4.2.1. Slowly rotating cube

To better understand the regime transitions and wake stability in the flow past a transversely rotating cube, it is important to first clarify the mechanism of vortex generation on the cube. In the RCF4 configuration, the transversely rotating cube has two primary angular positions: one with a face directly facing the incoming flow and the other with an edge (dividing two front faces) facing the flow. We consider an extreme case where the cube is fixed in the flow at these two angular positions and illustrate the corresponding streamwise vorticity $\omega _x$ generation in figure 5(a). Here, the choice of $\omega _x$ is motivated by its higher sensitivity to wake structure evolution, enabling more accurate and precise characterisation of the wake dynamics (Gai & Wachs Reference Gai and Wachs2023a ).

Figure 5.

Vortex generation mechanism on the surface and in the wake of the slowly rotating RCF4 cube. (a) Schematic of vortex generation on the front surface (left subpanels with black backgrounds) and in the wake of fixed cubes (right subpanels with white backgrounds), with one edge and one face facing the incoming flow ( $x^+$ direction). In all subpanels, red and blue regions indicate positive and negative $\omega _x$ , respectively. (b) Regime transition sequence for a slowly rotating cube at ${{R}e}=60$ and $\Omega = 0.02$ , compared with the regime transition in the flow past a fixed cube with an edge or a face facing the flow. The temporal evolution of the vorticity pattern $\omega _x$ in the particle wake is illustrated, along with side views of the isosurfaces $\omega _x = \pm 0.03$ .

As thoroughly discussed in our previous work (Gai & Wachs Reference Gai and Wachs2023d ), an inclined front face of the fixed angular particle generates a pair of counter-rotating streamwise vortices $\omega _x$ . For example, when the cube has an edge facing the flow, one pair of vortices ( $\omega _1$ , $\omega _2$ ) forms on the upper front face, while a second pair ( $\omega _3$ , $\omega _4$ ) forms on the lower front face. These two vortex pairs are then advected into the wake of the cube, maintaining a typical pattern of two vortex pairs (shown in blue and red) as depicted in figure 5(a). When a square face of the cube is oriented perpendicular to the flow, four pairs of oppositely signed streamwise vortices are generated. In addition to the two pairs formed on the upper and lower faces of the cube, two more vortex pairs appear on the side faces: the pair ( $\omega _5$ , $\omega _6$ ) on the left side face and the pair ( $\omega _7$ , $\omega _8$ ) on the right face. Although these side vortex pairs are also generated in the edge facing the flow configuration, their intensity is significantly lower and much less visible compared with the vortices produced on the upper and lower faces.

Now, we rotate the cube slowly at $\Omega =0.02$ around the $z$ -axis to observe how the vorticity pattern in the particle wake evolves with time. In figure 5(b), we present several snapshots of the temporal evolution of the $\omega _x$ pattern in the $y$ - $z$ plane at $x=x_p + 1.5$ ( $x_p$ denotes the $x$ coordinate of the particle centroid) for the case at ${{R}e}=60$ in the upper panel. The side views of the isosurface $\omega _x=\pm 0.03$ are illustrated in the bottom of the upper panel for better clarity. In the SV regime, the $\omega _x$ patterns are planar and symmetric, exhibiting a high similarity to those in the fixed edge cube case shown in figure 5(a). The two pairs of streamwise vortices are clearly visible in the side view. We know that in the case of a fixed edge cube in the multi-axis steady symmetry regime (MSS, ), the four vortex filaments in the wake are of equal length and remain steady. With transverse rotation, the length of the vortex pairs evolves in time. The length of the vortex pair ( $\omega _1$ , $\omega _2$ ) is clearly shorter than that of the vortex pair ( $\omega _3$ , $\omega _4$ ) in the first snapshot I. In snapshot II, the upper vortex pair ( $\omega _1$ , $\omega _2$ ) becomes longer instead. In the following, we will demonstrate how the reciprocal motion of vortex pairs and the evolution of their relative strength can explain the vortex suppression.

The similarity between the slowly rotating cube and a fixed edge cube can also be observed from the regime map. In the lower panel of figure 5(b), we present the regime map of flow past an edge and a face cube for ${{R}e}=40{-} 400$ . In place of the SV and UV regimes, the fixed cube exhibits the MSS ( ) and the planar steady symmetry regime (PSS, ), respectively. It is evident that the regime transition sequence of the slowly rotating cube shows very similar critical ${R}e$ for the first two regime transitions as those observed for the fixed edge cube, rather than those observed for the fixed face cube. At higher ${R}e$ , rotation causes flow instability to occur earlier (at ${{R}e}=300$ ) compared with the fixed edge cube (at ${{R}e}=340$ ).

4.2.2. Vortex merging induced by rotation

Unlike the multi-axis symmetry typically observed in the wake of a fixed cube, the particle rotation promotes the merging of surface vortices, resulting in a planar symmetry of vortex distribution, even at low Reynolds numbers such as ${{R}e}=60$ . The merging process occurs once the vortices are generated on the front surface of the cube. In figure 6, we present the $\omega _x$ distribution in the $y$ - $z$ plane at $x = x_p - 0.1$ for flows at ${{R}e}=60$ , $160$ , $350$ and $\Omega =0.02{-} 1$ . We highlight the vortex cores by setting the colour bar to $\omega _x \in [-1,1]$ , where regions with higher magnitude of $\omega _x$ are represented by red and blue, while areas with values close to $0$ appear in black.

Figure 6.

Visualisation of vortex merging on the front surface of the rotating RCF4 cube in the $y$ - $z$ plane at $x = x_p - 0.1$ , with red and blue regions representing positive and negative $\omega _x$ , respectively. Flow streamlines are included to enhance the visualisation of the wake vortices. The various flow regimes are highlighted in colour-coded boxes, and the cube projected outline on the $y$ - $z$ plane is sketched for reference.

In the case of ${{R}e}=60$ and $\Omega =0.02$ , we observe the generation of four pairs of streamwise vortices ( $\omega _1 - \omega _8$ ) similar to the fixed cube case in figure 5(a). Under the cube rotation in figure 6, the two side vortices $\omega _5$ and $\omega _8$ are fairly weak and get pushed away from the cube surface by their opposite-signed counter-parts $\omega _6$ and $\omega _7$ , respectively. This gives room for the vortex $\omega _6$ to spread on the left side of the cube and to reach its arm to its neighbour of the same positive sign, the stronger $\omega _4$ generated by the lower face. We observe the same mechanism between $\omega _7$ and $\omega _3$ . This process depicts the initiation of the surface vortex merging $\omega _4 \cup \omega _6$ and $\omega _3 \cup \omega _7$ that play key roles in the wake stability of the rotating cube. With the increase of the rotation rate, at ${{R}e}=60$ and $\Omega =0.1$ , we observe that the weak side vortices $\omega _5$ and $\omega _8$ , which are pushed away from the side faces, find their way to reach their neighbours of the same sign, $\omega _1$ and $\omega _2$ from the upper face, respectively. A second merging process takes place on the top of the cube, but slightly away from the surface, resulting in the vortex structure $\omega _1 \cup \omega _5$ and $\omega _2 \cup \omega _8$ as depicted at ${{R}e}=60$ and $\Omega = 0.3$ . We note that all the investigated cases exhibit these two types of rotation-induced vortex merging processes on the cube front surface, resulting in an inner pair and an outer pair of streamwise vortices propagating to the wake.

It is worth mentioning that some variations of the merging process may occur. At low rotating rate, such as $\Omega =0.02$ , we observe that the merging of the outer vortex may not happen, this is visible in the cases ${{R}e}=160$ and ${{R}e}=350$ in figure 6. In the HS regime, the periodic vortex shedding takes place, and we note that the roles of the inner and outer vortex pairs can swap in time. For instance, in the snapshot of the case at ${{R}e}=160$ and $\Omega =0.1$ , the merged vortex $\omega _1 \cup \omega _5$ appears on the cube surface while vortex $\omega _6$ is pushed away to meet $\omega _4$ , due to the larger inclination angle of the upper face. As the cube continues to rotate, we see that the merged vortex $\omega _1 \cup \omega _5$ moves aways from the side face, while the merged vortex $\omega _4 \cup \omega _6$ returns to the surface, similar to the snapshot at ${{R}e}=60$ and $\Omega =0.1$ . With the increase of the rotation rate, $\Omega \geqslant 0.4$ , the impact of the cube angular position diminishes, and the alternating role swap ceases. The merged vortices $\omega _4 \cup \omega _6$ and $\omega _3 \cup \omega _7$ consistently form the inner pair, as observed in the KH regime at ${{R}e} = 160$ , $\Omega = 0.8$ and $\Omega = 1$ . In the CS regime, even though the planar symmetry is broken on the front surface, the merging process still follows the same mechanism, as shown in the case ${{R}e}=350$ and $\Omega =0.8$ in figure 6.

4.2.3. Near-wake vortex structures

Now that we clarified the vortex generation and merging process, we present in figure 7 the $\omega _x$ distribution in the particle wake region in the $y$ - $z$ plane, right next to the rear face at $x = x_p + 0.6$ . At this point, the interaction between the vortices and the particle surface terminates. The evolution of the wake structure is driven solely by fluid dynamics and vortex interactions. Four Reynolds numbers ( ${{R}e} = 60 {-} 350$ ) are selected to illustrate the wake vortices across the seven flow regimes studied. The same colour scale ( $\omega _x \in [-1,1]$ ) as that used in figure 6 is used for consistency and better comparison. Regions in red and blue indicate higher vorticity magnitude, while black areas represent lower vorticity magnitude. In the case ${{R}e} = 60$ and $\Omega = 0.02$ , the vortex pattern appears blurred due to the low vortex strength.

Figure 7.

Visualisation of vortex pair generation near the rear surface of the rotating RCF4 cube in the $y$ - $z$ plane at $x = x_p + 0.6$ , with red and blue regions representing positive and negative $\omega _x$ , respectively. Flow streamlines are included to enhance the visualisation of the wake vortices. The various flow regimes are highlighted in colour-coded boxes, and the projected outline of the cube on the $y$ - $z$ plane is sketched for reference.

At low flow inertia, ${{R}e} = 60$ , the merged vortex pairs near the cube surface, $\omega _{46}$ (hereafter denoting the merged vortex $\omega _4 \cup \omega _6$ ) and $\omega _{37}$ (hereafter denoting the merged vortex $\omega _3 \cup \omega _7$ ), exhibit dominant magnitudes and maintain symmetry with respect to the $x$ - $y$ plane across the cube centroid. A higher $\Omega$ enhances $\omega _x$ , particularly in the VS regime, highlighted by the pink box in the first row of figure 7. In this regime, periodic vortex shedding is suppressed by the cube rotation, resulting in a steady vorticity distribution that only includes the vortex pairs $\omega _{46}$ and $\omega _{37}$ , as seen in the case ${{R}e} = 60$ and $\Omega = 0.6$ . At ${{R}e} = 160$ and ${{R}e} = 250$ , particularly in the UV and HS regimes, we observe the formation of the outer vortex pairs $\omega _{15}$ and $\omega _{28}$ , originating from the upper front surface of the cube, as shown in the cases at $\Omega = 0.3$ . In the HS regime, the periodic interactions between the inner and outer vortex pairs result in a variety of vorticity distribution patterns, as illustrated in figure 7.

At ${{R}e} = 160$ , increasing the rotation rate to $\Omega = 0.4$ results in relatively weak outer vortex pairs $\omega _{15}$ and $\omega _{28}$ , as they are pushed away from the wake, marking the onset of the VS regime. In the KH regime indicated by the blue box, at $\Omega = 0.8$ and $\Omega = 1$ , the magnitude of $\omega _{15}$ and $\omega _{28}$ increases again, and they tightly wrap around the inner vortex pair like tulip petals. A similar vorticity pattern is also observed in the DS regime at ${{R}e} = 250$ and ${{R}e} = 350$ , as shown in the green box in figure 7. In the DS regime, a second shear-induced vortical tail forms in the wake, with the vortex cores of this second tail clearly visible at $\Omega = 1$ for both ${{R}e} = 250$ and ${{R}e} = 350$ . Overall, the near-wake vortex structure maintains a planar or pseudo-planar symmetry with respect to the $x$ - $y$ plane, except in cases within the CS regime.

In summary, the generation and merging of vortex pairs on the particle surface dictate the near-wake vortex patterns, which in turn influence the formation, dynamics and stability of the wake vortical structures, thereby shaping the flow regimes. Our findings demonstrate that the formation of two vortex pairs – inner and outer – is a universal process across all flow regimes within the investigated parameter space.

4.3.1. Vortex shedding and suppression

We recall that behind a transversely rotating sphere, the particle rotation amplifies wake unsteadiness, leading to periodic hairpin vortex shedding at lower critical ${R}e$ compared with a fixed sphere. This shedding follows a characteristic build-up and release process. As $\Omega$ increases, the vortices in the recirculation region disappear, and vortex shedding is suppressed within $0.3 \leqslant \Omega \leqslant 0.5$ for ${{R}e} = 100 {-} 300$ (Giacobello et al. Reference Giacobello, Ooi and Balachandar2009). Beyond this suppression, at higher values of both $\Omega$ and ${R}e$ , the steady vortical tail becomes unsteady once again, driven by a Kelvin–Helmholtz instability.

Figure 8.

Vortical structures identified by $\lambda _2=-1$ in the wake of the rotating RCF4 cube (a) at ${{R}e}=160$ and (b) at $\Omega =0.3$ in the $x$ - $y$ plane. The red and blue regions indicate positive and negative disturbed velocity, $\tilde {u}_x \in [-0.1,0.1]$ , respectively. Flow streamlines based on $\boldsymbol{\tilde {u}}$ are plotted to enhance the clarity of the wake flow field. The different flow regimes are distinguished by colour-coded boxes on the left of each subpanel.

How different are the wake structures of a rotating angular particle compared with those of a rotating sphere? To investigate this, we present snapshots of the vortical structures behind a rotating cube in the $x$ - $y$ plane, as shown in figure 8. The vortex structures are identified using the isosurface $\lambda _2 = -1$ . Due to the dominance of the streamwise incoming velocity over the other velocity components, important details of the wake region may be obscured. To better capture the wake dynamics, we plot the streamlines of the disturbed flow velocity $\boldsymbol{\tilde {u}} = (\boldsymbol{u^*} - U^*_{ref}\boldsymbol{e_x}) / U^*_{ref}$ and colour these streamlines using the scale of $\tilde {u}_x \in [-0.1, 0.1]$ , to reveal more details of the wake region.

As an example, figure 8(a) illustrates the evolution of wake structures with increasing rotation rates $0.02 \leqslant \Omega \leqslant 1$ at ${{R}e} = 160$ . The flow regime transitions from UV to HS, VS and eventually KH, following a sequence similar to that observed for a sphere. At $\Omega = 0.02$ , the wake behind the rotating cube appears straight, with fragmented vortical structures shed from the cube at a low frequency. Beneath the vortical structures, a broad, elongated strip region ( $\tilde {u}_x \leqslant 0$ ) is visible in light blue, extending behind the cube. This impacted wake region, present in all subpanels of figure 8, encloses the wake vortical structures and defines the boundary of the disturbed wake flow. As the rotation rate increases to $\Omega = 0.1 {-}0.3$ , hairpin vortex shedding occurs with increasing frequency.

A notable difference from the rotating sphere case is that the vortex shedding frequency behind a rotating cube is significantly higher at the same ${R}e$ and $\Omega$ . Two distinct types of vortex structures are observed: upper hairpin vortices originate from the upper front face of the rotating cube, while lower hairpin vortices emerge from the lower front face and side faces. Compared with $\Omega = 0.2$ , the wake vortex structures become smaller at $\Omega = 0.3$ . In the highlighted yellow rectangles, it is clear that the size of the upper hairpin vortices decreases with increasing $\Omega$ , disappearing entirely at $\Omega = 0.4$ , where wake vortex shedding is suppressed and manifests as a steady filament. Some residual bumpy structures remain near the top rear face of the cube at $\Omega = 0.4$ , but they vanish completely at $\Omega = 0.6$ . At higher rotation rates, the fluctus structures resulting from the Kelvin–Helmholtz instability are observed along the steady filaments near the lower boundary of the impacted wake region at $\Omega = 0.8$ and $\Omega = 1$ .

In terms of flow inertia, we present the evolution of wake vortical structures in the $x$ - $y$ plane at the same $\Omega = 0.3$ while varying the Reynolds number in the range $60 \leqslant {{R}e} \leqslant 400$ in figure 8(b). We focus on the influence of flow inertia on wake instability, particularly in cases where vortex shedding is suppressed in the VS regime. At ${{R}e} = 60$ and ${{R}e} = 80$ , the $\lambda _2$ isosurface shows bumpy structures near the rear surface of the cube, but it stabilises into a steady filament in the far-wake region of the cube; this is not the case at $\Omega = 0.2$ and the same ${R}e$ . Thus, we classify these cases in the VS regime, where vortex suppression is effective. As ${R}e$ increases, vortex shedding reemerges, and the size of the vortical structures grows with ${R}e$ . At ${{R}e} = 120$ , small spine-like structures appear along the top boundary of the impacted wake region near the cube rear surface, highlighted in yellow rectangles in figure 8(b). As ${R}e$ increases, the spine structures grow in size and gradually evolve into upper hairpin vortices, as depicted in the HS regime at ${{R}e} = 200$ . This process reverses the vortex suppression highlighted in the yellow rectangles in figure 8(a). Specifically, the spines (or upper hairpin vortices) generated by the upper front face of the rotating cube are suppressed as $\Omega$ increases, but as flow inertia rises, these vortices reemerge through a reversed yet similar mechanism. Between ${{R}e} = 180$ and ${{R}e} = 200$ , the vortex shedding frequency noticeably decreases and remains constant up to ${{R}e} = 400$ . At this point, the hairpin vortical structures become more complex, while still maintaining periodicity and symmetry about the $x$ - $y$ plane.

From figure 8, it is clear that the sharp edges of the rotating cube, particularly those on the upper front face, play a critical role in both the suppression and reemergence of vortex shedding. These edges strongly influence the generation of vortices, which in turn drive flow instabilities through their interactions in the wake region. To better understand these interactions, we examine the evolution of streamwise vortex pairs $\omega _x$ in the $x$ - $z$ plane, visualised by isosurfaces at $\omega _x = \pm 0.03$ (with red indicating positive $\omega _x$ and blue indicating negative $\omega _x$ ) for $40 \leqslant {{R}e} \leqslant 200$ and $0.1 \leqslant \Omega \leqslant 0.8$ in figure 9. Our analysis focuses on the case at $\Omega = 0.3$ and ${{R}e} = 120$ , while reducing the transparency of other cases for enhanced clarity.

Figure 9.

Evolution of streamwise vortex pairs identified by isosurfaces of $\omega _x=\pm 0.03$ (red for positive $\omega _x$ and blue for negative $\omega _x$ ) in the $x$ - $z$ plane. Lower panel shows snapshots of temporal evolution of the core of the vortex pairs identified by $\omega _x=\pm 1$ within a single rotation period.

The flow past a rotating cube generates alternating vortex pairs of opposite sign in the $x$ - $z$ plane. As discussed in figure 7, one vortex pair, consisting of $\omega _{15}$ and $\omega _{28}$ , originates from the upper front face of the cube and is located in the upper region of the wake (relative to the $y$ -axis). The other pair, $\omega _{46}$ and $\omega _{37}$ , forms from the merging of vortices generated by the lower front and side faces, and resides in the lower region of the wake. These two vortex pairs are shed alternately into the wake, as highlighted in yellow at ${{R}e} = 120$ and $\Omega = 0.3$ in figure 9.

Considering ${{R}e} = 120$ , at a low rotation rate $\Omega = 0.1$ , the size of the upper vortex pair ( $\omega _{15}$ , $\omega _{28}$ ) is comparable to that of the lower vortex pair ( $\omega _{46}$ , $\omega _{37}$ ), leading to a standard periodic hairpin vortex shedding. As $\Omega$ increases, with the rotation vector aligned along the $z$ -axis, the first direct effect is a reduction in fluid–particle relative velocity on the upper side ( $y^+$ ) of the cube, while the relative velocity on the lower side ( $y^-$ ) increases (within the range $\Omega \leqslant 1$ in this study). This weakens the upper vortex pair ( $\omega _{15}$ , $\omega _{28}$ ) while strengthening the lower vortex pair ( $\omega _{46}$ , $\omega _{37}$ ). Another key point is that, during each rotation period, four faces of the cube contribute to the generation of the upper vortex pair, with each face producing one. These faces are referred to as effective faces, and their number, denoted as $N_{\mathcal{F}}$ , depends on the particle shape and the axis of rotation. The effective faces directly determine the primary shedding frequency of the vortex pairs, which is typically proportional to $N_{\mathcal{F}}$ , a distinctive feature of rotating angular particles. Hence, with $\Omega$ increasing from $0.1$ to $0.2$ in the HS regime, the shedding frequency doubles. As a result of these combined effects, at $\Omega =0.3$ , the lower vortex pair shedding becomes more frequent and stronger, resulting in vortex merging in the wake. In the upper wake, the upper vortex pair is too weak to pinch off its lower counterpart. This imbalance results in a long, steady vortical tail with two filaments in the VS regime, explaining the vortex suppression, which persists up to $\Omega =1$ at ${{R}e}=120$ .

In the lower panel of figure 9, we present six snapshots of the vortex pair shedding process over a single rotation period for the flow at ${{R}e}=120$ and $\Omega =0.3$ , shown from a side view. The positions of the upper and lower vortex pairs are indicated. To improve the clarity of the visualisation, the vorticity isosurface criterion has been increased to $\omega _x = \pm 1$ to better highlight the vortex cores. At $\Omega = 0.3$ , the upper vortex pair ( $\omega _{15}$ , $\omega _{28}$ ) is clearly weaker than the lower vortex pair ( $\omega _{46}$ , $\omega _{37}$ ), as seen in snapshot I. As the cube keeps rotating, snapshots IV and V show the interaction between the upper and lower vortex pairs. Due to their opposite signs, $\omega _{15}$ and $\omega _{46}$ repel each other in space, as do $\omega _{28}$ and $\omega _{37}$ . In snapshots V and VI, the upper vortex pair is pushed away and dissipates due to diffusion, while the two successive lower vortex pairs begin to merge. The presence of the weak upper vortex pair acts as wedges on the strong lower pair isosurface and leads to the formation of the spine structures observed in figure 8. As ${R}e$ increases, the upper vortex pairs strengthen and grow in size, significantly influencing the wake stability. At ${{R}e} = 160$ , their increased presence becomes substantial enough to prevent the merging of the lower vortex pairs, thereby leading to the HS regime. Consequently, the reemergence of periodic vortex shedding follows the same mechanism as previously discussed, but in reverse.

In summary, our results provide a clear explanation for vortex suppression and reemergence mechanism through the lens of vortex interaction. The alternating shedding of upper and lower vortex pairs characterises the periodic hairpin vortex formation, while the merging of the lower vortex pair is the primary mechanism behind vortex suppression. The weakened upper vortex pairs are responsible for the formation of spine structures in the wake. Compared with the smooth geometry of a sphere, the sharp edges and distinct faces of the rotating cube provide a clearer understanding of the roles played by the upper and lower vortex pairs in these processes.

4.3.2. Wake structures in high- ${R}e$ regimes

Figure 10.

Vortical structures identified by $\lambda _2=-1$ in the wake of the rotating RCF4 cube at (a) ${{R}e}=250$ , (b) ${{R}e}=300$ and (c) ${{R}e}=400$ . The red and blue regions indicate positive and negative disturbed velocity, $\tilde {u}_x \in [-0.1,0.1]$ , respectively. Flow streamlines based on $\boldsymbol{\tilde {u}}$ are plotted to enhance the clarity of the wake vortex patterns. The different flow regimes are distinguished by colour-coded boxes, corresponding to each rotation rate.

A remarkable feature of the flow past an angular particle is the wide variety of vortex shedding patterns that form in the wake. At high flow inertia, specifically when ${{R}e} \geqslant 250$ , the rotation of the cube introduces new dynamics in the wake structures, leading to distinct flow regimes. In figure 10, we present snapshots of these complex vortical structures and shedding patterns for ${{R}e} \geqslant 250$ in the $x$ - $y$ plane. The streamlines of the disturbed velocity field $\boldsymbol{\tilde {u}}$ are drawn to provide a clearer depiction.

At ${{R}e}=250$ , as shown in figure 10(a), vortices follow a classical periodic hairpin vortex shedding pattern in the HS regime for $0.02 \leqslant \Omega \leqslant 0.3$ . While the shedding frequency generally increases with $\Omega$ , a slight decrease is observed between $\Omega =0.2$ and $\Omega =0.3$ . Interestingly, at $\Omega =0.2$ , the upper and lower vortex pairs, which are similar in size, are shed in phase, leading to a double-symmetric hairpin vortex shedding (DHS regime), resembling the pattern observed in the flow past an edge tetrahedron (Gai & Wachs Reference Gai and Wachs2023b ). At $\Omega \leqslant 0.3$ , the impacted wake region ( $\tilde {u}_x\leqslant 0$ ) no longer appears as a wide strip, instead displaying blue regions tightly surrounding the vortical structures. As $\Omega$ increases, the flow transitions to the KH regime, and the impacted wake region once again takes on a long strip shape. In this regime, the upper vortex pairs are moderated by mechanisms discussed earlier but are not fully suppressed. The lower vortex pairs, due to high fluid inertia, are unable to merge. At $\Omega =0.4$ and $\Omega =0.6$ , we observe the formation of fluctus at the lower boundary of the impacted wake region, where red zones indicate the part of streamwise acceleration of the vortex cores. It is evident that the shedding frequency of the shear-induced vortices is higher than that of the vortices released from the recirculation region at $\Omega =0.3$ . As $\Omega$ increases further, the shear-induced vortex shedding stabilises at $\Omega =0.8$ and $\Omega =1$ . Interestingly, under high rotation rates, the upper vortex pairs become enhanced and tend to merge, forming a steady, elongated structure – referred to as the second tail – at ${{R}e}=250$ and $\Omega =1$ . Note that this secondary tail is located at the upper boundary of the impacted wake region, which also contains two filaments symmetric to the $x$ - $y$ plane. The emergence of the secondary tail marks the onset of the DS regime. Due to the interplay of the flow inertia and angular particle rotation, we obtain a steady flow at relatively high ${R}e$ and $\Omega$ .

Figure 10(b) presents the vortex shedding pattern at ${{R}e}=300$ , where the CS regime kicks in. Even when the cube slowly rotates at $\Omega =0.02$ , the wake vortex distribution behind the cube is no longer planar symmetric. As $\Omega$ increases, the wake structure reorganises and the HS regime is observed for $0.1 \leqslant \Omega \leqslant 0.3$ . Especially, the double-hairpin shedding is also observed for ${{R}e}=300$ and $\Omega =0.2$ . With further increase of $\Omega$ , we note that the shear-induced shedding becomes predominant at $\Omega =0.4$ and $\Omega =0.6$ . The upper vortex pairs are much enhanced by the flow inertia compared with the case ${{R}e}=250$ in figure 10(a). At $\Omega =0.8$ , the interaction between the upper and lower vortices becomes irregular, eliminating planar symmetry. Interestingly, at $\Omega =1$ , the planar symmetry is restored and the DS regime emerges. However, both tails at the upper and lower boundaries of the impacted wake region become unsteady. At ${{R}e}=400$ , as shown in figure 10(c), the wake structure loses its periodicity in the CS regime, except at $\Omega =0.2$ and $0.3$ . Compared with the wake patterns observed at ${{R}e}=250$ and $300$ , the impacted wake region at $\Omega =1$ is less inclined and more aligned with the cube. Although a second tail is still present, the vortex shedding pattern becomes fully chaotic and irregular.

To offer readers a more in-depth understanding of vortex shedding, suppression and wake instability, the wake structures in the flow past a rotating cube in the $x$ - $y$ plane are presented for additional selected cases in Appendix C.

4.4.1. Magnus effect and lift coefficient

We know that the flow interaction with the cube is predominantly influenced by two factors: the flow inertia ${R}e$ and the rotation rate $\Omega$ . To quantitatively assess the impact of these factors, we plot the evolution of the time-averaged lift coefficient, $\overline {C_{l,y}}$ , as a function of $\Omega$ for $40 \leqslant {{R}e} \leqslant 200$ , as shown in figure 11(a). For higher flow inertia ( ${{R}e} \geqslant 250$ ), the evolution of $\overline {C_{l,y}}$ is presented separately in figure 11(b), using the same colour bar for clarity. To ensure accurate temporal averaging, $\overline {C_{l,y}}$ is computed over a sufficiently long period once the flow has fully developed.

Figure 11.

(a) Evolution of the time-averaged lift coefficient $\overline {C_{l,y}}$ of the rotating RCF4 cube as a function of $\Omega$ at $40 \leqslant {{R}e} \leqslant 200$ with increment $\Delta {{R}e}=20$ . Our numerical simulation results (coloured circles) are compared with the empirical correlation proposed in (4.1) (dashed lines). (b) Evolution of $\overline {C_{l,y}}$ at higher Reynolds numbers, $250 \leqslant {{R}e} \leqslant 400$ with increment $\Delta {{R}e}=50$ . Darker circles indicate higher values of ${R}e$ .

In general, increasing $\Omega$ results in a higher lift force due to the enhanced Magnus effect. The rotation of the cube causes a direct deceleration of the flow velocity on the bottom side ( $y^-$ ) compared with the top side ( $y^+$ ). According to Bernoulli’s principle, the higher velocity on the top corresponds to lower pressure, while the lower velocity on the bottom creates higher pressure, generating lift pointing to the $y^+$ direction. At low flow inertia, such as ${{R}e}=40$ , $\overline {C_{l,y}}$ increases almost linearly with $\Omega$ . With the increase of inertia, however, the increase in $\overline {C_{l,y}}$ becomes constrained, causing the curves to bend downward in figure 11(a). The influence of ${R}e$ on $\overline {C_{l,y}}$ can be divided into two parts: for $\Omega \leqslant 0.4$ , the effect of ${R}e$ is small and tends to enhance $\overline {C_{l,y}}$ ; for $\Omega \gt 0.4$ , higher ${R}e$ leads to a lower $\overline {C_{l,y}}$ . A possible explanation is that the enhanced flow unsteadiness results in reducing the pressure difference between the top and bottom surfaces of the rotating cube, thereby diminishing the Magnus effect. Between $\Omega =0.3$ and $\Omega =0.4$ , we find a critical regime transition region where all the data points exhibit a very close value of $\overline {C_{l,y}}$ . In this region, the rotation-driven time scale becomes smaller than the vortex formation time scale. We observe the complete suppression of vortex shedding and a reorganisation of the wake. The wake flow fully transitions from the HS regime to the VS regime (KH regime for ${{R}e}=180$ and ${{R}e}=200$ ), as shown in figure 4. An extraordinary property of $\Omega =0.4$ is that the lift coefficient $\overline {C_{l,y}}$ becomes independent of ${R}e$ .

By incorporating ${R}e$ and $\Omega$ , we have developed an empirical correlation to predict $\overline {C_{l,y}}$ through multivariable regression analysis:

(4.1) \begin{equation} \overline {C_{l,y}} = 0.53 + {{R}e}^{-0.11}(2.02\Omega - 0.8)\exp (-0.017\Omega ^{1.08}{{R}e}^{0.86}). \end{equation}

Figure 11(a) shows that this correlation effectively captures the numerical results of $\overline {C_{l,y}}$ across the considered range of $40 \leqslant {{R}e} \leqslant 200$ and $0.02 \leqslant \Omega \leqslant 1$ . The high determination coefficient, $R^2 = 0.9989$ , underscores the accuracy of the prediction. Furthermore, the model predicts that $\overline {C_{l,y}}$ becomes independent of ${R}e$ at $\Omega \approxeq 0.4$ , where all curves converge to $\overline {C_{l,y}}\approxeq 0.53$ , indicating a common point of overlap.

At higher flow inertia, the lift coefficient exhibits a different behaviour with increasing ${R}e$ , potentially due to a bifurcation in the base flow regardless of rotation (Citro et al. Reference Citro, Tchoufag, Fabre, Giannetti and Luchini2016). The increased inertia brings flow instability to the near-wake region accompanied by pronounced pressure fluctuations. We present the evolution of $\overline {C_{l,y}}$ for ${{R}e} \geqslant 250$ in figure 11(b). A key distinction is that for $\Omega \leqslant 0.1$ , an increase in ${R}e$ leads to lower values of $\overline {C_{l,y}}$ . This effect becomes particularly pronounced as planar symmetry breaks down in the CS regime. For $\Omega \geqslant 0.2$ , increasing ${R}e$ continues to reduce $\overline {C_{l,y}}$ , following a mechanism similar to that observed for ${{R}e} \leqslant 200$ . The common point of overlap of $\overline {C_{l,y}}$ at ${{R}e} \geqslant 250$ shifts to a value lower than $\Omega =0.4$ observed at $40 \leqslant {{R}e} \leqslant 200$ , and a similar region emerges between $\Omega =0.2$ and $\Omega =0.3$ , where the variation of $\overline {C_{l,y}}$ with ${R}e$ is small. It is noteworthy that this region aligns closely with the onset of the KH regime, as shown in figure 4. When the wake flow transitions from the HS regime to the VS and KH regimes, the surface vortex structures generated on the rotating cube tend to stabilise. In this scenario, the increase in flow inertia affects both the upper and lower sides of the cube equally. This similar value of $\overline {C_{l,y}}$ suggests that the pressure difference is maintained at a constant level despite changes in flow inertia.

4.4.2. Drag coefficient and lift-to-drag ratio

Along the streamwise direction, we present the evolution of the time-averaged drag coefficient, $\overline {C_d}$ , and its relationship with $\overline {C_{l,y}}$ in figure 12. The numerical results indicate that increasing $\Omega$ consistently leads to higher values of $\overline {C_d}$ in figure 12(a). This occurs because the rotation reduces the magnitude of the negative pressure in the wake region and shifts it downward, while having minimal impact on the positive pressure on the front surface of the cube, resulting in the increased pressure difference. Same as the flow past a fixed cube, increasing flow inertia has the opposite effect, reducing $\overline {C_d}$ for all the $\Omega$ investigated. At ${{R}e} \geqslant 250$ , we observe a region of insensitivity in $\overline {C_d}$ at $\Omega =0.2$ , where $\overline {C_d}$ remains independent of further increases in ${R}e$ . One possible explanation is that the shedding frequency of the surface-generated vortices reaches its peak at this specific rotation rate. The rapid release of low-pressure vortex cores raises the time-averaged pressure in the wake region behind the rotating cube. This mechanism is further supported by the evolution of the recirculation region volume $V_r$ , where a local minimum is observed at $\Omega = 0.2$ due to high shedding frequency, as discussed in Appendix D. This high shedding frequency helps sustain the pressure difference between the front and rear faces of the cube, keeping $\overline {C_d}$ at a nearly constant level despite increases in flow inertia from ${{R}e}=250$ to $400$ . Interestingly, double-hairpin vortex shedding (DHS) is observed in this region at ${{R}e}=250$ and ${{R}e}=300$ , as shown in figure 10. To fully leverage our numerical dataset, we derived a novel empirical correlation for $\overline {C_d}$ in (E1) that builds upon the classical Schiller–Naumann formulation (Schiller & Naumann Reference Schiller and Naumann1935). The derivation and resulting expression are detailed in Appendix E.

Figure 12.

(a) Evolution of the time-averaged drag coefficient $\overline {C_d}$ of the rotating RCF4 cube as a function of $\Omega$ at $40 \leqslant {{R}e} \leqslant 400$ . The increasing darkness of the circles represents higher flow inertia. (b) Evolution of the lift-to-drag ratio, $\overline {C_{l,y}}/\overline {C_d}$ , as a function of $\Omega$ at $40 \leqslant {{R}e} \leqslant 400$ . The increasing darkness of the circles represents higher ${R}e$ .

Knowing the behaviour of both $\overline {C_{l,y}}$ and $\overline {C_d}$ , we now plot the ratio of these two coefficients, $\overline {C_{l,y}}/\overline {C_d}$ , in figure 12(b) to gain further insights into the hydrodynamic forces. A general trend is that the ratio $\overline {C_{l,y}}/\overline {C_d}$ increases with $\Omega$ . The effects of flow inertia tend to increase the ratio $\overline {C_{l,y}}/\overline {C_d}$ , particularly for ${{R}e} \leqslant 200$ . However, at ${{R}e} \geqslant 250$ , the flow inertia has the opposite effect, decreasing this ratio. We know that the flow inertia and the particle rotation are competing factors in vortex generation. For ${{R}e} \geqslant 250$ , the increasing flow inertia strengthens the vortex pairs on both the upper and lower surfaces of the rotating cube. This enhanced vortex activity moderates the strong lift forces generated by the Magnus effect, preventing further amplification of Magnus-induced lift and stabilising $\overline {C_{l,y}}/\overline {C_d}$ . The moderating effect of ${R}e$ on the ratio is particularly pronounced in both slow rotation ( $0.02 \leftarrow \Omega$ ) and fast rotation cases ( $\Omega \rightarrow 1$ ). One exception is the region of drag insensitivity at $\Omega =0.2$ for $250 \leqslant {{R}e} \leqslant 400$ , where we see the $\overline {C_{l,y}}/\overline {C_d}$ keeps increasing with ${R}e$ . It is important to highlight that in all the cases investigated, $\overline {C_{l,y}}/\overline {C_d}$ remains consistently bounded below $0.6$ , as depicted in figure 12(b). With increasing $\Omega$ , $\overline {C_{l,y}}/\overline {C_d}$ does not increase monotonically, but instead reaches a plateau, as the flow dynamics at high rotation rates restrain further amplification of lift relative to drag.

4.4.3. Rotation coefficient

Given the significant impact of $\Omega$ and ${R}e$ on the Magnus effect, we calculate the rotation coefficient, using (2.8), to quantify how effectively the rotation rate contributes to generating lift. A higher $\overline {C_{\Omega }}$ indicates a greater efficiency of Magnus effect for a given $\Omega$ . In figure 13, we show the evolution of the time-averaged rotation coefficient $\overline {C_{\Omega }}$ as a function of $\Omega$ , with marker darkness increasing with ${R}e$ .

Figure 13.

Evolution of the time-averaged rotation coefficient $\overline {C_\Omega }$ of the rotating RCF4 cube as a function of $0.06 \leqslant \Omega \leqslant 1$ at $40 \leqslant {{R}e} \leqslant 400$ . The increasing darkness of the circles represents higher ${R}e$ . The rotation coefficient of a rotating sphere, $C_{\Omega , S} = 0.41$ , is marked by a blue horizontal line for $\Omega \leqslant 0.5$ , as reported by Clanet (Reference Clanet2015), providing a comparative reference.

We observe that the efficiency of lift generation via the Magnus effect decreases monotonically with increasing $\Omega$ for low flow inertia cases ${{R}e} \leqslant 200$ . For example, at ${{R}e}=40$ , $\overline {C_{\Omega }}$ drops from $0.55$ at $\Omega =0.06$ to $0.4$ at $\Omega =1$ , indicating that the flow becomes less responsive to rotation. The effects of increasing ${R}e$ on $\overline {C_{\Omega }}$ are twofold. For $\Omega \leqslant 0.2$ , higher flow inertia enhances the interaction between the rotating cube and the fluid, strengthening the Magnus effect. However, for faster-rotating cubes at $\Omega \geqslant 0.3$ , increasing ${R}e$ reduces $\overline {C_{\Omega }}$ , as the enhanced flow unsteadiness decreases the pressure difference between the top and bottom of the particle, diminishing lift generation. As shown in figure 11, the decline in $\overline {C_{l,y}}$ with ${R}e$ at $\Omega \geqslant 0.4$ is a direct result of the diminished efficiency of the Magnus effect. At higher flow inertia, ${{R}e} \geqslant 250$ , $\overline {C_{\Omega }}$ generally decreases, except at $\Omega = 0.2$ . At low rotation rates, $\Omega \leqslant 0.1$ , increasing ${R}e$ reduces lift due to chaotic shedding, which disrupts vortex distribution and diminishes the lift force. However, at $\Omega = 0.2$ , $\overline {C_{\Omega }}$ shows a distinct trend, continuing to increase across the entire Reynolds number range $40 \leqslant {{R}e} \leqslant 400$ , indicating enhanced rotational efficiency. This behaviour is closely linked to the reorganisation of vortex pairs and their high shedding frequency at $\Omega = 0.2$ . A comparison is made with the characteristic rotation coefficient of a sphere $C_{\Omega , S} = 0.41$ for $\Omega \leqslant 0.5$ (Clanet Reference Clanet2015). The blue horizontal line in figure 13 serves as a reference to compare the rotating cube with a sphere under similar conditions. The cube demonstrates higher rotational efficiency, particularly at lower rotation rates, indicating a stronger Magnus effect due to its sharp edges. Analogous to the empirical correlation for $\overline {C_{l,y}}$ , we propose an equally valuable correlation for $\overline {C_{\Omega }}$ at $40 \leqslant {{R}e} \leqslant 200$ in (E2) (see Appendix E).

From a broader perspective, the force coefficients ( $\overline {C_d}$ , $\overline {C_{l,y}}$ and $\overline {C_{\Omega }}$ ) of the RCF4 provide essential insights on trajectories of settling angular particles. In our prior work (Gai & Wachs Reference Gai and Wachs2024), we demonstrated that the Magnus force and hydrodynamic torque play significant roles in shaping the trajectory of a freely settling cube, sustaining an intriguing helical motion in the HS regime. The Magnus force serves as the primary driver of the cube horizontal motion. Then, the resulting rotation-induced drift generates a secondary Magnus force that amplifies the settling drag. It is also demonstrated that the enhanced hydrodynamic torque triggers the transition from the oblique to the helical settling regimes. An additional analysis of the hydrodynamic torque $\overline {T_z}$ is presented in Appendix F, where we propose a simple scaling law to model $\overline {T_z}$ of the RCF4 over the range $200 \leqslant {{R}e} \leqslant 400$ , yielding a correlation with an $r^2=0.999$ .

In previous studies, the absence of accurate correlations necessitated the use of approximate values in force balance analysis to gain insight into the underlying physics. By providing more accurate lift, drag and torque coefficients, and correlations for a centre-fixed rotating cube, we offer an approach to overcome this limitation, paving the way for more robust and physically grounded analyses in future research.