3.1. Particle trajectories and field visualizations

In figure 3(a) a visualization example of the trajectory of a spheroid with $\alpha =2$ and an equivalent diameter $D_v=25 \eta$ over a relatively short time span, ${\sim }40$ dissipative times ($\tau _{\eta }$), obtained from the simulation is shown. The energy dissipation rate field in the fluid, $\epsilon$, is represented in colour over planes intersecting the spheroid. Such a quantity, that depends on the square of velocity gradients, allows us to foreground the effect the fluid–particle coupling. We observe that just a thin layer of large dissipation appears around the particle (see the zoom-in region of the slice at $t_1=8.7\tau _\eta$), while no wakes are observed at larger distances. This mild effect induced by the particle presence in the flow is better quantified by computing temporal fluid averages in the vicinity of the particle. This is numerically performed by seeding the flow with point-like Lagrangian probes whose positions are fixed in the particle frame of reference. Figure 3(b) shows the temporally and azimuthally averaged local dissipation rate around the spheroid ($\alpha =2$ and $D_v/\eta =25$). It appears that the dissipation rate close to the particle is enhanced by one order of magnitude with respect to the far-field value, furthermore, such an increase is uniformly distributed around the particle. Indeed, the contour line in the figure (black colour) marking where the particle influence reaches a saturation, indicates that the finite-size particle modifies the flow over a spheroidal region of approximately 1.5 times the linear size of the particle. Similar results for finite-size spheres have been observed in turbulence at $Re_\lambda =160$ (Cisse et al. Reference Cisse, Homann and Bec2013). The presence of the particle significantly modifies also the kinetic energy around the particle, the kinetic energy of the fluid is null at the particle interface and increases to the unperturbed value at approximately twice the particle size $D_v$, see figure 3(c). The same figure suggests that the slip velocity of neutrally buoyant particles is small, in particular $|\boldsymbol {u}-\boldsymbol {v}| \sim \boldsymbol {u}_{rms}$. This implies that the particle Reynolds number $Re_p = \langle |\boldsymbol {u}-\boldsymbol {v}| \rangle D_v / \nu \sim (D_v/\lambda ) Re_{\lambda } \sim O(10^{2})$. If the flow surrounding the particle were steady and laminar this would lead to a steady or periodically oscillating wake. In a turbulent flow, where the direction of motion changes frequently, such an effect is weakened. For the cases of spherical particles, this agrees with detailed study by Cisse et al. (Reference Cisse, Homann and Bec2013). Overall, the presence of a neutrally buoyant finite-sized non-spherical particle has weak feedback on the turbulent flow field. This is certainly very specific to the case of inclusions with the same density, the situation is very different in the case of heavy particles (Brändle de Motta et al. Reference Brändle de Motta, Estivalezes, Climent and Vincent2016) or large bubbles (Mathai et al. Reference Mathai, Huisman, Sun, Lohse and Bourgoin2018). This suggests that the picture of neutral inertial particles as essentially passive objects might be sound. We will now address this in more detail by looking at two quantities that are most sensitive to the fluid–particle one-way coupling, i.e. the fluid acceleration and later the angular velocity.

Figure 3.

(a) Example of trajectory for a spheroid with an equivalent diameter $D_v\approx 25\eta$ and aspect ratio $\alpha =2$ at $Re_\lambda =120$. The slices show the local dissipation rate $\epsilon _{\boldsymbol {x},t}$, normalized by $\epsilon$, at the planes through the spheroid centre at times $t=0,8.7,17.4,26.1,39.1$ in $\tau _\eta$. (b) Azimuthally and temporally averaged dissipation rate $\overline \epsilon$, normalized by $\epsilon$, around the spheroid with a diameter $D_v\approx 25\eta$ and aspect ratio $\alpha =2$ at $Re_\lambda =120$. The black line denotes the contour line of ${\rm log}_{10}(\bar {\epsilon }/\epsilon )=0$. (c) Azimuthally and temporally averaged square of $\boldsymbol {u}-\boldsymbol {v}$, normalized by the mean square of $\boldsymbol {u}$, around the spheroid with diameter $D_v= 25\eta$ and aspect ratio $\alpha =2$ at $Re_\lambda =120$. The black line denotes the contour line of $\overline {(\boldsymbol {u}-\boldsymbol {v})^{2}}/\langle \boldsymbol {u}^{2}\rangle ^{v,t}=1$. Here, $x_p$ and $y_p$ represent the coordinates in the particle frame.

3.2. Particle acceleration statistics: the VP model explains the size and shape dependencies in RP

We begin discussing the dependency of the particle acceleration intensity of large spheroids as a function of their shape and size and the physical mechanisms responsible for the observed dependencies. Figure 4(a) reports the temporal and ensemble average (denoted in the following $\langle \ldots \rangle$) of the single-Cartesian-component acceleration variance $\langle a_i^{2}\rangle$ of RP, normalized by the acceleration variance of fluid tracers $\langle a_i^{2}\rangle ^{tracer}$, as a function of the equivalent diameter in turbulent dissipative scale units $D_v/\eta$. Two observations are here in order. First, the acceleration variance decreases as the size of the spheroids increases. In the range $D_v/\eta \in [10,63]$ we measure the power law $\sim (D_v/\eta )^{-1.0 \pm 0.1}$ (at $Re_\lambda =120$). Second, we remark on a good collapse onto a single curve of all the data points corresponding to different aspect ratios (with $\alpha = [1/2,1,2,4]$). The observed scaling behaviour is consistent with the one observed for spherical particles at large $Re_\lambda$ values (Volk et al. Reference Volk, Calzavarini, Leveque and Pinton2011). Such a scaling deviates from $(D_v/\eta )^{-2/3}$ that can be guessed on the basis of dimensional reasoning relying on the K41 theory of turbulence (Voth et al. Reference Voth, Porta, Crawford, Alexander and Bodenschatz2002; Qureshi et al. Reference Qureshi, Bourgoin, Baudet, Cartellier and Gagne2007). This reasoning assumes that the particles are only one-way coupled to the fluid, i.e. they do not affect the fluid flow significantly through particle–fluid interactions. However, the existence of such a deviation has been already reported in studies on spherical particles. In particular, it is known that the scaling exponent has a Reynolds number dependence (Voth et al. Reference Voth, Porta, Crawford, Alexander and Bodenschatz2002; Qureshi et al. Reference Qureshi, Bourgoin, Baudet, Cartellier and Gagne2007). At lower $Re_\lambda$ the observed exponent is closer to the value $-4/3$, as observed in Homann & Bec (Reference Homann and Bec2010). The role of the pressure scaling dependence on $Re_\lambda$ has been conjectured to be responsible for such a behaviour. Our results at low Reynolds number, $Re_\lambda =32$ (see figure 11 in Appendix A) confirms such a dependence. On the other hand, the collapse of acceleration variance for different particle shapes (figure 4a) is a novel feature. It is remarkable because it implies that the hydrodynamical forces involved in the translational dynamics of spheroids are overall determined by the particle volume.

Figure 4.

(a) Normalized single component acceleration variance as a function of the normalized particle size $D_v/\eta$ at $Re_\lambda =120$ for real particles (RP: solid symbols) and volume averages (VA: open symbols). The dashed lines are the fitting lines for spheres of RP (exponent $-1.0 \pm 0.1$). Inset in (a) shows the ratio of acceleration variance of VA to acceleration variance of RP, $\langle a_i^{2} \rangle ^{VA}/\langle a_i^{2} \rangle$, as a function of $D_v/\eta$. (b) Normalized componentwise acceleration variance as a function of the normalized particle size $D_v/\eta$ at $Re_\lambda =120$ for real particles (RP: solid symbols) and virtual particles (VP: open symbols). The dashed lines are the fitting lines for spheres of RP. Inset in (b) shows the ratio of acceleration variance of VP to acceleration variance of RP, $\langle a_i^{2} \rangle ^{VP}/\langle a_i^{2} \rangle$, as a function of $D_v/\eta$. (c) The ratio of axial variance $\langle (\boldsymbol {a}\boldsymbol {\cdot } \boldsymbol {p})^{2}\rangle$ to lateral variance $\langle |\boldsymbol {a}\times \boldsymbol {p}|^{2}\rangle$ of particle acceleration as a function of aspect ratio at $Re_\lambda =120$ for RP (triangles), VA (square) and VP (circles), respectively. The black dash-dot line denotes the results of Jeffery tracers. The colour represents the normalized particle size $D_v/\eta$ in logarithmic scale.

In order to further substantiate these findings, we show the results of the numerical experiment with fluid volume averages (VA) and the so-called virtual particles (VP) in figure 4(a,b) (empty symbols). Similar scaling relations of acceleration variances are observed both for VA and VP in the inertial regime, whereas the magnitudes of the acceleration variances of VP and VA are not equal. In particular, the amplitude of accelerations fluctuations of VP particles is closer to that of RP particles (see insets in figure 4(a,b) where their ratio is plotted). Note that the acceleration variances of VA and VP represent respectively the volume averaging of the fluid flow acceleration in the fixed frame (Eulerian perspective) and in an advected frame (Lagrangian perspective). Therefore, these results stress that the scaling behaviour of the translational dynamics is dominated by the volume averaging instead of other surface forces such as e.g. the drag. However, an interesting question is: Why are VP capable of better approximating the statistics of real particles than VA? This might be due to preferential sampling of fluid structures along their Lagrangian trajectories, a fact that we tend to exclude because we know that these particles do not form clusters, or this might come from a preferential orientation in space (that is dynamical for the RP and VP cases and absent for VA). This is at odds with the expectation that particles of sizes well into the inertial range should become randomly oriented due to the averaging of turbulent fluid fluctuations that results from the spatial volume filtering.

In order to understand if any correlation exists between the particle acceleration and its orientation, we look at the ratio of the particle axial acceleration variance $\langle (\boldsymbol {a}\boldsymbol {\cdot }\boldsymbol {p})^{2}\rangle$ to the lateral acceleration variance $\langle |\boldsymbol {a}\times \boldsymbol {p}|^{2}\rangle$ as a function of their aspect ratio, see figure 4(c). For spherical particles ($\alpha =1$), no preferential alignment is expected between $\boldsymbol {a}$ and $\boldsymbol {p}$, even for various sizes, due to the full rotational symmetry of the body shape. For the non-spherical case, a value of the ratio different from the isotropic value 0.5 marks the existence of preferential orientation.

We find that the oblate particles preferentially align with their acceleration ($\langle (\boldsymbol {a}\boldsymbol {\cdot }\boldsymbol {p})^{2}\rangle /\langle |\boldsymbol {a}\times \boldsymbol {p}|^{2}\rangle >0.5$) and the prolate particles are predominantly perpendicular to the acceleration ($\langle (\boldsymbol {a}\boldsymbol {\cdot } \boldsymbol {p})^{2}\rangle /\langle |\boldsymbol {a}\times \boldsymbol {p}|^{2}\rangle <0.5$). It also turns out that the alignment of large particles is identical to that of JFT, which indicates that the alignment is insensitive to the particle's size. This is further confirmed by the measurements of the probability density functions (p.d.f.) of $|\cos (\theta _{\boldsymbol {a}, \boldsymbol {p}})|$ for various sizes and shapes (see figure 5), where $\theta _{ap}$ defines the angle between the acceleration vector $\boldsymbol {a}$ and orientation vector $\boldsymbol {p}$. The particular trend of $\langle (\boldsymbol {a}\boldsymbol {\cdot } \boldsymbol {p})^{2}\rangle /\langle |\boldsymbol {a}\times \boldsymbol {p}|^{2}\rangle$ as a function of $\alpha$ can then be understood in terms of the knowledge of JFT. It has been shown experimentally that fluid tracers have accelerations preferentially perpendicular to the local vorticity (Liberzon et al. Reference Liberzon, Luthi, Holzner, Ott, Berg and Jakob2012). It has also been theoretically demonstrated that prolate particles preferentially align with vorticity while oblate particles are preferentially perpendicular to it (Pumir & Wilkinson Reference Pumir and Wilkinson2011). The combination of these two pieces of evidence explain qualitatively the observed alignment, depending on the aspect ratio, between the particle orientation and the acceleration. The existence of these alignments indicates that the translation of finite-size neutrally buoyant particles in a flow is, in this respect, surprisingly similar to that of point-like inertialess axisymmetric fluid tracers, or conversely that the geometrical structure of flow at the inertial scale shows a similar structure to that of the velocity gradient, which occurs at the dissipation scale.

Figure 5.

The p.d.f.s of $|\cos (\theta _{\boldsymbol {a},\boldsymbol {p}})|$ for RP spheroids with different aspect ratios: (a) $\alpha =0.5$, (b) $\alpha =1$, (c) $\alpha =2$ and (d) $\alpha =4$ and for Jeffery fluid tracers.

We now turn our attention to the results for VA and VP. For VA, the values of the ratio $\langle (\boldsymbol {a}\boldsymbol {\cdot } \boldsymbol {p})^{2}\rangle /\langle |\boldsymbol {a}\times \boldsymbol {p}|^{2}\rangle$ are close to 0.5 for all considered sizes and aspect ratios, which is expected because VA is not aligned with the corresponding acceleration vector. Remarkably, it is found that the ratio $\langle (\boldsymbol {a}\boldsymbol {\cdot } \boldsymbol {p})^{2}\rangle /\langle |\boldsymbol {a}\times \boldsymbol {p}|^{2}\rangle$ of VP shows excellent agreement with that of RP (figure 4c). This suggests that the effect induced by the two-way coupling on the particle preferential alignment, if present, is weak. The similarity between RP and VP implies that the neutrally buoyant finite-size particle behaves roughly like a mass of fluid with the same particle size and shape. Furthermore, the size independence of the alignment indicates that the Lagrangian behaviour of a spheroidal fluid mass at the inertial scale is similar to the flow at the dissipative scale. The neutrally buoyant finite-size particles can then be regarded as probes of the Lagrangian coarse-grained field of turbulence.

In the next section, we will check if the similarity between real and virtual particles still holds when the rotational properties of particles are considered.

3.3. Particle rotation statistics: the VP model fails to describe the shape dependence in RP

As already mentioned in § 1, the rotations of tiny, $D_v \lesssim \eta$, neutrally buoyant particles in turbulent flows are determined by the evolution of the fluid velocity gradient along the Lagrangian particle trajectories. The Jeffery equation (2.12)–(2.13) well captures the angular dynamics of such particles. This has been recently proven in a series of combined experimental and numerical studies (Parsa et al. Reference Parsa, Calzavarini, Toschi and Voth2012; Byron et al. Reference Byron, Einarsson, Gustavsson, Voth, Mehlig and Variano2015; Calzavarini et al. Reference Calzavarini, Jiang and Sun2020; Jiang et al. Reference Jiang, Wang, Liu, Sun and Calzavarini2021). When it comes to larger particles, in particular inertial-scale particles, the physics is more complicated due to: (i) the non-smooth character of the flow field at the particle scale, (ii) the spatio-temporal filtering of flow fluctuations associated with the particle size and inertia and (iii) the two-way coupling with the fluid flow. This means that not only the shape (parametrized by $\alpha$) but also the particle sizes become relevant parameters of the problem. This is easily grasped in figure 6, where we show the mean square angular velocity, normalized by the value of the mean square angular velocity of point-like JFT with corresponding aspect ratio, as a function of the normalized diameter $D_v/\eta$ for RP (solid symbols). As the size increases, we observe that the mean square angular velocity decreases and scales approximately as $\langle {\varOmega }_i{\varOmega }_i\rangle \sim (D_v/\eta )^{-4/3}$ for $D_v \geqslant 10 \eta$. Since the angular velocity of JFT is nearly constant at changing the aspect ratio (Byron et al. Reference Byron, Einarsson, Gustavsson, Voth, Mehlig and Variano2015), the decrease of angular velocity is primarily an inertial term. Even if an offset between spherical and non-spherical particles is observed, the good collapse for prolate and oblate spheroids with different shapes substantiates the equivalent diameter $D_v$ as the relevant characteristic scale also for the rotational dynamics. We recall that the scaling exponent $-4/3$ can be dimensionally deduced using the relations of the K41 turbulence theory, specifically, by assuming that the rotation rate is related to the eddy turnover time at the particle scale, as confirmed in experiments with long slender fibres (Parsa & Voth Reference Parsa and Voth2014; Bounoua et al. Reference Bounoua, Bouchet and Verhille2018) and large cylinders (Bordoloi & Variano Reference Bordoloi and Variano2017; Pujara et al. Reference Pujara, Oehmke, Bordoloi and Variano2018).

Figure 6.

(a) Mean squared angular velocity, normalized by the results of Jeffery tracers with corresponding aspect ratio, as a function of $D_v/\eta$ at $Re_\lambda =120$. The open symbols denote the results of VA for spheres. The dashed line denotes the fitting line for spheres with scaling exponent $-4/3$. Inset shows the ratio of angular velocity variance of VA to that of RP, $\langle \varOmega _i\varOmega _i \rangle ^{VA}/\langle \varOmega _i\varOmega _i \rangle$, as a function of $D_v/\eta$ for spheres. (b) Mean squared angular velocity, normalized by the results of Jeffery tracers with corresponding aspect ratio, as a function of $D_v/\eta$ at $Re_\lambda =120$. The open symbols denote the results of VP. The dashed line denotes the fitting line for spheres with scaling exponent $-4/3$. Inset shows the ratio of angular velocity variance of VP to that of RP, $\langle \varOmega _i\varOmega _i \rangle ^{VP}/\langle \varOmega _i\varOmega _i \rangle$, as a function of $D_v/\eta$.

The VA and VP numerical experiments exhibit also a similar behaviour to RP, i.e. a decreasing trend with $D_v$ consistent with the $-4/3$ power law. However, the magnitude of the mean squared angular velocity is larger for VA and VP than that of RP (see again figure 6a,b). This suggests that the rotational dynamics of finite-size neutrally buoyant spheroids (RP) is also dominated by the volume filtering effect, as proven by its scaling behaviour with respect to $D_v$, while its amplitude must be affected by other physical effects (to be discussed later). From the insets of figure 6, we note that the angular velocity variance discrepancy between VA and RP and VP and RP is larger than the discrepancy of the acceleration variances. In conclusion, as far as angular velocity is concerned, we cannot say that VPs are a good approximation to RP. This discrepancy is even more important when one looks at the ratio between the intensity of axial as compared with lateral rotational components, that is to say, spinning and tumbling, as we will discuss in the following.

In figure 7(a), we show the ratio $\langle {\varOmega }_i^{s}{\varOmega }_i^{s}\rangle /\langle \dot {p}_i\dot {p}_i\rangle$ as a function of aspect ratio $\alpha$ for particles of different sizes. We first discuss the behaviour of sub-Kolmogorov-scale particles, whose dynamics is well described by the Jeffery equation. In this case, if the particle orientation vector is uncorrelated with the fluid velocity gradients, and $\boldsymbol {p}$ is an isotropic vector, upon time and ensemble averaging (2.13), one obtains

(3.1)\begin{equation} \frac{\langle {\varOmega}_i^{s}{\varOmega}_i^{s}\rangle}{\langle\dot{p}_i\dot{p}_i\rangle}= \frac{ \frac{1}{12}\langle \omega^{2} \rangle}{\frac{1}{6} \langle \omega^{2} \rangle + \frac{1}{5} \varLambda^{2} \langle \boldsymbol{S}:\boldsymbol{S} \rangle} = \frac{5}{10 + 6 \varLambda^{2}}. \end{equation}

Note that the last equality follows from the statistical properties of the HIT flow, where $\frac {1}{2} \langle \omega ^{2} \rangle = 2 \langle \boldsymbol{S}:\boldsymbol{S} \rangle = \epsilon /\nu$ (Parsa et al. Reference Parsa, Calzavarini, Toschi and Voth2012; Byron et al. Reference Byron, Einarsson, Gustavsson, Voth, Mehlig and Variano2015). This would produce only a mild variation of the spinning to tumbling ratio across the whole range of aspect ratios, as shown in figure 7(a) (blue solid curve). However, it is known that small particles develop strong correlations with velocity gradients that are responsible for a sharp deviation from this prediction. The present Jeffery model simulations clearly show this behaviour in figure 7(a) (spheres). Incidentally, we note that our results at $Re_\lambda =120$ agree with a previous numerical study (dashed line) at $Re_\lambda =433$ (Byron et al. Reference Byron, Einarsson, Gustavsson, Voth, Mehlig and Variano2015), underlying the weak $Re_\lambda$ dependence of this phenomenon. For spheres, the angular velocity is isotropic and the ratio $\langle {\varOmega }_i^{s}{\varOmega }_i^{s}\rangle /\langle \dot {p}_i\dot {p}_i\rangle$ stays around 0.5 for all cases. A marked asymmetry exists between oblate and prolate particle types, while the firsts tumble intensely the seconds are spin dominated, as they tend to align with the vorticity vector (Parsa et al. Reference Parsa, Calzavarini, Toschi and Voth2012; Voth & Soldati Reference Voth and Soldati2017). Furthermore, a saturation of these behaviours is observed in both cases for extreme aspect ratios, i.e. $\alpha \lesssim 1/4$ and $\alpha \gtrsim 1/4$.

Figure 7.

(a) The ratio of spinning rate squared to tumbling rate squared, $\langle \varOmega _i^{s}\varOmega _i^{s}\rangle /\langle \dot {p}_i\dot {p}_i\rangle$, as a function of aspect ratio $\alpha$ for RP. The colour bar represents the normalized particle size $D_v/\eta$ in the logarithmic scale. Spheres denote the results for Jeffery tracers. Leftward triangles denote the results at $Re_\lambda =32$ and rightward triangles denote the results at $Re_\lambda =120$. The arrows denote the direction in which the size of the particles increases. The dashed line shows the results of tracers at $Re_\lambda =433$ in Byron et al. (Reference Byron, Einarsson, Gustavsson, Voth, Mehlig and Variano2015). (b) The ratio of mean spinning rate squared to mean tumbling rate squared, $\langle {\varOmega }_i^{s}{\varOmega }_i^{s}\rangle /\langle \dot {p}_i\dot {p}_i\rangle$, as a function of size $D_v/\eta$. (c) The cartoon illustrates that the weight of the tumbling rate in the angular velocity of prolate spheroids becomes larger as the particle size increases. (d) The ratio of spinning rate squared to tumbling rate squared, $\langle \varOmega _i^{s}\varOmega _i^{s}\rangle /\langle \dot {p}_i\dot {p}_i\rangle$, as a function of aspect ratio $\alpha$ for VA and VP. The colour bar represents the normalized particle size $D_v/\eta$ in logarithmic scale.

What happens when particles have a finite size? A naive K41-based prediction would suggest that the tumbling scales with the length $\langle \dot {p}_i\dot {p}_i\rangle \sim l^{-4/3}$ and that the spinning scales with the particle diameter $\langle {\varOmega }_i^{s}{\varOmega }_i^{s} \rangle \sim 0.5 d^{-4/3}$ (where the 0.5 factor comes from the matching with the spherical case) so that

(3.2)\begin{equation} \frac{\langle {\varOmega}_i^{s}{\varOmega}_i^{s}\rangle}{\langle\dot{p}_i\dot{p}_i\rangle} \sim \frac{\alpha^{4/3}}{2} . \end{equation}

However, this prediction falls largely off the result of the simulations, where we see a marked decreasing $D_v$ dependence of the ratio of the mean square spinning to tumbling rate (see figure 7a,b). Larger particles in proportion spin less and tumble more. This is indeed true for all aspect ratios except for spheres (see figure 7d). It has been pointed out by Oehmke et al. (Reference Oehmke, Bordoloi, Variano and Verhille2021) that these types of dimensional reasonings ignore the decorrelation of the velocity field along the particle surface. The rotation rates stem from the integral of the velocity gradients along the particle surface and, as such, e.g. the spinning rate can be much attenuated in long fibres ($\alpha \gg 1$). This integral effect is possibly more important when both the diameter and the length are much larger than the dissipation scale. As a consequence, the scaling behaviours of the spinning and tumbling rates can deviate from K41-based scalings as another characteristic length scale is involved, and the ratio $\langle {\varOmega }_i^{s}{\varOmega }_i^{s}\rangle /\langle \dot {p}_i\dot {p}_i\rangle$ is likely a function of the particle size.

The experiments by Oehmke et al. (Reference Oehmke, Bordoloi, Variano and Verhille2021), the only ones to date to have measured both the spinning and tumbling of finite-size particles, find that, for prolate fibres, the variance of the spinning rate is always larger than the variance of the tumbling rate. These authors comment that the observed behaviour ‘surprisingly resembles that observed previously for sub-Kolmogorov fibers’. These observations are fully confirmed by our simulations (see Appendix B for a more detailed comparison). However, we additionally observe a new feature; the ratio $\langle {\varOmega }_i^{s}{\varOmega }_i^{s}\rangle /\langle \dot {p}_i\dot {p}_i\rangle$ for oblate and prolate spheroids shows systematic decreasing trends as the particle equivalent size increases, and this for both $Re_\lambda =32$ and $Re_\lambda =120$. For oblate spheroids ($\alpha =0.5$), the ratio becomes vanishingly small on increasing their size, which indicates that the oblate spheroids hardly spin when their size is large compared with the dissipation scale. Similarly, prolate spheroids of large size reduce their spinning to the point that it becomes smaller than tumbling (figure 7a).

Figure 7(b) shows the ratio $\langle {\varOmega }_i^{s}{\varOmega }_i^{s}\rangle /\langle \dot {p}_i\dot {p}_i\rangle$ as a function of the equivalent particle size $D_v/\eta$. It is found that the ratio $\langle {\varOmega }_i^{s}{\varOmega }_i^{s}\rangle /\langle \dot {p}_i\dot {p}_i\rangle \sim (D_v/\eta )^{-0.6}$ for all considered shapes. We remark that the existence of the observed scaling behaviour of the ratio $\langle {\varOmega }_i^{s}{\varOmega }_i^{s}\rangle /\langle \dot {p}_i\dot {p}_i\rangle$ is not certain and calls for future numerical and experimental confirmations. However, this emphasizes that the scaling exponents of the mean square tumbling rate and spinning rate cannot both be equal to $-4/3$. To our knowledge, no presently available model accounts for the observed behaviour.

In summary, the main feature connected to the finite size is that larger particles tend to tumble much more than spin (see cartoon figure 7c). Further, we observe that the VA and VP measurements do not have this property. While VA nearly keeps the value of the Jeffery randomly oriented prediction (3.1), VP do not sensibly vary the relative weight of spinning with respect to tumbling and apparently approximately follow the simplistic K41 prediction, (3.2), and (red line) 7(d).

In order to determine the leading physical mechanism responsible for the decrease of $\langle {\varOmega }_i^{s}{\varOmega }_i^{s}\rangle /\langle \dot {p}_i\dot {p}_i\rangle$, we carry out numerical experiments with particles whose location is fixed, either with the solid–fluid interface or without it, denoted respectively as RPFL and VPFL, as shown in figure 8. The rationale behind fixing the position is to rule out the role of preferential sampling of the flow position (Maxey Reference Maxey1987; Squires & Eaton Reference Squires and Eaton1991; Calzavarini et al. Reference Calzavarini, Kerscher, Lohse and Toschi2008). Indeed, even if this has never been observed, for finite-sized spherical particles (Fiabane et al. Reference Fiabane, Zimmermann, Volk, Pinton and Bourgoin2012), we can in principle not exclude it for non-spherical large particles. On the other hand, a direct measurement of preferential concentration in the present simulations is not an easy task due to the limited number of particles that we could place in the domain.

Figure 8.

Mean squared angular velocity, normalized by the results of tracers with corresponding aspect ratio, as a function of $D_v/\eta$ for the numerical experiments of (a) RPFL and (b) VPFL, compared with the results of RP. Insets show the ratio of the mean squared angular velocity of the numerical experiments to the results of RP as a function of $D_v/\eta$. The ratio of the mean spinning rate squared to the mean tumbling rate squared, $\langle \varOmega _i^{s}\varOmega _i^{s}\rangle /\langle \dot {p}_i\dot {p}_i\rangle$, as a function of aspect ratio $\alpha$ for the numerical experiments of (c) RPFL and (d) VPFL, respectively. The colour bar represents the normalized particle size $D_v/\eta$ in logarithmic scale.

We find that the mean normalized rotation rate variance of RPFL scales similarly as RP and has a similar amplitude (see figure 8a). Furthermore, $\langle {\varOmega }_i^{s}{\varOmega }_i^{s}\rangle /\langle \dot {p}_i\dot {p}_i\rangle$ decreases as the particle size increases (see figure 8c), which is consistent with RP. These results indicate that the possible preferential sampling is not responsible for the decrease of $\langle {\varOmega }_i^{s}{\varOmega }_i^{s}\rangle /\langle \dot {p}_i\dot {p}_i\rangle$. To further confirm this point, we show the rotational dynamics of VPFL in figure 8(b,d). Similar scaling behaviour of mean rotation rate squared as a function of particle size is observed, and $\langle {\varOmega }_i^{s}{\varOmega }_i^{s}\rangle /\langle \dot {p}_i\dot {p}_i\rangle$ remains constant at the value of the fixed tracers for the corresponding aspect ratio. Since VP do not have a solid interface and do not affect the flow, the results of these two additional numerical experiments suggest that the two-way coupling between the particles and the flow is responsible for the decrease of the ratio $\langle {\varOmega }_i^{s}{\varOmega }_i^{s}\rangle /\langle \dot {p}_i\dot {p}_i\rangle$.

3.3.1. The role of particle boundary layers

At this point, given the fact that fully resolved simulations allow us to virtually measure any fluid quantity, one may wonder if it is possible to better characterize the effect of the particle feedback on the flow. With this in mind, we measure the fluid vorticity $\boldsymbol {\omega }$ and the rate of strain tensor $\boldsymbol {S}$ in the boundary layers surrounding the particles. We then compare the relative importance of their intensities. This is performed by computing the variance ratio $\tfrac {1}{4}\overline {\boldsymbol {\omega }^{2}} / \overline {\boldsymbol {S}:\boldsymbol {S}}$, where the average is performed over time, ensemble, i.e. multiple trajectories, and over the azimuthal direction in order to improve the statistical convergence. We note that the ratio must have the value 1 in particle-free HIT flow.

Figure 9(a–c) shows three typical fields of the considered ratio around the particle up to the size of $1.5D_v$. It is found that $\tfrac {1}{4} \overline {\boldsymbol {\omega }^{2}} / \overline {\boldsymbol {S}:\boldsymbol {S}} \simeq 1.1$ (vorticity dominated) in a thin shell around the particle, while at immediately larger distances a wider region displays $\tfrac {1}{4} \overline {\boldsymbol {\omega }^{2}} / \overline {\boldsymbol {S}:\boldsymbol {S}} <1$ (strain dominated), finally beyond ${\sim } 1.5 D_v$ any effect has vanished.

Figure 9.

Field of the ratio $\tfrac {1}{4} \overline {\boldsymbol {\omega }^{2}} / \overline {\boldsymbol {S}:\boldsymbol {S}}$ at (a) $\alpha =1$, $D_v = 20$ (b) $\alpha =0.5$, $D_v = 25.1$ and (c) $\alpha =2$, $D_v =25.1$ at $Re_{\lambda } =120$. The colour bar indicates the value of $\tfrac {1}{4} \overline {\boldsymbol {\omega }^{2}} / \overline {\boldsymbol {S}:\boldsymbol {S}}$. (d) The ratio of $\langle \tfrac {1}{4} \overline {\boldsymbol {\omega }^{2}} \rangle |_{1.5D_v}$ to $\langle \overline {\boldsymbol {S}:\boldsymbol {S}} \rangle |_{1.5D_v}$ as a function of $D_v/\eta$.

The quantity $\langle \tfrac {1}{4} \overline {\boldsymbol {\omega }^{2}} \rangle |_{1.5D_v} / \langle \overline {\boldsymbol {S}:\boldsymbol {S}} \rangle |_{1.5D_v}$, where $\langle \ldots \rangle _{1.5D_v}$ is a volume average over shells up to $1.5 D_v$, for different aspect ratios and particle sizes is reported in figure 9(d). We can observe that the overall effect in the region affected by the particle presence is to decrease the importance of vorticity with respect to strain. This points to the fact that spinning is possibly reduced as a consequence of the reduction of vorticity in the flow surrounding a particle, in qualitative agreement with previous observations. However, we think that more sensitive tests remain to be performed in order to soundly quantify the effect of feedback on the flow.