MHD Turbulence at the Laboratory Scale: Established Ideas and New Challenges

The properties of MHD turbulence in the electrically conducting fluids available in the laboratory (where the magnetic Reynolds number is significantly smaller than unity) may be summarised as follows:(1) The Alfven waves, even under their degenerated form at this scale, are responsible for a tendency to two-dimensionality. Eddies tend to become aligned with the applied magnetic field and inertia tends to restore isotropy. The competition between these mechanisms results in a spectral law t^-2k^-3.(2) When insulating walls, perpendicular to the magnetic field, are present and close enough to each other, two-dimensionality can be established with a good approximation within the large scales, and the predominant mechanism is the inverse energy cascade.(3) These columnar eddies are nevertheless submitted to a dissipation within the Hartmann boundary layers present at their ends, whose time scale is independent of the wave number. When this damping effect is negligible, ordinary 2D turbulence is observed with k^-5/3 spectra. On the contrary when this (ohmic and viscous) damping is significant this 2D turbulence exhibits k^-3 spectra.Besides these homogeneous (except within the Hartmann layers) conditions, for instance in shear flows such as mixing layers, almost nothing is known except that two-dimensionality may be well established. The first results of a recent experimental investigation (still in development) are presented. Some challenging questions are raised, such as the interpretation of a surprising difference between the transport of momentum and the transport of a scalar quantity (heat) across that layer. A video was shown during the oral presentation of this paper, illustrating the energy transfer toward the large scales and the weakness of the dissipation suffered by this 2D velocity field.     

Thermal response of a periodic boundary layer near an axisymmetric stagnation point on a circular cylinder

Boundary layer solutions are provided to study the time-mean heat transfer characteristics in a laminar flow in the vicinity of an axisymmetric stagnation point. The velocity of the oncoming flow is assumed to oscillate relative to the body. Different solutions are constructed for the small and high values of the reduced frequency parameter. Numerical solutions for the temperature functions are presented, and the wall values of the thermal gradients are tabulated.     

Numerical solution of Navier–Stokes equations using multiquadric radial basis function networks

A numerical method based on radial basis function networks (RBFNs) for solving steady incompressible viscous flow problems (including Boussinesq materials) is presented in this paper. The method uses a ‘universal approximator’ based on neural network methodology to represent the solutions. The method is easy to implement and does not require any kind of ‘finite element‐type’ discretization of the domain and its boundary. Instead, two sets of random points distributed throughout the domain and on the boundary are required. The first set defines the centres of the RBFNs and the second defines the collocation points. The two sets of points can be different; however, experience shows that if the two sets are the same better results are obtained. In this work the two sets are identical and hence commonly referred to as the set of centres. Planar Poiseuille, driven cavity and natural convection flows are simulated to verify the method. The numerical solutions obtained using only relatively low densities of centres are in good agreement with analytical and benchmark solutions available in the literature. With uniformly distributed centres, the method achieves Reynolds number Re = 100 000 for the Poiseuille flow (assuming that laminar flow can be maintained) using the density of , Re = 400 for the driven cavity flow with a density of and Rayleigh number Ra = 1 000 000 for the natural convection flow with a density of . Copyright © 2001 John Wiley & Sons, Ltd.     

Reduced‐order modelling of an adaptive mesh ocean model

A novel proper orthogonal decomposition (POD) model has been developed for use with an advanced unstructured mesh finite‐element ocean model, the Imperial College Ocean Model (ICOM, described in detail below), which includes many recent developments in ocean modelling and numerical analysis. The advantages of the POD model developed here over existing POD approaches are the ability:

• 1. To increase accuracy when representing geostrophic balance (the balance between the Coriolis terms and the pressure gradient). This is achieved through the use of two sets of geostrophic basis functions where each one is calculated by basis functions for velocities u and v.
• 2. To speed up the POD simulation. To achieve this a new numerical technique is introduced, whereby a time‐dependent matrix in the discretized equation is rapidly constructed from a series of time‐independent matrices. This development imparts considerable efficiency gains over the often‐used alternative of calculating each finite element over the computational domain at each time level.
• 3. To use dynamically adaptive meshes in the above POD model.

Copyright © 2008 John Wiley & Sons, Ltd.     

Free convection from a vertical plate with nonuniform surface heat flux and embedded in a porous medium

An analysis is presented for the calculation of heat transfer due to free convective flow along a vertical plate embedded in a porous medium with an arbitrarily varying surface heat flux. By applying the appropriate coordinate transformations and the Merk series, the governing energy equation is expressed as a set of ordinary differential equations. Numerical solutions are presented for these equations which represent universal functions and several computational examples are provided.     

An experimental study of diabatic spiral vortex flow

In spiral vortex flow, between concentric cylinders with the inner cylinder rotating and the outer stationary, the addition of a thermal gradient across the gap is a known complicating factor. The present diabatic study for narrow and wide gaps (radius ratios N=0.955 and N=0.8), with a heated outer and adiabatic inner cylinder, was undertaken to investigate this problem. The heat transfer characteristics and the modes of transition have been investigated together with the relationship between them. Using standard on-line digital computer techniques, the onset of vortex flow and its higher transitions have been shown to cause a sharp increase in Nusselt number. At higher Taylor numbers, of the order of 10⁶, a marked change in the Nusselt number occurs with the onset of the transition to periodic turbulent vortex flow. Outer wall heating is seen to affect the modes of transition. Diabatic critical Taylor numbers are much higher than those for adiabatic conditions and are found to depend on the close approach of the vortices to the outer wall     

Heat transfer in tube assemblies under conditions of laminar axial,transverse and inclined flow

This paper considers laminar flow heat transfer in tube assemblies. The main interest is focused on the virtually unexplored cases of heat transfer under conditions of fully-developed flow inclined to the axes of the tubes and of purely transverse developing flow. The limiting cases of purely axial or purely transverse fully-developed flow are also examined. In all cases, the thermal boundary condition on the tubes is constant heat flux. Governing differential equations are expressed in terms of curvilinear-orthogonal coordinates and solved using finite-differences. Results are compared with available theoretical and experimental data. The effect of the transverse component of the flow on the temperature distribution is found to remain very strong even in nearly-axial flows and therefore considerably higher heat transfer coefficients are exhibited by a nearly-axial flow than a purely axial one.     

Laminar flow and heat transfer in confined channel flow past square bars arranged side by side

A numerical investigation was conducted to analyze the unsteady laminar flow field and heat transfer characteristics in a plane channel with two square bars mounted side by side to the approaching flow. A finite volume technique is applied with a fine grid and time resolution. The transverse separation distance between the bars (G/d) is varied from 0 to 5, whereas the bar height to channel height is d/H=1/8, and the channel length is L=5H. Different flow regimes develop in the channel due the interaction between the two mounted square bars, steady flow, flow with vortex shedding synchronization either in phase or in anti-phase, or biased flow with low frequency modulation of vortex shedding are found. Results show that the pressure drop increase and heat transfer enhancement are strongly dependent of the transverse separation distance of the bars and the channel Reynolds number.     

MHD effects on free convective flow over moving semi-infinite vertical cylinder with temperature oscillation

Numerical solutions of magnetodynamics(MHD) effects on the free convective flow of an incompressible viscous fluid past a moving semi-infinite vertical cylinder with temperature oscillation are presented.The dimensionless,unsteady,non-linear,and coupled governing partial differential equations are solved by using an implicit finite difference method of the Crank-Nicolson type.The velocity,temperature,and concentration profiles are studied for various parameters.The local skin-friction,the average skin-fr...