Instabilities and Transition in Magnetohydrodynamic Flows in Ducts with Electrically Conducting Walls

This Letter presents a numerical study of a magnetohydrodynamic flow in a square duct with electrically conducting walls subject to a uniform, transverse magnetic field. Two regimes of instability and transition of Hunt’s jets at the walls parallel to the magnetic field have been identified. The first one occurs for relatively low values of the Reynolds number Re and is associated with weak, periodic, counterrotating vortices discovered previously in linear stability studies. The second is a new regime taking place for higher values of Re. It is associated with trains of small-scale vortices enveloped into larger structures, and involves partial detachment of jets from parallel walls. Once this regime sets in, the kinetic energy of perturbations increases by 2 orders of magnitude.

Figure 1

Base velocity profile for a square duct with thin conducting walls for $c=0.5$ and for $\mathrm{Ha}=200$.

Figure 2

Kinetic energy density of perturbations averaged in time as a function of Re.

Figure 3

Mean axial velocity profiles for several values of Re.

Figure 4

Instantaneous contours of axial component of (a) the total axial velocity $u_1$ and (b) the transverse disturbance velocity $u_3^{\prime }$ for $\mathrm{Re}=5000$. (c) Instantaneous isolines of the $y$ component of the disturbance vorticity ${\omega }_2^{\prime }=(\mathbf{\nabla }\times {\mathbf{u}}^{\prime }{)}_2$ (red or gray and blue or dark gray lines) and mean velocity profile (black line). (c) is limited to the dashed box shown in (a).

Figure 5

Third-order moment ${\overline{u}}_1^{\prime 3}$ compared to mean velocity profile for $\mathrm{Re}=5000$.

Figure 6

Coherent structures in the outer jet region for $\mathrm{Re}=5000$, represented by isosurfaces of ${\lambda }_2=-0.015$.