Direct numerical simulations of a cylinder cutting a vortex

The interaction between a vortex and an impacting body which is oriented normally to it is complex due to the interaction of inviscid and viscous three-dimensional mechanisms. To model this process, direct numerical simulations of a thin cylinder intersecting a columnar vortex oriented normally to it are conducted. By varying the impact parameter and the Reynolds number, the two regimes of interaction mentioned in the literature are distinguished: the weak and strong vortex regimes. Low impact parameters, representing strong vortices, led to ejection and interaction of secondary vorticity from the cylinder's boundary layer, while high impact parameters, representing weak vortices, led to approximately inviscid interaction of the cylinder with the primary vortex through deformations. No significant effect of the Reynolds number in the overall phenomenology is found, even if larger Reynolds numbers lead to the formation of increasingly smaller and more intense vortex structures in the parameter range studied. Finally, the hydrodynamic force curves on the cylinder are analyzed, showing that intense forces could be locally generated for some parameter regimes, but that the average force on the cylinder does not substantially deviate from baseline cases where no vortex was present. Our results shed light on the underlying mechanisms of vortex-body interactions and their dependence on various parameters.

Figure 1

Schematic view of the computational domain. The $z$ direction is shortened for clarity.

Figure 2

Volume rendering visualization of the time evolution of $Q$ [Eq. (3)] for a simulation in the strong vortex regime with ${\mathrm{Re}}_{\mathrm{\Gamma }}=1000, I_P=0.05$. Regions in blue indicate regions of strongly positive $Q$. The wire is shown as a solid black object.

Figure 3

Volume rendering visualization of the time evolution of $Q$ [Eq. (3)] for a simulation in the weak vortex regime with ${\mathrm{Re}}_{\mathrm{\Gamma }}=1000, I_P=0.25$. Regions in blue indicate regions of strongly positive $Q$. The wire is shown as a solid black object.

Figure 4

Volume rendering visualization of $Q$ [Eq. (3)] after the wire has passed through the cylinder axis for ${\mathrm{Re}}_{\mathrm{\Gamma }}=1000$ and $I_P=0.05$, 0.1, 0.15, and 0.25 (left to right). Regions in blue indicate regions of strongly positive $Q$. The wire is shown as a solid black object.

Figure 5

Volume rendering visualization of the time evolution of $\left|\omega \right|$ during the early stages for a simulation in the strong vortex regime $I_P=0.05$ with ${\mathrm{Re}}_{\mathrm{\Gamma }}=1000$ (top row) and ${\mathrm{Re}}_{\mathrm{\Gamma }}=3000$ (bottom row). Regions shown in orange-brownish are regions of high vorticity.

Figure 6

Vorticity magnitude $\left|\omega \right|$ at the $z=0$ plane evolution just before impact in the strong vortex regime with $I_P=0.05$ at ${\mathrm{Re}}_{\mathrm{\Gamma }}=1000$ (top row) and ${\mathrm{Re}}_{\mathrm{\Gamma }}=3000$ (bottom row). Snapshots are at $t\mathrm{\Gamma }/D^2=940$ (left), 1140 (middle), and 1340 (right). Abbreviations: “DSV” detached secondary vorticity, “PVN” primary vortex necking.

Figure 7

Vorticity magnitude $\left|\omega \right|$ at the $y=0$ midplane evolution for a simulation in the strong vortex regime at $I_P=0.05$ with ${\mathrm{Re}}_{\mathrm{\Gamma }}=1000$ (top row) and with ${\mathrm{Re}}_{\mathrm{\Gamma }}=3000$ (bottom row). Snapshots are at $t\mathrm{\Gamma }/D^2=740$ (left), 940 (middle), 1340 (right). Abbreviations: “DSV,” detached secondary vorticity; “PVN,” primary vortex necking.

Figure 8

Volume visualization of the time evolution of $\left|\omega \right|$ during the late stages for a simulation in the strong vortex regime $I_P=0.05$ with ${\mathrm{Re}}_{\mathrm{\Gamma }}=1000$ (top row) and ${\mathrm{Re}}_{\mathrm{\Gamma }}=3000$ (bottom row). Regions shown in orange-brownish are regions of high vorticity.

Figure 9

Vorticity magnitude $\left|\omega \right|$ at the $y=0$ midplane evolution and $t\mathrm{\Gamma }/D^2=2140$ for a simulation in the strong vortex regime at $I_P=0.05$ with ${\mathrm{Re}}_{\mathrm{\Gamma }}=1000$ (left) and with ${\mathrm{Re}}_{\mathrm{\Gamma }}=3000$ (right). Abbreviation: “PVR,” primary vortex remnant.

Figure 10

Volume rendering visualization of the time evolution of $\left|\omega \right|$ during the early stages for a simulation in the weak vortex regime $I_P=0.25$ with ${\mathrm{Re}}_{\mathrm{\Gamma }}=1000$ (top row) and ${\mathrm{Re}}_{\mathrm{\Gamma }}=3000$ (bottom row). Regions shown in orange-brownish are regions of high vorticity.

Figure 11

Vorticity magnitude $\left|\omega \right|$ at the $z=0$ plane evolution just before impact in the weak vortex regime with $I_P=0.25$ at ${\mathrm{Re}}_{\mathrm{\Gamma }}=1000$ (top row) and ${\mathrm{Re}}_{\mathrm{\Gamma }}=3000$ (bottom row). Snapshots are at $t\mathrm{\Gamma }/D^2=220$ (left), 240 (middle), and 260 (right). Abbreviations: “PVN,” primary vortex necking; “DSV,” detached secondary vorticity.

Figure 12

Vorticity magnitude $\left|\omega \right|$ at the $y=0$ midplane evolution for a simulation in the weak vortex regime at $I_P=0.25$ with ${\mathrm{Re}}_{\mathrm{\Gamma }}=1000$ (top row) and ${\mathrm{Re}}_{\mathrm{\Gamma }}=3000$ (bottom row). Snapshots are at $t\mathrm{\Gamma }/D^2=220$ (left), 240 (middle), and 260 (right). Abbreviations: “DSV,” detached secondary vorticity; “OV,” opposite vorticity; “PV,” primary vortex.

Figure 13

Volume visualization of the time evolution of $\left|\omega \right|$ during the late stages for a simulation in the weak vortex regime $I_P=0.25$ with ${\mathrm{Re}}_{\mathrm{\Gamma }}=1000$ (top row) and ${\mathrm{Re}}_{\mathrm{\Gamma }}=3000$ (bottom row). Regions shown in orange-brownish are regions of high vorticity.

Figure 14

Vorticity magnitude $\left|\omega \right|$ at the $y=0$ midplane evolution and $t\mathrm{\Gamma }/D^2=400$ for a simulation in the weak vortex regime at $I_P=0.25$ with ${\mathrm{Re}}_{\mathrm{\Gamma }}=1000$ (left) and with ${\mathrm{Re}}_{\mathrm{\Gamma }}=3000$ (right).

Figure 15

Loss of circulation during the interaction quantified as the ratio between final and initial circulations as a function of $I_P$ for the four cases simulated at ${\mathrm{Re}}_{\mathrm{\Gamma }}=1000$.

Figure 16

Left: Pressure coefficient along the cylinder for ${\mathrm{Re}}_{\mathrm{\Gamma }}=1000, I_P=0.25$ at time $t\mathrm{\Gamma }/D^2=160$. Middle: Force coefficient along the cylinder axis for the same case. Left: Evolution of drag coefficient with time for the full cylinder (orange) and half the cylinder (blue).

Figure 17

Force coefficient along the cylinder axis at several time instances. Left column is cases at $I_P=0.05$, while right column is cases at $I_P=0.25$. Top row is at ${\mathrm{Re}}_{\mathrm{\Gamma }}=1000$, while bottom row is at ${\mathrm{Re}}_{\mathrm{\Gamma }}=3000$. Symbols: left column ($I_P=0.05$), light to dark: $t\mathrm{\Gamma }/D^2=700$ (light red), 1100 (dark red), 1500 (black), 1900 (dark blue), and 2300 (blue). Right column ($I_P=0.25$): $t\mathrm{\Gamma }/D^2=160$ (light red), 200 (dark red), 240 (black), 280 (dark blue), and 320 (light blue). The horizontal thin line represents the baseline value.

Figure 18

Vorticity magnitude for the $I_P=0.25$ and ${\mathrm{Re}}_{\mathrm{\Gamma }}=3000$ case at $t{\mathrm{\Gamma }}^2/D=220$ and $z=0$ (top), at $t{\mathrm{\Gamma }}^2/D=220$ and $y=0$ (middle) and at $t{\mathrm{\Gamma }}^2/D=400$ and $y=0$ (bottom). The left, middle, and right columns show the coarse, medium, and fine resolutions, respectively.

Figure 19

Force coefficient at $t{\mathrm{\Gamma }}^2/D=220$ for the $I_P=0.25$ and ${\mathrm{Re}}_{\mathrm{\Gamma }}=3000$ case for the three grids. A horizontal line shows the baseline value when no vortex is present. Symbols: blue, coarse grid; orange, medium grid; green, fine grid.