Transition from steady to oscillating convection rolls in Rayleigh-Bénard convection under the influence of a horizontal magnetic field

In this study we consider the effect of a horizontal magnetic field on the Rayleigh-Bénard convection in a finite liquid metal layer contained in a cuboid vessel $(200\times 200\times 40\mathrm{m}{\mathrm{m}}^3)$ of aspect ratio $\mathrm{\Gamma }=5$ . Laboratory experiments are performed for measuring temperature and flow field in the low melting point alloy GaInSn at Prandtl number $Pr\approx 0.03$ and in a Rayleigh number range $2.3\times {10}^4\le Ra\le 2.6\times {10}^5$. The field direction is aligned parallel to one pair of the two side walls. The field strength is varied up to a maximum value of 320 mT ($Ha=2470$, $Q=6.11\times {10}^6$, definitions of all nondimensional numbers are given in the text). The magnetic field forces the flow to form two-dimensional (2D) rolls whose axes are parallel to the direction of the field lines. The experiments confirm the predictions made by Busse and Clever [Busse and Clever, Stability of convection rolls in the presence of a horizontal magnetic field, J. de Méc. Théor. et Appl. 2, 495 (1983)] who showed that the application of the horizontal magnetic field extends the range in which steady 2D roll structures exist (Busse balloon) towards higher Ra numbers. A transition from the steady to a time-dependent oscillatory flow occurs when Ra exceeds a critical value for a given Chandrasekhar number Q, which is also equivalent to a reduction of the ratio $Q/Ra$. Our measurements reveal that the first developing oscillations are clearly of 2D nature, in particular a mutual increase and decrease in the size of adjacent convection rolls is observed without the formation of any detectable gradients in the velocity field along the magnetic field direction. At a ratio of $Q/Ra\approx 1$, the first 3D structures appear, which initially manifest themselves in a slight inclination of the rolls with respect to the magnetic field direction. Immediately in the course of this, there arise also disturbances in the spaces between adjacent convection rolls, which are advected along the rolls due to the secondary flow driven by Ekman pumping. The transition to fully developed 3D structures and then to a turbulent regime takes place with further lowering $Q/Ra$.

Figure 1

Schematic drawing of the experimental setup showing the convection cell and sensor configuration.

Figure 2

Flow regime diagram representing the parameter range Ra-Q covered within this study.

Figure 3

Flow measurements acquired by sensor $u_{\perp ,2}$ at $Ra=2.3\times {10}^4$, $Q=5.0\times {10}^6$, $Q/Ra=212$: (a) Spatiotemporal plot of the flow structure, and (b) time-averaged flow profile $v_y$ on the basis of 320 measured single profiles (error bars are plotted here for every 10th data point and result from the corresponding standard deviations).

Figure 4

(a) Detail of the time-averaged velocity profile shown in Figs. 3, (b) schematic drawing showing a general illustration of the flow pattern (according to Ref. [22]).

Figure 5

Time-averaged flow profiles $v_y$ for (a) $Ra=3.9\times {10}^4$ and varying Q, (b) for high Q and varying Ra.

Figure 6

Two-dimensional oscillations occurring at $Ra=7.8\times {10}^4$, $Q=3.8\times {10}^5$, $Q/Ra=4.82$: (a) Spatio-temporal plot of the flow structure perpendicular to the field acquired by sensor $u_{\perp ,2}$, (b) Power spectrum of velocity time series obtained by FFT, (c) Temperature time series, (d) Power spectrum of temperature time series obtained by FFT. The yellow line marks the position for determining the velocity time series presented in Fig. 8.

Figure 7

Flow measurements parallel to the field acquired by sensor $u_{\left|\right|,2}$ at $Ra=1.7\times {10}^5$, $Q=1.2\times {10}^5$, $Q/Ra=0.69$: (a) Spatio-temporal plot of the flow structure, and (b) time-averaged flow profile $v_x$. The yellow line marks the position for determining the velocity time series presented in Fig. 8. (error bars are plotted here for every 10th data point and result from the corresponding standard deviations).

Figure 8

Sinusoidal fits of time series of velocity recorded at $Ra=1.7\times {10}^5$, $Q=8.9\times {10}^5$, $Q/Ra=5.11$: (a) velocity $v_y$ perpendicular to the magnetic field (measuring depth $y=72\mathrm{mm}$), (b) velocity $v_x$ parallel to the magnetic field (measuring depth $x=40\mathrm{mm}$). The measuring positions are highlighted in Figs. 6 and 7, respectively.

Figure 9

Sections of instantaneous velocity profiles perpendicular to the magnetic field showing the varying roll size at different points in time: (a) UDV $u_{\perp ,2}$, (b) UDV $u_{\perp ,1}$. The labels roll2 and roll3 designate the second and third roll seen from the sensor position.

Figure 10

Characteristics of the velocity perpendicular to the magnetic field as first signs of a transition from a quasi-2D to a 3D flow recorded at $Ra=1.7\times {10}^5$, $Q=1.2\times {10}^5$, $Q/Ra=0.69$: (a) Sinusoidal fits of time series of velocity $v_y$ perpendicular to the magnetic field (measuring depth $y=72\mathrm{mm}$) [the measuring position is highlighted in Fig. 6], (b) comparison of the the time-averaged velocity profiles, $v_y$, for $u_{\perp ,2}$ and $u_{\perp ,3}$.

Figure 11

Characteristic properties of the velocity parallel to the magnetic field at $Ra=1.7\times {10}^5$, $Q=1.2\times {10}^5$, $Q/Ra=0.69$: (a) Spatio-temporal plot of the flow structure measured by sensor $u_{\left|\right|,2}$, (b) spatiotemporal plot of the flow structure measured by sensor $u_{\left|\right|,3}$, (c) Sinusoidal fits of time series of velocity $v_x$ parallel to the magnetic field (measuring depth $x=40\mathrm{mm}$), (d). Power spectrum of velocity time series obtained by FFT for sensor $u_{\left|\right|,3}$ at measuring positions $x=40\mathrm{mm}$ and $x=100\mathrm{mm}$. Both measuring positions are highlighted in (a) and (b).

Figure 12

(a) Time-averaged maximum velocity of the convection rolls as a function of Q, (b) corresponding values of the compensated Re number, $Re/R{a}^{0.5}$, plotted vs $Q/Ra$, (c) comparison of Q dependences of the compensated Re number and the extension of both the Hartmann and Bödewadt layers (d) comparison of Q dependences of Ha/Re and the boundary layer thicknesses for $Ra=7.8\times {10}^4$.

Figure 13

Dominant frequency of the oscillations $f_{os}$ and the ratio $f_{os}/f_{tu}$ vs Q for a selected Ra number $Ra=1.7\times {10}^5$.

Figure 14

Transitions in the parameter space: (a) critical parameter $R{a}_{os}\text{−}R{a}_c$ for the onset of oscillatory convection, (b) critical values $Q/Ra$ for both transitions from steady to oscillatory flow and from quasi-2D to 3D oscillations.