Linear stability of Hunt's flow

Abstract

We analyse numerically the linear stability of the fully developed flow of a liquid metal in a rectangular duct subject to a transverse magnetic field. The walls of the duct perpendicular to the magnetic field are perfectly conducting whereas the parallel ones are insulating. In a sufficiently strong magnetic field, the flow consists of two jets at the insulating walls and a near-stagnant core. We use a vector stream function formulation and Chebyshev collocation method to solve the eigenvalue problem for small-amplitude perturbations. Due to the two-fold reflection symmetry of the base flow the disturbances with four different parity combinations over the duct cross-section decouple from each other. Magnetic field renders the flow in a square duct linearly unstable at the Hartmann number Ha 5.7 with respect to a disturbance whose vorticity component along the magnetic field is even across the field and odd along it. For this mode, the minimum of the critical Reynolds number Re_c 2018, based on the maximal velocity, is attained at Ha ~ 10. Further increase of the magnetic field stabilises this mode with Re_c growing approximately as Ha. For Ha>40, the spanwise parity of the most dangerous disturbance reverses across the magnetic field. At Ha ~ 46 a new pair of most dangerous disturbances appears with the parity along the magnetic field being opposite to that of the previous two modes. The critical Reynolds number, which is very close for both of these modes, attains a minimum Re_c ~ 1130 at Ha ~ 70 and increases as Re_c ~ Ha^1/2 for Ha >> 1. The asymptotics of the critical wavenumber is k_c ~ 0.525Ha^1/2 while the critical phase velocity approaches 0.475 of the maximum jet velocity.Comment: 20 pages, 13 figures, production version, to appear in J. Fluid Mec

Linear stability of magnetohydrodynamic flow in a perfectly conducting rectangular duct

“…where E is the kinetic energy of perturbation averaged over the wavelength and the asterisk denotes the complex conjugate (Priede, Aleksandrova & Molokov 2010). At the moderate Hartmann number Ha = 15 considered above, most of the kinetic energy, i.e.…”

“…This is confirmed by the patterns of the critical perturbations, which are plotted over the duct cross-section in figure 6 for a moderate (Ha = 15) and a relatively strong (Ha = 100) magnetic field. As shown in our previous paper (Priede, Aleksandrova & Molokov 2010), the flow perturbation can be represented by the complex amplitudes of the streamwise (z) component of velocity (ŵ) and that of stream function (ψ z ), whose isolines are plotted in the left (x < 0) and the right (x > 0) sides of the cross-section, respectively. The real and imaginary parts of perturbations plotted at the top and bottom halves of the cross-section show the instant patterns shifted in time or in the stream-wise direction by a quarter of period or wavelength, respectively.…”

“…Linear stability of this flow is analysed using the same method as in our previous study (Priede, Aleksandrova & Molokov 2010). Since the method is based on a non-standard vector stream function formulation, it is briefly outlined in the Appendix.…”

“…The size of matrix for each eigenvalue problem is reduced by a factor of 16 in comparison to the original problem. Further details and the validation of the numerical method can be found in our previous paper (Priede, Aleksandrova & Molokov 2010).…”

“…1/2 andRe c ≈ 112 based on the maximum and average velocities, respectively (Priede, Aleksandrova & Molokov 2010). The low stability of Hunt's flow is due to two strong jets, which develop in a sufficiently strong magnetic field along the insulating walls and attain a velocity ∼ Ha relative to that of the core flow (Hunt 1965).…”