An oscillation with somewhere to go

Stand a cylinder in a tank and shake the water back and forth past it, sinusoidally, with no net flow in either direction. Sprinkle in some fine powder and watch. Within a minute the powder has organised itself into four persistent patterns, arranged symmetrically about the body, and there is a steady circulation carrying it round. Nothing is driving that circulation: the forcing reverses every half cycle, and its average is exactly zero.

This is steady streaming, and the number at the bottom of it is three-quarters.

The first-order flow, which averages to nothing #

Outside the layer the flow is inviscid and oscillates as $U(x)\cos\omega t$, with $U(x)$ whatever the body’s shape makes it — $2U_0\sin\theta$ round a cylinder, from the ordinary potential solution. Inside the layer the no-slip condition has to be met, and the classical solution is the Stokes layer this collection solves elsewhere:

$$u_1 = U(x)\left[\cos\omega t - e^{-\eta}\cos(\omega t - \eta)\right], \qquad \eta = y\sqrt{\frac{\omega}{2\nu}}.$$

Its thickness is $\sqrt{2\nu/\omega}$ and contains no free-stream velocity at all — a millimetre in water at a hertz, a tenth of that at a hundred hertz. Its average over a cycle is zero at every height, exactly, because every term in it is a cosine or a sine of $\omega t$.

The second-order forcing, which does not #

Write $u_1/U = a(\eta)\cos\omega t + b(\eta)\sin\omega t$, with $a = 1 - e^{-\eta}\cos\eta$ and $b = -e^{-\eta}\sin\eta$. Continuity supplies the transverse velocity $v_1$, which is $-U'(x)$ times the running integral of the same pair. Then the mean of the advective term follows from products of cosines and sines, all of which are elementary:

$$\left\langle u_1\frac{\partial u_1}{\partial x} + v_1\frac{\partial u_1}{\partial y}\right\rangle = \frac{U U'}{2}\Big[(a^2 + b^2) - (A a' + B b')\Big],$$

with $A$ and $B$ the running integrals of $a$ and $b$. Nothing in that expression is zero anywhere in the layer. The streamwise fluctuation and the transverse velocity it drives are correlated, and a correlation of two zero-mean quantities need not be zero — which is the same statement as the missing term in an averaged momentum equation, in a setting where it can be integrated.

The steady flow it drives satisfies

$$\nu\frac{d^2\bar{u}}{dy^2} = \left\langle u_1\frac{\partial u_1}{\partial x} + v_1\frac{\partial u_1}{\partial y}\right\rangle - \left\langle U\frac{dU}{dx}\right\rangle,$$

and the second term on the right is the reason this calculation gives a finite answer at all: the forcing has to be measured relative to its value at the edge of the layer, where the outer flow’s own mean advection is balanced by the outer mean pressure gradient. Measured against zero instead, the forcing tends to a constant far from the wall, the integral diverges, and the answer is a multiple of however deep the integration was taken. That is the trap this site recorded on a developing duct — a residual measured against the wrong thing — and it is waiting in every balance that closes.

Integrating twice, and getting three-quarters #

Two boundary conditions close the problem: no slip at the wall, and no stress at the top of the layer, since the outer flow is inviscid. Integrating the forcing once from infinity inwards gives the gradient, and again from the wall outwards gives the velocity, and the result approaches a constant:

$$\bar{u}_{\text{slip}} = -\frac{3}{4\omega}\,U\frac{dU}{dx}.$$

The 3/4 is not put in anywhere. It comes out of the integral, at $-0.74999953$ with the layer taken to sixty thicknesses on forty thousand points, and it does not move when the layer is taken to ninety on sixty thousand — which is the check that it is the answer rather than the truncation.

The slip velocity is not a slip in the ordinary sense: the fluid does satisfy no slip at the wall, and what the number describes is the steady velocity at the top of the oscillatory layer, which the outer flow feels as a boundary condition. The layer is thin, so from outside it looks exactly like a wall that is quietly moving fluid along itself.

Four cells, from nothing going anywhere #

Apply it to a cylinder. The outer velocity is $U = 2U_0\sin\theta$, so $U\,dU/dx = (4U_0^2/a)\sin\theta\cos\theta = (2U_0^2/a)\sin 2\theta$, and the slip velocity therefore goes as $\sin 2\theta$: it changes sign four times round the body.

Those four cells are what the powder finds. They are steady, they persist as long as the shaking does, and their sense reverses if the body is made to oscillate in still fluid instead of the fluid past a fixed body — a distinction that catches people out and which the sign of $U'$ settles.

The pattern is the classical one from acoustics, where it is seen as Kundt’s dust figures: a standing sound wave in a tube collects powder into regularly spaced heaps, and the spacing is set by the wave and the transport by exactly this streaming. Rayleigh’s 1884 paper is about that experiment.

What the forcing looks like, and where it lives #

The forcing curve in the hero figure is worth reading rather than passing over. It is largest within the first two or three layer thicknesses of the wall, changes sign once, and has decayed to nothing by about five — so the whole of the streaming is generated inside a region a fraction of a millimetre thick in most laboratory cases, and everything outside that region is simply carried along by what happened inside it.

That is the same structure as the vorticity source at a wall: everything is made in the layer and the interior of the flow merely redistributes it. A calculation that does not resolve the layer will not produce the streaming at all, and no amount of refinement in the outer flow supplies it — which is a familiar difficulty with a familiar remedy, since the slip velocity above is precisely the modelled boundary condition that lets an outer calculation have the effect without the layer.

Why it matters where it matters #

Streaming is small — second order in the amplitude — and it dominates transport in any situation where the first-order flow has nowhere to take anything.

In acoustics and ultrasonics. Acoustic streaming stirs the fluid near a transducer and is one of the principal mechanisms of ultrasonic cleaning and of heat transfer enhancement by sound. The velocities are millimetres a second and they never reverse, which beats a metre-a-second oscillation that goes nowhere.

In micro- and bio-fluidics. At small scales inertia is weak and mixing is notoriously difficult, because Stokes flow is reversible and stirring undoes itself. Steady streaming from an oscillating bubble or membrane is one of the few mechanisms that produce a genuine mean flow there, and it is used deliberately in acoustofluidic mixers.

In heat and mass transfer at an oscillating surface. A vibrating cylinder in still air develops the same four cells, and the heat transfer from it rises accordingly — an effect large enough to be a design variable in some heat exchangers, and one that no calculation carrying only the oscillatory flow can produce.

And in the ocean’s bottom boundary layer, where waves oscillate over the seabed and the streaming carries sediment in a direction that has nothing to do with any current. There it competes with the Stokes drift of the wave itself, which is the inviscid member of the same family and points the other way near the bed — so the direction sand travels under a wave is decided by which of two second-order terms is larger.

Two second-order effects, and which one moves a particle #

Kundt’s heaps are worth taking apart, because two different second-order mechanisms are at work in that tube and they act on different things. Separating them settles a question that decides whether a modern acoustofluidic device works.

The streaming computed above acts on the fluid, and anything carried by the fluid goes with it. A particle small enough to follow the flow is simply advected round the cells, at the streaming velocity and in the streaming’s pattern.

There is a second effect that acts on the particle directly. A particle in a standing sound field scatters the wave, and the scattered and incident fields interact to leave a steady force — the acoustic radiation force — which drives it towards a node or an antinode depending on how its density and compressibility compare with the fluid’s. That force is quadratic in the acoustic amplitude, like the streaming, and it exists in a fluid with no viscosity at all.

The two compete, and the competition is decided by size. The radiation force scales with the particle’s volume, as $a^3$; the drag the streaming exerts on it is Stokes drag, which scales with its radius, as $a$. So the ratio of the two goes as $a^2$, and there is a critical particle size below which the streaming wins outright.

$$\frac{F_{\text{radiation}}}{F_{\text{streaming drag}}} \propto a^2 .$$

For water at the megahertz frequencies these devices use, that crossover lands at a diameter of a couple of microns. Above it, particles are focused into sharp lines at the nodes; below it, they are stirred round the streaming rolls and focus into nothing at all.

That single number is the reason acoustophoretic separation is a mature technique for cells — a red blood cell is seven or eight microns and sits comfortably on the radiation-force side — and a difficult one for bacteria, viruses and vesicles, which sit on the other. It is why the literature on sorting sub-micron particles is largely a literature about suppressing streaming: raising the frequency, which shrinks the layer that generates it; shaping the channel so the slip velocity’s $U\,dU/dx$ is small where it matters; or arranging temperature gradients whose own second-order flow opposes the acoustic one.

The same split explains Kundt’s tube itself. The dust that collects into heaps is being carried by the streaming, and its spacing reports the wave’s half-wavelength because that is the spacing of the streaming cells — which is set by the standing wave and not by anything about the dust. A tube run with much larger particles would sort them by the radiation force instead, into a different pattern in different places, from the same sound field.

Two mechanisms of the same order in amplitude, with different powers of the particle size, and an experiment that sees only one of them at a time. It is a good example of why second order is not a synonym for small: both effects are quadratic in a quantity whose mean is zero, both are the entire transport in their own regime, and which of them a laboratory reports is settled by a choice nobody thinks of as part of the physics.

The other Reynolds number in the problem #

The derivation above assumes the streaming is weak enough not to disturb the layer that produces it, and that assumption has its own parameter. The streaming Reynolds number is $\mathrm{Re}_s = U_0^2/(\nu\omega)$ — the ratio of the streaming velocity to the viscous velocity scale of the layer — and the four-cell picture holds only while it is small.

At larger values the outer streaming develops its own boundary layer, the cells change shape, and at larger values still the whole pattern reverses direction. That reversal is well documented experimentally and is not a small effect; it means the sign of the answer above is a statement about a regime rather than about geometry.

So the honest statement of this essay’s result is: the slip velocity is $-(3/4\omega)U\,dU/dx$ in the limit of small amplitude and small streaming Reynolds number, and the second of those conditions is easier to violate than it looks.

A number that survives every closure #

One feature of the result deserves emphasis because it is unusual in this part of the subject: the $3/4$ contains no empirical constant, no closure and no fitted coefficient. It is a consequence of the Navier–Stokes equations, the no-slip condition and an expansion in amplitude, and every step is exact.

That is rare enough to be worth comparing with its neighbours. The law of the wall’s $\kappa$ is measured and has never been derived; every eddy-viscosity model carries five constants from experiment; the transition Reynolds number is a correlation. The streaming coefficient is in the other category, along with the separation angle of a Falkner–Skan layer and the exact value of the four-fifths law — results that are numbers rather than fits, because the problem they answer is linear once the expansion is made.

The trade is generality for exactness. Everything derived here holds only at small amplitude, and within that restriction it is as solid as anything in the subject.

What the picture cannot show #

The layer is flat and the body’s curvature is neglected. The boundary-layer approximation is used throughout, so the results hold where the layer is thin compared with the body — which for a cylinder in water at a few hertz means a body of centimetres or more.

The outer streaming is not computed here. What is computed is the slip velocity that drives it; the actual cellular pattern outside requires solving a steady flow with that slip as its boundary condition, and its extent depends on the streaming Reynolds number. The four-cell figure is the sign of the driving round the body rather than a solved outer field.

The frequency is single and the forcing is sinusoidal. A real acoustic field or a real sea state contains many frequencies, and the second-order term then contains cross-products between them, which produce steady flows and also difference-frequency components that neither frequency has on its own. Nothing here handles a spectrum of forcing, and the arithmetic of doing so is the arithmetic of what a spectrum does and does not contain.

And the whole calculation is second order in the amplitude. Third-order terms carry corrections that grow with the stroke length, and at amplitudes where the fluid particle’s excursion is comparable with the body, the flow separates and sheds vortices — at which point everything here is replaced by a different and much less tidy problem.

An estimate, so the size is not a mystery #

Numbers make the effect concrete. Take water shaken past a centimetre cylinder at ten hertz with a free-stream amplitude of ten centimetres a second. The layer thickness is $\sqrt{2\nu/\omega} = 0.13$ mm. The outer speed round the body reaches $2U_0 = 0.2$ m/s and its gradient along the surface is of order $2U_0/a = 20\ \mathrm{s}^{-1}$, so

$$\bar{u}_{\text{slip}} \approx \frac{3}{4}\cdot\frac{0.2 \times 20}{63} \approx 0.05\ \text{m/s},$$

which is a quarter of the oscillation’s own peak speed and enormously more than any diffusion in the problem. The streaming Reynolds number is $U_0^2/\nu\omega = 0.01/(10^{-6}\times 63) \approx 160$, which is well past the small-parameter regime — so the four-cell picture is qualitatively right and the coefficient is not to be trusted at that amplitude.

That pair of numbers is the honest summary of the subject: the effect is easy to produce and hard to predict, because the regime in which it can be computed exactly is quieter than the regime in which anybody wants it.

Who found it, and when #

Rayleigh gave the $3/4$ in 1884, in the paper On the circulation of air observed in Kundt’s tubes, and his derivation is essentially the one above. Schlichting extended it to bodies of general shape in 1932, which is why the near-wall pattern is often called Schlichting streaming, with the outer cells called Rayleigh streaming. The large-streaming-Reynolds-number reversal was mapped by Riley and others from the 1960s.

The surprising connection is with the site’s account of what a boundary condition is. An oscillating outer flow imposes no mean condition on the fluid at all, and the fluid produces one anyway: the layer generates a mean slip that the outer flow must then satisfy. A boundary condition has been manufactured by the interior of a layer, out of a forcing whose average is zero, which is the neatest available demonstration that an averaged description carries structure the instantaneous one does not obviously contain.

Where the ladder goes next #

Beside this rung sits the inviscid member of the family — a wave’s Stokes drift, where the same second-order arithmetic moves parcels rather than driving a current — and the two together are the whole of what an oscillation with no mean can transport. Above it lies the streaming at large amplitude, where the cells reverse, which needs a solver this collection does not have.