3.1. Vortex structures at $60 \leqslant {\textit{Re}}_{\!D} \leqslant 300$ with $0 \leqslant N \leqslant 10$

The wake generated by the moving sphere continues to propagate downstream after the sphere undergoes an impulsive stop, forming a more complex vortex system due to the presence of the stationary sphere. Vortices appear in the sphere wake, exhibiting an axisymmetric and steady behaviour at $20 \lt {\textit{Re}}_{\!D} \lt 210$ , a steady planar symmetric behaviour between $210 \lt {\textit{Re}}_{\!D} \lt 270$ and an unsteady planar symmetric behaviour at ${\textit{Re}}_{\!D} \gt 270$ (Johnson & Patel Reference Johnson and Patel1999). The primary flow after the impulsive stop is aligned with the direction of the sphere’s prior motion, namely the positive Z axis, so the negative Z zone is referred to as the upstream region and the positive Z zone is referred to as downstream, as shown in figure 3(a). At the moment of the impulsive stop, the vortex closest to the sphere, defined as the primary vortex, interacts immediately with the sphere. At short time scales after the impulsive stop, the primary vortex still locates on the upstream side of the sphere and pulls streamlines away from the sphere surface, promoting the development of the recirculation zone, as shown in figure 3(b). At long time scales, when ${\textit{Re}}_{\!D}$ is large enough, the primary vortex propagates to the downstream side, as shown in figure 3(c). The streamlines are pushed close to the sphere surface due to the primary vortex located on the downstream side. The influence of the primary vortex on the recirculation gradually shifts from enhancement to suppression as it moves from the upstream to downstream side. Moreover, the presence of the primary vortex downstream facilitates the shedding of the vortex in the recirculation zone due to the induced velocity. Meanwhile, the vortices gradually move away from the sphere surface owing to inertia and the self-induced velocity of the vortex ring.

The unsteady nature of the flow after the stop leads to a time-dependent evolution of the wake structures. A fixed dimensionless time of $t^* = 6$ is used to characterise and compare the flow features in the following analysis. At this moment, the primary vortex propagates downstream across most Reynolds numbers and interaction parameters. To elucidate the influence of a transverse magnetic field, the flow structures at $N=0$ are first discussed here. The instantaneous streamlines, the vortex structures identified by the Q-criterion and the corresponding skin friction lines on the downstream surface of the sphere are shown in figure 4. Lighthill (Reference Lighthill1963) pointed out that skin friction lines converge at separation regions in three-dimensional flow, providing a clear representation of flow traces around the sphere, enabling a more effective analysis of flow structures. At ${\textit{Re}}_{\!D}=100$ , no vortex ring forms downstream and the skin friction lines converge at a single central point. This indicates that the flow prior to the impulsive stop lacks sufficient energy to generate a recirculation zone, instead merely following the sphere surface downstream as presented in the streamlines plot. At ${\textit{Re}}_{\!D}=200$ , $250$ and $300$ , a downstream vortex ring is clearly observed, which is actually the primary vortex propagating downstream. The corresponding skin friction lines form a closed convergence line, indicating the presence of a recirculation zone near the downstream side of the sphere. Moreover, the vortex system exhibits axisymmetry at ${\textit{Re}}_{\!D}=100$ and $200$ , but transitions to planar symmetry at ${\textit{Re}}_{\!D}=250$ and $300$ , which is consistent with the flow past a stationary sphere. As a result, the vorticity on the symmetry plane becomes asymmetric between the two sides of the sphere and exhibits alternating dominance at higher Reynolds numbers. This asymmetry leads to unequal recirculation strengths on both sides, causing the divergence point to deviate from the centre point. At ${\textit{Re}}_{\!D}=300$ , the asymmetry becomes more pronounced, resulting in skin friction lines that converge into a non-circular closed line, in contrast to the nearly circular convergence line at ${\textit{Re}}_{\!D}=200$ and $250$ . Nevertheless, the symmetry of the skin friction lines remains consistent with that of the vortex ring.

Figure 3.

Schematic illustrating the flow evolution process following the impulsive stop of the sphere. The upstream and downstream regions are demarcated by the plane passing through the sphere centre and oriented perpendicular to the sphere motion direction. The vortex structures might not be symmetrical at higher Reynolds numbers. The sphere velocity before the impulsive stop is from top to bottom. (a) Before the impulsive stop; (b) short time scales after the impulsive stop; (c) long time scales after the impulsive stop.

Figure 4.

Vortex structures and corresponding skin friction lines on the downstream surface of the sphere without a magnetic field for various ${\textit{Re}}_{\!D}$ . The top row shows instantaneous streamlines on the symmetry plane. The middle row presents vortex structures identified by the Q-criterion and the bottom row shows skin friction lines. All plots are from the data at $t^*=6$ . Results are shown for (a) ${\textit{Re}}_{\!D}=100$ , (b) ${\textit{Re}}_{\!D}=200$ , (c) ${\textit{Re}}_{\!D}=250$ , (d) ${\textit{Re}}_{\!D}=300$ .

The wake of the sphere influenced by a transverse magnetic field with different strengths can be categorised into three main states: the steady double-plane symmetric state, the steady plane symmetric state and the unsteady plane symmetric state (Pan et al. Reference Pan, Zhang and Ni2019b ). Due to the different initial states at the instant the sphere impulsively stops, various vortex structures are generated and developed under the influence of a magnetic field with various strengths. Pan et al. (Reference Pan, Zhang and Ni2018, Reference Pan, Zhang and Ni2019b ) reported a ‘reversion phenomenon’ in the study of wake structure behind a sphere, which means that the topological structure with a higher ${\textit{Re}}_{\!D}$ and a higher $N$ is similar to the case with lower ${\textit{Re}}_{\!D}$ and $N$ . Furthermore, the magnetic field typically counteracts the inertia effects of the fluid, exerting a stabilising influence at lower Reynolds numbers and interaction parameters (Zheng, Pan & Ni Reference Zheng, Pan and Ni2023). Consequently, within the range $60\leqslant {\textit{Re}}_{\!D}\leqslant 300$ and $0\leqslant N\leqslant 10$ , the flow behaviour at ${\textit{Re}}_{\!D}=300$ with varying interaction parameters may exhibit greater complexity. Therefore, the cases at ${\textit{Re}}_{\!D}=300$ for $0\leqslant N\leqslant 10$ are discussed first. For comparison, the instantaneous streamlines at the fully developed state with different $N$ are provided in Appendix A. The vortex structures and corresponding skin friction lines on the downstream surface of the sphere at ${\textit{Re}}_{\!D}=300$ with corresponding $N$ are presented in figure 5. At $N=0.05$ , the vortex near the sphere propagates downstream and detaches, forming a primary vortex structure identified by the Q-criterion. With increasing $N$ , the formation of the primary vortex structure becomes increasingly incomplete and is eventually confined to the upstream region. Consequently, four distinct flow regimes emerge, each associated with a characteristic topology of skin friction lines, as shown in the bottom row.

Figure 5.

Vortex structures and corresponding skin friction lines on the downstream surface of the sphere at ${\textit{Re}}_{\!D}=300$ with various $N$ at $t^* = 6$ . The top row presents vortex structures identified by the Q-criterion and the bottom row shows skin friction lines. Results are shown for (a) $N=0.05$ , (b) $N=0.2$ , (c) $N=0.5$ , (d) $N=4$ .

Figure 6.

Skin friction lines on the downstream surface of the sphere and azimuthal vorticity contours at ${\textit{Re}}_{\!D}=300$ with $N=0.01$ for different impulsive stop instants within a vortex-shedding period. Results are shown for (a,b) $t_{\textit{stop}}=0T$ and (c,d) $t_{\textit{stop}}=1/2T$ . Azimuthal vorticity contours are shown on the symmetry plane normal to the Yaxis. From left to right is the positive X direction.

At ${\textit{Re}}_{\!D}=300$ and $N=0.05$ , the magnetic field is relatively weak, so the flow around the sphere resembles that in the absence of a magnetic field as shown in figures 4(d) and 5(a). Periodic vortex shedding occurs for $t^*\lt 0$ and a primary vortex ring forms as the wake continues moving downstream and interacts with the sphere. When the sphere is impulsively stopped at different times within the shedding period, slight variations in the skin friction lines arise. Figure 6 compares the flow fields at two representative stop times, $t_{\textit{stop}} = 0T$ and $t_{\textit{stop}} = 1/2T$ , where $T$ is the vortex-shedding period prior to the stop. A prominent accompanying vortex appears near the sphere for $t_{\textit{stop}}=1/2T$ , which corresponds to the vortex shed from the sphere prior to the impulsive stop. For $t_{\textit{stop}}=0T$ , a weak accompanying vortex also emerges on the upstream side of the sphere, while it is much weaker as it had not yet been shed completely prior to the impulsive stop. However, a more pronounced vortex is observed farther away from the sphere. This distant vortex gradually moves closer to the sphere due to inertial effects while simultaneously dissipating, as indicated by the azimuthal vorticity contour in figure 6(b). Differences in the skin friction lines are also observed between the two impulsive stop instances, as shown in figure 6(a,c). When $t_{\textit{stop}} = 1/2T$ , a closed circular convergence line forms on the downstream surface of the sphere, indicating a recirculation zone extending continuously in the azimuthal direction. However, at $t_{\textit{stop}} = 0T$ , the convergence line of the skin friction lines is not closed, suggesting the absence of recirculation on the right-hand side, as shown in figure 6(a). As illustrated in figure 3, the upstream vortex pulls streamlines away from the sphere, promoting recirculation, while the downstream vortex pushes streamlines toward the sphere, suppressing recirculation. In both cases, the wake retains sufficient residual energy to sustain recirculation after the impulsive stop. For $t_{\textit{stop}} = 0T$ , the primary vortex on the right-hand side retains enough energy to suppress recirculation, preventing the formation of a closed convergence line. However, for $t_{\textit{stop}} = 1/2T$ , a strong upstream accompanying vortex closer to the sphere facilitates the formation of recirculation, counteracting the suppression from the downstream primary vortex. As a result, a convergence line also forms on the right-hand side, completing a closed convergence loop, as shown in figure 6(d). Although the skin friction lines differ slightly between the two cases, the underlying vortex structures remain fundamentally similar. Accordingly, both cases are classified as model V in figure 8.

Figure 7.

Effect of a directionally anisotropic Lorentz force damping on wake structural dynamics. (a) Isosurface of azimuthal vorticity at ${\textit{Re}}_{\!D}=300$ , $N=0.2$ . (b) Distribution of the main current lines around the sphere. (c) The Z component of the Lorentz force contour in a section 0.5 $D$ away from the sphere centre. The circle of the black solid line represents the vertical projection of the sphere along the Zaxis on the plane. The direction perpendicular to the paper and outward is the positive direction of the Z axis.

At ${\textit{Re}}_{\!D} = 300$ and $N = 0.2$ , the anisotropic influence of the transverse magnetic field on the vortex structure gradually emerges, promoting a transition toward a steady wake characterised by double-plane symmetry in the fully developed stage. At $t^*\lt 0$ , vortex shedding is entirely suppressed, resulting in a flow field that is perfectly symmetric on the plane perpendicular to the magnetic field. At $t^*\gt 0$ , the wake propagates downstream and develops into a primary vortex ring, as illustrated in figure 5(b). In contrast to the streamwise magnetic field configuration, a transverse magnetic field disrupts the circumferential symmetry of the primary vortex. This symmetry breaking originates from the misalignment between the velocity vector of the vortex ring and the direction of the applied transverse magnetic field. An isolated view of the primary vortex from the negative Z-axis perspective, as presented in figure 7(a). The distinctive vortex ring size results from spatial variations in the Lorentz force across different flow regions. To elucidate the Lorentz force distribution, the spatial distribution of current lines around the stopped sphere at $t^*=6$ is presented in figure 7(b). According to Ohm’s law (2.3), the electric current comprises two components: one induced by electromotive effects and the other arising from electrostatic potential. The former is generated by the motion of conducting fluid within a magnetic field, while the latter results from electric potential gradients. A competition between these two components determines the local distribution of electric current within the computational domain.

In the present flow configuration, the sphere moves with a constant velocity for $t^*\lt 0$ , causing the fluid near the sphere to flow downstream, while the fluid farther away remains nearly stationary. As a result, electromotive effects dominate the electric current in the vicinity of the sphere, whereas electrostatic potential contributions prevail in the region far from the sphere. The sphere moves at a constant velocity along the positive Z axis for $t^*\lt 0$ , inducing a downstream fluid motion with velocity vectors predominantly aligned in the same direction for $t^*\gt 0$ . Consequently, the flow interacts with the applied magnetic field oriented along the positive Y axis, generating electric currents along the negative X axis near the sphere, in accordance with Ohm’s law. In the region far from the sphere, the flow velocity is significantly weaker. Therefore, the current direction is governed by the electric potential induced by the currents near the sphere, resulting in a net current flowing along the positive Xaxis. Meanwhile, due to the charge conservation law and the electrically insulating condition of the sphere, current lines are closed and bend along the sphere surface, as shown in figure 7(b).

Figure 8.

A simple map of skin friction line patterns on the downstream surface of an impulsively stopped sphere in the plane $\{N,{\textit{Re}}_{\!D}\}$ . The direction of the magnetic field is transverse with the motion direction of a sphere. Marks of various shapes represent calculated data points, of which only a subset is shown for clarity. A summary of the simulation cases is provided in table 2 in Appendix B. The patterns are divided into five regimes labelled I, II, III, IV and V, corresponding to five schematic diagram models on the right-hand side, respectively. Model I: one straight convergent line; model II: one convergent point; model III: one circular convergent line with one divergent point located at the centre; model IV: one circular convergent line with two divergent points located at two sides of the centre; model V: one circular convergent line with one off-centre convergent point. Here $\circ$ and $\square$ respectively represent the convergence point and the divergence point.

The currents, interacting with the applied magnetic field oriented along the positive Y axis, generate Lorentz forces directed along the positive Z axis in the region far from the sphere and along the negative Zaxis near the sphere, as illustrated in figure 7(c). The Lorentz forces suppress the flow along the positive Zaxis near the sphere, and again damp the flow in the upper and lower regions farther away. As a result, the vorticity near the Y–Z plane is more strongly suppressed than that near the X–Z plane. Accordingly, the isosurface of azimuthal vorticity becomes progressively thinner from the Y–Z plane to the X–Z plane along the circumferential direction, as shown in figure 7(a). Moreover, the wake retains sufficient energy to form recirculation around the sphere surface, resulting in a closed circular convergence line in the skin friction lines, as presented in figure 5(b). The stronger primary vortex on the downstream side of the sphere along the X direction pushes the streamline closer to the sphere surface, ultimately delaying flow separation on the X–Z plane. Consequently, the convergence line of skin friction lines in figure 5(b) develops an elliptic shape with its major axis aligned along the Y direction. Nevertheless, a circular convergence line and a divergence point appear in the skin friction lines in this class of flow pattern, which is categorised as model III in figure 8.

At ${\textit{Re}}_{\!D}=300$ and $N=0.5$ , the difference in the damping effect of the transverse magnetic field between the regions near the Y–Z plane and the X–Z plane enhances. The wake still contains enough energy to form the recirculation zone downstream. However, at $t^*=6$ , the primary recirculations appear on both sides and have not yet developed to be adjacent. After passing the two primary recirculation zones, the flow remains closely attached to the sphere surface. When the streamlines from both sides meet near the downstream centre, the fluid tends to move along the magnetic field direction (Yaxis) because the damping in other directions is stronger. Ultimately, this seems like the separation of a single divergence point into two, and the line connecting them is perpendicular to the magnetic field, as shown in figure 5(c), which is concluded as model IV in figure 8. With further increases in magnetic field strength, the flow perpendicular to the magnetic field direction weakens dramatically, becoming insufficient to generate a recirculation zone. Consequently, the closed convergence lines in the skin friction line diagram vanish, transforming into a single convergence line driven primarily by the flow parallel to the magnetic field direction. Although the flow intensity in the Y-axis direction remains relatively high, the transverse magnetic field causes the flow near the bottom of the sphere to align parallel to the magnetic field, preventing the occurrence of recirculation. As a result, the skin friction lines form a straight line crossed by the centre at ${\textit{Re}}_{\!D}=300$ with $N=4$ , as shown in figure 5(d), which is concluded as model I in figure 8. Moreover, the straight and convergent skin friction line is perpendicular to the direction of the transverse magnetic field due to the preferred orientation of the vortex symmetry plane at $t^*\lt 0$ , reported by Pan et al. (Reference Pan, Zhang and Ni2019b ).

At lower Reynolds numbers, the wake lacks sufficient energy to sustain recirculation, primarily due to the limited residual energy from the wake prior to the impulsive stop. Meanwhile, when the transverse magnetic field is relatively weak, the anisotropy damping effect is not pronounced. Therefore, the skin friction lines exhibit nearly axisymmetric patterns and converge at a centre point on the sphere surface, resembling the topological structure observed in the absence of a magnetic field at ${\textit{Re}}_{\!D}=100$ . This flow pattern can be categorised as model II in figure 8. Moreover, as the interaction parameters varies, flow patterns corresponding to models III, IV and V can also be observed for $150 \leqslant {\textit{Re}}_{\!D} \leqslant 300$ . Due to the different initial states mentioned above before the impulsive stop of the sphere and the influence of the transverse magnetic field, there are a total of five primary flow types at $60 \leqslant {\textit{Re}}_{\!D} \leqslant 300$ with $0 \leqslant N \leqslant 10$ , as shown in the skin friction lines diagram in figure 8. After a series of numerical simulations for nearly 200 cases, flow structures following the impulsive stop of the sphere are categorised into five regimes in the $\{N,{\textit{Re}}_{\!D}\}$ phase plot as shown in figure 8, based on the diagrams of skin friction lines. In the phase plot, marks of various shapes represent calculated data points, of which only a subset is shown for clarity. The cases summary is provided in table 2 in Appendix B.

3.2. Vortex structures around an impulsively stopped sphere at ${\textit{Re}}_{\!D}=1000$

Most previous studies have focused on ${\textit{Re}}_{\!D}\leqslant 300$ , whereas cases at slightly higher Reynolds numbers can exhibit more complex flow structures. For example, Tomboulides & Orszag (Reference Tomboulides and Orszag2000) reported that the wake at ${\textit{Re}}_{\!D}=1000$ becomes weakly turbulent due to the presence of the Kelvin–Helmholtz instability. To explore this regime, the case of ${\textit{Re}}_{\!D}=1000$ with $0\leqslant N\leqslant 10$ is investigated in this subsection. In all cases, the sphere is brought to a impulsive stop after travelling a sufficient distance to reach a fully developed state. When $N\gt 0$ , the imposed magnetic field is present from the beginning of the simulation.

Figure 9.

Temporal evolution of vortex structures visualised using the Q-criterion ( $Q_{\textit{criterion}}=1$ ) at ${\textit{Re}}_{\!D}=1000$ without a magnetic field. Results are shown for (a) $t^*=-12$ , (b) $t^*=0$ , (c) $t^*=1$ , (d) $t^*=5$ .

Figure 10.

Vortex structures visualised using the Q-criterion ( $Q_{\textit{criterion}}=1$ ) at ${\textit{Re}}_{\!D}=1000$ with $N=0.3$ at $t^*=6$ . Panels (a) and (b) represent the views at the symmetry plane perpendicular to the Yaxis and the X axis, respectively.

In the absence of a magnetic field, complex wake structures comprising small-scale vortices due to the Kelvin–Helmholtz instability are observed at $t^* \lt 0$ , as shown in figure 9. A prominent vortex-shedding period is present, indicating the formation of fundamental vortex structures in the wake. Meanwhile, as shown in figure 2(b), a secondary and slightly pronounced vortex-shedding frequency is also detected, suggesting the presence of numerous secondary induced vortices appear among the primary vortices as illustrated in figure 9(a). Compared with the ${\textit{Re}}_{\!D}=300$ cases, more complex vortex structures are revealed at ${\textit{Re}}_{\!D}=1000$ when visualised using the same Q-criterion threshold, indicating the generation of a stronger wake. Following the impulsive stop ( $t^* \gt 0$ ), the residual kinetic energy remains sufficiently large for the downstream vorticity to detach from the sphere surface, forming a sequence of vortex structures, including a leading vortex structure (LVS), a first vortex structure (VS-1) and a second vortex structure (VS-2). Details of their formation and interaction are provided in Appendix A. During the wake evolution, a primary vortex structure develops downstream and a series of accompanying vortices interact with the sphere simultaneously, as shown in figure 9(d). The overall interaction pattern is broadly similar to that observed at ${\textit{Re}}_{\!D}=300$ , except that numerous small-scale vortices coexist with the fundamental vortex structures.

With the application of a transverse magnetic field, the wake structure before the impulsive stop tends to develop a double-plane symmetric state as $N$ increases. As demonstrated in figure 7, the Lorentz force induces anisotropy, enhancing vorticity generation in the X–Z plane and confining vortex shedding within this plane, unlike the helical shedding without a magnetic field. After the stop, a primary vortex forms and, under the anisotropic magnetic influence, an additional vortex-pair structure (VSP-2) develops as shown in figure 10, arising from the combined effects of VS-1 induction and the residual wake energy. A detailed visualisation of these vortex structures is provided in Appendix A. Due to the anisotropic damping, vortices aligned with the magnetic field exhibit greater intensity, particularly near the Y–Z plane. The downstream vortex structure VS-1 suppresses recirculation, causing the convergence line of skin friction lines near the X–Z plane to shift closer to the central point, indicating a delayed onset of recirculation. This produces a gourd-like closed convergent line, as presented in figure 11(a), deviating from circular or elliptical shapes but consistent with model III. Similar to cases with lower Reynolds numbers, as $N$ increases further, the vortex-shedding period approaches infinity, indicating the emergence of a completely steady wake behind the sphere. Moreover, the kinetic energy of the flow around the sphere decays progressively, eventually becoming insufficient to sustain a recirculation zone. As a consequence, the skin friction lines exhibit the characteristics described in model I, featuring only a straight convergent line perpendicular to the magnetic field direction as presented in figure 11(c). Further details about the influence of the transverse magnetic field on the wake evolution are provided in Appendix A. With the increment of $N$ at ${\textit{Re}}_{\!D}=1000$ , the pattern of skin friction lines undergoes a sequence of transitions, model III $\to$ IV $\to$ I, which is similar to cases at ${\textit{Re}}_{\!D}=300$ . This consistent transition pathway reaffirms the presence of the ‘reversion phenomenon’.

Figure 11.

Skin friction lines on the downstream surface of the sphere at $t^*=6$ for ${\textit{Re}}_{\!D}=1000$ with various interaction parameters. Results are shown for (a) $N=0.3$ , (b) $N=0.9$ , (c) $N=5$ .

4.1. Inspiration in the Stokes flow regime

To better understand the influence of vortex evolution on drag following the impulsive stop of a sphere, insight is first drawn from the classical Stokes flow. In the classical first Stokes problem, an infinite plate impulsively starts from rest and moves at a constant velocity parallel to its surface. The velocity profile as a function of time $t$ and the wall-normal coordinate $z$ can be obtained by solving the governing equation using the method of the separation of variables. Analogously, for an infinite plate that impulsively stops from a constant velocity, the velocity profile $u(z, t)$ can be derived as

(4.3) \begin{equation} u(z, t) = u_0 {\textit{erf}}(\eta ) = u_0 \frac {2}{\sqrt {\pi }} \int _{0}^{\eta } {\rm e}^{-\lambda ^2} \, {\rm d}\lambda ,\end{equation}

where the dimensionless variable $\eta = {z}/{2\sqrt {\nu t}}$ and $u_0$ is the velocity of the plate prior to the stop. The shear force per unit area acting on the plate can be expressed as

(4.4) \begin{equation} f_{\textit{plate}} = \rho \nu \left . \frac {\partial u}{\partial z} \right |_{z=0} = \frac {\rho u_0 \sqrt {\nu }}{\sqrt {\pi t}}. \end{equation}

Based on the fluid density $\rho$ and the initial plate velocity $u_0$ , the force can be non-dimensionalised into a force coefficient defined as

(4.5) \begin{equation} C_{d, {\textit{plate}}} = \frac {f_{\textit{plate}}}{\frac {1}{2} \rho u_0^2} = \frac {2}{\sqrt {\pi }}\frac {\nu }{u_0 \sqrt {\nu t}} = \frac {2}{\sqrt {\pi }}\frac {1}{{\textit{Re}}} ,\end{equation}

where ${\textit{Re}}$ is the Reynolds number based the characteristic length scale $\sqrt {\nu t}$ . Following the impulsive stop of the plate, vorticity is generated at the surface and diffuses outward in the wall-normal direction, forming a thin vorticity-concentrated boundary layer whose thickness grows as $\sqrt {\nu t}$ . This diffusive length scale characterises the unsteady flow field and naturally serves as the basis for constructing a time-dependent drag scaling law.

Inspired by the classical first Stokes problem and (4.5), the temporal evolution of the drag coefficient following an impulsive stop can be inferred by adopting the vorticity diffusion scale, $\sqrt {\nu t^*}$ , as the characteristic length. In the Stokes flow regime the drag coefficient of a sphere is given by $ {24}/{{\textit{Re}}_{\!D}}$ . At short time scales following the impulsive stop of the sphere, the vorticity is generated on the sphere surface and diffuses outward. By substituting the vorticity diffusion length scale into the drag expression after the impulsive stop, a temporal scaling of $C_d^{\textit{Stokes}} \sim {{\textit{Re}}}^{-1} \sim {t^*}^{-1/2}$ is obtained, in agreement with the classical result rigorously derived from the Stokes equations by Basset (Reference Basset1888). This scaling behaviour is further validated by the present numerical simulation at ${\textit{Re}}_{\!D}=0.1$ as shown in figure 12, where the temporal evolution of the drag coefficient agrees well with both the present scaling prediction and the analytical solution in Basset (Reference Basset1888), confirming the validity of the proposed characteristic length scale and the underlying assumptions of the scaling analysis. Moreover, the flow around the sphere retains the characteristics as Stokes flow at both short time scales and long time scales following the impulsive stop, as shown in figure 12(b,c). However, the drag decay exhibits two distinct scaling laws: ${t^*}^{-1/2}$ at short time scales and ${t^*}^{-1}$ at long time scales. The former can be predicted well by the present scaling analysis, whereas the latter has been derived based on the global mass and momentum conservation principles (Lawrence & Mei Reference Lawrence and Mei1995). In Lawrence & Mei (Reference Lawrence and Mei1995) the flow is assumed to be fully developed both before and after the impulsive stop, and the ${t^*}^{-1}$ decay law is obtained by analysing the decay of the characteristic velocity at long time scales.

Figure 12.

Verification at low Reynolds numbers ${\textit{Re}}_{\!D}=0.1$ . (a) The time history of the normalised drag coefficient; (b) instantaneous streamlines on the symmetry plane at short time scales; (c) instantaneous streamlines on the symmetry plane at long time scales.

4.2. Extension to inertia and MHD damping effects

The previously discussed scaling law, derived from the Stokes flow, is valid only in the low-Reynolds-number regime where the inertial effect is negligible. However, as the Reynolds number increases, inertial effects become significant and the presence of a transverse magnetic field introduces additional damping on the flow through the Lorentz force. To extend the drag decay analysis to this more general regime, appropriate corrections must be incorporated. The corrections can be systematically derived within the scaling analysis framework, which is inspired in the Stokes regime through substituting the characteristic length scale in drag coefficient expressions.

The drag on a sphere has been extensively studied, with the inertia correction introduced to the classical Stokes drag at high Reynolds numbers. At low but finite Reynolds numbers, Oseen’s theory refines the classical Stokes drag by incorporating the convective inertia correction, yielding the Oseen drag ${24}/{{\textit{Re}}_{\!D}}(1 +({3}/{16}) {\textit{Re}}_{\!D})$ . The second term in this expression represents the first-order inertial correction and is valid within the low-Reynolds-numbers regime. At higher Reynolds numbers, the semi-empirical formula $C_d = {24}/{{\textit{Re}}_{\!D}}(1 + 0.15{\textit{Re}}_{\!D}^{0.687})$ is commonly used across a wide range of Reynolds numbers (Clift, Grace & Weber Reference Clift, Grace and Weber2005). Analogous to the Oseen drag, the second term can also be considered as a correction part, introducing inertia effects at high Reynolds numbers. Consequently, the sphere drag can be considered as comprising two parts: the Stokes part $C_d^{\textit{Stokes}} \sim {{\textit{Re}}^{-1}}$ and a correction part $C_d^{\textit{correction}} \sim {{\textit{Re}}^{-1/3}}$ . The Stokes part can be derived as $C_d^{\textit{Stokes}} \sim {{t^*}^{-1/2}}$ , as illustrated in § 4.1. For the correction part, we need to consider two components: the inertia effect and MHD effect. At high Reynolds numbers, convection induced by the inertia effect becomes increasingly significant, indicated by the formation and propagation of vortex structures, which are both relevant to the vorticity diffusion scale. Inspired by the Stokes part in § 4.1, the vorticity diffusion distance is adopted as the characteristic length scale of the inertia correction component, deriving $C_d^{\textit{inertia}} \sim {({t^*}^{1/2})^{-1/3}} \sim {t^*}^{-1/6}$ .

Figure 13.

Drag decomposition model. The drag decay following an impulsive stop of a sphere is decomposed into three contributions: the original Stokes effect, an inertia correction and a MHD correction. Based on the vorticity diffusion scale and the drag coefficient formulation, the temporal scalings associated with each contribution are derived and summarised here.

Furthermore, the scaling law analysis of drag is extended to account for the presence of a transverse magnetic field here. Due to the increasingly significant dissipation induced by the transverse magnetic field as the interaction parameter increases, the vorticity magnitude decays over time. This decay can be quantitatively characterised by the decay of the angular momentum $\varGamma \sim {t^*}^{-1/2}$ (Davidson Reference Davidson2001). Since the viscous forces scale proportionally with angular momentum and appear in the denominator part of ${\textit{Re}}$ , the correction to the drag coefficient induced by the MHD effect can be expressed as $C_d^{\textit{MHD}} \sim {{\textit{Re}}^{-1/3}} \sim ({\textit{inertia}\, \textit{force}}/{{\textit{viscous}\, \textit{force}}})^{-1/3} \sim ( {1}/{{t^*}^{-1/2}})^{-1/3} \sim {t^*}^{-1/6}$ . The Stokes effect, as originally derived from theoretical analyses (Basset Reference Basset1888; Lawrence & Mei Reference Lawrence and Mei1995), along with the present extensions incorporating for inertia and MHD effects, are schematically illustrated and summarised in figure 13. Notably, although partial flow separation occurs after the impulsive stop, the sphere remains predominantly enveloped by the surrounding flow, representing the baseline Stokes contribution to the drag. The inertia effect and the MHD damping effect both influence the enveloped flow, further influencing the drag decay. Therefore, the inertia and MHD corrections are superimposed on the baseline Stokes flow, and the coupled decay law is obtained through a multiplicative combination of these effects rather than a simple additive relation. The drag coefficient for $t^*\gt 0$ is influenced by these three components in different ways, depending on the Reynolds number, the interaction parameter and the characteristic time scale.

For short time scales $t^* \ll 1$ , as presented in figure 14, the drag coefficient decays according to the scaling law ${t^*}^{-2/3}$ for small interaction parameters, while a slower decay is observed for large interaction parameters. The decay scaling law of the drag coefficient without a magnetic field at short time scales has been reported as ${t^*}^{-1/2}$ by Basset (Reference Basset1888) in the case of an impulsively started sphere without a magnetic field, whereas our previous work found ${t^*}^{-2/3}$ in the impulsively stopped case (Cai et al. Reference Cai, Pan and Ni2022). The discrepancy originates from the formation of a primary vortex ring following the impulsive stop, which induces an outward tendency in the upstream region near the sphere, thereby reducing the surface shear stress and ultimately leading to a faster attenuation of the drag coefficient. In the presence of a streamwise magnetic field, the attenuation rate of the drag coefficient remains comparable to the non-magnetic cases at smaller $N$ , but significantly decreases at larger $N$ (Cai et al. Reference Cai, Pan and Ni2022). Similarly, under a transverse magnetic field, the decay also follows ${t^*}^{-2/3}$ at smaller $N$ but slows down at larger $N$ , as shown in figure 14. The fact that the decay scaling laws remain identical under both streamwise and transverse magnetic fields can be explained qualitatively by the flow structure. Although the direction of the magnetic field is different, the vortex before the stop is still present and the damping effect of the magnetic field is significantly weak at smaller $N$ , indicating that the vortex structure is similar to the case without a magnetic field. The formation and diffusion of the vortex can be demonstrated by the instantaneous streamlines on the symmetry plane, as shown in figure 15. The primary vortex persists both at $t^*=0$ and $t^*=1$ , and the new vorticity is generated around the sphere at $0 \lt t^* \lt 1$ , as shown in figure 15(d) by streamlines of the velocity difference. Therefore, the temporal behaviour of the drag coefficient is the same as in the case without a magnetic field, involving two components: the Stokes effect and the inertia effect. These two effects together determine the decay scaling law ${t^*}^{-1/2}{t^*}^{-1/6} \sim {t^*}^{-2/3}$ for $t^* \ll 1$ . As a result, at shorter time scales with small $N$ , the decay rate of the normalised drag coefficient remains ${t^*}^{-2/3}$ , which is nearly identical regardless of the presence and the direction of the magnetic field.

Figure 14.

The time history of $C_d^*$ at ${\textit{Re}}_{\!D} \leqslant 300$ with a transverse magnetic field of different intensity. Results are shown for (a) ${\textit{Re}}_{\!D} = 300$ with smaller $N$ , (b) small $N$ with different ${\textit{Re}}_{\!D}$ , (c) ${\textit{Re}}_{\!D}=300$ with larger $N$ .

Figure 15.

Instantaneous streamlines on the X–Z plane at ${\textit{Re}}_{\!D}=270$ with $N=0.1$ . The blank part represents the impulsively stopped sphere. Results are shown for (a) $t^*=0$ , (b) $t^*=1$ , (c) $t^*=15$ . (d) Difference between $t^*=0$ and $t^*=1$ . (e) Difference between $t^*=14$ and $t^*=15$ .

Figure 16.

Instantaneous streamlines on two symmetry planes at ${\textit{Re}}_{\!D}=300$ with $N=1$ at different time scales. The blank part represents the impulsively stopped sphere. Results are shown for (a) the X–Z plane at $t^*=1$ , (b) the Y–Z plane at $t^*=1$ , (c) the X–Z plane at $t^*=10$ , (d) the Y–Z plane at $t^*=10$ .

With the increment of the interaction parameter, the damping effect of the magnetic field is strong enough to suppress thoroughly the vortex shedding before the stop. Meanwhile, the double-plane symmetry of the flow structure is enhanced by the stronger transverse magnetic field. On the X–Z symmetry plane, vortices remain visible on the upstream side, as shown in figure 16(a). The formation mechanism is similar to that described in figure 3. Unlike in low- $N$ cases, however, the upstream vortices remain nearly stationary, without inducing significant inertial convection effects due to their lack of motion. Nonetheless, the MHD effect due to the dissipation of vortex intensity still exists. As a result, the flow pattern on the X–Z plane reflects the combined influence of the origin Stokes part and the MHD part in figure 13, yielding a drag decay scaling law $C_d^* \sim {t^*}^{-1/2}{t^*}^{-1/6} \sim {t^*}^{-2/3}$ . On the Y–Z symmetry plane, no vortices are observed in figure 16(b), owing to the stronger damping effect as illustrated in figure 7. Consequently, both the convective and MHD contributions are absent, and the drag decays at a slower rate and tends to a ${t^*}^{-1/2}$ scaling with the Stokes effect as the sole governing mechanism. As the interaction parameter increases, the flow pattern on the Y–Z plane, characterised by the Stokes-like behaviour, gradually dominates the overall dynamics, leading the drag decay scaling law to transition from ${t^*}^{-2/3}$ to ${t^*}^{-1/2}$ , as shown in figure 14(c).

For long time scales $t^* \gg 1$ , as shown in figure 14(a,b), the scaling law $C_d^* \sim {t^*}^{-4/3}$ under a transverse magnetic field is observed for small interaction parameters. For weak magnetic field intensities and ${\textit{Re}}_{\!D}=300$ , vortex shedding still occurs from the sphere for $t^*\lt 0$ , although the shedding period increases with the magnetic field intensity, as reported by Pan et al. (Reference Pan, Zhang and Ni2019b ). The detached vortex at $t^*\lt 0$ moves downstream at $t^*\gt 0$ due to inertia, interacting with the sphere and leading to oscillations in $C_d^*$ , as shown in figure 14(a). The oscillations of $C_d^*$ at $t^* \approx 1$ and $t^* \gg 1$ arise from the same underlying mechanism: the contrary influence of the vortex depending on its position relative to the sphere. When the vortex resides on the upstream side, it pulls streamlines away from the sphere surface, reducing the local shear stress. As the vortex propagates downstream, it pushes streamlines closer to the sphere surface, enhancing the shear stress. In addition, the low-pressure region associated with the vortex also travels from the upstream to the downstream side, further contributing to the oscillations in $C_d^*$ . However, despite the presence of oscillations, the scaling law $C_d^* \sim {t^*}^{-4/3}$ remains clearly evident. When the magnetic field is slightly stronger or when ${\textit{Re}}_{\!D}\lt 270$ , no vortex shedding occurs from the sphere at $t^*\lt 0$ . As a result, the oscillations in $C_d^*$ at long time scales disappear, as shown in figure 14(b). The propagation and evolution of the vortex at long time scales is illustrated in figure 15(c,e). The primary vortex appearing at short time scales propagates downstream and gradually dissipates. However, the residual wake gives rise to new vortices on the upstream side of the sphere, which dominates the variation of the drag coefficient. Meanwhile, the damping effect of the transverse magnetic field causes the vorticity magnitude to decay over time. Therefore, all three components in figure 13 need to be considered, i.e. the origin Stokes effect ${t^*}^{-1}$ at long time scales reported by Lawrence & Mei (Reference Lawrence and Mei1995), the inertia effect and the MHD effect. Consequently, the scaling law at $t^* \gg 1$ can be derived as $C_d^* \sim {t^*}^{-1}{t^*}^{-1/6}{t^*}^{-1/6} \sim {t^*}^{-4/3}$ .

Figure 17.

Difference of streamlines at long time scales, at ${\textit{Re}}_{\!D}=200$ , with and without a transverse magnetic field. (a) Without a magnetic field; (b) with a transverse magnetic field, $N=0.1$ ; (c) contour of the X component of the Lorentz force at $t^*=1$ .

In the absence of a magnetic field, the drag decay scaling law at long time scales within the range $60 \leqslant {\textit{Re}}_{\!D} \leqslant 250$ is previously found to follow a temporal decay of ${t^*}^{-1}$ (Cai et al. Reference Cai, Pan and Ni2022). In contrast, when a transverse magnetic field is applied here, both the inertia and MHD effects are incorporated as corrections to the original Stokes contribution in the drag decomposition model at long time scales. The MHD effect arises naturally from the imposition of the magnetic field. However, the inertia effect, which is negligible in non-magnetic cases, becomes significant under the influence of the transverse magnetic field. This difference is primarily attributed to the presence of vortex structures on the local upstream side of the sphere, which persist due to the Lorentz force and consequently influence the incoming flow after the impulsive stop. The instantaneous streamlines on the symmetry plane at long time scales, at ${\textit{Re}}_{\!D}=200$ , with and without a transverse magnetic filed, are shown in figure 17(a,b). After the sphere is impulsively stopped, the residual wake is convected downstream and away from the centreline due to the presence of the sphere. As a result, no vortex structures appear in the local inflow zone marked by the red rectangle in the absence of a magnetic field. However, the Lorentz force under a transverse magnetic field confines the wake closer to the centreline at short time scales, as shown in figure 17(c), allowing the upstream vortices in the local inflow zone to influence the flow impacting on the sphere at long time scales. Consequently, the inertia effect associated with the convection effect of vortices must be accounted for when applying a transverse magnetic field with small $N$ . At long time scales, both the inertia and MHD effects are coupled with the original Stokes effect, leading to the drag decay scaling $C_d^* \sim {t^*}^{-1}{t^*}^{-1/6}{t^*}^{-1/6} \sim {t^*}^{-4/3}$ , which is consistent with simulation results as shown in figure 14(b).

Under a stronger magnetic field, the vortex near the X–Z plane lacks sufficient energy to propagate to the downstream side. In contrast to cases with smaller interaction parameters, the primary vortex appears to be effectively trapped on the upstream side due to the pronounced damping induced by the Lorentz force. As a consequence, the vortex remains upstream even at long time scales, as illustrated in figure 16(c), leading to the absence of oscillations in the time evolution of the normalised drag coefficient $C_d^*$ . Moreover, the temporal decay of $C_d^*$ at $t^* \gg 1$ follows the same decay scaling law, $C_d^* \sim {t^*}^{-2/3}$ , as that observed at $t^* \ll 1$ in figure 14(c). This behaviour can be qualitatively understood from the perspective of flow structure evolution. The damping effect of the transverse magnetic field is strong enough to suppress the formation of a complete vortex ring at short time scales. Meanwhile, at long time scales, no vortex in the wake moves downstream. Therefore, the flow structure remains similar at both short and long time scales. As a result, the scaling law governing the decay of $C_d^*$ remains consistent across different time scales.

4.3. Extension to higher ${\textit{Re}}$

Figure 18.

The time history of $C_d^*$ at ${\textit{Re}}_{\!D}=500$ and $1000$ without a magnetic field. The average value of $C_d^*$ is averaged for different stopped times. Results are shown for (a) ${\textit{Re}}_{\!D}=500$ and (b) ${\textit{Re}}_{\!D}=1000$ .

At higher Reynolds numbers, increasingly complex vortex structures emerge in the wake of the sphere, and the applicability of the formerly constructed drag decomposition model is examined. At ${\textit{Re}}_{\!D}=500$ and $1000$ , the large-scale wake structures remain clearly observable and the drag coefficient continues to follow a power-law decay. Considering the combined contributions of the Stokes effect and inertia effects, as illustrated in figure 13, the temporal evolution of the normalised drag coefficient follows $C_d^* \sim {t^*}^{-1/2}{t^*}^{-1/6} \sim {t^*}^{-2/3}$ at short time scales and $C_d^* \sim {t^*}^{-1}{t^*}^{-1/6} \sim {t^*}^{-7/6}$ at long time scales. These scaling predictions show better agreement with the simulation results than the original ones, as shown in figure 18. As mentioned in § 3, in the absence of a magnetic field, the flow field exhibits unsteady vortex shedding at ${\textit{Re}}_{\!D}=1000$ during the fully developed stage prior to the impulsive stop of the sphere. Therefore, to analyse the general evolution, drag coefficients from cases with varying stop times are averaged. Eight stop times are selected, uniformly distributed within a single primary vortex-shedding period. Similar to the cases with lower Reynolds numbers and interaction parameters, the decay scaling behaviour ${t^*}^{-2/3}$ remains essentially unchanged, as presented in figure 18. Notably, a significant fluctuation in $C_d^*$ occurs near $t^*=1$ , including a reversal in its direction. This behaviour results from the exclusion of negative values on the log–log scale. After the impulsive stop of the sphere, the upstream fluid continues moving forward and interacts with the sphere. Due to the acceleration effect caused by the sphere’s curved surface, the flow velocity near the upstream-facing side of the sphere increases, forming a low-pressure region. The low-pressure region continues to move downstream and gradually dissipates due to viscous and magnetic damping effects, as shown in figure 19. As a result, a significant fluctuation occurs in the pressure drag, which in turn causes a fluctuation in the total drag. At long time scales, the complex wake structures cause the drag decay to exhibit rather chaotic behaviours. However, the averaged overall trend still remains more consistent with the scaling laws derived from the drag decomposition framework than with the original ones.

Figure 19.

Pressure contours of $t^*=1$ and $t^*=2$ at ${\textit{Re}}_{\!D}=1000$ without a magnetic field. The stopped instant $t_{\textit{stop}}=0.75T$ .

Figure 20.

The time history of $C_d^*$ at ${\textit{Re}}_{\!D}=1000$ with different interaction parameters. (a) Relatively large $N$ ; (b) relatively larger $N$ .

Figure 21.

Instantaneous streamlines on the symmetry plane X–Z at ${\textit{Re}}_{\!D}=1000$ with $N=0.3$ . The blank part represents the impulsively stopped sphere. Results are shown for (a) $t^*=1$ and (b) $t^*=15$ .

Figure 22.

Instantaneous streamlines on two symmetry planes at $t^*=15$ at ${\textit{Re}}_{\!D}=1000$ with $N=10$ . The blank part represents the impulsively stopped sphere. Results are shown for the (a) X–Z plane and (b) the Y–Z plane.

When a transverse magnetic field is applied, the temporal decay of the drag coefficient follows a ${t^*}^{-5/6}$ scaling for large $N$ and a ${t^*}^{-2/3}$ scaling for larger $N$ , as shown in figure 20. As shown in figure 10, when the magnetic field is not very strong, vortex shedding still occurs from the sphere surface before the impulsive stop. Under the influence of the transverse magnetic field, the flow around the sphere exhibits double-plane symmetry. The streamlines on the X–Z symmetry plane, as shown in figure 21, reveal the presence of persistent upstream vortex structures that propagate downstream due to inertia. As a result, the inertial convection induced by these vortex structures remains significant. Meanwhile, due to the damping effect of the transverse magnetic field on vortices, the MHD effect must also be accounted for. Consequently, the drag is corrected by both the inertia effect and MHD effect based on the Stokes effect. Therefore, the scaling law becomes $C_d^* \sim {t^*}^{-1/2}{t^*}^{-1/6}{t^*}^{-1/6} \sim {t^*}^{-5/6}$ by incorporating the three effects. At long time scales, the main vortex structures concentrate on the upstream side and remain similar to that at short time scales as shown in figure 21. As a result, the decay rate of $C_d^*$ at long time scales is consistent with that observed at short time scales. However, when the magnetic field is sufficiently strong to suppress the downstream propagation of vortices, the vortex structures appear frozen on the upstream side, as illustrated by the instantaneous streamlines at long time scales in figure 22(a). Consistent with the case of ${\textit{Re}}_{\!D}=300$ , the streamlines indicate that the flow around the Y–Z plane is similar to the Stokes flow as shown in figure 22(b). Thus, the MHD effect ${t^*}^{-1/6}$ combines with the Stokes contribution ${t^*}^{-1/2}$ , resulting in an overall scaling law of ${t^*}^{-2/3}$ , as shown in figure 20(b). Although the vortex structures become increasingly complex at higher Reynolds numbers, the reasonable extension of the drag decomposition model gives the explanation for the drag decay with three effects in figure 13 across various parameters. The derived scaling laws exhibit better consistency with the simulation results than the original scaling laws.

4.4. Extension to prolate spheroid

The alteration of particle shape can lead to increasing flow complexity. Specifically, the geometric anisotropy of a prolate spheroid introduces additional asymmetry into the flow field after an impulsive stop. In this context, the applicability of the established drag decomposition model is further examined. In the moderate Reynolds number regime, the drag coefficient formula of a prolate spheroid is given by the correlation proposed in Fröhlich et al. (Reference Fröhlich, Meinke and Schröder2020):

(4.6) \begin{align} C_d^{\theta } = \frac {24}{{{\textit{Re}}}_{d_e}} g_{\theta }(\beta )\left (1+0.15 {{{\textit{Re}}}_{d_e}^{0.687}} + c_{\theta ,1}(\log \beta )^{c_{\theta ,2}} {{{\textit{Re}}}_{d_e}}^{c_{\theta ,3} + c_{\theta ,4} \log \beta }\right )\! .\end{align}

Here $\theta$ is the inclination angle between the free-stream velocity and the spheroid’s long axis, $\beta$ is the aspect ratio between the long axis and the short axis, and $g_{\theta }(\beta )$ is a function relevant to the geometric shape parameters. The coefficients $c_{\theta ,i}$ are empirical fitting parameters. In the general definition of ${{\textit{Re}}}_{d_e}$ , the volume-equivalent diameter $d_e$ is adopted as the characteristic length. It is noted that $c_{\theta , 1}$ is generally at least one order of magnitude smaller than the preceding term coefficient $0.15$ . Moreover, $g_{\theta }$ depends solely on the shape parameters and is independent of the Reynolds number. Consequently, the drag decay scaling law remains the same behaviour as that for a sphere, regardless of the aspect ratio $\beta$ or the inclination angle $\theta$ .

Figure 23.

The time history of $C_d^*$ for a prolate spheroid with different configuration parameters. The legend in the plot, for example, `AR4an90Re200T1’ indicates that $\beta =4$ , $\theta =90^\circ$ , ${{\textit{Re}}}_{d_e}$ = 200 and $t_{\textit{stop}}=1/4T$ , where $T$ is the vortex-shedding period prior to the impulsive stop. (a) Relatively low ${{\textit{Re}}}_{d_e}$ , no oscillations appearing at $t^* \gg 1$ ; (b) relatively high ${{\textit{Re}}}_{d_e}$ , oscillations appearing at $t^* \gg 1$ .

Figure 24.

Temporal evolution of vortex structures around a prolate spheroid ( $\beta =4$ , $\theta =90^\circ$ ) at ${{\textit{Re}}}_{d_e}=100$ and ${{\textit{Re}}}_{d_e}=200$ . Results are shown for (a) ${{\textit{Re}}}_{d_e}=100$ , visualised using the instantaneous streamlines on the symmetry plane (Y–Z plane); (b) ${{\textit{Re}}}_{d_e}=200$ , visualised using the Q-criterion ( $Q_{\textit{criterion}}=0.1$ ).

The time history of $C_d^*$ after the impulsive stop for a prolate spheroid with different $\beta$ , $\theta$ and ${{\textit{Re}}}_{d_e}$ is presented in figure 23. For short time scales $t^* \ll 1$ , all configurations exhibit a drag decay following the scaling law $C_d^* \sim {t^*}^{-2/3}$ . For long time scales $t^* \gg 1$ , the drag decays as ${t^*}^{-1}$ at relatively low ${{\textit{Re}}}_{d_e}$ and as ${t^*}^{-7/6}$ at relatively high ${{\textit{Re}}}_{d_e}$ where unsteady vortex shedding is present prior to the impulsive stop. As $\beta$ approaches unity, the spheroid geometry approaches that of a sphere, while larger $\beta$ introduces greater anisotropy. Therefore, cases with $\beta = 4$ are selected to visualise the vortex structures and examine the validity of the drag decomposition model. The temporal evolution of the vortex structures around an impulsively stopped prolate spheroid with $\beta =4$ and $\theta =90^\circ$ is presented in figure 24. At ${{\textit{Re}}}_{d_e}=200$ , the wake exhibits periodic vortex shedding in the fully developed stage prior to the impulsive stop. To capture this vortex-shedding behaviour, the vortex evolution is visualised using the Q-criterion. However, at ${{\textit{Re}}}_{d_e}=100$ , the wake remains steady and exhibits double-plane symmetry prior to the impulsive stop, so the evolution process is visualised by streamlines on the symmetry plane. The X-Z plane is chosen for comparison, as the vortex shedding at ${{\textit{Re}}}_{d_e}=200$ primarily develops around this plane. At short time scales, the primary vortex forms upstream near the spheroid for both ${{\textit{Re}}}_{d_e}=100$ and $200$ . Therefore, the inertia effect must be incorporated with the Stokes effect in the drag decomposition model, leading to ${C_d}^* \sim {t^*}^{-1/2}{t^*}^{-1/6} \sim {t^*}^{-2/3}$ , which is consistent with the simulation results shown in figure 23 at $t^* \ll 1$ .

At long time scales, the drag decay at moderate ${{\textit{Re}}}_{d_e}$ is surprisingly found to still follow the same scaling law as that derived by Lawrence & Mei (Reference Lawrence and Mei1995) for a sphere at the low-Reynolds-numbers regime, following the temporal decay of ${t^*}^{-1}$ . The flow around a prolate spheroid involves both the long-axis and short-axis sides. Due to its geometry, thicker in the middle and tapered at both ends, the majority of the drag arises from the abdominal region. Therefore, the vortex structures developing near the short-axis side have a more pronounced influence on the overall drag evolution. After the impulsive stop, the primary vortex propagates downstream and radially outward from the centreline, driven by its self-induced radial velocity. Due to the smaller cross-section on the short-axis side and the steady flow prior to the impulsive stop, no upstream vortex structure forms in the local region near the upstream side of the spheroid at long time scales, as shown in figure 24(a). As a result, the upstream flow impacting on the spheroid at long time scales resembles that in the low-Reynolds-number regime for a sphere. This suggests that the Stokes effect can still be defined as ${t^*}^{-1}$ at long time scales in the drag decomposition model. At ${{\textit{Re}}}_{d_e}=200$ , a sequence of accompanying vortices move downstream as shown in figure 24(b), influencing the inflow impacting on the spheroid. As a result, the inertia effect must again be combined with the Stokes effect, giving ${C_d}^* \sim {t^*}^{-1}{t^*}^{-1/6} \sim {t^*}^{-7/6}$ , which aligns well with the simulation results presented in figure 23(b) for $t^* \gg 1$ . To further examine the inertia effect ${t^*}^{-1/6}$ in the drag decomposition model at moderate $\theta$ , the time history of $C_d^*$ and evolution of vortex structures for a prolate spheroid with $\beta =4$ and $\theta =75^\circ$ are presented in figure 25. Under this configuration, the wake prior to the impulsive stop exhibits periodic vortex shedding. After the impulsive stop, a primary vortex forms upstream at short time scales and a sequence of accompanying vortices on the upstream side impacts on the spheroid at long time scales. Therefore, the drag decay must incorporate the inertia effect with the Stokes effect both at short and long time scales, leading to the decay law ${C_d}^* \sim {t^*}^{-1/2}{t^*}^{-1/6} \sim {t^*}^{-2/3}$ for $t^*\ll 1$ and ${C_d}^* \sim {t^*}^{-1}{t^*}^{-1/6} \sim {t^*}^{-7/6}$ for $t^*\gg 1$ . Although the spheroid is notably prolate with $\beta =1.5, 2.5, 4$ and inclined with $\theta =0^\circ , 75^\circ , 90^\circ$ , the drag decomposition model presented in figure 13 successfully predicts the drag decay scaling law by reasonably extending the framework to incorporate the evolution of the vortex structures. At $t^* \sim 1$ , the vortex structures around the spheroid are significantly influenced by its shape, but the coefficient of the shape-correction term is relatively small compared with the inertial contribution and was therefore neglected. Nevertheless, the proposed scaling laws still capture the overall decay trend more accurately than the uncorrected laws.

Figure 25.

The time history of $C_d^*$ and temporal evolution of vortex structures for a prolate spheroid with $\beta =4$ and $\theta =75^\circ$ at ${{\textit{Re}}}_{d_e}=250$ . (a) Time history of $C_d^*$ ; (a) vortex structures at different $t^*$ , visualised using the Q-criterion ( $Q_{\textit{criterion}}=0.1$ ).

4.5. Extension to magnetic fields with different directions

In realistic situations the magnetic field is not necessarily aligned or perpendicular to the primary flow direction. Therefore, it is essential to extend the analysis to magnetic fields with different orientations. Before introducing an inclined magnetic field, the streamwise configuration is re-examined here after our previous work (Cai et al. 2022). The slight difference between the streamwise and transverse cases about the drag decay exists in the small and medium interaction numbers regime. However, for strong magnetic fields, a new drag decay law at long time scales is identified.

The time history of $C_d^*$ after the impulsive stop for different interaction parameters is shown in figure 26. When $N$ is relatively small, the streamwise magnetic field is too weak to noticeably influence the vortex structure and vorticity transport. Accordingly, the drag decay follows the same scaling laws as those without a magnetic field, only including the Stokes effect and the inertia effect. This leads to the decay scaling law ${C_d}^* \sim {t^*}^{-1/2} {t^*}^{-1/6} \sim {t^*}^{-2/3}$ at short time scales and ${C_d}^* \sim {t^*}^{-1} {t^*}^{-1/6} \sim {t^*}^{-7/6}$ at long time scales. The difference between the streamwise and transverse cases arises primarily from the angle between the magnetic field and the main flow direction. In the transverse configuration the nearly perpendicular alignment causes the MHD effect to appear even at small interaction parameters, whereas the nearly parallel configuration delays the effect under the same parameters. When $N$ reaches moderate values, the drag decay scaling law is consistent with that at smaller Reynolds numbers without a magnetic field, as presented in figure 26(b). In this regime, the inertia effect must be considered at short time scales but it becomes negligible at long time scales. While no vortex structures were observed in the key region at long time scales in figure 17 (§ 4.2), a primary vortex does exist in the upstream zone at short time scales, as shown in figure 27. As a result, both the Stokes effect and inertia effect are included at short time scales, whereas only the Stokes effect remains at long time scales, leading to ${C_d}^* \sim {t^*}^{-1/2} {t^*}^{-1/6} \sim {t^*}^{-2/3}$ at short time scales and ${C_d}^* \sim {t^*}^{-1}$ at long time scales.

Figure 26.

The time history of $C_d^*$ at ${\textit{Re}}_{\!D} = 300$ with a streamwise magnetic field of different intensity. (a) Smaller $N$ ; (b) medium $N$ ; (c) larger $N$ .

Figure 27.

Instantaneous streamlines on the symmetry plane X–Z at ${\textit{Re}}_{\!D}=300$ with $N=1$ . The blank part represents the impulsively stopped sphere. Results are shown for (a) $t^*\approx 1$ and (b) $t^*=10$ .

Figure 28.

Instantaneous streamlines on the symmetry plane X–Z at ${\textit{Re}}_{\!D}=300$ with $N=9$ when $t^*=10$ , representing longer time scales. The blank part represents the impulsively stopped sphere. (a) Streamwise magnetic field; (b) transverse magnetic field.

When $N$ is large, the drag decays as a different law, exhibiting an exponential decay law at long time scales, as shown in figure 26(c). In applications of the drag formulations, the drag force is often expressed dimensionally as $F_d=3\pi \rho \nu U D g({\textit{Re}})$ , where $U$ is the real local velocity of the fluid around the sphere and $g({\textit{Re}})$ is the drag coefficient as a function of the real local Reynolds number, ${\textit{Re}}=UL / \nu$ . The local velocity can be set as $U=\gamma (t) U_0$ and then ${\textit{Re}}=\gamma (t) {\textit{Re}}_0$ . When $\gamma (t) \ll 1$ , the drag can be expanded at zero as

(4.7) \begin{equation} F_d \sim \gamma (t) g(\gamma {\textit{Re}}_0) \sim \gamma (t)g(0) + \gamma ^2(t) g'(0) {\textit{Re}}_0 + {\cdots} \sim \gamma (t), \quad \gamma (t) \ll 1. \end{equation}

Consequently, the drag decay follows the same functional form as the velocity decay, both exhibiting either a power-law decay or an exponential behaviour. When the magnetic field is aligned with the flow direction, the streamlines are confined with a narrow region in the wake. In the wake region the electric current arising from electromotive effects plays a more significant role than electrostatic potential effects. The electric current density is dominated by the partial Ohm’s law, $\boldsymbol{{J}} \approx \boldsymbol{{u}} \times \boldsymbol{{B}}_0$ . At sufficiently large values of $N$ , the contribution of the Lorentz force becomes more important than other terms in the momentum equation. Accordingly, the relation between the fluid velocity and the Lorentz force can be approximated as $ {\partial \boldsymbol{{u}}}/{\partial t} \sim N (\boldsymbol{{J}} \times \boldsymbol{{B}}_0 ).$ Substituting the partial Ohm’s law and following a straightforward derivation, the simplified velocity decay can be expressed as

(4.8) \begin{equation} {|\boldsymbol{{u}}|} \sim U_0 e^{-f\left (N\right )t} \sim \gamma (t) U_0. \end{equation}

At long time scales, the velocity magnitude becomes small and $\gamma (t) \ll 1$ is satisfied. Hence, the drag decay follows the same exponential form as the velocity, as presented in figure 26. The difference between the streamwise and transverse cases arises primarily from the different vortex structures as presented in figure 28. In MHD flows, a strong magnetic field prompts vortices to tend to be parallel to its direction, so the upstream vortex observed in the transverse case does not form for a streamwise magnetic field. Moreover, the strong streamwise magnetic field confines the vorticity to a narrow region around the sphere. The mechanism of vorticity transport dominating the drag decay gradually loses the leading role. As a result, the drag decay law in the streamwise case is dominated by the velocity decay rather than by the vortex-induced effects.

Figure 29.

The time history of $C_d^*$ at $\textit{Re}_D=300$ and $N=0.1$ for a magnetic field with different directions.

After examining the two extreme configurations for the direction of a magnetic field, cases with a magnetic field of any direction were then also investigated. The angle between the applied magnetic field and the transverse direction is set to $0^\circ$ , $30^\circ$ , $45^\circ$ , $60^\circ$ and $90^\circ$ , as presented in figure 29. For $N=0.1$ , the drag decay exhibits different behaviours between the streamwise and transverse configurations. Therefore, the drag decay at ${\textit{Re}}=300$ and $N=0.1$ is examined for different magnetic field orientations. As shown in figure 29, the drag decay as ${t^*}^{-2/3}$ at short time scales, which is the same as the two extreme configurations. At long time scales, as the magnetic field orientation changes from transverse to streamwise, the drag decay law transitions from ${t^*}^{-4/3}$ to ${t^*}^{-1}$ . As expected, the changing tendency of the drag decay law is the same as the orientation angle of the magnetic field. The closer the orientation is to either extreme configuration, the more closely its drag decay follows the corresponding scaling law. Although the magnetic field might take any orientation in realistic situations, the drag decay behaviour remains bounded by the two limiting cases.