than smoothed over #

A model of a model. The dissipation equation is empirical and its transient is its own. The existence and order of the lag are robust; the 520 per cent and the third of a turnover are the model’s.

Homogeneous shear, and no walls. Every complication that makes a real flow interesting is absent: no inhomogeneity, no transport, no pressure-strain redistribution, no anisotropy beyond what one scalar carries.

A step is not a flow. Nothing steps its strain rate discontinuously. A step is the input that separates a response from its forcing most sharply, which is why it is used, and a real flow’s forcing is spread across frequencies — which is what the next essay’s frequency response is for.

One step, one direction. The strain is increased. Decreasing it puts the imbalance the other way — dissipation exceeding production, energy falling — and the recovery is slower, because the turnover time lengthens as the energy drains rather than shortening. Nothing here computes that case.

And the frozen-length coefficient is a construction. Holding the integral scale at its pre-step value is a way of exhibiting the lag, not a measurement of a length. A real integral scale changes during the transient, more slowly than the small scales and faster than frozen.