Convection in an imposed magnetic field. Part 1. The development of nonlinear convection

1981 ◽  
Vol 108 ◽  
pp. 247-272 ◽  
Author(s):  
N. O. Weiss
Keyword(s):  
Magnetic Field ◽  
Thermal Diffusivity ◽  
Rayleigh Number ◽  
Numerical Experiments ◽  
Dynamical Regime ◽  
Two Dimensional ◽  
Nonlinear Convection ◽  
Boussinesq Fluid ◽  
Chandrasekhar Number ◽  
Steady Convection

Nonlinear two-dimensional magnetoconvection in a Boussinesq fluid has been studied in a series of numerical experiments with values of the Chandrasekhar number Q ≤ 4000 and the ratio ζ of the magnetic to the thermal diffusivity in the range 1 ≥ ζ ≥ 0·025. If the imposed field is strong enough, convection sets in as overstable oscillations which give way to steady convection as the Rayleigh number R is increased. In the dynamical regime that follows, magnetic flux is concentrated into sheets at the sides of the cells, from which the motion is excluded.

Convection in an imposed magnetic field. Part 2. The dynamical regime

1981 ◽  
Vol 108 ◽  
pp. 273-289 ◽  
Author(s):  
N. O. Weiss
Keyword(s):  
Magnetic Field ◽  
Steady State ◽  
Thermal Diffusivity ◽  
Large Scale ◽  
Time Dependent ◽  
Dynamical Regime ◽  
Two Dimensional ◽  
Chandrasekhar Number ◽  
Steady Convection ◽  
Active Flux

Nonlinear, two-dimensional magnetoconvection has been investigated numerically for a fixed Rayleigh number of 104, with the ratio ζ of the magnetic to the thermal diffusivity in the range 0·4 ≥ ζ ≥ 0·05. As the Chandrasekhar number Q is decreased, convection first sets in as overstable oscillations, which are succeeded by steady convection with dynamically active flux sheets and, eventually, with kinematically concentrated fields. In the dynamical regime spatially asymmetrical convection, with most of the flux on one side of the cell, is preferred. As Q increases, these asymmetrical solutions become time-dependent, with oscillations about the steady state which develop into large-scale oscillations with reversals of the flow. Although linear theory predicts that narrow cells should be most unstable, the nonlinear results show that steady convection occurs most easily in cells that are roughly twice as wide as they are deep.

Horizontal convection is non-turbulent

2002 ◽  
Vol 466 ◽  
pp. 205-214 ◽  
Author(s):  
F. PAPARELLA ◽  
W. R. YOUNG
Keyword(s):  
Thermal Diffusivity ◽  
Energy Dissipation ◽  
Rayleigh Number ◽  
Prandtl Number ◽  
Numerical Solutions ◽  
Inviscid Limit ◽  
Two Dimensional ◽  
Uniform Temperature ◽  
Horizontal Convection ◽  
Boussinesq Fluid

Consider the problem of horizontal convection: a Boussinesq fluid, forced by applying a non-uniform temperature at its top surface, with all other boundaries insulating. We prove that if the viscosity, ν, and thermal diffusivity, κ, are lowered to zero, with σ ≡ ν/κ fixed, then the energy dissipation per unit mass, κ, also vanishes in this limit. Numerical solutions of the two-dimensional case show that despite this anti-turbulence theorem, horizontal convection exhibits a transition to eddying flow, provided that the Rayleigh number is sufficiently high, or the Prandtl number σ sufficiently small. We speculate that horizontal convection is an example of a flow with a large number of active modes which is nonetheless not ‘truly turbulent’ because ε→0 in the inviscid limit.

Методика практического восстановления параметров формы поверхностных двухмерных дефектов с учетом нелинейных свойств ферромагнетика

2021 ◽  
pp. 46-55
Author(s):  
А.В. Никитин ◽  
А.В. Михайлов ◽  
А.С. Петров ◽  
С.Э. Попов
Keyword(s):  
Magnetic Field ◽  
Free Surface ◽  
Numerical Experiments ◽  
Input Data ◽  
Metal Plate ◽  
Two Dimensional ◽  
Nonlinear Properties ◽  
Data Errors ◽  
The Magnetic Field

A technique for determining the depth and opening of a surface two-dimensional defect in a ferromagnet is presented, that is resistant to input data errors. Defects and magnetic transducers are located on opposite sides of the metal plate. The nonlinear properties of the ferromagnet are taken into account. The components of the magnetic field in the metal were reconstructed from the measured components of the magnetic field above the defect-free surface of the metal. As a result of numerical experiments, the limits of applicability of the method were obtained. The results of the technique have been verified experimentally.

Natural Convection With Non-Newtonian Shear-Thinning Power Law Fluids in Inclined Two Dimensional Rectangular Cavities

2009 ◽  
Author(s):  
Lyes Khezzar ◽  
Dennis Siginer
Keyword(s):  
Natural Convection ◽  
Numerical Experiments ◽  
Newtonian Fluids ◽  
Flow Mode ◽  
Mode Transition ◽  
Two Dimensional ◽  
Power Law Fluids ◽  
Side Walls ◽  
Differential Scheme ◽  
Boussinesq Fluid

Steady two-dimensional natural convection in rectangular cavities has been investigated numerically. The conservation equations of mass, momentum and energy under the assumption of a Newtonian Boussinesq fluid have been solved using the finite volume technique embedded in the Fluent code for a Newtonian (water) and three non Newtonian carbopol fluids. The highly accurate Quick differential scheme was used for discretization. The computations were performed for one Rayleigh number, based on cavity height, of 105 and a Prandtl number of 10 and 700, 6,000 and 1.2×104 for the Newtonian and the three non-Newtonian fluids respectively. In all of the numerical experiments, the channel is heated from below and cooled from the top with insulated side-walls and the inclination angle is varied. The simulations have been carried out for one aspect ratio of 6. Comparison between the Newtonian and the non-Newtonian cases is conducted based on the behaviour of the average Nusselt number with angle of inclination. Both Newtonian and non-Newtonian fluids exhibit similar behavior with a sudden drop around an angle of 50° associated with flow mode transition from multi-cell to single-cell mode.

The breakdown of steady convection

1988 ◽  
Vol 188 ◽  
pp. 47-85 ◽  
Author(s):  
T. B. Lennie ◽  
D. P. Mckenzie ◽  
D. R. Moore ◽  
N. O. Weiss
Keyword(s):  
Numerical Experiments ◽  
Steady Motion ◽  
Period Doubling ◽  
Chaotic Regime ◽  
Spatial Structures ◽  
Internal Heating ◽  
Periodic Behaviour ◽  
Boussinesq Fluid ◽  
Considerable Detail ◽  
Steady Convection

Two-dimensional convection in a Boussinesq fluid with infinite Prandtl number, confined between rigid horizontal boundaries and stress-free lateral boundaries, has been investigated in a series of numerical experiments. In a layer heated from below steady convection becomes unstable to oscillatory modes caused by the formation of hot or cold blobs in thermal boundary layers. Convection driven by internal heating shows a transition from steady motion through periodic oscillations to a chaotic regime, owing to the formation of cold blobs which plunge downwards and eventually split the roll. The interesting feature of this idealized problem is the interaction between constraints imposed by nonlinear dynamics and the obvious spatial structures associated with the sinking sheets and changes in the preferred cell size. These spatial structures modify the bifurcation patterns that are familiar from transitions to chaos in low-order systems. On the other hand, even large-amplitude disturbances are constrained to show periodic or quasi-periodic behaviour, and the bifurcation sequences can be followed in considerable detail. There are examples of quasi-periodic behaviour followed by intermittency, of period-doubling cascades and of transitions from quasi-periodicity to chaos, associated with a preference for narrower rolls as the Rayleigh number is increased.

scholarly journals Linear and Nonlinear Convection with an Aligned Magnetic Field

1990 ◽  
Vol 142 ◽  
pp. 135-136
Author(s):  
N. Rudraiah ◽  
I S Shivakumara ◽  
P Geetavani
Keyword(s):  
Magnetic Field ◽  
Fluid Layer ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Horizontal Magnetic Field ◽  
Nonlinear Convection ◽  
Linear And Nonlinear ◽  
Horizontal Fluid Layer ◽  
Aligned Magnetic Field

The effect of horizontal magnetic field on the onset of three-dimensional convection in a horizontal fluid layer is studied. It is found that the two-dimensional solutions are unstable to three-dimensional disturbances. A detailed bifurcation study is reported.

Nonlinear convection in a layer with nearly insulating boundaries

1980 ◽  
Vol 96 (2) ◽  
pp. 243-256 ◽  
Author(s):  
F. H. Busse ◽  
N. Riahi
Keyword(s):  
Thermal Conductivity ◽  
Prandtl Number ◽  
Fluid Layer ◽  
Three Dimensional ◽  
Thermal Conductance ◽  
Convection Flow ◽  
Surprising Result ◽  
Two Dimensional ◽  
Nonlinear Convection ◽  
Steady Convection

A general class of solutions is studied describing three-dimensional steady convection flows in a fluid layer heated from below with boundaries of low thermal conductivity. Non-linear properties of the solutions are analysed and the physically realizable convection flow is determined by a stability analysis with respect to arbitrary three-dimensional disturbances. The most surprising result is that square-pattern convection is preferred in contrast to two-dimensional rolls that represent the only form of stable convection in a symmetric layer with highly conducting boundaries. The analysis is carried out in the limit of infinite Prandtl number and for a particular boundary configuration. But it is shown that the results hold for arbitrary Prandtl number to the order to which they have been derived and that other assumptions about the boundaries require only minor modifications as long as their thermal conductance remains low.

Oscillations in double-diffusive convection

1981 ◽  
Vol 109 ◽  
pp. 25-43 ◽  
Author(s):  
L. N. Da Costa ◽  
E. Knobloch ◽  
N. O. Weiss
Keyword(s):  
Rayleigh Number ◽  
Period Doubling ◽  
Thermosolutal Convection ◽  
Two Dimensional ◽  
Double Diffusive Convection ◽  
Dimensional Problem ◽  
Diffusive Convection ◽  
Steady Solutions ◽  
Steady Convection ◽  
Double Diffusive

We have studied the transition between oscillatory and steady convection in a simplified model of two-dimensional thermosolutal convection. This model is exact to second order in the amplitude of the motion and is qualitatively accurate for larger amplitudes. If the ratio of the solutal diffusivity to the thermal diffusivity is sufficiently small and the solutal Rayleigh number, RS, sufficiently large, convection sets in as overstable oscillations, and these oscillations grow in amplitude as the thermal Rayleigh number, RT, is increased. In addition to this oscillatory branch, there is a branch of steady solutions that bifurcates from the static equilibrium towards lower values of RT; this subcritical branch is initially unstable but acquires stability as it turns round towards increasing values of RT. For moderate values of RS the oscillatory branch ends on the unstable (subcritical) portion of the steady branch, where the period of the oscillations becomes infinite. For larger values of RS a birfurcation from symmetrical to asymmetrical oscillations is followed by a succession of bifurcations, at each of which the period doubles, until the motion becomes aperiodic at some finite value of RT. The chaotic solutions persist as RT is further increased but eventually they lose stability and there is a transition to the stable steady branch. These results are consistent with the behaviour of solutions of the full two-dimensional problem and suggest that period-doubling, followed by the appearance of a strange attractor, is a characteristic feature of double-diffusive convection.

scholarly journals Numerical Solution of Natural Convective Heat Transfer Under Magnetic Field Effect

2019 ◽  
Vol 13 (1) ◽  
pp. 23-29
Author(s):  
Serpil Şahin ◽  
Hüseyin Demir
Keyword(s):  
Magnetic Field ◽  
Heat Transfer ◽  
Rayleigh Number ◽  
Convective Heat Transfer ◽  
Convective Heat ◽  
Field Effect ◽  
Time Derivative ◽  
Magnetic Field Effect ◽  
Stability Properties ◽  
Chandrasekhar Number

Abstract In this study, non-Newtonian pseudoplastic fluid flow equations for 2-D steady, incompressible, the natural convective heat transfer are solved numerically by pseudo time derivative. The stability properties of natural convective heat transfer in an enclosed cavity region heated from below under magnetic field effect are investigated depending on the Rayleigh and Chandrasekhar numbers. Stability properties are studied, in particular, for the Rayleigh number from 104 to 106 and for the Chandrasekhar number 3, 5 and 10. As a result, when Rayleigh number is bigger than 106 and Chandrasekhar number is bigger than 10, the instability occurs in the flow domain. The results obtained for natural convective heat transfer problem are shown in the figures for Newtonian and pseudoplastic fluids. Finally, the local Nusselt number is evaluated along the bottom wall.

Thermoconvective instabilities in a porous medium bounded by two concentric horizontal cylinders

1976 ◽  
Vol 76 (2) ◽  
pp. 337-362 ◽  
Author(s):  
Jean-Paul Caltagirone
Keyword(s):  
Porous Medium ◽  
Finite Elements ◽  
Rayleigh Number ◽  
Critical Rayleigh Number ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Dimensional Model ◽  
Different Types ◽  
Rayleigh Numbers ◽  
Steady Convection

The study of natural convection in a saturated porous medium bounded by two concentric, horizontal, isothermal cylinders reveals different types of evolution according to the experimental conditions and the geometrical configuration of the model. At small Rayleigh numbers the state of the system corresponds to a regime of pseudo-conduction. The isotherms are coaxial with the cylinders. At larger Rayleigh numbers a regime of steady two-dimensional convection sets in between the two cylinders. Finally, for Rayleigh numbers above the critical Rayleigh number Ra*c the phenomena become three-dimensional and fluctuating. The appearance of these different regimes depends, moreover, on the geometry considered and, in particular, on two numbers: R, the ratio of the radii of the cylinders, and A, the ratio of the length of the cylinders to the radius of the inner one. In order to approach these experimental observations and to obtain realistic theoretical models, several methods of solving the equations have been used.The perturbation method yields information about the thermal field and the heat transfer between the cylinders under conditions close to the equilibrium state.A numerical two-dimensional model enables us to extend the range of investigation and to represent properly the phenomena when steady convection appreciably modifies the temperature distribution and the velocities within the porous layer.Neither of these models allows account to be taken of the instabilities observed experimentally above a critical Rayleigh number Ra*c. For this reason, a study of stability has been carried out using a Galerkin method based on equations corresponding to an initial state of steady convection. The results obtained show the importance of three-dimensional effects for the onset of fluctuating convection. The critical transition Rayleigh number Ra*c is thus determined in terms of the ratio of the radii R by solving an eigenvalue problem.A numerical three-dimensional model based on the method of finite elements has thus been developed in order to point out the different types of evolution with time. Steady two-dimensional convection and fluctuating three-dimensional convection have been actually found by calculation. The solution of the system of equations by the method of finite elements is briefly described.The experimental and theoretical results are then compared and a general physical interpretation is given.

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