Toroidal Square Research Articles

Curvature effect on the instabilities of lid-driven flow in a toroidal square cavity

The effect of curvature on the three-dimensional instabilities of lid-driven flow in a toroidal square cavity is numerically studied using the linear stability analysis based on the spectral element method. The critical stability curves are determined, and four types of instability modes are predicted. For small curvatures (δ ≤ 0.2), the most dangerous mode is a steady short-wavelength mode (λc ≈ 0.4), which is consistent with that in the straight square cavity. In a narrow range of 0.22 ≤ δ ≤ 0.28, the critical mode is oscillatory and of medium wavelength (0.84 < λc < 1.07). For δ ≥ 0.29, two types of steady long-wavelength modes (λc > 1.26) become the critical mode. The disturbance energy analyses show that all the instabilities are caused by the centrifugal instability mechanism.

Effect of radius of toroidal square duct on the transition of electromagnetically driven liquid metal flow

Toroidal magnetohydrodynamic flows are important for fusion technology because of the blanket surrounding the core of the fusion machine. The effect of the radius of toroidal square ducts on flow transition of an electromagnetically driven liquid metal is assessed. The flow governing equations are solved numerically using direct numerical simulation. Critical Reynolds (Re) numbers where transition occurs are obtained for Hartmann (Ha) numbers in the range of 3 < Ha < 500. The results are compared with previous experimental and numerical results of toroidal square ducts of various radii. It is shown that, independent of the toroidal duct mean radius (r), the transition occurs in three regimes: in the sub-magnetohydrodynamic regime for very low Ha numbers, in a plateau regime for 10 < Ha < 20 and in a higher Ha regime. By increasing the radius of the torus, a transition occurs at higher Reynolds numbers as turbulent fluctuations are developing inside the boundary layers of the cylindrical side walls. The results indicate that transition occurs at RerH̃a-3.39 in the sub-magnetohydrodynamic regime and at RerH̃a-1.4 in the magnetohydrodynamic one. The boundary layer thickness at the Hartmann walls were found to scale as Ha-0.8.

Dimer and fermionic formulations of a class of colouring problems

We show that the number Z of q-edge-colourings of a simple regular graph of degree q is deducible from functions describing dimers on the same graph, namely the dimer generating function or equivalently the set of connected dimer correlation functions. Using this relationship to the dimer problem, we derive fermionic representations for Z in terms of Grassmann integrals with quartic actions. Expressions are given for planar graphs and nonplanar graphs embeddable (without edge crossings) on a torus. We discuss exact numerical evaluations of the Grassmann integrals using an algorithm by Creutz and present an application to the 4-edge-colouring problem on toroidal square lattices, comparing the results with numerical transfer matrix calculations and a previous Bethe ansatz study.

Critical phenomena in discrete-time interconnection networks

We present theoretical and numerical results for the performance of a multiprocessor network modeled as a ring and as a toroidal square lattice of nodes with local processors that generate messages for output ports/buffers. The output buffers are assumed to have infinite capacity, and the service time is deterministic. Two models are considered. One assumes that every processor generates messages with rate λ per time slot and per output port/buffer. The other model considers that the generation rate of a node depends on the intensity of the flow of arriving messages. Explicit expressions for the distribution of queue lengths, the average number of messages in the buffers, the average latency, and the critical network load depending on the distance between the source and the destination are obtained. Simulation results show excellent agreement with theoretical predictions based on the assumption of independent queues.

First- and higher-order effects of curvature and torsion on the flow in a helical rectangular duct

A series expansion method is employed to determine the first-order terms in curvature ε and torsion η of fully developed laminar flow in helical square ducts and in helical rectangular ducts of aspect ratio two. The first-order solutions are compared to solutions of the full governing equations. For toroidal square ducts with zero pitch, the first-order solution is fairly accurate for Dean numbers,De=Reε1/2, up to about 20, and for straight twisted square ducts the first-order solution is accurate for Germano numbers,Gn= ηRe, up to at least 50 whereReis the Reynolds number. Important conclusions are that the flow in a helical duct with a finite pitch or torsion to the first order (i.e. with higher-order terms in ε and η neglected) is obtained as a superposition of the flow in a toroidal duct with zero pitch and a straight twisted duct; that the secondary flow in helical non-circular ducts for sufficiently smallReis dominated by torsion effects; and that for increasingRe, the secondary flow eventually is dominated by effects due to curvature. Torsion has a stronger impact on the flow for aspect ratios greater than one. A characteristic combined higher-order effect of curvature and torsion is an enlargement of the lower vortex of the secondary flow at the expense of the upper vortex, and also a shift of the maximum axial flow towards the upper wall. For higher Reynolds numbers, bifurcation phenomena appear. The extent of a few solution branches for helical ducts with finite pitch or torsion is determined. For ducts with small torsion it is found that the extent of the stable solution branches is affected little by torsion. Physical velocity components are employed to describe the flow. The contravariant components are found useful when describing the convective transport in the duct.

The use of Markov Random Fields as models of texture

We propose Markov Random Fields (MRFs) as probabilistic models of digital image texture where a textured region is viewed as a finite sample of a two-dimensional random process describable by its statistical parameters. MRFs are multidimensional generalizations of Markov chains defined in terms of conditional probabilities associated with spatial neighborhoods. We present an algorithm that generates an MRF on a finite toroidal square lattice from an independent identically distributed (i.i.d.) array of random variables and a given set of independent real-valued statistical parameters. The parametric specification of a consistent collection of MRF conditional probabilities is a general result known as the MRF-Gibbs Random Field (GRF) equivalence. The MRF statistical parameters control the size and directionality of the clusters of adjacent similar pixels which are basic to texture discrimination and thus seem to constitute an efficient model of texture. In the last part of this paper we outline an MRF parameter estimation method and goodness of fit statistical tests applicable to MRF models for a given unknown digital image texture on a finite toroidal square lattice. The estimated parameters may be used as basic features in texture classification. Alternatively these parameters may be used in conjunction with the MRF generation algorithm as a powerful data compression scheme.