Alban Pothérat

Numerical simulations of a cylinder wake under a strong axial magnetic field

We study the flow of a liquid metal in a square duct past a circular cylinder in a strong externally imposed magnetic field. In these conditions, the flow is quasi-two-dimensional, which allows us to model it using a two-dimensional (2D) model. We perform a parametric study by varying the two control parameters Re and Ha (Ha2 is the ratio of Lorentz to viscous forces) in the ranges [0…6000] and [0…2160], respectively. The flow is found to exhibit a sequence of four regimes. The first three regimes are similar to those of the non-magnetohydrodynamic (non-MHD) 2D circular wake, with transitions controlled by the friction parameter Re∕Ha. The fourth one is characterized by vortices raising from boundary layer separations at the duct side walls, which strongly disturbs the Karman vortex street. This provides the first explanation for the breakup of the 2D Karman vortex street first observed experimentally by Frank, Barleon, and Muller [Phys. Fluids 13, 2287 (2001)]. We also show that, for high values of Ha (Ha⩾1120), the transition to the fourth regime occurs for Re∝0.56Ha, and that it is accompanied by a sudden drop in the Strouhal number. In the first three regimes, we show that the drag coefficient and the length of the steady recirculation regions located behind the cylinder are controlled by the parameter Re∕Ha4∕5. Also, the free shear layer that separates the recirculation region from the free stream is similar to a free MHD parallel layer, with a thickness of the order of Ha−1∕2 that is quite different to that of the non-MHD case, and therefore strongly influences the dynamics of this region. We also present one case at Re=3×104 and Ha=1120, where this layer undergoes an instability of the Kelvin–Helmholtz-type.We study the flow of a liquid metal in a square duct past a circular cylinder in a strong externally imposed magnetic field. In these conditions, the flow is quasi-two-dimensional, which allows us to model it using a two-dimensional (2D) model. We perform a parametric study by varying the two control parameters Re and Ha (Ha2 is the ratio of Lorentz to viscous forces) in the ranges [0…6000] and [0…2160], respectively. The flow is found to exhibit a sequence of four regimes. The first three regimes are similar to those of the non-magnetohydrodynamic (non-MHD) 2D circular wake, with transitions controlled by the friction parameter Re∕Ha. The fourth one is characterized by vortices raising from boundary layer separations at the duct side walls, which strongly disturbs the Karman vortex street. This provides the first explanation for the breakup of the 2D Karman vortex street first observed experimentally by Frank, Barleon, and Muller [Phys. Fluids 13, 2287 (2001)]. We also show that, for high values of Ha (H...

Numerical simulations of an effective two-dimensional model for flows with a transverse magnetic field

This paper presents simulations of the two-dimensional model developed by Poth´ erat et al. (2000) for MHD flows between two planes with a strong transverse homogeneous and steady magnetic field, accounting for moderate inertial effects in Hartmann layers. We first show analytically how the additional terms in the equations of motion accounting for inertia soften velocity gradients in the horizontal plane, and then we implement the model in a code to carry out numerical simulations to be compared with available experimental results. This comparison shows that the new model can give very accurate results as long as the Hartmann layer remains laminar. Both experimental velocity profiles and global angular momentum measurements are recovered, and local and global Ekman recirculations are shown to alter significantly the shape of the flow as well as the global dissipation.

Formation mechanism of hairpin vortices in the wake of a truncated square cylinder in a duct

We investigate the laminar shedding of hairpin vortices in the wake of a truncated square cylinder placed in a duct, for Reynolds numbers around the critical threshold of the onset of vortex shedding. We single out the formation mechanism of the hairpin vortices by means of a detailed analysis of the flow patterns in the steady regime. We show that unlike in previous studies of similar structures, the dynamics of the hairpin vortices is entwined with that of the counter-rotating pair of streamwise vortices, which we found to be generated in the bottom part of the near wake (these are usually referred to as base vortices). In particular, once the hairpin structure is released, the base vortices attach to it, forming its legs, so these are streamwise, and not spanwise as previously observed in unconfined wakes or behind cylinders of lower aspect ratios. We also single out a trail of Omega-shaped vortices, generated between successive hairpin vortices through a mechanism that is analogous to that active in near-wall turbulence. Finally, we show how the dynamics of the structures we identified determine the evolution of the drag coefficients and Strouhal numbers when the Reynolds number varies.

Direct numerical simulations of low-Rm MHD turbulence based on the least dissipative modes

We present a new spectral method for the direct numerical simulation of magnetohydrodynamic turbulence at low magnetic Reynolds number. The originality of our approach is that instead of using traditional bases of functions, it relies on the basis of eigenmodes of the dissipation operator, which represents viscous and Joule dissipation. We apply this idea to the simple case of a periodic domain in the three directions of space, with a homogeneous magnetic field in the e z direction. The basis is then still a subset of the Fourier space, but ordered by growing linear decay rate |λ| (i.e. according to the least dissipative modes). We show that because the lines of constant energy tend to follow those of constant |λ| in the Fourier space, the scaling for the smallest scales |λ max | in a forced flow can be expressed, using this single parameter, as a function of the Reynolds number as , where k f is the forcing wavelength, or as a function of the Grashof number G f , which gives a non-dimensional measure of the forcing, as |λ max | 1/2 /(2π k f ) ≃ 0.47 G f 0.20 . This scaling is also found to be consistent with heuristic scalings derived by Alemany et al . ( J. Mec. , vol. 18, 1979, pp. 277–313) and Potherat & Alboussiere ( Phys. Fluids , vol. 15, 2003, pp. 3170–3180) for interaction parameter S ≳ 1, and which we are able to numerically quantify as k ⊥ max / k f ≃ 0.5 Re 1/2 and k z max / k f ≃ 0.8 k f Re / Ha . Finally, we show that the set of least dissipative modes gives a relevant prediction for the scale of the first three-dimensional structure to appear in a forced, initially two-dimensional turbulent flow. This completes our numerical demonstration that the least dissipative modes can be used to simulate both two- and three-dimensional low- Rm magnetohydrodynamic (MHD) flows.

Dimensionality, secondary flows and helicity in low-Rm MHD vortices

In this paper, we examine the dimensionality of a single electrically driven vortex bounded by two no-slip and perfectly insulating horizontal walls a distance h apart. The study was performed in the weakly inertial limit by means of an asymptotic expansion, which is valid for any Hartmann number. We show that the dimensionality of the leading order can be fully described using the single parameter l(z)(nu)/h, where l(z)(nu) represents the distance over which the Lorentz force is able to act before being balanced by viscous dissipation. The base flow happens to introduce inertial recirculations in the meridional plane at the first order, which are shown to follow two radically different mechanisms: inverse Ekman pumping driven by a vertical pressure gradient along the axis of the vortex, or direct Ekman pumping driven by a radial pressure gradient in the Hartmann boundary layers. We demonstrate that when the base flow is quasi-2D, the relative importance of direct and inverse pumping is solely determined by the aspect ratio eta/h, where eta refers to the width of the vortex. Of the two mechanisms, only inverse pumping appears to act as a significant source of helicity.

Three-dimensionality in quasi?two-dimensional flows: Recirculations and Barrel effects

A scenario is put forward for the appearance of three-dimensionality both in quasi-2D rotating flows and quasi-2D magnetohydrodynamic (MHD) flows. We show that 3D recirculating flows and currents originate in wall boundary layers and that, unlike in ordinary hydrodynamic flows, they cannot be ignited by confinement alone. They also induce a second form of three-dimensionality with quadratic variations of velocities and current across the channel. This scenario explains both the common tendency of these flows to two-dimensionality and the mechanisms of the recirculations through a single formal analogy covering a wide class of flows including rotating and MHD flows. These trans-disciplinary effects are thus active in atmospheres, oceans or the cooling blankets of nuclear fusion reactors.

Spatial evolution of a quasi-two-dimensional Kármán vortex street subjected to a strong uniform magnetic field

A vortex decay model for predicting spatial evolution of peak vorticity in a wake behind a cylinder is presented. For wake vortices in the stable region behind the formation region, results have shown that the presented model has a good capability of predicting spatial evolution of peak vorticity within an advecting vortex across 0.1 ≤ β ≤ 0.4, 500 ≤ H ≤ 5000, and 1500 ≤ ReL ≤ 8250. The model is also generalized to predict the decay behaviour of wake vortices in a class of quasi-two-dimensional magnetohydrodynamic duct flows. Comparison with published data demonstrates remarkable consistency.

A shallow water model for magnetohydrodynamic flows with turbulent Hartmann layers

We establish a shallow water model for flows of electrically conducting fluids in homogeneous static magnetic fields that are confined between two parallel planes where turbulent Hartmann layers are present. This is achieved by modelling the wall shear stress in these layers using the Prandtls mixing length model, as did the authors of Albousssiere \& Lingwood (Phys. Fluids, 2000). The idea for this new model arose from the failure of previous shallow water models that assumed a laminar Hartmann layer to recover the correct amount of dissipation found in some regimes of the MATUR experiment. This experiment, conducted by the authors of Messadek \& Moreau (J. Fluid Mech. 2002), consisted of a thin layer of mercury electrically driven in differential rotation in a transverse magnetic field. Numerical Simulations of our new model in the configuration of this experiment allowed us to recover experimental values of both the global angular momentum and the local velocity up to a few percent when the Hartmann layer was in a sufficiently well developed turbulent state. We thus provide an evidence that the unexplained level of dissipation observed in MATUR in these specific regimes was caused by turbulence in the Hartmann layers. A parametric analysis of the flow, made possible by the simplicity of our model, also revealed that turbulent friction in the Hartmann layer prevented quasi-2D turbulence from becoming more intense and limited the size of the large scales.

Onset of plane layer magnetoconvection at low Ekman number

We study the onset of magnetoconvection between two infinite horizontal planes subject to a vertical magnetic field aligned with background rotation. In order to gain insight into the convection taking place in the Earths tangent cylinder, we target regimes of asymptotically strong rotation. The critical Rayleigh number Ra-c and critical wavenumber a(c) are computed numerically by solving the linear stability problem in a systematic way, with either stress-free or no-slip kinematic boundary conditions. A parametric study is conducted, varying the Ekman number E (ratio of viscous to Coriolis forces) and the Elsasser number. (ratio of the Lorentz force to the Coriolis force). E is varied from 10(-9) to 10(-2) and. from 10(-3) to 1. For a wide range of thermal and magnetic Prandtl numbers, our results verify and confirm previous experimental and theoretical results showing the existence of two distinct unstable modes at low values of E-one being controlled by the magnetic field, the other being controlled by viscosity (often called the viscous mode). It is shown that oscillatory onset does not occur in the range of parameters we are interested in. Asymptotic scalings for the onset of these modes are numerically confirmed and their domain of validity is precisely quantified. We show that with no-slip boundary conditions, the asymptotic behavior is reached for E < 10(-6) and establish a map in the (E, Lambda) plane. We distinguish regions where convection sets in either through the magnetic mode or through the viscous mode. Our analysis gives the regime in which the transition between magnetic and viscous modes may be observed. We also show that within the asymptotic regime, the role played by the kinematic boundary conditions is minimal

Controlling the dimensionality of low-Rm MHD turbulence experimentally

This paper introduces an experimental apparatus, which drives turbulence electrically in a liquid metal pervaded by a high magnetic field. Unlike past magnetohydrodynamic setups involving a shallow confinement, the experiment presented here drives turbulence whose dimensionality can be set anywhere between three-dimensional and quasi two-dimensional. In particular, we show that the dimensionality and componentality of the turbulence thus generated are in fact completely fixed by the single parameter