Inertia

Pull a canoe paddle hard through the water and lift it out, and the stroke leaves two small whirlpools on the surface where it passed. They turn opposite ways and glide off side by side in the direction the paddle pushed, though the paddle is already out of the water. A smoke ring or a ring of steam that flies on for a while does the same. Moving water keeps moving unless something stops it (inertia), and as it moves it carries its motion somewhere else. That is why, in chapter 0, the water kept moving for a while after the stirring stopped. This chapter is about the last term of the equation, the one by which a flow carries itself along. It is the term highlighted in the equation bar.

Velocity carried along

In chapter 1, the acceleration $\mat{D\mathbf{u}/Dt}$ seen by following one parcel of water (chapter 1’s fluid particle) was the sum of two parts: the change seen from a fixed point, $\partial\mathbf{u}/\partial t$, and the change the parcel meets by moving to where the velocity is different.

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

With no pressure, no viscosity and no outside force, nothing pushes the parcel, so it goes on with the velocity it has: the left-hand side is zero. Yet from a fixed point the velocity there changes, because parcels with other velocities flow in.

$$\frac{\partial \mathbf{u}}{\partial t} = -\adv{(\mathbf{u}\cdot\nabla)\mathbf{u}}$$

The flow carries velocity along, and the velocity that was over there arrives here. That is what the advection term does. The velocity appears in it twice: the velocity does the carrying, and the velocity is what is carried. That is why, as chapter 1 said, the term is nonlinear, and why a flow can change its own shape as it carries itself along.

The experiment below holds a pair of vortices turning opposite ways: the upper one anticlockwise, the lower one clockwise. Gold dye marks the two vortices, and the dots are particles carried by the flow. The Advection term switch turns this term off and on.

Side note

Chapter 1 said that chapter 5 would switch this term on alone. Strictly, the pressure projection stays on too. Without it the water would crowd together in places and thin out in others, the fake fluid of chapter 2’s experiment with the projection off. The pressure only keeps the water from being compressed; the advection term alone moves the pattern of swirling. The viscosity is zero and the outside force is off, so you cannot stir this one. The stage’s left and right edges join, so a pair that leaves on the right comes back on the left. Above and below are slip walls, 0.4 from the vortices. Each vortex has a circulation of 0.1, and the centres are 0.2 apart (dimensionless values). In a browser without WebGPU, a recorded video plays instead, and the video cannot switch the advection term.

Nothing pushes the pair, and still it moves to the right. The water between the two vortices runs to the right, dragged that way by the upper vortex and by the lower one alike, and that stream carries each vortex: the upper one rides on the flow the lower one makes, and the lower one on the flow the upper one makes.

A vortex’s strength is its circulation $\Gamma$: the speed at which the water goes round a circle about the vortex, times the length of the circle (where the speed varies round the circle, take its average). Treat the vortex as a point with no size, and every circle about it has the same circulation, so the water a distance $r$ away goes round at $\Gamma/(2\pi r)$. Two vortices a distance $d$ apart each ride on the other’s flow at that speed, so the pair travels at:

$$V = \frac{\Gamma}{2\pi d}$$

In this experiment that is 0.08. The pair on screen is slower: between 1 and 4 seconds it covers about 0.06 a second, about three quarters of the estimate. The walls above and below hold it back, and the engine’s arithmetic spreads the vortices and slows it further (see the side note). Switch the advection term off and the pair stops where it is. The dye and the particles do not stop: water keeps streaming between the two vortices, and the dye winds into the vortices and stretches longer and longer. The pattern of velocity stays in place while the water flows through it. What has stopped is the moving of the pattern: the flow carrying its own velocity along. Switch the term back on and the pair moves on. In three dimensions the same thing happens in the shape of a ring, which is why a smoke ring flies along by itself.

Side note

Although the viscosity is zero, the pair is slower than the estimate, and it slows further as it goes, for two reasons. One is the walls above and below. Water cannot flow through a wall, so the flow is as if a mirror-image vortex, turning the other way, sat beyond each wall, and those images push the pair back. Between these walls even a pair of point vortices would travel at 0.87 of $\Gamma/(2\pi d)$ (the vortices’ size slows it by only 0.4 %). The other is the way the engine splits each step into stages. It first carries the velocity along with the flow (the advection step), then corrects it with pressure. In real water, pressure keeps bending the circling water round the circle, but the advection step carries each velocity along without turning it. So by the end of a step the velocity inside a vortex leans slightly outward, and when the pressure step removes that outward part, some of the swirl goes with it. The fast-turning middle loses most, so this numerical error, piling up step after step, spreads the vortex out. A longer step makes the error bigger; a finer grid hardly changes it. A single vortex on its own, which a flow without viscosity would leave unchanged, loses half the strength at its middle in 9 seconds with the advection step on, and none with it off. One with half the circulation, turning more slowly, loses only about a quarter of the strength at its middle in the same 9 seconds. In the pair, over the 12 seconds the advection term was on, the circulation fell by only 5 % but the strength at the middle of each vortex by about half, and a pair spread out like that moves more slowly: between 1 and 4 seconds it travelled at 0.74 of the estimate, and after the term came back on it moved about 15 % slower than at the start (while the term is off, nothing changes). These are readings on the author’s computer, on the 480×270 grid; on a phone’s 242×136 grid the speed was nearly the same (0.73).

Inertia against viscosity: the Reynolds number

Which is larger, the advection term, by which a flow carries itself along, or the viscosity term, which evens out velocity between neighbours (chapter 4), makes a great difference to the shape of a flow. Let us estimate the size of each. Call the flow’s typical speed $U$ and the distance over which the velocity changes a lot $L$. The velocity gradient is then about $U/L$, and the gradient of that gradient about $U/L^2$.

$$\left|\adv{(\mathbf{u}\cdot\nabla)\mathbf{u}}\right| \sim \frac{U^2}{L}$$ $$\left|\vis{\nu\nabla^2\mathbf{u}}\right| \sim \frac{\nu U}{L^2}$$

Their ratio is the Reynolds number.

$$\mathrm{Re} = \frac{U^2/L}{\nu U/L^2} = \frac{UL}{\nu}$$

A large Reynolds number means inertia shapes the flow; a small one means viscosity does. It is an estimate of size, not an exact value, and it changes with the speed and length you choose, so a Reynolds number always comes with the speed and the length behind it. For a cylinder, they are the speed of the oncoming water and the cylinder’s diameter. Take chapter 4’s channel: with its mean speed of 0.2, its width of 1 and its $\nu$ of 0.02, the Reynolds number is 10, a flow shaped by viscosity. Two bodies of the same shape, even of different sizes, in flows of the same Reynolds number see flows of the same shape. That is why a small model in a wind tunnel can stand in for a real aircraft. The experiment below keeps the speed and the diameter and changes only $\nu$, so that the Reynolds number climbs from 1 to 40, 150 and 1000. The Reynolds number switch picks a rung, and you can drag the mouse (or swipe a finger on a phone) to stir the water.

Side note

All values are dimensionless. The water comes in at the left at a speed $U$ of 0.5, and the cylinder’s diameter $D$ is 0.1; $\nu = UD/\mathrm{Re}$ sets each rung’s viscosity. Above and below are slip walls, and the cylinder blocks a tenth of the channel. The grid is 240×135 on every device, 13.5 cells across the diameter. At Re 1 the large $\nu$ makes $\nu\,\Delta t/h^2$ 15.2, so the engine solves the viscosity with 160 sweeps a step (20 from Re 40 on); the answer of such a step matched the answer of 3,000 sweeps within 0.011 % of $U$. The cylinder sits 0.002 above the middle of the stage. Set exactly on the middle line, the cylinder leaves the wake at Re 150 nearly symmetric for more than 10 seconds, where in a real flow small disturbances break the symmetry at once. The first two rungs show the flow with streamlines, the last two with the Vorticity layer, whose colour shows how fast the water spins in place and which way: orange water turns anticlockwise, sky blue clockwise. In a browser without WebGPU, a recorded video plays instead, and the video cannot pick a rung or be stirred.

At Re 1, viscosity wins. The flow stays on the cylinder, wraps round it and closes up behind it, and nothing runs backward. Front and back are not quite alike, though: the slowed water trails a long way behind. Along the middle line the water runs below 90 % of $U$ for 2.4 diameters ahead of the cylinder and for 7.1 diameters behind it. A little inertia is still there. The reach ahead is this short partly because the inlet, where the water comes in at $U$, is only 4 diameters ahead of the cylinder.

Side note

A flow with no inertia at all (a Stokes flow) would be the same in front and behind, but in the plane no such flow past a cylinder can be found: the cylinder’s influence reaches out without end. This is the Stokes paradox. In a real flow, far enough away, inertia competes with viscosity however small the Reynolds number, and that resolves the paradox. Still, the influence reaches far, so at Re 1 the flow reaches the walls above and below as well. On this stage the walls are 5 diameters from the cylinder’s centre and the inlet 4 diameters ahead of the cylinder, so the flow is not quite the one without walls or inlet.

At Re 40 the flow comes away from the cylinder on its rear flanks. Behind it a pocket forms that the water outside cannot enter, and inside it the water turns in two eddies, one above the other, in place. About 2 seconds after the step up the pocket has settled, and after that it hardly changes. The flow first comes away from a cylinder at a Reynolds number of about 5 to 7, and the two eddies stay in place until it reaches about 47. The pocket grows with the Reynolds number: at Re 40 without walls it reaches about 2.2 to 2.3 diameters behind the cylinder. In this experiment it reached 2.1, a little shorter, most likely because of the nearby walls and the coarse grid. The dark tail on screen, where the streamlines and the particles leave a gap, is much longer: behind the pocket the water still moves slowly for a while, and only the first 2.1 diameters behind the cylinder run backward.

At Re 150 the pocket stretches, starts to sway, and a few seconds later eddies break away from the top and the bottom in turn. They drift off in two staggered rows: a Kármán vortex street. The plot at the top right is the vertical speed at one point behind the cylinder, and it rises and falls once for each pair of eddies, one from each side, that passes. The number of swings a second $f$, combined with the diameter and the speed, is the Strouhal number.

$$\mathrm{St} = \frac{fD}{U}$$

In this experiment it was 0.177. Measured in real flows, it is about 0.18 at Re 150. Several things go into the small difference: the nearby walls, the coarse grid, and the engine’s numerical error, which acts like a little extra viscosity (the error of splitting each step into stages and the smearing by interpolation; see the appendix). The engine’s own check, on a finer grid (480×270) at Re 100, read 0.179, 9 % above the value of about 0.164 measured in real flows at that Reynolds number. Nearby walls raise the Strouhal number; numerical error that makes the flow act as if its Reynolds number were lower brings it down. This experiment did not measure the two apart.

At Re 1000 the eddies roll up tighter and closer to the cylinder, and drift off as distinct blobs. The flow on screen is still an even vortex street, with a Strouhal number of 0.196. Real water behind a cylinder at Re 1000 is not like this. A real flow starts to twist in three dimensions along the cylinder once the Reynolds number passes about 190, and at Re 1000, while eddies go on breaking away, the wake behind them is turbulent; the Strouhal number is about 0.2. This engine is two-dimensional and cannot hold that twisting, and its grid is too coarse to draw the thin layers that break away from the cylinder properly. The Re 1000 on screen is not turbulence. It shows only this much: as viscosity falls, the layers that break away get thinner and the eddies roll up closer to the cylinder.

The Reynolds numbers where a flow changes belong to its shape. The 47 and the 190 are the cylinder’s. Flow in a round pipe turns turbulent at a Reynolds number of about 2,000, made from the mean speed and the pipe’s diameter: another shape and another Reynolds number. A threshold of the Reynolds number always comes with the shape of the flow it belongs to.

Two layers sliding

In the sky, clouds sometimes curl over in a row like breaking waves where fast wind slides over slower air. Unless something holds it, the boundary between two layers sliding at different speeds cannot stay flat. In chapter 4’s first experiment, too, fast water slid over still water along the top and bottom edges of the middle layer. That layer was the same all the way from left to right, so the advection term had nothing to do there (chapter 4 had switched it off as well), and the layer only spread by viscosity. In the experiment below, the upper layer slides to the right and the lower one to the left, with a very small wave on the boundary between them, and this time every term but the outside force is on. Gold dye marks the lower layer. The Fluid switch picks water or honey.

Side note

The two layers move at 0.2 and −0.2, and the velocity between them changes along a tanh curve of half-thickness $\delta = 0.03$. A very small wave sits on the boundary at the start: a wave that rises and falls four times across the stage with a vertical speed of 0.003, mixed with a few other waves of up to three tenths of that size from a fixed random seed. The water’s and the honey’s $\nu$ are those of chapter 4’s first experiment, 0.00001 and 0.001. Made from half the difference in speed and the half-thickness, this layer’s Reynolds number is 600 in water and 6 in honey. The outside force is off, so you cannot stir it. Each pick of Fluid starts the two layers again from the same wave. The stage’s left and right edges join, and above and below are slip walls. In a browser without WebGPU, a recorded video plays instead, and the video cannot switch fluids.

In water the wave on the boundary grows on its own. Where the boundary bulges up, the upper layer carries it one way; where it sags down, the lower layer carries it the other way. So the wave tilts as it grows taller, and by about 3 seconds its crests topple over and roll up. While the wave is low, it grows by the same factor in each stretch of time. Once this experiment’s wave gets going, it grows about threefold every second; as it grows tall its growth slows, and at 5 seconds it is 41 times its first size. By then the boundary has rolled up into four vortices, one for each rise and fall the first wave made across the stage. A little later neighbouring vortices circle round each other and merge, and by about 12 seconds there are two. This rolling up of layers sliding at different speeds is called the Kelvin–Helmholtz instability.

Not every wave grows alike. Put twelve waves of one size on this experiment’s boundary together and measure them, and the ones 15 to 20 half-thicknesses long grow fastest, while those shorter than 8 half-thicknesses shrink instead. The theory of a tanh layer without viscosity puts the fastest-growing wave at about 14 half-thicknesses and the shortest wave that grows at $2\pi$ half-thicknesses. This experiment’s wave, close to that length, grew at 86 % of the rate the theory gives, and a wave a little longer than $2\pi$ half-thicknesses shrank. This experiment did not sort out what makes these differences. Viscosity alone cannot explain them: water’s is far too small to damp waves that fast. Since the fastest wave’s length goes with the layer’s thickness, a thinner layer rolls up into smaller, closer vortices.

In honey the same wave does not grow. Viscosity quickly evens out the difference in speed across the boundary, the layer thickens, and the wave on it shrinks away instead. Even after 8 seconds the boundary only thickens and smears; it does not roll up. The same two layers roll up at a Reynolds number of 600, and at 6 they do not within this time. Inertia against viscosity decides the shape of the flow here too.

One misconception

“Turbulence is just random,” people say. Turbulence is hard to predict and changes from moment to moment, but it does not move any old way. It holds eddies of many sizes, one inside another, and in three dimensions the large eddies give rise to smaller ones and hand their energy down to them. In a flow in the plane it goes the other way: neighbouring vortices merge and grow, as the four vortices of the sliding layers became two. Pairing is not limited to flows in the plane, though: in a real mixing layer, where two streams at different speeds meet and mix, the large eddies also grow by pairing up with their neighbours. Experiments in the 1970s showed that even when such a layer between two streams of gas is turbulent, a clear row of large rolled-up eddies stands out, just like the one in this chapter’s third experiment. What this chapter shows on screen is not turbulence but the structures a flow makes by itself on its way there: layers rolling up and eddies breaking away in turn. What makes those structures is the advection term, the flow carrying itself along.

With that, every term of the equation is in place. In chapter 6’s lab you switch every term on and off and make flows yourself.

In one sentence

The advection term $\adv{(\mathbf{u}\cdot\nabla)\mathbf{u}}$ is the flow carrying its own velocity along, and when the Reynolds number $UL/\nu$, which weighs it against the viscosity term, is large, inertia shapes the flow: a vortex pair moves on its own, eddies break away behind a cylinder, and sliding layers roll up.