Linear global stability of a downward flow of liquid metal in a vertical duct under strong wall heating and transverse magnetic field

The linear global stability of a downward flow of liquid metal in a vertical duct under strong wall heating and a transverse magnetic field is examined numerically. The two-dimensional steady-state magnetohydrodynamic (MHD) mixed convection with an upward reverse flow is first computed by the finite-element method. Then linear global stability equations of the MHD mixed convection are derived and solved by the spatial discretization of the Taylor-Hood finite element and an implicitly restarted Arnoldi algorithm for the resulted generalized eigenvalue problem. Elevator and oscillatory unstable modes are revealed through the eigenspetrum computation of linear global stability equations. The elevator mode is found to be always unstable and independent of the basic flow profile, though a large magnetic field may suppress its growth rate. The unstable oscillatory mode is directly related to the basic upward reverse flow and first occurs at the specific flow structure which has an upward reverse flow near the heating wall and a downward flow near the opposite wall. The critical curves of the Grashof number with respect to the Hartmann number for the three-dimensional oscillatory mode are plotted and reveal that larger Hartmann numbers have larger critical Grashof numbers. Energy budget analyses are also performed and show that the shear Kelvin-Helmholtz instability due to the existence of an inflection point is the key instability mechanism of the three-dimensional oscillatory mode. The appearance of the unstable oscillatory mode may be regarded as an alternative physical explanation of the high-amplitude, low-frequency pulsations of temperature in the experiments and related numerical simulations.

Figure 1

Flow configuration for liquid metal flowing downward in a vertical duct under one-wall heating and a transverse magnetic field.

Figure 2

Field distributions of (a, c) the streamwise velocity $W$ and (b, d) the temperature $\mathrm{\Theta }$ along the cross section for the steady state of the downward mixed convection flow in a vertical duct at $\mathrm{Re}=5000, \mathrm{Ha}=200, \mathrm{Gr}=2\times {10}^6$, and $\mathrm{Pr}=0.0321$. Solid and dashed lines in (a) and (b) represent positive and negative values, respectively.

Figure 3

Field distributions of the streamwise velocity $W$ and temperature $\mathrm{\Theta }$ along the middle line $x=0$ of the cross section for the steady state of the downward mixed convection flow in a vertical duct with different Hartmann numbers at $\mathrm{Re}=5000, \mathrm{Gr}=2\times {10}^6$, and $\mathrm{Pr}=0.0321$.

Figure 4

Transition curves Gr-${\mathrm{Ha}}_t$ by which the parameter region is divided into two parts and two different flow structures are distinguished with different Reynolds number where $\mathrm{Pr}=0.0321$.

Figure 5

The growth rate of the most unstable elevator mode as a function of the Grashof number $\mathrm{Gr}$ with different Hartmann numbers at $\mathrm{Re}=5000$ and $\mathrm{Pr}=0.0321$.

Figure 6

The spatial structure of the unstable elevator mode with respect to the streamwise velocity $\widehat{w}$ and temperature $\widehat{\theta }$ in the cross section at $\mathrm{Re}=5000, \mathrm{Ha}=200, \mathrm{Gr}=2\times {10}^6$, and $\mathrm{Pr}=0.0321$.

Figure 7

The linear growth rate ${\lambda }_r$ and angular frequency ${\lambda }_i$ of the three-dimensional oscillatory unstable mode as a function of the wave number $k$ for different Hartmann numbers at $\mathrm{Re}=5000, \mathrm{Gr}=2\times {10}^6$, and $\mathrm{Pr}=0.0321$.

Figure 8

(a) The critical Grashof number ${\mathrm{Gr}}_c$ and (b) the critical wave number $k_c$ as a function of the Hartmann number for the three-dimensional oscillatory mode of the MHD mixed convection flow for different Reynolds numbers with $\mathrm{Pr}=0.0321$. Here, thick lines represent critical curves, and thin blue lines transition curves in Fig. 4.

Figure 9

Spatial structure of the three-dimensional oscillatory mode in the midplane section $x=0$ of the vertical duct for the perturbations of the transverse velocity in the $y$ direction (top row), the streamwise velocity in the $z$ direction (second row), the vorticity component in the $x$ direction (third row), and the temperature (bottom row) at a critical point of $k_c=0.4358, \mathrm{Gr}=3530593.55$ with $\mathrm{Re}=5000, \mathrm{Ha}=200$, and $\mathrm{Pr}=0.0321$.

Figure 10

Spatial distribution of the local streamwise shear of the basic flow $E_{\mathrm{sw}}^{\prime }$ and local buoyancy $E_b^{\prime }$ at a critical point of the three-dimensional oscillatory unstable mode with $\mathrm{Re}=5000, \mathrm{Ha}=200, {\mathrm{Gr}}_c=3530593.55, k_c=0.436$, and $\mathrm{Pr}=0.0321$.

Figure 11

Critical curve Ha-${\mathrm{Gr}}_c$ of the oscillatory mode (thick solid line) and contour lines of the linear growth rate of the elevator mode (thin red lines), where $\mathrm{Re}=5000$ and $\mathrm{Pr}=0.0321$.