3.1.1. Film boiling without a MF

We begin the presentation of our results by analysing the film-boiling phenomenon without MF and compare with existing boiling mechanisms and empirical models. Figure 3 shows the temporal evolution of the space-averaged $Nu$. It can be seen that $Nu$ oscillates periodically around the correlation proposed by Klimenko (Reference Klimenko1981), given that $Nu_k=0.19(Gr\,Pr)^{{1}/{3}}\times (0.89Ja^{-{1}/{3}})$, this agreement indicates the credibility of the numerical simulation results. This periodic oscillation corresponds to the shedding of bubbles. The evolution of the liquid/vapour interface for the fifth bubble cycle (from $t=51.9$ to $t=63.0$), corresponding to the blue region in figure 3, is depicted in figure 4, while figure 5 presents the corresponding streamline and temperature field distributions within the vertical cross-section ($XOY$). It is noteworthy that, during the fifth bubble cycle, the bubble generated in the fourth bubble cycle has just detached, resulting in the presence of a rising bubble in figure 4(a–d). The film-boiling process within one bubble cycle can be divided into four stages. Stage one: from $t=51.9$ to $t=53.7$, the phase change causes the expansion of the vapour phase volume, and the bulge grows rapidly. Stage two: from $t=53.7$ to $t=55.6$, the Rayleigh–Taylor instability occurs, leading to the formation of a significant bulge and vortices. Stage three: from $t=55.6$ to $t=59.3$, the neck connecting the thin film and bubble starts to contract, leading to a mushroom-shaped bubble and the vortex structure. Stage four: from $t=59.3$ to $t=63.0$, the bubble pinches off, and a new bubble cycle initiates. Furthermore, within one bubble cycle, the vapour film thickness initially increases due to the expansion of the vapour phase volume and the growth of the bulge. Later, it decreases when the neck phenomenon occurs. Conversely, the Nusselt number decreases first and then increases within one bubble cycle, as explained by (2.21). The temperature profile reveals a linear distribution from the overheated wall to the liquid/vapour interface throughout the film boiling process. A smaller distance between the liquid/vapour interface and the overheated wall corresponds to a more significant temperature gradient, resulting in a higher $Nu$. In contrast, a larger distance leads to a smaller $Nu$. These findings align with previous studies (Juric & Tryggvason Reference Juric and Tryggvason1998; Zhang & Ni Reference Zhang and Ni2018). Next, we investigate the influence of a vertical MF on the film-boiling dynamics.

Figure 3.

Temporal evolution of the space-averaged $Nu$ for the single-mode film boiling without a MF. The dashed line represents the theoretical results of Klimenko (Reference Klimenko1981).

Figure 4.

Vapour bubble generation during the film-boiling flow without a MF at different dimensionless times, which correspond to the blue region in figure 3. Panels show (a) $t=51.9$, (b) $t=53.7$, (c) $t= 55.6$, (d) $t=57.4$, (e) $t=59.3$ and (f) $t=63.0$.

Figure 5.

Temperature profile (contours) and velocity streamlines (black arrow lines) in the vertical cross-section ($XOY$) of the computational domain. The time instants from panels (a–f) correspond to those displayed in figure 4(a–f).

3.1.2. Effect of the vertical MF

We examine the influence of the vertical MF applied along the $x$ direction on film-boiling flows. The horizontal directions perpendicular to the applied MF are represented by $y$ and $z$. Thus, it is essential to highlight that, under the vertical MF, the MHD effect is expected to be isotropic in any horizontal plane parallel to $YOZ$. Consequently, the results in two vertical cross-sections, namely the $XOY$ and $XOZ$ planes, are the same. Therefore, we only illustrate findings for the $XOY$ cross-section. We vary the MF intensities with ${B = 0.05\,\textrm {T}}$, ${B = 0.1\,\textrm {T}}$, ${B = 0.15\,\textrm {T}}$, ${B = 0.2\,\textrm {T}}$ and ${B = 0.4\,\textrm {T}}$. According to (2.23), the corresponding interaction numbers (Hartmann numbers) are $N=0.7$ ($Ha=1.9$), $N=2.8$ ($Ha=3.8$), $N=6.3$ ($Ha=5.7$), $N=11.1$ ($Ha=7.6$) and $N=44.5$ ($Ha=15.3$), respectively. The MF intensity choice considers a spectrum ranging from inertia being dominant to the Lorentz force being dominant.

To illustrate the MHD effect, we compare the liquid/vapour interface patterns within four bubble generation cycles, and the results are presented in figure 6. From the first and second columns, it can be seen that the vertical MF inhibits the necking phenomenon and delays the detachment of the bubble. For instance, at $t=16.7$, the bubble is about to fall off when $N=0$, while the necking phenomenon does not appear when $N = 44.5$. Besides, as the MF strength increases, the periodic detachment of the bubble is gradually suppressed, and the vapour jet emerges. At $N=0$, the periodic detachment of the bubble caused by Rayleigh–Taylor instability can be observed, and the bubble detachment locations for each bubble cycle are almost the same. However, when $N=6.3$ and $N=11.1$, this phenomenon can only be observed in the previous short period. After the boiling flow completely develops, the vapour jet arises, and a small bubble is released from the top of the jet, corresponding to the onset of vapour jet instability. Also, the bubble detachment location increases, implying that the MF suppresses the onset of Rayleigh–Taylor instability. When $N$ increases to $44.5$, the detachment of the bubble is barely observable, and boiling takes place in the form of a steady vapour column after the flow completely develops. In this stage, small amplitude surface waves can still be observed at the interface, indicating that the high-intensity MF further inhibits the occurrence of jet instability.

Figure 6.

Temporal evolution of the bubble shapes during the film-boiling flow without and with the vertical MFs. Note that in the pictures, the long arrow represents the direction of the MF, while the same holds in the following figures of the present study. Panels show (a) $N=0$ ($B = 0$ T), (b) $N=6.3$ ($B = 0.15$ T), (c) $N=11.1$ ($B = 0.2$ T) and (d) $N=44.5$ ($B = 0.4$ T).

To investigate the impact of a vertical MF on film-boiling flow and its consequent influence on the boiling pattern, we initially compare the streamlines for cases $N=0$, $N=11.1$ and $N=44.5$, as depicted in figure 7(a–c). It can be seen that, with an increase in MF intensity, the suppression of the vortex structure, owing to the MHD effect, becomes more pronounced. In figure 7(a), where $N=0$, vortices are observed both at locations where the bubble has detached and where the bubble is still connected to the thin film. However, in figure 7(b i), where $N=11.1$, vortices are only present at the location where the bubble has detached. Vortices no longer appear when $N$ increases to 44.5, as depicted in figure 7(c i). Additionally, close inspection of the bubble shapes reveals that the interface curvature at the bottom of the bulge decreases, and the necking phenomenon is thus weakened.

Figure 7.

(a–c) Comparison of the temperature field and streamline patterns in the vertical cross-section ($XOY$) under different vertical MFs. Panels (b ii,c ii) show the comparison of the Lorentz force (black arrows) and horizontal velocity distribution ($u_y$, the positive direction: from left to right) (contours). (d,e) The distribution of the current line. Panels show (a) $N=0$, $t=57.4$; (b) and (d) $N=11.1$, $t=53.7$; (c) and (e) $N=44.5$, $t=40.7$.

Now, we aim to elucidate the mechanisms underlying the aforementioned phenomenon. First, the distribution of the current line surrounding the liquid/vapour interface is plotted in figure 7(d,e). It can be seen that the current lines form closed loops in two cases, indicating that the computation of the induced current is conserved. Figure 7(b ii,c ii) presents the distribution of the Lorentz force (black arrows) and the horizontal component of velocity (contours, $u_y$ in the $XOY$ cross-section). It is essential to clarify that, due to a significantly large ratio of electric conductivities ($\sigma _{e, l}/\sigma _{e, v}$), the Lorentz forces in the vapour phase can be neglected so that they only manifest in the liquid phase. It is evident that the Lorentz force induced by the vertical MF acts in the opposite direction to $u_y$. Notably, under moderate MF intensity (figure 7b), the Lorentz force near the upper part of the bubble points towards the inside of the bubble, while at the lower part, it points towards the outside. This results in a torque opposite to the direction of the local vortex, as indicated by the purple circle with arrows in the diagram. Consequently, the horizontal velocity is weakened, consistent with the colour scale results in figure 7(a–c). This explains the suppression of vortex formation, and such decay in vortices counteracts the pressure difference across the vapour–liquid interface. Thus, the surface tension, which balances this pressure difference in the form of $\sigma \kappa = p_{v}-p_{l}$, should be reduced correspondingly. The magnitude of surface tension is positively correlated with the curvature of the interface, and thus, the presence of the Lorentz force reduces interface curvature, making the bubble more challenging to pinch off. Simultaneously, due to the rapid evaporation of the liquid, the continuous generation of vapour converges towards the bulge, ultimately leading to the formation of a vapour jet under the influence of buoyancy. For vapour jets, bubble detachment is triggered by the amplification of instabilities and surface waves (Lin & Reitz Reference Lin and Reitz1998). Then, in the scenario with a large MF intensity ($N=44.5$, figure 7c), the Lorentz forces entirely suppress the vortices and lead to uniformly distributed $u_y$, with the flow directing toward the centre of the vapour column and upward flow. In this context, the Lorentz force also plays a role in suppressing the growth of surface waves while concurrently sustaining boiling in the form of a vapour column. In § 3.1.4, an additional case based on the large MF intensity case $(N=44.5)$ will be further introduced to validate this finding.

The temporal evolution of the space-averaged Nusselt number ($Nu$) under different vertical MFs is shown in figure 8(a). It can be observed that, as the vertical MF intensity increases, $Nu$ transitions from approximately constant amplitude oscillation ($N \leq 2.8$) to damped oscillation ($N \geq 6.3$). This observation is consistent with the variations in the bubble detachment pattern under the influence of a vertical MF, suggesting that the vertical MF serves as a damping force, restraining bubble pinch-off behaviour. Specifically, for cases with $N=6.3$, $N=11.1$ and $N=44.5$, the initially oscillating $Nu$ eventually stabilizes at a constant value. This stabilization occurs because, at these MF intensities, film boiling tends to maintain a stable jet form after the full development of boiling flow, as depicted in figure 6. Bubble detachment occurs at locations far from the overheated wall or even without bubble detachment, keeping the interface near the overheated wall relatively stable. Given that, during saturated film boiling, the temperature field in the vapour phase exhibits a nearly linear distribution, the temperature distribution remains unchanged, as observed in figures 6 and 7.

Figure 8.

(a) Temporal evolution of the space-averaged $Nu$ in the single-mode film boiling under different vertical MF intensities. The dashed line represents the theoretical results of Klimenko (Reference Klimenko1981). (b) Variation of the space- and time-averaged $Nu$ ($\overline {Nu}$) with vertical MF intensity in the case of single-mode film boiling. The dimensional parameter $B$ is used on the horizontal axis to enhance clarity in presenting the value distribution.

Figure 8(b) illustrates the space- and time-averaged Nusselt number ($\overline {Nu}$) calculated using (2.22) for different MF intensities. In cases of constant amplitude oscillation ($N=0$, $N=0.7$ and $N=2.8$), $Nu$ is averaged over time from the second trough to the seventh trough in figure 8(a), in order to eliminate the influence of the first bubble cycle. The final constant value is utilized in the case of damped oscillation, where $N=6.3$, $N=11.1$ and $N=44.5$. The dimensional parameter $B$ is used on the horizontal axis to enhance clarity in presenting the distribution of $\overline {Nu}$ under different MF intensities. The chart is demarcated with an orange dashed line based on the pattern of bubble detachment and $Nu$ variation. The left-hand side of the dashed line represents the periodic bubble detachment area, where $Nu$ exhibits constant amplitude oscillation. In this region, the vertical MF results in a slightly higher $\overline {Nu}$ than the case without a MF. On the right-hand side of the dashed line is the vapour column boiling area, where $Nu$ exhibits damped oscillation. The value of $\overline {Nu}$ sharply declines in the transition zone. After that, $\overline {Nu}$ gradually increases with the increase in MF intensity.

To understand the reason behind the increase in $\overline {Nu}$ with the rise in MF intensity in this region ($N \geq 6.3$), we compare the bubble shapes at $t=70.4$ under three different MF intensities, as shown in figure 9(a). Note that, in the same figure, the embedded panel illustrates the vapour interface in the vapour film region ($y > 0.2$), and figure 9(b) presents the temporal evolution of the minimum distance between the interface and the overheated wall, and the development of the corresponding $Nu$ is also depicted. It can be observed that the vapour thickness decreases with the increase in MF intensity, leading to an increase in $Nu$ and $\overline {Nu}$. Now, the question arises: Why does the vapour thickness decrease by generating a vapour jet as the MF intensity rises? Figure 10(a) explains the reasons by plotting the variation of the vertical component of velocity $(u_x)$ with height on the axisymmetric line near the overheated wall (as shown by the black dashed line in figure 10b–d) under three different MF intensities ($N=6.3$, $N=11.1$ and $N=44.5$) at $t=70.4$. This figure shows that, the greater the MF intensity, the higher the vertical velocity component near the overheated wall on the axisymmetric line. This allows vapour generated through evaporation to flow quickly from the near-wall region into the vapour jet, resulting in a smaller minimum distance between the vapour interface and the overheated wall. Analysing the vertical component of the velocity contour ($u_x$) plot (figure 10b–d), it is observed that the vertical velocity is highest at the necking point of the jet, leading to a blocking effect that there is a sharp decrease in vertical velocity at its lower end (marked in the red box in the pictures). This decrease suppresses the process of upstream vapour entering the jet interior. Previous analyses have indicated that an increase in MF intensity suppresses jet instability, causing the necking point of the jet to move farther from the overheated wall. Therefore, the necking phenomenon has a weaker inhibitory effect on the velocity near the wall on the symmetry axis. In summary, when boiling occurs in the form of a jet, the location of bubble detachment influences the velocity at which vapour enters the centre of the jet. The higher the bubble detachment position, the greater the vapour velocity in the jet, making vapour overflow more quickly. Thus, a higher MF intensity decreases the vapour film thickness near the overheated wall, promoting $Nu$ and $\overline {Nu}$.

Figure 9.

(a) Comparison of the bubble shapes at $t=70.4$ under three MF intensities ($N=6.3$, $N=11.1$ and $N=44.5$). (b) Temporal evolution of the space-averaged Nusselt number ($Nu$) and the minimum distance between the liquid/vapour interface and the overheated wall over time after boiling reaches a steady state under these three MF intensities.

Figure 10.

(a) Variation of the vertical component of velocity ($u_x$, the positive direction: from bottom to top) with dimensionless height ($x/\lambda _d$) on the axis line near the overheated wall (see black dashed line in panel b) under three different MF intensities ($N=6.3$, $N=11.1$ and $N=44.5$) at $t=70.4$. Panels (b–d) show the distribution of $u_x$ for $N=6.3$, $N=11.1$ and $N=44.5$ at $t=70.4$, respectively. (e) The distribution of $u_x$ for $N=0$ at $t=57.4$.

The abrupt cliff-like decrease in $\overline {Nu}$ during the transition region in figure 8(b) can be explained through the same analytical method. Figure 10(e) presents the distribution of $u_x$ for $N=0$ at $t=57.4$, when the necking phenomenon caused by Rayleigh–Taylor instability is distinctly visible. When film boiling occurs in the form of a periodic bubble detachment, the location of bubble necking is near the vapour film and close to the overheated wall. Thus, the necking phenomenon brings about a sharp increase in vertical velocity, directly translating into an abrupt rise in vapour overflow velocity, as shown by the colour scale in figure 10(e). This facilitates the rapid influx of vapour generated by evaporation into the bulge, significantly reducing the vapour film thickness during the bubble necking stage. As a result, $Nu$ increases, enhancing $\overline {Nu}$.

3.1.3. Effect of the horizontal MF

Next, we investigate the effect of a horizontal MF on the single-mode boiling configuration. Here, the MF is applied in the $y$ direction. Thus, the vertical $XOY$ plane is parallel to the MF, while the vertical $XOZ$ plane is perpendicular to the MF. The choice of MF strengths is the same as those used in the case of vertical MF configuration, namely $N=0.7$ $\textrm {(B = 0.05 T)}$, $N=2.8$ ${(B = 0.1\,\textrm {T})}$, $N=6.3$ ${(B = 0.15\,\textrm {T})}$, $N=11.1$ ${(B = 0.2\,\textrm {T})}$ and $N=44.5$ ${(B = 0.4\,\textrm {T})}$.

Figure 11 depicts the evolution of the liquid/vapour interface patterns under different horizontal MF intensities. Anisotropy is observed in the boiling flow when a horizontal MF is applied. For instance, in the latter part of the process ($t > 30.6$) for $N=2.8$, columnar bulges out at the thin film grow along the MF direction, while the detaching bubble is also non-axisymmetric. With further increases in the MF intensity ($N=6.3$, $N=11.1$ and $N=44.5$), this characteristic manifests much earlier that a 2-D boiling flow forms and the vapour features as a thin sheet along the direction of the MF after complete development. Furthermore, after the vapour sheet grows to a certain height (see those scenarios at $t = 74.1$), the amplification of minor disturbances can lead to the instability of the vapour sheet. In the study of sheet instability (Hagerty & Shea Reference Hagerty and Shea1955), two modes of instability for sheets have been identified: the sinuous mode and the varicose mode. From figure 11, it is evident that the flow corresponds to the sinuous mode.

Figure 11.

Temporal evolution of the vapour bubble shapes during the film-boiling flow without and with the horizontal MFs. Panels show (a) $N=2.8$ ${(B = 0.1\,\textrm {T})}$, (b) $N=6.3$ ${(B = 0.15\,\textrm {T})}$, (c) $N=11.1$ ${(B = 0.2\,\textrm {T})}$ and (d) $N=44.5$ ${(B = 0.4\,\textrm {T})}$.

To understand those scenarios caused by the horizontal MFs, we examine the temperature and streamline distribution in two vertical cross-sections: the $XOY$ plane parallel to the MF and the $XOZ$ plane perpendicular to the MF. These distributions are plotted for two different MF intensities ($N=6.3$ and $N=11.1$) at $t=27.8$, as illustrated in figures 12(a) and 12(b), respectively. The corresponding distribution of Lorentz forces and vertical velocity component ($u_x$) are presented in figure 12(c,d). Figure 12(e) shows the distribution of current lines under the horizontal MF. It can be observed that the current lines are not closed. This occurs because we set periodic boundary conditions around the computational domain, causing the current lines to pass through the boundaries. By comparing the velocity streamline distribution (see figure 12(a,b), it becomes evident that the horizontal MF primarily influences the flow vortices in the direction perpendicular to the MF, which suppresses the vortex on the side of the bubble near the wall. When $N$ increases to 11.1, the vortex completely disappears, and no downward flow is observed on the $XOY$ plane. Then, the distribution of Lorentz force, as displayed in figure 12(c,d), explains the disappearance of the vortex perpendicular to the MF. From figure 12(c i,d i), it is observed that, in the $XOY$ plane, the direction of the Lorentz force is upward on the side of the bubble near the wall, while the Lorentz force is directed vertically downward at the top of the bubble. This results in a torque opposite to the direction of the local vortex, as indicated by the orange circle with arrows in the diagrams. So, the horizontal MF tends to suppress vortices perpendicular to the MF. As a result, vortices gradually disappear, and the necking phenomenon weakens, causing the interface to move away from the overheated wall under the influence of continuously generated vapour. Eventually, a columnar bulge along the MF direction is formed. In contrast, for the $XOZ$ plane perpendicular to the MF (see figure 12(c ii,d ii), the direction of the Lorentz force is uniformly vertical downward, and the flow field structure does not exhibit significant changes with variations in MF intensity. Thus, the necking phenomenon still occurs in this plane, forming a vapour sheet. These conclusions align with the theoretical prediction of MHD effects (Sommeria & Moreau Reference Sommeria and Moreau1982; Davidson Reference Davidson1995), demonstrating that the Lorentz force tends to decrease the velocity gradient parallel to the MF to reduce Joule dissipation. Furthermore, the temperature distribution depicted in figure 12(a,b) exhibits a pattern similar to that of the vertical MF cases. It can be seen that, as the MF intensity increases, the temperature inside the bulge becomes lower. This trend is in line with the distribution of the vertical component of velocity $(u_x)$ shown in figure 12(c,d), and the underlying reason is analogous to the vertical MF situation. Specifically, with the increase in MF intensity, the weakening of the necking phenomenon leads to a reduction in the velocity of vapour entering the bulge, making heat transfer more challenging. Consequently, the internal temperature becomes lower.

Figure 12.

(a,b) Comparison of temperature field and streamline patterns in two vertical cross-sections ($XOY$: parallel to the MF, in (a i,b i); $XOZ$: perpendicular to the MF, in (a ii,b ii)) under different horizontal MF intensities. (c,d) Comparison of the Lorentz forces (black arrows) and the distribution of the vertical component of velocity ($u_x$, contours) corresponding to panels (a,b). (e) Distribution of the current lines. Panels show (a,c) $N=6.3$, $t=27.8$; (b,d,e) $N=11.1$, $t=27.8$.

Figure 13(a) presents the temporal evolution of $Nu$ under varying horizontal MF intensities. At lower MF intensities $(N<6.3)$, the $Nu$ variation demonstrates approximate periodic oscillations, aligning with the periodic detachment of bubbles. Notably, when $N=2.8$, a considerable reduction in amplitude occurs, attributed to the initiation of the horizontal MF effect. This effect induces the formation of a columnar bulge along the MF direction at the vapour film's bottom, leading to an increased vapour layer thickness and subsequently causing a decline in $Nu$. As the MF intensity increases $(N \geq 6.3)$, $Nu$ undergoes a brief period of periodic oscillations in the initial stages due to bubble detachment. However, once a stable vapour sheet is established, a transition to a more stable state occurs, and higher MF intensities accelerate this transition.

Figure 13.

(a) Temporal evolution of the space-averaged $Nu$ under different horizontal MF intensities in single-mode film boiling. The dashed line represents the theoretical results of Klimenko (Reference Klimenko1981). (b) Variation of the space- and time-averaged $Nu$ ($\overline {Nu}$) with horizontal MF intensity in the case of the single-mode film boiling. The dimensional parameter $B$ is used on the horizontal axis to enhance clarity.

The value of $\overline {Nu}$ for the aforementioned cases is plotted in figure 13(b). Similar to the vertical MF, for small MF intensity cases $(N<6.3)$, the average of $Nu$ from the second trough to the seventh trough in figure 13(a) is selected, while for large MF intensity cases, the final value is employed. Based on the final character of the film boiling, the chart is divided into two regions. The left-hand side represents the periodic detachment region of the bubble. In this region, $\overline {Nu}$ experiences a sharp decrease after the MF reaches a certain intensity. This is consistent with the previously mentioned reduction in the amplitude of $Nu$, both attributed to the appearance of the elongated bulge in the vapour film. The right-hand side is the vapour sheet boiling region, where $\overline {Nu}$ increases as the MF intensity increases. To investigate the reason, we compare the shapes of the interface in the $XOZ$ plane at the initial formation of the vapour sheet (figure 14a) and when the vapour sheet stably exists (figure 14b) under these three different MF intensities. It is clear that there is a significant difference in the thickness of the thin film near the overheated wall, particularly away from the vapour sheet side, as indicated by the red square in the diagram. The higher the MF strength, the smaller the thickness, resulting in a larger $Nu$ and $\overline {Nu}$. From figure 12(c,d), we know that, in the $XOZ$ plane, the Lorentz force is consistently directed downward. Therefore, we hypothesize that the Lorentz force might suppress the expansion of the vapour film near the overheated wall. To verify this hypothesis, we output the vertical component of the Lorentz force ($F_{l,x}$) on the near-wall interface corresponding to figure 14, as depicted in figure 15. It can be observed that, as the MF intensity increases, the downward effect of the Lorentz force becomes more pronounced. In other words, the suppression of the expansion of the vapour film near the overheated wall is stronger, resulting in a smaller thickness of the vapour film and a larger $\overline {Nu}$.

Figure 14.

Comparison of the bubble shapes in the vertical cross-section ($XOZ$ plane) perpendicular to the MF under three different MF intensities ($N=6.3$, $N=11.1$ and $N=44.5$). In panel (a), the red, green and blue lines depict the results for $N=6.3$ at $t=42.6$, $N=11.1$ at $t=33.3$ and $N=44.5$ at $t=22.2$, respectively. In panel (b), the red, green and blue lines depict the results for $N=6.3$ at $t=57.4$, $N=11.1$ at $t=50.0$ and $N=44.5$ at $t=40.7$, respectively.

Figure 15.

Comparison of the vertical component of the Lorentz force ($F_{l,x}$) on the liquid/vapour interface under different MF intensities corresponding to figure 14. In panel (a), the red, green and blue lines depict the results for $N=6.3$ at $t=42.6$, $N=11.1$ at $t=33.3$ and $N=44.5$ at $t=22.2$, respectively. In panel (b), the red, green and blue lines depict the results for $N=6.3$ at $t=57.4$, $N=11.1$ at $t=50.0$ and $N=44.5$ at $t=40.7$, respectively.

After examining the MHD effect on the growth of vapour bubbles, it is useful to compare our findings with those of previous studies. The most relevant work is by Chandrasekhar (Reference Chandrasekhar2013), who used linear stability analysis to investigate the effects of vertical and horizontal MFs on Rayleigh–Taylor instability in inviscid interfacial flows. Specifically, the exponential growth rate $n$ of disturbances for a flat interface between inviscid fluids in ideal MHD, with a vertical MF applied to the interface, is given by: $n^{3}+n^{2} k B_{0}(\sqrt {\rho _l/\rho _0}+\sqrt {\rho _v/\rho _0})+n(k^{2} B_{0}^{2}-(\rho _l-\rho _v)/(\rho _l+\rho _v) g k)=2 (\rho _l-\rho _v)/(\rho _l+\rho _v) g k^{2} B_{0}/(\sqrt {\rho _l/\rho _0}+\sqrt {\rho _v/\rho _0})$, where $g$ is the gravitational acceleration, $k$ is the wavenumber, $B_0 = B / \sqrt {\rho _0 \mu _0}$, $\rho _0 = (\rho _l + \rho _v) / 2$ and $\mu _0$ is the vacuum magnetic permeability. Notably, there is no critical wavelength, so all modes are unstable, but a larger $B_0$ enhances the stability of the interface. This aligns with our results, which show that the vapour bubble grows more slowly under a stronger vertical MF. However, the Rayleigh–Taylor (RT) instability in boiling flow cannot be completely eliminated, even with a very strong MF. For the horizontal MF parallel to the interface, linear stability analysis also reveals anisotropic effects induced by the external MF. Horizontal disturbances transverse to the $\boldsymbol {B}$-lines are unaffected by the horizontal MF, while disturbances parallel to the $\boldsymbol {B}$-lines have an exponential growth rate $n$ given by: $n^2=gk(\rho _l-\rho _v)/(\rho _l+\rho _v)-k^2{B_0}^2$, with $k$ denoting the wavenumber of the disturbances in that direction. This shows that a sufficiently strong MF can suppress Rayleigh–Taylor instability in the direction parallel to the $\boldsymbol {B}$-lines but has little impact in the direction transverse to the $\boldsymbol {B}$-lines, consistent with our numerical results. We also acknowledge the importance of future studies addressing these challenges by incorporating a phase-change model into the framework of linear stability analysis for a more comprehensive understanding of boiling coupled RT instability under MHD conditions.

3.1.4. What happens after MF removal?

To further understand the MHD effect on film boiling, in this section, we investigate what happens when the MF is suddenly removed after film boiling reaches a steady state under MF (vapour column boiling for vertical MF scenario and vapour sheet boiling for horizontal MF scenario).

Initially, we examine the scenario based on the vertical MF with large intensity $(N=44.5)$. The MF is suddenly removed when the film boiling reaches a relatively stable state $(t=44.4)$. The results are depicted in figure 16. It is observed that, after the removal of the MF, the amplitude of surface waves progressively increases, and vortices appear, ultimately leading to the detachment of small bubbles. This observation supports the findings discussed in § 3.1.2 that the Lorentz force plays a vital role in suppressing the growth of surface waves and the subsequent detachment of the vapour bubble. The comparison of the temporal evolution of $Nu$ with the MF and upon its removal is illustrated in figure 17(a). After removing the vertical MF, a noticeable decrease in $Nu$ can be observed. This is attributed to the fact that, upon the removal of the vertical MF, the stable columnar boiling pattern can no longer be sustained, leading to the detachment of bubbles at the top of the vapour column (see figure 16). As analysed in § 3.1.2, the bubble detachment behaviour inhibits the vapour influx into the columnar bulge, causing the near-wall vapour film to thicken and reducing $Nu$. Figure 17(b) compares the evaporation rate (here, we measure the evaporation rate using the mass flow at the interface ($\dot {m} S_{\varGamma }$)) between the cases, revealing a significant decrease in the evaporation rate after the removal of the MF, which is also caused by the increase in thickness of the near-wall vapour film. Note that the oscillation of $\dot {m} S_{\varGamma }$ after removing the MF is caused by the reappeared detachment of the vapour bubble. The change in evaporation rate provides another perspective on the reason for bubble detachment. After removing the MF, the decrease in evaporation rate prevents film boiling from supplying sufficient vapour for stable columnar bulge.

Figure 16.

Temporal evolution of the temperature profile (colour contours) and velocity streamline pattern (black arrow lines) in the vertical cross-section ($XOY$) after removing the MF at $t=44.4$ for $N=44.5$. Panels show (a) $t=44.4$, (b) $t=50.0$,(c) $t=51.9$, (d) $t=54.6$, (e) $t=56.5$ and (f) $t=59.3$. It can be seen that the removal of the MF leads to vapour jet instability, resulting in the reappearance of bubble detachment.

Figure 17.

Comparison of the temporal evolution of (a) the space-averaged Nusselt number ($Nu$) and (b) the evaporation rate ($\dot {m} S_{\varGamma }$) with the vertical MF and after the removal of the MF at $t=44.4$ for $N=44.5$. Note that the oscillation of the curves, corresponding to the MF removal, is due to the reappearance of vapour bubble detachment.

Now, we examine the scenario based on a horizontal MF with the intensity of $N=22.2$. When film boiling reaches a relatively stable vapour sheet boiling at $t=57.4$, the MF is suddenly removed. However, intriguingly, we observe a very interesting result: film boiling in the vapour sheet state does not appear to be affected after removing the horizontal MF, as depicted in figure 18. Furthermore, by comparing $Nu$ and the evaporation rate (figure 19), it is observed that, with and without the presence of a horizontal MF, both $Nu$ and the evaporation rate are almost identical. This indicates that the vapour sheet formed under a horizontal MF is initially stable, in contrast to the vapour column generated under a vertical MF. To explain this, we should remember that the vertical MF eliminates the vortices around the vapour bubble in an axisymmetric manner, and such a controlling effect leads to the generation of an isotropic vapour column. Therefore, the axisymmetric Lorentz forces, directing radial-wards, maintain the force balance around the vapour column. Consequently, if the Lorentz forces are removed to break such force balance, the downward flow must develop due to any disturbance that emerges on the vapour column, resulting in an expanding vortex region and, hence, a detached bubble. Nevertheless, the horizontal MF suppresses the vortices in the parallel plane ($XOY$) but has little influence on the vortices in the perpendicular plane ($XOZ$). Besides, figure 12(c,d) reveals that, corresponding to the stable vapour sheet, the Lorentz forces direct almost downwards in terms of $F_{l,x} \propto u_x{\cdot } B_y^2$, because any $z-$ component of $\boldsymbol {F}_{l}$ requires the participation of $u_z$, which almost disappears for a stable and 2-D vapour sheet. It indicates that, for the vapour sheet scenario under horizontal MF, the Lorentz forces play little role in the force balance in the radial direction, and thus, removing it causes little influence on the vapour pattern.

Figure 18.

(a–c) The temporal evolution of the liquid/vapour interface after removing the horizontal MF at $t=44.4$ for $N=44.5$. (d–f) Temperature profile (colour contours) and velocity streamline pattern (black arrow lines) in the vertical cross-section ($XOZ$) corresponding to panels (a–c). Panels (a,d), (b,e) and (c,f) correspond to $t=59.3$, $t=63.0$ and $t=66.7$, respectively.

Figure 19.

Comparison of the temporal evolution of (a) the space-averaged Nusselt number ($Nu$) and (b) the evaporation rate ($\dot {m} S_{\varGamma }$) with the horizontal MF and after the removal of the MF at $t=57.4$ for $N=22.2$. We observe that, in (a,b), the values are almost identical before and after removing the horizontal MF.

3.2.1. Effect of the vertical MF

Figure 20 depicts the evolution of the liquid/vapour interface under three different vertical MFs. In the initial stage $(t<60.0)$ of film boiling, as observed in the first two columns of the figure, the distribution of bubbles aligns with the theory proposed by Zuber (Reference Zuber1958) for the most dangerous wavelength ($\lambda _d$). The distance between adjacent bubbles correlates with $\lambda _d$ even if the initial disturbances are randomly imposed. Additionally, in the early stage, the interaction between bubbles is not pronounced, allowing for a clear observation that, under the multi-bubble model, the impact of a vertical MF on bubble shapes follows the conclusions drawn from the single-mode film boiling: as the MF strength increases, the film boiling transitions from the periodic detachment of bubbles from the bottom to the vapour jet releasing small bubbles from the top. This evolution ultimately leads to vapour columnar boiling without any bubble detachment, a pattern confirmed by figure 21, which presents the vertical cross-section of a single bubble in the multi-mode, illustrating the distribution of the Lorentz force, the $z-$ component of velocity $(u_z)$, the temperature field and the velocity streamlines. A comparison with figure 7 reveals that, in multi-mode boiling, the effect of the vertical MF aligns with the observations in single-mode boiling. The evolution of $Nu$ also mirrors the single-mode cases in the early stages. As the MF strength increases, the Nusselt number shifts from approximate periodic oscillations to damped oscillations. A slight increase in the averaged value of $Nu$ for large $N$ is also observed, as shown in figure 22.

Figure 20.

Temporal evolution of the bubble shapes during multi-mode film boiling without and with the vertical MFs. Panels show (a) $N=0$ ($B=0.0$ T), (b) $N=11.1$ ($B=0.2$ T) and (c) $N=100.1$ ($B=0.6$ T).

Figure 21.

Distribution of the temperature field, the streamline patterns, the Lorentz forces and the horizontal component of velocity ($u_z$, the positive direction: from left to right), in the vertical cross-section of a single bubble in multi-mode film boiling under different vertical MFs. Panels show (a) $N=11.1$, $t=47.2$ and(b) $N=100.1$, $t=37.0$.

Figure 22.

Temporal evolution of the space-averaged $Nu$ under different vertical MF intensities in the case of the multi-mode film boiling. The dashed line represents the theoretical results of Klimenko (Reference Klimenko1981).

Next, let us focus on the later stages of the film-boiling process under vertical MFs. In the last two columns ($t > 64$) of figure 20, it becomes evident that the positions of bubbles during boiling undergo displacement in both scenarios – with and without MF. Particularly, such a boiling pattern comprising vapour columns under strong vertical MF becomes unstable too. This displacement is primarily attributed to the non-uniform growth of bubbles generated by the initial random perturbation, leading to amplified interactions between bubbles over time. This phenomenon is depicted in figure 23, chosen with $N=11.1$ as an example since this behaviour is observed consistently regardless of the presence of the MF. In figure 23, two adjacent bubbles (A and B) are selected for observation, and the evolution of their shapes and velocity streamlines in the vertical plane ($XOY$) at different time instants is presented. At $t=51.9$, the detachment behaviours of bubbles A and B differ, resulting in asymmetry in the flow field on either side of bubble B, as indicated by regions R1 and R2 in the figure. Vortices are present in region R1, while region R2 lacks vortices. As is well known, regions with vortices exhibit lower pressure. Consequently, bubble B tends to approach bubble A. At $t=63.0$, the two bubbles are very close together, and there is almost no horizontal flow in region R1, while horizontal flow persists in region R2. This causes bubble B to approach bubble A further, eventually leading to their fusion and resulting in a chaotic flow state. Additionally, when bubbles merge, vapour overflows into new positions to form bubbles. The entire boiling system is no longer in a state where bubbles simultaneously detach and generate, but rather, the formation, detachment and merging of bubbles occur simultaneously. Consequently, at this stage, the height fluctuation of the near-wall thin film is no longer significant. Therefore, the $Nu$ variation curve shown in figure 22 exhibits small amplitude oscillations around the correlation proposed by Klimenko (Reference Klimenko1981) during the latter stage.

Figure 23.

Interaction of two neighbouring bubble sites and the evolution of the streamline patterns at different time instants from $t=51.9$ to $t=66.7$ under the vertical MF with $N=11.1$.

3.2.2. Effect of the horizontal MF

Figure 24 presents the evolution of the liquid/vapour interface under three different horizontal MFs. In the initial stage of boiling flow, as shown in figure 24(a i,b i,c i), the relatively weak interaction between bubbles results in the effect of the horizontal MF on each bubble in the multi-mode cases being consistent with the conclusion drawn for the single-mode situation. With an increase in MF strength, the boiling flow tends to become two-dimensional along the MF direction. For small MF strength ($N=2.8$ in figure 24), boiling still occurs in the form of bubble detachment. For large MF strength ($N=11.1$ and 100.1 in figure 24), boiling takes place in the form of a vapour sheet. Figure 25, using $N=2.8$ and $N=11.1$ as examples, illustrates the distribution of the flow field, temperature, Lorentz forces and vertical velocity component $(u_x)$ in two cross-sections of a single bubble in the multi-mode case during the initial stage of film boiling. The results align with the single-mode situation (figure 12). In other words, the mechanism of the horizontal MF cases is consistent with that of single-mode situations. Additionally, the variation of $Nu$, as shown in figure 26, is also similar to the single-mode cases before $t=60.0$.

Figure 24.

Temporal evolution of the bubble shapes during the multi-mode film-boiling flow without and with the horizontal MFs. Panels (a–c) show $N=2.8$ ${(B=0.1\,\textrm {T})}$, $N=11.1$ ${(B=0.2\,\textrm {T})}$ and $N=100.1$ ${(B=0.6\,\textrm {T})}$, respectively.

Figure 25.

Distribution of the temperature field, the streamline patterns, the Lorentz forces and the vertical component of velocity $(u_x)$, in two vertical cross-sections ($XOY$: parallel to the MF and $XOZ$: perpendicular to the MF) of a single bubble in the multi-mode film boiling under the horizontal MFs. Panels show (a) $N=2.8$, $t=42.6$ and (b) $N=11.1$, $t=31.5$.

Figure 26.

Temporal evolution of the space-averaged $Nu$ under different horizontal MF intensities in the case of multi-mode film boiling. The dashed line represents the theoretical results of Klimenko (Reference Klimenko1981).

Similar to the case with a vertical MF, as boiling progresses to a certain extent, the flow becomes chaotic due to the influence of bubble interactions, as shown in ($t > 57.0$) figure 24(a iii,a iv,b iii,b iv,c iii,c iv). Moreover, the flow exhibits more chaotic behaviour in horizontal MF cases. This is attributed to the fact that, under the horizontal MF, each single column of bubbles along the direction of the MF forms a vapour sheet. As boiling proceeds, the interaction between vapour sheets intensifies compared with bubbles and vapour columns, ultimately resulting in larger-scale rupture or fusion of vapour sheets, exerting a more significant influence on the entire flow system. Figure 27 illustrates the evolution of streamline patterns in the vertical cross-section perpendicular to the MF under the condition of $N=100.1$. It can be observed that, as boiling progresses, the asymmetry of vortices on both sides of the four vapour sheets (A, B, C, D) becomes more pronounced. The vapour sheets are then twisted, eventually leading to the fusion of vapour sheets B, C and D at the base and fragmentation at the top, as shown in figure 27 at $t=48.2$. The fragments produce small bubbles, making the flow system more chaotic. After fusion, new vapour sheets E and F are generated, as shown in figure 27 at $t=66.7$, corresponding to the sudden increase in $Nu$ in figure 26. For other cases where $N<100.1$, the variation in $Nu$ exhibits smaller fluctuations in the later stages. This is because, in these situations, the merging and generation of bubbles or vapour sheets occur almost simultaneously, resulting in a smaller fluctuation in the distance between the interface and the overheated wall.

Figure 27.

Dynamics of vapour sheets and the evolution of the streamline pattern for different time instants ranging from $t=22.2$ to $t=66.7$, with the horizontal MFs $(N=100.1)$.

Moreover, it is essential to note that bubbly flow in multi-mode film boiling eventually transitions into turbulence or pseudo-turbulence, making the accurate resolution of bubble coalescence and collisions a significant area of ongoing research. The complexity of these interactions is beyond the scope of the present work. For example, VOF methods often simplify the coalescence process (Scardovelli & Zaleski Reference Scardovelli and Zaleski1999). Additionally, recent work by Innocenti et al. (Reference Innocenti, Jaccod, Popinet and Chibbaro2021) suggests that a spatial resolution of $\Delta = 2d/Ar$ (where $Ar = \rho _l g^{1/2} d^{3/2}/\mu _l$, and $d$ is the typical size of the bubble) is sufficient for convergence in the spectra of liquid velocity fluctuations in bubbly flows. In our study, the multi-mode film-boiling flows are characterized by an Archimedes number of $Ar \approx 120$. The spatial resolution at grid level 7, which corresponds to $\Delta = d/96$, meets the convergence criteria suggested by Innocenti et al. While this does not constitute a full grid convergence verification, it provides confidence in the resolution used to capture the essential physics of heat and mass transfer during multi-mode film boiling. Future research will focus on a more detailed study of vapour bubbly flows under external MFs, including a comprehensive statistical analysis and examination of spectra of kinetic energy and velocity fluctuations on various grids, to rigorously address these complex interactions in the presence of external MFs.