Transition between quasi-two-dimensional and three-dimensional Rayleigh-Bénard convection in a horizontal magnetic field

Magnetohydrodynamic Rayleigh-Bénard convection was studied experimentally and numerically using a liquid metal inside a box with a square horizontal cross section and an aspect ratio of 5. Applying a sufficiently strong horizontal magnetic field converts the convective motion into a flow pattern of quasi-two-dimensional (quasi-2D) rolls arranged parallel to the magnetic field. The aim of this paper is to provide a detailed description of the flow field, which is often considered as quasi-2D. In this paper, we focus on the transition from a quasi-two-dimensional state toward a three-dimensional flow occurring with decreasing magnetic-field strength. We present systematic flow measurements that were performed by means of ultrasound Doppler velocimetry. The measured data provide insight into the dynamics of the primary convection rolls, the secondary flow induced by Ekman pumping, and they reveal the existence of small vortices that develop around the convection rolls. New flow regimes have been identified by the velocity measurements, which show a pronounced manifestation of three-dimensional flow structures as the ratio $\mathrm{Ra}/Q$ increases. The interaction between the primary swirling motion of the convection rolls and the secondary flow becomes increasingly strong. Significant bulging of the convection rolls causes a breakdown of the original recirculation loop driven by Ekman pumping into several smaller cells. The flow measurements are completed by direct numerical simulations. The numerical simulations have proven to be able to qualitatively reproduce the newly discovered flow regimes in the experiment.

Figure 1

Experimental setup: top view (a) and side view (b). The dimensions are in mm. Schematic drawing of the counter-rotating convection rolls and UDV measuring lines (c).

Figure 2

(a) Regime diagram of the liquid-metal RBC exposed to a horizontal magnetic field replotted from [14], where the gray line represents the expected neutral stability curve for the vessel with an aspect ratio of 5 based on the result in [3]; abbreviations: OS, oscillation; Tr, transition (transition means here the switching between different roll number states). (b) Parameter points of flow measurements carried out within this study, whereas the circled symbols mark the measurements to be presented in the next paragraph. The overlap between both diagrams is marked by the gray rectangle in (a).

Figure 3

Flow measurement of a five-roll steady regime at $\mathrm{Ra}/Q=5.1$ ($Q=3.4\times {10}^3,\mathrm{Ra}=1.7\times {10}^4$). (a) (Ch2): Fluid velocity perpendicular to the magnetic field, (b) (Ch7), and (c) (Ch8): fluid velocity measured along the magnetic-field lines.

Figure 4

Flow measurement of a five-roll regime showing short-period oscillations at $\mathrm{Ra}/Q=6.9$ ($Q=8.0\times {10}^3$ and $\mathrm{Ra}=5.5\times {10}^4$). (a) (Ch2) Fluid velocity perpendicular to the magnetic field, (b) (Ch6) fluid velocity measured along the magnetic-field lines.

Figure 5

Flow measurement of a four-roll regime showing advective transport of flow structures along the magnetic-field direction at $\mathrm{Ra}/Q=33$ ($Q=7.3\times {10}^2$ and $\mathrm{Ra}=2.4\times {10}^4$). (a) (Ch2) Fluid velocity perpendicular to the magnetic field, (b) (Ch7) fluid velocity measured along the magnetic-field lines.

Figure 6

Flow measurement of a four-roll regime showing the occurrence of standing waves in both velocity components at $\mathrm{Ra}/Q=203$ ($Q=3.3\times {10}^2$ and $\mathrm{Ra}=6.7\times {10}^4$). (a) (Ch2) Fluid velocity perpendicular to the magnetic field, (b) (Ch6) fluid velocity measured along the magnetic-field lines.

Figure 7

Time series of the temperature (a)–(d) and temperature spectra (e)–(h) recorded simultaneously to the four flow measurements presented in Sec. 3a: (a,e) $\mathrm{Ra}/Q=5.1$, (b,f) $\mathrm{Ra}/Q=6.9$, (c,g) $\mathrm{Ra}/Q=33$, and (d,h) $\mathrm{Ra}/Q=203$ (thermocouple position 2 in Fig. 1).

Figure 8

Illustration of the axial flow forming due to Ekman-pumping along the convection rolls.

Figure 9

(a) Variation of the thickness of the Bödewadt layer ${\delta }_{\mathrm{B}}$ and the Hartmann layer ${\delta }_{\mathrm{H}}$ vs the $Q$ number; (b) characteristic velocities $u_e,{\overline{u}}_{y,\max }$ and the ratio $u_e/{\overline{u}}_{y,\max }$ vs the $Q$ number; and time-averaged profiles of (c) the velocity perpendicular to the magnetic field, ${\overline{u}}_y\left(y\right)$ (Ch1), and (d) parallel to the magnetic field, ${\overline{u}}_x\left(x\right)$ (Ch4), for the case $Q=2.0\times {10}^3$ and $\mathrm{Ra}=2.1\times {10}^4$.

Figure 10

Numerical simulation at $Q=5.7\times {10}^3,\mathrm{Ra}=3.0\times {10}^4$ ($\mathrm{Ra}/Q=5.3$, five-roll short OS regime). (a) Isosurface of $Q_{3\mathrm{D}}=0$; (b) flow in the direction parallel to the magnetic field ($u_x)$ on the plane $z=3/4$; (c) velocity perpendicular to $B$ on the $y$ line at $x=5/2$ and $z=3/4$; (d) velocity parallel to $B$ on the $x$ line at $y=5/4$ and $z=3/4$; the location of the measuring line for (d) is indicated by a thin black frame in (a) and (b); see the Supplemental Material [26] for a movie of (a).

Figure 11

As in Fig. 10, but obtained at $Q=3.23\times {10}^3, \mathrm{Ra}=3.0\times {10}^4$ ($\mathrm{Ra}/Q=9.3$, five-roll advective regime); see the Supplemental Material [26] for a movie of (a).

Figure 12

As in Fig. 10, but obtained at $Q=1.76\times {10}^3, \mathrm{Ra}=3.0\times {10}^4$ ($\mathrm{Ra}/Q=17.0$, five-roll advective regime); see the Supplemental Material [26] for a movie of (a).

Figure 13

As in Fig. 10, but obtained at $Q=6.25\times {10}^2, \mathrm{Ra}=3.0\times {10}^4$ ($\mathrm{Ra}/Q=64.0$, four-roll standing-wave regime); see the Supplemental Material [26] for a movie of (a).