3. Results and discussion

A visualisation of two typical trajectories of prolate and oblate particles from the experiments is shown in figure 1(a,b). The evolution of the particle centre of mass gives an indication of the large-scale circulation which occurs in the RB cell in the present conditions. The particle orientation, also visually rendered in figure 1(a,b), identified by the unit vector $\boldsymbol {p}(t)$, evolves smoothly in the bulk and changes abruptly mostly in correspondence of the top and bottom regions (although particle–wall collisions do not occur). A quantitative measurement of these changes is obtained via the p.d.f. for the instantaneous normalised quadratic tumbling rate, $\dot {p}_i \dot {p}_i/\overline {\dot {p}_i \dot {p}_i}$. The p.d.f.s, figure 1(c,d), show exceptionally large fluctuations, up 60 times the mean quadratic tumbling value. For comparison we display on the same figure the expected p.d.f. for a set of initially random uniformly oriented particles of the corresponding aspect ratios tumbling in a stationary plane shear flow, i.e. performing so-called Jeffery orbits. Note that the latter p.d.f. has a finite support due to the periodic evolution of the particle orientation, and it has a much shorter tail. Additionally, we remark that the shape of these curves (i) does not seem to carry a particle aspect ratio dependence; (ii) agrees well with previous experimental measurements (Parsa et al. Reference Parsa, Calzavarini, Toschi and Voth2012; Marcus et al. Reference Marcus, Parsa, Kramel, Ni and Voth2014) performed in developed turbulent flows at much higher Reynolds numbers; and (iii) are in excellent agreement with the DNS. The latter point gives us confidence in relying on numerical results for further analysis.

Figure 1. Visualisation of two trajectories of a prolate $(a)$ and oblate $(b)$ particle from experiments (EXP). Their duration is $215 \tau _{\eta }$ for $\alpha =6$ (rod) and $183\tau _{\eta }$ for $\alpha =1/6$ (disk). Probability density function (p.d.f.) of the square tumbling rate for rods $(c)$ and disks $(d)$: RB experiments (empty circles); DNS results (red line); Jeffery orbits in a steady plane shear flow (black line). For comparison, experimental results in HIT from Parsa et al. (Reference Parsa, Calzavarini, Toschi and Voth2012) at $\alpha =5, Re_{\lambda } =161$$(c)$and Marcus et al. (Reference Marcus, Parsa, Kramel, Ni and Voth2014) at $\alpha =1/10, Re_{\lambda } = 91$$(d)$ are shown.

We now look at the dependence of the mean quadratic tumbling as a function of the aspect ratio; the symbol $\overline {\vphantom {M}\ldots }$ denoting in the following, global volume and time averages. Figure 2(a) presents the measurements from numerics in the aspect ratio range $\alpha \in [10^{-2},\ 10^{2}]$ together with the ones from the experiments for oblate and prolate particles, which again nicely agree with each other. More importantly the behaviour is compared with the measurements of the same quantity in HIT, here DNS at $Re_{\lambda }=32$ and results from Parsa et al. (Reference Parsa, Calzavarini, Toschi and Voth2012) in the same conditions. The tumbling rate in RB is for all the aspect ratios slower than the one observed in the HIT flow, pointing to the existence of either a stronger preferential alignment with the underlying flow or to an intrinsic smaller variability of the velocity gradient fluctuations.

Figure 2. Mean squared tumbling rate as a function of aspect ratio $\alpha$ with global energy dissipation rate $\overline {\epsilon }$ as normalisation scale. $(a)$ Global measurements for RB DNS at $Ra=10^{9}$ and $Pr=40$ (blue diamond); HIT DNS at $Re_\lambda =32$ (black circles); and for comparison, HIT DNS at $Re_\lambda =31$ from Parsa et al. (Reference Parsa, Calzavarini, Toschi and Voth2012) (grey dashed-dotted line). Filled symbols are experimental measurements in RB for $\alpha =6$ and $1/6$. $(b)$ Local measurements in bulk and near-wall subdomains with the same normalisation as in panel $(a)$. Error bars on experimental data points are determined from the absolute difference of the same measurement in two equal data subsets.

A major difference between RB turbulence and HIT concerns the non-homogeneity caused by the presence of geometrical boundaries. Therefore, following the decomposition approach in Grossmann & Lohse (Reference Grossmann and Lohse2000) we divide the volume in two subsets, a central cubic volume of side $H/3$ denoted as ‘bulk’ and the complement of it denoted as ‘near-wall’ region (see cartoon in figure 2b). Figure 2(b) shows the computed mean tumbling rates in the two regions both from experiments and simulations (note that we use the symbol $\langle \cdots \rangle$ to indicate a local-in-space average). Two observations are in order. First, this analysis confirms the strong inhomogeneity of the RB flow. The global tumbling rate appears to be dominated by the near-wall region, where it is twice as strong as in the bulk. Second, we observe very different trends for the quadratic tumbling rate versus $\alpha$ in the two regions.

In order to make sense of these two behaviours we normalise the mean tumbling rate with respect to the local dissipation scales in the two subregions, i.e. the energy dissipation rate $\langle \epsilon \rangle$ is here computed either in bulk, or near-wall domains, see figure 3. For the bulk we observe a remarkable match with the measurements in HIT, proving that the flow gradient properties in the core of the RB system, at this $Ra$ number, are the same as in nearly homogeneous and isotropic turbulent flows. Interestingly, when the local energy dissipation rate rescaling is adopted, the near-wall tumbling rate falls below the bulk one. This behaviour can be contrasted with the one of an initially isotropic set of particles evolving in a steady plane shear flow, denoted as Jeffery orbits in the same figure. (Note that the behaviour for Jeffery orbits in figure 3 corrects the one provided in Parsa et al. (Reference Parsa, Calzavarini, Toschi and Voth2012, figure 4), which was affected by a normalisation error, hence the curve mentioned in that work should be multiplied by a factor 4/3. The mean normalised tumbling rate for Jeffery orbits is well approximated by $\alpha /(3\alpha ^{2}+3)$.)

Figure 3. Local measurements of mean squared tumbling rate in bulk and near-wall regions with the respective local energy dissipation rates normalisations ($\langle \epsilon \rangle$ standing here either for $\langle \epsilon \rangle _{{bulk}}$ or $\langle \epsilon \rangle _{{near\text {-}wall}}$) as a function of aspect ratio $\alpha$. For comparison, we show HIT DNS at $Re_\lambda =32$ (black circles), Jeffery orbits (violet line) and a best fit of the linear-combination of the two latter curves: $a\times [ \langle \dot {p}_i \dot {p}_i \rangle \nu / \langle \epsilon \rangle ]_{{HIT}} + (1-a)\times [\langle \dot {p}_i \dot {p}_i \rangle \nu / \langle \epsilon \rangle ]_{\textit{Jeffery}}$ with coefficient $a=0.5 \pm 0.08$.

It is now tempting to speculate that the behaviour observed in the near-wall (and indeed globally dominant in the RB system) is a superposition of turbulence and steady shear. A test of this guess, based on a simplistic linear combination of the two mentioned tumbling rate behaviours does not fall far from the actual shape (see shaded region in figure 3). The linear combination fit provides an estimate for the background shear intensity comparable to the one of turbulent fluctuations. More importantly, it accounts for the observed local maximum that occurs for aspect ratios just below the unit value.

In order to support the hypothesis that a background shear affects the observed global tumbling rate, we study particle orientation with respect to the flow velocity. Figure 4(a) shows by means of DNS data that $\alpha =6$ rods align with the velocity in the near-wall region, while $\alpha =1/6$ disks are orthogonal to it; on the contrary, in the bulk any preferential alignment vanishes. Note that the p.d.f. of the modulus of the cosine is considered here for symmetry reasons: due to the fore-and-aft symmetry of particles the parallel or antiparallel orientation with respect to a vector are equivalent. This qualitatively agrees with the Jeffery orbits phenomenology in a shear flow (Jeffery Reference Jeffery1922). Additionally, we also verified that rods/disks are biased towards parallel/perpendicular orientations to the cell walls, in agreement with the near-wall dynamics in turbulent channel flows (Marchioli et al. Reference Marchioli, Fantoni and Soldati2010; Challabotla et al. Reference Challabotla, Zhao and Andersson2015), see appendix 1 for details.

Figure 4. $(a)$ The p.d.f. of the absolute value of the cosine of the angle of particle orientation with the fluid velocity, $|{\rm cos}\, \theta _{\boldsymbol {u},\boldsymbol {p}}| = |\boldsymbol {p} \boldsymbol {\cdot } \boldsymbol {u}|/\|\boldsymbol {u}\|$; $(b)$ the same for the temperature gradient, $|{\rm cos}\,\theta _{\boldsymbol {\nabla }T,\boldsymbol {p}}| = |\boldsymbol {p} \boldsymbol {\cdot } \boldsymbol {\nabla } T|/\|{\nabla }T\|$.

In figure 4(b) we also show that disk-like particles get strongly aligned with the temperature gradient, $\boldsymbol {\nabla } T$, while rod-like particles stay weakly but preferentially orthogonal to it. This feature occurs equally well in the bulk and in near-wall and it is related to a similarity between the orientation equation for the particle and the evolution equation for the gradient of a scalar field advected by the fluid, as first proposed in Calzavarini et al. (Reference Calzavarini, Jiang and Sun2020). Indeed, (2.5) can be rewritten in terms of the evolution of an auxiliary non-unit vector, $\boldsymbol {q}(t)$, (Szeri Reference Szeri1993):

(3.1a,b)\begin{equation} \boldsymbol{p}(t)=\frac{\boldsymbol{q}(t)}{\|\boldsymbol{q}(t)\|}, \quad \dot{\boldsymbol{q}}(t) = \left( \boldsymbol{\varOmega} + \frac{\alpha^{2}-1}{\alpha^{2}+1} \boldsymbol{\mathcal{S}} \right)\boldsymbol{q}. \end{equation}

The limit of a thin disk, $\alpha \to 0$, in (3.1a,b) leads to an equation that apart from the diffusive term is formally identical to the one of the gradient of a scalar field, $T$, following the advection diffusion equation

(3.2a,b)\begin{equation} \dot{\boldsymbol{q}}_{\alpha=0}(t) = - \boldsymbol{\nabla}\boldsymbol{u}^{T}\ \boldsymbol{q}_{\alpha=0},\quad \dot{\boldsymbol{\nabla}T} = - \boldsymbol{\nabla}\boldsymbol{u}^{T}\ \boldsymbol{\nabla}T + \kappa \Delta \boldsymbol{\nabla}T. \end{equation}

Similarly the opposite limit of a rod, $\alpha \to \infty$, as first pointed out by Pumir & Wilkinson (Reference Pumir and Wilkinson2011), shares an analogous similarity with the vorticity ($\boldsymbol {\omega }$) equation of motion

(3.3a,b)\begin{equation} \dot{\boldsymbol{q}}_{\alpha=\infty}(t) = \boldsymbol{\nabla}\boldsymbol{u}\ \boldsymbol{q}_{\alpha=\infty}, \quad \dot{\boldsymbol{\omega}} = \boldsymbol{\nabla}\boldsymbol{u}\ \boldsymbol{\omega} + \nu \Delta \boldsymbol{\omega}. \end{equation}

Note that while the former equivalence becomes exact in the limit of $Pr \to \infty$, the latter one occurs in the opposite limit $Pr \to 0$. As a consequence one should expect a correlation between the rotational dynamics of the temperature gradient with the orientation of a disk; such a correlation is only approximate owing to the presence of thermal dissipation and to the different initial conditions for the particle and thermal gradient orientations. However, the correlation effect is particularly evident in the present flow due to the high value of $Pr$ which makes the decorrelating effect of thermal diffusivity negligible. We also observe that $\boldsymbol {q}_{\alpha =\infty }\boldsymbol {\cdot } \boldsymbol {q}_{\alpha =0} = const.$, suggesting that rod-like particles would be most likely not be in line with the thermal gradient direction. Appendix 2 reports complementary measurements on particle preferential alignment with the vorticity and rate-of-strain eigenvalues both in RB and HIT.

The measurements presented so far support the hypothesis that the global tumbling rate in RB is affected by a persistent background shear flow coexisting with turbulent fluctuations. In order to better study the effect of this superposition and the sensitivity of particle tumbling to the relative intensity of shear as compared with turbulence, we build a synthetic velocity gradient field as follows:

(3.4)\begin{equation} \boldsymbol{\nabla} \boldsymbol{u}_{s}(t) = \frac{1}{\sqrt{1+s^{2}}}\left( \boldsymbol{\nabla} \boldsymbol{u}(t) + \left( \begin{array}{ccc} 0 & s\ \tau_{\eta}^{-1} & 0 \\ 0 & 0 & 0\\ 0 & 0 & 0 \end{array}\right) \right). \end{equation}

Here $\boldsymbol {\nabla } \boldsymbol {u}(t)$ comes from a DNS record of the evolution of the gradient tensor along Lagrangian trajectories in HIT, while $s$ is an adjustable shear intensity coefficient in $\tau _{\eta }$ units. Note that the overall normalisation is constructed in such a way that the mean dissipative time ($\tau _{\eta }$) of the resulting gradient field $\boldsymbol {\nabla } \boldsymbol {u}_{s}$ is the same as the one of the original HIT field $\boldsymbol {\nabla } \boldsymbol {u}(t)$. This allows us to compare the shear intensity with the amplitude of the global field. As a caveat, we note that (3.4) is a rough approximation of a realistic turbulent shear flow, where velocity gradient fluctuations have specific signatures (Pumir Reference Pumir2017), and obviously also as a simplistic approximation of an RB flow. The mean tumbling rate of an initially random uniformly oriented set of particles evolved along many of such gradient trajectories is reported in figure 5. The family of curves, corresponding to different values of $s$, show similar features to the ones of the quadratic tumbling rate measured in the RB cell. Interestingly, this model provides an estimate, $s^{*} = 2 \pm 0.3$, of the background shear-rate intensity as compared with the turbulent velocity gradient fluctuations. The estimated associated time scale ($\tau _{s}$ for shear) expressed in the dissipative time of the synthetic global field ($\tau _{\eta }$) turns out to be $\tau _{s} = \tau _{\eta } \sqrt {1+s^{*2}}/s^{*} \simeq 1.1 \pm 0.1 \tau _{\eta }$.

Figure 5. Mean quadratic tumbling rate versus the particle aspect ratio as obtained evolving a particle ensemble with (3.4) where the intensity of the background shear field is $s$ times the average intensity of the HIT velocity gradient fluctuations. The pure HIT case $s=0$ (black line); the pure shear case, Jeffery orbits $s=\infty$ (violet line); and the measurements in RB from DNS (diamonds) are also drawn.

We note that this intensity is incompatible with a persistent shear-rate determined by the large-scale circulation turnover time, because here $T=H/u_{rms} \simeq 24 \tau _{\eta }$, while it is much closer to the time scale associated with the kinetic BL $\tau _{BL} = \delta /u_{rms} = T Re^{-1/2} \simeq 1.6 \tau _{\eta }$.

Finally, it is tempting to ask what will be the fate of the mean quadratic tumbling intensity as a function of $\alpha$ with varying Rayleigh number. This can be guessed dimensionally by considering (see Shraiman & Siggia (Reference Shraiman and Siggia1990) and Grossmann & Lohse (Reference Grossmann and Lohse2000) for the estimates) that $\langle \epsilon \rangle _{BL} \sim \nu (u_{rms}/\delta )^{2}$ and occupies a volume $V_{BL}=(H^{3} -(H-2\delta )^{3})$ while $\langle \epsilon \rangle _{bulk}\sim u_{rms}^{3}/(H-2\delta )$ occurs in a volume $V_{bulk}=(H-\delta )^{3}$ (where bulk denotes here the total volume minus the one occupied by BLs). With this we evaluate $s^{*} \sim \sqrt {\langle \epsilon \rangle _{BL} V_{BL}}/\sqrt {\langle \epsilon \rangle _{bulk} V_{bulk}} \sim Re^{-1/2}$, which is bound to decrease to zero as $Ra$ increases (because $Re\sim Ra^{1/2}$). Similarly, $(\tau _{s}/\tau _{\eta })^{2} \simeq (\overline {\epsilon } H^{3})/(\langle \epsilon \rangle _{BL} V_{BL})$ increases with $Ra$, meaning that the relative importance of the mean shear will reduce with respect to turbulence. However, we remark that the above dimensional argument is speculative and shall be put under scrutiny by future experimental/numerical studies capable of accessing higher $Ra$ number conditions.