Linear stability of magnetohydrodynamic flow in a square duct with thin conducting walls

摘要

This study is concerned with numerical linear stability analysis of liquidmetal flow in a square duct with thin electrically conducting walls subject toa uniform transverse magnetic field. We derive an asymptotic solution for thebase flow which is valid not only for high but also moderate magnetic fields.This solution shows that for low wall conductance ratios $c\ll1,$ an extremelystrong magnetic field with the Hartmann number $Ha\sim c^{-4}$ is required toattain the asymptotic flow regime considered in the previous studies. We use avector stream function/vorticity formulation and a Chebyshev collocation methodto solve the eigenvalue problem for three-dimensional small-amplitudeperturbations in ducts with realistic wall conductance ratios $c=1,0.1,0.01$and Hartmann numbers up to $10^{4}.$ As for similar flows, instability in asufficiently strong magnetic field is found to occur in the side-wall jets withthe characteristic thickness $\delta\sim Ha^{-1/2}.$ This results in thecritical Reynolds number and wavenumber increasing asymptotically with themagnetic field as $Re_{c}\sim110Ha^{1/2}$ and $k_{c}\sim0.5Ha^{1/2}.$ Therespective critical Reynolds number based on the total volume flux in a squareduct with $c\ll1$ is $\bar{Re}_{c}\approx520.$ Although this value is somewhatlarger than$\bar{Re}_{c}\approx313$ found by Ting et al. (1991) for theasymptotic side-wall jet profile, it still appears significantly lower than theReynolds numbers at which turbulence is observed in experiments as well as indirect numerical simulations of this type of flows.