Steady does not mean nothing is happening

A flow is called steady when nothing at any fixed point changes with time. Put a probe in the stream, leave it there, and the reading never moves. Photograph the whole field twice, an hour apart, and the two photographs are identical.

It is very easy to read that as nothing is happening, and almost everybody does at first. It is not what it says.

The bands show how hard the fluid is being accelerated. They are largest right where the flow is most obviously doing something — at the nose, where the oncoming air has to stop, and round the shoulders, where it has to be turned. The two small holes on the horizontal axis are the stagnation points, and they are holes because the acceleration there is genuinely zero.

So the field is unchanging and the fluid in it is not. Reconciling those two facts is the material derivative, and it is the piece of notation that separates people who can do fluid mechanics from people who have read about it.

Two different questions #

The confusion comes from two ways of asking what the flow is doing, and they have different answers.

The first fixes attention on a point in space and asks how the velocity there changes with time. This is what a probe bolted to a wind-tunnel wall measures, and it is what “steady” is about. Write it ∂u/∂t. In a steady flow it is exactly zero, everywhere, forever.

The second fixes attention on a parcel of fluid and asks how its velocity changes as it travels. This is what a small drifting balloon would measure, and it is what Newton’s second law is about, because it is a parcel that has mass and a parcel that is being pushed on. Write it Du/Dt.

These are not the same quantity and there is no reason they should be. A probe at the nose of the cylinder reads a constant zero; the parcel that arrives at that probe has just been decelerated from the full free-stream speed. The probe is right and the parcel is right.

The everyday version of this is a river running into a narrows. Standing on the bank, the water at any chosen spot always moves at the same speed; the gauge never changes. A leaf dropped in from upstream is nonetheless accelerated as it enters the narrows and decelerated as it leaves. Nobody finds that puzzling about the leaf. It becomes puzzling only when the same situation is written down with a partial derivative in it and the derivative is read as though it were the whole story.

Where the difference comes from #

The bridge between the two is short enough to write in one line, and worth deriving rather than quoting.

A parcel’s velocity depends on where it is, and where it is depends on time. So the rate of change of its velocity has two sources: the field may be changing under it, and the parcel may be moving to a place where the field is different. Chain rule:

$$\frac{D\mathbf{u}}{Dt} = \frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u}\cdot\nabla)\mathbf{u}$$

The first term is the local rate of change — the probe’s answer. The second is the convective term, and it is the one that survives when the flow is steady. It says the parcel is accelerating because it is being carried into a region where the velocity is different, not because the velocity anywhere has changed.

That term is also the reason fluid mechanics is hard. It is nonlinear: the velocity appears twice, multiplied by itself. Almost every difficulty in the subject — turbulence, the impossibility of general solutions, the fact that adding two flows together only works when that term has been thrown away — traces back to it.

Its size is worth a moment. In the flow drawn above, with the free-stream speed set to one and the cylinder radius set to one, the convective acceleration reaches about 4.4 in the same units. That is not a small correction to anything; it is the entire dynamics. A treatment that dropped it would not be an approximation to this flow, it would be a description of a different one — a fluid drifting past a cylinder without noticing it was there.

There is a second thing the term explains, and it is the reason the sums in this subject so rarely come out in closed form. Because the velocity multiplies its own derivative, doubling the free-stream speed does not double the answer to anything; it quadruples the accelerations and therefore the pressure differences, which is why every pressure coefficient in aerodynamics is defined by dividing out a ½ρU². The definition looks like a convention and is actually the nonlinearity being tidied away so that the remaining numbers can be compared between speeds.

Something has to be doing the pushing #

A parcel that accelerates has been pushed, and in a fluid with no viscosity there is exactly one candidate: a difference in pressure between one side of it and the other. So wherever the first figure is dark, the pressure field must have a gradient, and the two pictures ought to be readable against each other.

They match, and the match is not a coincidence but the momentum equation with the viscous and unsteady terms deleted:

$$(\mathbf{u}\cdot\nabla)\mathbf{u} = -\frac{1}{\rho}\nabla p$$

Reading it left to right, it says a parcel carried into a faster region must have been pushed there by a pressure that falls in that direction. Reading it right to left, it says any pressure difference in a steady inviscid flow is accompanied by an acceleration. There is no third possibility: a steady flow with pressure differences and no acceleration would be a fluid with a net force on it going nowhere.

This is worth holding on to, because the popular account of fast air and low pressure usually presents the two as simply correlated — where it is fast it is low — without saying which causes which or what the mechanism is. The mechanism is this equation, and it makes the causal direction a matter of choice: the pressure field and the velocity field are two descriptions of one arrangement, and neither is prior to the other.

What the solver computed, and how it was checked #

The acceleration field above is not sketched. At each sample point the exact velocity field of the cylinder solution is differentiated by central differences and contracted with the velocity itself, which is the convective term written out.

Checking a field of derivatives is harder than checking a field of velocities, because there is no obvious thing it has to equal. The check used here comes from a property that has to hold and was not built in.

For a flow that is steady and irrotational, the convective acceleration is the gradient of half the speed squared. That means its line integral along any path is just the difference in ½q² between the ends. So the acceleration field can be integrated along a path, the speeds at the two ends can be measured separately, and the two numbers have to agree.

Along the streamline running into the nose, integrating the computed acceleration gives −0.394 31, and the change in ½q² between the same two points is −0.394 31. They agree to five figures, and nothing in the code makes them.

That identity has a name. It is Bernoulli’s equation, arrived at from the other end: not postulated as a conservation law but obtained by adding up the work the acceleration does along the way. The site’s other essay on it, where Bernoulli applies, approaches it as a hypothesis with conditions attached; this is the same statement produced by integration.

The check has one property worth stating, because it is what makes it a check rather than a tautology. It walks an arbitrary path, not a streamline. Along a streamline the identity holds even for a rotational flow, because the term it drops is perpendicular to the velocity — so a test that only ever walked streamlines would pass on solid-body rotation, which is precisely the case it most needs to refuse. Handed a radial path through solid-body rotation, it refuses.

The two frames have names #

The distinction is old enough to have picked up labels, and they are used inconsistently enough to be worth pinning down.

Watching fixed points is the Eulerian description. Watching parcels is the Lagrangian one. Both names are slightly wrong historically — Euler used both and Lagrange introduced neither — which is normal for this subject.

Almost all of fluid mechanics is done in the Eulerian frame, and for a good reason: the Lagrangian one requires keeping track of where every parcel started, and in any flow more interesting than a straight line, parcels that began as neighbours do not stay neighbours. A field defined on fixed coordinates does not have that problem. The price is the convective term, and it is a price worth paying.

The exception is where the parcel’s history is the answer. Whether a pollutant reaches a town, whether a fuel droplet has had time to evaporate, how long a parcel of air spends in the hot part of a combustor — these are Lagrangian questions, and answering them from an Eulerian field means integrating along paths, which is exactly what the second figure above does.

The parcel that never arrives #

There is one consequence of the two frames disagreeing that is worth working out in full, because it is exact, it is startling, and it is invisible in every Eulerian quantity on this page.

Follow a parcel down the axis towards the nose. As it approaches, the velocity ahead of it falls off in proportion to how far it still has to go — that is what a smooth field with a zero in it looks like from close up. So its distance to the stagnation point obeys a rate of change proportional to that distance, which is exponential decay, and exponential decay never reaches zero.

A fluid particle on the stagnation streamline takes infinite time to arrive at the body. Nothing in the field is singular; the velocity is finite everywhere, the acceleration is finite everywhere, and the picture is perfectly ordinary. The transit time diverges anyway, and it diverges logarithmically, which is slow enough that the divergence is easy to miss and certain enough that no numerical integrator ever gets there.

The same argument at the other end says the parcels leaving the rear stagnation point have been in transit forever. So the dividing streamline drawn round the body in every figure of this collection is a curve no particle of fluid traverses, in either direction, at any speed. It is a boundary between two families of paths rather than a path.

Two practical readings follow. A total-pressure probe works precisely because of this: the fluid it samples has had unlimited time to decelerate, so the deceleration is complete and the reading is the genuine stagnation value.

And anything that does strike the nose of a body — a raindrop, a supercooled droplet, an insect — got there by not following the flow. Only something with enough inertia to leave its streamline can arrive at all, which is why an aircraft’s leading edge is where the ice forms and the massless tracer is not.

What the picture cannot show #

The first figure has a limitation that is easy to miss, and it is the same limitation the whole Eulerian frame has.

It shows the magnitude of the acceleration and not its direction. A parcel at the nose is being decelerated — pushed backwards against its own motion — and a parcel at the shoulder is being turned almost entirely sideways, with very little change in speed at all. Both appear as bands of similar weight. Two completely different mechanical situations, drawn identically.

That is not laziness in the figure. It is what happens whenever a vector field is reduced to a scalar for drawing, and a reader looking at any contoured quantity should ask what was thrown away to make it contourable. Here it was the whole direction.

The second figure fixes it for one parcel and only one. Speed and acceleration are plotted against distance travelled, so a reader can see the parcel slow, accelerate, and slow again — but only along the single streamline that was chosen, and the choice was made by the author rather than by the physics.

Where the model stops #

Everything above is exact for this flow, which is the ideal one: inviscid, irrotational, incompressible, steady. Each of those does work.

Drop steady and the local term comes back, and the two contributions can be comparable or can cancel. A wing pitching rapidly has a flow where the local term dominates and where quantities derived on steady assumptions — including the lift curve — are quantitatively wrong.

Drop irrotational and the neat identity used for the check no longer holds off a streamline, because the discarded term u × ω is no longer zero. Bernoulli then applies along each streamline separately with a different constant on each, which is the distinction most misapplications of it fail to notice.

Drop inviscid and there is a further force on the parcel that has nothing to do with pressure, and the acceleration inside the boundary layer is dominated by it. The figure above shows nothing at all about the last millimetre next to the surface, where the parcel is being decelerated by friction rather than by pressure.

The generalisation #

The material derivative is not about velocity. It applies to any quantity carried by the fluid.

Temperature, for instance: DT/Dt is the rate at which a parcel’s temperature changes, and it splits the same way into a local term and a convective one. A steady temperature field with a flow running through it heats and cools parcels continuously while every thermometer in it reads a constant value. The same is true of concentration, of salinity, of anything a fluid carries.

This is the structure behind an observation that seems unrelated. A river is at a steady temperature at every point along it, and yet the water in it is being warmed as it travels. Both are true, and someone who has only the Eulerian picture will find the second surprising.

There is a connection here worth noticing, and it is the surprise this essay is built around. The convective term is the only reason a steady flow can do work on the fluid in it. In a genuinely uniform stream the term vanishes, nothing accelerates, and no pressure differences are needed anywhere. Every pressure difference in this subject — including the one that holds a wing up — exists because parcels are being carried into regions where the velocity differs from where they came from. Lift is a convective-term phenomenon.

Who found it, and when #

The chain-rule decomposition is Euler’s, in the 1750s, and it appears in the paper where the equations of inviscid flow were first written down. It arrived before the word “acceleration” had settled into its modern meaning and before anybody had a reason to care about the distinction, which is a common shape for a good idea.

The notation D/Dt is much later — nineteenth century, and variously attributed — and the phrase “material derivative” later still. Stokes called it the derivative “following the motion of the fluid”, which is longer and clearer than either.

What is striking is how long it took for the distinction to become standard teaching. Well into the twentieth century, textbooks were writing “the flow is steady, therefore the acceleration is zero” — a statement that is wrong in a way that does not show up until somebody asks what force is holding the fluid on its curved path round the shoulder of a cylinder. The answer is a pressure gradient, and the pressure gradient is there precisely because the acceleration is not zero.

Where the ladder goes next #

Next rungs on this anchor: the acceleration of a parcel in an unsteady flow, where the two terms can be made to cancel exactly and the parcel travels in a straight line through a field that curves; the rotation rate of a parcel, which is vorticity and which splits from the deformation rate in the same way this splits from the local rate; and the strain rate, which is what a viscous stress actually responds to.

Then across to what a flow is, which is the assumption that lets a parcel be spoken of at all, and to mass conservation, which is the other statement about parcels that the whole subject rests on.