Compressible magnetoconvection in three dimensions: planforms and nonlinear behaviour

1995 ◽  
Vol 305 ◽  
pp. 281-305 ◽  
Author(s):  
P. C. Matthews ◽  
M. R. E. Proctor ◽  
N. O. Weiss
Keyword(s):  
Magnetic Field ◽  
Shear Flow ◽  
Three Dimensional ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Nonlinear Behaviour ◽  
Two Dimensional ◽  
Horizontal Shear ◽  
Mean Flow ◽  
The Magnetic Field

Convection in a compressible fiuid with an imposed vertical magnetic field is studied numerically in a three-dimensional Cartesian geometry with periodic lateral boundary conditions. Attention is restricted to the mildly nonlinear regime, with parameters chosen first so that convection at onset is steady, and then so that it is oscillatory.Steady convection occurs in the form of two-dimensional rolls when the magnetic field is weak. These rolls can become unstable to a mean horizontal shear flow, which in two dimensions leads to a pulsating wave in which the direction of the mean flow reverses. In three dimensions a new pattern is found in which the alignment of the rolls and the shear flow alternates.If the magnetic field is sufficiently strong, squares or hexagons are stable at the onset of convection. Both the squares and the hexagons have an asymmetrical topology, with upflow in plumes and downflow in sheets. For the squares this involves a resonance between rolls aligned with the box and rolls aligned digonally to the box. The preference for three-dimensional flow when the field is strong is a consequence of the compressibility of the layer- for Boussinesq magnetoconvection rolls are always preferred over squares at onset.In the regime where convection is oscillatory, the preferred planform for moderate fields is found to be alternating rolls - standing waves in both horizontal directions which are out of phase. For stronger fields, both alternating rolls and two-dimensional travelling rolls are stable. As the amplitude of convection is increased, either by dcereasing the magnetic field strength or by increasing the temperature contrast, the regular planform structure seen at onset is soon destroyed by secondary instabilities.

Instability of pressure-driven gas–liquid two-layer channel flows in two and three dimensions

2018 ◽  
Vol 849 ◽  
pp. 1-34 ◽  
Author(s):  
Lennon Ó Náraigh ◽  
Peter D. M. Spelt
Keyword(s):  
Flow Pattern ◽  
Linear Theory ◽  
Three Dimensional ◽  
Two Dimensions ◽  
Channel Flows ◽  
Three Dimensions ◽  
Nonlinear Behaviour ◽  
Two Dimensional ◽  
Vorticity Field ◽  
Flow Pattern Map

We study unstable waves in gas–liquid two-layer channel flows driven by a pressure gradient, under stable stratification, not assumed to be set in motion impulsively. The basis of the study is direct numerical simulation (DNS) of the two-phase Navier–Stokes equations in two and three dimensions for moderately large Reynolds numbers, accompanied by a theoretical description of the dynamics in the linear regime (Orr–Sommerfeld–Squire equations). The results are compared and contrasted across a range of density ratios $r=\unicode[STIX]{x1D70C}_{liquid}/\unicode[STIX]{x1D70C}_{gas}$. Linear theory indicates that the growth rate of small-amplitude interfacial disturbances generally decreases with increasing $r$; at the same time, the cutoff wavenumbers in both streamwise and spanwise directions increase, leading to an ever-increasing range of unstable wavenumbers, albeit with diminished growth rates. The analysis also demonstrates that the most dangerous mode is two-dimensional in all cases considered. The results of a comparison between the DNS and linear theory demonstrate a consistency between the two approaches: as such, the route to a three-dimensional flow pattern is direct in these cases, i.e. through the strong influence of the linear instability. We also characterize the nonlinear behaviour of the system, and we establish that the disturbance vorticity field in two-dimensional systems is consistent with a mechanism proposed previously by Hinch (J. Fluid Mech., vol. 144, 1984, p. 463) for weakly inertial flows. A flow-pattern map constructed from two-dimensional numerical simulations is used to describe the various flow regimes observed as a function of density ratio, Reynolds number and Weber number. Corresponding simulations in three dimensions confirm that the flow-pattern map can be used to infer the fate of the interface there also, and show strong three-dimensionality in cases that exhibit violent behaviour in two dimensions, or otherwise the development of behaviour that is nearly two-dimensional behaviour possibly with the formation of a capillary ridge. The three-dimensional vorticity field is also analysed, thereby demonstrating how streamwise vorticity arises from the growth of otherwise two-dimensional modes.

scholarly journals INCREASING THE EFFICIENCY OF MULTY-VARIANT CALCULATIONS OF ELECTROMAGNETIC FIELD DISTRIBUTION IN MATRIX OF A POLYGRADIENT SEPARATOR

2021 ◽  
Author(s):  
Jasim Mohmed Jasim Jasim ◽  
Iryna Shvedchykova ◽  
Igor Panasiuk ◽  
Julia Romanchenko ◽  
Inna Melkonova
Keyword(s):  
Magnetic Field ◽  
Field Distribution ◽  
Three Dimensional ◽  
Magnetic Potential ◽  
Geometric Parameters ◽  
Two Dimensional ◽  
Air Gaps ◽  
Vector Magnetic Potential ◽  
Electromagnetic Separator ◽  
The Magnetic Field

An approach is proposed to carry out multivariate calculations of the magnetic field distribution in the working gaps of a plate polygradient matrix of an electromagnetic separator, based on a combination of the advantages of two- and three-dimensional computer modeling. Two-dimensional geometric models of computational domains are developed, which differ in the geometric dimensions of the plate matrix elements and working air gaps. To determine the vector magnetic potential at the boundaries of two-dimensional computational domains, a computational 3D experiment is carried out. For this, three variants of the electromagnetic separator are selected, which differ in the size of the working air gaps of the polygradient matrices. For them, three-dimensional computer models are built, the spatial distribution of the magnetic field in the working intervals of the electromagnetic separator matrix and the obtained numerical values of the vector magnetic potential at the boundaries of the computational domains are investigated. The determination of the values of the vector magnetic potential for all other models is carried out by interpolation. The obtained values of the vector magnetic potential are used to set the boundary conditions in a computational 2D experiment. An approach to the choice of a rational version of a lamellar matrix is substantiated, which provides a solution to the problem according to the criterion of the effective area of the working area. Using the method of simple enumeration, a variant of the structure of a polygradient matrix with rational geometric parameters is selected. The productivity of the electromagnetic separator with rational geometric parameters of the matrix increased by 3–5 % with the same efficiency of extraction of ferromagnetic inclusions in comparison with the basic version of the device

Small-scale magnetic fields in turbulence: Saffman's approximation revisited

1980 ◽  
Vol 99 (3) ◽  
pp. 481-493
Author(s):  
Ralph Baierlein
Keyword(s):  
Magnetic Field ◽  
Numerical Evaluation ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Small Scale ◽  
Fluid Element ◽  
Relative Time ◽  
Fourier Decomposition ◽  
The Subject ◽  
The Magnetic Field

The subject is the small-scale structure of a magnetic field in a turbulent conducting fluid, ‘small scale’ meaning lengths much smaller than the characteristic dissipative length of the turbulence. Philip Saffman developed an approximation to describe this structure and its evolution in time. Its usefulness invites a closer examination of the approximation itself and an attempt to place sharper limits on the numerical parameters that appear in the approximate correlation functions, topics to which the present paper is addressed.A Lagrangian approach is taken, wherein one makes a Fourier decomposition of the magnetic field in a neighbourhood that follows a fluid element. If one construes the viscous-convective range narrowly, by ignoring magnetic dissipation entirely, then results for a magnetic field in two dimensions are consistent with Saffman's approximation, but in three dimensions no steady state could be found. Thus, in three dimensions, turbulent amplification seems to be more effective than Saffman's approximation implies. The cause seems to be a matter of geometry, not of correlation times or relative time scales.Strictly-outward spectral transfer is a characteristic of Saffman's approximation, and this may be an accurate description only when dissipation suppresses the contributions from inwardly directed spectral transfer. In the spectral region where dominance passes from convection to dissipation, one can generate expressions for the parameters that arise in Saffman's approximation. Their numerical evaluation by computer simulation may enable one to sharpen the limits that Saffman had already set for those parameters.

FERMION THRESHOLD ENERGY STATES

1992 ◽  
Vol 06 (24) ◽  
pp. 1531-1534
Author(s):  
CHANGHONG ZHU
Keyword(s):  
Magnetic Field ◽  
Magnetic Flux ◽  
Three Dimensional ◽  
Threshold Energy ◽  
Index Theorem ◽  
Equation Of Motion ◽  
Field Configuration ◽  
Two Dimensional ◽  
Magnetic Field Configuration ◽  
The Magnetic Field

We show that for a three-dimensional non-relativistic spinor confined on a plane, the spin-up component obeys the same equation of motion as a two-dimensional spinor. Threshold energy solution is investigated when the electron is moving in the vortex field. It can be proved from the index theorem that the existence of the threshold states depends on the magnetic flux only, not on the magnetic field configuration.

On the stability of parallel flows with parallel magnetic fields

1966 ◽  
Vol 293 (1434) ◽  
pp. 342-358 ◽  
Keyword(s):  
Magnetic Field ◽  
Magnetic Fields ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Parallel Magnetic Field ◽  
Parallel Flows ◽  
The Mean ◽  
The Stability ◽  
The Magnetic Field ◽  
Parallel Magnetic Fields

The first part of the paper is a physical discussion of the way in which a magnetic field affects the stability of a fluid in motion. Particular emphasis is given to how the magnetic field affects the interaction of the disturbance with the mean motion. The second part is an analysis of the stability of plane parallel flows of fluids with finite viscosity and conductivity under the action of uniform parallel magnetic fields. We show that, in general, three-dimensional disturbances are the most unstable, thus disagreeing with the conclusion of Michael (1953) and Stuart (1954). We show how results obtained for two-dimensional disturbances can be used to calculate the most unstable three-dimensional disturbances and thence we prove that a parallel magnetic field can never completely stabilize a parallel flow.

[Ni(cyclam)(OCOR)2], a finite molecular complex: hydrogen-bonded supramolecular aggregation in one, two and three dimensions, and coordination polymers in one and two dimensions

2001 ◽  
Vol 58 (1) ◽  
pp. 78-93 ◽  
Author(s):  
Choudhury M. Zakaria ◽  
George Ferguson ◽  
Alan J. Lough ◽  
Christopher Glidewell
Keyword(s):  
Hydrogen Bonding ◽  
Coordination Polymer ◽  
Hydrogen Bonds ◽  
Three Dimensional ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Two Dimensional ◽  
Compound I ◽  
Supramolecular Aggregation ◽  
Linear Coordination

In the complexes [Ni(cyclam)(OCOR)2] (cyclam = 1,4,8,11-tetraazacyclotetradecane), where (RCOO)− is 2-naphtho-ate [bis-(2-naphthoato)-1,4,8,11-tetraazacyclotetradecanenickel(II), (I), monoclinic P21/c, Z′ = 0.5], 3,5-dinitrobenzoate [bis-(3,5-dinitrobenzoato)-1,4,8,11-tetraazacyclotetradecanenickel(II), (II), triclinic P\bar 1, Z′ = 0.5], 4-nitrobenzoate [bis-(4-nitrobenzoato)-1,4,8,11-tetraazacyclotetradecanenickel(II), (III), monoclinic P21/n, Z′ = 0.5], 3-hydroxybenzoate [bis-(3-hydroxybenzoato)-1,4,8,11-tetraazacyclotetradecanenickel(II), (IV), monoclinic P21/c, Z′ = 0.5] and 4-aminobenzo-ate [bis-(4-aminobenzoato)-1,4,8,11-tetraazacyclotetradecanenickel(II), (V), monoclinic C2/c, Z′ = 0.5], the Ni lies on a centre of inversion with monodentate carboxylato ligands occupying trans sites. Compound (I) consists of isolated molecules. In (II) and (III), N—H...O hydrogen bonds link the complexes into chains. Compounds (IV) and (III) form two- and three-dimensional structures generated entirely by hard hydrogen bonds. The 5-hydroxyisophthalate(2−) anion forms a hydrated complex, [Ni(cyclam)(5-hydroxyisophthalate)(H2O)]·4H2O {[aqua-(5-hydroxyisophthalato)-1,4,8,11-tetraazacyclotetradecanenickel(II)] tetrahydrate, (VI), monoclinic Cc, Z′ = 1}, in which the monodentate carboxylato ligand and a water molecule occupy trans sites at Ni: extensive hydrogen bonding links the molecular aggregates into a three-dimensional framework. The terephthalate(2−) anion forms a hydrated linear coordination polymer {catena-poly[terephthalato-1,4,8,11-tetraazacyclotetradecanenickel(II)] monohydrate, (VII), monoclinic C2/c, Z′ = 0.5}. In 1,2,4,5-benzenecarboxylate tris[1,4,8,11-tetraazacyclotetradecanenickel(II)] diperchlorate hydrate (VIII), [Ni(cyclam)]3·[1,2,4,5-benzenetetracarboxylate(4−)]·[ClO4]2·-[H2O]3, there are two distinct Ni sites: [Ni(cyclam)]2+ and centrosymmetric [C10H2O8]4− units form a two-dimensional coordination polymer, whose sheets are linked by centrosymmetric [Ni(cyclam)(H2O)2]2+ cations.

scholarly journals Theory of magnetic reconnection in solar and astrophysical plasmas

2012 ◽  
Vol 370 (1970) ◽  
pp. 3169-3192 ◽  
Author(s):  
David I. Pontin
Keyword(s):  
Magnetic Field ◽  
Magnetic Reconnection ◽  
Three Dimensional ◽  
Three Dimensions ◽  
Astrophysical Plasmas ◽  
Current State ◽  
Fundamental Process ◽  
Recent Developments ◽  
The Magnetic Field ◽  
Theoretical Results

Magnetic reconnection is a fundamental process in a plasma that facilitates the release of energy stored in the magnetic field by permitting a change in the magnetic topology. In this paper, we present a review of the current state of understanding of magnetic reconnection. We discuss theoretical results regarding the formation of current sheets in complex three-dimensional magnetic fields and describe the fundamental differences between reconnection in two and three dimensions. We go on to outline recent developments in modelling of reconnection with kinetic theory, as well as in the magnetohydrodynamic framework where a number of new three-dimensional reconnection regimes have been identified. We discuss evidence from observations and simulations of Solar System plasmas that support this theory and summarize some prominent locations in which this new reconnection theory is relevant in astrophysical plasmas.

scholarly journals 3D Magnetic Reconnection

2010 ◽  
Vol 6 (S271) ◽  
pp. 227-238 ◽  
Author(s):  
Clare E. Parnell ◽  
Rhona C. Maclean ◽  
Andrew L. Haynes ◽  
Klaus Galsgaard
Keyword(s):  
Magnetic Reconnection ◽  
Energy State ◽  
Single Point ◽  
Three Dimensional ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Wide Range ◽  
Field Lines ◽  
Magnetohydrodynamic Models ◽  
The Magnetic Field

AbstractMagnetic reconnection is an important process that is prevalent in a wide range of astrophysical bodies. It is the mechanism that permits magnetic fields to relax to a lower energy state through the global restructuring of the magnetic field and is thus associated with a range of dynamic phenomena such as solar flares and CMEs. The characteristics of three-dimensional reconnection are reviewed revealing how much more diverse it is than reconnection in two dimensions. For instance, three-dimensional reconnection can occur both in the vicinity of null points, as well as in the absence of them. It occurs continuously and continually throughout a diffusion volume, as opposed to at a single point, as it does in two dimensions. This means that in three-dimensions field lines do not reconnect in pairs of lines making the visualisation and interpretation of three-dimensional reconnection difficult.By considering particular numerical 3D magnetohydrodynamic models of reconnection, we consider how magnetic reconnection can lead to complex magnetic topologies and current sheet formation. Indeed, it has been found that even simple interactions, such as the emergence of a flux tube, can naturally give rise to ‘turbulent-like’ reconnection regions.

Velocity distribution function for a dilute granular material in shear flow

1997 ◽  
Vol 340 ◽  
pp. 319-341 ◽  
Author(s):  
V. KUMARAN
Keyword(s):  
Distribution Function ◽  
Shear Flow ◽  
Velocity Distribution ◽  
Scaling Laws ◽  
Three Dimensional ◽  
Particle Diameter ◽  
Velocity Distribution Function ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Binary Collisions

The velocity distribution function for the steady shear flow of disks (in two dimensions) and spheres (in three dimensions) in a channel is determined in the limit where the frequency of particle–wall collisions is large compared to particle–particle collisions. An asymptotic analysis is used in the small parameter ε, which is naL in two dimensions and n2L in three dimensions, where n is the number density of particles (per unit area in two dimensions and per unit volume in three dimensions), L is the separation of the walls of the channel and a is the particle diameter. The particle–wall collisions are inelastic, and are described by simple relations which involve coefficients of restitution et and en in the tangential and normal directions, and both elastic and inelastic binary collisions between particles are considered. In the absence of binary collisions between particles, it is found that the particle velocities converge to two constant values (ux, uy) =(±V, 0) after repeated collisions with the wall, where ux and uy are the velocities tangential and normal to the wall, V=(1−et) Vw/(1+et), and Vw and −Vw are the tangential velocities of the walls of the channel. The effect of binary collisions is included using a self-consistent calculation, and the distribution function is determined using the condition that the net collisional flux of particles at any point in velocity space is zero at steady state. Certain approximations are made regarding the velocities of particles undergoing binary collisions in order to obtain analytical results for the distribution function, and these approximations are justified analytically by showing that the error incurred decreases proportional to ε1/2 in the limit ε→0. A numerical calculation of the mean square of the difference between the exact flux and the approximate flux confirms that the error decreases proportional to ε1/2 in the limit ε→0. The moments of the velocity distribution function are evaluated, and it is found that 〈u2x〉→V2, 〈u2y〉 ∼V2ε and − 〈uxuy〉 ∼ V2εlog(ε−1) in the limit ε→0. It is found that the distribution function and the scaling laws for the velocity moments are similar for both two- and three-dimensional systems.

Magnetic damping of jets and vortices

1995 ◽  
Vol 299 ◽  
pp. 153-186 ◽  
Author(s):  
P. A. Davidson
Keyword(s):  
Magnetic Field ◽  
Reynolds Number ◽  
Angular Momentum ◽  
Kinetic Energy ◽  
Lorentz Force ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Magnetic Damping ◽  
Field Lines ◽  
The Magnetic Field

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.

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