Evolution of wake structure around an impulsively stopped sphere with a streamwise magnetic field at $60\le \text{Re}\le 300$

With a streamwise magnetic field, the evolution of wake structure in a flow around an impulsively stopped sphere in an incompressible viscous fluid is investigated. The research parameter range is $60\le \mathrm{Re}\le 300$ and $0\le N\le 10$, where $\mathrm{Re}$ and $N$ are the Reynolds number and the interaction parameter, respectively. For present cases, the flow is fully developed before the sphere stopped and its symmetrical feature of the wake will be preserved after the sphere stopped. A complicated vortex structure system including a primary vortex ring, a fragmented secondary vortex, and an accompanying vortex is formed, which is summarized in a {$N,\mathrm{Re}$} phase diagram. A scaling law of the drag force is found after the sphere stopped. It decays as $t^{*-2/3}$ at a small timescale and $t^{*-1}$ or $t^{*-7/6}$ for different Reynolds numbers at a large timescale in the absence of a magnetic field, where $t^{*}$ is the dimensionless time after abruptly stopping the sphere. When the magnetic field is applied, the decay rate of drag force is slower at a small timescale but faster at a large timescale. Meanwhile, peak azimuthal vorticity at the primary vortex ring core is shown to decay as $t^{*-4/3}$ at a large timescale when the flow is axisymmetric. It will decay faster under the influence of a streamwise magnetic field.

Figure 1

Geometry configuration of flow. (a) The location of the sphere starting and stopping are shown by dashed and solid spheres, respectively. (b) The comoving frame is fixed in the center of the traveling sphere with the same translation velocity ${\mathit{u}}_s$. The radius of spherical computational domain is set as $r=15D$.

Figure 2

Azimuthal vorticity ${\omega }_{\theta }$ isosurface of vortex structures and corresponding vorticity contours in the symmetric plane at $t^{*}>0$. The primary vortex ring, secondary vortex $B$, and accompanying vortex $A$ are represented by red, blue, and green, respectively. The stopped sphere is expressed by a white geometry. (a) $\mathrm{Re}=200$; (b) $\mathrm{Re}=250$; (c) $\mathrm{Re}=300$, $t_{\mathrm{stop}}=0T$; (d) $\mathrm{Re}=300$, $t_{\mathrm{stop}}=2/4T$.

Figure 3

A scheme diagram model for vortex structures near the sphere at $60\le \mathrm{Re}\le 300$. (a) Legend; (b) Model I: no vortex structure; (c) Model II: axisymmetric primary vortex ring; (d) Model III: inclined primary vortex ring; (e) Model IV: inclined primary vortex ring with secondary vortex $B$; (f) Model V: inclined primary vortex ring with vortex $B$ and $A$.

Figure 4

The time evolution of primary vortex ring core trajectory for radial and axial displacements in the symmetric plane. The left and right cores are shown with dashed lines and solid lines, respectively. (a) $100\le \mathrm{Re}\le 200$; (b) $220\le \mathrm{Re}\le 260$; (c) $\mathrm{Re}=300$ and $t_{\mathrm{stop}}=0,1/4T,2/4T,3/4T$.

Figure 5

Azimuthal vorticity ${\omega }_{\theta }$ contours in the symmetric plane with Reynolds number $\mathrm{Re}=300$ at various time instances when the magnetic field is applied. (a) $\mathrm{Re}=300$, $N=0.1$; (b) $\mathrm{Re}=300$, $N=0.25$; (c) $\mathrm{Re}=300$, $N=0.5$; (d) $\mathrm{Re}=300$, $N=0.75$; (e) $\mathrm{Re}=300$, $N=1$; (f) $\mathrm{Re}=300$, $N=6.25$.

Figure 6

The radial Lorentz force ${\text{LF}}_r$ contours at $\mathrm{Re}=300$. The direction of Lorentz force in the orange red region is indicated as a black arrow (away from the sphere), and that in the blue region is shown as a red arrow (close to the sphere). (a) $\mathrm{Re}=300$, $N=0.25$, ${\text{LF}}_r$; (b) $\mathrm{Re}=300$, $N=0.75$, ${\text{LF}}_r$.

Figure 7

The time evolution of primary ring core trajectory for radial and axial displacements in the symmetric plane with different interaction number $N$ at $\mathrm{Re}=300$. The left and right cores are described by dashed lines and solid lines, respectively. (a) Trajectories; (b) time-varying displacement.

Figure 8

A simple map with wake patterns around a stopped sphere in the plane {$N,\mathrm{Re}$}. It is divided into five regimes called I, II, III, IV, and V. They correspond to the five flow patterns in Fig. 3. The flow patterns in regimes I, II, III, IV, and V belong to Models I, II, III, IV, and V, respectively.

Figure 9

The variation of normalization drag coefficient $\widetilde{C_d}=C_d/C_d^i$ over time at different $\mathrm{Re}$ and $N$ ranges. The drag coefficient $C_d$ is normalized with the average drag coefficient $C_d^i$ at $t^{*}\le 0$. (a) $60\le \mathrm{Re}\le 250$ with $N=0$; (b) $\mathrm{Re}=300,0T\le t_{\mathrm{stop}}\le 3/4T$ with $N=0$; (c) $\widetilde{C_d}$ with different $N$; (d) instantaneous streamline at $\mathrm{Re}=200$, $N=0$; (e) instantaneous streamline at $\mathrm{Re}=200$, $N=2.25$.

Figure 10

The time evolution of dimensionless peak azimuthal vorticity at the center of the primary ring core in the symmetric plane. (a) $100\le \text{Re}\le 200$ with $N=0$; (b) $220\le \text{Re}\le 300$ with $N=0$; (c) $\text{Re}=200$ with different $N$; (d) $\text{Re}=300$ with different $N$.