Dynamics, wakes, and regime transitions of a fixed angular particle in an unbounded inertial flow. II. From tetrahedron to sphere

In this study, we investigate an unbounded Newtonian fluid flow past a stationary Platonic polyhedron. To reveal the effects of the particle angularity and angular position on the flow regime transitions, we select the three following different angular positions of the Platonic polyhedrons: a face facing the flow (F), an edge facing the flow (E), and a vertex facing the flow (V). We consider all five Platonic polyhedrons whose sphericity increases with the number of faces and a sphere as an asymptotic polyhedron of sphericity 1 featuring an infinite number of faces. Two well-known regime transitions are studied as a function of the particle sphericity, the particle angular position and the particle Reynolds number Re: the loss of symmetry in the particle wake region and the loss of stationarity of the flow. In the multiplanar symmetry regime, the flow symmetry in the particle wake region is highly related to the particle front surface. The number and orientation of particle front surface edges determine the wake vorticity pattern. With increasing Re, the steady flow past an angular particle transitions to a planar symmetry regime for all particle shapes and all particle angular positions. The plane of symmetry in the planar symmetry regime selects one of the axis in the multiplanar symmetry regime, and its direction may change in time and/or with Re. In the unsteady regime, we notice a wide variety of periodic hairpin vortex shedding modes. These vortex shedding modes are determined by the shape of the flow recirculation region, which itself is a reflection of the angularity and the angular position of the particle. Furthermore, we present an in-depth analysis of the hydrodynamic forces exerted on the Platonic polyhedrons at the vertex angular position (V) for a wide range of Reynolds numbers $100\le \text{Re}\le 500$. We attempt to understand and explain the significant variations in the flow regime transitions brought by the particle angularity and the particle angular position.

Figure 1

Numerical setup: an inertial flow past a fixed Platonic polyhedron located in a cubic computational domain of side length $L=40$, streamwise direction is $x^{+}$, i.e., fluid flows from left to right.

Figure 2

Wake structures identified by the isosurface of ${\lambda }_2=-1$ and colored by $u_x$ in the flow past the CV at $\mathrm{Re}=500$ with adaptive grid in the $x\text{−}y$ plane at $z=z_p$.

Figure 3

Boundary points distribution on the surface of the Platonic polyhedrons.

Figure 4

Geometric features of the Platonic polyhedrons at the three angular positions (A.P.): E (edge), F (face), and V (vertex) with orthogonal projection in the $y\text{−}z, x\text{−}y$, and $x\text{−}z$ planes.

Figure 5

(a) $\overline{C_d}$ of the CV as a function of Re ($\blacksquare$), comparison with Richter's correlation ($—$); (b) relative error $\varepsilon$ between our computed results and the Richter's correlation for $100\le \mathrm{Re}\le 500$ ($▾$).

Figure 6

Regime transition map of the flow past a Platonic polyhedron at the three angular positions (A.P.): multiplanar symmetry ($▬$), planar symmetry ($▬$), periodic vortex shedding ($▬$), chaotic vortex shedding ($▬$), and double-hairpin vortex shedding ($▬$); regime transition of the flow past a sphere: axisymmetric regime ($▬$).

Figure 7

Symmetry of the wake vortex structure identified by streamwise vorticity ${\omega }_x$ (positive in red, negative in blue, and near-zero in white) of the flow in the MSS regime past a Platonic polyhedron in the $y\text{−}z$ plane located at $x=x_p+1.5$.

Figure 8

LIC streamlines of the flow at $\mathrm{Re}=100$ in the near-wake region of the Platonic polyhedron in the $y\text{−}z$ plane at $x=x_p+1$. LIC streamlines are colored by the flow velocity magnitude (from min in dark blue to max in dark red) and particle edges are projected as white lines.

Figure 9

LIC streamlines of the flow at $\mathrm{Re}=100$ in the far-field region of the Platonic polyhedron in the $y\text{−}z$ plane at $x=x_p+10$. LIC streamlines are colored by the flow velocity magnitude (from min in dark blue to max in dark red, same color map as in Fig. 8) and particle edges are projected as white lines.

Figure 10

Distribution of streamwise vorticity ${\omega }_x$ (positive in orange, negative in blue, and near-zero in black) on the rear surface in the $y\text{−}z$ plane located at $x\approx x_p+0.5$ of the Platonic polyhedron at the three angular positions in the flow in the MSS regime and in the PSS regime. (a) Edge, (b) Face, and (c) Vertex.

Figure 11

Symmetry of the wake vortex structure identified by streamwise vorticity ${\omega }_x$ (positive in red, negative in blue, and near-zero in white) of the flow in the PSS regime past a Platonic polyhedron in the $y\text{−}z$ plane located at $x=x_p+1.5$.

Figure 12

Wake structures identified by the isosurface ${\lambda }_2=-1$ and colored by $u_x$ in the flow in the HS/DHS regime in the $x\text{−}z$ plane (left) and in the $x\text{−}y$ plane (right).

Figure 13

LIC streamlines of the flow past a Platonic polyhedron at the three angular positions in the HS/DHS regime in the $x\text{−}y$ plane and in the $x\text{−}z$ plane. LIC streamlines are colored by the velocity magnitude (from min in blue to max in red).

Figure 14

LIC streamlines of the flow in the near-wake region of the Platonic polyhedron at the three angular positions in the HS/DHS regime in the $y\text{−}z$ plane located at $x=x_p+1$. LIC streamlines are colored by the velocity magnitude (from min in dark blue to max in dark red) and particle edges are projected as white lines.

Figure 15

LIC streamlines of the flow at $\mathrm{Re}=200$ in the far field region of the TE in the $y\text{−}z$ plane at different distances from the particle: (a) $x=x_p+7.2$, (b) $x=x_p+7.4$, (c) $x=x_p+7.6$, (d) $x=x_p+7.8$, (e) $x=x_p+8.0$, (f) $x=x_p+8.1$, (g) $x=x_p+9.2$, and (h) $x=x_p+11.6$. LIC streamlines are colored by the flow velocity magnitude (from min in dark blue to max in dark red, same color map as in Fig. 14) and particle edges are projected as white lines.

Figure 16

LIC streamlines of the flow at $\mathrm{Re}=300$ in the far field region of the DV in the $y\text{−}z$ plane at different distances from the particle: (a) $x=x_p+5.5$, (b) $x=x_p+5.6$, (c) $x=x_p+6.2$, (d) $x=x_p+7.8$, (e) $x=x_p+8.8$, (f) $x=x_p+9.2$, (g) $x=x_p+11.2$, and (h) $x=x_p+12$. LIC streamlines are colored by the flow velocity magnitude (from min in dark blue to max in dark red, same color map as in Fig. 14) and particle edges are projected as white lines.

Figure 17

(a) St of Platonic polyhedrons at the vertex angular position as a function of Re; (b) $L_r$ as a function of Re in the wake of a Platonic polyhedron at the vertex angular position: TV ($▴$), CV ($\blacksquare$), OV ($♦$), DV ($⬟$), and IV ($⬢$); markers highlighted with flow regimes: MSS ($▬$), PSS ($▬$), HS ($▬$), and CS ($▬$).

Figure 18

Recirculation region and vortex shedding frequency: (a) size of the recirculation region in the OF case at $\mathrm{Re}=350$; (b) St as a function of the aspect ratio ${\chi }_r$ of the recirculation region at $x=x_p+1$: T ($•$), C ($•$), O ($•$), D ($•$), and I ($•$).

Figure 19

$\overline{C_d}$ of Platonic polyhedrons at the vertex angular position as a function of Re: TV ($▴$), CV ($\blacksquare$), OV ($♦$), DV ($⬟$), and IV ($⬢$), and the sphere ($•$).

Figure 20

Time evolution of $C_{l,y}$ and $C_{l,z}$ of the CV and of the OV during the transition from the MSS regime to the HS regime.

Figure 21

Time evolution of $C_{l,y}$ and $C_{l,z}$ of the DV and of the IV during the transition from the MSS regime to the HS regime.

Figure 22

$C_{l,y}\text{−}{C}_{l,z}$ phase diagram of Platonic polyhedrons at the vertex angular position, comparison between the PSS regime and the HS regime. Time evolution given in rainbow color from purple (start time) to red (end time).

Figure 23

Wake structures identified by the isosurface ${\lambda }_2=-1$ and colored by $u_x$ in the flow past a Platonic polyhedron in the CS regime in the $x\text{−}z$ plane (left) and in the $x\text{−}y$ plane (right).

Figure 24

$C_{l,y}\text{−}{C}_{l,z}$ phase diagram of Platonic polyhedrons at the vertex angular position, comparison between the HS regime and the CS regime. Time evolution given in rainbow color from purple (start time) to red (end time).

Figure 25

Vortex shedding frequency $f$ spectrum in the HS regime ($▬$) and in the CS regime ($▬$). $I$ denotes the dimensionless frequency intensity obtained by FFT.

Figure 26

$\overline{C_l}/\overline{C_d}$ of Platonic polyhedrons at the vertex angular position as a function of Re: TV ($▴$), CV ($\blacksquare$), OV ($♦$), DV ($⬟$), and IV ($⬢$); markers highlighted with flow regimes: MSS ($▬$), PSS ($▬$), HS ($▬$), and CS ($▬$).

Figure 27

Pressure distribution colored from min in dark blue to max in dark red in the $x\text{−}y$ plane at $z=z_p$ around the Platonic polyhedron at different angular positions in the flow at $\mathrm{Re}=100$

Figure 28

Three-dimensional view of the wake structures identified by the isosurface of ${\lambda }_2=-1$ and colored by $u_x$ in the flow past a Platonic polyhedron in the HS/DHS regime. (a) Edge, (b) Face, and (c) Vertex.

Figure 29

$C_{l,y}\text{−}{C}_{l,z}$ phase diagram in the OV case in the HS regime. Time evolution given in rainbow color from purple (start time) to red (end time).