2.1 Experimental set-up

To run our experiments, we manufactured a Plexiglas trough $L_{x}=190.5~\text{mm}$ long, $L_{y}=25.4~\text{mm}$ broad and $69.6~\text{mm}$ deep. An aluminium piece attached to the top couples the trough to an electromechanical shaker (Dynamic Systems VTS-80). A function waveform generator (Rigol DG-1022) and a bipolar amplifier (NF HSA-4011) generate a sinusoidal signal with tuneable amplitude and frequency which properly feeds the shaker. The acceleration of the system is precisely measured using an accelerometer (PCB Piezotronics 340A65) and a lock-in amplifier (Stanford Research SR830), synchronised with the shaker input signal. A second channel of the function waveform generator is used to create a synchronisation signal that triggers the imaging system at a programmable phase.

Figure 1.

Experimental set-up. A Plexiglas trough (a) is vertically driven by an electromagnetic shaker (b). A double-pulsed laser (c) and an optical array (d) generate a light sheet (e) which passes through the trough. A high-speed camera (f) and a long-pass filter (g) acquire the images for PIV.

The upside-down configuration chosen for the experiment allows the system to be illuminated from the bottom, which is the only way to avoid undesired light refractions for the PIV imaging. A scheme of the set-up is displayed in figure 1. Further details can be found in Gordillo & Mujica (Reference Gordillo and Mujica2014). The PIV imaging system consists of a double-pulsed Nd:YAG laser (Quantel Evergreen 70) and a high-speed camera (Phantom v641), properly synchronised. The laser beam passes through an optical system which projects a $2~\text{mm}$ thick laser sheet into the trough. The optical system, mounted on a translational stage, allows the position of the illuminated plane to be freely adjusted in the $y$ direction.

In all of our runs, we fill the trough up to $h=20~\text{mm}$ with an aqueous solution of potassium bromide (KBr). The concentration, $13.7\,\%$ , is such that the density of the solution matches the density of the seeding particles (Kanomax Fluostar, $\varnothing =15~\unicode[STIX]{x03BC}\text{m}$ ) i.e. $\unicode[STIX]{x1D70C}=1.1~\text{g}~\text{cm}^{-3}$ (cf. tables for aqueous solutions in Lide Reference Lide2004). The seeding particles are fluorescent, a feature that is used to create sharp images. A long-pass filter placed in front of the objective lens filters off any reflection of the laser sheet on the free surface, and only the light emitted by the fluorescent particles reaches the camera sensor. A small amount of wetting agent, $2~\text{ml}$ of Kodak Photoflo, is also added to the solution. The inclusion of this agent has been shown to improve wall wettability in this kind of system (Wu, Keolian & Rudnick Reference Wu, Keolian and Rudnick1984). Some ethyl alcohol (less than $1~\text{ml}$ ) was used to previously dissolve the PIV seeding particles into water, avoiding the formation of clusters. The mixture of seeding particles was then kept in a syringe and added manually to the water until the concentration of particles in images was suitable for PIV (four particles in an $8\times 8$  pixel window, Raffel et al. Reference Raffel, Willert, Wereley and Kompenhans2007).

Faraday waves are generated at the free surface of the fluid when the forcing acceleration $\unicode[STIX]{x1D6E4}$ is increased above a threshold $\unicode[STIX]{x1D6E4}_{0}$ , which is a function of the forcing frequency $2f$ . The waves are subharmonic, i.e. they oscillate at half of the forcing frequency $f$ . The wavelength $\unicode[STIX]{x1D706}$ and wavenumber $k=2\unicode[STIX]{x03C0}/\unicode[STIX]{x1D706}$ are related to the natural frequencies of the surface $f_{m,n}=\unicode[STIX]{x1D714}_{m,n}/(2\unicode[STIX]{x03C0})$ through the dispersion relation for closed basins,

The quantity $k$ is the modulus of the wavenumber $\boldsymbol{k}=(k_{x},k_{y})=(m\unicode[STIX]{x03C0}/L_{x},n\unicode[STIX]{x03C0}/L_{y})$ , and $m,n\in \mathbb{Z}$ are the numbers of nodes in each direction (for more details, cf. Miles Reference Miles1976). For a forcing frequency $2f$ close to $2f_{m,n}$ , the systems responds with a wavelength given by (2.1). The sign of the detuning $\unicode[STIX]{x0394}f=f-f_{m,n}$ defines whether the onset of the Faraday waves is subcritical ( $\unicode[STIX]{x0394}f<0$ ) or supercritical ( $\unicode[STIX]{x0394}f>0$ ), as shown by Douady (Reference Douady1990).

2.2 Observation of the streaming patterns

At first glance, the flow beneath the free surface is very simple: the fluid is carried from one crest to the neighbouring ones and then back through a cycle (Van Dyke Reference Van Dyke1982). However, particles do not come back exactly to the same position after each cycle. This fact can be easily observed when taking images stroboscopically. An overexposed image with strobe flashing shows the slow tracks followed by particles after several cycles (see figure 2). These tracks will be referred to as streaming patterns, and their study, as the streaming flow or velocity field related to it, is the main goal of this work.

Figure 2.

Streaming patterns beneath purely longitudinal Faraday waves for (a) $2f=7.6~\text{Hz}$ , (b) $2f=8.9~\text{Hz}$ , (c) $2f=9.9~\text{Hz}$ and (d) $2f=10.8~\text{Hz}$ in the midplane in the crosswise direction ( $y=L_{y}/2$ ). The images were obtained by strobe illumination. The number of nodes of the standing Faraday wave is respectively $m=4,5,6,7$ ( $n=0$ ). The surface-wave amplitude $\unicode[STIX]{x1D701}$ for each case is respectively (a) $3.1~\text{mm}$ , (b) $1.9~\text{mm}$ , (c)  $3.3~\text{mm}$ , (d) $3.4~\text{mm}$ .

Although the streaming patterns were observed indirectly by Faraday (Reference Faraday1831) through the accumulation of particles on the bottom of the trough, the first visualisation is due to Douady (Reference Douady1990), who used Kalliroscope particles for this purpose.

The streaming patterns are generated by a secondary flow which is very small compared with the main oscillatory flow. In our experiments, the streaming velocity field is typically around 20 times weaker than the oscillatory velocity field (Gordillo & Mujica Reference Gordillo and Mujica2014). Despite its weakness, the streaming patterns can be easily observed using a strobe light as a consequence of the fact that the main flow (the dominant velocity signal) is periodical whereas the streaming flow is mainly steady. Hence, the motion that particles undergo after a complete cycle due to the oscillatory field is zero, and only the contribution of the streaming velocity field is observed. It should be noted that particles under the free surface of a standing wave, which is the case of Faraday waves, have a Stokes drift (Stokes Reference Stokes1847) equal to zero after a complete cycle (Gordillo Reference Gordillo2012).

The streaming patterns also show a weak dependence on the phase at which the images are captured. Naturally, the patterns are distorted with the deformation of the free surface at a given phase. For the sake of simplicity, all of the quantitative measurements of the streaming field are performed at the phase at which the free surface is flat and only on longitudinal Faraday waves (no crosswise component, i.e. $n=0$ in (2.1)). For the same reason, all of the measurements shown here are restrained to the midplane in the $y$ direction.

The images in figure 2 also show that the structure of the streaming patterns is highly three-dimensional, even for Faraday waves with no crosswise component on the surface. Particles may enter and exit the illuminated plane, which explains the observed spiral trajectories followed by the particles.

2.3 Measurement protocol

To characterise the dependence of the streaming patterns on the amplitude and the frequency, we performed several experimental runs. The protocol was automatised using MATLAB, which controlled, via several interfaces (GPIB, USB, RS232 and Ethernet), all of the devices involved in the experiment. Each set of measurements analysed a single forcing frequency. The protocol was as follows. First, the amplitude was increased until stationary Faraday waves with large amplitude emerged. A downward-ramp scheme in amplitude was then followed. For each amplitude, we recorded a first series of 36 images, which completed a cycle, to determine the shape of the interface in terms of the phase (or time). Immediately afterwards, the shape of the free surface was detected using a standard algorithm for edge detection. After the flat-state phase was determined, the synchronisation signal was shifted and a second run consisting of 64 stroboscopic flat-state images was recorded. Measurements were made cyclically every four minutes, following the downward ramp until the Faraday waves on the surface vanished.

In table 1, we display a list of the experiments performed. In all cases, the wave at the surface was stationary and purely longitudinal. (The only slight deviation in the transverse direction is due to wetting between the fluid and the solid wall, which does not introduce a significant perturbation to the flow.) The runs include different frequencies and different wavenumbers. We also studied different nonlinear scenarios, i.e. subcritical and supercritical behaviour. Measurement ramps lasted between 1 and 4 h, depending on the initial amplitude and the ramp step size.

Table 1.

Set of measurements for the streaming patterns. Each line corresponds to a ramp in amplitude. The set includes four different wavelengths. Faraday waves destabilise through subcritical or supercritical bifurcations depending on the frequency detuning. Both subcritical and supercritical behaviours are explored. The last column shows the observed type of streaming pattern.

2.4 Experimental results

Our sequences of images were processed by means of two different techniques. In a very simple way, the sequences can be used to detect the trajectories followed by the seed particles. A synthetic overexposed image is obtained by maximising the grey value of each pixel along several frames. This is actually how the images in figure 2 were obtained. This strategy allows us to rapidly probe the streaming patterns and provides a powerful qualitative tool to analyse the data on the run. The second technique is PIV, which generates precise quantitative data. For this purpose, each sequence of images was analysed using our own PIV code. After preparing masks and background subtraction, we used time averaging in the correlation-function space to improve the signal-to-noise ratio (Meinhart, Wereley & Santiago Reference Meinhart, Wereley and Santiago2000). This allowed us to obtain neat velocity fields with a high spatial resolution ( $8\times 8$  pixel window size, equivalent to $0.64\times 0.64~\text{mm}^{2}$ resolution) and no spatial averaging (0 % overlapping).

Figure 3.

Velocity fields of the streaming patterns shown in figure 2 (increasing wavenumber from (a) to (d)). The background shows the shape of the interface $20\,\text{ms}$ after the flat phase. Panels (a,c,d) show ordered patterns with a spatial period equal to half of the surface waves. Panel (b) shows a distorted streaming pattern. The values for the wave amplitude $\unicode[STIX]{x1D701}$ are the same as in figure 2.

Figures 2 and 3 show respectively the streaming pattern and the velocity field $(u,w)$ in the $(x,z)$ -plane for four different forcing frequencies. The presented cases are a good representation of the whole set of collected data. In all of our measurements, we observe (i) a sequence of counter-rotating structures with a wavelength equal to half of the free-surface wavelength, (ii) that the velocity field strength decreases with depth and (iii) that the fluid is pushed downward at the free-surface antinodes and upward at the nodes. However, there are clear differences between the flow structures in the panels presented in figures 2 and 3. Based on our results and qualitative criteria, we identify three types of streaming pattern.

Type I:

sequence of counter-rotating rolls (c and d).

Type II:

sequence of moustache-shaped swirls (a).

Type III:

irregular patterns (b).

Although only types I and II show ordered structures, type III displays well-shaped rolls next to the lateral walls at $x=0$ and $x=L_{x}$ . Types I and II share another common feature: both display squeezed counter-rotating structures at the bottom.

To illustrate quantitatively the classification described above, we chose two quantities that characterise the state of the system. The first one is the peak amplitude $\unicode[STIX]{x1D701}$ of the Faraday wave and is used to quantify the state of the free surface. This quantity can be obtained from the sequence of images used for determining the flat state (see § 2.3). As the second parameter, intended to represent the state of the flow, we chose the amplitude (along the $x$ coordinate) of the vertically averaged velocity $w$ , i.e.

Since the lateral walls induce strong coherent structures for all of the types, the domain $D$ is chosen such that the first and last rolls are removed, i.e. $D:[\unicode[STIX]{x1D706}/4,L_{x}-\unicode[STIX]{x1D706}/4]$ . The quantity ${\hat{w}}$ can be easily obtained from PIV data (the discretised $\text{d}z$ is $0.64~\text{mm}$ ). Averaging along $z$ cancels noise induced by PIV detection.

Figure 4.

Bifurcation diagram for streaming patterns. The diagram shows all of the collected data which collapse into two branches. The upper branch contains type-I streaming patterns (roll structures) and the lower one type-II swirl patterns. Type-III (filled symbols) irregular patterns can only be observed for very low amplitudes.

The two chosen quantities are plotted against each other in figure 4 for all of our measurements. For values of $\unicode[STIX]{x1D701}$ below some threshold $\unicode[STIX]{x1D701}_{c}\approx 1.5~\text{mm}$ , ${\hat{w}}$ remains almost equal to zero, which is related to the absence of periodic structures. Beyond this threshold, ${\hat{w}}$ bifurcates into two branches, each containing a large number of collapsed measurement points. All type-I (rolls) streaming patterns collapse to the upper branch, while type-II (swirls) patterns collapse to the lower one. The upper branch displays ${\hat{w}}$ values twice as strong as than the lower one. Type-III (irregular) patterns can be observed only for low ${\hat{w}}$ values, which gives a hint to understand their existence. A reasonable explanation is that the coherent periodic structures induced by the Faraday waves are so weak that they compete with weak parasitic large-scale eddies. The latter are presumably generated by surface tension effects and persist for long times, even after the vibration is switched off.

Figure 4 reveals that subcritical or supercritical behaviour does not have any relevant effect on streaming patterns: series (c,d) form a continuous line for $m=6$ ; the same stands for series (e,f), where $m=7$ . More strikingly, type-I (rolls) and type-II (swirls) patterns may emerge regardless the value of $\unicode[STIX]{x1D706}$ . The evidence for $m=7$ is remarkable, because both types are observed with the same forcing parameters: type I, runs (e,f,h) and type II, runs (i,j). The mixture in the trough was the same and most of the runs were taken during the same day. We could not find a way to choose which streaming-flow pattern (I or II) was going to be selected during an experiment: the system spontaneously chooses one and keeps it until the wave on the surface vanishes. Multistability of viscous modes has been observed in other flows; e.g. Coles (Reference Coles1965) observed more than 20 different states in Taylor–Couette flow under the same forcing parameters.

As a closing for this section, we would like to readdress the effect of the streaming patterns at long time scales on particles in the liquid. Consider a particle subjected to the flow under Faraday waves. Besides the oscillating forces due to the rapidly oscillating velocity field, our particle is subjected to a viscous steady force due to the streaming flow. Hence, at long time scales, the particle will be slowly drifted by this force until it reaches a stagnation point. Particles heavier than the fluid will accumulate at stagnation points at the bottom of the trough, creating the heaps observed by Faraday in his seminal experiments. Particles lighter than the fluid will float at the free surface and accumulate into rafts, as has been shown in experiments by Falkovich et al. (Reference Falkovich, Weinberg, Denissenko and Lukaschuk2005) and Sanli et al. (Reference Sanli, Lohse and van der Meer2014). Streaming flows are thus a relevant extra ingredient to understand the motion induced on particles on Faraday waves. A step forward is to build a model able to predict the streaming patterns observed in our experiments. The next section is devoted to this task.

3.1 Viscous streaming due to oscillatory boundary layers

3.2 Streaming in the bulk

The final step is to determine how the streaming due to the boundary layers induces a streaming in the bulk. For this purpose, consider the general Navier–Stokes equation for the bulk in a comoving frame of reference,

where $a(t)=-g+\unicode[STIX]{x1D6E4}\cos 2\unicode[STIX]{x1D714}t$ is the vertical acceleration of the system. To simplify the notation, we omit the underline. We decompose the velocity and pressure field into oscillatory and steady terms, $\boldsymbol{u}(\boldsymbol{x},t)=\widetilde{\boldsymbol{u}}(\boldsymbol{x},t)+\overline{\boldsymbol{u}}(\boldsymbol{x})$ and $p(\boldsymbol{x},t)=\widetilde{p}(\boldsymbol{x},t)+\overline{p}(\boldsymbol{x})$ . For most Faraday-wave experiments, since $\unicode[STIX]{x1D706},h,L_{x},L_{y}\gg \unicode[STIX]{x1D6FF}$ , the flow under the waves is potential. Thus, if at dominant order $\widetilde{\boldsymbol{u}}$ scales as some small parameter $\unicode[STIX]{x1D716}\ll 1$ , the oscillatory vorticity will be a correction of higher order. Likewise, we expect that $\overline{\boldsymbol{u}}=O(\unicode[STIX]{x1D716}^{2})$ . To proceed, we write the convective term in (3.14) in terms of $\unicode[STIX]{x1D74E}$ , $(\boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D735})\boldsymbol{u}=\unicode[STIX]{x1D735}((1/2)\boldsymbol{u}^{2})-\boldsymbol{u}\times \unicode[STIX]{x1D74E}$ . By applying curl to (3.14), and selecting oscillatory terms, we obtain an equation for the oscillatory vorticity,

which shows that $\widetilde{\unicode[STIX]{x1D74E}}=O(\unicode[STIX]{x1D716}^{3})$ at leading order. On the other hand, time averaging of (3.14) leads to

where we have kept the term $\unicode[STIX]{x2202}_{t}\overline{\boldsymbol{u}}$ to account for a weak slow time dependence of the steady flow. It should be noted now that for standing waves,

up to order $O(\unicode[STIX]{x1D716}^{4})$ , since the temporal part of the integral does not yield steady terms. (In non-standing waves, the latter expression yields extra terms such as a Stokes drift, as shown in full detail by Nicolás & Vega (Reference Nicolás and Vega2003).) Hence, we obtain for the streaming field in the bulk

By defining $\overline{p}^{\prime }\equiv \overline{p}+\unicode[STIX]{x1D70C}gz+(1/2)\unicode[STIX]{x1D70C}\langle \tilde{\boldsymbol{u}}^{2}\rangle$ , the equation for the steady part of $\boldsymbol{u}$ becomes identical to a Navier–Stokes equation for a steady flow with no external forcing. The equation is equivalent to the one derived by Nicolás & Vega (Reference Nicolás and Vega2003) in the particular case of non-propagating waves. On the other hand, it can be easily shown that the incompressibility condition for the steady field is the usual one,

Equations (3.18) and (3.19) plus the boundary conditions found in §§ 3.1.1 and 3.1.2 contain all of the information required to determine the steady streaming in the bulk. Remarkably, the contribution due to the local oscillatory velocity field only appears in the effective pressure in (3.20) and has null effect on $\overline{\boldsymbol{u}}$ at leading order. The steady component of the flow stemming from the nonlinearities in the bulk is thus negligible in comparison with the streaming generated by the boundary layer. Henceforth, we expect that the boundary layers are mainly responsible for the appearance of streaming in quasi-inviscid stationary waves.

However, the framework we have presented still lacks an explanation to describe how the streaming flow evolves in time. For instance, the streaming field has non-vanishing vorticity: where does this vorticity come from? Batchelor (Reference Batchelor2000, see § 5.13, p. 361) gave an answer to this question: the steady streaming is only the final stage of a transient process, where vorticity is slowly transferred from the boundary layers into the volume. Considering that the transient process is much slower than any oscillating dynamics (e.g. the wave period or the transient to establish the surface wave), we can modify (3.18) to account for a slow temporal dependence of the streaming flow. The result is strikingly familiar,

which is analogous to a full Navier–Stokes equation. On the other hand, the incompressibility condition naturally remains unchanged during the transient process.

So far, we have shown that the evolution equation for the streaming velocity field for non-propagating waves can be solved at dominant order by using full Navier–Stokes equations for the steady fields (an apparently naïve choice, but not so). Although there is a modified pressure and different boundary conditions, their application remains stunningly simple.

As a final comment, equation (3.20) stands for the average Eulerian velocity. For standing waves, both the Eulerian and the Lagrangian velocity are equal in the bulk (see Batchelor Reference Batchelor2000, pp. 353–364). However, the same is not valid inside the boundary layers, as shown by (3.2) (cf. Longuet-Higgins Reference Longuet-Higgins1953, figure 5), which leads to the inversion of the streaming direction. A remarkable example of streaming inversion has been reported in air-driven Chladni patterns in van Gerner et al. (Reference van Gerner, van der Weele, van der Hoef and van der Meer2011).

3.3 Classical Faraday waves

4.1 Numerical methods

To run simulations of the streaming flow inside the incompressible vibrating fluid, we implemented our own code. The code directly integrates the Navier–Stokes equations with the divergence-free condition (3.20) in a parallelepiped domain delimited at the top by the interface of the fluid at rest. The flat top interface is a fair approximation for the time-averaged position of low-amplitude Faraday waves. In (3.20), the gravity and the vertical vibration are gathered with the pressure to form a time-dependent effective pressure and do not have any effect on the velocity fields since a single fluid of constant density is involved in this model. In the code, the fields are discretised by finite-difference schemes using the MAC (marker and cell) disposition on a regular staggered mesh. The pressure is computed at the centre of each cell and the components of the velocity are located at the faces corresponding to their direction. The incompressible Navier–Stokes equations are solved by a projection method (Chorin Reference Chorin1968) and the pressure is computed incrementally with a BiCGStab (stabilised biconjugate gradient) method. The spatial discretisation is made with standard centred schemes of second order except for the advection terms, for which ENO (essentially non-oscillatory) schemes are used. The temporal stepping involves a first-order forward Euler discretisation with adaptive time steps to ensure numerical stability.

The simulations use the equations from table 2 as matched boundary conditions, with the three quantities $k$ , $\unicode[STIX]{x1D714}$ and $A$ fixed. We recall that for longitudinal waves, the wavenumber is defined as $k=m\unicode[STIX]{x03C0}/L_{x}$ , with $m$ an integer, and $\unicode[STIX]{x1D714}$ is deduced from $k$ by means of (2.1) in consistency with the experimental observations. We use as control parameter the velocity-potential amplitude $A$ (see (3.21)), which is related to the amplitude of Faraday waves via

obtained by using the trivial dynamic boundary condition at the surface. Therefore, using (4.1) and the definitions for the Strouhal and Reynolds numbers from § 3, we can write the Reynolds number as $Re\equiv \unicode[STIX]{x1D706}\unicode[STIX]{x1D714}\unicode[STIX]{x1D701}/2\unicode[STIX]{x1D708}=\unicode[STIX]{x1D706}\unicode[STIX]{x1D714}^{2}A/2g\unicode[STIX]{x1D708}$ and the wave steepness as $St^{-1}\equiv \unicode[STIX]{x1D701}/\unicode[STIX]{x1D706}=\unicode[STIX]{x1D714}A/g\unicode[STIX]{x1D706}$ . We emphasise some features about the theoretical model and the simulations. First, the fact that the coefficients $k$ , $\unicode[STIX]{x1D714}$ and $A$ are set up manually renders the model and the simulations independent of the bifurcation criticality, a feature consistent with the experimental results. Second, all of the matched boundary conditions for the streaming flow are proportional to $A^{2}/\unicode[STIX]{x1D714}$ . Hence, cases with altered $\unicode[STIX]{x1D714}$ are also swept by varying $A$ ; the complete model could be rewritten as a function of the single variable $A^{\prime }=A/\sqrt{\unicode[STIX]{x1D714}}$ instead of $\unicode[STIX]{x1D714}$ and $A$ separately, reducing the dimensions of the parameter space.

Figure 5.

Streamlines in the $y$ midplane for $m=7$ with a clean liquid–air interface. The amplitude $A$ decreases from (a) to (d): (a) $35.5$ , (b) $20$ , (c) $11.3$ and (d) $0.2~\text{cm}^{2}~\text{s}^{-1}$ . Accordingly, the dimensionless parameters $(Re,St^{-1})$ for each case are (a) $12250,0.222$ , (b) $6890,0.125$ , (c) $3880,0.07$ and (d) $69,0.001$ .

Figure 6.

Streaming velocity field in the $y$ midplane for $m=7$ and varying $A$ . The interface is assumed to be clean. Amplitudes (as well as dimensionless parameters) are the same as in figure 5.

Figure 7.

Streamlines in the $y$ midplane for $m=7$ and varying $A$ . The interface is fully contaminated. Amplitudes and dimensionless parameters are the same as in figure 5.

Figure 8.

Streaming velocity field in the $y$ midplane for $m=7$ and varying $A$ . The interface is fully contaminated. Amplitudes and dimensionless parameters are the same as in figure 5.

4.2 Numerical results

In all of our simulations, we set the liquid properties and geometric parameters to match those in the experiment: the density $\unicode[STIX]{x1D70C}=1.1~\text{g}~\text{cm}^{-3}$ , the dynamic viscosity $\unicode[STIX]{x1D708}=9.09\times 10^{-3}~\text{cm}^{2}~\text{s}^{-1}$ , the dimensions of the fluid layer $L_{x}\times L_{y}\times h=(19.05\times 2.54\times 2)~\text{cm}^{3}$ and the number of nodes in the longitudinal direction $m=4,5,6,7$ . The corresponding frequencies of Faraday waves given by the dispersion relation (2.1) are $f=3.8$  Hz, 4.45 Hz, 4.95 Hz and 5.4 Hz respectively (the frequency of vibration of the container corresponds to $2f$ , as shown in table 1). The velocity-potential amplitude $A$ is varied between $0.2~\text{cm}^{2}~\text{s}^{-1}$ and $63~\text{cm}^{2}~\text{s}^{-1}$ , where the lowest bound is the value below which the streaming flow remains qualitatively unchanged and the upper bound is the value at which the code outcomes diverge. With these amplitudes, the Reynolds number ranges from $60$ to $2\times 10^{4}$ , and the steepness goes from $5\times 10^{-4}$ to $0.4$ (for specific values, see the caption to figure 5). All of our values largely meet the theoretical requirements of § 3, i.e. large $Re$ and small wave steepness $St^{-1}$ . In our simulations, we used a regular discretised mesh that contained $240\times 80\times 80$ cells. Simulations were performed with both clean and fully contaminated boundary conditions at the interface, cf. table 2.

In figures 5 and 6, we first show the streamlines and the associated streaming velocity field in the $y$ midplane. We specifically set $m=7$ , $f=5.4$ Hz on a clean interface as $A$ is varied. Likewise, in figures 7 and 8, we plot respectively the streamlines and the associated streaming velocity fields for the same values of $m$ and $\unicode[STIX]{x1D714},$ but for a fully contaminated liquid–air interface.

Comparison of figures 5 and 6 with 7 and 8 shows that contamination induces very unlike streaming flows at low amplitudes. The low-amplitude fully contaminated Faraday streaming flows are very similar to the type-I patterns (rolls) found in the experiments, with weak spiralling loops at the top of the domain and small recirculations at the bottom. The fact that the streaming flows do not form perfectly closed loops is related to the presence of transverse variations of the velocity component $v$ . The features of the flow are then expected to differ from those obtained with the two-dimensional theory, even on the $y$ midplane. The experimental patterns of type I (figures 2 and 3, patterns (c) and (d)) display the strongest similarities with the simulated fields of figures 7 and 8 at moderate amplitude $A=11.3~\text{cm}^{2}~\text{s}^{-1}$ . Therein, both the streamlines and the velocity fields have very similar shapes.

The clean-interface Faraday streaming flows look completely different from the fully contaminated ones and contrast with all of the patterns observed in the experiments. However, they display similarities to the type-II patterns (swirls), namely the presence of accumulation points in the vicinity of the interface, suggesting again that the flow is fairly three-dimensional on the $y$ midplane. In contrast, there are no loops at the top of the domain. Intermediate degrees of contamination on the interface, which yield matched boundary conditions between (3.31) and (3.32), have been introduced by Martin et al. (Reference Martin, Martel and Vega2002) for 2D flows and might improve the resemblance between experiments and simulations.

When the amplitude goes beyond $A\approx 25~\text{cm}^{2}~\text{s}^{-1}$ , ordered streaming structures destabilise, giving rise to unsteady patterns (see panel (a) in figures 5–8). The patterns oscillate on time scales that are still much slower than the oscillation period, in agreement with the hypotheses of § 3.2. Although a full description of the onset of this instability is out of the scope of this work, we point out that these dynamical regimes appear regardless of the interface contamination and for all of the wavenumbers that we explored.

On the other hand, varying the wavenumber $m$ in the boundary conditions does not have any dramatic effect on the flow structure. Besides the expected change of wavelength of the streaming flows that occurs, the loops at the walls are overcome by the central patterns as $m$ is decreased, particularly in the clean case.

To see whether the two-dimensional approximation used in previous works is good enough to explain the streaming field qualitatively, we exploit the three-dimensional data obtained from the numerical simulations. First, we compare the three-dimensional streaming patterns of our numerical simulations in the $y$ midplane (figures 5–8) with their two-dimensional counterparts (figure 9) at low amplitude, $A=0.2~\text{cm}^{2}~\text{s}^{-1}$ , and for $m=7$ . For this purpose, we ran a two-dimensional simplified version of the code following § 3.3.3. The main qualitative differences between the streaming fields of figures 5–8 and the two-dimensional ones of figure 9 stem from the fact that streamlines become perfectly closed loops when the flow is purely two-dimensional, as expected. The differences are localised near the walls of the domain and are especially striking in the clean case (compare figure 9 a with figures 5 d and 6 d). The differences remain visible when the interface is contaminated at the frontier between two superposed loops, but to a lesser extent (compare figure 9 b with figures 7 d and 8 d). Nevertheless, the main differences between the two-dimensional streaming flows and the three-dimensional ones are located in regions where the streaming velocity is weak.

Figure 9.

Velocity fields (arrows) and streamlines (contours) of the two-dimensional streaming flow for $m=7$ and $A=0.2~\text{cm}^{2}~\text{s}^{-1}$ ; (a) clean and (b) fully contaminated interface.

Figure 10.

Velocities of the streaming flow in the horizontal plane $z=-1$  cm for $m=7$ and $A=0.2~\text{cm}^{2}~\text{s}^{-1}$ . Arrows represent the horizontal velocity components ( $u,v$ ) while contours represent $w$ ; (a) clean and (b) fully contaminated interface.

In our three-dimensional simulations, the transverse effects are more visible in horizontal planes, for instance at $z=-1~\text{cm}$ , as shown in figure 10. They are highlighted by significant gradients of $w$ in the transverse direction. Likewise, perceptible deviations of the velocity from or towards the midplane $y=L_{y}/2$ are observed. The transverse effects are more marked near the walls and are exacerbated with increasing wave amplitude $A$ and decreasing trough width $L_{y}$ .

Further insights about three-dimensional effects can be gained by tracking small fluid parcels inside the bulk. For the sake of simplicity, we restrict ourselves to the lowest-amplitude patterns, $A=0.2~\text{cm}^{2}~\text{s}^{-1}$ . Figures 11 and 12 depict the trajectories of fluid parcels in a longitudinal plane at $x=9.8$ cm (close to the $x$ midplane) in the clean and fully contaminated cases respectively. On the other hand, close to the sides of the channel (not shown here), the trajectories are qualitatively similar to those shown with a contaminated interface (figure 12).

Figure 11.

Trajectories of fluid parcels starting from various points of the plane $x=9.80$  cm (central cell) for $m=7$ , $2f=10.8$  Hz and $A=0.2~\text{cm}^{2}~\text{s}^{-1}$ with a clean interface. (a) Viewed from the face $x=L_{x}$ . (b) Slanted view. Each colour denotes the trajectory of the same particle in plots (a) and (b). The starting points of the paths are represented by disks.

Figure 12.

Trajectories of fluid parcels starting from various points of the plane $x=9.80$  cm (central cell) for $m=7$ , $2f=10.8$  Hz and $A=0.2~\text{cm}^{2}~\text{s}^{-1}$ with a fully contaminated interface. Views and colour conventions are the same as in figure 11.

Figure 13.

Diagram analogous to figure 4, obtained in this case from numerical simulations: the vertical streaming velocity ${\hat{w}}$ , defined in (2.2), increases as a function of the wave amplitude $\unicode[STIX]{x1D701}$ . Red symbols represent Faraday streaming flows obtained with a fully contaminated interface, which share most properties of the type-I patterns (rolls). Black symbols stand for Faraday streaming flows obtained with a clean interface that are roughly similar to type-II patterns (swirls). The symbols $\times$ , ▿, $+$ and ○ correspond to $m=4$ , 5, 6 and 7 respectively. The same figure is shown in the inset with a log–log representation. The dashed line at the bottom of the inset has a slope of 2.

In both figures, the $y$ midplane, which is a symmetry plane of the problem, plays the role of separatrix between the two halves of the domain: fluid parcels starting within a half stay indefinitely there. The $x$ midplane plays the same role of separatrix. A more detailed inspection shows that all of the planes of the equation $x=m^{\prime }L_{x}/(2m)$ are separatrices too, $m^{\prime }$ being an integer, except those at $m^{\prime }=1$ and $m^{\prime }=m-1$ in the clean case. Hence, the motion of a fluid particle remains confined to small portions of a box of length $L_{x}/(2m)$ and width $L_{y}/2$ , depending on its starting point. The separatrices are also visible in figures 5–7 at the locations of vertical streamlines as well as in figure 10 separated by the orthogonal planes of null $u$ or $v$ components. It is important to emphasise that odd $m^{\prime }$ values correspond to nodes of the interface vibration whereas even $m^{\prime }$ values correspond to the antinodes. Outside the separatrices, the complexity of the trajectories is mainly illustrated by the paths in the upper part of the domain. For instance, the loops formed by the trajectories in figures 11 and 12 clearly differ: while clean conditions display loops parallel to the $x=0$ plane, fully contaminated conditions yield loops parallel to the $y=0$ plane. In the case of figure 11, the motion displays a transverse component stronger than the longitudinal one. This is due to the absence of constraints on the velocity field at the top (free tangential-stress/Neumann condition) and the distance from the lateral boundaries. The separatrices disappear when the amplitude increases and the patterns become disordered. In that case, the trajectories of all fluid parcels span the whole domain. All of these assessments indicate that three-dimensional effects on Faraday streaming flows are noticeable, even for longitudinal waves. This contrasts with the weak and local effect of walls on Faraday surface waves.

Finally, we focus on more quantitative aspects of Faraday streaming flows to compare our simulations with the experimental results presented in figure 4. Taking the same definition of the average velocity ${\hat{w}}$ as in § 2.4, we plot in figure 13 our numerical results for $\widehat{w}$ versus $\unicode[STIX]{x1D701}$ as we did for our experiments in figure 4. Different symbols correspond to different $m$ values (see caption), while different colours distinguish between patterns obtained with a contaminated surface (in red) and those obtained with a clean surface (in black). We recall that clean-interface patterns share some properties with type-II patterns (swirls), while the contaminated ones remarkably resemble type-I patterns (rolls).

Figures 4 and 13 show interesting similarities. In both cases, ${\hat{w}}$ displays small values for $\unicode[STIX]{x1D701}<0.15~\text{cm}$ . We also observe fair agreement, as the clean-interface streaming flows generate lower vertically averaged velocities than contaminated streaming flows. The values of ${\hat{w}}$ obtained in simulations are remarkably close to those measured in the experiments. However, there is still a qualitative discrepancy between the numerical and experimental diagrams. In experiments, we observe the collapse of the branches of ${\hat{w}}$ above the common threshold $\unicode[STIX]{x1D701}=\unicode[STIX]{x1D701}_{c}$ . Instead, simulations show scattered branches dependent on $m$ , where ${\hat{w}}$ increases as $\unicode[STIX]{x1D701}^{2}$ , as shown in the inset of figure 13. This feature is directly linked to the boundary-condition dependence on $A$ . The origin of this disagreement between numerical and experimental diagrams remains as an open and interesting question. It may come from insufficiently controlled surface properties in the experiment, as well as from the neglect of capillary effects or parasitic circulations in the model.