3.1 Preferential sampling of fluid velocities

Figure 1.

Preferential sampling of the vertical component of fluid velocity by gyrotactic, rod-like swimmers ( $\unicode[STIX]{x1D6FC}=1$ ). (a) Average vertical component of the fluid velocity measured along cell trajectories. Curves are computed at $Re_{\unicode[STIX]{x1D706}}=173$ . (b,c) Distributions of swimmers for populations marked in (a) with $\unicode[STIX]{x1D6F9}=1$ , $\unicode[STIX]{x1D6F7}=5$ (b) and $\unicode[STIX]{x1D6F9}=10$ , $\unicode[STIX]{x1D6F7}=20$ (c) plotted on a colour map of the vertical velocity $u_{z}$ . Slower (b) or faster (c) swimmers sample preferentially regions with negative (blue) or positive (red) vertical velocity.

The vertical displacement in the water column of a gyrotactic swimmer following (2.1) can be affected by fluid transport even in a flow with vanishing average velocity. Indeed, although the Eulerian average of the fluid velocity vanishes, the average swimmer velocity $\langle \boldsymbol{v}\rangle =\langle \boldsymbol{u}\rangle _{S}+\unicode[STIX]{x1D6F7}\langle p_{z}\rangle \hat{\boldsymbol{z}}$ depends on the mean value of $\boldsymbol{u}$ computed along particle trajectories (notice that in the above expression symmetry implies $\langle p_{x,y}\rangle =0$ ). While for the horizontal components one has $\langle u_{x}\rangle _{S}=\langle u_{y}\rangle _{S}=0$ for symmetry reasons, gravitaxis breaks the up–down symmetry so that one can have $\langle u_{z}\rangle _{S}\neq 0$ (Fouxon & Leshansky Reference Fouxon and Leshansky2015). Since the largest swimming speeds of motile algae are typically of the order of $u_{\unicode[STIX]{x1D702}}$ or smaller and $u_{rms}\gg u_{\unicode[STIX]{x1D702}}$ , preferential sampling of regions of upwards or downwards flow can in principle strongly enhance or hinder the swimmer’s drift towards the surface. Figure 1(a) shows the average value of the vertical component of the fluid velocity along swimmer trajectories $\langle u_{z}\rangle _{S}$ . The different curves are computed for rod-like cells ( $\unicode[STIX]{x1D6FC}=1$ ) and $\langle u_{z}\rangle _{S}$ is plotted for several values of $\unicode[STIX]{x1D6F9}$ as a function of the swimming parameter, $\unicode[STIX]{x1D6F7}$ . It is clear that slow swimmers tend to preferentially sample downwards fluid velocities (figure 1 b), while faster ones (figure 1 c) spend more time where $u_{z}>0$ .

Figure 2.

Small- $\unicode[STIX]{x1D6F7}$ behaviour of preferential sampling of fluid velocities and small-scale clustering, compared with the theoretical predictions in Gustavsson et al. (Reference Gustavsson, Berglund, Jonsson and Mehlig2016) (dashed lines). (a) Vertical component of fluid velocity conditioned on swimmer trajectories for different elongations as labelled at $\unicode[STIX]{x1D6F9}=0.1$ . Inset: same as main panel but for rods ( $\unicode[STIX]{x1D6FC}=1$ ) at different $\unicode[STIX]{x1D6F9}$ . (b) Accumulation index $N$ rescaled with the stochastic prediction. Inset: the same curves without rescaling.

Figure 3.

Average of the vertical fluid velocity, $\langle u_{z}\rangle _{S}$ , along swimmer trajectories for rod-like ( $\unicode[STIX]{x1D6FC}=1$ ) cells as a function of $\unicode[STIX]{x1D6F7}=v_{s}/u_{\unicode[STIX]{x1D702}}$ (a–c), and of $\unicode[STIX]{x1D6F7}_{L}=v_{s}/u_{rms}$ (d–f) for different $Re_{\unicode[STIX]{x1D706}}$ , as labelled in (a). Whereas the small- $\unicode[STIX]{x1D6F7}$ behaviour in (a–c) rescales consistently, the transition to preferential sampling of positive values of $u_{z}$ is better rescaled by $\unicode[STIX]{x1D6F7}_{L}$ .

For a more quantitative analysis, we focus at first on small values of $\unicode[STIX]{x1D6F7}$ . In this limit the behaviour of the curves is consistent with what observed is in (Durham et al. Reference Durham, Climent, Barry, De Lillo, Boffetta, Cencini and Stocker2013), in that the average fluid velocity seen by the cells is increasingly negative as cell velocity grows. This behaviour has been discussed for a $d$ -dimensional, random, single-scale flow in Gustavsson et al. (Reference Gustavsson, Berglund, Jonsson and Mehlig2016), where it was predicted that in the small- $\unicode[STIX]{x1D6F7}$ limit, $\langle u_{z}\rangle _{S}\propto -\unicode[STIX]{x1D6F7}G_{d}(\unicode[STIX]{x1D6FC})K(\unicode[STIX]{x1D6F9})$ , where $G_{d}(\unicode[STIX]{x1D6FC})=[d(1-\unicode[STIX]{x1D6FC})+2]/d$ and $K(\unicode[STIX]{x1D6F9})=\unicode[STIX]{x1D6F9}/(2\unicode[STIX]{x1D6F9}+1)$ . That prediction is remarkably verified in our simulations, as shown in figure 2, where the average fluid velocity along the trajectories is plotted for different values of the shape parameter $\unicode[STIX]{x1D6FC}$ , showing that the factor due to particle geometry $G_{d}(\unicode[STIX]{x1D6FC})$ is indeed correct. When the shape parameter $\unicode[STIX]{x1D6FC}$ is fixed, the dependence on $\unicode[STIX]{x1D6F9}$ is also fairly well satisfied (see inset). A similar prediction is confirmed also on the small- $\unicode[STIX]{x1D6F7}$ properties of clustering as shown in figure 2(b), which will be discussed in the next section. In general, we can also conclude that the small- $\unicode[STIX]{x1D6F7}$ behaviour of this observable is independent of $Re_{\unicode[STIX]{x1D706}}$ . Figure 3(a–c) show the vertical fluid speed curves for fixed $\unicode[STIX]{x1D6F9}$ at various $Re_{\unicode[STIX]{x1D706}}$ . Notice that in our non-dimensional units, based on the Kolmogorov velocity, the curves collapse in the small- $\unicode[STIX]{x1D6F7}$ region for each value of $\unicode[STIX]{x1D6F9}$ (see also Durham et al. Reference Durham, Climent, Barry, De Lillo, Boffetta, Cencini and Stocker2013, for the case of spherical cells).

We now consider larger values of the swimming velocity. As noted above, fast swimming cells tend to preferentially sample regions of upwards flow. As shown by figure 3, the critical speed $\unicode[STIX]{x1D6F7}_{c}$ of transition from downwelling to upwelling regions (defined as the point at which $\langle u_{z}\rangle _{S}=0$ ) is smaller for larger values of $\unicode[STIX]{x1D6F9}$ . However, numerical results show that $\unicode[STIX]{x1D6F7}_{c}$ saturates to a finite value for large $\unicode[STIX]{x1D6F9}$ . This has profound implications for the statistics of velocity and velocity gradients along particle trajectories. While the correlation time of gradients is of the order of $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}$ , the time it takes for a swimmer to cross the correlation length of gradients, $\unicode[STIX]{x1D702}$ , is of the order of $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6F7}}=\unicode[STIX]{x1D702}/v_{s}=\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}/\unicode[STIX]{x1D6F7}$ , so that $\unicode[STIX]{x1D6F7}>1$ implies that the gradients correlation time as seen by a swimmer is $\unicode[STIX]{x1D70F}_{corr}=\min (\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6F7}},\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}})=\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6F7}}$ (Fouxon & Leshansky Reference Fouxon and Leshansky2015). As a consequence, the preferential sampling of upwards velocities cannot be understood in terms of a quasiequilibrium solution, at variance with the $\langle u_{z}\rangle _{S}<0$ case for spherical particles. However, an even more important difference is revealed if the actual observed values of $\unicode[STIX]{x1D6F7}_{c}$ are considered. Indeed, when $\langle u_{z}\rangle _{S}$ is plotted at fixed $\unicode[STIX]{x1D6F9}$ upon changing the extension of the inertial range with simulations at different resolutions, one can see that $\unicode[STIX]{x1D6F7}_{c}$ increases with $Re_{\unicode[STIX]{x1D706}}$ , and the corresponding values of the swimming velocity are of the order of (or larger than) the large-scale velocity of the flow $u_{rms}$ . Figure 3(d–f) shows the $\langle u_{z}\rangle _{S}$ as a function of a large-scale swimming parameter $\unicode[STIX]{x1D6F7}_{L}=v_{s}/u_{rms}$ . In this parameterization, the critical values $\unicode[STIX]{x1D6F7}_{L,c}(\unicode[STIX]{x1D6F9})$ are independent of $Re_{\unicode[STIX]{x1D706}}$ . For large $\unicode[STIX]{x1D6F9}$ , the critical $\unicode[STIX]{x1D6F7}_{L}$ appears to asymptotically converge from above to $\unicode[STIX]{x1D6F7}_{L,\infty }\approx 0.86$ , which can be obtained as the $\unicode[STIX]{x1D6F9}\gg 1$ limit of the analytical expression in Gustavsson et al. (Reference Gustavsson, Berglund, Jonsson and Mehlig2016). This numerical result is summarized in figure 4 and is consistent with a picture in which the transition to upwards velocity sampling is essentially controlled by the integral scale of the flow.

Figure 4.

Critical value of $\unicode[STIX]{x1D6F7}_{L}$ (main panel) and $\unicode[STIX]{x1D6F7}$ (inset) for various $Re_{\unicode[STIX]{x1D706}}$ as a function of $\unicode[STIX]{x1D6F9}$ . Dashed line: theoretical asymptotic value of $\unicode[STIX]{x1D6F7}_{L,c}$ (see text).

The fact that the inversion in preferential sampling is controlled by the large-scale velocity has not been noted before, to our knowledge. Indeed it can be observed only if a sufficient separation between large and small scales is present in the velocity field. It is therefore important to note that this scale separation effect is not in contradiction to the stochastic results in Gustavsson et al. (Reference Gustavsson, Berglund, Jonsson and Mehlig2016), and indeed the asymptotic value of the critical swimming parameter defined with respect to $u_{rms}$ is in quantitative agreement with the prediction found in the same paper (dashed line in figure 4). The techniques adopted in that paper relied on a flow with a single scale, so that both gradients and velocities have the same correlation times. It is remarkable that those techniques are able to capture a qualitative behaviour that is observed in a multiscale flow. The observation that those features, in high- $Re_{\unicode[STIX]{x1D706}}$ flows, are found for very large swimming velocities (of the order of $u_{rms}$ instead of $u_{\unicode[STIX]{x1D702}}$ ) can perhaps suggest why the qualitative agreement between stochastic theory and simulations is so good. Indeed, for very fast swimmers, as observed above, fluid fields can be described as noise with correlation time inversely proportional to the swimming speed, losing the details of the multiscale structure of the flow.

3.2 Fractal clustering

We now focus on the small-scale, fractal clustering properties of gyrotactic suspensions by varying the shape parameter, $\unicode[STIX]{x1D6FC}$ , with emphasis on the spherical ( $\unicode[STIX]{x1D6FC}=0$ ) and rod-like ( $\unicode[STIX]{x1D6FC}=1$ ) limits. We start by considering the limit of stable ( $\unicode[STIX]{x1D6F9}\ll 1$ ) and slowly swimming ( $\unicode[STIX]{x1D6F7}\ll 1$ ) cells.

In such limit, the perturbative approach originally developed in Durham et al. (Reference Durham, Climent, Barry, De Lillo, Boffetta, Cencini and Stocker2013) for spherical cells (see § 2) can be extended to generic shapes by imposing stationarity in (2.2) (Gustavsson et al. Reference Gustavsson, Berglund, Jonsson and Mehlig2016). At the first order in $\unicode[STIX]{x1D6F9}$ one obtains that gyrotactic cells behave as tracers in an effective velocity field $\boldsymbol{v}$

(3.1) $$\begin{eqnarray}\dot{\boldsymbol{x}}=\boldsymbol{v}=\boldsymbol{u}+\unicode[STIX]{x1D719}\boldsymbol{p}^{\ast },\quad \text{with }\boldsymbol{p}^{\ast }=[\unicode[STIX]{x1D6F9}(\unicode[STIX]{x1D714}_{y}+2\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D61A}_{13}),\unicode[STIX]{x1D6F9}(-\unicode[STIX]{x1D714}_{x}+2\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D61A}_{23}),1].\end{eqnarray}$$

The resulting velocity field, $\boldsymbol{v}$ , is compressible with divergence

(3.2) $$\begin{eqnarray}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{v}=-\unicode[STIX]{x1D6F7}\unicode[STIX]{x1D6F9}[(1+\unicode[STIX]{x1D6FC})\unicode[STIX]{x2202}_{z}^{2}u_{z}+(1-\unicode[STIX]{x1D6FC})(\unicode[STIX]{x2202}_{x}^{2}+\unicode[STIX]{x2202}_{y}^{2})u_{z}].\end{eqnarray}$$

For rods, this expression reduces to $\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{v}=-2\unicode[STIX]{x1D6F7}\unicode[STIX]{x1D6F9}\unicode[STIX]{x2202}_{z}^{2}u_{z}$ , while for spheres it gives $\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{v}=-\unicode[STIX]{x1D6F7}\unicode[STIX]{x1D6F9}\unicode[STIX]{x1D6FB}^{2}u_{z}$ . Tracers advected by a weakly compressible flow field $\boldsymbol{v}(\boldsymbol{x},t)$ form clusters of fractal codimension $3-D\sim \langle \unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{v}\rangle _{S}$ (Falkovich, Fouxon & Stepanov Reference Falkovich, Fouxon and Stepanov2002). Thus one can infer that swimmers should form clusters with codimension $3-D\sim (\unicode[STIX]{x1D6F7}\unicode[STIX]{x1D6F9})^{2}$ . For a generic $\unicode[STIX]{x1D6FC}$ , in a $d$ -dimensional stochastic Gaussian flow $\langle \unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{v}\rangle _{S}$ can be computed exactly at first order of a perturbative approach based on a Kubo expansion (Gustavsson et al. Reference Gustavsson, Berglund, Jonsson and Mehlig2016), obtaining the prediction

(3.3) $$\begin{eqnarray}3-D\sim (\unicode[STIX]{x1D6F7}\unicode[STIX]{x1D6F9})^{2}C_{d}(\unicode[STIX]{x1D6FC})\quad [C_{d}(\unicode[STIX]{x1D6FC})=[(d+2)(d+4)-2d(d+4)\unicode[STIX]{x1D6FC}+(4+2d+d^{2})\unicode[STIX]{x1D6FC}^{2}]/d].\end{eqnarray}$$

We measured the fractal dimension of the clusters in terms of the correlation dimension, $D_{2}$ , ruling the small-scale scaling behaviour of the probability to find a pair of cells with distance less than $r$ , i.e. $p_{2}(r)\sim r^{D_{2}}$ when $r\rightarrow 0$ . We estimated $D_{2}$ from the local slopes of such probability. When $D_{2}\approx 3$ , as expected in the limit $\unicode[STIX]{x1D6F9},\unicode[STIX]{x1D6F7}\ll 1$ here considered, this kind of measure is affected by large errors, so we also measured the accumulation index $N$ which is obtained as follows. The volume is divided in cubes of side $\ell$ , and we count the average number of particles in such cubes, $\langle n\rangle _{\ell }$ , and the standard deviation, $\unicode[STIX]{x1D70E}^{2}=\langle n^{2}\rangle _{\ell }-\langle n\rangle _{\ell }^{2}$ . Then we define $N=(\unicode[STIX]{x1D70E}-\unicode[STIX]{x1D70E}_{P})/\langle n\rangle _{\ell }$ , where $\unicode[STIX]{x1D70E}_{P}=\sqrt{\langle n\rangle _{\ell }}$ is the standard deviation of a Poissonian process having the same mean number of particles. As discussed in Durham et al. (Reference Durham, Climent, Barry, De Lillo, Boffetta, Cencini and Stocker2013), one can show that if $\ell$ is small enough (here we have chosen $\ell \approx 2\unicode[STIX]{x1D702}$ ) then $N\propto 3-D_{2}$ .

In figure 2(b) we show $N$ as a function of $(\unicode[STIX]{x1D6F9}\unicode[STIX]{x1D6F7})^{2}C_{d}(\unicode[STIX]{x1D6FC})$ for different values of $\unicode[STIX]{x1D6F9},\unicode[STIX]{x1D6F7}$ and $\unicode[STIX]{x1D6FC}=0,1/2,1$ . We observe a fair collapse of the data (compare with the inset where the geometric constant $C_{d}(\unicode[STIX]{x1D6FC})$ is removed) onto a curve that displays a linear behaviour for small values of the abscissa, as predicted by (3.3), providing again (see also the discussion of figure 2 a) strong evidence that the first-order Kubo expansion of Gustavsson et al. (Reference Gustavsson, Berglund, Jonsson and Mehlig2016) reproduces the small- $\unicode[STIX]{x1D6F9}$ and small- $\unicode[STIX]{x1D6F7}$ limits of gyrotactic cells also in turbulent multiscale flows. We observe that the linear limiting behaviour seems to be more persistent for the spherical shapes. Similarly to inertial particles, which display the maximal clustering for Stokes number of order $1$ (Bec Reference Bec2003), gyrotactic cells also have a more pronounced clustering for $\unicode[STIX]{x1D6F9}\sim 1$ (Durham et al. Reference Durham, Climent, Barry, De Lillo, Boffetta, Cencini and Stocker2013). Zhan et al. (Reference Zhan, Sardina, Lushi and Brandt2014) showed that spherical cells are generally more clustered than elongated swimmers, except for very large $\unicode[STIX]{x1D6F9}$ . In figure 5 we compare vis a vis the behaviour of $D_{2}$ (estimated from the local slopes of $p_{2}(r)$ ) for spherical ( $\unicode[STIX]{x1D6FC}=0$ ) and rod-like ( $\unicode[STIX]{x1D6FC}=1$ ) as a function of the swimming number $\unicode[STIX]{x1D6F7}$ for different values of $\unicode[STIX]{x1D6F9}\geqslant 1$ . By comparing figure 5(a,c), it is clear that spherical cells cluster more than elongated ones when $\unicode[STIX]{x1D6F9}$ is small enough, while the opposite is true for unstable cells $\unicode[STIX]{x1D6F9}\gg 1$ . The latter limiting behaviour can be explained by noticing that at $\unicode[STIX]{x1D6F9}\rightarrow \infty$ the dynamics of (2.1)–(2.2) in the ( $\boldsymbol{x},\boldsymbol{p}$ )-phase space has an average volume contraction rate equal to $-\unicode[STIX]{x1D6FC}(d+2)\langle \,\boldsymbol{p}\boldsymbol{\cdot }\unicode[STIX]{x1D64E}\,\boldsymbol{p}\rangle _{S}$ , which is identically zero for spherical cells. In other terms, the dynamics of spherical cells with large $\unicode[STIX]{x1D6F9}$ preserves volumes in phase space, and so does its projection in position space (i.e. $D_{2}=3$ ), while elongated swimmers obey a dissipative dynamics which leads to fractal clustering. For intermediate values of $\unicode[STIX]{x1D6F9}$ (figure 5 b) a non-trivial role is played by the swimming speed: slow spherical swimmers cluster more than elongated ones, but the opposite is true for fast swimmers.

Figure 5.

Correlation dimension of clusters for $\unicode[STIX]{x1D6F9}=0.5$ (a), 1.0 (b) and 10 (c) as a function of the swimming parameter $\unicode[STIX]{x1D6F7}$ . Spherical swimmers ( $\unicode[STIX]{x1D6FC}=0$ ) concentrate more than elongated cells (in this case, ideal rods) for small values of $\unicode[STIX]{x1D6F9}$ (a), whereas rod-like swimmers maintain observable clustering even for large $\unicode[STIX]{x1D6F9}$ (c). For intermediate values of $\unicode[STIX]{x1D6F9}$ , slow (fast) elongated swimmers cluster less (more) than spherical ones (b).