The pressure a parcel remembers keeps it finite

A gradient left to itself followed the velocity gradient of a single fluid parcel with the one simplification that makes the problem solvable: of the pressure’s second derivatives, which the gradient equation needs, keep only the part that makes the gradient stay trace-free — the isotropic part — and throw the rest away. What is left is the restricted Euler equation, and it can be solved completely. Every trajectory is an explicit curve in the plane of the gradient’s two invariants, the vorticity lines up with the middle axis of strain exactly as turbulence measurements show, and every gradient, three hundred random ones out of three hundred, blows up in finite time. The essay ended by asking how little of the discarded pressure is needed to stop that, and said that whatever answers it will be a closure.

This essay takes the smallest answer that has been proposed and works: a pressure that remembers. The idea, due to Chevillard and Meneveau, is that the part of the pressure the restricted equation discards is the part set by the parcel’s neighbours, which a single parcel cannot see, and that the most a single parcel can know about its neighbours is how it has itself been deformed recently. The question is whether that much memory is enough to stop the runaway, and what the ensemble of parcels it produces looks like against real turbulence.

What the restricted equation threw away #

The velocity gradient $A_{ij} = \partial u_i/\partial x_j$ following a parcel obeys an exact equation: its rate of change is minus its own square, minus the pressure’s second-derivative tensor — the pressure Hessian — plus viscous terms. What a parcel does first split a gradient into its strain and its rotation; the equation says how both change. The pressure Hessian is not local: it is set by the whole flow through a Poisson equation, which is why pressure has no speed and why a parcel’s equation cannot be closed on the parcel alone.

Its trace is local — incompressibility fixes it as minus the trace of $A^2$ — and the restricted Euler equation keeps exactly that: an isotropic pressure Hessian, the same in every direction, just large enough to keep the gradient trace-free. Everything anisotropic is discarded, and the anisotropic part is what, in real turbulence, redirects the strain before it can run away.

A pressure isotropic a moment ago #

The recent-fluid-deformation closure keeps the restricted equation’s form and changes where the isotropy is imposed. The pressure Hessian is taken to be isotropic not in the present coordinates but in the coordinates the parcel’s neighbourhood had one Kolmogorov time earlier. Transported to the present, an isotropic tensor in old coordinates becomes proportional to the inverse Cauchy–Green tensor $C^{-1}$ of the deformation since then — its directions are the directions the parcel has been stretched and squeezed along — and its size is fixed, as before, by incompressibility. The deformation over one Kolmogorov time is estimated from the present gradient alone, as $C = e^{\Gamma A}e^{\Gamma A^T}$, where $\Gamma$ is the Kolmogorov time over the large-eddy time. The viscous term is treated the same way, as a Laplacian in the recent coordinates, which makes it a damping proportional to the trace of $C^{-1}$.

Two further ingredients make an ensemble: a random forcing, standing for the large eddies that keep supplying gradient, and many parcels, each followed with its own seeded noise. The model has one parameter, $\Gamma$, which falls as the Reynolds number rises, and the matrix exponential it needs is computed by scaling and squaring, checked against an eigen-decomposition to three parts in $10^{15}$. With $\Gamma$ set to zero the memory disappears, $C^{-1}$ becomes the identity, and the closure reduces exactly to the restricted Euler equation with a linear damping — the check that the new term is a correction to the old equation and not a different one.

One gradient, with and without its memory #

The first figure is the whole effect on one parcel. A gradient of size about twelve, in units of the inverse large-eddy time, is started and left alone. Under the restricted Euler equation it grows faster and faster and reaches infinity at 0.83 large-eddy times: the finite-time singularity of the essay before. With the remembered pressure the same gradient grows by a twelfth over the first third of a large-eddy time and then turns over and decays.

The turning over is the anisotropic pressure at work. As the gradient runs towards the restricted equation’s singularity — a sheet being flattened, one direction compressed faster and faster — the deformation over the last Kolmogorov time grows along exactly those directions, $C^{-1}$ grows along the compressed one, and the remembered pressure pushes back hardest where the runaway is going. It is a restoring force built out of the runaway’s own history.

The ensemble never blows up #

Forced by noise and followed for fifteen large-eddy times after five of burn-in, a hundred and fifty parcels at a memory of a tenth produce twenty-two thousand samples and not one blow-up; nor did any ensemble at any memory tried, down to four-hundredths. The second figure plots their gradients in the plane of the two invariants that the restricted equation’s trajectories live in: $Q$, which is positive where rotation dominates strain, and $R$, whose sign says whether the strain is stretching or compressing more.

The cloud is the teardrop that simulations and measurements of turbulence find. It crowds along the right-hand branch of Vieillefosse’s curve, in the quadrant where strain dominates and the flow is making sheets — the branch every restricted-Euler trajectory runs off along to infinity, here populated but not escaped. And it spreads upwards on the left, where rotation dominates and vortices are being stretched. The two quadrants the cloud avoids, upper right and lower left, are the ones real turbulence avoids too. The restricted equation had the teardrop’s direction; the remembered pressure gives it a boundary.

What the teardrop is made of #

The cloud can be counted by quadrant, and the count says both what the closure gets right and where it leans. Of the twenty-two thousand samples at a memory of a tenth, 48 per cent lie in the lower right, where strain dominates rotation and the strain is making sheets; 22 per cent in the upper left, where rotation dominates and vortices are being stretched; and 15 per cent in each of the other two. The ordering is the measured one — the sheet-forming and vortex-stretching quadrants are the populated pair in every simulation — but the closure puts too many parcels in the first and too few in the second. That is the same imbalance the fifth figure measures directly as too much strain, seen from the other side.

The quantity $Q$ plotted up the side is also the quantity a vortex-identification criterion thresholds: positive $Q$ is where every snapshot says vortex. The ensemble’s parcels spend about a third of their lives there and two-thirds where strain wins, which is roughly the proportion of a turbulent volume that the criterion would call a vortex — and a reminder that most of a turbulent flow, by volume, is being strained rather than spun.

The vorticity’s preferred axis #

The third figure is the other signature. Vorticity in turbulence is not aligned with the direction of greatest stretching, as a naive picture of spin being wound up would suggest, but preferentially with the middle axis, the one whose strain is weakest. The restricted Euler equation predicts that alignment exactly, at its singularity, which was one of its triumphs. The closure keeps it without the singularity: the vorticity’s mean squared cosine with the middle axis is 0.51 against a third for no preference, with the most stretching axis 0.28 and with the most compressing 0.20. Simulations of isotropic turbulence give much the same ordering.

A shorter memory is a higher Reynolds number #

The memory $\Gamma$ is the ratio of the Kolmogorov time to the large-eddy time, and it falls as the square root of the Reynolds number, so running the ensemble at shorter memories is running it at higher Reynolds numbers. The fourth figure shows two statistics of a longitudinal gradient — a velocity’s derivative along its own direction — against the memory. The flatness, which measures how heavy the distribution’s tails are, grows from 2.9 at a memory of three-tenths to about five at five-hundredths: the gradients become more intermittent, with rarer and more violent events, as the Reynolds number rises, which is what real turbulence does.

The skewness stays near −0.3 at every memory. Real turbulence has a skewness near −0.5, nearly independent of the Reynolds number, and it is not a detail: the skewness is proportional to the rate at which the gradients produce vorticity by stretching, which is the energy cascade’s own signature in a single derivative — the small-separation end of the four-fifths law. The closure produces too little of it, and at the shortest memories the vorticity’s alignment with the middle axis overshoots too, to 0.70 at four-hundredths. The model is known to lose realism as the Reynolds number rises, and these are the places it shows.

Why one Kolmogorov time #

The closure’s one choice is how far back the parcel’s memory reaches, and the choice is the Kolmogorov time for a physical reason. The pressure Hessian at a point is set by the gradients in a neighbourhood a few Kolmogorov lengths across, and a neighbourhood that size is reorganised by its own strain in about a Kolmogorov time; measured along a parcel’s path, the pressure Hessian stops resembling its earlier self on that time scale. A memory much shorter would give the restricted equation back, and its blow-up; a memory much longer would impose an isotropy from coordinates the neighbourhood no longer has.

The memory also sets the model’s Reynolds number, which is the part that makes it more than a device. The large eddies force the gradient on the large-eddy time; the remembered pressure and viscosity act on the Kolmogorov time; and their ratio is the only number in the problem. That the flatness grows as the ratio shrinks is therefore a prediction, not a fit: the intermittency of the small scales grows with the Reynolds number in the model for the same reason it grows in real turbulence, because the gradients have longer to be driven between the forcing’s kicks and the memory’s corrections.

The identity the closure cannot keep #

The fifth figure is the most instructive failure. In any homogeneous turbulence the mean squared strain rate is exactly half the mean squared vorticity — the average of $Q$ is zero, because $Q$ is the divergence of a vector and the average of a divergence over a homogeneous field vanishes. No closure can be right about turbulence and wrong about that. The ensemble’s ratio sits between 1.07 and 1.23, too strained at every memory.

The reason is structural, and it is the same reason the restricted equation needed a closure at all. The identity is a statement about many parcels at once: it holds because neighbouring gradients are tied together by the velocity field they share. A single parcel’s equation, however clever its pressure, knows nothing of its neighbours, and nothing in it forces their average to come out right. The remembered pressure supplies the local effect of the neighbours — redirection of the strain — and not the global constraint they impose. The part of the pressure it still misses is the part that is genuinely non-local.

Memory, and a closure with none #

It is worth setting this closure beside a closure with no memory at all, which found the simplest model of turbulent stresses failing precisely because it responds to the present strain and not to its history. The two are the same lesson at two scales. At the scale of the mean flow a closure without memory gets the stresses wrong wherever the strain changes quickly; at the scale of a single parcel’s gradient, a closure without memory — the restricted equation — blows up, and one Kolmogorov time of memory is enough to prevent it. In both, the history of the deformation carries information the present state does not.

The comparison also says how a closure should be judged. Any closure is a statement about what a simpler equation has thrown away and what replaces it, and what averaging costs set out why no closure is derived: it is chosen, and then tested against the statistics it was not built to reproduce. The remembered pressure was built to stop a blow-up. That it also gives the teardrop, the alignment and a flatness that grows with Reynolds number is evidence it has caught something real about the neighbours’ effect; that it misses the skewness and the homogeneity identity is evidence of what it has not. Both halves are information, and a closure that matched every statistic it was tested on would be one that had not been tested hard enough.

What was checked #

The sixth figure is the ledger. The matrix exponential of a symmetric matrix agrees with the one built from its eigen-decomposition to three parts in $10^{15}$. The closure’s right-hand side keeps a trace-free gradient trace-free to two parts in $10^{16}$ across fifty random gradients, as incompressibility requires. With no memory it equals the restricted Euler right-hand side minus the gradient, exactly. And two ensembles with independent noise, eighty parcels each, give skewnesses of −0.33 and −0.36 — the size of the sampling error on every statistic quoted.

What the ensemble cannot show #

Space. Each parcel is alone; the ensemble is many independent parcels, not a flow. Nothing here has a spectrum, an eddy or a neighbour.

The forcing is an assumption. The large eddies are a Gaussian noise of fixed strength, uncorrelated in time, acting on both strain and rotation equally; the gradients’ statistics depend on that choice, and the forcing of real small scales by large ones is neither Gaussian nor white.

The deformation is estimated, not followed. The Cauchy–Green tensor is computed from the present gradient as though it had been constant for a Kolmogorov time. A parcel whose gradient has just changed sharply is given the wrong history.

Statistics from finite ensembles. Every moment carries a sampling error of a few hundredths, and the flatness, which weights the tails, more.

The convention the numbers depend on #

Time is in large-eddy times; $\Gamma$ is the Kolmogorov time over the large-eddy time. $Q = -\tfrac12\mathrm{tr}A^2$ and $R = -\tfrac13\mathrm{tr}A^3$, scaled by the mean enstrophy $\langle Q_w\rangle = \langle\omega^2\rangle/4$. The skewness and flatness are of the diagonal components of the gradient — longitudinal derivatives — pooled over the three directions. Alignments are mean squared cosines between the vorticity and the unit eigenvectors of the strain-rate tensor, ordered from most stretching to most compressing.

Who found it, and when #

Vieillefosse found the restricted Euler equation’s singularity in 1982 and Cantwell mapped its solutions in 1992. The alignment of vorticity with the intermediate strain axis was found in simulations by Ashurst, Kerstein, Kerr and Gibson in 1987. Chevillard and Meneveau proposed the recent-fluid-deformation closure in 2006, and showed it reproducing the teardrop and the alignment and losing realism at high Reynolds number; Meneveau’s review of 2011 sets it among the other closures of the gradient equation.

Still open: memory that is followed, not estimated #

The closure estimates the parcel’s recent deformation as though its present gradient had been acting for a Kolmogorov time, which is the approximation that lets a single parcel be followed at all — and it is exactly wrong for a parcel whose gradient has just changed. The next calculation follows the deformation instead: each parcel carries its own Cauchy–Green tensor, integrated along its path with the past gradients and relaxed over a Kolmogorov time, so that the pressure really does remember. It asks whether a genuine memory raises the skewness towards −0.5 and brings the strain-to-vorticity ratio back to one, or whether those two misses are the neighbours’ and no memory of one parcel’s own history can supply them.