Liquid-metal MHD flow in rectangular ducts with thin conducting or insulating walls: laminar and turbulent solutions

Abstract

This paper treats the steady, fully-developed flow of a liquid metal in a rectangular duct of constant cross-section with a uniform, transverse magnetic field. Thin conducting wall boundary conditions at the top/bottom walls (perpendicular to the magnetic field) are extended to allow electrical currents to return through either the wall or the Hartmann layers. Hence, a unified analysis of flows in ducts with wall conductance ratios in the range of interest of fusion blanket applications, namely, from thin conducting to insulating wall ducts, is conducted. The flow in laminar and turbulent regimes is investigated through a composite core-side-layer spectral collocation solution which explicitly resolves the flow in the side layers (parallel to the magnetic field) even for very large Hartmann numbers. Turbulent profiles are obtained through an iterative scheme in which turbulence is introduced through an eddy viscosity model from the renormalization group theory of turbulence [Yakhot, V. and Orsag, S.A., J. Sci. Comput., 1986, 1 (1), 3]. The transition from a flow in a duct with thin conducting walls to one with insulating walls is clearly displayed by varying the wall conductance ratio from 0.05 to 0 for Hartmann numbers in the range 10 3–10 5. In turbulent regime, Reynolds numbers vary in the range 5 × 10 4–5 × 10 5. For thin conducting wall duct flows, turbulence is concentrated in the increased side layers while the core remains unperturbed. In insulating wall ducts, the flow remains in the laminar regime within the considered range of Reynolds numbers.