4.1. Visualizations

The top and bottom of the domain are taken to be rigid; we impose no-slip boundary conditions on the velocity field. Three types of boundary conditions have been tested on the lateral walls: no-slip, free-slip and periodic (figures 3 a–c respectively). Although our parameters were originally chosen for a different purpose, we find that the patterns that we observe are strikingly similar to those observed in earlier experiments by Douady & Fauve (Reference Douady and Fauve1988), figure 1(b), and Douady (Reference Douady1990), figures 10(b), 11(b) and 15(b), with different parameters: water, with ${\it\rho}_{1}=1000~\text{kg}~\text{m}^{-3}$ , ${\it\mu}_{1}=9\times 10^{-4}~\text{kg}~\text{m}^{-1}~\text{s}^{-1}$ and ${\it\sigma}=0.03~\text{kg}~\text{s}^{-2}$ , in a container of dimensions $80~\text{mm}\times 80~\text{mm}\times 5~\text{mm}$ oscillated with $f=37~\text{Hz}$ or $f=70~\text{Hz}$ .

For all three boundary conditions, we observe the superlattice-like patterns shown in figure 3. The interface in the $x$ – $y$ plane consists of four large square sub-blocks, each composed of smaller squares whose sides have approximate length ${\it\lambda}_{c}$ . In each case, the overall pattern has symmetry $D_{2}$ , i.e. it is invariant under reflections through the two diagonals. Each of the four sub-blocks is in phase opposition with its two adjacent neighbours. In figure 3, the patterns in (a) and (c) have periodicity length $L$ while (b) has a period of $2L$ . Although one would expect the periodic case (c) to be the simplest to analyse, we observe that the blocks in the free-slip case (b) are more homogeneous than those in cases (a) and (c); we will therefore focus on this case.

Figure 2.

The calculation domain of size $L\times L\times L/3$ ( $L=132~\text{mm}$ ), divided into $8\times 8\times 4$ subdomains. (a) Interface profile. (b) An immersed solid cylinder of diameter $D=L$ and height $h=L/3$ surrounding the flow. The flow outside of the cylinder is not computed. For both (a) and (b) the Cartesian resolution per subdomain is $64^{3}$ , which gives a global resolution of $512\times 512\times 256$ .

Figure 3.

Snapshots of the top view of the interface coloured by the vertical velocity for (a) no-slip, (b) free-slip and (c) periodic boundary conditions.

Figure 4 displays the temporal evolution of the interface at saturation. (This evolution is seen in greater detail in supplementary movie 1 to this article available at http://dx.doi.org/10.1017/jfm.2015.213.) After one period of forcing oscillation $T$ , troughs are replaced by crests (see figure 4 a,c) as a consequence of the subharmonic behaviour of the interface. This subharmonic behaviour is also displayed by the large sub-blocks, since the appearance of each block is transformed into that of its two adjacent neighbours after time $T$ and then returns to its initial shape after time $2T$ has passed. The pattern is thus invariant under the combined operations of rotation by ${\rm\pi}/2$ and time translation by $T$ , a spatio-temporal symmetry.

Figure 4.

Interface profiles at times separated by intervals of $T/2$ , where $T$ is the forcing period and $2T$ is the subharmonic response period. In (a) and (c), the small squares consist of wells surrounded by ridges with peaks at each corner, while in (b), low central peaks are surrounded by four higher peaks. In (a), a weak diagonal ‘bridge’ connects the large squares on the bottom left and top right; in (c), the bridge links the top left and bottom right. (See supplementary movie.)

Figure 5 shows the interface and the velocity on a vertical slice at the same three instants. The contrasting behaviour of the velocity in the left and right halves of the slice is displayed. The interface of figure 5(a) has a maximum (minimum) at the left (right) boundary, and vice versa for (c), illustrating again that the field is not periodic in a domain of length $L$ .

Figure 5.

Velocity profiles on the plane $y=3L/4$ , corresponding to each of the three instants shown in figure 4. The vectors are coloured according to their vertical component.

4.2. Fourier spectra

For free-slip boundary conditions, the pattern is periodic when it is doubled horizontally by reflection at the boundaries. The doubled pattern is invariant under the symmetry group $D_{4}$ of the square, consisting of reflections through the $x$ and $y$ coordinate axes and rotations of multiples of ${\rm\pi}/2$ . We decompose the doubled interface height profile ${\it\zeta}(x,y,t)$ into spatial Fourier modes, and then into spatio-temporal modes as follows:

where $\boldsymbol{k}_{l,m}\equiv (({\rm\pi}l)/L,({\rm\pi}m)/L)$ , $\boldsymbol{x}=(x,y)$ and $k_{l,m}=|\boldsymbol{k}_{l,m}|$ . The square symmetry of the doubled interface implies that the eight modes in an octet $\hat{{\it\zeta}}_{\pm l,\pm m}$ and $\hat{{\it\zeta}}_{\pm m,\pm l}$ all have the same amplitude. Figure 6 shows two such octets, with representative elements $\hat{{\it\zeta}}_{23,1}$ and $\hat{{\it\zeta}}_{23,3}$ , located on circles which we will call $\mathscr{C}_{1}$ and $\mathscr{C}_{3}$ respectively, i.e. with $l=23$ and $m=1$ or $m=3$ .

Figure 6.

Spatial spectrum of a snapshot of the interface in the doubled domain. Circles $\mathscr{C}_{1}$ (dark blue) and $\mathscr{C}_{3}$ (dark brown) have slightly different radii. The circle $\mathscr{C}_{1}$ contains the fundamental mode $\hat{{\it\zeta}}_{23,1}$ and its images under reflection and rotation (dark blue dots surrounded by circles). The circle $\mathscr{C}_{3}$ contains the highest third-harmonic mode $\hat{{\it\zeta}}_{23,3}$ and its images (dark brown dots surrounded by squares). Only these modes and, to a lesser extent, the weaker second harmonic $\hat{{\it\zeta}}_{46,0}$ (orange) have significant amplitude. (a) Restriction to $[-25,25]^{2}$ highlights the square symmetry of the pattern. (b) Restriction to the first quadrant $[0,50]^{2}$ includes $\hat{{\it\zeta}}_{46,0}$ .

The circle $\mathscr{C}_{1}$ has radius $k_{c}$ , the critical wavenumber. The dominant wavevector octet of our pattern, represented by $\boldsymbol{k}_{23,1}$ , is in accordance with the experimental observations of Douady & Fauve (Reference Douady and Fauve1988), who scanned over the forcing frequency in order to vary the selected wavenumbers and patterns. They noted that the range of existence was greatest for pairs $(l,m)$ with a small ratio $m/l$ and was maximum for $m=1$ , as in our case, i.e. for wavevectors that are almost parallel to the walls.

The modes on $\mathscr{C}_{3}$ can be seen as the result of cubic interactions between modes of $\mathscr{C}_{1}$ :

Modes on $\mathscr{C}_{3}$ are fed continually and damped slowly, if at all, since $|k_{23,3}-k_{c}|/k_{c}=0.75\,\%$ . Generalizing (4.2) to any $m\ll l$ shows that such cubic interactions lead to $\boldsymbol{k}_{l,3m}$ , with $|k_{l,3m}-k_{c}|/k_{c}\sim 4(m/l)^{2}$ . Due to its proximity to the critical circle, this mode is weakly damped and should often be present when Faraday waves are excited in a square domain that is much larger than the critical wavelength. A second harmonic of $\mathscr{C}_{1}$ , arising from the quadratic interaction

is seen in figure 6(b) along with its image under ${\rm\pi}/2$ rotation. These modes occur in a quartet rather than an octet because of their reflection symmetry in the coordinate axes and are of smaller amplitude than the modes on $\mathscr{C}_{1}$ or $\mathscr{C}_{3}$ . The experimental techniques of Douady & Fauve (Reference Douady and Fauve1988) did not give them access to amplitudes of Fourier modes other than the dominant one.

According to the terminology of Simonelli & Gollub (Reference Simonelli and Gollub1989), the four components $\hat{{\it\zeta}}_{\pm l,\pm m}$ comprise one pure mode, while $\hat{{\it\zeta}}_{\pm m,\pm l}$ comprise another pure mode. Douady & Fauve (Reference Douady and Fauve1988) observed mixed modes (which they called symmetric) for small $m/l$ , and pure modes (which they called dissymmetric) for larger $m/l$ . Pure modes may be combined to form two types of mixed modes, the in-phase mode in which all of the components have the same sign (as in our figure 6) and the out-of-phase mode in which the components of the two pure modes are of opposite sign. Silber & Knobloch (Reference Silber and Knobloch1989) showed that these two types of mixed modes are equivalent if $l$ and $m$ have opposite parities. Crawford (Reference Crawford1991) showed that this was also true for $l$ and $m$ with the same parity (as in our case) by invoking the concept of hidden symmetries. In a square container with Neumann boundary conditions, the doubled pattern formed by reflecting at each boundary, as we have done above, satisfies periodic boundary conditions, whose inherited (hidden) translation invariances are incorporated into the normal form.

For each of $\mathscr{C}_{1}$ or $\mathscr{C}_{3}$ , the resulting pattern has a superlattice structure, in which squares of different sizes coexist, as shown in figure 7(a,b). Although shown in domains of length $L$ , the periodicity length is $2L$ . The smallest squares have length scale $2{\rm\pi}/k_{23,m}$ for $m=1,3$ . The patterns formed by modes on $\mathscr{C}_{3}$ have an additional intermediate length scale $2L/3$ . Close examination of figure 7(b) shows that adjacent medium squares are almost, but not exactly, identical. Figure 7(c) shows a superposition of the two patterns of figure 7(a,b), weighted by the amplitudes of their respective Fourier components. During most of the time, the interface resembles this superposition; compare with figure 4. When the amplitudes of modes on $\mathscr{C}_{1}$ and $\mathscr{C}_{3}$ approach zero, which occurs twice during one subharmonic period (see figure 8 a and figure 4 b), it is necessary to take modes like $\hat{{\it\zeta}}_{46,0}$ into account.

Figure 7.

Patterns constructed from Fourier modes in (a) $\mathscr{C}_{1}$ , (b) $\mathscr{C}_{3}$ and (c) a superposition of the two. Lighter (darker) zones correspond to the peaks (troughs) of the interface. Superlattices, consisting of smaller and larger squares, are observed in all of the plots; the smallest boxes are approximately of size ${\it\lambda}_{c}$ . (b) Medium squares of size $2L/3$ are present (long-dashed box). (c) Reconstructed pattern, made by combining patterns in (a) and (b) weighted by the corresponding Fourier coefficients $\hat{{\it\zeta}}_{23,1}$ and $\hat{{\it\zeta}}_{23,3}$ . Compare with figure 4(c).

Figure 8.

(a) Temporal evolution of the three main spatial modes. Modes $\hat{{\it\zeta}}_{23,1}$ (blue curve and circles) and $\hat{{\it\zeta}}_{23,3}$ (red curve and $+$ symbols) are subharmonic: their period of oscillation is twice that of the forcing. Mode $\hat{{\it\zeta}}_{46,0}$ (black curve and $\times$ symbols) is harmonic. (b) Temporal spectrum of the main modes. The modes $\hat{{\it\zeta}}_{23,1}$ , $\hat{{\it\zeta}}_{23,3}$ contain only odd temporal harmonics while $\hat{{\it\zeta}}_{46,0}$ has mainly even harmonics. Each spatial mode has a finite $n=0$ component and hence a non-zero temporal mean.

The temporal evolution of $\hat{{\it\zeta}}_{23,1}$ , $\hat{{\it\zeta}}_{23,3}$ and $\hat{{\it\zeta}}_{46,0}$ is displayed in figure 8(a) and the spatio-temporal spectrum is shown in figure 8(b). The mode amplitudes $\hat{{\it\zeta}}_{23,1}$ and $\hat{{\it\zeta}}_{23,3}$ are related by a multiplicative constant and contain only subharmonic temporal components (i.e.  $n$ odd in (4.1)). The temporal component $n=1$ corresponding to ${\it\Omega}={\it\omega}/2$ dominates the temporal evolution. Both $\hat{{\it\zeta}}_{23,1}$ and $\hat{{\it\zeta}}_{23,3}$ evolve in phase opposition, crossing zero at the same time and yielding patterns such as that in figure 4(b). In contrast, $\hat{{\it\zeta}}_{46,0}$ is harmonic and is dominated by two temporal components $n=0$ and $n=2$ , as expected from the quadratic resonance of $\hat{{\it\zeta}}_{23,1}$ with itself or with $\hat{{\it\zeta}}_{23,3}$ . Each spatial mode has a finite $n=0$ mean component; the offset of $\hat{{\it\zeta}}_{46,0}$ is especially noticeable.

4.3. Simulation in a cylindrical container

We performed simulations under the same conditions, but in a cylindrical container of diameter $D=L$ and height $h=L/3$ (see figure 2 b). The lateral boundary conditions are free-slip and the advancing and receding contact angles are fixed at $100^{\circ }$ and $60^{\circ }$ respectively. Figure 9 shows snapshots of the interface every $T/2$ . (This evolution is seen in greater detail in supplementary movie 2 to this article.) The pattern has the spatio-temporal symmetry of invariance under combined time evolution by $T/2$ and rotation by ${\rm\pi}/2$ , while each instantaneous pattern is reflection-symmetric about both diagonal axes. At the centre, we observe squares whose characteristic wavelength remains close to ${\it\lambda}_{c}$ of table 1 for $f=30~\text{Hz}$ . Closer to the boundary, five-sided cells can be seen as the pattern organizes itself to fit in the circular domain. Aside from this, the pattern is relatively insensitive to the circular shape of the domain because $L/{\it\lambda}_{c}=11.7\gg 1$ , in contrast to small-radius experiments and theory. In most of these (e.g. Ciliberto & Gollub Reference Ciliberto and Gollub1985; Crawford et al. Reference Crawford, Knobloch and Riecke1990; Das & Hopfinger Reference Das and Hopfinger2008; Batson et al. Reference Batson, Zoueshtiagh and Narayanan2013), the interface takes the form of Bessel functions, including axisymmetric modes, while Rajchenbach et al. (Reference Rajchenbach, Clamond and Leroux2013) report intriguing patterns of the form of stars and pentagons.

Figure 9.

Interface profile in the cylinder at time intervals of $T/2$ . The central part of the domain is still occupied by squares, but five-sided cells are present near the boundaries, allowing the pattern to fit inside a circle. (See supplementary movie.)