2.1. Dimensionless numbers

The governing equations (A1)–(A2) are non-dimensionalised using the following characteristic scales. The reference length is chosen as $L_{\textit{ref}}^* = D_{\textit{sph}}^*$ , where $D_{\textit{sph}}^*$ denotes the diameter of the volume-equivalent sphere (defined in (2.1)). The reference velocity is set to the free-stream flow velocity, $U_{\textit{ref}}^* = U_0^*$ . Thus, the characteristic time scale is $t_{\textit{ref}}^* = L_{\textit{ref}}^* / U_{\textit{ref}}^*$ . Several rotation-related quantities are expressed in terms of the imposed rotation rate $\varOmega ^*$ to enhance the physical clarity of the resulting dimensionless measures.

The dimensionless parameters examined in this study are: (i) input quantities, including the particle Reynolds number $\textit{Re}$ , the non-dimensional rotation rate $\varOmega$ , the rotational Reynolds number ${\textit{Re}}_{\varOmega }$ , the particle sphericity $\phi$ and the number of rotational symmetry folds $n_{\varOmega }$ ; and (ii) output measures, such as the lift and drag coefficients ( $C_l$ , $C_d$ ), the streamwise torque $T_x$ , the Strouhal number $\mathcal{S}tr$ characterising vortex shedding, the normalised vorticity field $\boldsymbol{\omega }$ , the helicity density $h$ and other derived quantities of interest.

Since all angular particles have the same volume, the particle Reynolds number is defined using the diameter of the volume-equivalent sphere,

(2.1) \begin{align} {\textit{Re}} = \frac {\rho _{\kern-1pt f}^* U_{\textit{ref}}^* D_{\textit{sph}}^*}{\mu _{\kern-1pt f}^*}, \quad \mathrm{with} \ D_{\textit{sph}}^* = \left ( \frac {6 V_{{p}}^*}{\pi } \right )^{1/3}, \end{align}

where $V_{{p}}^*$ is the particle volume.

Particle sphericity, $\phi$ , measures how closely an angular particle approximates a perfect sphere and is defined as the ratio of the surface area of the corresponding volume-equivalent sphere, $S_{\textit{sph}}^*$ , to that of the angular particle, $S_p^*$ , i.e.

(2.2) \begin{align} \phi = \frac {S_{\textit{sph}}^*}{S_p^*}, \end{align}

A value of $\phi$ between $0$ and $1$ characterises non-spherical particles, with $\phi = 1$ corresponding to a perfect sphere. Non-sphericity can also be described by the angularity (defined as the inverse of $\phi$ ), such that the angularity level increases with decreasing $\phi$ .

The non-dimensional rotation rate is defined as

(2.3) \begin{align} \varOmega = \frac {\varOmega ^* D_{\textit{sph}}^*}{2U_{\textit{ref}}^*}, \end{align}

where $\varOmega ^*=||\boldsymbol{\varOmega ^*}||$ is the magnitude of the prescribed rotation rate and $\boldsymbol{\varOmega }^*$ denotes the rotation vector of the angular particle (about the streamwise $x$ -axis). The number of rotational symmetry folds, $n_{\varOmega }$ , specifies the number of equal angular increments by which the particle can be rotated and still appear identical to its original angular position.

The rotation Reynolds number is then defined as

(2.4) \begin{align} {\textit{Re}}_{\varOmega } = \frac {\rho _{\kern-1pt f}^* \varOmega ^*D_{\textit{sph}}^{*,2}}{2\mu _{\kern-1pt f}^*} = \varOmega {\textit{Re}}. \end{align}

The drag coefficient $C_d$ and lift coefficients $C_{l,y}$ and $C_{l,z}$ are defined as

(2.5) \begin{align} C_d = \frac {8F_x^*}{\rho _{\kern-1pt f}^*U_{\textit{ref}}^{*,2} \pi D_{\textit{sph}}^{*,2}}, \quad C_{l,y} = \frac {8F_y^*}{\rho _{\kern-1pt f}^*U_{\textit{ref}}^{*,2} \pi D_{\textit{sph}}^{*,2}}, \quad C_{l,z} = \frac {8F_z^*}{\rho _{\kern-1pt f}^*U_{\textit{ref}}^{*,2} \pi D_{\textit{sph}}^{*,2}}, \end{align}

where $\boldsymbol{F}^*$ is the hydrodynamic force exerted on the particle.

Similar to the forces, the dimensionless torques are calculated by

(2.6) \begin{align} T_x = \frac {8T_x^*}{\rho _{\kern-1pt f}^* U_{\textit{ref}}^{*,2} \pi D_{\textit{sph}}^{*,3}}, \quad T_y = \frac {8T_y^*}{\rho _{\kern-1pt f}^* U_{\textit{ref}}^{*,2} \pi D_{\textit{sph}}^{*,3}}, \quad T_z = \frac {8T_z^*}{\rho _{\kern-1pt f}^* U_{\textit{ref}}^{*,2} \pi D_{\textit{sph}}^{*,3}}, \end{align}

where $T_i^*$ denotes the $i$ th component of the hydrodynamic torque $\boldsymbol{T^*}$ for $i \in \{x, y, z\}$ . In the DLM/FD method (Seyed-Ahmadi & Wachs Reference Seyed-Ahmadi and Wachs2020), $\boldsymbol{F}^*$ and $\boldsymbol{T}^*$ are calculated through the integration of Lagrange multipliers over $\mathcal{P}$ , such that

(2.7) \begin{align} \boldsymbol{F}^*= \int _{\mathcal{P}} \boldsymbol{\lambda }^*\,{\rm d}\boldsymbol{x}, \,\quad \boldsymbol{T^*}= \int _{\mathcal{P}} \boldsymbol{r}\times \boldsymbol{\lambda }^*\,{\rm d}\boldsymbol{x}. \end{align}

To capture the characteristic frequency of the hydrodynamic force $f_c^*$ , the Strouhal number is defined by normalising $f_c^*$ with the inverse of the advective time scale $U_{\textit{ref}}^*/D_{\textit{sph}}^*$ ,

(2.8) \begin{align} \mathcal{S}tr = \frac {f_c^* D_{\textit{sph}}^*}{U_{\textit{ref}}^*}. \end{align}

The dimensionless vorticity, providing a measure of the local fluid element rotation, is defined as

(2.9) \begin{align} \omega _x = \frac {\omega ^*_x D_{\textit{sph}}^*}{U_{\textit{ref}}^*}, \quad \omega _y = \frac {\omega ^*_y D_{\textit{sph}}^*}{U_{\textit{ref}}^*}, \quad \omega _z = \frac {\omega ^*_z D_{\textit{sph}}^*}{U_{\textit{ref}}^*}, \end{align}

where $\omega _x^*$ , $\omega _y^*$ and $\omega _z^*$ denote the vorticity components along the $x$ -, $y$ - and $z$ -directions, respectively.

2.2. Numerical setup

We present in figure 1 the computational domain $\mathcal{D}$ , a cubic box of side length $L_0 = 40$ filled with a Newtonian fluid. The centroid of the angular particle is located at $x_p=(10,\,20,\,20)$ and rotates about the streamwise $x$ -axis. Figure 1 highlights the wake generated by a rotating cube whose front face is perpendicular to the free stream and exhibits a fourfold rotational symmetry (hereafter referred to as sRCF). We focus on the face-facing-flow configuration, as it is the stable angular position of freely settling cubes in vertical regimes and serves as a canonical case for flow interaction with angular particles (Saha Reference Saha2004; Gai & Wachs Reference Gai and Wachs2024). Vortical structures are identified using the $\lambda _2=-1$ criterion (Jeong & Hussain Reference Jeong and Hussain1995), to extract coherent vortical regions. The snapshot corresponds to ${\textit{Re}} = 200$ and $\varOmega = 0.4$ , with the region of interest indicated by a pink box. The substantial difference in length scale between the particle and the computational domain is clearly visible.

Figure 1.

Computational domain and representative wake structures for the flow past a rotating cube (sRCF) at $\varOmega =0.4$ and ${\textit{Re}}=200$ , with the fourfold rotational symmetry aligned with the $x$ -axis. Vortical structures, identified using the $\lambda _2=-1$ criterion, are coloured by the streamwise velocity $u_x$ (red indicates high velocity and blue low velocity). The right-hand panel illustrates the adaptive mesh refinement on the $x$ – $z$ plane, with two slices perpendicular to the flow highlighting the streamwise vorticity distribution (red, $\omega _x\gt 0$ , and blue, $\omega _x\lt 0$ ) in the near wake of the rotating cube (blue box, $x=x_p + 0.8$ , and green box, $x=x_p + 1.6$ ).

Figure 1. Long description

Panel A: A 3D visualization of fluid flow around a rotating cube, showing vortical structures colored by streamwise velocity. The cube is positioned in the center of a grid, with arrows indicating the direction of flow. Panel B: A zoomed-in view of the vortical structures around the cube, highlighting the streamwise vorticity distribution in the near wake. The vortical structures are colored by streamwise velocity, with red indicating high velocity and blue indicating low velocity. Panel C: Two slices perpendicular to the flow, showing the streamwise vorticity distribution in the near wake of the rotating cube. The slices are highlighted with red and blue colors representing different vorticity values.

An enlarged view of the octree mesh is shown in the upper-right panel, displaying an $x$ – $z$ slice at $y = 19$ . This close-up highlights the locally refined mesh required to resolve the boundary-layer dynamics and the complex helical structures developing in the wake. Two additional slices perpendicular to the flow direction are presented in the lower-right panel in figure 1. These cross-sections reveal the evolution of the streamwise vorticity as the sampling plane moves downstream from the particle rear face. Although the detailed vortex patterns vary along the wake, they retain characteristic features such as rotational symmetry and the number of vortex pairs.

The cubic computational domain has six boundaries: left and right in the $x$ -direction, top and bottom in the $y$ -direction and front and back in the $z$ -direction, such that $\varGamma = \varGamma _{\textit{left}} \cup \varGamma _{\textit{right}}\cup \varGamma _{\textit{top}} \cup \varGamma _{\textit{bottom}}\cup \varGamma _{\textit{front}} \cup \varGamma _{\textit{back}} .$ The velocity field $\boldsymbol{u}^*$ satisfies Dirichlet conditions on the left boundary and a homogeneous Neumann condition is imposed on the right boundary. A no-slip condition is applied on the particle surface $\partial \mathcal{P}$ . On the remaining boundaries, symmetry conditions are imposed for the velocity, together with homogeneous Neumann conditions for the pressure, consistent with the default Basilisk implementation. A complete description of the boundary and initial conditions is provided in Appendix A.

We employ Basilisk’s Cartesian octree adaptive grid, in which each parent cell is recursively subdivided into eight child cells to refine regions of interest (Popinet Reference Popinet2015; Selçuk et al. Reference Selçuk, Ghigo, Popinet and Wachs2021). The grid adapts dynamically at every time step: cells are refined where velocity gradient variations are large and coarsened where the flow is smooth. The particle surface is maintained at the highest refinement level to ensure accurate resolution of the boundary layer and near-wake structures. The octree hierarchy imposes a factor of two between successive refinement levels, giving a minimum cell size $\Delta x = L_0/2^{n_l}$ , where $n_l$ is the maximum refinement level. Here, we use $n_l = 12$ , corresponding to about $102$ grid points per $D_{\textit{sph}}^*$ , which is found to be adequate for the flow regimes considered. Additional details and mesh-convergence results are available in Gai & Wachs (Reference Gai and Wachs2024, Reference Gai and Wachs2025).

Figure 2.

Distribution of Lagrange multipliers (white dots) on the surfaces of two Platonic polyhedrons: a cube ( $\phi =0.806$ ) and an octahedron ( $\phi =0.846$ ). A sphere ( $\phi =1.0$ ) is included for reference. Specific dual angular positions and their rotation axes are marked with colour-coded arrows: green for (sRCF, sROV) and white for (sRCV, sROF). The associated numbers of rotational symmetry folds of the rotation axes ( $n_{\varOmega }$ ) are depicted.

The rigid-body motion constraints within the DLM/FD solver are enforced using a collocation point method as illustrated in figure 2 (Glowinski et al. Reference Glowinski, Pan, Hesla and Joseph1999; Wachs et al. Reference Wachs, Hammouti, Vinay and Rahmani2015; Gai & Wachs Reference Gai and Wachs2023a ). The collocation set consists of interior points (grid nodes lying inside the particle) and surface points, which are uniformly distributed along directions parallel to the edges, ensuring a uniform coverage of the particle surface. Although the shapes are isometric, the flow structures and hydrodynamic responses depend strongly on the particle angular position. We select several rotation axes with different symmetry folds $n_{\varOmega }$ : a fourfold axis for the dual pair (sRCF, sROV) and a threefold axis for the dual pair (sRCV, sROF), as shown in figure 2. Here, sRCV denotes a rotating cube with one vertex facing the incoming flow, whereas sROF and sROV denote rotating octahedrons with, respectively, a front face and a vertex facing the incoming flow. Also, sRS denotes a sphere rotating about the streamwise axis. See § 6.1 for precise definition of these angular positions. To enhance the resolution of sharp edges and corners, a uniform point-to-point spacing of $l_{pp} = 2\Delta x$ is adopted, consistent with the smallest grid cell size around the particle. As the refinement level $n_l$ increases, more surface points are introduced accordingly. We ensure that the cube edge, the shorter of the two volume-equivalent Platonic polyhedrons, is resolved with 82 grid points. This strategy provides good spatial accuracy (Wachs et al. Reference Wachs, Hammouti, Vinay and Rahmani2015) and has been applied successfully to angular particle flows (Gai & Wachs Reference Gai and Wachs2023a , Reference Gai and Wachs2023b ).

The dimensionless time step is adapted to the rotation rate, from $\Delta t = 5\times 10^{-3}$ at $\varOmega \lt 0.1$ down to $10^{-4}$ at $\varOmega \geq 0.8$ , under a Courant–Friedrichs–Lewy (CFL) constraint evaluated over all cells; the full scheme is given in Appendix A. Approximately 260 PR-DNS simulations are performed in this study to obtain a comprehensive understanding of the streamwise rotation of angular particles in inertial flow. Appendix B presents a physical validation of the time-averaged drag coefficient $\overline {C_d}$ for a streamwise-rotating sphere. Further details on time-convergence validations are provided in Gai & Wachs (Reference Gai and Wachs2025).

4.1. Slowly streamwise-rotating cube

We first elucidate the vortex generation on the surface of a slowly rotating sRCF at ${\textit{Re}} = 200$ in figure 4. Panels I–IV trace the emergence and downstream evolution of counter-rotating vortex pairs that ultimately form the cross-shaped structure shown in panel V. In the $y$ – $z$ plane at $x = x_p - 0.1$ (panel I), four pairs of opposite-signed vortices develop along the front edges of the cube. As the flow convects downstream (II–III), a central negative vortex $\omega _0$ forms due to the rotation of the rear face, while the four negative lobes shed from the front edges migrate inwards and progressively merge into the central rear-face vortex $\omega _0$ . By $x = x_p + 1.0$ (IV), these lobes have coalesced into a stable cross-shaped pattern dominated by the streamwise vortex $\omega _0$ , consistent with the pattern in panel V. Compared with the fixed cube (CF; cube with a face facing the flow) wake shown in panels VI–VII, slow rotation produces a distinct negative cross surrounded by four isolated positive petals. The sign of this pattern is set by the direction of rotation: reversing the direction of rotation reverses the sign of this distribution. As summarised in the lower diagram of figure 4, the slowly rotating cube at $\varOmega = 0.02$ follows the same sequence of regime transitions as the fixed cube (Gai & Wachs Reference Gai and Wachs2023b ), with the SV, WF, PS and CS regimes aligning with their fixed-cube counterparts at comparable critical $\textit{Re}$ . Slow rotation therefore preserves the fixed-cube transition scenario.

Figure 4.

The vortex generation process on the surface and in the near wake of a slowly rotating sRCF cube at ${\textit{Re}}=200$ and $\varOmega =0.02$ . Panels I–IV show the evolution of surface vortices with increasing distance from the rear face, leading to the streamwise vorticity pattern in panel V. The vorticity field of a fixed cube at ${\textit{Re}}=200$ is shown in panels VI–VII. Red and blue regions indicate positive and negative $\omega _x$ . The lower diagram summarises the regime transition sequence for the rotating sRCF at $\varOmega =0.02$ compared with the fixed cube.

4.2. Vortex merging mechanism and near-wake structures

To clarify the mechanisms of vortex generation and interaction for a streamwise-rotating sRCF, we examine the distribution of $\omega _x$ in the near wake at $x = x_p + 0.6$ . Four representative $\textit{Re}$ are selected to span all regimes identified in figure 3. At low inertia ( ${\textit{Re}} = 140$ ), the SV regime exhibits a vorticity pattern consisting of four pairs of opposite-signed vortices ( $\omega _1$ – $\omega _8$ ) along the front edges, together with a central vortex $\omega _0$ generated by the rear face rotation. As the rotation rate increases to $\varOmega = 0.1$ , the negative petals $\omega _{1,3,5,7}$ begin to migrate inwards and merge with $\omega _0$ . By $\varOmega = 0.2$ , this process yields a strengthened central negative structure, denoted $\omega ^{\ominus }$ , which is a defining feature of the SF and SH regimes.

In contrast, the positive petals remain separated on the lateral flanks of the wake, as illustrated at $\varOmega = 0.4$ . As only the negative vortices merge at this stage, the wake develops a distinctive spearhead structure: helical strands generated by the positive vortices appear in the near wake, while a long, straight filament associated with $\omega ^{\ominus }$ extends downstream. At higher rotation rates, the positive vortices ( $\omega _{2,4,6,8}$ ) begin to connect (e.g. at $\varOmega = 0.6$ ), eventually forming a coherent positive envelope $\omega ^{\oplus }$ surrounding the negative core. This two-layer vortex system ( $\omega ^{\ominus }$ and $\omega ^{\oplus }$ ) stabilises the near wake and signals the re-emergence of the SF regime at high $\varOmega$ . A similar two-layer topology is observed behind streamwise-rotating smooth bluff bodies such as spheres, the limiting case of angular particles with infinitely many faces. In figure 5, the rotating cube therefore provides a clear, mechanistic interpretation of the origin of this characteristic two-layer vortex structure.

Figure 5.

Visualisation of streamwise vortex generation and merging in the near wake of the rotating sRCF cube, shown in the $y$ – $z$ plane at $x = x_p + 0.6$ . Red and blue indicate positive and negative streamwise vorticity $\omega _x$ and streamlines are superimposed to highlight the wake topology. Flow regimes are delineated by colour-coded boxes, and the projected outline of the cube on the $y$ – $z$ plane is included for reference.

Figure 5. Long description

The image contains multiple panels arranged in a grid, each showing a visualization of flow patterns around rotating polyhedral particles. The panels are organized by different Reynolds numbers (Re) and rotation rates (Ω). Each panel depicts the streamwise vortex generation and merging in the near wake of the rotating particle, with red and blue indicating positive and negative streamwise vorticity. Streamlines are superimposed to highlight the wake topology. The flow regimes are delineated by color-coded boxes, and the projected outline of the cube is included for reference. The panels are labeled with specific values of Re and Ω, showing how these parameters affect the wake dynamics and hydrodynamic responses.

At ${\textit{Re}} = 260$ and $\varOmega =0.02$ , the wake enters the WF regime, consistent with the PSS regime of the fixed cube (Gai & Wachs Reference Gai and Wachs2023b ). The residual planar symmetry is only weakly stable under rotation, giving rise to the characteristic waviness of the filament. As $\varOmega$ increases, the wake reorganises into the SF regime, and periodic vortex shedding emerges at $\varOmega =0.4$ . At ${\textit{Re}}=340$ , the wake transitions from the PS to the CS regime. A notable feature at $\varOmega =0.4$ is the appearance of a hollow central region devoid of strong $\omega _x$ , which disrupts the merging between $\omega _0$ and the edge vortices.

At ${\textit{Re}} = 460$ , the wake becomes predominantly chaotic except near $\varOmega = 0.4$ . In the WCS regime, the edge-generated vortex pairs remain attached to their counterparts and barely merge with the central vortex, producing a pattern reminiscent of the PS regime at ${\textit{Re}} = 340$ and $\varOmega = 0.4$ . Thus, despite increasingly prominent small-scale chaotic structures, the wake retains some large-scale organisation. Remarkably, even at ${\textit{Re}}=460$ , the two-layer structure ( $\omega ^{\ominus }$ , $\omega ^{\oplus }$ ) persists at high rotation rates $\varOmega = 0.8$ and $1$ . This persistence indicates that rotational motion exerts a stronger influence on the wake than the translational flow. At $\varOmega = 1$ , additional internal paired vortices begin to develop within the chaotic core region.

4.3. Vortex street and far-wake behaviours

Beyond the near-wake dynamics, we present the far-wake vortex streets generated by the sRCF, with representative cases depicted in figure 6. A clear distinction emerges among the SV, SF and WF regimes: in the SV regime ( ${\textit{Re}}=220$ , $\varOmega =0.02$ ), no coherent structures persist into the far wake; in the SF regime ( ${\textit{Re}}=100$ , $\varOmega =0.4$ ), the wake forms a long, straight vortical tube; and in the WF regime ( ${\textit{Re}}=220$ , $\varOmega =0.1$ ), this tube becomes wavy and twisted. The SH regime appears at slightly higher $\textit{Re}$ and $\varOmega$ , characterised by a pronounced helical structure in the near wake that transitions into a steady filament farther downstream. This regime results from the partial merging of rear-face rotation-induced vortex and edge vortices, as shown in figure 5. At ${\textit{Re}}=220$ , the typical spearhead structure is evident at $\varOmega =0.4$ (figure 6 a); compared to its counterpart at $\varOmega =0.2$ , the helical core becomes noticeably more compact as the rotation rate increases. Further increasing $\varOmega$ leads to fragmentation of the vortex filament and the onset of periodic shedding downstream.

Figure 6.

Vortical structures identified by $\lambda _2=-1$ in the wake of the sRCF: (a) at ${\textit{Re}}=220$ and (b) at $\varOmega =0.4$ , shown in the $x$ – $y$ plane. Red and blue denote positive and negative streamwise disturbance velocity, $\tilde {u}_x=u_x-U_{\textit{ref}}\in [-0.1,0.1]$ , respectively. In-plane streamlines highlight the three-dimensional wake topology, colour-coded boxes mark different flow regimes, and the downstream region spans $L_0/2$ .

At $\varOmega =0.4$ and ${\textit{Re}}=260$ (figure 6 b), the wake exhibits an intermediate state between the SH and PS regimes: a compact spearhead structure forms in the near wake, but the onset of filament undulation is already visible. Increasing the flow inertia to ${\textit{Re}}=340$ disrupts this structure. At moderate $\textit{Re}$ and $\varOmega$ , the translational and rotational forcings compete, producing a shift from the rotation-induced helical filament to alternating hairpin vortices, characteristic of flow past a fixed cube. This competition explains the progressive fragmentation of the wake, consistent with the observation that, at ${\textit{Re}}=380$ , the vortex street increasingly resembles a sequence of alternately shed hairpin vortices. At higher inertia, the hairpin pattern becomes dominant. The PS wake can be pictured as a family of vortex loops shed at the face-passing period: at low $\varOmega$ they resemble the alternating hairpin loops of a fixed sphere wake, and increasing rotation progressively twists them into helical strands wound around the central filament, until at high $\varOmega$ the wake reduces to a pair of intertwined helices. Disordered tangles appear in the WCS and CS regimes at ${\textit{Re}}=460$ and ${\textit{Re}}=500$ (figure 6 b). These tangled structures appear to result from sporadic, substantial releases of near-wake vortical fluid embedded within an already chaotic background, arising from the competition between rotational forcing and translational advection.

Overall, the wake structures observed behind a streamwise-rotating cube differ markedly from those behind smooth bluff bodies such as spheres or cylinders. A comprehensive overview of the wake vortex structures, in terms of $\omega _x$ and $\lambda _2$ isosurfaces for the sRCF over the $({\textit{Re}},\varOmega )$ range investigated, is presented in Appendix C for completeness.

5.1. Drag coefficient

We now turn from wake topology in § 4 to the resulting hydrodynamic responses. Figure 7(a) shows the time-averaged drag coefficient $\overline {C_d}$ as a function of $\textit{Re}$ and $\varOmega$ . Increasing $\varOmega$ systematically enhances $\overline {C_d}$ . Streamwise rotation modifies the separation pattern around the cube surface, producing a more laterally expanded wake, which is consistent with a reduction in wake pressure and a higher net drag. In contrast, increasing flow inertia reduces $\overline {C_d}$ for all rotation rates considered, in agreement with the behaviour of a fixed cube. As the drag coefficient plays a central role in fluid–particle coupling, we develop an empirical correlation for $\overline {C_d}$ of an sRCF over $60 \leq {\textit{Re}} \leq 500$ and $0.02 \leq \varOmega \leq 1$ . Within this parameter range, the correlation is obtained by least-squares regression of the PR-DNS data. It adopts the functional form of the classical Schiller–Naumann drag law (Schiller & Naumann Reference Schiller and Naumann1935) as a fitting skeleton, with an additional rotational correction, yielding

(5.1) \begin{align} \overline {C_d} = \frac {24}{{\textit{Re}}}\big (1 + 0.02{\textit{Re}}^{0.93}\big )\big (2 + 0.62\varOmega ^2\big ). \end{align}

The first factor represents the Stokes contribution, the second accounts for inertial corrections, and the third captures the enhancement induced by streamwise rotation. The rotational contribution is inherently nonlinear and follows an $\varOmega ^{2}$ scaling, which remains invariant under the transformation $\varOmega \rightarrow -\varOmega$ . Moreover, we verify that, in the joint limit of low $\textit{Re}$ and low $\varOmega$ , (5.1) is consistent with $\overline {C_d}$ for flow past a fixed CF cube (Gai & Wachs Reference Gai and Wachs2023b , Reference Gai and Wachs2023c ). Figure 7(a) compares this correlation with the numerical data over the full range of $\textit{Re}$ and $\varOmega$ investigated. Equation (5.1) reproduces the simulation results with high fidelity, yielding a coefficient of determination of $r^{2} = 0.9764$ . Figure 7(b) provides a complementary view by showing the drag coefficient $\overline {C_d}$ as a function of $\textit{Re}$ . The curves flatten progressively at large $\textit{Re}$ , showing that the drag becomes less sensitive to further increases in translational inertia in this regime. Meanwhile, the persistent separation between the different $\varOmega$ curves indicates that rotational forcing remains an important control parameter throughout the range considered. The correlation of (5.1) is also shown in figure 7(b), providing a complementary comparison with the PR-DNS dataset.

Figure 7.

(a) Time-averaged drag coefficient $\overline {C_d}$ versus $\varOmega$ for flow past an sRCF at $60 \leq {\textit{Re}} \leq 500$ ; darker circles denote larger $\textit{Re}$ , and dashed curves show the corresponding correlations. (b) Plot of $\overline {C_d}$ versus $\textit{Re}$ over $0.02 \leq \varOmega \leq 1$ ; darker circles denote larger $\varOmega$ .

Figure 7. Long description

Panel A: A scatter plot shows the time-averaged drag coefficient (C_d) versus the dimensionless rotation rate (Omega). The x-axis represents the dimensionless rotation rate (Omega) ranging from 0 to 1. The y-axis represents the time-averaged drag coefficient (C_d) ranging from 0 to 2.5. Darker circles denote larger Reynolds numbers (Re), and dashed curves show the corresponding correlations. The plot indicates a positive correlation between the drag coefficient and the rotation rate, with a higher density of data points at lower rotation rates. Panel B: Another scatter plot shows the time-averaged drag coefficient (C_d) versus the Reynolds number (Re). The x-axis represents the Reynolds number (Re) ranging from 0 to 500. The y-axis represents the time-averaged drag coefficient (C_d) ranging from 0.9 to 1.8. Darker circles denote larger dimensionless rotation rates (Omega). The plot shows a decreasing trend in the drag coefficient as the Reynolds number increases, with different curves for various rotation rates.

5.2. Lift coefficients

Another key aspect of the hydrodynamic response of the rotating cube is the lift force, which reflects the lateral asymmetry of the wake and its vortex dynamics. We consider the norm of the time-averaged lift as a function of $\varOmega$ in figure 8(a), which is defined as

(5.2) \begin{align} \Vert \overline {C_l}\Vert _2 = \big(\overline {C_{l,y}}^{2} + \overline {C_{l,z}}^{2} \big)^{1/2}. \end{align}

Here $\Vert \overline {C_l}\Vert _2$ quantifies the presence of a persistent lateral force bias, and is directly relevant to the prediction of particle drift and dispersion in inertial flows. At low flow inertia ( $60 \le {\textit{Re}} \le 240$ ), the lift remains nearly indistinguishable from zero in figure 8(a). As indicated by the regime map shown in figure 3, the wake over the low- $\textit{Re}$ regimes retains a nearly axisymmetric near-wake structure with only weak helical deviations, thus yielding $\Vert \overline {C_l}\Vert _2 \approx 0$ . At higher flow inertia, non-zero lift values begin to emerge, producing the W-shaped evolution. The inset of figure 8(a) isolates representative cases to clarify the effects of regime transition in shaping the lift evolution. From ${\textit{Re}} = 260$ , a first local maximum appears at $\varOmega = 0.02$ , due to the onset of unsteadiness and asymmetry at slow rotation rate. Moderate rotation then suppresses these instabilities, resulting in a reduction of $\Vert \overline {C_l}\Vert _2$ for $\varOmega \geq 0.2$ up to ${\textit{Re}} = 320$ . From ${\textit{Re}} = 340$ , a pronounced maximum forms near $\varOmega = 0.4$ , where helical structures in the near wake become sufficiently developed to generate a significant persistent lateral force. At $\varOmega = 0.8$ , all cases exhibit a clear minimum associated with reorganisation of the wake vortex system.

Figure 8.

(a) Norm of the mean lift, $\Vert \overline {C_l}\Vert _2$ , as a function of $\varOmega$ for flow past a rotating sRCF cube at $60 \leq {\textit{Re}} \leq 500$ . Representative cases of $\overline {C_l}$ evolution across the regime transitions are given in the inset. (b) Time-averaged magnitude of the instantaneous lift, $\overline {\Vert C_l\Vert _2}$ . The increasing darkness of the circles represents higher $\textit{Re}$ .

Apart from the persistent lateral force bias, we also examine the time-averaged magnitude of the instantaneous lift, shown in figure 8(b), and defined as

(5.3) \begin{align} \overline {\Vert {C}_l \Vert _2} = \overline {\big(C_{l,y}^{2} + C_{l,z}^{2} \big)^{1/2}} . \end{align}

With this definition, the orientation of the lift force is disregarded and only its intensity is retained. At low inertia ${\textit{Re}} \lt 260$ , the lift magnitude increases with $\varOmega$ , reflecting the progressive enhancement of wake asymmetry induced by the sharp edges. In the case of a sphere at ${\textit{Re}}=100$ and over the same range of $\varOmega$ , $\overline {\Vert {C}_l \Vert _2}$ remains close to zero (Kim & Choi Reference Kim and Choi2002). From ${\textit{Re}}=260$ , the lift magnitude increases at low rotation rates with the onset of the WF regime, as illustrated by the golden yellow curve. At higher $\textit{Re}$ , $\overline {\Vert {C}_l \Vert _2}$ remains significant over the entire range $0.02 \leq \varOmega \leq 1$ , and its dependence on $\varOmega$ becomes non-monotonic. In particular, a clear W-shaped evolution is observed, with a local maximum near $\varOmega = 0.4$ and a local minimum around $\varOmega = 0.8$ .

5.3. Recirculation region

To better understand the hydrodynamic forces, we study the recirculation region in the wake of the sRCF by identifying the zones of negative streamwise velocity ( $u_x\lt 0$ ), as illustrated in figure 9(a). The streamwise length of this region, $L_r$ (measured from the rear face of the cube to the downstream reattachment point), is reported in figure 9(b).

Figure 9.

Evolution of the recirculation region with $\varOmega$ for $60 \leq {\textit{Re}} \leq 500$ . (a) Recirculation region (green) at ${\textit{Re}} = 160$ for $\varOmega = 0.02$ and $\varOmega = 0.6$ in the $x$ – $y$ plane, with vortex structures identified by $\omega _x = \pm 1$ (red and blue) and $\lambda _2 = -1$ (semi-transparent white) shown for reference. (b) Recirculation length $L_r$ ; darker circles denote larger $\textit{Re}$ . Black stars: $L_r$ of sRS at ${\textit{Re}} = 100$ (solid), $250$ (dashed) and $300$ (dotted) digitised from Kim & Choi (Reference Kim and Choi2002). (c) In-plane streamlines in the $y$ – $z$ cut-plane at $x = x_p + 0.1$ within the sRCF recirculation region for $\varOmega = 0.4$ and $\varOmega = 0.8$ at ${\textit{Re}} = 380$ (a,c) and ${\textit{Re}} = 440$ (b), coloured by $\tilde {u}_x$ (red, $\tilde {u}_x\gt 0$ ; blue, $\tilde {u}_x\lt 0$ ). The corresponding three-dimensional wake structures are also shown for reference.

Figure 9. Long description

Panel A: Two visualizations of the recirculation region in the x-y plane for different values of Ω. The green region represents the recirculation area, with vortex structures identified by red and blue colors and semi-transparent white regions shown for reference. Panel B: A scatter plot showing the recirculation length (Lr) as a function of Ω. The circles represent different values of Re, with darker circles denoting larger Re. Black stars represent data from Kim & Choi (2002) for different values of Re. Panel C: In-plane streamlines within the recirculation region for different values of Re and Ω, colored by velocity (red for high velocity and blue for low velocity). The corresponding three-dimensional wake structures are also shown for reference.

For $60 \leq {\textit{Re}} \leq 200$ , $L_r$ increases monotonically with $\textit{Re}$ , indicating the strengthening and elongation of the recirculation zone with rising flow inertia. At higher $\textit{Re}$ , however, $L_r$ begins to decrease at high rotation rates, $\varOmega \geq 0.6$ .

A notable feature in figure 9(b) is the presence of a local minimum at $\varOmega = 0.4$ and a local maximum at $\varOmega = 0.8$ . The minimum at $\varOmega = 0.4$ reflects a short but strongly asymmetric recirculation region, consistent with the emergence of pronounced helical structures and the corresponding peak in the lift evolution (figure 8 a). In contrast, the maximum at $\varOmega = 0.8$ corresponds to a more elongated yet more symmetric wake, which limits lateral imbalance and therefore produces a low persistent lateral force. For reference, figure 9(b) also includes the recirculation length of the sRS (Kim & Choi Reference Kim and Choi2002). Over the entire $\textit{Re}$ and $\varOmega$ range, the sphere baseline lies below the corresponding cube data. The sharp-edge separation alone produces a substantially longer recirculation region than the sphere, before any rotation effect comes into play. Interestingly, the sphere at ${\textit{Re}} = 300$ also exhibits a local minimum of $L_r$ at intermediate rotation rates. For the cube, however, the extrema are directly tied to the asymmetry of the near wake and its lateral-force signature discussed in § 5.2.

Figure 9(c) shows in-plane streamlines within the recirculation region at ${\textit{Re}} = 380$ and $440$ in a $y$ – $z$ cut-plane. At both $\textit{Re}$ , the cross-sectional flow is markedly asymmetric and unsteady at $\varOmega = 0.4$ , with helical structures discernible in the near wake, but becomes far more organised at $\varOmega = 0.8$ , even though the flow then belongs to the CS regime. This contrast demonstrates that the extrema of the lateral force are governed by the degree of asymmetry in the recirculation region.

In particular, the observation that the near-wake flow is more ordered in the CS regime than in the PS and WCS regimes is somewhat counterintuitive. This apparent paradox reflects the fact that near-wake organisation and global wake chaos are distinct concepts. The nonlinear coupling between translational advection and rotation triggers a chaotic vortex only farther downstream, as evidenced by the three-dimensional vortex isosurfaces.

5.4. Streamwise hydrodynamic torque

Beyond drag and lift, hydrodynamic torque is crucial in governing the dynamics of freely moving particles (Fernandes et al. Reference Fernandes, Ern, Risso and Magnaudet2008; Ern et al. Reference Ern, Risso, Fernandes and Magnaudet2009; Gai & Wachs Reference Gai and Wachs2024). Figure 10(a) shows the evolution of the time-averaged streamwise component of torque $\overline {T_x}$ , computed from (2.6). Since $\varOmega$ is prescribed, $\overline {T_x}$ measures the resistive response of the fluid to the imposed rotation. The results show that $\overline {T_x}$ increases strongly with $\varOmega$ , while its magnitude decreases with increasing $\textit{Re}$ , due to the progressive reduction of viscous shear.

Figure 10.

(a) Time-averaged torque $\overline {T_x}$ as a function of $\varOmega$ of the rotating sRCF cube over $0.02 \leq \varOmega \leq 1$ at $60 \leq {\textit{Re}} \leq 500$ . (b) Scaling of the streamwise torque, shown as $\varOmega ^{2}/\overline {T_x}$ plotted against ${\textit{Re}}_{\varOmega }$ . The colour of the symbols indicates the flow Reynolds number $\textit{Re}$ . The golden yellow line represents the empirical correlation, with a coefficient of determination $r^2 = 0.9944$ .

Dividing $\overline {T_x}$ by $\varOmega ^{2}$ yields the following scaling with ${\textit{Re}}_{\varOmega }$ :

(5.4) \begin{align} \frac{\overline {T_x}}{\varOmega ^2} = 0.28 + 1.2\frac {24}{{\textit{Re}}_{\varOmega }}. \end{align}

The two coefficients are fitted to the PR-DNS data for an sRCF over $60 \leq {\textit{Re}} \leq 500$ and $0.02 \leq \varOmega \leq 1$ . All datasets collapse onto a single master curve in figure 10(b), demonstrating the dominant role of ${\textit{Re}}_{\varOmega }$ on streamwise torque in the parameter range investigated. For visual clarity, we represent the data in its inverted form, $\varOmega ^2/\overline {T_x}$ , as a function of ${\textit{Re}}_\varOmega$ . The correlation reproduces the numerical results with excellent accuracy ( $r^{2}=0.9944$ ). Consistent with the drag correlations discussed earlier, the quadratic scaling $\varOmega ^{2}$ also appears in the evolution of $\overline {T_x}$ . At low ${\textit{Re}}_{\varOmega }$ , the correlation reduces to a linear dependence of $\overline {T_x}$ on $\varOmega$ , consistent with the classical Stokes-flow analysis for rotating bodies (Happel & Brenner Reference Happel and Brenner1965). The low ${\textit{Re}}_{\varOmega }$ here reflects weak rotation at moderate $\textit{Re}$ , not a viscous-flow limit; the linear prefactor should therefore not be identified as a Stokes torque coefficient.

In addition to the hydrodynamic forces and torques, we present in Appendix D a brief analysis of the helicity generated by a streamwise-rotating cube. Its generation may have important implications for turbulence dynamics.

5.5. Frequency characteristics of the hydrodynamic forces

To examine the frequency signature of the hydrodynamic forces acting on the sRCF, we apply a fast Fourier transform (FFT) to the temporal evolution of the lift force $F_y$ once the flow is fully developed. We use $F_y$ as a representative signal, which provides clear and robust spectral content. The resulting frequency spectra are presented in figure 11. Three Reynolds numbers, ${\textit{Re}} = 100$ , $220$ and $500$ , are selected as representative cases of low, moderate and high flow inertia in figure 11(a)–11(c).

Figure 11.

(a–c) Frequency spectra of vortex shedding from the rotating sRCF cube at ${\textit{Re}}=100$ , $220$ and $500$ . Here $I$ is the normalised FFT spectral amplitude, shown on a linear scale in each panel. (d) Strouhal numbers $\mathcal{S}tr$ of the primary and secondary peaks over the considered $\varOmega$ and $\textit{Re}$ ranges. The blue curve gives the model prediction for the primary peak, $\mathcal{S}tr_1=4\varOmega /\pi$ , equation (5.5); the red curve shows the imposed rotation rate, $y=\varOmega$ ; the grey curve indicates the first harmonic of the primary peak; and the yellow curve corresponds to the fixed-CF-cube value $\mathcal{S}tr_0=0.11$ (Gai & Wachs Reference Gai and Wachs2023b ). These characteristic frequencies are marked by dashed vertical lines of the corresponding colours in panels (a–c).

Figure 11. Long description

Panel A: Three line graphs show the frequency spectra of vortex shedding from a rotating cube at a Reynolds number of 100. Each graph represents a different rotation rate (Ω) ranging from 0.02 to 1.00. The x-axis represents the frequency (f), and the y-axis represents the normalized FFT spectral amplitude (I). Panel B: Three line graphs show the frequency spectra of vortex shedding from a rotating cube at a Reynolds number of 220. Each graph represents a different rotation rate (Ω) ranging from 0.02 to 1.00. The x-axis represents the frequency (f), and the y-axis represents the normalized FFT spectral amplitude (I). Panel C: Three line graphs show the frequency spectra of vortex shedding from a rotating cube at a Reynolds number of 500. Each graph represents a different rotation rate (Ω) ranging from 0.02 to 1.00. The x-axis represents the frequency (f), and the y-axis represents the normalized FFT spectral amplitude (I). Panel D: A line graph shows the Strouhal numbers of the primary and secondary peaks over the considered Reynolds and rotation rates. The x-axis represents the rotation rate (Ω), and the y-axis represents the Strouhal number (S_r). The graph includes multiple lines representing different models and conditions, with dashed vertical lines indicating characteristic frequencies.

At ${\textit{Re}}=100$ and $220$ , the frequency of the primary spectral peak increases systematically with the rotation rate $\varOmega$ , indicating that the wake response is strongly organised by the periodic forcing imposed by the rotating cube. The spectra also reveal clear harmonic peaks, highlighting the nonlinear nature of the flow–particle interaction in the inertial regime. At high inertia ( ${\textit{Re}}=500$ ), the wake becomes highly chaotic and the primary peaks are no longer dominated by the imposed particle rotation, although secondary peaks show that rotation still leaves a measurable spectral imprint.

To characterise the global evolution of the shedding frequencies, we perform an FFT on all sRCF simulations and compute the Strouhal number $\mathcal{S}tr$ using (2.8) from the primary and secondary spectral peaks. The resulting values are shown as a function of $\varOmega$ in figure 11(d). The dark- and light-blue symbols correspond to the Strouhal numbers associated with the primary ( $\mathcal{S}tr_1$ ) and secondary ( $\mathcal{S}tr_2$ ) spectral peaks, respectively. In our earlier study of transversely rotating particles (Gai & Wachs Reference Gai and Wachs2025), we proposed a linear relationship for the dominant shedding frequency,

(5.5) \begin{align} \mathcal{S}tr_1 = \frac {n_{\mathcal{F}}}{\pi }\varOmega . \end{align}

Here $n_{\mathcal{F}}$ denotes the number of effective faces intersecting the plane perpendicular to the rotation axis. Unlike the drag and torque correlations, this scaling contains no fitted coefficients or adjustable parameters. Here, the sRCF cube has four effective faces, $n_{\mathcal{F}} = n_{\varOmega } = 4$ , which yields the line $y = 4\varOmega /\pi$ . As shown in figure 11(d), the scaling successfully captures the dominant frequency trend for low and moderate $\textit{Re}$ . Physically, this scaling describes a forcing frequency rather than an intrinsic wake frequency: as the particle rotates, it imposes a periodic kinematic forcing on the near wake at the face-passing frequency $n_{\mathcal{F}}\varOmega /\pi$ , analogous to the blade-passing frequency of rotating machinery. Primary peaks that follow $\mathcal{S}tr_1 = 4\varOmega /\pi$ therefore indicate vortex shedding locked to this geometric forcing, whereas peaks clustering near $\mathcal{S}tr_0 = 0.11$ reflect the intrinsic response of the wake, which at high $\textit{Re}$ competes with the rotational forcing and reverts to the natural shedding frequency of the fixed cube. Occasionally, primary peaks shift to the first harmonic $\mathcal{S}tr_1 = 8\varOmega /\pi$ , while the secondary peaks preferentially populate this same harmonic. These harmonics represent the nonlinear response of the wake to the periodic rotational forcing, reflecting symmetry-induced modulation and enhanced vortex interactions at large $\varOmega$ .

6.1. Symmetry folds and duality

Among the particles considered in this study, two rotational symmetry classes arise: those with threefold rotational symmetry $n_{\varOmega }=3$ (sRCV, sROF) and those with fourfold rotational symmetry $n_{\varOmega }=4$ (sRCF, sROV). Polyhedral duality provides a natural framework to analyse these configurations. In geometry, duality interchanges the roles of vertices and faces while preserving the same symmetry. More generally, every polyhedron is associated with a dual, wherein the vertices of one correspond to the faces of the other, and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other (Wenninger Reference Wenninger1974).

The (CF, OV) and (CV, OF) form two sets of geometrically dual angular positions as depicted in figure 13(a). In the CF case, the cube has four inclined faces $\mathcal{F}_{\textit{CF},1-4}$ that are not normal to the incoming flow and therefore contribute to streamwise vorticity generation. By duality, the OV configuration of the octahedron similarly presents four inclined front faces $\mathcal{F}_{\textit{OV},1-4}$ with the same fourfold rotational symmetry. The same correspondence holds for the (CV, OF) pair. This dual framework enables a direct comparison of how equivalent symmetry constraints control vortex generation and the resulting flow-regime transitions.

Without streamwise rotation, dual pairs in inertial flow generate vortex structures that share the same symmetry yet can differ markedly in strength and topology. A clear example is shown in figure 13(b) for the fixed (CF, OV) and (CV, OF) dual pairs at ${\textit{Re}}=100$ . By superimposing the two configurations and their associated streamwise vortex structures (isosurfaces $\omega _x=\pm 0.8$ ), we observe that the vortex pairs generated on the front edge of the CF ( $\omega _{\textit{CF},1}$ , $\omega _{\textit{CF},2}$ ) are substantially weaker compared with their counterparts on the OV ( $\omega _{\textit{OV},1}$ , $\omega _{\textit{OV},2}$ ), even though both particles share the same fourfold rotational symmetry. The CV–OF dual pair, however, exhibits the same symmetry and comparable vorticity strength, as depicted in the right panel of figure 13(b).

6.2. Flow-regime map and wake structures

To provide an overview of the effects of particle shape on wake structures, we present the regime map for these angular particles in figure 14. Since a universal metric for quantifying angular position is lacking, the configurations are grouped into the two dual pairs (sRCF, sROV) and (sRCV, sROF). For the dual pair (sRCF, sROV), the flow-regime sequences are similar at low flow inertia ${\textit{Re}} \leq 100$ . At higher flow inertia, the PS regime emerges for the sROV at ${\textit{Re}} = 200$ and $\varOmega \leq 0.4$ , yet the wakes of the sRCF remain largely stable. At ${\textit{Re}} = 300$ , both cases exhibit the transition from the PS to the CS regime, with the critical $\varOmega$ higher in the sRCF case. For the (sRCV, sROF) dual pair, the agreement in regime transition is striking: they display an identical sequence for ${\textit{Re}} = 100$ . At ${\textit{Re}} \geq 200$ , however, the two configurations diverge markedly. Figure 14 shows that particle duality imposes a robust constraint on wake organisation: dual polyhedrons tend to exhibit identical regime sequences at low inertia, even though their shapes differ in angularity.

Figure 13.

(a) Dual symmetry of the (CF, OV) and (CV, OF) pairs, with corresponding dual faces shown in the same colours. (b) Vorticity generation by the (CF, OV) and (CV, OF) dual pairs at ${\textit{Re}}=100$ in flow along the $x$ -axis. Fixed angular particles and their dual counterparts are superimposed and centred at dual orientations. Coloured vortical regions denote positive and negative $\omega _x=\pm 0.8$ generated by each particle, with the colour code shown for the two dual angular positions. Axes are shown locally for each view.

Figure 14.

Flow-regime map for streamwise-rotating angular particles at ${\textit{Re}}=100$ , $200$ and $300$ over $0.02 \leq \varOmega \leq 1$ . Horizontal bars give the wake regimes for a given particle shape and orientation, with colours denoting the corresponding regimes. Configurations are arranged as dual angular pairs: (sRCF, sROV) and (sRCV, sROF).

Figure 15.

Wake vortical structures of the dual pair (sRCV, sROF) at ${\textit{Re}}=100$ and $0.1 \leq \varOmega \leq 0.6$ coloured by the streamwise velocity $u_x$ with the corresponding values of $\varOmega$ indicated above each case.

Will dual particles produce dual wake structures? Often yes, but not universally. The geometric symmetry of the particle has a significant impact on the symmetry of the near-wake vortex structures (see Appendix E for detailed examples). Figure 15 shows the vortex structures of the dual pair (sRCV, sROF) over $0.1 \leq \varOmega \leq 0.6$ at ${\textit{Re}}=100$ . Within this parameter range, both configurations traverse the same SH–SF regime sequence and display nearly identical vortex morphologies. This close resemblance arises from the high geometric symmetry shared by the dual pair and the comparable strength of their surface-generated vortices. However, such similarity does not persist across all $\textit{Re}$ for the (sRCV, sROF) pair, nor does it hold for the other dual pairs at identical ( ${\textit{Re}}, \varOmega$ ) conditions.

Figure 16.

Vortex pair formation near the rear surface of the dual pairs (sRCV, sROF) and (sRCF, sROV), shown in the $y$ – $z$ plane at $x=x_p+0.6$ . Red and blue denote positive and negative $\omega _x$ , respectively, with in-plane streamlines superimposed to highlight the wake structures. Colour-coded boxes outline the corresponding flow regimes, and the projected angular particle contour in the $y$ – $z$ plane is shown for reference.

Figure 16. Long description

The image contains a grid of images showing vortex formations around different shapes with varying parameters. The grid is organized into four rows and eight columns. Each row represents a different shape: sRCV, sROF, sRCF, and sROV. Each column represents a different value of the parameter Ω, ranging from 0.02 to 1.0. Each cell in the grid shows a visual representation of the vortex formations around the shape for the corresponding value of Ω. The shapes are outlined in color-coded boxes, and the vortex formations are depicted with streamlines and color gradients indicating positive and negative vorticity.

Figure 16 provides a detailed view of the near-wake evolution for the dual pairs (sRCV, sROF) and (sRCF, sROV). For the sRCV, the front face consists of three inclined facets, which generate three pairs of opposite-signed streamwise vortices at ${\textit{Re}}=100$ and $\varOmega =0.02$ . As $\varOmega$ increases, the merging pathway induced by streamwise rotation follows the same mechanism as that of the sRCF (figure 5). At $\varOmega =0.1$ , a central vortex $\omega _0$ forms from the rear face and attracts the negative petals $\omega _{1,3,5}$ shed from the front faces. Streamwise rotation progressively compacts these structures, leading to helical strands increasingly aligned with the streamwise axis. By $\varOmega =0.3$ , these structures coalesce into a distinct central vortex $\omega ^{\ominus }$ . At higher rotation rates (e.g. $\varOmega =0.8$ ), the outer positive petals merge to form the envelope vortex $\omega ^{\oplus }$ . Its dual sROF undergoes the same sequence of merging events, which explains the identical regime transitions for the (sRCV, sROF) pair.

Similarly, the (sRCF, sROV) dual pair exhibits a close sequence of regime transitions at ${\textit{Re}}=100$ , as shown in figure 16. The fourfold rotational symmetry of the wake is preserved in both the sRCF and sROV cases. However, the merging of the negative vortices $\omega ^{\ominus }$ is less complete in the sROV case: even at $\varOmega =1$ , the four edge-induced vortices remain distinguishable within the core region. At ${\textit{Re}} \geq 200$ , this subtle difference becomes strongly amplified, resulting in markedly different flow-regime sequences in figure 14.

6.3. Hydrodynamic responses

Having examined the generation and interaction of vortical structures, we now turn to the hydrodynamic forces acting on streamwise-rotating angular particles. Overall, particle angularity and angular position play key roles in shaping the force responses, as illustrated in figure 17. We begin with the norm of the time-averaged lift $\Vert \overline {C_l}\Vert _2$ at ${\textit{Re}}=100$ for all angular positions and for a streamwise-rotating sphere (figure 17 a). Unlike the sphere, which maintains negligible lift over the entire rotation range, the angular particles develop modest lift at high rotation rates $\varOmega \ge 0.8$ at ${\textit{Re}}=100$ , with the exception of the sRCF. As discernible in figure 16, the wake of the sRCF retains a high degree of rotational symmetry at ${\textit{Re}}=100$ . At ${\textit{Re}}=200$ , we observe a significant increase in the lift at $\varOmega \leq 0.4$ , while the lift at $\varOmega \geq 0.6$ remains moderate, as depicted in figure 17(b). The sRCV and sROV exhibit higher lift than their face-facing (the flow) counterparts. In figure 16, the $\omega _x$ patterns in the wakes of the sRCV and the sROV configurations display enlarged and more finely structured vortical features within the wake core, compared to the sRCF and the sROF. The vertex-facing (the flow) orientation enhances the lateral expansion of the wake and, together with the incomplete vortex merging, leads to a breakdown of wake symmetry.

Figure 17.

Effects of particle shape and angular position on the hydrodynamic forces: (a, b) Norm of the time-averaged lift coefficient $\Vert \overline {C_l}\Vert _2$ at ${\textit{Re}}=100$ and ${\textit{Re}}=200$ , respectively. (c, d) Time-averaged drag coefficient $\overline {C_{d}}$ at ${\textit{Re}}=100$ and ${\textit{Re}}=200$ . Results are shown for the four face/edge angular positions (OF, CF, OV, and CV). The response for a streamwise-rotating sphere is plotted for reference (black dashed line).

Figures 17(c) and 17(d) show the evolution of $\overline {C_d}$ at ${\textit{Re}}=100$ and ${\textit{Re}}=200$ . Clearly, all angular particles exhibit higher drag than the sphere over $0.02 \leq \varOmega \leq 1$ . The sRCV produces the highest values of $\overline {C_d}$ . The vertex-facing configurations sRCV and sROV yield higher drag than their face-facing counterparts, sRCF and sROF. The higher $\overline {C_d}$ observed for vertex-facing configurations can be attributed in part to their stronger lateral wake expansion. Furthermore, as shown in figure 13(b), the sROV presents a larger projected frontal area than that of the sRCF in the streamwise direction, which also contributes to a higher $\overline {C_d}$ . In general, particle angularity enhances the hydrodynamic lift and drag with a substantial effect of the angular position. By contrast, neither geometric duality nor the rotational symmetry fold has a discernible influence on the mean lift or drag.

One may further ask whether, at sufficiently high rotation rates, the cube approaches the behaviour of a smooth body of revolution such as a cylinder. In Appendix F, the sRCF cube is compared with a volume-equivalent sphere and three cylinders of different aspect ratios. The hydrodynamic responses of the cube differ from those of smooth bodies as $\varOmega$ increases. The edge-generated vortical structures persist at all rotation rates.

We further examine the recirculation length $L_r$ for streamwise-rotating angular particles in figure 18. At ${\textit{Re}} = 100$ , $L_r$ increases with $\varOmega$ for all shapes, with the sRCV and the sROV developing a noticeably longer recirculation region than their face-facing counterparts. All angular particles sustain a substantially longer recirculation region than the sRS. The enlarged recirculation zone in these vertex-facing configurations enhances the pressure difference between the front and the rear of the particle, thereby contributing to the higher values of $\overline {C_d}$ reported in figure 17. At higher $\textit{Re}$ , $L_r$ continues to increase for most cases, although the influence of particle orientation becomes less pronounced.

Figure 18.

Evolution of the recirculation length $L_r$ behind streamwise-rotating angular particles at ${\textit{Re}} = 100$ over $0.02 \leq \varOmega \leq 1$ . The solid line denotes the streamwise-rotating sphere (sRS) as a reference.

6.4. Frequency characteristics

In figure 19 we present the $\mathcal{S}tr$ associated with the primary and secondary spectral peaks of the lift force $F_y$ for all angular particle configurations. A key observation is that particle duality plays an essential role in shaping the frequency signature of the hydrodynamic forces. The sROV case in figure 19(a) shows a frequency distribution closely resembling that of its dual, the sRCF, in figure 11(d). The primary peaks are distributed along the fundamental frequency branch $\mathcal{S}tr_1 = 4\varOmega /\pi$ and the fixed-octahedron value $\mathcal{S}tr_0 = 0.12$ , with several secondary peaks appearing at the first harmonic $\mathcal{S}tr_2 = 8\varOmega /\pi$ . Note that, for the sROV, the number of effective faces $n_{\mathcal{F}}$ is identical to $n_{\varOmega }=4$ . The distribution of the primary peaks agrees well with the prediction of (5.5).

Figure 19.

The Strouhal number of the primary and secondary peaks as a function of $\varOmega$ for (a) sROV, (b) sRCV and (c) sROF. The blue curve represents the model prediction (5.5) for the primary peak $\mathcal{S}tr_1$ , while the red curve corresponds to the prescribed particle rotation rate $\mathcal{S}tr = \varOmega$ . The grey/olive curves indicate the higher-order harmonics of the primary peak. The yellow curve denotes the characteristic $\mathcal{S}tr_0$ of the corresponding fixed particle: a fixed octahedron $\mathcal{S}tr_0=0.12$ in panels (a) and (c), and a fixed cube $\mathcal{S}tr_0=0.11$ in panel (b) (Gai & Wachs Reference Gai and Wachs2023b ).

The sRCV and the sROF expose six effective faces to the flow (faces whose normals are not aligned with the flow direction), $n_{\mathcal{F}} = 2n_{\varOmega } = 6$ . In the case of transverse rotation, all six effective faces act successively to push fluid into the wake, which leads to a dominant frequency of $\mathcal{S}tr_1=6\varOmega /\pi$ (Gai & Wachs Reference Gai and Wachs2024). Under streamwise rotation, however, each effective face has an opposite counterpart that produces a nearly identical contribution to the lateral force but of opposite sign. The contributions of opposite faces largely cancel, so that, although the geometric forcing occurs at the face-passing frequency $n_{\mathcal{F}}\varOmega /\pi = 6\varOmega /\pi$ , only half of the effective faces contribute independently to the lateral force. This halving is specific to cases where the effective faces form two symmetry-equivalent subsets and opposite-face pairing removes the alternation between them. The dominant frequency thus follows (5.5) with the reduced count $n_{\mathcal{F}}/2$ , $\mathcal{S}tr_1 = ({n_{\mathcal{F}}}/{2\pi })\varOmega = 3\varOmega /\pi$ , as depicted in figures 19(b) and 19(c). In addition, the presence of many interacting faces enhances nonlinear coupling and strengthens higher-order harmonic responses, leading to a shift of the primary peaks to $\mathcal{S}tr_1=9\varOmega /\pi$ at high rotation rates. Note that this is the second harmonic when considering $3\varOmega /\pi$ as the fundamental frequency. This behaviour is clearly observed for both the sRCV and the sROF.

Overall, the frequency content of the lateral forces on streamwise-rotating angular particles reflects the competition between rotational forcing and intrinsic wake response. When the wake locks onto the rotation, the dominant frequency is set by the geometric forcing of (5.5), with the full effective face count $n_{\mathcal{F}}$ for the sRCF and sROV and the reduced count $n_{\mathcal{F}}/2$ for the sRCV and sROF, where opposite-face contributions cancel in the lateral force.