A parcel goes straight while the streamlines curve

Steady does not mean nothing is happening took the flow past a cylinder, a picture that never changes, and found every parcel in it being braked, hauled round and braked again. The acceleration a parcel feels has two halves — the rate at which the field changes at a fixed point, and the rate at which the parcel carries itself to a place where the field is different — and in a steady flow the first is zero and the second does all of it.

That essay named the opposite case as the next question on its list: an unsteady flow in which the two halves cancel exactly, so that every parcel travels in a straight line at a constant speed while the field it travels through changes and its streamlines curve. How long the fluid has been in there took the acceleration in another direction, to the age a parcel carries. This essay builds the cancelling flow, and asks what it costs — because a parcel that does not accelerate is a parcel no force acts on, and that is a demand on the whole field.

A pattern that slides past #

The simplest such flow is a uniform stream carrying a pattern with it. Let the fluid move across the page at a speed $U$, and let it also carry a transverse velocity that varies along the stream, $v = g(x - Ut)$. The pattern of $v$ slides along at exactly the speed of the fluid carrying it.

Every parcel then keeps its transverse velocity for ever: whatever $v$ it started with is the $v$ of the part of the pattern it is riding on, and that part of the pattern moves with it. So every parcel moves in a straight line, at a constant velocity $(U, g(x_0))$ set by where it started. The flow is incompressible, since $\partial u/\partial x + \partial v/\partial y = 0 + 0$, and it satisfies Euler’s equations exactly with no pressure gradient at all. Its two accelerations are $\partial v/\partial t = -U g'$ and $u\,\partial v/\partial x = U g'$: equal and opposite at every point and every instant.

The first figure draws it with $U = 1$ and $g = 0.6\sin$. The streamlines at an instant — the curves everywhere tangent to the velocity at that instant — are waves, rising and falling with slopes as large as 0.6. The paths of five parcels over the next six time units are straight lines, some rising, some falling, one level. Integrated step by step through the changing field by fourth-order Runge–Kutta, with no knowledge of the answer, every path stays on its straight line to a part in $10^{13}$.

This is the sharpest form of what streamlines are not paths established. There the two sets of curves differed in an unsteady flow; here they differ as much as they can. No parcel ever follows a streamline for longer than an instant, and the curvature of every streamline is the curvature of a pattern moving past, not of anything moving along it.

Taylor’s frozen pattern, made exact #

The swept wave is a familiar idea in an exact form. Every measurement of turbulence from a fixed probe relies on Taylor’s frozen-turbulence hypothesis: that the pattern of eddies is carried past the probe by the mean wind faster than it changes, so a record in time can be read as a picture in space. The span that takes away the infinity used it to turn a gust spectrum along a flight path into one across a wing. In a real turbulent flow the hypothesis is an approximation, because the eddies’ own velocities make them accelerate and change as they go. The swept wave is the case in which it is exact: the transverse velocity is a pattern frozen into the fluid and carried at exactly the fluid’s speed, and nothing in it evolves.

That tells in both directions. Where Taylor’s hypothesis holds exactly, the parcels do not accelerate, and the pressure has nothing to do; where a real flow’s pressure field is doing work — sweeping small eddies round large ones, stretching and turning them — the hypothesis fails by exactly that much. The frozen pattern is the pressure-free limit of the thing every hot-wire record assumes.

Two accelerations, each large, that sum to nothing #

The second figure follows one parcel and measures the two halves separately. The local rate at which the transverse velocity changes at the parcel’s position swings between $\pm 0.6$; the convective rate swings between $\mp 0.6$; the sum, computed by central differences on the field, is zero to six parts in $10^{12}$ of either. Both halves are large and neither is an approximation: a probe held at a fixed point would record the velocity oscillating, and a probe dragged along the stream at speed $U$ would record it constant.

The faint curve is the contrast that makes the point. Freeze the same pattern in place — a steady flow with wavy streamlines — and a parcel released in it follows a streamline, since in a steady flow the two sets of curves coincide. Its local acceleration is zero and its convective one is not: it is thrown from side to side along the wave, which is the steady essay’s situation exactly. The same instantaneous picture carries parcels on wavy paths if it stands still and on straight ones if it moves with the stream.

What cancelling costs: no pressure anywhere #

A parcel that does not accelerate feels no net force. With no viscosity and no gravity the only force is the pressure gradient, so a flow in which every parcel keeps its velocity is a flow with no pressure gradient anywhere. That is a strong demand, because pressure in an incompressible flow is not a free field: pressure has no speed — it is decided everywhere at once by the velocity field, through a Poisson equation whose source is the velocity gradient’s square,

$$\nabla^2 p = -\rho\,\mathrm{tr}(A^2), \qquad A = \nabla\mathbf u.$$

For a two-dimensional incompressible flow the gradient $A$ has no trace, and a traceless $2\times2$ matrix has $\mathrm{tr}(A^2) = -2\det A$ — an identity checked here on two hundred random matrices to rounding. So the pressure’s Laplacian is $2\rho\det A$, which in terms of the local strain rate $s$ and vorticity $\omega$ is $\rho(\omega^2 - 4s^2)/2$. A pressure-free flow needs this to vanish at every point: the rotation and the strain must balance exactly, everywhere.

The third figure is the map. Where spin wins over strain the pressure’s Laplacian is positive — a pressure minimum, as in a vortex’s core. Where strain wins it is negative — a saddle of pressure, as at a stagnation point. Only on the diagonal can a flow be pressure-free, and the diagonal is a pure shear: a velocity gradient with one nonzero entry, whose strain and rotation are equal. Every point of the swept wave lies on it, since its only gradient is $\partial v/\partial x$.

That is the whole family, in two dimensions. A pressure-free incompressible flow is a pure shear at every point, oriented however the flow likes and varying from place to place as long as the pressures it would need all cancel. Such a flow’s gradient is nilpotent — its square is zero — and so the Jacobian of its flow map, $\det(I + tA)$ for a gradient that is constant along each path, is one for all time. The paths never converge and never cross. This is also the boundary on which every snapshot says vortex draws its line: the criterion that sorts a flow into rotating and straining regions by the sign of the same determinant puts a pure shear exactly on the fence.

A cloud that can compress pays later #

Remove incompressibility and the demand disappears. A cloud of spray, a stream of dust, a dilute gas of particles that do not collide has no pressure to speak of, and there the same rule — every parcel keeps its velocity — holds with any initial velocity field at all. The flow map is $x = a + t\,u_0(a)$: the initial field carried along straight lines.

The fourth figure is the one-dimensional version. Parcels start evenly spaced with velocity $-A\sin a$, so the parcels near $a = 0$ are converging fastest. Every path is straight. The spacing between neighbours shrinks as $1 - tA\cos a$, and at $t^* = 1/A$ it reaches zero at $a = 0$: two neighbouring parcels arrive at the same place at the same time, the density there is infinite, and the paths cross. It is a caustic. After it the cloud is three streams deep over a widening interval — at each point, three parcels with three different velocities — and no single-valued velocity field describes it any more.

The Jacobian that vanishes is not an abstraction. Mass conservation says that a parcel’s density is its starting density divided by the factor by which the flow map has stretched it, so the density of the cloud is $\rho_0/\det F$ — mass that has nowhere to go piling up where the map squeezes. In the incompressible swept wave the factor is one for all time and the density never changes; in the cloud it falls to zero along a curve, and the density along that curve goes to infinity. The caustic is where the Lagrangian description — follow each parcel — is still perfectly well behaved while the Eulerian one — a velocity at each point — has broken, since at a caustic two parcels arrive at one point with two velocities.

That is the price the compressible pressure-free flow pays for its freedom, and it is the same event every compression that becomes a shock meets in a gas — the characteristics crossing — except that in a gas the pressure intervenes first and turns the crossing into a shock, and in a cloud nothing does.

In two dimensions the cloud folds into sheets #

The fifth figure takes the cloud into two dimensions, with an initial velocity made from six seeded waves. The map’s Jacobian is now $\det(I + tA_0)$, and it first vanishes at $t^* = -1/\lambda_{\min}$, where $\lambda_{\min}$ is the most negative eigenvalue of the initial velocity gradient anywhere in the square — the place where the cloud is converging fastest along its fastest-converging direction. Found by searching the eigenvalues on one grid, the first caustic is at 8.37 time units; found independently, by differencing the flow map itself on a finer, offset grid and marching until its Jacobian first touches zero, it is at 8.37 again, to four parts in $10^4$.

At nine-tenths of that time the parcels are visibly bunching into bands. At one and a half times it they have crossed along curves and piled into thin, dense sheets with emptier cells between them. Zel’dovich proposed exactly this, in 1970, as the first stage of the cosmic web: matter falling freely in the expanding universe, with gravity supplying the initial velocities and nothing supplying a pressure, collapses first into sheets — his “pancakes” — along the directions of fastest convergence, and later into filaments where sheets cross.

Three flows, one rule #

The three settings share one statement and differ in what they do with it. A parcel keeps its velocity exactly where the pressure gradient vanishes, and whether a whole flow can do that depends on whether the fluid is allowed to compress:

• Incompressible: only as a pure shear at every point — a strong restriction, satisfied by the swept wave, by any steady parallel flow, and by little else. The parcels’ paths are straight and never cross, and the flow can last for ever.
• Compressible but pressure-free: with any initial velocity, freely, until the first caustic at $-1/\lambda_{\min}$.
• A real gas: the pressure is not zero, the parcels do accelerate, and a converging flow steepens into a shock before its characteristics cross.

One viscous flow belongs on the diagonal too, and it is the most familiar shear of all. In plane Couette flow — fluid between two plates, one sliding — the velocity rises linearly across the gap, the gradient is a single constant entry, and it is a pure shear exactly. Its viscous stress is the same everywhere, so it exerts no net force on any parcel; its pressure is uniform; and every parcel moves in a straight line at a constant speed. It is a solution of the Navier–Stokes equations as well as Euler’s, and it is why the film that heats itself could begin from a shear whose kinematics are trivial and put all its difficulty into the heat. Adding a pressure gradient to drive the flow — the parabolic profile of a channel — keeps every point on the diagonal, since a parabolic profile is still a pure shear, and that is the useful warning: the diagonal is necessary and not sufficient. The Poisson equation fixes the pressure’s Laplacian, and a uniform pressure gradient has none. In a channel the parcels still do not accelerate, because viscosity balances the gradient; in an ideal fluid the same uniform gradient would accelerate every parcel equally, and the swept wave’s pattern would be carried along by a stream that was speeding up.

A shower’s spray, a meteor stream, a cloud of sand in a gust and the large-scale matter of the universe are all close to the middle case for part of their lives, and all of them form the caustics that case predicts.

What was checked #

The ledger holds four checks: the swept wave’s local and convective accelerations cancelling to $6\times10^{-12}$ of either; paths integrated through its changing field staying straight to $10^{-13}$; the traceless identity on two hundred random gradients; and the two-dimensional cloud’s first caustic found by its own map’s Jacobian within $4\times10^{-4}$ of the eigenvalue formula.

What pressure-free flow leaves out #

Viscosity. A viscous fluid has a stress besides pressure, and a pure shear carries one; the swept wave is an exact solution of Euler’s equations, not of the Navier–Stokes equations, and viscosity would smooth its pattern while it slid.

Collisions and interaction. A real cloud’s particles collide, drag on the air around them and, in the cosmological case, attract one another. The caustic is the point at which the pressure-free model breaks, not the point at which the physical cloud does anything infinite: collisions or a gas pressure take over near it.

Three dimensions. The incompressible restriction in three dimensions is weaker than pure shear — a traceless $3\times3$ gradient can square to a matrix of zero trace without being nilpotent — and the family of three-dimensional pressure-free incompressible flows is correspondingly richer.

The convention: the material derivative #

The acceleration of a parcel is the material derivative, $D\mathbf u/Dt = \partial\mathbf u/\partial t + (\mathbf u\cdot\nabla)\mathbf u$, the local rate plus the convective rate. The swept wave’s lengths are in units of its wavelength over $2\pi$ and its times in units of that length over the stream speed. $A$ is the velocity gradient, $A_{ij} = \partial u_i/\partial x_j$; $s$ is the largest principal strain rate and $\omega$ the vorticity.

Who worked it out #

Euler’s equations and the split of the acceleration into its local and convective parts are Euler’s, from 1757. The pressure Poisson equation’s source as the invariant of the velocity gradient, and the division of a flow into strain-dominated and rotation-dominated regions by its sign, were made into a tool by Okubo in 1970 and Weiss in 1991. The free-streaming cloud and its caustics are the approximation Zel’dovich published in 1970 for the formation of large-scale structure; the same crossing of characteristics is Riemann’s of 1860 in a gas.

Still open: a cloud that feels the air it moves through #

The pressure-free cloud crossed itself because nothing resisted the convergence. A cloud of droplets or sand is not free: each particle drags on the air, relaxing towards the air’s velocity over its own response time, and the air, if it is still, pulls every particle towards rest. The next calculation gives the cloud a linear drag with a stated response time and asks when that is enough to prevent the caustic altogether — the answer should be a number of the kind the Stokes number measures, the response time against $-1/\lambda_{\min}$ — and, where it is not enough, how much later and how much less dense the first sheet forms.