André Thess

Numerical study of the instability of the Hartmann layer

(Received 7 March 2003 and in revised form 1 July 2003) Direct numerical simulation is applied to investigate instability and transition to turbulence in the flow of an electrically conducting incompressible fluid between two parallel unbounded insulating walls affected by a wall-normal magnetic field (the Hartmann flow). The linear stability analysis of this flow provided unrealistically high critical Reynolds numbers, about two orders of magnitude higher than those observed in experiments. We propose an explanation based on the streak growth and breakdown mechanism described earlier for other shear flows. The mechanism is investigated using a two-step procedure that includes transient growth of two-dimensional optimal perturbations and the subsequent three-dimensional instability of the modulated streaky flow. In agreement with recent experimental investigations the calculations produce a critical range between 350 and 400 for the Hartmann thickness based Reynolds number, where the transition occurs at realistic amplitudes of two- and three-dimensional perturbations.

Theory of the Lorentz force flowmeter

A Lorentz force flowmeter is a device for the contactless measurement of flow rates in electrically conducting fluids. It is based on the measurement of a force on a magnet system that acts upon the flow. We formulate the theory of the Lorentz force flowmeter which connects the measured force to the unknown flow rate. We first apply the theory to three specific cases, namely (i) pipe flow exposed to a longitudinal magnetic field, (ii) pipe flow under the influence of a transverse magnetic field and (iii) interaction of a localized distribution of magnetic material with a uniformly moving sheet of metal. These examples provide the key scaling laws of the method and illustrate how the force depends on the shape of the velocity profile and the presence of turbulent fluctuations in the flow. Moreover, we formulate the general kinematic theory which holds for arbitrary distributions of magnetic material or electric currents and for any velocity distribution and which provides a rational framework for the prediction of the sensitivity of Lorentz force flowmeters in laboratory experiments and in industrial practice.

Instabilities in two‐dimensional spatially periodic flows. Part I: Kolmogorov flow

The linear stability of the parallel flow ψ0=sin(y) (Kolmogorov flow) is considered, taking into account viscosity, linear friction, and confinement (lateral walls). The computations provide neutral stability curves in the parameter space, wave numbers, and wave speeds, as well as the spatial structure of first unstable modes. Evidence is presented that stability parameters depend nonuniformly on the confinement. It is shown that already weak transverse confinement significantly decreases the longitudinal wavelength of perturbations at instability onset. Strong confinement changes the character of the instability into an oscillatory one instead of a purely exponential growing mode, which is obtained for weakly confined systems. Theoretical predictions of critical parameters are in reasonable agreement with experimental results in electromagnetically driven flows of conducting fluids.

Mean temperature profiles in turbulent Rayleigh–Bénard convection of water

We report an investigation of temperature profiles in turbulent Rayleigh–Benard convection of water based on direct numerical simulations (DNS) for a cylindrical cell with unit aspect ratio for the same Prandtl number Pr and similar Rayleigh numbers Ra as used in recent high-precision measurements by Funfschilling et al . ( J. Fluid Mech ., vol. 536, 2005, p. 145). The Nusselt numbers Nu computed for Pr = 4.38 and Ra = 10 8 , 3 × 10 8 , 5 × 10 8 , 8 × 10 8 and 10 9 are found to be in excellent agreement with the experimental data corrected for finite thermal conductivity of the walls. Based on this successful validation of the numerical approach, the DNS data are used to extract vertical profiles of the mean temperature. We find that near the heating and cooling plates the non-dimensional temperature profiles Θ( y ) (where y is the non-dimensional vertical coordinate), obey neither a logarithmic nor a power law. Moreover, we demonstrate that the Prandtl–Blasius boundary layer theory cannot predict the shape of the temperature profile with an error less than 7.9% within the thermal boundary layers (TBLs). We further show that the profiles can be approximated by a universal stretched exponential of the form Θ( y ) ≈ 1 − exp(− y − 0.5 y 2 ) with an absolute error less than 1.1% within the TBLs and 5.5% in the whole Rayleigh cell. Finally, we provide more accurate analytical approximations of the profiles involving higher order polynomials in the approximation.

Structure of the wake of a magnetic obstacle.

We use a combination of numerical simulations and experiments to elucidate the structure of the flow of an electrically conducting fluid past a localized magnetic field, called magnetic obstacle. We demonstrate that the stationary flow pattern is considerably more complex than in the wake behind an ordinary body. The steady flow is shown to undergo two bifurcations (rather than one) and to involve up to six (rather than just two) vortices. We find that the first bifurcation leads to the formation of a pair of vortices within the region of magnetic field that we call inner magnetic vortices, whereas a second bifurcation gives rise to a pair of attached vortices that are linked to the inner vortices by connecting vortices.

Transition from two-dimensional to three-dimensional magnetohydrodynamic turbulence

We report a theoretical investigation of the robustness of two-dimensional inviscid magnetohydrodynamic (MHD) flows at low magnetic Reynolds numbers with respect to three-dimensional perturbations. We use a combination of linear stability analysis and direct numerical simulations to analyse three problems, namely the flow in the interior of a triaxial ellipsoid, and two unbounded flows: a vortex with elliptical streamlines and a vortex sheet parallel to the magnetic field. The flow in a triaxial ellipsoid is found to present an exact analytical model which demonstrates both the existence of inviscid unstable three-dimensional modes and the stabilizing role of the magnetic field. The nonlinear evolution of the flow is characterized by intermittency typical of other MHD flows with long periods of nearly two-dimensional behaviour interrupted by violent three-dimensional transients triggered by the instability. We demonstrate, using the second model, that motion with elliptical streamlines perpendicular to the magnetic field becomes unstable with respect to the elliptical instability once the magnetic interaction parameter falls below a critical magnitude whose value tends to infinity as the eccentricity of the streamlines increases. Furthermore, the third model indicates that vortex sheets parallel to the magnetic field, which are unstable for any velocity and any magnetic field, emit eddies with vorticity perpendicular to the magnetic field. Whether the investigated instabilities persist in the presence of small but finite viscosity, in which case two-dimensional turbulence would represent a singular state of MHD flows, remains an open question.

Experimental study of liquid metal channel flow under the influence of a nonuniform magnetic field

We present an experimental study of a liquid metal flow in a rectangular channel under the influence of an inhomogeneous magnetic field. This is a fundamental problem of liquid metal magnetohydrodynamics that is relevant to the technique of electromagnetic braking in the process of continuous casting of steel as well as for Lorentz force velocimetry. Based on local velocity and electric potential measurements we identify three distinct flow regions; namely (i) a turbulence suppression region, (ii) a vortical region, and (iii) a wall jet region. It is shown that in region (i) the applied inhomogeneous magnetic field brakes the incoming flow in its central part and transforms the velocity profile, which is initially flat, into an M-shaped form. In the central part of the flow, the intensity of the velocity fluctuations is found to decrease strongly. In region (ii) where the magnetic field is strongest, the flow is characterized by large-scale vortical structures which are time dependent for certain values o...

Large-scale intermittency of liquid-metal channel flow in a magnetic field

We predict a novel flow regime in liquid metals under the influence of a magnetic field. It is characterized by long periods of nearly steady, two-dimensional flow interrupted by violent three-dimensional bursts. Our prediction has been obtained from direct numerical simulations in a channel geometry at low magnetic Reynolds number and translates into physical parameters which are amenable to experimental verification under laboratory conditions. The new regime occurs in a wide range of parameters and may have implications for metallurgical applications.

Lorentz force sigmometry: A contactless method for electrical conductivity measurements

The present communication reports a new technique for the contactless measurement of the specific electrical conductivity of a solid body or an electrically conducting fluid. We term the technique “Lorentz force sigmometry” where the neologism “sigmometry” is derived from the Greek letter sigma, often used to denote the electrical conductivity. Lorentz force sigmometry (LoFoS) is based on similar principles as the traditional eddy current testing but allows a larger penetration depth and is less sensitive to variations in the distance between the sensor and the sample. We formulate the theory of LoFoS and compute the calibration function which is necessary for determining the unknown electrical conductivity from measurements of the Lorentz force. We conduct a series of experiments which demonstrate that the measured Lorentz forces are in excellent agreement with the numerical predictions. Applying this technique to an aluminum sample with a known electrical conductivity of rAl ¼ 20:4MS=m and to a copper sample with rCu ¼ 57:92MS=m we obtain rAl ¼ 21:59MS=m and rCu ¼ 60:08MS=m, respectively. This demonstrates that LoFoS is a convenient and accurate technique that may find application in process control and thermo-physical property measurements for solid and liquid conductors. V C 2012 American Institute of Physics.

Lorentz force velocimetry based on time-of-flight measurements

Lorentz force velocimetry (LFV) is a contactless technique for the measurement of liquid metal flowrates. It consists of measuring the force acting upon a magnetic system and arising from the interaction between an external magnetic field and the flow of an electrically conducting fluid. In this study, a new design is proposed so as to make the measurement independent of the fluid’s electrical conductivity. It is made of one or two coils placed around a circular pipe. The forces produced on each coil are recorded in time as the liquid metal flows through the pipe. It is highlighted that the auto- or cross-correlation of these forces can be used to determine the flowrate. The reliability of the flowmeter is first investigated with a synthetic velocity profile associated with a single vortex ring, which is convected at a constant speed. This configuration is similar to the movement of a solid rod and enables a simple analysis of the flowmeter. Then, the flowmeter is applied to a realistic three-dimensional ...