Thierry Alboussiere

Experimental study of the instability of the Hartmann layer

Hartmann layers are a common feature in magnetohydrodynamics, where they organize the electric current distribution in the flow and hence the characteristics of the velocity field. In spite of their importance their stability properties are not well understood, mainly because of the scarcity of experimental data. In this work we investigated experimentally the transition to turbulence in the Hartmann layers that arise in magnetohydrodynamic flows in ducts. From measurements of the friction factor a well-marked transition to turbulence was found at a critical Reynolds number, based on the laminar Hartmann layer thickness, of approximately 380, valid also for laminarization and for a wide range of intensities of the magnetic field. The sensitivity of this result to the roughness characteristics of the walls along which the Hartmann layers develop confirms that these layers are related to the transition observed and provides more information on its stability properties.

Experimental study of a vortex in a magnetic field

It is well-known that magnetohydrodynamic (MHD) flows behave differently from conventional fluid flows in two ways: the magnetic field makes the flow field anisotropic in the sense that it becomes independent of the coordinate parallel to the field; and the flow of liquid across the field lines induces an electric current, leading to ohmic damping. In this paper, an experimental study is presented of the long-time decay of an initially three-dimensional flow structure subject to a steady magnetic field, when the ratio of the electromagnetic Lorentz forces to the nonlinear inertial forces, quantified by the magnetic interaction parameter, N 0 , takes large as well as moderate values. This investigation is markedly different from previous studies on quasi-two-dimensional MHD flows in thin layers of conducting fluids, where only Hartmann layer friction held the key to the dissipation of the flow. The initial ‘linear’ phase of decay of an MHD flow, characterized by dominant Lorentz forces and modelled extensively in the literature, has been observed for the first time in a laboratory experiment. Further, when N 0 is large compared to unity, a distinct regime of decay of a vortex follows this linear phase. This interesting trend can be explained in terms of the behaviour of the ratio of the actual magnitudes of the Lorentz to the nonlinear inertial forces – the true interaction parameter – which decreases to a constant of order unity towards the end of the linear phase of decay, and remains invariant during a subsequent ‘nonlinear’ phase.

Weakly nonlinear stability of Hartmann boundary layers

Abstract By means of a weakly nonlinear stability analysis it is shown that the Hartmann boundary layer presents subcritical instability in the proximity of the minimum linear critical Reynolds number. This gives further support to earlier speculations that finite amplitude effects account for the discrepancies between the results of the linear stability analysis and experimental evidence on laminarisation.

Evolution of a vortex in a magnetic field

Abstract The evolution of a single vortex in an electrically-conducting liquid, subject to a uniform magnetic field acting parallel to the axis of the vortex, is investigated by an order-of-magnitude analysis and a numerical model. The non-linear phase of decay, wherein the Lorentz and the inertial forces are of the same order of magnitude, is studied in detail. As the kinetic energy decays primarily due to Joule dissipation, the vortex evolves in such a way that the component of angular momentum parallel to the direction of the magnetic field is conserved. If the true interaction parameter, N t , which denotes the actual ratio of the Lorentz to the inertial forces, is assumed to be a constant of order unity in the non-linear regime, the evolution of the vortex can be fully described. The above assumption is proven to be correct not only from the values of N t obtained in the numerical simulation, but also from the good agreement between the theoretical and numerically-obtained energy decay laws for the non-linear phase, at finite time. In addition, the true interaction parameter proves to be useful in estimating the minimum magnetic field strength required for stable evolution of a swirling vortex.

Quasi characteristic MHD flows

Abstract At small magnetic Reynolds number, a two-dimensional model is proposed for MHD flows in a nonuniform magnetic field and in a cavity of nonuniform depth in the direction of the magnetic field. The characteristic surfaces appear when deriving the model and play a crucial role in the resulting solutions. A new type of free shear layers are found for the first time, developing along such surfaces, of thickness of order Ha −1/4 .

Rayleigh–Bénard stability and the validity of quasi-Boussinesq or quasi-anelastic liquid approximations

The linear stability threshold of the Rayleigh-Benard configuration is analyzed with compressible effects taken into account. It is assumed that the fluid obeys a Newtonian rheology and Fouriers law of thermal transport with constant, uniform (dynamic) viscosity and thermal conductivity in a uniform gravity field. Top and bottom boundaries are maintained at different constant temperatures and we consider here boundary conditions of zero tangential stress and impermeable walls. Under these conditions, and with the Boussinesq approximation, Rayleigh (1916) first obtained analytically the critical value 27pi^4/4 for a dimensionless parameter, now known as the Rayleigh number, at the onset of convection. This manuscript describes the changes of the critical Rayleigh number due to the compressibility of the fluid, measured by the dimensionless dissipation parameter D and due to a finite temperature difference between the hot and cold boundaries, measured by a dimensionless temperature gradient a. Different equations of state are examined: ideal gas equation, Murnaghans model and a generic equation of state, which can represent any possible equation of state. We also consider two variations of this stability analysis. In a so-called quasi-Boussinesq model, we consider that density perturbations are solely due to temperature perturbations. In a so-called quasi-anelastic liquid approximation, we consider that entropy perturbations are solely due to temperature perturbations. In addition to the numerical Chebyshev-based stability analysis, an analytical approximation is obtained when temperature fluctuations are written as a combination of only two modes. This analytical expression allows us to show that the superadiabatic critical Rayleigh number quadratic departure in a and D from 27pi^4/4 involves the expansion of density up to the degree three in terms of pressure and temperature.