P. A. Davidson

Swirling recirculating flow in a liquid metal column generated by a rotating magnetic field

In this paper we consider theoretical and experimental aspects of axisymmetric, swirling flow which is generated in a column of liquid metal by a rotating magnetic field. Two cases are discussed, one in which there is no axial variation in the stirring force, and one where the body force is restricted to a relatively short length of the column. The latter case is of considerable practical interest in continuous casting. One-dimensional stirring, where the swirl is independent of z and θ, is well understood. The magnetic body force is balanced by shear, all inertial forces being zero (except for the centripetal acceleration). However, in two-dimensional axisymmetric stirring, the axial variation in swirl drives a strong secondary poloidal flow. The principal local force balance is between the magnetic torque and inertia. The body force spins up the fluid as it passes through the forced region and the secondary flow sweeps this angular momentum into the unforced region. Consequently, the size and distribution of the swirl is controlled by the secondary flow. The role of wall friction is considered and shown to control the length of the recirculating eddy. An approximate solution of the inviscid equations of motion, based on the angular momentum integral, is derived for the flow in the forced region. This is compared with the results of numerical experiments. The analysis predicts that the swirl velocity scales on { B (σ/ρω) ½ }ω R , has a maximum at the bottom of the driven region, and penetrates an axial distance of the order ℝ R away from the forced region. (For turbulent flow the Reynolds number ℝ must be based on an effective eddy viscosity.) All these features were reproduced experimentally.

Is grid turbulence Saffman turbulence

There has been a longstanding debate as to whether the large scales in grid turbulence should be classified as of the Batchelor or Saffman type. In the former, the integral scales, u and l, satisfy u 2 l 5 ≈ constant , while in Saffman turbulence we have u 2 l 3 = constant . For strictly homogeneous turbulence the energy decay rates in these two types of turbulence differ, with u 2 ~ t −10/7 in Batchelor turbulence and u 2 ~ t −6/5 in Saffman turbulence. We present high-resolution measurements of grid turbulence taken in a large wind tunnel. The particularly large test section allows us to measure energy decay exponents with high accuracy. We find that the turbulence behind the grid is almost certainly of the Saffman type, with u 2 l 3 = constant . The measured energy decay exponent, however, is found to lie slightly below the theoretical prediction of u 2 ~ t −1.2 . Rather we find u 2 ~ t − n , with n = 1.13±0.02. This discrepancy is shown to arise from a weak temporal decay of the dimensionless energy dissipation coefficient, el/ u 3 , which is normally taken to be constant in strictly homogeneous turbulence, but which varies very slowly in grid turbulence.

Swirling flow in an axisymmetric cavity of arbitrary profile, driven by a rotating magnetic field

We investigate the swirling flow of liquid metal in an axisymmetric cavity of arbitrary profile, generated by a rotating magnetic field. In addition to the primary swirling motion, a recirculation is generated by the Bodewadt-like boundary layers on the inclined sides of the cavity. As in the classic problem of «spin-up» in a cylinder, this secondary flow has a dominating effect over the distribution of angular momentum. In the inviscid core, the angular momentum is independent of z, the axial coordinate, and the applied body force is balanced by the Coriolis force

Magnetic damping of jets and vortices

It is well known that the imposition of a static magnetic field tends to suppress motion in an electrically conducting liquid. Here we look at the magnetic damping of liquid-mental flows where the Reynolds number is large and the magnetic Reynolds number is small. The magnetic field is taken as uniform and the fluid is either infinite in extent or else bounded by an electrically insulating surface S . Under these conditions, we find that three general principles govern the flow. First, the Lorentz force destroys kinetic energy but does not alter the net linear momentum of the fluid, nor does it change the component of angular momentum parallel to B . In certain flows, this implies that momentum, linear or angular, is conserved. Second, the Lorentz force guides the flow in such a way that the global Joule dissipation, D , decreases, and this decline in D is even more rapid than the corresponding fall in global kinetic energy, E . (Note that both D and E are quadratic in u ). Third, this decline in relative dissipation, D / E , is essential to conserving momentum, and is achieved by propagating linear or angular momentum out along the magnetic field lines. In fact, this spreading of momentum along the B -lines is a diffusive process, familiar in the context of MHD turbulence. We illustrate these three principles with the aid of a number of specific examples. In increasing order of complexity we look at a spatially uniform jet evolving in time, a three-dimensional jet evolving in space, and an axisymmetric vortex evolving in both space and time. We start with a spatially uniform jet which is dissipated by the sudden application of a transverse magnetic field. This simple (perhaps even trivial) example provides a clear illustration of our three general principles. It also provides a useful stepping-stone to our second example of a steady three-dimensional jet evolving in space. Unlike the two-dimensional jets studied by previous investigators, a three-dimensional jet cannot be annihilated by magnetic braking. Rather, its cross-section deforms in such a way that the momentum flux of the jet is conserved, despite a continual decline in its energy flux. We conclude with a discussion of magnetic damping of axisymmetric vortices. As with the jet flows, the Lorentz force cannot destroy the motion, but rather rearranges the angular momentum of the flow so as to reduce the global kinetic energy. This process ceases, and the flow reaches a steady state, only when the angular momentum is uniform in the direction of the field lines. This is closely related to the tendency of magnetic fields to promote two-dimensional turbulence.

Structure formation in homogeneous freely decaying rotating turbulence

One of the most striking features of rotating turbulence is the inevitable appearance of large-scale columnar structures. Whilst these structures are frequently observed, the processes by which they are created are still poorly understood. In this paper we consider the emergence of these structures from freely decaying, rotating turbulence with Ro ∼ 1. Our study follows the conjecture by Davidson, Staplehurst & Dalziel ( J. Fluid Mech. , vol. 557, 2006, p. 135) that the structure formation may be due to linear inertial wave propagation, which was shown to be consistent with the growth of columnar eddies in inhomogeneous turbulence. Here we extend that work and consider the case of homogeneous turbulence. We describe laboratory experiments where homogeneous turbulence is created in a rotating tank. The turbulence is generated with Ro ∼ 1, and as the energy decays, the formation of columnar vortices is observed. The axial growth of these columnar structures is then measured using two-point correlations and in all cases the results are consistent with structure formation via linear inertial wave propagation. In particular, we obtain a self-similar collapse of the two-point correlations when the axial coordinate is normalized by Ω tb , where b is a measure of the integral scale in the horizontal plane and Ω is the rotation rate. Although our results do not exclude the possibility of significant nonlinear dynamics, they are consistent with the conjecture of Davidson et al . (2006) that linear dynamics play a strong guiding hand in structure formation.

Freely decaying, homogeneous turbulence generated by multi-scale grids

We investigate wind-tunnel turbulence generated by both conventional and multi-scale grids. Measurements were made in a tunnel which has a large test section, so that possible side wall effects are very small and the length ensures that the turbulence has time to settle down to a homogeneous shear-free state. The conventional and multi-scale grids were all designed to produce turbulence with the same integral scale, so that a direct comparison could be made between the different flows. Our primary finding is that the behaviour of the turbulence behind our multi-scale grids is virtually identical to that behind the equivalent conventional grid. In particular, all flows exhibit a power-law decay of energy, u 2 ~ t − n , where n is very close to the classical Saffman exponent of n = 6/5. Moreover, all spectra exhibit classical Kolmogorov scaling, with the spectra collapsing on the integral scales at small k , and on the Kolmogorov microscales at large k . Our results are at odds with some other experiments performed on similar multi-scale grids, where significantly higher energy decay exponents and turbulence levels have been reported.

The role of angular momentum in the magnetic damping of turbulence

Landau & Lifshitz showed that Kolmogorovs E ∼ t −10/7 law for the decay of isotropic turbulence rests on just two physical ideas: ( a ) the conservation of angular momentum, as expressed by Loitsyanskys integral; and ( b ) the removal of energy from the large scales via the energy cascade. Both Kolmogorovs original analysis and Landau & Lifshitzs reinterpretation in terms of angular momentum are now known to be flawed. The existence of long-range velocity correlations means that Loitsyanskys integral is not an exact representation of angular momentum, nor is it strictly conserved. However, in practice the long-range velocity correlations are weak and Loitsyanskys integral is almost constant, so that the Kolmogorov/Landau model provides a surprisingly simple and robust description of the decay. In this paper we redevelop these ideas in the context of MHD turbulence. We take advantage of the fact that the angular momentum of a fluid moving in a uniform magnetic field has particularly simple properties. Specifically, the component parallel to the magnetic field is conserved while the normal components decay exponentially on a time scale of τ=ρ/σ B 2 . We show that the counterpart of Loitsyanskys integral for MHD turbulence is ∫ x 2 ⊥ Q ⊥ d x , where Q ij is the velocity correlation. When the long-range correlations are weak this integral is conserved. This provides an estimate of the rate of decay of energy. At low values of magnetic field we recover Kolmogorovs law. At high values we find E ∼ t −1/2 , which is a result derived earlier by Moffatt. We also show that ∫ x 2 ⊥ Q ∥ d x decays exponentially on a time scale of τ. We interpret these results in terms of the behaviour of isolated vortices orientated normal and parallel to the magnetic field.

Anisotropy of magnetohydrodynamic turbulence at low magnetic Reynolds number

Turbulent fluctuations in magnetohydrodynamic flows are known to become anisotropic under the action of a sufficiently strong magnetic field. We investigate this phenomenon in the case of low magnetic Reynolds number using direct numerical simulations and large eddy simulations of a forced flow in a periodic box. A series of simulations is performed with different strengths of the magnetic field, varying Reynolds number, and two types of forcing, one of which is isotropic and the other limited to two-dimensional flow modes. We find that both the velocity anisotropy (difference in the relative amplitude of the velocity components) and the anisotropy of the velocity gradients are predominantly determined by the value of the magnetic interaction parameter. The effects of the Reynolds number and the type of forcing are much weaker. We also find that the anisotropy varies only slightly with the length scale.

The logarithmic structure function law in wall-layer turbulence

The k - 1 spectral law for near-wall turbulence has received only limited experimental support, the most convincing evidence being that of Nickels et al. (Phys. Rev. Lett. vol. 95, 2005, 074501.1). The real-space analogueof this law is a logarithmic dependence on r of the streamwise longitudinal structure function. We show that, unlike the k - 1 law, the logarithmic law is readily seen in the experimental data. We argue that this difference arises from the finite value of Reynolds number in the experiments. Reducing the Reynolds number is equivalent to restricting the range of eddy sizes which contribute to the k - 1 , or lnr, laws. While the logarithmic law is relatively insensitive to a truncation in the range of eddy sizes (it continues to hold over the relevant range of eddy sizes), it turns out that the k - 1 law is not. This is a direct consequence of the so-called aliasing problem associated with one-dimensional spectra, whereby energy is systematically and artificially displaced to small wavenumbers.

Stability of Interfacial Waves in Aluminium Reduction Cells

We have developed a simple model of reduction cell instabilities which highlights the critical role played by the single grouping J o B z /hH, where J o is the current density in the cell, B z is the vertical component of the background magnetic field and h and H are the depths of electrolyte and aluminium. We discuss the implications of this model for the stability of real cells and suggest means of increasing the stability threshold of cells without reducing their performance.