Why, how, and when, MHD turbulence becomes two-dimensional
TL;DR: In this article, a description of MHD turbulence at low magnetic Reynolds number and large interaction parameter is proposed, in which attention is focussed on the role of insulating walls perpendicular to a uniform applied magnetic field.
Abstract: A description of MHD turbulence at low magnetic Reynolds number and large interaction parameter is proposed, in which attention is focussed on the role of insulating walls perpendicular to a uniform applied magnetic field. The flow is divided in two regions: the thin Hartmann layers near the walls, and the bulk of the flow. In the latter region, a kind of electromagnetic diffusion along the magnetic field lines (a degenerate form of Alfv6n waves) is displayed, which elongates the turbulent eddies in the field direction, but is not sufficient to generate a two-dimensional dynamics. However the normal derivative of velocity must be zero (to leading order) at the boundaries of the bulk region (as at a free surface), so that when the length scale 1, perpendicular to the magnetic field is large enough, the corresponding eddies are necessarily two-dimensional. Furthermore, if I, is not larger than a second limit, the Hartmann braking effect is negligible and the dynamics of these eddies is described by the ordinary Navier-Stokes equations without electromagnetic forces. MHD then appears to offer a means of achieving experiments on two-dimensional turbulence, and of deducing velocity and vorticity from measurements of electric field.
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TL;DR: In this paper, the authors present a review of the available information on two-dimensional turbulence, emphasizing on aspects accessible to experiment, and outlining contributions made on simple flow configurations, and open questions are made explicit.
449 citations
TL;DR: In this article, a quantitative experimental study of the two-dimensional inverse energy cascade is presented, where the flow is electrically driven in a horizontal layer of mercury and three-dimensional perturbations are suppressed by means of a uniform magnetic field.
Abstract: A quantitative experimental study of the two-dimensional inverse energy cascade is presented. The flow is electrically driven in a horizontal layer of mercury and three-dimensional perturbations are suppressed by means of a uniform magnetic field, so that the flow can be well approximated by a two-dimensional Navier–Stokes equation with a steady forcing term and a linear friction due to the Hartmann layer. Turbulence is produced by the instability of a periodic square network of 36 electrically driven alternating vortices. The inverse cascade is limited at large scales, either by the linear friction or by the finite size of the domain, depending on the experimental parameters. In the first case, spectra are measured and the corresponding two-dimensional Kolmogorov constant is in the range 3–7. In the second case, a condensation of the turbulent energy in the lowest mode, corresponding to a spontaneous mean global rotation, is observed. Such a condensation was predicted by Kraichnan (1967) from statistical thermodynamics arguments, but without the symmetry breaking. Random reversals of the rotation sense, owing to turbulent fluctuations, are more and more sparse as friction is decreased. The lowest mode fluctuations and the small scales are statistically independent.
371 citations
TL;DR: In this article, the authors provide a critical summary of recent work on turbulent flows from a unified point of view and present a classification of all known transfer mechanisms, including direct and inverse energy cascades.
315 citations
TL;DR: Magnetic fields can be used to melt, pump, stir, and stabilize liquid metals as mentioned in this paper, which provides a nonintrusive means of controlling the flow of metal in commercial casting and refining operations.
Abstract: ▪ Abstract Magnetic fields can be used to melt, pump, stir, and stabilize liquid metals. This provides a nonintrusive means of controlling the flow of metal in commercial casting and refining operations. The quest for greater efficiency and more control in the production of steel, aluminum, and high-performance superalloys has led to a revolution in the application of magnetohydrodynamics (MHD) to process metallurgy. Three typical applications are described here, chosen partially on the basis of their general interest to fluid dynamicists, and partially because of their considerable industrial importance. We look first at magnetic stirring, where a rotating magnetic field is used to agitate and homogenize the liquid zone of a partially-solidified ingot. This is a study in Ekman pumping. Next, we consider magnetic damping, where an intense, static magnetic field is used to suppress fluid motion. In particular, we look at the damping of jets, vortices, and turbulence. We conclude with a discussion of the ma...
223 citations
TL;DR: In this paper, a magnetohydrodynamic (MHD) body-force modified turbulent boundary layer was created by the interaction of a permanent magnetic field and an applied electric field from a magnet/electrode array integral to the surface of the plate.
Abstract: Single‐component velocity field measurements, mean and fluctuating wall shear stress measurements, and photographic flow visualizations have been made of a magnetohydrodynamic (MHD) body‐force modified turbulent boundary layer. The turbulent boundary layer flowed over a flat plate in salt water at zero pressure gradient; the MHD force was created by the interaction of a permanent magnetic field and an applied electric field from a magnet/electrode array integral to the surface of the plate. A MHD force, when applied to an electroconducting fluid and acting in a streamwise direction, can generate a near‐wall jet, decreasing the boundary layer thickness and suppressing the intensity of the turbulent fluctuations across the boundary layer. At very high interactions, the force causes an increase in mean wall shear and in turbulence; in the zero free‐stream velocity limit, the force acts as a pump. An increase in local skin friction, however, is offset by a grain in thrust due to the force. At moderate interac...
171 citations
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TL;DR: In this paper, it was shown that in spatially homogeneous two-dimensional turbulence, the mean square vorticity is unaffected by convection and can only decrease under the action of viscosity.
Abstract: Two‐dimensional and three‐dimensional turbulence have different properties, but both contain the two basic ingredients of randomness and convective nonlinearity, and some of the statistical hypotheses which have been proposed for three‐dimensional turbulence should be applicable to two‐dimensional motion. This justifies a numerical integration of the unaveraged equations of motion in two dimensions with random initial conditions as a means of testing the soundness of ideas such as those leading to the Kolmogoroff equilibrium theory. In spatially homogeneous two‐dimensional turbulence, the mean‐square vorticity is unaffected by convection and can only decrease under the action of viscosity. Consequently the rate of dissipation of energy tends to zero with the viscosity (v). On the other hand, the mean‐square vorticity gradient is increased by convective mixing, and it seems likely that the rate of decrease of mean‐square vorticity tends to a nonzero limit κ as ν → 0. This suggests the existence of a “casca...
901 citations
01 Jan 1953
TL;DR: In this paper, the steady motion of an electrically conducting, viscous fluid along channels in the presence of an imposed transverse magnetic field when the walls do not conduct currents was studied.
Abstract: This paper studies the steady motion of an electrically conducting, viscous fluid along channels in the presence of an imposed transverse magnetic field when the walls do not conduct currents. The equations which determine the velocity profile, induced currents and field are derived and solved exactly in the case of a rectangular channel. When the imposed field is sufficiently strong the velocity profile is found to degenerate into a core of uniform flow surrounded by boundary layers on each wall. The layers on the walls parallel to the imposed field are of a novel character. An analogous degenerate solution for channels of any symmetrical shape is developed. The predicted pressure gradients for given volumes of flow at various field strengths are finally compared with experimental results for square and circular pipes.
442 citations
TL;DR: In this paper, the authors used the test-field model of turbulence to compute the growth of prediction error for inertial-range turbulence in both three and two dimensions, and found that initial uncertainty in high wavenumbers spreads through the entire inertial range according to a similarity behavior.
Abstract: The test-field model of turbulence is used to compute the growth of prediction error for inertial-range turbulence in both three and two dimensions. It is found that initial uncertainty in high wavenumbers spreads through the entire inertial range according to a similarity behavior. For the energy inertial range, the time required for error to reach wavenumber k from very high wavenumbers is t=Aϵ−1/3k−2/3, where ϵ is the rate of energy transfer per unit mass and A≈10 in three dimensions or A≈2.5 in two dimensions. For the enstrophy inertial range in two dimensions the time for error to propagate from k′ down to k≪k′ (k and k′ both in the inertial range) is t≈4η−1/2{[ln(k′/ k1)]2/3−[ln(k/ k1)]2/3}, where η is the rate of enstrophy transfer and k1 marks the bottom of the enstrophy inertial range. Error growth is also computed for a two-dimensional spectrum that fits the energy spectrum of planetary waves in the atmosphere. An initial state determined with a horizontal resolution feasible with a sat...
275 citations