3.2.1. Unstable modes and their critical curves

The stability property of the steady, axisymmetric base flow to three-dimensional small perturbations is explored by solving the eigenvalue equations (2.8)–(2.10). Flow stability analysis relies on an eigenvalue calculation. According to previous research on hydrodynamics flows past a sphere (Natarajan & Acrivos Reference Natarajan and Acrivos1993; Pier Reference Pier2008; Meliga et al. Reference Meliga, Chomaz and Sipp2009), bubble (Tchoufag et al. Reference Tchoufag, Magnaudet and Fabre2013) and disk (Tchoufag, Fabre & Magnaudet Reference Tchoufag, Fabre and Magnaudet2014), it has been proved that the non-axisymmetric mode with an azimuthal wavenumber of $m=1$ is the most unstable. Therefore, the eigenvalues of $Re=150$, 200, 300 and 400 at different interaction numbers $N$ are first examined with $m$ restricted to 1. The results of the global eigenvalue spectra are presented in figure 10. For hydrodynamics ($N=0$), Natarajan & Acrivos (Reference Natarajan and Acrivos1993) reported the occurrence of regular and Hopf bifurcations at critical Reynolds numbers $Re_c^I\approx 210$ and $Re_c^{II}\approx 277$, respectively. When considering a streamwise magnetic field, an unstable stationary mode appears at $Re=200$, $N=20$. This suggests that a strong magnetic field can promote flow instability, as the flow is stable in the absence of a magnetic field when $Re < Re_c^I$. At the same time, for $Re > Re_c^{II}$, the unstable oscillating mode is suppressed by the magnetic field at $Re=300$ and $400$. It means that no Hopf bifurcation or vortex shedding occurs. Therefore, the magnetic field enhances flow stability. These findings demonstrate that a streamwise magnetic field has a dual effect on the stability of flow past a sphere.

Figure 10. The global eigenvalue $\lambda$ spectra at $m=1$ with different interaction numbers, $N=0$, 1, 5 and 20, for different Reynolds numbers: (a) $Re=150$, (b) $Re=200$, (c) $Re=300$, (d) $Re=400$.

Yang & Prosperetti (Reference Yang and Prosperetti2007) investigated a LSA of the flow past a spheroidal bubble. They observed an unstable oscillating mode with $m=2$ when the aspect ratio $\chi =2.5$ and $Re=660$. A large aspect ratio can enlarge the recirculation region behind the bubble in the base flow. It is worth noting that a strong magnetic field has a similar effect on the recirculation behind the sphere in § 3.1. Hence, the effect of azimuthal wavenumber $m$ on the stability of a flow past a sphere under the influence of a streamwise magnetic field should also be examined. Since unstable modes are more easily identified at higher Reynolds numbers, eigenvalue $\lambda$ spectra at $Re=400$ with different $m$ and different $N$ are checked, as shown in figure 11. No unstable mode is observed in the cases of $m=0$, 3, 4, while an unstable oscillating mode at $N=0$ and an unstable stationary mode at $N=20$ are observed in the case of $m=2$. The angular frequency $\lambda _i$ of the unstable mode in figure 11(b) is different from that with $m=1$ in figure 10(d), which infers that vortex shedding at $Re=400$, $N=0$ exhibits multiple frequencies. Citro et al. (Reference Citro, Siconolfi, Fabre, Giannetti and Luchini2017) reported that the angular frequency of the unstable oscillating mode provided an estimation of the vortex shedding frequency. In fact, Mittal (Reference Mittal1999) confirmed that the wake became irregular and lost planar symmetry with the occurrence of multi-frequency vortex shedding when $Re>355$.

Figure 11. The global eigenvalue $\lambda$ spectra at $Re=400$ with different interaction numbers, $N=0$, 1, 5 and 20, for different azimuthal wavenumbers: (a) $m=0$, (b) $m=2$, (c) $m=3$, (d) $m=4$.

Unstable stationary modes and oscillating modes with different $m$ are observed in figures 10 and 11. In order to understand the structures of these unstable modes, spatial distributions of the streamwise perturbation velocity $u_x$ at $Re=400$ are shown in figure 12. For the stationary mode with $m=1$, the distribution of $u_x$ is predominantly located within the separation bubble when $N=0$, while it extends downstream as $N$ increases to 1. Furthermore, the intensity of $u_x$ is reduced within the separation bubble when $N=5$ and 20. The distribution of $u_x$ is significantly affected by the strength of the magnetic field, as it shifts from the recirculation to the layer between the recirculation and the outflow (hereafter called the shear layer) with increasing $N$. For the stationary mode with $m=2$ under a strong magnetic field, the distribution of $u_x$ exhibits a shorter extent downstream compared with the case with $m=1$. For the oscillating mode in the absence of a magnetic field, the distribution shows a spatially periodic downstream structure for both $m=1$ and 2. It is noted that the distributions of $u_x$ for the stationary and oscillating modes with weak magnetic fields resemble those presented by Natarajan & Acrivos (Reference Natarajan and Acrivos1993) and Meliga et al. (Reference Meliga, Chomaz and Sipp2009) at the thresholds of regular and Hopf bifurcations, respectively.

Figure 12. Spatial distribution of the streamwise perturbation velocity $u_x$ of unstable modes observed in figures 10(d) and 11(b). The black line is the outline of the separation bubble. (a) Stationary mode at $Re=400$, $N=0$, $m=1$. (b) Oscillating mode at $Re=400$, $N=0$, $m=1$. (c) Stationary mode at $Re=400$, $N=1$, $m=1$. (d) Stationary mode at $Re=400$, $N=5$, $m=1$. (e) Stationary mode at $Re=400$, $N=20$, $m=1$. ( f) Oscillating mode at $Re=400$, $N=0$, $m=2$. (g) Stationary mode at $Re=400$, $N=20$, $m=2$.

In addition to the unstable modes, a series of eigenvalues with growth rates less than zero are also observed in figures 10 and 11. By focusing on eigenvalues where $\lambda _r$ is close to but less than zero, it is possible to determine whether the eigenvalues continuously shift to an unstable state ($\lambda _r > 0$) by changing parameters $Re$ and $N$. The distributions of $u_x$ of the global modes for the second largest growth rate, which is next to the most unstable mode (the largest growth rate), are examined. The cases $(Re=400,m=1,N=1)$, $(Re=400,m=1,N=5)$, $(Re=400,m=1,N=20)$ and $(Re=400,m=2,N=20)$ are analysed, and the results reveal that the distribution at the rear of the sphere is similar with its most unstable mode. But its amplitude is continuously amplified downstream, as it is far away from the sphere. Such an unphysical spatial structure for the second largest growth rate makes us believe that this eigenvalue is spurious. Natarajan & Acrivos (Reference Natarajan and Acrivos1993) also reported these interior eigenvalues, which were considered spurious and did not have a physical implication. Hence, the eigenvalues associated with unstable modes are separated from these spurious eigenvalues. Furthermore, Garnaud et al. (Reference Garnaud, Lesshafft, Schmid and Huerre2013) reported that the presence of a spurious mode was affected by the domain size, outflow boundary condition and finite precision of the computer arithmetic.

To investigate the detailed effect of the magnetic field, figure 13 shows the overall variations of the growth rate for different unstable global modes as a function of the interaction number $N$. The position of the local maximum growth rate is represented by a hollow circle. The behaviour of the growth rate reflects the influence of the magnetic field on the evolution of perturbations, where the magnetic field can either damp or promote perturbation growth depending on whether the growth rate decreases or increases with increasing $N$. Figure 13(a) provides the results of stationary modes with $m=1$ at $Re=150$, 200, 300 and 400. When the magnetic field strength is weak, it promotes perturbation growth if $Re< Re_c^I$, while it dampens perturbation growth if $Re>Re_c^I$. As the magnetic field strength varies from moderate to strong (e.g. in the moderate range $N\in (4, 25)$ and strong range $N\in (25, 40)$ for $Re=300$), the influence of the magnetic field on perturbation growth transitions from promotion to damping. It is clearly shown that the variation of growth rate is more sensitive to a weak magnetic field compared with a strong one. The overall instability/stability of the flow is determined by the positive/negative sign of the growth rate of the global mode. Thus, inspecting the growth rate can predict the instability threshold. In the range $N\in (0, 40)$ the flow is always stable for $Re=150$, while it is always unstable for $Re=400$. The states of flow instability/stability for $Re=200$ and $Re=300$ exhibit one ($N\approx 14$) and two thresholds ($N\approx 0.6, N\approx 4$), respectively. It is interesting to note that with increasing $N$ at $Re=300$, the growth rate initially decreases from positive to negative and then returns to positive. The magnetic field shows a stabilising effect at weak strengths and a destabilising effect at strong strengths. Such a stabilising–destabilising effect of the magnetic field is a research topic of interest.

Figure 13. Variation of the growth rate of the unstable global mode with the interaction number $N$. The hollow circle represents the position of the local maximum growth rate. (a) Stationary modes at different Reynolds numbers and $m=1$. (b) Oscillating modes at $Re=300$, $Re=400$ and $m=1$. (c) Stationary modes with different azimuthal wavenumbers at $Re=400$.

Figure 13(b) compares the growth rates of the oscillating mode with those of the stationary mode at the same azimuthal wavenumber $m=1$. The growth rate of the oscillating mode decreases more rapidly with $N$ and reaches a negative value at a smaller $N$, which means that the weak magnetic field has a stronger suppressing effect on the oscillating mode than the stationary mode. The result also infers that vortex shedding will be suppressed at a relatively weak magnetic field. This finding is consistent with the experimental results in Yonas (Reference Yonas1967), whose work reported that a relatively weak magnetic field was able to completely suppress the dominant frequencies. Figure 13(c) plots the variation of the growth rate of the stationary mode with $N$ at different azimuthal wavenumbers $m=0$, 1, 2, 3, 4 and $Re=400$. The growth rates of $m=0$ and 4 are always less than zero, while the growth rates of $m=2$ and 3 increase from negative values to positive values at threshold values of $N\approx 7$ and $22$, respectively. It indicates that the strong magnetic field can induce instability of the higher $m$th mode with larger critical values of $N$ at $Re=400$.

So far, five unstable modes are observed, which consist of stationary modes with $m=1,2,3$ and oscillating modes with $m=1,2$. Their critical curves are obtained by computing the critical Reynolds number $Re_c$ corresponding to different $N$ in figure 14(a). Since the Hartmann number $Ha$ is also an important dimensionless parameter, the critical curves in the $\{Re, Ha\}$ plane are also given in figure 14(b). The critical curve for the first regular bifurcation caused by the stationary unstable mode of $m=1$ is marked by the wine red dotted line. Other critical curves associated with stationary unstable modes of $m=2$ and $m=3$, oscillating unstable modes of $m=1$ and $m=2$ are denoted by green, purple, orange and blue dotted lines, respectively. For the first regular bifurcation, the critical Reynolds number experiences a rapid increase from 212.7 to 386.3, then decreases to 179.1 and slightly increases again to 179.2 as $N$ increases from 0 to 40. There are two inflection points, one is $Re=386.3$, $N=1.8$ marked by the $P_2$ green solid circle and the other is $Re=179.1$, $N=38$ marked by the red solid circle in figure 14(a). These two points are referred to as balance points in the subsequent section. According to the change of the critical Reynolds number, the critical curve clearly demonstrates multiple effects of the magnetic field on the unstable stationary mode with $m=1$, including the stabilising effect of a weak magnetic field ($N<1.8$), the destabilising effect of a strong magnetic field ($1.8< N<38$) and the re-stabilising effect of a much stronger magnetic field ($N>38$).

Figure 14. Five critical curves in the parameter plane with ${\unicode{x2460}}$ the regular bifurcation of $m=1$, ${\unicode{x2461}}$ the Hopf bifurcation of $m=1$, ${\unicode{x2462}}$ the regular bifurcation of $m=2$, ${\unicode{x2463}}$ the regular bifurcation of $m=3$ and ${\unicode{x2464}}$ the Hopf bifurcation of $m=2$. There are two inflection points on the critical curve of the regular bifurcation with $m=1$, which are $Re=386.3$, $N=1.8/Ha=26.4$ and $Re=179.1$, $N=38/Ha=82.5$, respectively.(a) The $\{Re, N\}$ plane with four critical points $P_1$ ($Re=271.1, N=0.4$), $P_2$ ($Re=386.3, N=1.8$), $P_3$ ($Re=216.2, N=10$) and $P_4$ ($Re=180.1, N=30$); (b) the $\{Re, Ha\}$ plane.

For the regular bifurcations resulting from stationary unstable modes with $m=2$ and $m=3$, $Re_c$ of the former one decreases from 400 to 256.6 as $N$ increases from 6.63 to 40, while $Re_c$ of the latter one decreases from 400 to 367.1 as $N$ increases from 22.74 to 40. These results indicate that the strong magnetic field has a destabilising effect on both of these stationary modes. In contrast, for the Hopf bifurcations resulting from oscillating unstable modes with $m=1$ and $m=2$, $Re_c$ of the former one increases from 280.8 to 396 as $N$ increases from 0 to 0.28, while $Re_c$ of the latter one increases from 396.2 to 400 as $N$ increases from 0 to 0.008. These findings suggest that the weak magnetic field exhibits a stabilising effect on these oscillating modes. The variation in the critical Reynolds number as $N$ increases indicates that the critical Reynolds number is particularly sensitive to weak magnetic fields. This implies that the flow instability can be more easily altered in an environment with a weak magnetic field.

These five critical curves divide the $\{Re,N\}$ plane or the $\{Re,Ha\}$ plane into six regimes in figure 14, which describes the transition scenario in the wake of the sphere intuitively by LSA. The wake structures in regimes $I, II_1, II_2$ and $III$ have been described in detail in our previous DNS study by Pan et al. (Reference Pan, Zhang and Ni2018). Pan et al. (Reference Pan, Zhang and Ni2018) studied wake structures affected by the magnetic field and provided the map of regimes for wake patterns in the $\{Re,N\}$ plane in the range of $Re\leq 300$ and $N\leq 10$, which also revealed the first regular bifurcation and the second Hopf bifurcation. But their critical curve of the first regular bifurcation is discontinuous between the weak and strong magnetic fields, and the re-stabilising effect of a much stronger magnetic field is not reported. The present study concerns the flow instability and expends the parameter range of the Reynolds number up to 400 and the interaction number up to 40. Complete critical curves of the first regular and second Hopf bifurcations are obtained and other regular bifurcations resulting from unstable stationary modes with $m=2$ and $m=3$ are also reported.

Ghidersa & Dušek (Reference Ghidersa and Dušek2000) and Pier (Reference Pier2008) have previously discussed the spatial structure of the $m$th mode, which consists of $m$ pairs of counter-rotating vortices resulting from an azimuthal perturbation velocity. It is well established that the steady axisymmetric wake undergoes a transition to a steady plane symmetric wake characterized by a pair of counter-rotating vortices at the first regular bifurcation, which is driven by the stationary unstable mode with $m=1$. Subsequently, the wake transitions to a plane symmetric wake with periodic vortex shedding at the second Hopf bifurcation, caused by the oscillating unstable mode with $m=1$. This indicates that the wake after bifurcation has the same structural characteristics as the unstable mode with $m=1$. Since the structural characteristic of the $m$th mode can provide insight into the wake structure, unstable stationary modes with $m=2$ and $m=3$ may predict a new wake structure with two and three pairs of counter-rotating vortices in regimes $IV$ and $V$, respectively. However, it should be noted that the present LSA is based on an assumption of axisymmetric base flow and does not consider the interaction between different unstable modes. Therefore, the predicted wake structures in regimes $IV$ and $V$ obtained through LSA may differ from those observed in DNS. To confirm the wake structures in regimes $IV$ and $V$ and to examine the wake structures in other regimes, the DNS method in Pan et al. (Reference Pan, Zhang and Ni2018) is adopted here.

Figure 15 illustrates five representative wake patterns at the rear of the sphere at $Re=300$ and 400 with different interaction parameters. Dye lines in experiment (Johnson & Patel Reference Johnson and Patel1999) revealed shedding of large-amplitude hairpin vortices at $Re=300$ without a magnetic field, for which the wake kept a plane symmetric state. Under the influence of the magnetic field at $N=0.01$ in figure 15(a), the plane symmetric shedding vortex is preserved, its amplitude of hairpin vortices is damped by the magnetic field as reported by Pan et al. (Reference Pan, Zhang and Ni2018). With increasing magnetic field, vortex shedding is completely suppressed and the wake structure transitions to a steady plane symmetric state. As shown in figure 15(b) at $N=0.4$, a tilted recirculation exists behind the sphere with a double-threaded wake consisting of two steady streamwise vortices, which are opposite in sign with unequal strength and shape. By further increasing the magnetic field at $N=1$ (figure 15c), the wake structure transitions to a steady axisymmetric state with an attached separation bubble. The wake patterns shown in figures 15(a), 15(b) and 15(c) correspond to the wake structures in regimes $III$, $II_1$ and $I$, respectively.

Figure 15. Five wake patterns behind the sphere at $Re=300$ and 400 with different interaction parameters.(a) Unsteady plane symmetric wake with vortex sheddings at $Re=300$, $N=0.01$. Isosurfaces of the streamwise vorticity with $\omega _z \pm 0.3$. (b) Steady plane symmetric wake with a tilt recirculation at $Re=300$, $N=0.4$. Pressure contours and streamlines in the symmetry plane. (c) Steady axisymmetric wake with an attached separation bubble at $Re=300$, $N=1$. (d) Steady plane symmetric wake without a recirculation at $Re=400$, $N=5$. Upper part: pressure contours and streamlines in the symmetry plane. Lower part: skin friction lines on the sphere surface viewed from the rear of the sphere. (e) Steady double-plane symmetric wake without a recirculation at $Re=400$, $N=20$.

For strong magnetic fields at $Re=400$ with $N=5$ and 20, the wake structure within the recirculation is dismissed and the flow pattern shows a steady plane symmetric state or a steady double-plane symmetric state. Lighthill (Reference Lighthill1963) noted that convergence of skin friction lines was a criterion of three-dimensional flow separation, which helps us analyse the wake structure by showing flow traces. Skin friction lines in figures 15(d) and 15(e) clearly exhibit the single or double-plane symmetric characteristic, which corresponds to the structures of regimes $II_2$ and $IV$, respectively. This indicates that the $m=1$ and $m=2$ unstable modes in LSA can predict $m$ pairs of counter-rotating vortical structures, as confirmed by DNS. However, the results in figures 15(b) and 15(d) indicate that a pair of counter-rotating vortical structures associated with the $m=1$ mode actually has two different wake patterns, such as the steady plane symmetric wake with a tilt recirculation in pattern $II_1$ at weak magnetic fields and the steady plane symmetric wake without a recirculation in $II_2$ at strong magnetic fields. Furthermore, DNS are conducted for other cases with different values of $N$ at $Re=400$ to examine the appearance of a double-plane symmetric wake and the existence of a three-plane symmetric wake. The results demonstrate that the plane symmetric wake exists for the cases with $N=5$, 10 and 15, while the double-plane symmetric wake exists for the cases with $N=20$, 30 and 40. It is found that the threshold for the appearance of the double-plane symmetric wake is between $N=15$ and $N=20$ predicted by DNS, which differs from the appearance of the $m=2$ unstable mode at $N=6.63$ predicted by LSA. Moreover, the three-plane symmetric wake with three pairs of counter-rotating vortical structures, as predicted by the $m=3$ unstable mode, is not observed by DNS in the current parameter range.

In LSA, when the base flow is axisymmetric and stationary, a generic perturbation can be decomposed into Fourier modes with azimuthal wavenumber $m$, where linear stability equations are decoupled for each value of mode $m$. For a given $m$ mode, its asymptotic behaviour of the base flow to three-dimensional small perturbations is entirely dictated by the leading mode or the mode with the fastest global growth rate. Furthermore, DNS directly solves the nonlinear Navier–Stokes equations. The DNS result consists of various asymptotic states of $m$ modes in the linear Navier–Stokes equations, as well as the interactions among them. Hence, the DNS result may correspond to a specific asymptotic state from the LSA, which is determined by the competition of leading modes among different $m$ modes. Szaltys et al. (Reference Szaltys, Chrust, Przadka, Goujon-Durand, Tuckerman and Wesfreid2012) performed the modal decomposition of the streamwise vorticity at $Re=300$ obtained by experiments for the flow past a sphere. Axial vorticity of modes $m=1$, $m=2$ and $m=3$ were observed and the $m=1$ mode was dominant, which indicates a pair of counter-rotating streamwise vortical structure. As shown in figure 13(c), with a strong magnetic field, $\lambda _{r,m=1} > \lambda _{r,m=2} > 0$, the leading mode at the $m=1$ mode is dominant and its asymptotic state predicts a pair of counter-rotating vortical structures behind the sphere, which is consistent with the DNS result in figure 15(d). With a much stronger magnetic field, $\lambda _{r,m=2} > \lambda _{r,m=1} > \lambda _{r,m=3}$, the largest leading mode $\lambda _{r,m=2}$ predicts two pairs of counter-rotating vortical structures behind the sphere, which now controls the flow pattern in DNS, as shown in figure 15(e).

3.2.2. Energy budget analysis

To provide further understanding of the instability mechanism, a balance analysis of the perturbation kinetic energy budget at the critical thresholds is carried out, which is a classical approach in LSA (Pralits, Brandt & Giannetti Reference Pralits, Brandt and Giannetti2010). This analysis involved multiplying the complex conjugate of the perturbation velocity ${\boldsymbol {u}}^*$ with the linear stability equations and integrating in the whole computational domain $\varOmega$. Since the domain is sufficiently large to neglect disturbances at the boundaries, divergence terms in this equation give little contribution and so have been dropped. After performing simplifications, the equation of perturbation kinetic energy can be expressed as

(3.1)\begin{gather} \frac{{\rm d}}{{\rm d}t} \int_\varOmega \frac{1}{2} {\boldsymbol{u}}^* \boldsymbol{\cdot} {\boldsymbol{u}} \,{\rm d}\varOmega ={-} \underbrace{\int_\varOmega {\boldsymbol{u}}^* \boldsymbol{\cdot} (\boldsymbol{\nabla} {\boldsymbol{U}}_0 \boldsymbol{\cdot} {\boldsymbol{u}}) \,{\rm d} \varOmega}_{E_p} - \underbrace{\frac{1}{Re} \int_\varOmega \boldsymbol{\omega}^* \boldsymbol{\cdot} \boldsymbol{\omega} \,{\rm d} \varOmega}_{E_v} - \underbrace{N \int_\varOmega {\boldsymbol{j}}^* \boldsymbol{\cdot}{\boldsymbol{j}} \,{\rm d} \varOmega}_{E_j} , \end{gather}

where $\boldsymbol {\omega }=\boldsymbol {\nabla } \times {\boldsymbol {u}}$ and ${\boldsymbol {j}}=-\boldsymbol {\nabla } \phi + {\boldsymbol {u}} \times {\boldsymbol {e}}_x$ represent the perturbation vorticity and electric current, respectively. The first term on the right-hand side is the production of the perturbation kinetic energy, denoted as $E_p$, which accounts for the energy transferring from the base flow to the disturbance. This term includes both base-flow shear and strain conversion, the shear of $\partial U_{0x}/\partial r$, namely the gradient of the axial velocity along the radial direction, has the most significant contribution among them in $E_p$. The remaining two terms are the viscous dissipation $E_v$ and Joule dissipation $E_j$ of the perturbation kinetic energy, respectively. These terms have negative values and serve to dampen the growth of disturbances, thereby exerting stabilising effects on the perturbation kinetic energy.

To gain a visual understanding of the impact of different terms on the perturbation kinetic energy equation, the energy production and Joule dissipation terms are normalized by the viscous dissipation term. Figure 14(a) reveals four critical points in the $Re$–$N$ plane for $Re=300$. Table 3 presents the kinetic energy budgets at these critical points, considering the production of perturbation, viscous force and Lorentz force. It is evident that in all cases the production of the perturbation kinetic energy term is balanced with the viscous dissipation term and the Joule dissipation term. The viscous dissipation term is the dominant stabilising term. A similar conclusion is also found in a MHD duct flow by Hu (Reference Hu2021). It is noted that the proportion of $E_j$ at $N=0.053$ is much larger than that at $N=0.61$, which implies that the magnetic field has a greater stabilising effect on the oscillating mode of $m=1$ than the stationary one of $m=1$.

Table 3. Kinetic energy budgets by production of perturbation, viscous force and Lorentz force at the critical points of four unstable modes at $Re=300$.

According to (3.1), the perturbation Joule dissipation effect is related to the perturbation electric current ${\boldsymbol {j}}$. Figure 16 illustrates the spatial distribution of the axial perturbation electric current at critical points of $Re=300$, which reveals a periodic distribution in the wake for the oscillation mode, whereas the stationary mode exhibits a primarily localized distribution around the sphere. The periodic distribution in the oscillatory mode results in a higher proportion of $E_j$ compared with the stationary mode. Consequently, the decrease of growth rate of the oscillatory mode affected by the magnetic field reaches a negative value more rapidly, as demonstrated in figure 13(b). Hence, as the magnetic field increases for $Re>Re^{II}_c$, a transition occurs from an unsteady vortex shedding wake to a steady plane symmetric wake at the Hopf bifurcation of $m=1$. It is noted that the hairpin structures being shed at $Re=300$ without a magnetic field are coherent both in time and space, which leads to a single dominant vortex shedding frequency. However, when $Re>350$, the imbalance in supply, storage and emission of energy of hairpin structures will cause vortex dislocations, which breaks the plane symmetric state and shows other vortex shedding frequencies besides the dominant one (Sakamoto & Haniu Reference Sakamoto and Haniu1990). These other frequencies are suppressed by the smaller magnetic field and transitions to a single vortex shedding frequency, which correspond to the Hopf bifurcation of $m=2$, as depicted in the forth and fifth critical curves of figure 14(a). With further increases in $N$, the perturbation Joule dissipation term has an increasingly significant role in the kinetic energy budget by exerting magnetic damping effects. However, the effect of the magnetic field on flow instability changes from stable to unstable. Further investigations concerning the destabilising effect of the magnetic field are given in the next sensitivity analysis section.

Figure 16. Spatial distribution of the axial perturbation electric current at critical points for $Re=300$ with (a) $N=0.053$, (b) $N=0.61$, (c) $N=4.29$, (d) $=14.4$.

3.3.1. Sensitivity to base-flow modifications

The stability problem resulting from base-flow modifications is considered first. The sensitivity to base-flow modifications $\boldsymbol {\nabla }_{{\boldsymbol {U}}_0} \lambda$ defined in (2.34) is computed for critical points $P_1, P_2, P_3, P_4$ on the first critical curve in figure 14. This sensitivity includes two parts, the real part $\boldsymbol {\nabla }_{{\boldsymbol {U}}_0} \lambda _r$ and the imaginary part $\boldsymbol {\nabla }_{{\boldsymbol {U}}_0} \lambda _i$, which correspond to the growth rate sensitivity and the frequency sensitivity, respectively. Since the present stability problem for these critical points is steady and two-dimensional axisymmetric, only the growth rate sensitivity is considered. The growth rate sensitivity to radial and streamwise base-flow modifications is two-dimensional real vector fields and it is represented by streamlines to convey the local orientation of the sensitivity field, while contours indicate its magnitude in figure 17. Far from the sphere, the sensitivity decays to zero due to the spatial separation of the direct and adjoint global modes (Marquet et al. Reference Marquet, Sipp and Jacquin2008). The maximum magnitude of the sensitivity is located within the recirculation region and near the axis of symmetry for $P_1$, while it exists in both the recirculation region and the shear layer for $P_2$, and solely at the shear layer for $P_3$, $P_4$. This sensitivity analysis identifies specific flow regions where the growth rate of the global mode is most sensitive to base-flow modifications. Figure 17 reveals that such a region transfers from the recirculation to the shear layer with increasing magnetic field. It is worth mentioning that according to the first critical curve in figure 14, the effect of the magnetic field on flow instability undergoes a transition from stabilisation to destabilisation.

Figure 17. Growth rate sensitivity to base-flow modifications $\boldsymbol {\nabla }_{{\boldsymbol {U}}_0} \lambda _r$ of the leading eigenvalue for critical points on the first critical curve: (a) $P_1: Re=271.1, N=0.4$; (b) $P_2: Re=386.3, N=1.8$; (c) ${P_3: Re=216.2}, N=10$; (d) $P_4: Re=180.1, N=30$. The dashed line denotes the separation bubble. The solid streamlines with arrows represent the orientation of the sensitivity field.

Marquet et al. (Reference Marquet, Sipp and Jacquin2008) considered uncertain base-flow modifications $\delta {\boldsymbol {U}}_0$. However, in the present research a small-amplitude modification of the Lorentz force in the flow induces specific base-flow modifications. Such base-flow modifications are determined by increasing the magnetic field strength through a variation of $\delta N=0.01$ for critical points, which are calculated with (2.17)–(2.19), as illustrated in figure 18. In these four situations, the upstream flow of the sphere is slowed down by the magnetic field, which leads to the formation of a high-pressure stagnation zone in front of the sphere, as shown in figure 9. Focusing on the head part (separation line) and the tail part (reattachment point) of the separation bubble at the rear of the sphere, it can be observed that as the magnetic field increases, base-flow modifications $\delta {\boldsymbol {U}}_0$ make the flow accelerate in the entire recirculation region for point $P_1$. Since the flow within the separation bubble follows a clockwise vortex pattern, the separation line will move downstream, while the reattachment point will move upstream. As a result, the separation angle of the separation bubble increases, and the recirculation length of the separation bubble decreases. For point $P_2$, the separation angle continues to increase, but the base-flow modifications $\delta {\boldsymbol {U}}_0$ start to change direction at the tail part of the separation bubble, which results in flow deceleration. Consequently, the reattachment point moves downstream, which increases the recirculation length. In the case of points $P_3$ and $P_4$, the base-flow modifications $\delta {\boldsymbol {U}}_0$ create a counterclockwise vortex within the separation bubble, which reduce the shear strength in the recirculation region due to its counter-rotating nature with $\delta {\boldsymbol {U}}_0$. Additionally, these modifications decelerate the flow near the outline of the separation bubble. Hence, the separation line moves upstream with decreasing the separation angle, while the reattachment point moves downstream with increasing the recirculation length. In summary, an increase in magnetic field strength causes the recirculation length to decrease under weak magnetic fields, while it increases under strong magnetic fields. Conversely, the separation angle behaves in the opposite manner. The base-flow modifications induced by increasing the magnetic field strength through a variation $\delta N=0.01$ agree with results shown in figures 6(a) and 6(b).

Figure 18. Base-flow modifications $\delta {\boldsymbol {U}}_0$ induced by increasing the magnetic field strength through a variation $\delta N=0.01$ for critical points: (a) $P_1$, (b) $P_2$, (c) $P_3$, (d) $P_4$. Their orientations are represented by the solid streamlines with arrows. The red dashed line is the separation bubble.

The contribution of the specific base-flow modifications $\delta {\boldsymbol {U}}_0$ induced by increasing magnetic field strength to the global growth rate variation $\delta \lambda _r$ is analysed. Recall that $\delta \lambda _r$ is obtained by integrating the integrand $(\boldsymbol {\nabla }_{{\boldsymbol {U}}_0} \lambda _r)^\ast \boldsymbol {\cdot } \delta {\boldsymbol {U}}_0$ over the entire space. This integrand helps identify regions responsible for the stabilisation or destabilisation of the global mode. For example, when the base-flow modifications $\delta {\boldsymbol {U}}_0$ in figure 18 are oriented in the same direction as the growth rate sensitivity $\boldsymbol {\nabla }_{{\boldsymbol {U}}_0} \lambda _r$ in figure 17, the integrand is positive, which indicates flow destabilisation. Figure 19 depicts the spatial distribution of this integrand, where positive/negative values at local positions indicate contributions of local base-flow modifications $\delta {\boldsymbol {U}}_0$ to the destabilisation/stabilisation of the global mode. Three regions with dominant contributions can be distinguished in figure 19: the region inside the separation bubble mainly contributing to the stabilisation of the global mode in figures 19(a) and 19(b), the region near the tail of the separation bubble mainly contributing to the destabilisation in figure 19(c), and the region at the shear layer mainly contributing to the destabilisation in figure 19(d). These regions demonstrate the complex effect of increasing the magnetic field strength on the variation of the growth rate $\delta \lambda _r$, and also suggest that the stabilising/destabilising effect of the magnetic field can not be captured solely through local stability/destability analysis. Their dominant contributions to the global growth rate variation are reflected in the integrated values over the entire space, which are given in table 4. Accordingly, the region inside the separation bubble is identified as responsible for the stabilising effect under weak magnetic fields, while the base-flow modifications in this region reduce the size of the separation bubble. The regions near the tail and at the shear layer of the separation bubble are responsible for the destabilising effect under strong magnetic fields, while the base-flow modifications in these regions increase the length of the recirculation region. As noted in Marquet et al. (Reference Marquet, Sipp and Jacquin2008), the sensitivity to the Lorentz force is closely related to the sensitivity to the base-flow modification. The latter one can be seen as a specific base-flow modification induced by the Lorentz force. These two sensitivity analyses examine the flow instability from different perspectives. Hence, the effect of the magnetic field will be continuously discussed in the Lorentz force sensitivity analysis. The sensitivity to base-flow modifications is coupled with the sensitivity to the Lorentz force method to help explain how the magnetic field affects flow instability.

Figure 19. Spatial distribution of the growth rate variation $(\boldsymbol {\nabla }_{{\boldsymbol {U}}_0} \lambda _r)^\ast \boldsymbol {\cdot } \delta {\boldsymbol {U}}_0$ induced by increasing the magnetic field strength through a variation $\delta N=0.01$ for critical points: (a) $P_1$, (b) $P_2$, (c) $P_3$, (d) $P_4$. The dashed line is the separation bubble.

Table 4. Variation of the global growth rate computed by $\delta \lambda _r=\langle \boldsymbol {\nabla }_{{\boldsymbol {U}}_0} \lambda _r, \delta {\boldsymbol {U}}_0 \rangle$ induced by increasing the magnetic field strength through a variation $\delta N=0.01$ for different critical points.

3.3.2. Sensitivity to Lorentz force

To determine the influence of the magnetic field on flow instability characteristics, the sensitivity function of the eigenvalue to Lorentz force $\boldsymbol {\nabla }_{\boldsymbol {F}} \lambda _r$ defined in (2.33) is considered. This sensitivity function helps identify the significant regions in the flow where changes in the Lorentz force have the greatest impact on the global eigenvalue. Similarly to the sensitivity to base-flow modifications, only growth rate sensitivity analysis is performed here for the critical points $P_1, P_2, P_3, P_4$. As a vector function, its streamlines represented by a black line with arrows are displayed in figure 20, which provides the local orientation of the sensitivity function field. For weak magnetic fields, a closed streamline structure is predominantly located in the recirculation region behind the sphere. As the magnetic field strength increases, the streamline structure elongates in the direction of the magnetic field and moves towards the shear layer region.

Figure 20. Spatial distribution of the growth rate variation $(\boldsymbol {\nabla }_{\boldsymbol {F}} \lambda _r)^\ast \boldsymbol {\cdot } \delta {\boldsymbol {F}}$ induced by increasing the magnetic field strength through a variation $\delta N=0.01$ for critical points $P_1, P_2, P_3, P_4$ in the first critical curve. The solid black lines with arrows represent the streamlines of $\boldsymbol {\nabla }_{\boldsymbol {F}} \lambda _r$. The red dashed line represents the outline of the separation bubble. Results are shown for (a) $P1$, (b) $P2$, (c) $P3$, (d) $P4$.

When considering a variation in the Lorentz force induced by changing the magnetic field strength, it can be represented as $\delta {\boldsymbol {F}} =\delta N (-\boldsymbol {\nabla } \varPhi _0+ {\boldsymbol {U}}_0 \times {\boldsymbol {e}}_x) \times {\boldsymbol {e}}_x=-\delta N U_{0r} {\boldsymbol {e}}_r$, where $\delta N$ is set to 0.01 in the present study and $U_{0r}$ is the radial velocity of the base flow that is mainly distributed around the sphere. Similarly to the sensitivity to base-flow modifications, the integrand $(\boldsymbol {\nabla }_{{\boldsymbol {F}}} \lambda _r)^\ast \boldsymbol {\cdot } \delta {\boldsymbol {F}}$ indicates a local variation in the growth rate associated with the variation in the Lorentz force, the spatial distribution of which is displayed in figure 20. The red region with positive values contributes to an increase in the growth rate, which is referred to as the local destabilising region. Conversely, the blue region with negative values contributes to a decrease in the growth rate, which is referred to as the local stabilising region. For a variation in the magnetic field strength, the stability of the global mode in these sensitivity regions will be affected most significantly.

The variation of the growth rate $\delta \lambda _r=\langle \boldsymbol {\nabla }_{\boldsymbol {F}} \lambda _r,\delta {\boldsymbol {F}} \rangle$ is determined by the competition between local stabilising and destabilising effects in the entire space. The results are presented in table 5. It is noted that $\delta \lambda _r$ reflects the global stabilising or destabilising effect of a small variation in the magnetic field strength for a selected point on the first critical curve. When $N<1.8$, $\delta \lambda _r$ is negative. The contribution from the local stabilising effect in the blue regions is dominant, which mainly locates inside the recirculation region behind the sphere and the tail of the separation bubble in figure 20(a). Therefore, the magnetic field exhibits a stabilising effect on the flow. On the other hand, when $N > 1.8$, $\delta \lambda _r$ is positive. The contribution from the local destabilising effect in the red regions is dominant, which locates at the shear layer region in figures 20(c) and 20(d). The magnetic field now demonstrates a destabilising effect. These conclusions are consistent with analyses of sensitivity to base-flow modifications. Furthermore, since $\delta {\boldsymbol {F}}$ only has a radial component, the streamwise component of the sensitivity to Lorentz force $\boldsymbol {\nabla }_{\boldsymbol {F}} \lambda _r$ does not contribute to the fluid instability. For a strong magnetic field, the closed streamline structure of $\boldsymbol {\nabla }_{\boldsymbol {F}} \lambda _r$ is stretched nearly along the streamwise direction. This indicates that the change in the Lorentz force has little impact on the flow instability under strong magnetic fields. In other words, the critical Reynolds number is not sensitive to strong magnetic fields as $N$ increases, as illustrated by the first critical curve in figure 14 with $N>30$. Additionally, when comparing the magnitude of legends between figures 20(c) and 20(d), it is noteworthy that the dominance of the destabilising effect reduces at a strong magnetic field.

Table 5. Variation of the growth rate computed by $\delta \lambda _r=\langle \boldsymbol {\nabla }_{\boldsymbol {F}} \lambda _r, \delta {\boldsymbol {F}} \rangle$ induced by increasing the magnetic field strength through a variation $\delta N=0.01$ for different critical points.

3.4.1. Weak magnetic field situations

The dominant stabilising regions identified by present sensitivity analyses are located inside or at the tail of the separation bubble for weak magnetic fields. It supports the view that the stabilising effect of a weak magnetic field corresponds to a reduction in the separation bubble. A similar conclusion was reported by Boujo & Gallaire (Reference Boujo and Gallaire2014) in a study of a two-dimensional steady flow past a circular cylinder case using linear sensitivity analysis. They found that zones where the control configuration, such as a small cylinder, reduced the recirculation length corresponded to the regions where they had a stabilising effect on the most unstable global mode associated with vortex shedding at moderate Reynolds numbers. Since the application of a weak magnetic field changes the flow field towards its hydrodynamics state with a lower value of $Re$, a macroscopical analogy is given to understand the influence of a weak magnetic field effect on the flow instability. The viscous dissipation and Joule dissipation terms in the perturbation kinetic energy equation (3.1) show same stabilising influence on the flow. Furthermore, the Lorentz force in the momentum equation can be reformulated as a stress form ${\boldsymbol {J}}\times {\boldsymbol {B}} = \boldsymbol {\nabla }\boldsymbol {\cdot } \boldsymbol {\tau }$, where Maxwell stress is $\tau _{ij}=(B_iB_j/\mu )-{\boldsymbol {B}}^2/2\mu \delta _{ij}$. Here, $\mu$ is the fluid magnetic permeability. The role of the Maxwell stress is equivalent to viscous stress, which shows a magnetic damping effect. Hence, the action of the Lorentz force on the flow can be seen as an effective viscosity. Increasing the magnetic field is equivalent to introducing an effective viscosity into its corresponding hydrodynamics situation, which is equivalent to decreasing the effective Reynolds number. For example, increasing $N$ at $Re=300$ is equivalent to a decreasing effective Reynolds number less than $300$ with $N=0$, for which the flow will experience the Hopf bifurcation and the regular bifurcation corresponding to a wake transition from an unsteady vortex shedding state to a steady plane symmetric state then to a steady axisymmetric state.

3.4.2. Strong magnetic field situations

As discussed in § 3.4.1, a constant magnetic field exhibits a damping effect on flow instability, which has been used in many metallurgical applications (Chandrasekhar Reference Chandrasekhar2013). However, the Lorentz force has an anisotropic effect along the magnetic field, especially at a strong magnetic field. When the magnetic field is strong, increasing its strength leads to an elongation of the separation bubble. Now, the influence of the magnetic field on the flow instability shows a destabilising effect. Although the contribution of Joule dissipation to the balance of the kinetic energy budget increases as $N$ increases, the magnetic field alters the structure of the base flow, which also affects the flow instability. The Lorentz force aligns the flow streamlines parallel to the magnetic field. Consequently, the azimuthal vorticity is confined within the shear layer region, as depicted in figure 8. Furthermore, the dominant destabilising regions identified by sensitivity analyses are also located in this region. Hence, the value of $\partial U_{0x}/\partial r$ in the shear layer region has the most significant contribution in the production of perturbation kinetic energy $E_p$, which results in a destabilising effect of magnetic field on the global mode. Since such an effect exists in the shear layer region, it is called a ‘shear destabilising effect’.

In order to explain the mechanism of wake transition $I \rightarrow II_2$ under strong magnetic fields, a schematic is presented in figure 21. Based on the sensitivity analysis in figure 20(c), the dominant destabilising region is located over the separation line of the attached separation bubble, where perturbation production results in a wake transition. Figure 21(a) shows the position for a maximal current perturbation in a front view. Its corresponding Lorentz force pulls the flow away from the axisymmetric plane. Under the influence of convection by upstream outflow, such a perturbation effect propagates along the outline of the recirculation, as depicted in figure 21(b). The distribution of perturbation along the outline of the separation bubble is consistent with the sensitivity to base-flow modifications in figure 19(c). It is noted that the recirculation is stretched by strong magnetic fields. The enclosed flow in the recirculation is weakened by the Lorentz force as demonstrated in figure 18(c), which is hard to maintain the stretched recirculation. Hence, beyond the critical values of $Re$ and $N$ at the first critical curve, recirculation 1 in figure 21(b) will be broken by the perturbation effect. Recirculation 1 is dismissed then the upstream fluid convects from the outer region of recirculation 2 because of inertia and meets with the outer upstream fluid at position A, as shown in figure 21(c). Both recirculations 1 and 2 initially exist in an axisymmetric state. Since recirculation 1 is dismissed, the enclosed flow in recirculation 2 can not be maintained. The outflow in figure 21(c) will reduce the size of recirculation 2 until it is also dismissed. Such a procedure can be traced by skin friction lines in figure 22, where the skin friction lines are adopted to analyse flow traces at the rear of the sphere. The movement of the stagnation point from the middle of the sphere surface to the B position in the upper part of figure 22 means that during the time evolution, recirculation 2 is reduced and then dismissed. At last, recirculations 1 and 2 are successively dismissed. As shown in the upper part of figure 22, the upstream fluid flows over the sphere near the B position, which is squeezed in the A–B symmetry plane and meets with the other side outer upstream fluid near the A position. Then the fluid will flow along the azimuthal direction to downstream.

Figure 21. Schematic for wake transitions at strong magnetic fields. (a) The generation of perturbation.(b) Propagation of the perturbation effect. (c) Wake transition from an axisymmetric state to a plane symmetric state. (d) Wake transition from an axisymmetric state to a double-plane symmetric state.

Figure 22. Time evolution of the skin friction lines on the sphere surface during the wake transition with strong magnetic fields. Results are shown for (a–d) $Re=400$, $N=5$; (e–h) $Re=400$, $N=20$.

Furthermore, for a much stronger magnetic field, it is harder to maintain the recirculation. As a consequence of the elimination of recirculation 1, recirculation 2 will be disrupted. The upstream outer flow on the C and D sides will feed the flow on the A and B sides, as illustrated in figure 21(d). Similarly, the skin friction lines in the lower part of figure 22 exhibit the transition process from a steady axisymmetric wake to a steady double-plane symmetric wake, which is predicted by the LSA from $I \rightarrow IV$ in figure 14.

3.4.3. Global view

A global view of wake transitions in the $Re$–$N$ plane of figure 14 can be given here. In the case of weak magnetic fields, the wake transition $III \rightarrow II_1 \rightarrow I$ experiences a Hopf bifurcation of $m=1$ and regular bifurcation of $m=1$. For strong magnetic fields, the wake transition $I \rightarrow II_2$ experiences a regular bifurcation of $m=1$. In the case of much stronger magnetic fields, the wake transition $I \rightarrow IV$ experiences a regular bifurcation of $m=2$. It is noted that with a much stronger magnetic field, there is a competition of leading mode among various unstable azimuthal modes, as shown in figure 13(c). The largest leading mode, $m=2$ mode in this case, will determine the DNS flow pattern. In the present DNS cases the magnetic field is imposed at the beginning and the flow field is axisymmetric at the initial stage, which is similar to the present axisymmetric base-flow assumption in LSA. Different intensities of the magnetic field correspond to distinct simulation cases. However, for a fixed simulation case, e.g. $Re=300$ in the critical curves of figure 14, an increasing magnetic field is considered in this case, it is found that the wake transition $III \rightarrow II_1 \rightarrow I \rightarrow II_2$ is continuous but it is not the case for wake transition $II_2 \nrightarrow IV$. The former wake transition relates to the $m=1$ mode, while the later one relates to the $m=2$ mode. These two modes will interact with each other and affect the final DNS wake structure. When increasing $N$, a wake transition $I \rightarrow II_2$ occurs. By further increasing $N$, the prior development of mode $II_2$ can change the base flow, which will in turn change its stability and prevent the growth of mode $IV$.

The stabilising effect and shear destabilising effect of the magnetic field always compete with each other. The former one is dominant for weak magnetic fields, while the latter one is dominant for strong magnetic fields. A balance point between these two effects occurs at $Re=386.3$, $N=1.8$ at the first critical curve. However, in the case of a much stronger magnetic field, the sensitivity analysis of the Lorentz force reveals that the shear destabilizing effect is no longer sensitive to the increase in magnetic field strength. Since the shear is confined to the shear layer region, changes in the magnetic field have minimal impact on the flow shear behind the sphere, which means that the production of perturbation kinetic energy $E_p$ is not sensitive to the magnetic field variation. On the other hand, the Joule dissipation of perturbation kinetic energy $E_j$ is proportional to $N$. Hence, the stabilising effect will again be dominant for much stronger magnetic fields. A new balance point between stabilising effect and shear destabilizing effect locates at $Re=179.1$, $N=38$ at the first critical curve. With further increases in the magnetic field, as the shear destabilising effect is not sensitive to the magnetic field variation, the stabilising effect is now dominant, which will lead to a damping behaviour on the growth rate of perturbation in figure 13(a) and a slight turning right of the first critical curve when $N>38$.