The line the dye actually draws

The rung below this one set out three curves, said correctly that the third is what nearly every published flow photograph actually shows, and then drew the first two.

The reason was not editorial. The routine that draws that family keeps a list of what it knows how to make, the list has two entries in it, and the router that dispatches to it has three — so asking for a streakline threw an error at the point of use. The essay was written round a gap in its own machinery, and it is a good example of the class: an option that reads as supported, refuses when called, and is never called because the essays that would have called it are the ones not yet written.

A flow chosen so that nothing has to be integrated #

The field used here is uniform in space and periodic in time:

$$u = U_0 + A\sin\omega t, \qquad v = V_0 + B\cos\omega t.$$

That looks like an evasion and is the opposite of one. It is the simplest field that is genuinely unsteady — unsteady in the strong sense, since whether a flow is steady is partly a choice of frame and no frame makes this one steady — it has every property the argument needs, and — the point — every curve through it can be integrated by hand:

$$x(t;s) = x_0 + U_0(t-s) - \frac{A}{\omega}\left(\cos\omega t - \cos\omega s\right),$$

with the same shape for $y$. So a streakline here is an expression rather than the output of a stepper. Its self-intersections are exact rather than apparent, its closure is exact rather than within a tolerance, and the assertions behind these figures compare a computed curve against a formula rather than against a finer version of itself. That last distinction is the one that matters: comparing a solver with a finer run of the same solver tests the step size and not the method.

The price is that the field has no spatial structure at all, so it has no body in it, no boundary layer and no vorticity. What it has is time dependence, and time dependence is the entire subject. It is also, for the same reason, a flow in which mass has nowhere to go trivially: a field that is the same everywhere has no divergence to speak of.

What a streakline is parameterised by #

The three curves are usually distinguished by a sentence each, and the sentences make them sound like three examples of one thing. They are not, and the difference is visible in what runs along each.

A streamline is parameterised by arc length at a fixed instant. A pathline is parameterised by time, for a fixed particle. A streakline is parameterised by release time, at a fixed instant, for a fixed place — and release time is not a coordinate of the flow, it is a coordinate of the experiment.

That is why a streakline is a record rather than a solution. Nothing satisfies a differential equation along it. The curve is an assembly of the current positions of a set of particles that have nothing in common except where they entered, and its shape encodes the history of the flow at that one point over the interval it spans — which makes it a relative of a scalar field, whose value is a record of where its fluid was rather than of what the velocity is now.

And it therefore has a length in time as well as in space. A streakline is not a curve until somebody says how long the dye has been running, and the same experiment with the injector opened five seconds earlier gives a longer curve with a different shape at the far end. Neither a streamline nor a pathline has a parameter like that: the first is fixed by the instant and the second by the particle.

Crossing itself, which a streamline cannot do #

The clearest way to see that a streakline is a different kind of object is to make it do something the other curves are forbidden to.

A streamline cannot cross itself, and the reason is a single sentence: two tangent directions at one point would require the velocity to have two values there. A pathline in a steady flow cannot cross itself either, for the same reason applied at the same instant. Both are curves of the field, and the field is single-valued.

A streakline is not a curve of the field, so nothing stops it. Where it crosses, two particles released at different times happen to be in the same place at once — which is impossible for material points and perfectly possible for a dye filament, because a filament is a tube of finite width and the two pieces are at different depths, or have mixed. The crossing is a fact about the picture and not about the fluid — the same distinction the count on a pattern turns on, where what is being counted belongs to the drawing rather than to the flow, and it is exactly the feature that makes a smoke photograph of an oscillating flow hard to read.

The condition for it is arithmetic rather than judgement. The filament folds when the fluid reverses, the fluid reverses when the oscillation’s amplitude exceeds the mean, and both quantities are printed in the figure. Below that threshold the curve is single-valued in $x$ and a reader may treat it as a line; above it, a reader tracing the curve from one end is tracing backwards in release time through regions they have already passed.

The case that separates the three completely #

The demonstration worth the essay is the one with no mean flow at all.

Set $U_0 = V_0 = 0$. Then a particle released at time $s$ has

$$x(t) = \frac{A}{\omega}\left(\cos\omega s - \cos\omega t\right), \qquad y(t) = \frac{B}{\omega}\left(\sin\omega t - \sin\omega s\right),$$

which is an ellipse of semi-axes $A/\omega$ and $B/\omega$ centred at a point that depends on when it was let go. Every particle in the flow traces a congruent ellipse, and each one closes: after one period every particle is exactly back where it started, to fifteen decimal places rather than nearly.

The streakline at time $t$, obtained by varying $s$ instead of $t$, is an ellipse of the same two semi-axes centred at yet another place. So here are three statements that are all true at once:

• every pathline is a closed curve of a certain size and shape;
• the streakline is a closed curve of exactly that size and shape;
• no particle anywhere in the flow is travelling along the streakline.

The last one is checked rather than asserted. At the instant drawn, the velocity is the same everywhere, so every particle sitting on the streakline is moving in one fixed direction; the streakline’s own tangent varies round the ellipse; and the two agree at the two points where a closed curve must be tangent to any fixed direction and nowhere else. The assertion counts the agreements and refuses if there are more than the geometry requires.

A curve made entirely of particles, none of which is going along it. That is as sharp as the distinction gets, and it cannot be made in a steady flow, where the three curves coincide and the question has no content.

What this says about a photograph #

The rung below listed which laboratory technique produces which curve. This rung can say what follows for a reader trying to get a number out of one.

A streakline’s shape is not the flow’s shape, it is the flow’s history at a point. Two experiments with identical instantaneous velocity fields and different pasts give different streaklines, so a streak photograph carries information the field at that instant does not contain — which is genuinely useful, and is not what the picture is usually read for.

Its length is set by the operator. How much of the curve exists is how long the injector has been open, which means the visual weight of a feature in a smoke photograph is partly a decision somebody made about a valve. Every instrument does this to some degree — the exposure is chosen and rarely stated — and this one does it to the extent of the curve rather than to its sharpness.

And a curve that crosses itself is not a defect in the visualisation. It is the honest output of the technique in a flow that reverses, and reading it as a tangle to be tidied up loses the one feature that says the flow reversed. A photograph showing something smooth proves nothing, and this is the converse: a photograph showing something messy may be proving a great deal.

There is a fourth case worth naming because it is where the confusion does real damage. In a flow that is nearly steady — a wake shedding slowly, a slowly accelerating tunnel — the streakline and the streamline are close, and the temptation is to treat the difference as experimental error. It is not error; it is a measurement of the unsteadiness, and its size is the flow’s timescale divided by the release interval, which is a reduced frequency and behaves like every other reduced frequency in this collection.

The measurement a streakline is genuinely good for #

A curve that is the wrong tool for reading a velocity field is the right tool for something, and it is worth saying what, because the streakline’s reputation as the mislabelled curve undersells it.

It measures the history at a point, which nothing else does. The velocity field at an instant contains no information about what the flow was doing a second ago. A streakline released from a fixed probe contains exactly that information, encoded along its length, with the oldest fluid at the far end and the newest at the source. Reading it backwards from the far end is reading the record forwards in time.

The arithmetic of that reading is short in this flow and is the same in any flow. A point of the streakline at parameter $s$ is a particle that has been travelling for $t - s$, so the position of that point is the integral of the velocity the particle has met over that interval. Differentiating along the curve with respect to release time gives, after one cancellation, the difference between the velocity now at that point and the velocity at the source when the particle left. So the streakline’s own tangent carries a comparison between two instants — which is a strange quantity, and is why the curve is hard to read, and is also exactly the quantity a history contains.

The practical use is the frequency. Count the folds along a streak photograph and divide by the release interval, and the flow’s own oscillation frequency comes out without measuring a velocity anywhere. That is how a shedding frequency was measured before hot wires, and it is still how an unsteadiness is first noticed in a tunnel: somebody opens a smoke valve and the filament visibly wiggles at a rate a person can count.

And it is the only one of the four a person can produce with a match. The technique needs a source of marked fluid and no instrumentation at all, which is why it is old, why it is everywhere, and why the curve it makes is the one the subject’s iconography is built from.

Why the confusion is durable #

Three reasons, and they are worth separating because only one of them is a mistake.

Steady flows are the ones people photograph. A tunnel is run at a fixed speed and a picture is taken when the flow has settled, and in a steady flow the three curves genuinely coincide. So the overwhelming majority of streak photographs are correctly readable as streamlines, and a reader who treats them that way is right nearly all the time. A habit that works nearly always is very hard to break with an argument.

The failure is not visible in the picture. An unsteady streak photograph looks like a steady one: smooth curves, plausibly arranged, with nothing in the frame announcing that the flow moved while the plate was exposed. There is no artefact to notice.

And the vocabulary encourages it. “Streamline” is the word in general use and it does duty for all three, so a caption reading “streamlines over a cylinder” is not usually a claim about which curve was produced. It is a claim that the picture is of a flow.

The one of those that is a genuine mistake is the second, and the repair is a habit rather than an argument: before reading a flow photograph as a field, ask what the flow’s own timescale was and how long the exposure lasted. If the ratio is not small, the picture is a record and not a snapshot.

Where the four curves stand after this #

It is worth putting the family in one place, since this rung has changed what the collection can draw.

Streamline. Tangent to the velocity at one instant. In this field, a straight line always, because the velocity has no spatial dependence — which is a good check on the reader’s understanding: the streamlines of a wildly unsteady flow can be as simple as it is possible for a curve to be.

Pathline. One particle’s history. Closed ellipses here, and the curve a parcel followed round a flow traces.

Streakline. Everything released so far from one place, now. The subject of this essay.

Timeline. Everything that was on one line at one moment, now. Not drawn here, and it is the one of the four that behaves least like the others — a material line carried by a flow is a curve whose distortion is the measurement rather than its position.

Which of the four is a property of the flow #

There is a classification worth making once, because it separates the four curves by something more useful than how they are produced.

A streamline is a property of the field. Give somebody the velocity field at one instant and they can draw every streamline; nothing else is needed. It contains no history and no experiment.

A pathline is a property of the field and one initial condition. Give the field over an interval and a starting point, and the curve follows. Two of them from different starting points are different curves of the same flow.

A streakline is a property of the field, a place and an interval. Three inputs, of which two are decisions somebody made about apparatus — where the injector is and how long it has been open.

And a timeline is a property of the field and a whole line of initial conditions, which is the most of any of them.

That ordering is the useful thing. The further down the list a curve is, the more of the picture is a statement about the experiment rather than about the flow — and the more information the picture contains that the instantaneous field does not. Those are the same property seen from two sides, and which one it looks like depends on what is being asked.

A reader trying to recover the field from a picture wants the top of the list and is frustrated by the bottom. A reader trying to recover the flow’s history wants the bottom and gets nothing from the top. Almost every complaint about flow visualisation in this collection is one of those two readers holding the other’s picture.

The frame, one more time #

The rung below has a section observing that steady is a choice of frame, and this rung’s machinery makes a sharper version of that available which is worth spending a paragraph on.

Take the oscillating stream and change to a frame moving at the mean velocity $(U_0, V_0)$. The field becomes $u = A\sin\omega t$, $v = B\cos\omega t$ — no mean at all — which is exactly the case whose streakline closes into an ellipse.

So the closed streakline and the drifting one are the same flow in two frames. Every curve in the figures changes: the pathlines go from cycloids to closed ellipses, the streakline from a wave to a closed loop, and the streamline’s direction is different at every instant.

What does not change is that the three are different from each other. A frame change is a change of what each curve looks like and not of whether they coincide — because coincidence requires the flow to be steady, and no frame makes this one steady.

That is the precise statement the rung below’s section is reaching for. Steadiness is frame-dependent; the distinction between the curves is not. A flow that is unsteady in every frame has four genuinely different curve families in every frame, and a flow that is steady in some frame has them coinciding in that frame and separating in every other.

And it identifies the case the whole ladder is easiest to get wrong on: a travelling wave. A wave moving at a constant speed is unsteady in the laboratory and steady in a frame moving with it, so its streaklines and streamlines differ in the photograph and coincide in the analysis — which is why a picture of a wave and a calculation of one so often disagree about what the flow is doing.

What the picture cannot show #

No fluid dynamics. The field is prescribed and satisfies no equation of motion. It is divergence-free trivially, being uniform in space, and it is irrotational for the same reason, so nothing about vorticity, pressure or viscosity can be read from any of it.

No spatial structure. There is no body, no wall and no shear, which means the one thing this flow cannot demonstrate is the case where the field varies and the curves separate because of that rather than because of time. That case needs an integrator and the integrator carries an error, which is exactly the trade this essay declined.

A filament has a width and this curve has none. A real dye line diffuses, so its far end — the oldest fluid — is broad and faint while its near end is sharp. The curves here are of zero thickness and the age structure along them is invisible, which loses the one cue an experimenter uses to tell which end is which.

And the crossings are not events. Two pieces of a real filament arriving at one place mix, or pass at different depths, and either way the subsequent history is not what these curves show. The model is kinematic and its particles are permitted to occupy the same point.

Who separated them, and when #

The three-way distinction is old and its clean statement is usually credited to the nineteenth-century tradition that produced the rest of the kinematics on this site — the vocabulary is settled by the time of Lamb’s Hydrodynamics. What is more recent is the recognition that the distinction is routinely lost in practice, which came with photography: a technique that produces streaklines cheaply, in quantity, and with captions written by whoever developed the plate.

The closed-streakline example has the flavour of a textbook exercise and is one, appearing in various forms in kinematics chapters since at least the 1960s. Its point is not usually pressed as far as it goes. The version normally set asks the student to show that the three curves differ; the version here asks which particle is travelling along the streakline, and the answer is none of them.

Where the ladder goes next #

The rung above is the fourth curve, and it is the one that behaves least like the other three. A timeline is a set of particles that lay on a straight line at one instant, and what it measures is not where fluid went but how fast neighbouring fluid was going relative to its neighbour — a gradient rather than a trajectory. It is the technique behind hydrogen-bubble visualisation, it draws a boundary-layer profile directly, and it comes with two systematic errors that are inside the photograph rather than beside it.

The one beside it is the instrument question this essay keeps touching and does not settle: given a photograph and no knowledge of how it was made, what can be recovered? The answer turns on the exposure relative to the flow’s own timescale, and it is the same ratio that decides whether a particle image is a point or a streak.