3.1.1. Tensor eigendecomposition

To separate the isotropic and anisotropic contributions of the particle cloud geometry, the normalised anisotropy tensor (a symmetric and trace-free tensor) is introduced as

(3.9) \begin{align} \boldsymbol{A} = \frac {\boldsymbol{G}}{\textrm{tr}(\boldsymbol{G})} - \frac {1}{3}\,\boldsymbol{I} \kern-1pt , \end{align}

where $\boldsymbol{I}$ denotes the identity matrix and $\textrm{tr}(\boldsymbol{G})$ is the trace of the gyration tensor, used to normalise it. Then the eigendecomposition of $\boldsymbol{A}$ can be expressed as

(3.10) \begin{align} \boldsymbol{A} = \boldsymbol{E}\,\boldsymbol{\varLambda }\,\boldsymbol{E}^{\top } \kern-1pt, \end{align}

where $\boldsymbol{E}\in \mathbb{R}^{3\times 3}$ is the orthonormal matrix whose columns are the eigenvectors of $\boldsymbol{A}$ , and $\boldsymbol{\varLambda } = \textrm{diag}(\lambda _1,\lambda _2,\lambda _3)$ is the diagonal matrix of real eigenvalues, ordered such that $\lambda _1 \geqslant \lambda _2 \geqslant \lambda _3$ and satisfying $\textrm{tr}(\boldsymbol{\varLambda }) = 0$ . The eigenvalues $\lambda _i$ describe the shape of the particle cloud, whereas the eigenvectors define the principal orientation of the cluster in space. The resulting eigendecomposition reformulates the gyration tensor as

(3.11) \begin{align} \boldsymbol{G} = \textrm{tr}(\boldsymbol{G})\,\boldsymbol{E}\!\left (\boldsymbol{\varLambda } + \frac {1}{3}\boldsymbol{I}\right )\!\boldsymbol{E}^{\top } \kern-1pt. \end{align}

This representation isolates the trace, $\textrm{tr}(\boldsymbol{G})$ , which quantifies the overall spatial extent of the particle cloud independent of shape and orientation. The deviatoric component then characterises the directional distribution of this extent; described by the eigenvalues and eigenvectors of the anisotropy tensor, it encodes the cloud geometry to distinguish between rod-like, disk-like and isotropic morphologies.

3.1.2. Barycentric & polar maps

Analysing in detail the real eigenvalues emerging from the anisotropic tensor decomposition, three limiting states exist: (i) one-component (rod-like), where $2/3 = \lambda _{1} \gt \lambda _{2} = \lambda _{3} = -1/3$ ; (ii) two-component axisymmetric (disk-like) with $1/6 = \lambda _{1} = \lambda _{2} \gt \lambda _{3} = -1/3$ ; and (iii) three-component (isotropic) characterised by $\lambda _{1} = \lambda _{2} = \lambda _{3} = 0$ . The anisotropy tensor shapes and their limiting states can be visualised, for instance, in terms of the barycentric map (Banerjee et al. Reference Banerjee, Krahl, Durst and Zenger2007; Jofre et al. Reference Jofre, Domino and Iaccarino2019), which relies on the fact that any anisotropic state is a convex combination of the three limiting states. In a Euclidean space, these can be represented as the vertices of an equilateral triangle with coordinates $\boldsymbol {x}_{1c} = (0,0)$ , $\boldsymbol {x}_{2c} = (1,0)$ and $\boldsymbol {x}_{3c} = (1/2,\sqrt {3}/2)$ . A graphical representation of the map and the different anisotropy shapes is illustrated in figure 1(b) where the limiting cases occupy the corners of the triangle while their combinations are included in the interior. One of the main advantages is that it provides a linear relation between anisotropy eigenvalues and Euclidean space through the projection

(3.12) \begin{align} \boldsymbol {x} = \boldsymbol {x}_{1c}\left (\lambda _{1} - \lambda _{2} \right ) + 2\boldsymbol {x}_{2c} \left (\lambda _{2} - \lambda _{3}\right ) + \boldsymbol {x}_{3c}\left (3 \lambda _{3} + 1\right )\!, \end{align}

which, together with the requirement that the eigenvalues sum to zero, is a unique invertible linear mapping.

In addition to the eigenvalue-based description provided by the diagonal matrix $\boldsymbol{\varLambda }$ , a complementary analysis addresses the orientation of the eigenvectors contained in the orthonormal matrix $\boldsymbol{E}$ with respect to the reference coordinate system, as schematically illustrated in figure 1(c). This aspect is particularly relevant in the context of rotating turbulence, where anisotropic forcing imposes preferred directions (most notably the rotation axis) on the particle distribution. To quantify such orientation statistics, a polar-map representation is employed where each eigenvector $\boldsymbol{e} = (e_{x}, e_{y}, e_{z})^{\textrm {T}}$ is expressed in spherical coordinates. Following standard physics conventions, the polar angle $\theta = \arccos (|e_{z}|)$ measures the inclination of the eigenvector with respect to the vertical axis, while the azimuthal angle $\phi = \operatorname {atan2}(e_{y}, e_{x})$ specifies the direction of its projection onto the horizontal plane. Since the eigenvectors are normalised ( $|\boldsymbol{e}| = 1$ ), their orientation is fully characterised by the pair $(\theta ,\phi )$ , allowing the 3-D principal directions of the particle cloud to be visualised on a 2-D polar map. In this framework, isotropic configurations are characterised by equal eigenvalues ( $\lambda _1=\lambda _2=\lambda _3$ ), corresponding to the isotropic vertex of the realisability triangle, and by uniformly distributed eigenvector orientations over the unit sphere. The latter implies a probability measure $ \textit{dP} = (4\pi )^{-1}\sin \theta \,\text{d}\theta \,\text{d}\phi$ , which, upon integration over the azimuthal angle and adopting the director definition $\theta =\arccos (|\boldsymbol{e}\boldsymbol{\cdot }\hat {\boldsymbol{z}}|)$ , yields the isotropic reference $P_{\textit{iso}}(\theta )=\sin \theta$ over $\theta \in [0,\pi /2]$ . Deviations from isotropy manifest as both a displacement of the barycentric coordinate (reflecting shape anisotropy) and as directional clustering in the polar map, indicating preferential alignment of the principal axes. Collectively, these two representations provide a unified description of particle-cluster anisotropy in terms of both shape and orientation.

4.3.1. Qualitative characterisation of the particle field

Figure 5 provides a qualitative comparison of the various configurations by displaying instantaneous particle distributions alongside their corresponding turbulent structures. Both the carrier flow and the particle field, partitioned into adjacent halves of the domain for visual clarity, are visualised through isosurfaces and point clouds, respectively, coloured by their normalised kinetic energy to provide an intuitive measure of the local intensity of motion. The spatial distribution of the particle kinetic energy, $E^{\textit{p}}_{\textit{kin}}(\boldsymbol{x}_{p})$ , is observed to mirror the underlying flow structure except in the most extreme cases, indicating that inertial particle clustering remains closely coupled to the carrier turbulence. In the HIT reference case (panels a–c), tracers ( $ \textit{St}_{\eta }=0$ ) are distributed nearly uniformly (panel a), whereas particles with $ \textit{St}_{\eta }=10^{0}$ exhibit the most pronounced clustering. These particles form elongated filamentary structures consistent with the preferential concentration mechanism driven by the competition between local strain and vorticity (Bec et al. Reference Bec, Biferale, Cencini, Lanotte, Musacchio and Toschi2007; Ireland et al. Reference Ireland, Bragg and Collins2016), as seen in panel (b). For higher inertia ( $ \textit{St}_{\eta }=10^{1}$ , panel c), the clusters appear more diffuse, as the particle response time significantly exceeds the characteristic eddy turnover time. Additionally, the range of particle kinetic energy is found to be sensitive to the particle inertia.

Figure 5.

Three-dimensional visualisations of the carrier-flow structure and dispersed particle field at $t/T_{L_0}=65$ . The flow and particle fields are shown via the normalised kinetic energy $E_{\textit{kin}}$ and $E^{\textit{p}}_{\textit{kin}}(\boldsymbol{x}_{p}) = |\boldsymbol{v}_{p}(\boldsymbol{x}_{p})|^{2}/2$ . Rows correspond to increasing rotation: HIT (a–c), $ \textit{Ro}=10^{0}$ (d–f), $ \textit{Ro}=5\times 10^{-1}$ (g–i), and $ \textit{Ro}=10^{-1}$ (j–l). Columns denote $ \textit{St}_{\eta }=0$ (left), $10^{0}$ (centre) and $10^{1}$ (right).

Figure 5. Long description

Panel A: A 3-D visualization showing the carrier-flow structure and dispersed particle field with no rotation. The flow and particle fields are represented by the normalized kinetic energy. Panel B: A 3-D visualization showing the carrier-flow structure and dispersed particle field with low rotation. The flow and particle fields are represented by the normalized kinetic energy. Panel C: A 3-D visualization showing the carrier-flow structure and dispersed particle field with moderate rotation. The flow and particle fields are represented by the normalized kinetic energy. Panel D: A 3-D visualization showing the carrier-flow structure and dispersed particle field with high rotation. The flow and particle fields are represented by the normalized kinetic energy. Panel E: A 3-D visualization showing the carrier-flow structure and dispersed particle field with very high rotation. The flow and particle fields are represented by the normalized kinetic energy. Panel F: A 3-D visualization showing the carrier-flow structure and dispersed particle field with extremely high rotation. The flow and particle fields are represented by the normalized kinetic energy. Panel G: A 3-D visualization showing the carrier-flow structure and dispersed particle field with no rotation. The flow and particle fields are represented by the normalized kinetic energy. Panel H: A 3-D visualization showing the carrier-flow structure and dispersed particle field with low rotation. The flow and particle fields are represented by the normalized kinetic energy. Panel I: A 3-D visualization showing the carrier-flow structure and dispersed particle field with moderate rotation. The flow and particle fields are represented by the normalized kinetic energy. Panel J: A 3-D visualization showing the carrier-flow structure and dispersed particle field with high rotation. The flow and particle fields are represented by the normalized kinetic energy. Panel K: A 3-D visualization showing the carrier-flow structure and dispersed particle field with very high rotation. The flow and particle fields are represented by the normalized kinetic energy. Panel L: A 3-D visualization showing the carrier-flow structure and dispersed particle field with extremely high rotation. The flow and particle fields are represented by the normalized kinetic energy.

The most distinctive and novel features emerge with the introduction of rotation, as the particle organisation deviates significantly from the behaviour observed in the HIT case and begins to reflect the anisotropic structure of the carrier flow. In the moderately rotating case ( $ \textit{Ro}=10^{0}$ , panels d–f), a clear and systematic trend emerges as particle inertia increases. Even tracer particles ( $ \textit{St}_{\eta }=0$ ) exhibit signatures of anisotropy in response to the columnar vortices, exhibit the imprint of flow anisotropy in their particle-carried kinetic energy, reflecting the underlying columnar vortices. Particles with $ \textit{St}_{\eta }=10^{0}$ show even more pronounced exclusion, being expelled from the vortex cores and accumulating along the periphery, where centrifugal ejection and Coriolis-induced trapping reach a dynamical balance. At higher inertia ( $ \textit{St}_{\eta }=10^{1}$ ), this effect is further intensified; as a result of centrifugal ejection, the vortices become almost entirely devoid of particles. These particles migrate toward the annular regions encircling the vortex columns, forming sharply defined bands of high concentration.

An intermediate regime is observed at $ \textit{Ro} = 5 \times 10^{-1}$ (panels g–i), where the influence of rotation becomes more pronounced while retaining features of the moderate rotation case. Compared with $ \textit{Ro}=10^{0}$ , the flow reorganises into multiple large-scale vortical structures, which begin to interact and populate the domain more densely. This transition is accompanied by a clear amplification of rotation-induced particle dynamics. Tracer particles (panel g) already exhibit enhanced columnar appearance, where the structure of particle energy resembles that of the columnar vortices. For finite inertia, the centrifugal expulsion mechanism becomes more effective: particles with $ \textit{St}_{\eta }=10^{0}$ are more systematically excluded from the vortex cores and accumulate in elongated structures between neighbouring columns while at higher inertia (panel i), this effect is further intensified, leading to the formation of more extended particle-depleted regions within the vortices and sharper, more coherent clustering in the interstitial zones. Overall, the $ \textit{Ro}=5 \times 10^{-1}$ configuration represents a transitional regime in which the key rotational mechanisms (centrifugal expulsion, preferential concentration and the emergence of void regions) are strengthened relative to $ \textit{Ro}=10^{0}$ , while not yet reaching the fully columnar, quasi-2-D organisation observed at lower Rossby numbers.

A distinct phenomenology emerges in the strongly rotating regime ( $ \textit{Ro} = 10^{-1}$ , panels j–l), where the vortices exhibit smaller horizontal scales and are organised into long, vertically coherent columnar structures as a consequence of the increased rotation. In this case, the tracers within the vortex cores exhibit kinetic energy levels distinct from those in the surrounding fluid, as evidenced by the contrasting colour bands. Concurrently, inertial particles no longer merely skirt the periphery of isolated vortices; instead, they align into slender, vertically extended bands that closely follow the dominant columnar structures. This progressive transition (from nearly isotropic filamentary clustering in HIT to vertically coherent columnar alignment at low Rossby numbers) reflects the interplay between inertial filtering (the reduced responsiveness of particles to high-frequency fluid fluctuations due to their finite response time) and rotational anisotropy in shaping the morphology of particle concentrations. Such an extreme regime, in which centrifugal forcing dominates particle dynamics and leads to rapid radial expulsion and vertically coherent organisation, has also been documented by Maity (Reference Maity2023). In that framework, particles are centrifuged out of the computational domain while still experiencing a statistically equivalent carrier flow due to the imposition of periodic boundary conditions.

Interestingly, with the exception of the ‘degenerate’ cases (figure 5 k,l), the normalised kinetic energy levels (referenced via the left colourbar in figure 5) reveal that tracer particles attain the highest kinetic energy levels, reflecting their capacity to faithfully follow the most intense fluid fluctuations. As the Stokes number increases, the maximum particle kinetic energy decreases; specifically, for $ \textit{St}_{\eta } \gt 10^{0}$ , the dynamics is dominated by inertial filtering, where particles are unable to respond to the most rapid fluid fluctuations. This results in $E_{\textit{kin}}^{\textit{p}} \lt E_{\textit{kin}}$ on average, which is consistent with observations from prior studies (Ireland et al. Reference Ireland, Bragg and Collins2016; Brandt & Coletti Reference Brandt and Coletti2022). Furthermore, particles tend to accumulate in relatively low-energy regions where they decelerate, thereby increasing their residence time in these zones (Pietrzyk et al. Reference Pietrzyk, Horwitz, Najjar and Minich2022). Although these mechanisms have been primarily established in the context of HIT, the rotating cases analysed here demonstrate that a similar interplay between preferential sampling and inertial filtering persists, albeit fundamentally reshaped by the anisotropic, columnar architecture of the carrier flow.

Figure 6.

(a) Ratio between the average particle kinetic energy and the average carrier-flow kinetic energy $E_{\textit{kin}}$ for HIT, rotating turbulence. The solid black line denotes the Abrahamson–Ireland inertial filtering model. (b) Ratio between the average fluid kinetic energy sampled along particle trajectories for the same cases. Tracer values are fixed at unity by construction. The two panels separate the effects of inertial filtering and preferential sampling on the particle kinetic energy budget.

To substantiate these qualitative observations, the particle and the fluid kinetic energy sampled along particle trajectories are examined below. These quantities make it possible to quantify both the attenuation of particle kinetic energy due to inertial filtering and the tendency of inertial particles to preferentially sample regions of lower-than-average carrier-flow kinetic energy (Ireland et al. Reference Ireland, Bragg and Collins2016). With the aim to obtain statistically independent estimates, the analysis is performed over seven representative flow–particle corresponding snapshots separated by $2T_{L0}$ , a time interval sufficiently large to reduce temporal correlations, while collectively covering the full steady-state portion of the simulation. Figure 6(a) shows the ratio between the ensemble-average particle kinetic energy, $E_{\textit{kin}}^{p}(St_\eta )$ , and the average carrier-flow kinetic energy, $E_{\textit{kin}}$ . In HIT, $E_{\textit{kin}}^{p}/E_{\textit{kin}}$ decreases monotonically with $ \textit{St}_\eta$ , manifesting the inertial filtering phenomenon: finite-inertia particles are unable to follow the fastest carrier-flow fluctuations and, therefore, retain less kinetic energy. The solid line reports the Abrahamson model (Abrahamson Reference Abrahamson1975; Ireland et al. Reference Ireland, Bragg and Collins2016),

(4.15) \begin{align} \frac {E_{\textit{kin}}^{p}}{E_{\textit{kin}}}\approx \frac {1}{1+\tau _{\kern-1pt p}/T_L} = \frac {1}{1+ St_{\eta } \tau _{\eta }/T_L}, \end{align}

where $T_L$ denotes the fluid Lagrangian integral time scale, determined from the tracer velocity autocorrelation in the HIT case. From the present DNS, a value of $T_L / \tau _{\eta } \approx 15$ is obtained, which is in good agreement with the value $T_L / \tau _{\eta } \approx 12.5$ predicted by the analytical expression of Zaichik et al. (Reference Zaichik, Simonin and Alipchenkov2003), $T_{L}/\tau _{\eta } = 2 (Re_{\kern-1pt \lambda } + C_{1})/(\sqrt {15}\, C_{0})$ , evaluated at $ \textit{Re}_{\kern-1pt \lambda } = 90$ using the semi-empirical constants $C_{0} = 7$ and $C_{1} = 32$ . This agreement indicates that the model captures the overall attenuation trend and provides a consistent non-rotating reference. The rotating cases show a substantially stronger reduction of $E_{\textit{kin}}^{p}/E_{\textit{kin}}$ , especially at $ \textit{Ro}=5 \times 10^{-1}$ , indicating that background rotation enhances the effective filtering of the carrier-flow fluctuations.

Since the Abrahamson–Ireland formulation assumes isotropic exponential decorrelation with a single scalar $T_L$ , it is not applied directly to the rotating cases, and is shown only as an HIT reference; however, it is nevertheless instructive to report the corresponding values obtained from the present DNS. At $ \textit{Ro}=10^{0}$ , the horizontal components remain approximately equal, $T_{L}/ \tau _{\eta } \simeq T_{L}/ \tau _{\eta } \approx 10$ , while the vertical component is reduced, $T_{L,z} \lt T_{L,x,y}$ . For $ \textit{Ro}=5 \times 10^{-1}$ , the horizontal components remain approximately equal, $T_{L}/\tau_{\eta} \simeq T_{L}/\tau_{\eta} \approx 6$ , and all components exhibit a significant reduction in magnitude, with a tendency toward reduced directional disparity, reflecting the increasing dominance of coherent columnar structures as observed in recent works (Maity Reference Maity2023; Polanco et al. Reference Polanco, Arun and Naso2023).

The analysis is further complemented by a quantification of sampling bias. Figure 6(b) isolates preferential sampling through the ratio $E_{\textit{kin}}(\boldsymbol{x}_{\kern-1pt p})/E_{\textit{kin}}$ , defined as the mean carrier-flow kinetic energy sampled at particle positions normalised by the Eulerian domain average. While tracers recover unity by construction, inertial particles systematically sample regions of lower-than-average kinetic energy. This effect is weak in HIT but becomes much stronger as rotation increases, with the largest deficit observed for $ \textit{Ro}=5 \times 10^{-1}$ and $ \textit{St}_\eta =10^{1}$ . The results therefore show that the reduction of particle kinetic energy in rotating turbulence is controlled by the combined action of enhanced inertial filtering and stronger preferential sampling of low-energy regions. From a physical perspective, rotation expels inertial particles from the energetic cores of the coherent columnar vortices and promotes residence in their outer layers and in the surrounding inter-vortex regions, where the local kinetic energy is lower on average.

The scenario that emerges is that rotating turbulence follows an intuitive mechanism. Whereas increasing $ \textit{Re}_{\kern-1pt \lambda }$ would enrich the range of turbulent time scales larger than the particle response time and, therefore, would weaken inertial filtering, increasing rotation (viz. decreasing $ \textit{Ro}$ ) suppresses small-scale turbulent fluctuations and narrows the range of energetic scales able to drive particle motion. This enhances the kinetic energy deficit of the dispersed phase. In parallel, stronger rotation amplifies preferential sampling by promoting particle residence in low-energy regions, such as the shear layers and outer peripheries of coherent columnar vortices, rather than within their energetic cores. This effect, related to the centrifugal expulsion discussed in the next section, becomes increasingly pronounced with particle inertia.

Figure 7.

Instantaneous visualisation of the flow field and particle spatial distribution for (a) HIT, (b) rotating turbulence at $ \textit{Ro}=10^{0}$ , (c) rotating turbulence at $ \textit{Ro}=5 \times 10^{-1}$ , and (d) rotating turbulence at $ \textit{Ro} = 10^{-1}$ . For each case, the top left subpanel shows the normalised planar kinetic energy field, defined as $E_{\textit{kin},\perp }=u_x^2 + u_y^2$ , within an $x$ – $y$ slice, while the accompanying subpanels depict the instantaneous particle spatial distribution within a slab of finite thickness ( $2\times \eta$ ) centred on the same plane.

To complement the 3-D visualisations and concentrate on the dynamics most characteristic of rotating turbulence, the focus is now placed on particle organisation within the horizontal $(x,y)$ plane. Both particle inertia and the structure of the underlying flow leave a distinct imprint on the spatial distribution, as seen in the planar snapshots of figure 7. This planar visualisation approach, frequently employed in studies of inertial suspensions (Bec et al. Reference Bec, Biferale, Boffetta, Celani, Cencini, Lanotte, Musacchio and Toschi2006; Monchaux et al. Reference Monchaux, Bourgoin and Cartellier2012), facilitates a direct comparison with the classical phenomenology of preferential concentration in HIT. Consistent with previous DNS results (Chen et al. Reference Chen, Goto and Vassiliscos2006; Goto & Vassilicos Reference Goto and Vassilicos2006; Yoshimoto & Goto Reference Yoshimoto and Goto2007) and laboratory experiments (Eaton & Fessler Reference Eaton and Fessler1994), tracers remain nearly uniformly distributed, exhibiting no significant tendency to accumulate. As inertia increases (panels b and c), particles deviate from fluid trajectories and form increasingly heterogeneous patterns. Specifically, for $ \textit{St}_{\eta }=10^{0}$ , clustering is most pronounced, giving rise to filamentary and cellular-void cluster structures. For $ \textit{St}_{\eta }=10^{1}$ , the distribution becomes more diffuse as inertial filtering suppresses the response to small-scale turbulent motions. These trends align with the well-established properties of inertial clustering in HIT (Bec et al. Reference Bec, Biferale, Boffetta, Celani, Cencini, Lanotte, Musacchio and Toschi2006; Calzavarini et al. Reference Calzavarini, Kerscher, Lohse and Toschi2008).

Remarkably, these visualisations extend the canonical picture of preferential concentration to the rotating regime, revealing novel organisational mechanisms induced by background rotation. The underlying physics is inherently complex, as rotation not only modifies the local force balance acting on the particles but also reorganises the carrier flow into large-scale coherent vortices that modulate inertial segregation. For the moderate rotation case ( $ \textit{Ro}=10^{0}$ , panel b), the flow self-organises into columnar vortices aligned with the axis of rotation. In this regime, the particle field develops annular concentration patterns encircling the vortex cores, arising from the interplay between centrifugal ejection and Coriolis-induced confinement. As illustrated in figure 7(b), the tracer spatial distribution displays depleted zones coinciding with the vortex interiors, while inertial particles are progressively expelled toward the peripheries as $ \textit{St}_{\eta }$ increases, forming dense rings at the vortex boundaries. Consequently, this radial migration mechanism becomes increasingly dominant with rising particle inertia.

At the intermediate rotation rate, $ \textit{Ro}=5\times 10^{-1}$ (panel c), the flow already exhibits a clear departure from the single-vortex organisation observed at weaker rotation. Multiple coherent vortical structures coexist in the horizontal plane, producing a more complex spatial pattern of particle segregation. The particle field responds directly to this multi-vortex organisation: tracers remain broadly distributed while reflecting the underlying vortical topology, whereas inertial particles are increasingly expelled from the vortex interiors and accumulate along the interfaces between neighbouring structures. As $ \textit{St}_{\eta }$ increases, these inter-vortex regions become preferential accumulation zones, leading to more localised and filamentary concentrations around the vortex borders, thereby enhancing the clustering effect in such low levels of kinetic energy.

Finally, under strong rotation ( $ \textit{Ro}=10^{-1}$ ), the phenomenological landscape changes significantly. The flow becomes quasi-2-D, dominated by persistent vortical columns, hallmarks of the split energy cascade and emergent medium-scale coherent structures that are smaller than the integral scale $L_{0}$ . The dispersed phase responds accordingly: tracer particles organise anisotropically within the horizontal planes, forming inhomogeneous patterns that reflect the underlying columnar structures. Conversely, inertial particles with $ \textit{St}_{\eta }=10^{0}$ and $ \textit{St}_{\eta }=10^{1}$ align into vertically coherent columns. This emergence of vertically aligned particle bands highlights the prevalence of rotational anisotropy and the suppression of turbulent dispersion along the axis of rotation, an effect that is analysed in detail in § 4.4.

Figure 8.

Evolution of particle acceleration components normalised by the reference Stokes drag acceleration $a_{ref}$ . The trends highlight the relative contributions of Stokes drag (solid), Coriolis (dashed) and centrifugal (dotted) terms across a range of Stokes numbers, $\textit{St}_{\eta}$ . (b) Regime diagram in the $\textit{St}_{\eta}$ – $ \textit{Ro}$ parameter space. The dashed–dotted line represents the marginal stability curve delineating the transition between the particle accumulation given by the centrifugal components; symbols indicate the specific simulation cases investigated in the present study.

Figure 8. Long description

Panel A: Two line graphs depict the evolution of particle acceleration components normalized by the reference Stokes drag acceleration. The x-axis represents time normalized by a reference time scale (t/T_L0) ranging from 50 to 64. The y-axis represents the normalized acceleration (a_rms/a_ref) on a logarithmic scale from 10^-2 to 10^2. The graphs compare different Rossby numbers (Ro) and Stokes numbers (St_η). The trends highlight the relative contributions of Stokes drag (solid lines), Coriolis (dashed lines), and centrifugal (dotted lines) terms. The top graph shows data for St_η = 10^0, and the bottom graph shows data for St_η = 10^1. Panel B: A scatter plot represents a regime diagram in the parameter space of Stokes number (St_η) on the y-axis and Rossby number (Ro) on the x-axis, both on logarithmic scales. The dashed-dotted line represents the marginal stability curve delineating the transition between particle accumulation given by the centrifugal components. Symbols indicate specific simulation cases investigated in the present study, with colors representing the ratio of the square of the Coriolis acceleration to the square of the root mean square (rms) particle acceleration.

A fundamental question arises: Why do different rotation regimes lead to such distinct spatial organisations of the dispersed phase, even when the underlying flow structures appear qualitatively similar? The contrasting behaviours observed between the moderate and strong rotation cases can be elucidated by examining the particle acceleration within a rotating reference frame. Building upon the MRG equation, the total particle acceleration $\boldsymbol{a}_{p}^{\textit{tot}}$ is decomposed as

(4.16) \begin{align} \begin{cases} a_{\textit{rms}}^{*,\textit{tot}} = \left \langle \left | \displaystyle \frac {d \boldsymbol{v}^{*}_{p}}{d t^{*}} \right |^{2} \right \rangle _{\textit{prts}}^{1/2} = a_{\textit{rms}}^{*,\textit{St}} + a_{\textit{rms}}^{*,\textit{Co}} + a_{\textit{rms}}^{*,\textit{Ce}} \: , \\a_{\textit{rms}}^{*,\textit{St}} = \displaystyle \frac {1}{St_{\eta }} \Big \langle \big | \boldsymbol{u}^{*}_{p} - \boldsymbol{v}^{*}_{p} \big |^{2} \Big \rangle _{\textit{prts}}^{1/2} \: , \\a_{\textit{rms}}^{*,\textit{Co}} = \displaystyle \frac {1}{Ro} \Big \langle \big | \boldsymbol{\varOmega }^{*} \times \boldsymbol{v}^{*}_{p} \big |^{2} \Big \rangle _{\textit{prts}}^{1/2} \:, \\a_{\textit{rms}}^{*,\textit{Ce}} = \displaystyle \frac {1}{4Ro^{2}} \Big \langle \big | \boldsymbol{\varOmega }^{*} \times \big [ \boldsymbol{\varOmega }^{*} \times \big ( \boldsymbol{x}^{*}_{p} - \boldsymbol{x}^{*}_{0} \big ) \big ] \big |^{2} \Big \rangle _{\textit{prts}}^{1/2} \: , \end{cases} \end{align}

where $\langle \,\boldsymbol{\cdot }\,\rangle _{\textit{prts}}$ denotes an ensemble average over all particles and superscripts $\textit{tot}$ , $\textit{St}$ , $\textit{Co}$ and $\textit{Ce}$ denote the total acceleration, Stokes drag, Coriolis and centrifugal contributions, respectively. Notably, from the scaling of the MRG equation, the Coriolis term is proportional to $ \textit{Ro}^{-1}$ , whereas the centrifugal term increases more steeply, proportional to $ \textit{Ro}^{-2}$ , suggesting that it should eventually dominate the particle dynamics as rotation intensifies. This theoretical trend is confirmed by the time series of these accelerations shown in figure 8(a), where the HIT curves serve only as a reference for the non-rotating baseline. For the moderately rotating case ( $ \textit{Ro} = 10^{0}$ ; blue curves), the hierarchy of forces remains well ordered: Stokes drag is the leading contribution, followed by Coriolis acceleration, and finally, the centrifugal term, whose magnitude is relatively small. This balance explains why particles with $ \textit{St}_{\eta }=10^{0}$ preferentially orbit around vortex peripheries: drag keeps them tied to the local flow, while the weaker rotational effects only slightly modify their cross-stream migration. For $ \textit{St}_{\eta }=10^{1}$ , this separation between Stokes, Coriolis and centrifugal terms is reduced, yet Stokes drag still regulates the dynamics, preventing strong radial expulsion effects. This trend continues at the intermediate rotation rate of $ \textit{Ro}=5\times 10^{-1}$ (red curves). The force hierarchy remains closer to the moderate rotation case than to the extreme $ \textit{Ro}=10^{-1}$ regime: Stokes drag is still larger than the rotational contributions for $ \textit{St}_{\eta }=10^{0}$ , while for $ \textit{St}_{\eta }=10^{1}$ , the centrifugal term becomes slightly larger than the Stokes contribution. The Coriolis acceleration nevertheless remains subdominant, indicating that particle dynamics at this Rossby number are still primarily governed by drag, which prevents the strong radial ejection mechanism observed at $ \textit{Ro}=10^{-1}$ .

A qualitatively different regime emerges at strong rotation ( $ \textit{Ro} = 10^{-1}$ ; green curves). In this case, centrifugal acceleration becomes the dominant contribution for both Stokes numbers, exceeding Coriolis and drag by more than an order of magnitude. The relative importance of the remaining forces depends on inertia: for $ \textit{St}_{\eta }=10^{0}$ , the Stokes and Coriolis terms remain comparable, whereas for $ \textit{St}_{\eta }=10^{1}$ , Coriolis clearly surpasses Stokes drag, reflecting the weakened particle–flow coupling at large inertia. This force imbalance produces a strong radial ejection mechanism that drives particles away from vortex cores and into thin, vertically coherent shells, consistent with the extreme clustering patterns observed in figures 5(k), 5(l) and 7(d). As a result of rotation being centred within the domain, the periodic remapping of centrifugally expelled particles leads to vertically aligned bands that accumulate near the edges and corners of the computational box.

The scaling of the acceleration terms in (4.16) further suggests the existence of a transition in the $(St_{\eta },Ro)$ parameter space. In particular, comparing the Stokes and centrifugal contributions yields the condition

(4.17) \begin{align} \frac { \left\langle a_{\textit{rms}}^{*,\textit{Ce}} \right\rangle _{{t}}}{ \big\langle a_{\textit{rms}}^{*,\textit{St}} \big\rangle _{t}} \sim \frac {St_{\eta }}{Ro^{2}}, \end{align}

indicating that centrifugal effects become dominant when $ \textit{St}_{\eta } \gtrsim Ro^{2}$ . This balance defines a marginal curve separating regimes in which particle dynamics remain primarily controlled by drag from those in which rotational forcing governs particle transport. The regime diagram reported in figure 8(b) confirms this scaling and provides a useful and efficient interpretation of the different clustering pathways. To better populate the transition region in the $(St_{\eta },Ro)$ parameter space, three additional strongly rotating simulations at $ \textit{Ro}=10^{-1}$ were performed for low-inertia particles, $ \textit{St}_{\eta }=10^{-1}$ , $2.5\times 10^{-1}$ and $5.0\times 10^{-1}$ . These cases were used to evaluate the acceleration–balance ratio $a^{\textit{Ce}}/a^{\textit{St}}$ and to confirm the progressive approach to the marginal condition $ \textit{St}_{\eta }\sim Ro^{2}$ . In particular, the slender, vertically aligned particle structures observed in figures 5(k) and 5(l) emerge only when both sufficient inertia and strong rotation are present, corresponding to the region $ \textit{St}_{\eta } \gtrsim Ro^{2}$ . In contrast, intermediate cases below this threshold exhibit qualitatively distinct and less organised clustering patterns, demonstrating that varying $ \textit{St}_{\eta }$ and $ \textit{Ro}$ does not lead to equivalent dynamical states unless this combined condition is satisfied.

The visual patterns observed so far point to a profound reorganisation of particle clusters under rotation, yet their quantitative characterisation requires analyses beyond visual inspection. Thus, the following subsection turns to the gyration-tensor formalism presented in § 3.1 to rigorously assess the size, shape and anisotropy of these clusters.

4.3.2. Spatial distribution and particle clustering

The gyration tensor, introduced in § 3.1, provides a robust statistical framework for characterising the local spatial distribution of particle clusters. Its trace, $\textrm{tr}(\boldsymbol{G}) = G_{\alpha \alpha }$ , serves as a measure of the overall spatial dispersion (or squared radius of gyration) of a cluster. Simultaneously, its eigenvalues quantify the intrinsic geometry (distinguishing between spherical, filamentary or sheet-like morphologies), while the corresponding eigenvectors define the principal axes of orientation. Compared with traditional diagnostics such as RDFs or Voronoi tessellations, the gyration tensor offers a more comprehensive description by concurrently capturing size, shape and alignment. This multidimensional characterisation is particularly advantageous for analysing the anisotropic clustering and vertically coherent structures typical of rotating turbulent flows.

The statistical analysis is performed over seven instantaneous snapshots extracted from the statistically steady state, separated by approximately $2\,T_{L0}$ in order to ensure sufficient temporal decorrelation. At each snapshot, all particles are considered as potential reference points. Candidate clusters are defined as spherical sampling volumes of radius $R$ centred on each particle and characterised by a centroid (see § 3.2). However, for small occupancies, estimates of anisotropy and orientation can be dominated by discretisation noise, leading to spurious geometrical classifications. To ensure the statistical robustness of the gyration-tensor analysis, a two-stage selection and declustering protocol is adopted. First, the occupancy criterion: in three dimensions, resolving the principal axes of an equivalent ellipsoidal structure requires a number of samples exceeding the minimal degrees of freedom of the tensor, $N \geq d(d+1)/2$ , which provides a minimum value of six particles. Nonetheless, this represents only a lower bound and does not guarantee stability of the inferred geometry. Thus, in analogy with established analyses of turbulent structures (Moisy & Jiménez Reference Moisy and Jiménez2004), where reliable identification of filamentary and sheet-like morphologies requires adequately resolved point sets, a more stringent threshold of $N_{min }=20$ is adopted. Therefore, only samples satisfying $N \geqslant N_{min }$ are retained, ensuring convergence of the second-order moments entering the gyration tensor and providing robust estimates of its eigenvalues and eigenvectors. Second, to avoid oversampling the same physical cluster and to ensure statistical independence of the retained samples, a declustering procedure is applied. Specifically, once candidate clusters are selected, by satisfying the occupancy criterion, they are accepted only if their centroids are separated by a minimum distance $d_{min } = 2R$ from previously accepted clusters, using a periodic distance metric. This criterion effectively enforces a non-overlapping sampling of the particle field at the scale $R$ , preventing artificial bias due to multiple counting of the same structure. For completeness, Appendix C examines the sensitivity of the selection protocol to its two control parameters, the occupancy threshold $N_{min }$ and the minimum centroid separation $d_{min }$ . This analysis shows that, although these parameters affect the absolute number of retained clusters, the relative ordering among cases and the range of radii yielding statistically robust gyration-tensor statistics remain unchanged. The same cluster-based framework is also used to estimate the local mass loading in the most critical configurations, providing an order-of-magnitude assessment of the conditions under which particle feedback may become locally relevant.

Figure 9.

Number of accepted sampling spheres, $N_{\mathcal{S}_{\kern-1pt p}(R)}$ , normalised by the total number of particles $N_{\kern-1pt p}$ , as a function of the sampling radius $R/\eta$ . Samples are retained only if they satisfy the occupancy threshold $N \geqslant N_{min }=20$ and the declustering criterion. The shaded yellow region denotes the range of radii for which the number of accepted sampling spheres satisfies $N_{S_{p}(R)} \gt 10^2$ , corresponding to a statistically well-converged ensemble.

The impact of this selection criterion is illustrated in figure 9, which reports the number of accepted sampling spheres, $N_{\mathcal{S}_{\kern-1pt p}(R)}$ , normalised by the total number of particles, as a function of the sampling radius. For small radii, the number of retained samples is strongly reduced, except for the $ \textit{Ro} = 5 \times 10^{-1}$ and $ \textit{St} = 10^{1}$ case, reflecting the fact that the local particle population frequently falls below the threshold $N_{min }$ , and thus, these scales are dominated by undersampled configurations. As the radius increases, the number of accepted clusters rapidly grows and reaches a maximum at intermediate scales of approximately $10 \eta$ , corresponding to well-resolved clusters that satisfy the occupancy criterion and provide statistically robust estimates of the gyration tensor. At larger radii, the number of independent samples decreases again due to the combined effects of geometric overlap and the progressive loss of locality of the sampling volumes. This figure also confirms the rotation-induced displacement of the preferential concentration peak toward larger inertia: while in HIT the $ \textit{St}_{\eta }=10^{0}$ case yields more accepted clusters than $ \textit{St}_{\eta }=10^{1}$ over the relevant radii, increasing rotation progressively reverses this tendency, with the $ \textit{Ro}=5\times 10^{-1}$ , $ \textit{St}_{\eta }=10^{1}$ case remaining populated over the broadest range of $R/\eta$ . Thus, because only the radii within the shaded yellow area are considered (ensuring a minimum of $100$ sampling spheres for the analysis), the adopted threshold effectively filters out noise-dominated configurations at small scales while retaining a sufficiently large ensemble of statistically independent clusters at intermediate radii. Consequently, the gyration-tensor statistics presented in the subsequent section are not affected by undersampling, and the observed oscillatory features can be attributed to genuine physical clustering rather than finite-sample effects.

Figure 10.

The PDFs of the normalised trace of the gyration tensor, $\log _{10}(G_{\alpha \alpha }/R^2)$ , for various analysis radii $R$ (scaled by the Kolmogorov length $\eta$ ). Columns from left to right represent tracers ( $\textit{St}_{\eta}=0$ ) and inertial particles with $\textit{St}_{\eta}=10^{0}$ and $\textit{St}_{\eta}=10^{1}$ , respectively. Rows from top to bottom correspond to the HIT reference case (a–c), moderate rotation at $ \textit{Ro}=10^{0}$ (d–f), and strong rotation at $ \textit{Ro}=5 \times 10^{-1}$ (g–i). Curves of different colours indicate increasing analysis radii, as labelled. These distributions quantify the degree of spatial concentration and the multiscale compactness of the local particle neighbourhood as a function of both inertia and rotational intensity.

The trace of the gyration tensor, a scalar quantity that quantifies the spatial spread of particle occupancy within the spherical neighbourhood surrounding each reference particle serves as a measure of the local concentration within the prescribed analysis volume. Since the components of the gyration tensor represent squared distances, the trace is normalised by $R^{2}$ , where $R$ denotes the radius of the spherical sampling volume used to sample the local particle ensemble. As illustrated in figure 1(a), this radius establishes the characteristic length scale of the cluster associated with each reference particle. Consequently, the dimensionless quantity $\log _{10}(G_{\alpha \alpha }/R^{2})$ characterises the local compactness of the particle cloud: negative values signify dense, tightly packed clusters, whereas values approaching zero correspond to more diffuse spatial distributions. In this framework, the trace analysis provides a multiscale assessment of particle concentration, offering a perspective analogous to that of the classical RDF.

Figure 10 displays the probability density functions (PDFs) of $\log _{10}(G_{\alpha \alpha }/R^{2})$ for several analysis radii, as indicated by the colour-coded curves. In the HIT reference case, the tracer distribution (panel a) exhibits sharp, narrow peaks centred around −0.22, confirming that these particles populate the flow almost uniformly across all sampled scales. This reference value follows from the second moment of a uniform distribution within a spherical sampling volume of radius $R$ . In this case,

(4.18) \begin{align} \langle r^2 \rangle = \frac {\int _{0}^{R} r^2 (4\pi r^2)\,\text{d}r}{\int _{0}^{R} (4\pi r^2)\,\text{d}r} = \frac {\int _{0}^{R} r^{4}\,\text{d}r}{\int _{0}^{R} r^{2}\,\text{d}r} = \frac {R^{5}/5}{R^{3}/3} = \frac {3}{5}R^{2}, \end{align}

so that

(4.19) \begin{align} \frac {G_{\alpha \alpha }}{R^{2}} = \frac {3}{5}, \quad \log _{10}\!\left (\frac {G_{\alpha \alpha }}{R^{2}}\right ) \simeq -0.22. \end{align}

This provides a natural reference state corresponding to homogeneous spatial filling of the sampling volume.

As particle inertia increases (figure 10 b,c), the PDFs progressively broaden and develop pronounced asymmetric tails toward more negative values, particularly at small radii. This behaviour reflects the emergence of strongly inhomogeneous particle distributions, characterised by compact clusters embedded within extended void regions. Notably, the case $ \textit{St}_{\eta }=10^{0}$ (panel b) exhibits the strongest clustering signature: a wider spread toward negative values together with a larger number of statistically converged radii, extending down to $R/\eta \simeq 5$ . This indicates a high probability of encountering compact particle aggregates at scales approaching the Kolmogorov length. In contrast, the higher-inertia case $ \textit{St}_{\eta }=10^{1}$ (panel c) shows a comparatively narrower distribution, signalling a reduced efficiency of small-scale clustering. Altogether, these observations provide scale-resolved evidence of the well-known peak in preferential concentration at $ \textit{St}_{\eta }=O(1)$ , where the coupling between particle response time and the smallest turbulent time scales maximises clustering intensity (Elgobashi Reference Elgobashi1994; Gibert et al. Reference Gibert, Xu and Bodenschatz2012). This confirms that preferential concentration is strongest near $ \textit{St}_{\eta }\simeq 1$ , where inertial filtering and turbulent strain dynamics are optimally balanced.

The nature of particle clustering undergoes a profound transformation in the presence of rotation. Indeed, as observed in the previous sections, rotation imprints a large-scale geometrical organisation onto the flow, characterised by the emergence of coherent columnar vortices, that fundamentally reshapes the spatial distribution of the dispersed phase. These vortical cores and their surrounding shear layers promote the formation of compact particle clusters even at moderate or high inertia, creating structures that are absent in the HIT regime. This influence is clearly reflected in the inertial particle statistics. In the moderate rotation regime ( $ \textit{Ro}=10^{0}$ , figure 10e–f), the PDFs exhibit a systematic broadening toward more negative values as particle inertia increases, indicating a monotonic enhancement of clustering intensity from $ \textit{St}_{\eta }=10^{0}$ to $ \textit{St}_{\eta }=10^{1}$ . In particular, the $ \textit{St}_{\eta }=10^{1}$ case displays the widest distributions together with statistically robust curves extending down to smaller analysis radii ( $R/\eta \simeq 3$ ), demonstrating an increased probability of finding highly compact particle aggregates at near-dissipative scales. This behaviour contrasts with the HIT case, where preferential concentration peaks at $ \textit{St}_{\eta }=O(1)$ , and instead suggests that rotation alters the clustering mechanism by progressively favouring higher-inertia particles. The observed shift is consistent with the tendency of inertial particles to be expelled from rotating coherent structures and accumulate in their strain-dominated peripheries, leading to enhanced small-scale segregation as inertia increases.

By increasing the rotation ( $ \textit{Ro} = 5 \times 10^{-1}$ , figure 10 g–i), the trends observed at moderate rotation intensify further. While the tracer distributions remain largely unchanged, inertial particles exhibit increasingly broad and highly skewed PDFs as $ \textit{St}_{\eta }$ increases, indicating a systematic and monotonic amplification of clustering. In particular, the distributions extend progressively toward more negative values of $\log _{10}(G_{\alpha \alpha }/R^{2})$ , revealing the formation of extremely compact particle clusters. The persistence of these tails across multiple radii, together with their extension to the smallest accessible scales, suggests that clustering occurs down to near- and even sub-Kolmogorov scales. The present results strengthen the concept that rotation not only enhances clustering intensity but also shifts its maximum toward higher inertia, leading to a regime in which the strongest preferential concentration is attained at $ \textit{St}_{\eta }=10^{1}$ . Overall, the combined effects of rotational constraint and particle inertia produce a strongly heterogeneous spatial distribution characterised by tightly packed clusters embedded within large-scale voids. This dual behaviour suggests that rotation introduces a multiscale clustering mechanism: a large-scale organisation dictated by columnar vortices and a small-scale preferential concentration driven by particle inertia. The synergy between these mechanisms results in maximum spatial heterogeneity within the high-Stokes and strong rotation regime. Taken together, the distributions reveal a clear functional division: inertia governs dissipative-scale clustering through preferential concentration, whereas rotation dictates large-scale segregation by imposing coherent vortical organisation and geometric confinement.

4.3.3. Geometric characterisation of cluster morphology

The trace of the gyration tensor quantifies the spatial extent and compactness of particle clusters, yet it remains agnostic to their specific geometric architecture. To further delineate cluster morphology, the anisotropy of the tensor is analysed via the barycentric map representation introduced in § 3.1 (3.12). The projection of the eigenvalues of the normalized anisotropy tensor onto a 2-D barycentric triangle facilitates a direct visualisation of the relative prevalence of one-dimensional (prolate/rod-like), 2-D (oblate/disk-like) and 3-D (spherical/isotropic) geometries within the particle ensemble. This approach enables a comprehensive assessment of the manner in which external forcing, such as background rotation, reshapes the fundamental topology of particle clusters.

Figure 11.

Barycentric-map PDFs characterising the geometric anisotropy of particle clusters. Rows correspond to the HIT reference case (a–c), moderate rotation at $ \textit{Ro}=10^{0}$ (d–f), and strong rotation at $ \textit{Ro}=5 \times 10^{-1}$ (g–i). The three vertices represent pure rod-like ( $x_{1c}$ ), disk-like ( $x_{2c}$ ) and isotropic ( $x_{3c}$ ) cluster geometries, respectively (refer to figure 1 for the geometric definitions).

The barycentric maps depicted in figure 11 are constructed at a representative neighbourhood scale $R/\eta$ , corresponding to the radius at which the PDF of the trace distribution $\log _{10}(G_{\alpha \alpha }/R^{2})$ attains its peak (see figure 10). This choice ensures that the maps capture the statistically dominant geometrical configuration of the particle clouds. For each panel, the red marker denotes the location of the maximum of the joint PDF, while the contour lines correspond to the $25\,\%, 50\,\% \text{ and } 75\,\%$ probability levels, highlighting the progressive concentration of the distribution around its most probable state. In the HIT reference case, the barycentric distribution shows that tracers are confined to a narrow region in the vicinity of the isotropic vertex, reflecting the spatial uniformity of their local environments.

The introduction of particle inertia breaks this symmetry; the barycentric clouds shift toward the oblate (disk-like) boundary, signalling a tendency for particle groups to organise into flattened, sheet-like arrangements. This behaviour is consistent with the established phenomenology of inertial clustering in homogeneous turbulence, whereby particles with $ \textit{St}_{\eta } \sim 10^0$ are centrifuged from vortex cores and accumulate within high-strain regions to form thin, elongated structures (Monchaux et al. Reference Monchaux, Bourgoin and Cartellier2012; Bec et al. Reference Bec, Gustavsson and Mehlig2024).

In the moderately rotating regime ( $ \textit{Ro}=10^{0}$ , figure 11 d–f), the barycentric distributions migrate toward the oblate (disk-like) even more. This behaviour illustrates the influence of developing columnar vortices on cluster geometry, which produces structures characterised by simultaneous transverse compression and axial elongation. At intermediate inertia ( $ \textit{St}_{\eta }=10^{0}$ , panel e), the distribution shifts toward the oblate limit, indicating a predominance of quasi-planar sheets organised along the periphery of the columnar vortices. With a further increase in inertia ( $ \textit{St}_{\eta }=10^{1}$ , panel f), the probability density concentrates significantly near the oblate side. This shift reflects the enhanced confinement of high-inertia particles along vortex filaments and the axis of rotation, a result that is entirely consistent with the pronounced axial alignment observed in the 3-D visualisations.

Finally, in the strongly rotating regime ( $ \textit{Ro}=5 \times 10^{-1}$ , figure 11g–i), the barycentric distributions undergo a marked reorganisation driven by particle inertia. While tracers remain tightly clustered near the isotropic vertex, inertial particles exhibit a progressive broadening and displacement of the distribution toward the 2-D side region of the map. In particular, for increasing $ \textit{St}_{\eta }$ , the probability density becomes more dispersed and shifts away from the isotropic state, indicating a transition toward elongated, anisotropic cluster geometries. At the highest inertia, the distribution no longer forms a compact region but instead displays significant scatter across the barycentric space, reflecting a highly intermittent organisation of particle clouds. The concentration of probability toward the oblate side, together with the loss of a well-defined peak, suggests that strong rotation promotes the formation of columnar clusters, while simultaneously enhancing spatial variability. This behaviour is consistent with the underlying flow anisotropy, where rotational effects suppress transverse fluctuations and favour particle accumulation along strain-dominated, vertically coherent structures.

4.3.4. Orientation of particle clusters

The analysis of cluster orientation relative to the rotation axis complements the characterisation of cluster geometry. As with the barycentric analysis, polar maps are constructed from the most representative clusters, those corresponding to the peak values of $G_{\alpha \alpha }/R^{2}$ identified in figure 10. The focus of this analysis is placed on the polar angles $\theta _{1}$ , $\theta _{2}$ and $\theta _{3}$ , which quantify the inclination of the three principal eigenvectors of the gyration tensor with respect to the vertical $z$ axis. This axis coincides with the rotation axis and defines the primary direction of anisotropy within the flow. Consequently, the probability distributions of $\theta _{i}$ provide a direct measure of cluster alignment: peaks near $\theta = 0$ signify vertical orientation, whereas concentrations near $\theta = \pi /2$ correspond to horizontal alignment. These orientation statistics are presented as smoothed polar histograms of normalised probability density, providing a clear visualisation of the directional anisotropy within the tensor eigenspace.

Figure 12.

Polar distributions of angles $\theta _{1}$ , $\theta _{2}$ and $\theta _{3}$ of the gyration-tensor eigenvectors. The isotropic reference (black dotted line) corresponds to a uniformly distributed director, yielding $P_{\textit{iso}}(\theta )=\sin \theta$ . Columns correspond to increasing particle inertia: $ \textit{St}_{\eta }=0$ , $ \textit{St}_{\eta }=10^{0}$ and $ \textit{St}_{\eta }=10^{1}$ . Rows correspond to HIT (a–c), rotating turbulence at $ \textit{Ro}=10^{0}$ (d–f), and strongly rotating turbulence at $ \textit{Ro}=5 \times 10^{-1}$ (g–i).

The polar distributions of $\theta _{1}$ , $\theta _{2}$ and $\theta _{3}$ are presented in figure 12 together with the isotropic reference, corresponding to a uniformly oriented direction in three dimensions. Within the canonical HIT regime (top row), the three eigenvectors exhibit nearly uniform angular distributions across the entire polar range, as expected for an isotropic flow devoid of a preferred direction. For tracer particles (panel a), the probability density closely follows the isotropic reference. In contrast, for particles with finite inertia (panels b–c), a slight deviation from isotropy is observed, with a weak preference for orientations around $\theta \approx 90^{\circ }$ . This behaviour arises from a preferential concentration, whereby inertial particles accumulate in strain-dominated regions and avoid vortical structures, thereby biasing the local deformation field they sample. As a result, the principal axes tend to lie within planes perpendicular to the reference direction, leading to a mild enhancement of probability near $\theta \approx \pi /2$ . This corresponds to a statistically planar, yet otherwise random, orientation of cluster axes within the horizontal plane. Overall, in the absence of rotation, particle-cluster orientations remain only weakly correlated with the flow coordinates.

The orientation landscape transitions markedly under moderate rotation ( $ \textit{Ro}=10^{0}$ , figure 12 d–f), where the distributions deviate slightly from isotropy. The eigenvectors associated with the smallest eigenvalues ( $\theta _{3}$ ) exhibits enhanced probability near $\theta \approx 90^{\circ }$ , indicating that the minor axes preferentially reside within the horizontal plane orthogonal to the rotation axis. Such configurations represent particle clusters organised into planar sheets; likely the horizontal cross-sections of large-scale columnar vortices. This alignment is particularly pronounced for $ \textit{St}_{\eta }=10^{0}$ , a regime where the interplay between inertial clustering and rotational confinement is optimised.

In the strong rotation regime ( $ \textit{Ro}=5 \times 10^{-1}$ , figure 12 g–i), the spatial organisation of the dispersed phase becomes even more anisotropic. For tracer particles (panel g), the angle of the major axis $\theta _{1}$ remains predominantly horizontal, indicating the persistence of vortical structures and their effective coupling with the particle motion, as illustrated in the schematics accompanying each panel. With increasing inertia however, the distribution of $\theta _{1}$ undergoes a sharp collapse toward $\theta \approx 0^{\circ }$ , signifying that the major principal axis aligns with the axis of rotation. Simultaneously, the secondary and tertiary axes ( $e_2$ and $e_3$ ) concentrate around $\theta \approx 90^{\circ }$ , corresponding to the transverse plane. This configuration characterises rod-like or columnar clusters whose the angle of the dominant axis $\theta _{1}$ follows the rotational direction, while the smaller dimensions $\theta _{2}$ and $\theta _{3}$ define the cross-sectional plane of the filament. Such structures, sketched in the lower portion of each polar map, reflect the strong quasi-two-dimensionalisation of the turbulent field under rapid rotation. Furthermore, the combined influence of the flow topology and centrifugal acceleration constrains the particles into vertically elongated clusters, effectively suppressing the random, isotropic configurations observed in the HIT case.

4.4.1. Lagrangian acceleration and intermittency

Lagrangian acceleration represents a fundamental quantity in turbulence research, as it encapsulates the multiscale coupling between dissipative fluctuations and large-scale flow organisation. This variable is widely recognised as a primary observable for quantifying intermittency, extreme events and the nonlinear response of particles to turbulent forcing (Voth et al. Reference Voth, Crawford, Alexander and Bodenschatz2001; Biferale et al. Reference Biferale, Boffetta, Celani, Lanotte and Toschi2004). From a theoretical perspective, the acceleration variance are intrinsically linked to the local energy dissipation rate via the classical Heisenberg–Yaglom framework (Toschi & Bodenschatz Reference Toschi and Bodenschatz2009). Deviations from these theoretical predictions serve to quantify the degree of small-scale intermittency within the flow (Wang et al. Reference Wang, Wan and Biferale2022). In the case of inertial particles, acceleration statistics further encode the complex interplay between dissipative eddies and particle inertia, providing a direct diagnostic of preferential concentration and clustering dynamics (Bec et al. Reference Bec, Biferale, Boffetta, Celani, Cencini, Lanotte, Musacchio and Toschi2006; Mathai et al. Reference Mathai, Calzavarini, Brons, Sun and Lohse2016). Consequently, the characterisation of acceleration distributions is essential for elucidating the underlying nature of particle clustering.

Figure 14.

The PDFs of the normalised Lagrangian acceleration components for various Stokes numbers (columns) and Rossby numbers (colours). The top row depicts the vertical component $a_{z}$ (parallel to the rotation axis), while the bottom row displays the transverse component $a_{\perp } = (a_{x} + a_{y})/2$ within the plane of rotation. All distributions are normalised by their respective r.m.s. acceleration, $a_{\textit{rms}}$ , to facilitate a direct comparison of their intermittent tails.

The PDFs of the normalised Lagrangian acceleration components are presented in figure 14 for various particle inertias and flow configurations. The top panels depict the vertical component $a_{z}/a_{z,\textit{rms}}$ , while the bottom panels show the acceleration within the transverse plane, $a_{\perp }/a_{\perp ,\textit{rms}}$ . Here, $a_{i,\textit{rms}} \equiv \sqrt {\langle a_{i}^{2}\rangle }$ denotes the r.m.s. value of either the vertical ( $i=z$ ) or transverse ( $i=\perp \equiv (a_x + a_y)/2$ ) component. Decomposing the total acceleration into vertical and horizontal contributions highlights the distinct dynamical behaviours emerging from the anisotropy imposed by rotation. The HIT cases (black curves in figure 14) serve as a benchmark for the subsequent analysis of rotating turbulence. In the tracer limit ( $ \textit{St}_{\eta }=0$ ), the acceleration distributions are isotropic, with both components exhibiting nearly identical variances and pronounced stretched-exponential tails. Such tails signify the emergence of extreme events, a well-known feature of Lagrangian intermittency (Voth et al. Reference Voth, Crawford, Alexander and Bodenschatz2001, Reference Voth, La Porta, Crawford, Alexander and Bodenschatz2002). Even at the moderate Reynolds number of the present simulations ( $ \textit{Re}_{\kern-1pt \lambda }\approx 90$ ), the probability of high-acceleration events remains significant, confirming the persistent nature of small-scale intermittency. These intense acceleration bursts originate from the entrapment of fluid particles within coherent vortical structures, where local velocity gradients and centripetal accelerations are maximal (Biferale et al. Reference Biferale, Boffetta, Celani, Lanotte and Toschi2004). As particle inertia increases, the acceleration PDFs progressively narrow (black curves, moving from left to right panels), indicating a systematic attenuation of extreme events. This trend is consistent with the understanding of particle inertia as a temporal low-pass filter of underlying turbulent fluctuations (Ayyalasomayajula et al. Reference Ayyalasomayajula, Gylfason, Collins, Bodenschatz and Warhaft2006; Bec et al. Reference Bec, Biferale, Boffetta, Celani, Cencini, Lanotte, Musacchio and Toschi2006; Cencini et al. Reference Cencini, Bec, Biferale, Boffetta, Celani, Lanotte, Musacchio and Toschi2006). Physically, this reduction in intermittency is driven by preferential concentration: inertial particles are centrifuged away from intense vortical regions and preferentially sample strain-dominated zones. For larger Stokes numbers ( $ \textit{St}_{\eta } = 10^{0}$ ), this filtering effect dominates; the finite response time of the particles smooths out rapid, small-scale accelerations, leading to a further suppression of the non-Gaussian tails and a shift toward more moderate fluctuations.

The introduction of rotation profoundly alters the Lagrangian dynamics, revealing several distinct physical mechanisms. First, a pronounced anisotropy in the particle acceleration emerges. The PDFs of the axial acceleration (figure 14 a–c) are markedly narrower than those of the transverse components for $ \textit{Ro}=10^{0}$ and $ \textit{Ro}=5 \times 10^{-1}$ . This suggests that background rotation suppresses vertical intermittency while simultaneously enhancing horizontal fluctuations. Such behaviour is in close agreement with the experimental observations of Del Castello & Clercx (Reference Del Castello and Clercx2011), who reported a reduction in intense acceleration events along the rotation axis alongside an amplification of transverse accelerations. The underlying mechanism is attributed to the reorganisation of the flow into columnar vortices aligned with the rotation axis; this structural shift attenuates axial velocity gradients while reinforcing transverse strain and centrifugal effects at the vortex peripheries. Consequently, particles experience diminished accelerations in the axial direction but more intense, intermittent forcing within the transverse plane. Second, this anisotropic signature persists across the investigated range of Stokes numbers, though its magnitude is modulated by particle inertia. For moderate inertia ( $ \textit{St}_{\eta }=10^{0}$ ), the transverse acceleration displays pronounced non-Gaussian tails, reflecting localised bursts of intense forcing as particles interact with coherent vortical columns. At higher inertia ( $ \textit{St}_{\eta }=10^{1}$ ), the distributions narrow progressively, as particle response times exceed the characteristic time scales of the smallest turbulent eddies, effectively filtering the acceleration signal. This trend aligns with the theoretical framework of Biferale et al. (Reference Biferale, Bonaccorso, Mazzitelli, van Hinsberg, Lanotte, Musacchio, Perlekar and Toschi2016), wherein rotation promotes preferential sampling: heavy particles diffuse more efficiently in the transverse plane, while their axial motion is increasingly constrained. Thus, acceleration anisotropy not only quantifies the quasi-two-dimensionalisation of the Lagrangian dynamics but also provides a dynamical basis for the planar clustering and axial alignment discussed in § 4.3.2. Third, the dependence of acceleration statistics on the Rossby number reveals a non-monotonic behaviour. Under moderate rotation ( $ \textit{Ro}=10^{0}$ , blue curves), the transverse acceleration PDFs exhibit broader tails than those in the HIT case, indicating enhanced horizontal intermittency. In contrast, under stronger rotation ( $ \textit{Ro}=5 \times 10^{-1}$ , red curves), both transverse and axial acceleration PDFs narrow substantially, indicative of the stringent dynamical constraints imposed by intense rotation. This mirrors the findings of Del Castello & Clercx (Reference Del Castello and Clercx2011), who noted that while moderate rotation intensifies lateral intermittency, rapid rotation leads to a global reduction in fluctuations.

Figure 15.

Statistical moments of the Lagrangian acceleration components as a function of the Stokes number $ \textit{St}_{\eta }$ for HIT (black diamonds) and rotating turbulence at $ \textit{Ro}=10^{0}$ (blue circles) and $ \textit{Ro}=5 \times 10^{-1}$ (red stars). The top row depicts the axial acceleration $a_{z}$ , while the bottom row displays the transverse magnitude $a_{\perp }$ . From left to right, panels show the skewness $S_{a}$ (a,d), the flatness $F_{a}$ (b,e), and the normalised r.m.s. acceleration $a_{\textit{rms}}/a_{0}$ (c,f).

The probability distributions discussed above are complemented by a quantitative analysis of their statistical moments. Specifically, the r.m.s. value, skewness and flatness provide compact measures of acceleration intensity, asymmetry and intermittency, respectively. These metrics represent standard benchmarks for characterising Lagrangian acceleration statistics in turbulent flows (Voth et al. Reference Voth, La Porta, Crawford, Alexander and Bodenschatz2002; Del Castello & Clercx Reference Del Castello and Clercx2011). The following classical definitions are adopted for a generic acceleration component $a_i$ :

(4.20) \begin{align} S_{a}(i) = \frac {\langle a_{i}^{3}\rangle }{\langle a_{i}^{2}\rangle ^{3/2}}, \quad F_{a}(i) = \frac {\langle a_{i}^{4}\rangle }{\langle a_{i}^{2}\rangle ^{2}}. \end{align}

The flatness factor $F_{a}$ characterises the relative contribution of extreme events to the tails of the probability distribution. Specifically, values significantly exceeding the Gaussian baseline of $F_{a}=3$ signify pronounced intermittency. Simultaneously, the skewness $S_{a}$ quantifies the degree of asymmetry between positive and negative acceleration fluctuations, providing insight into the directional bias of the underlying forces.

Results presented in figure 15 corroborate the trends observed in the acceleration PDFs and quantify the influence of rotation and inertia on higher-order statistics. In the HIT regime (black symbols), the skewness of all acceleration components remains negligible, consistent with the statistical isotropy of the flow. Simultaneously, the flatness reaches values of $F_{a} \simeq 15$ , which aligns with canonical results for HIT at moderate Reynolds numbers (Voth et al. Reference Voth, La Porta, Crawford, Alexander and Bodenschatz2002). These elevated values reflect the pronounced intermittency of Lagrangian acceleration associated with intense vortical events. Conversely, the systematic decrease in $F_{a}$ with increasing Stokes number indicates the expected attenuation of extreme fluctuations due to inertial filtering. The introduction of rotation yields more complex statistical trends. For moderate rotation ( $ \textit{Ro}=10^{0}$ ), the skewness of the transverse acceleration $a_{\perp }$ remains small, indicating an absence of preferential planar bias. However, the axial component $a_{z}$ exhibits a mild asymmetry that remains invariant with $ \textit{St}_{\eta }$ . Rotational anisotropy is clearly manifested in the flatness: the transverse components (bottom row) display significantly higher values at $ \textit{Ro}=10^{0}$ , signifying enhanced intermittency and a higher frequency of extreme acceleration events within the plane orthogonal to the rotation axis. For both axial and transverse components, the flatness decreases as $ \textit{St}_{\eta }$ increases, confirming the suppression of small-scale intermittency as inertia becomes the dominant factor. In the enhanced-rotation regime ( $ \textit{Ro}=5 \times 10^{-1}$ ), $F_{a}(a_{\perp })$ declines, suggesting that extreme acceleration events are quenched as the flow approaches a quasi-2-D state dominated by stable columnar vortices. Collectively, these findings demonstrate that rotation redistributes acceleration intermittency from the axial to the transverse direction – reinforcing planar dynamics under moderate rotation while globally suppressing fluctuations in the rapid-rotation regime.

Finally, values of r.m.s. acceleration provide a direct measure of the characteristic force intensity and serve as a quantitative benchmark for assessing deviations from classical turbulence theory. In the right-hand column of figure 15, the acceleration components are normalised by the reference acceleration $a_{0} = (\varepsilon ^{3} / \nu )^{1/4}$ , consistent with the Heisenberg–Yaglom scaling (Voth et al. Reference Voth, Crawford, Alexander and Bodenschatz2001; Bec et al. Reference Bec, Biferale, Boffetta, Celani, Cencini, Lanotte, Musacchio and Toschi2006) for HIT. Experimental and numerical investigations indicate that the normalised acceleration magnitude, $|a_{\textit{rms}}|/a_{0}$ , exhibits a weak dependence on the Taylor-scale Reynolds number $ \textit{Re}_{\kern-1pt \lambda }$ , typically approaching an asymptotic value between 3 and 7 at sufficiently high $ \textit{Re}_{\kern-1pt \lambda }$ (Voth et al. Reference Voth, Crawford, Alexander and Bodenschatz2001; Bec et al. Reference Bec, Biferale, Boffetta, Celani, Cencini, Lanotte, Musacchio and Toschi2006; Wang et al. Reference Wang, Wan and Biferale2022). Furthermore, in rotating flows the transverse acceleration components are consistently more energetic than the axial ones; a clear signature of persistent small-scale anisotropy even in high- $ \textit{Re}_{\kern-1pt \lambda }$ regimes (Del Castello & Clercx Reference Del Castello and Clercx2011). In the present HIT simulations, the r.m.s. values of both vertical and horizontal accelerations decrease systematically with increasing Stokes number. This trend reflects the inertial filtering of small-scale fluctuations, as documented in previous experimental (Voth et al. Reference Voth, La Porta, Crawford, Alexander and Bodenschatz2002; Ayyalasomayajula et al. Reference Ayyalasomayajula, Gylfason, Collins, Bodenschatz and Warhaft2006) and computational (Bec et al. Reference Bec, Biferale, Boffetta, Celani, Cencini, Lanotte, Musacchio and Toschi2006) studies, and confirms that particle inertia suppresses the fluctuating intensity of the Lagrangian dynamics. The inset in the lower-right panel of figure 15 shows the total normalised acceleration $a_{\textit{rms}}/a_{0}$ , which falls within the range reported in the literature for comparable $ \textit{Re}_{\kern-1pt \lambda }$ (Voth et al. Reference Voth, Crawford, Alexander and Bodenschatz2001; Bec et al. Reference Bec, Biferale, Boffetta, Celani, Cencini, Lanotte, Musacchio and Toschi2006). Overall, these results demonstrate that the present simulations accurately reproduce the canonical Heisenberg–Yaglom scaling in the HIT limit, providing a robust reference for analysing rotational effects. For the rotating cases, the r.m.s. acceleration generally decreases monotonically with the Stokes number for both axial and transverse components. This confirms that the filtering effect of inertia on small-scale fluctuations persists under the influence of background rotation.

4.4.2. Curvature and torsion of Lagrangian particle trajectories

The Lagrangian characterisation of particle dynamics is completed by an analysis of trajectory geometry in terms of curvature, $\mathscr{K}\kern1pt$ , and torsion, $\mathscr{T}\kern1pt$ . These quantities characterise the degree of bending and twisting of the particle path, respectively, following the formal definitions provided in § 3.2. Their PDFs provide a concise statistical description of the Lagrangian geometry, complementing the acceleration analysis by linking kinematic intermittency to the underlying flow topology (Scagliarini Reference Scagliarini2011; Kadoch et al. Reference Kadoch, Bassenne, Esmaily-Moghadam, Schneider, Farge and Bos2016). To disentangle short-term fluctuations caused by flow reversals from the more persistent bending induced by coherent vortices, the curvature and torsion signals are temporally filtered along each trajectory. This filtering is performed over windows of $\Delta t/\tau _{\eta } = n$ , with $n = 1, 5, 25$ . The unfiltered case ( $n=0$ ) represents the instantaneous statistics, whereas increasing $n$ facilitates the averaging of local geometric properties over trajectory segments spanning one to several Kolmogorov time scales.

Figure 16.

The PDFs of trajectory curvature $\mathscr{K}\kern1pt$ (top row, a–c) and torsion $\mathscr{T}\kern1pt$ (bottom row, d–f) for various Stokes numbers ( $ \textit{St}_{\eta }=0, 10^{0}, 10^{1}$ , left to right) and flow regimes: HIT (black), moderate rotation $ \textit{Ro}=10^{0}$ (blue) and strong rotation $ \textit{Ro}=5 \times 10^{-1}$ (green). All geometric quantities are normalised by the Kolmogorov length scale $\eta$ . Different symbols denote temporal filtering windows of $\Delta t/\tau _{\eta } = 1, 5, 25$ applied along the particle paths. Reference power-law scalings $(\mathscr{K}\kern1pt \eta )^{1}$ and $(\mathscr{K}\kern1pt \eta )^{-5/2}$ for curvature, and $(\mathscr{T}\kern1pt \eta )^{0}$ and $(\mathscr{T}\kern1pt \eta )^{-3}$ for torsion, are indicated by dashed and dotted red lines, respectively.

The curvature PDFs for the HIT cases (black symbols in figure 16) reproduce the established dual-power-law behaviour: scaling as $(\mathscr{K}\kern1pt \eta )^{1}$ for small curvatures and $(\mathscr{K}\kern1pt \eta )^{-5/2}$ for large curvatures. This result is consistent with theoretical predictions derived from independent Gaussian velocity-acceleration statistics and aligns with previous experimental observations (Cencini et al. Reference Cencini, Bec, Biferale, Boffetta, Celani, Lanotte, Musacchio and Toschi2006; Xu et al. Reference Xu, Ouellette and Bodenschatz2007; Hengster et al. Reference Hengster2024). Temporal filtering predominantly affects the lower tail, where the probability of small-curvature events is reduced. In contrast, the high-curvature tail remains largely invariant, confirming that extreme bending events correspond to short lived yet intense interactions with vortical structures. Similarly, the torsion PDFs exhibit the anticipated scalings of $(\mathscr{T}\kern1pt\eta )^{0}$ and $(\mathscr{T}\kern1pt\eta )^{-3}$ for small and large values of $\mathscr{T}\kern1pt$ , respectively. While the low- $\mathscr{T}\kern1pt$ plateau remains unaffected by filtering, large- $\mathscr{T}\kern1pt$ events are progressively attenuated. This behaviour stems from the geometric relationship between the two parameters: since torsion is inversely proportional to the square of the curvature magnitude, the suppression of low- $\mathscr{K}\kern1pt$ (quasi-rectilinear) segments naturally reduces the incidence of extreme torsion values (Hengster et al. Reference Hengster2024).

Background rotation systematically modulates these geometric statistics in a physically consistent manner. For tracers (left column), the curvature and torsion distributions under moderate rotation ( $ \textit{Ro} = 10^{0}$ ) remain nearly identical to those of the HIT case, indicating the robustness of the power-law tails and the weak influence of rotation on trajectory geometry in this regime. However, a pronounced shift toward higher values of both $\mathscr{K}\kern1pt$ and $\mathscr{T}\kern1pt$ emerges under strong rotation at $ \textit{Ro} = 5 \times 10^{-1}$ . This behaviour is consistent with the development of axial columnar vortices (see figure 7), which constrain particle trajectories to circulate around coherent structures characterised by smaller length scales. An analogous trend was reported by Alards et al. (Reference Alards, Rajaei, Del Castello, Kunnen, Toschi and Clercx2017) for tracer particles in rapidly rotating Rayleigh–Bénard convection, where the onset of vertically aligned plumes produces a comparable shift of the curvature peak toward larger values. In both systems, the increased curvature reflects the dominance of coherent columnar motions and the corresponding reduction in the effective radius of curvature as trajectories are confined within these vortical plumes.

Table 2.

Interpretation of trajectory topology based on curvature and torsion.

Table 2. Long description

A table categorizing trajectory regimes based on curvature and torsion. The table has three columns: Regime, K and T, and Physical interpretation. It contains six rows, each describing a different regime. Row 1: Quasi-rectilinear, Low K and Low T, Weakly bent trajectories. Row 2: Twisting (no strong bending), Low K and High T, Rapid reorientation in 3-D. Row 3: Vortex core (planar trapping), High K and Low T, Tight in-plane circling. Row 4: Vortex border (intermittent), High K and Intermediate T, Intermittent interaction with vortical structures. Row 5: Helical-like, High K and High T, Fully 3-D winding.

The interplay between rotation and inertia further enriches the particle dynamics, leading to divergent trends in curvature and torsion statistics. Regarding curvature, the sensitivity to the imposed temporal filtering persists across both HIT and rotating regimes. Independently, increasing particle inertia further smooths the trajectory by filtering rapid flow fluctuations. However, this effect is attenuated at higher $ \textit{St}_{\eta }$ , where the overall intermittency of the particle motion is inherently reduced. In contrast, torsion displays a distinct response to inertia. In the HIT regime, inertial particles exhibit a clear filtering characteristic: the probability distributions of $\mathscr{T}\kern1pt$ narrow progressively with increasing $ \textit{St}_{\eta }$ , particularly at small torsion ( $\mathscr{T}\kern1pt\eta \ll 1$ ). This trend reflects the suppression of rapid 3-D directional changes due to particle inertia. Under moderate rotation however, the torsion PDFs for various $ \textit{St}_{\eta }$ exhibit only a marginal shift without significant narrowing of the distribution. This behaviour indicates that the trajectory geometry is primarily governed by large-scale rotational constraints rather than small-scale inertial filtering. Such a statistical collapse suggests that the coherent columnar vortices inherent to rotating turbulence impose a persistent, organised structure on particle motion, effectively homogenising the trajectory geometry across different levels of inertia.

While these statistics provide quantitative information on the geometric complexity of particle trajectories, their physical interpretation in terms of trajectory shapes is not immediately evident. Table 2 provides a compact mapping between curvature–torsion statistics and trajectory topology. In particular, large curvature combined with weak torsion corresponds to strongly bent but nearly planar trajectories, characteristic of tight in-plane circling around vortical structures. Conversely, simultaneous enhancement of curvature and torsion identifies genuinely 3-D helical-like motion (the pure helix is given by $\mathscr{T}\kern1pt/\mathscr{K}\kern1pt=\textrm {constant}$ ), where particles wind around vortices while drifting along them. This interpretation is consistent with classical differential geometry, where curvature and torsion uniquely determine the local shape of a space curve (Millman & Parker Reference Millman and Parker1977), and with Lagrangian turbulence studies showing that the ratio $\mathscr{T}\kern1pt/\mathscr{K}\kern1pt$ governs the transition from planar to helical trajectories, which is closely associated with vortex-induced spiral motion (Choi et al. Reference Choi, Park and Lee2010). This interpretation is further supported by the joint PDFs of curvature and transverse acceleration shown in figure 17. These statistics provide a mechanistic perspective on how rotation reshapes the geometry of Lagrangian trajectories.

Figure 17.

Joint PDFs of trajectory curvature and transverse acceleration, $\textrm{PDF}(\mathscr{K}\kern1pt\eta , a_{\perp }/a_{\perp ,\textit{rms}})$ . Curvature is normalised by the Kolmogorov length scale $\eta$ , while transverse acceleration is normalised by its r.m.s. value; the colour scale denotes log-probability density. Columns represent Stokes numbers $ \textit{St}_{\eta }=0$ , $10^{0}$ and $10^{1}$ (from left to right). Rows correspond to the different flow regimes: homogeneous isotropic turbulence (top), moderately rotating turbulence at $ \textit{Ro}=10^{0}$ (middle) and strongly rotating turbulence at $ \textit{Ro}=5 \times 10^{-1}$ (bottom).

Interestingly, figure 16(c,f) reveal a further distinctive feature at high inertia ( $ \textit{St}_{\eta }=10^{1}$ ) and $ \textit{Ro}=5 \times 10^{-1}$ , namely the emergence of an enhanced probability at finite curvature accompanied by a corresponding modification of the torsion statistics. The curvature PDF exhibits increased levels for $\mathscr{K}\kern1pt \eta \gtrsim 10^{-1}$ , indicating that particle trajectories preferentially bend over a finite range of length scales rather than exhibiting a purely scale-free behaviour. A dynamical interpretation of this behaviour can be obtained by relating curvature to the underlying force balance. Since the Frenet–Serret expression gives $\mathscr{K}\kern1pt=a_n/u^2$ , with $a_{n} \approx a_{\perp}$ for moderate to strong rotation, a transverse acceleration dominated by the Coriolis contribution, $a_{\perp } \sim 2\varOmega u$ , leads to a characteristic curvature $\mathscr{K}\kern1pt \sim 2\varOmega /u$ , corresponding to a radius of curvature $R_c \sim u /(2\varOmega )$ , analogous to inertial circular motion. The observed peak in the curvature distribution is therefore consistent with the emergence of a preferred bending scale imposed by the competition between inertial and rotational effects, although trajectories remain strongly perturbed by drag and turbulence and do not form ideal circular orbits.

In order to further investigate the relation between planar bending and the dynamical feature of the particle, in figure 17 is shown the joint PDF of the curvature and perpendicular acceleration. In the HIT regime, the joint PDFs of $(a_{\perp }, \mathscr{K}\kern1pt)$ exhibit a broad, positively sloped ridge – a direct kinematic consequence of the Frenet–Serret relationship $\mathscr{K}\kern1pt=a_{n}/u^{2}$ , which implies that sharp trajectory bends coincide with intense normal accelerations. These results indicate that curvature and transverse acceleration originate from the same physical mechanism: intermittent small-scale strain and vorticity that induce bursts of normal acceleration and rapid changes in trajectory direction. Rotation fundamentally reorganises this relationship. At an intermediate Rossby number ( $ \textit{Ro}=10^{0}$ ), the joint PDFs of $(a_{\perp }, \mathscr{K}\kern1pt)$ broaden toward higher values of $a_{\perp }/a_{\perp ,\textit{rms}}$ while $\mathscr{K}\kern1pt\eta$ remains nearly invariant. This discrepancy signifies a decoupling between trajectory bending and the underlying forcing; transverse accelerations become increasingly dominated by Coriolis and centrifugal contributions rather than by local vorticity-induced curvature. The persistence of non-zero curvature alongside diminishing torsion identifies trajectories that orbit columnar structures along arcs that are significantly curved yet weakly twisted. Under strong rotation ( $ \textit{Ro}=5 \times 10^{-1}$ ), the Lagrangian geometry becomes distinctly 2-D. The associated $(a_{\perp }, \mathscr{K}\kern1pt)$ distributions develop a pronounced vertical feature, where extremely large transverse accelerations coexist with only modest curvature. This represents the Lagrangian signature of a regime where Coriolis and centrifugal forces outweigh turbulent gradients, producing intense planar accelerations along smooth, large-radius paths. The suppression of torsion and the amplification of $a_{\perp }$ directly mirror the anisotropic acceleration moments discussed previously; characterised by enhanced $a_{\perp }$ and quenched $a_{z}$ . Increasing the particle inertia from $ \textit{St}_{\eta }=0$ to $10$ amplifies these effects. Inertial filtering attenuates the smallest-scale bends, reducing the high- $\mathscr{K}\kern1pt$ tail, while rotational forces increasingly dictate the amplitude and direction of transverse acceleration. Consequently, inertial particles under strong rotation follow trajectories that are ‘bent, planar and dynamically forced’, where torsion is nearly eliminated and $a_{\perp }$ is governed by global rotation dynamics rather than by turbulent dissipation.