Dynamics, wakes, and regime transitions of a fixed angular particle in an unbounded inertial flow. I. Regular tetrahedron angular position

We investigate the unbounded inertial flow of a Newtonian fluid past a fixed regular tetrahedron, the Platonic polyhedron with the lowest sphericity, in the range of Reynolds number $100\le \text{Re}\le 500$. Three angular positions of the tetrahedron are considered: a face facing the flow (TF), an edge facing the flow (TE), and a vertex facing the flow (TV). We analyze and determine the two well-known regime transitions: loss of symmetry of the wake structures and loss of stationarity of the flow as a function of the Reynolds number Re. In the steady regime, we show that the symmetry of the wake structure is closely related to the number of edges on the particle front surface. For the unsteady flows, we find a new symmetric double-hairpin vortex shedding regime in the edge facing the flow (TE) cases due to its unique angular position. The dominant frequency of this double-hairpin vortex shedding is twice that of the single-hairpin vortex shedding in the flow past a sphere, a cube, or a tetrahedron at other angular positions. Furthermore, we present a force analysis attempting to explain the wake transitions in connection to the drag and lift force evolution. Eventually, we provide the critical Reynolds numbers for the regime transitions from steady to unsteady flows, which are highly dependent on the angular position of the tetrahedron.

Figure 1

Numerical setup: A fixed tetrahedron located in a cubic computational domain of side length $L=40$, the streamwise direction is $x$ and the flow is from left to right.

Figure 2

Snapshot of wake structures in the early transient of the flow, identified by the isosurface of ${\lambda }_2=-1$ and colored by $u_x$ in the flow past the TE tetrahedron at $\mathrm{Re}=500$ with adaptive octree grid shown in the $x\text{−}y$ plane located at $z=20$.

Figure 3

Outline, surface, and Lagrangian point distribution of the tetrahedron.

Figure 4

Definition of rotation angles $\theta , \phi$, and $\xi$ with respect to the three coordinate axes.

Figure 5

Geometric features in the $y\text{−}z$ plane of the tetrahedron in its three angular positions: 3D view, projection of edges (Edges), projection of outline (Proj.), and crosswise cross-sectional surface area ratio $S_{\text{pp},x}=S_{\text{pp},x}^{*}/S_{\text{sph}}^{*}$.

Figure 6

Geometric features in the $x\text{−}z$ and $x\text{−}y$ planes of the tetrahedron in its three angular positions: projection of outline and lateral cross-sectional surface area ratios $S_{\text{pp},y}=S_{\text{pp},y}^{*}/S_{\text{sph}}^{*}, S_{\text{pp},z}=S_{\text{pp},z}^{*}/S_{\text{sph}}^{*}$.

Figure 7

Time-time-averaged drag coefficient $\overline{C_d}$ as a function of Re in the flow past a fixed sphere at $1\le \mathrm{Re}\le 500$ ($•$), comparison with numerical results of Selçuk et al. [51] ($▾$) and the Schiller-Naumann correlation ($\_\_\_\_$).

Figure 8

Wake structures identified by the isosurface of ${\lambda }_2=-1$ and colored by $u_x$ in the flow past a fixed sphere: (a) axisymmetric regime ($\mathrm{Re}=210$), (b) beginning of planar-symmetric regime ($\mathrm{Re}=220$), (c) end of planar-symmetric regime ($\mathrm{Re}=270$), and (d) periodic hairpin vortex shedding regime ($\mathrm{Re}=280$).

Figure 9

Spatial convergence of $\overline{C_d}$ as a function of the minimal grid size $\mathrm{\Delta }x$ for the flow past a pyramid tetrahedron: simulation results ($•$) and fitting curve ($\_\_\_\_$).

Figure 10

Symmetry of the vortex structure (identified by ${\omega }_x, {\omega }_y$, and ${\omega }_z$, positive in red, negative in blue, and near-zero in white) in the wake region of the TE tetrahedron in the $y\text{−}z$ plane located at $x=x_p+1.5$.

Figure 11

Symmetry of the vortex structure (identified by ${\omega }_x, {\omega }_y$, and ${\omega }_z$, positive in red, negative in blue, and near-zero in white) in the wake region of the TF tetrahedron in the $y\text{−}z$ plane located at $x=x_p+1.5$.

Figure 12

Symmetry of the vortex structure (identified by ${\omega }_x, {\omega }_y$, and ${\omega }_z$, positive in red, negative in blue, and near-zero in white) in the wake region of the TV tetrahedron in the $y\text{−}z$ plane located at $x=x_p+1.5.$

Figure 13

Side view of the wake structures identified by the isosurfaces ${\omega }_x=\pm 0.03$ as a function of Re. (a) TE tetrahedron, (b) TF tetrahedron, and TV tetrahedron.

Figure 14

Wake structures of the flow past a TE, colored by $u_x$ (low velocity in blue and high velocity in red), as a function of Re, identified by the isosurface ${\lambda }_2=-1$; views in the $x\text{−}z$ plane (left) and the $x\text{−}y$ plane (right).

Figure 15

Wake structures of the flow past a TF, colored by $u_x$ (low velocity in blue and high velocity in red), as a function of Re, identified by the isosurface ${\lambda }_2=-1$; views in the $x\text{−}z$ plane (left) and the $x\text{−}y$ plane (right).

Figure 16

Wake structures of the flow past a TV, colored by $u_x$ (low velocity in blue and high velocity in red), as a function of Re, identified by the isosurface ${\lambda }_2=-1$; views in the $x\text{−}z$ plane (left) and the $x\text{−}y$ plane (right).

Figure 17

Symmetric double-hairpin vortex shedding process within a period in the flow at $\mathrm{Re}=200$ past a TE; $\left(R_1\right)$ isosurface of ${\lambda }_2=-1$ in the $x\text{−}y$ plane at $z=z_p$ colored by $u_x$; $\left(R_2\right)$ LIC of the velocity field in the $x\text{−}y$ plane at $z=z_p$, from blue to red denoting the low to high flow velocity; $\left(R_3\right)$: streamwise vorticity (positive in red, negative in blue, and ${\omega }_x\approx 0$ in white) in the particle wake region in the $y\text{−}z$ plane at $x=x_p+1.5$, indicated in a white horizontal line in the row $R_2$.

Figure 18

Countours of $u_x$ in the $x\text{−}y$ and $x\text{−}z$ planes to identify the recirculation region (blue) in the flow past the TE tetrahedron at $\mathrm{Re}=200$.

Figure 19

Time-averaged $\overline{C_d}$ as a function of Re of the flow past a tetrahedron at the three angular positions: TE ($▴, ▴, ▴$), TF ($▾$), and TV ($•$).

Figure 20

Time-averaged $\overline{C_l}$ as a function of Re of the flow past a tetrahedron at the three angular positions: TE ($▴$), TF ($▾$) and TV ($•$): multiaxis steady symmetry ($▬$), planar steady symmetry ($▬$), periodic vortex shedding ($▬$), and chaotic vortex shedding ($▬$); the standard deviation $\sigma \left(\overline{C_l}\right)$ is represented by the shaded light colored area. (a) TE, (b) TF, and (c) TV.

Figure 21

Flow past the TE tetrahedron: (a), (b) time evolution of $C_d$ and $C_{l,z}$ at $\mathrm{Re}=330$ ($\_\_\_\_$) and $\mathrm{Re}=340$ ($\_\_\_\_$), and (c), (d) maximal lift coefficient ($▾$) and minimal lift coefficient ($▴$) ($C_{l,y}$ and $C_{l,z}$) as a function of Re.

Figure 22

Time evolution of $C_{l,y}$ in the flow past the TF tetrahedron at $\mathrm{Re}=110$ ($\_\_\_\_$), $\mathrm{Re}=120$ ($\_\_\_\_$), $\mathrm{Re}=130$ ($\_\_\_\_$), $\mathrm{Re}=140$ ($\_\_\_\_$), $\mathrm{Re}=150$ ($\_\_\_\_$), $\mathrm{Re}=160$ ($\_\_\_\_$), $\mathrm{Re}=230$ ($\_\_\_\_$), and $\mathrm{Re}=240$ ($\_\_\_\_$).

Figure 23

Time evolution of $C_{l,y}$ in the flow past the TV tetrahedron at $\mathrm{Re}=140$ ($\_\_\_\_$), $\mathrm{Re}=150$ ($\_\_\_\_$), $\mathrm{Re}=160$ ($\_\_\_\_$), $\mathrm{Re}=170$ ($\_\_\_\_$), $\mathrm{Re}=180$ ($\_\_\_\_$), $\mathrm{Re}=190$ ($\_\_\_\_$), $\mathrm{Re}=260$ ($\_\_\_\_$), and $\mathrm{Re}=270$ ($\_\_\_\_$).

Figure 24

$C_{l,y}\text{−}{C}_{l,z}$ phase diagrams at the three angular positions. Time evolution shown in rainbow colors from purple (start time) to red (end time).

Figure 25

Dimensionless vortex shedding frequency $f$ spectrum in the periodic HS/DHS regime and the CS regime. $I$ denotes the dimensionless intensity of the frequency obtained through FFT.

Figure 26

St as a function of Re at the three angular positions: TE ($▴$), TF ($▾$), and TV ($•$).

Figure 27

Recirculation region length $L_r$ as a function of Re of flow past a tetrahedron at the three angular positions: TE ($▴$), TF ($▾$), TV ($•$). Inset: 3D U-shape recirculation region (red) and $u_x=0.5$ isosurface (blue) in the wake of the TE at $\mathrm{Re}=200$.

Figure 28

Side view of wake structures of the flow past a TV, colored by $u_x$ (low velocity in blue and high velocity in red), as a function of Re, identified by the isosurface ${\lambda }_2=-1$. (a) TE, (b) TF, and (c) TV.