Destabilization of a liquid metal by nonuniform Joule heating

We study the effect of an impressing AC magnetic field at the bottom of a liquid metal layer of thickness $h$. In this situation the fluid is set in motion by the buoyancy forces caused by internal heat sources. The heat sources, caused by the Joule effect induced by the AC field, present an exponentially decaying profile, with characteristic length $\delta$. As the magnetic field is horizontal, the Lorentz force has no influence on the dynamics of the system since it contributes only to the magnetic pressure. We propose an analysis of both the transient and fully developed regimes using linear stability analysis (LSA) and direct numerical simulations (DNSs). The transient period is governed by the temporal evolution of the temperature field as well as the development of the convective instability, which can be concomitant and therefore requires adopting a transient LSA algorithm to track these two effects. The DNSs have been performed for various distributions of the heat sources and various total heat input. This corresponds to independently varying $\delta /h$ in the range $0.04\le \delta /h\le 0.45$ and a Rayleigh number $1.1\times {10}^4\le \text{Ra}\le 1.2\times {10}^5$. We observe the relaxation of the temperature up to the steady conductive profile before the transition to the nonlinear regime when $\text{Ra}$ is small, whereas for larger $\text{Ra}$, nonlinear effects appear during the relaxation of the temperature profile. The unsteadiness of the temperature field significantly alters the development of the instability because of a much smaller growth rate. Surprisingly, we observe that $\delta /h$ has only a limited influence on averaged quantities as well as on the patterns for both the linear and nonlinear regimes. This comes with the fact that the profiles present an apparent reflectional symmetry, despite the asymmetry of the governing equations.

Figure 1

Studied configuration: an unbounded horizontal layer of liquid metal confined by two walls. The bottom wall is insulating, and the top wall is a perfect conductor. $\mathbf{g}$ is the acceleration of gravity. For the DNS, the infinite extent of the domain is approximated by periodic boundary conditions.

Figure 2

Nondimensional Joule term (solid lines) profile computed from averaging Eq. (9). The dashed lines represent the semi-infinite solution given by Eq. (10).

Figure 3

Time evolution of kinetic energy. Both time and kinetic energy are normalized by the thermal diffusion scale.

Figure 4

Time and $(x,y)$ direction averaged temperature profiles for the three $\text{Ra}$ in the steady-state regime. The circles represent $T_{\infty }$ from (15) for $\delta /h=0.45$ (blue), 0.14 (red), and 0.04 (black).

Figure 5

$T_{\mathrm{mean}}$ vs $\text{Ra}$. The dashed line is a linear fit of the data.

Figure 6

Average over time and $(x,y)$ direction of (a) the velocity tangential to the walls $\sqrt{\overline{U^2+V^2}}$ and of (b) the vertical velocity magnitude $\sqrt{\overline{W^2}}$.

Figure 7

Color maps of instantaneous $W$ velocity (red is upward and blue is downward) in a plane parallel to gravity at $\text{Ra}=1.1\times {10}^4$ (a) in the linear regime and (b) in the steady regime. The horizontal dashes show the skin depth: (a1 and b1) $\delta /h=0.45$, (a2 and b2) $\delta /h=0.14$, and (a3 and b3) $\delta /h=0.04$. The width of each panel is $1/3$ of the total width. The top panels are normalized by the instantaneous $W$ amplitude, and the bottom panels share the same color scale indicated in panel (b).

Figure 8

Normalized energy spectral density (ESD) for the case $\delta /h=0.14$ and $\text{Ra}=3.7\times {10}^4$ in the linear regime (blue), at maximum kinetic energy (orange), and in the stationary regime (green). The wave number $k$ in the legend indicates the main peak location.

Figure 9

Comparison of $s$ computed with LSA (lines) and DNS (symbols).

Figure 10

Contours of $s\left(t=\infty \right)$ in the $(\text{Ra},\delta /h)$ plane. The circles represent the DNS. The black line is the marginal stability. The gray triangle is the critical Rayleigh ${\text{Ra}}_c(\delta /h)$ computed by Tasaka and Takeda [11]. The gray squares correspond to the limiting cases $\delta /h=0$ and $\delta /h\to \infty$, as given by Goluskin [12].

Figure 11

Evolution of velocity profiles (top panel, DNS: lines; LSA: circles) and temperature gradients computed from (15) and (16) (bottom panel), for (a) $\delta /h=0.45$, (b) $\delta /h=0.14$ and (c) $\delta /h=0.04$ at $\mathrm{Ra}=1.1\times {10}^4$. The triangles in the top panel represent the maximum of $\sqrt{W^2}$. The velocity profiles are normed by their integral. The dashed line is $T_{\infty }$ from (15).