On the finite-amplitude steady convection in rotating mushy layers

2001 ◽  
Vol 437 ◽  
pp. 337-365 ◽  
Author(s):  
PETER GUBA
Keyword(s):  
Three Dimensional ◽  
Finite Amplitude ◽  
Taylor Number ◽  
Mushy Layer ◽  
Hexagonal Symmetry ◽  
Uniform Rotation ◽  
Mushy Layers ◽  
Solid Phases ◽  
Steady Convection ◽  
Liquid And Solid Phases

This study concentrates on a relatively simple model of a mushy layer originally proposed by Amberg & Homsy (1993) and later studied in further detail by Anderson & Worster (1995). We extend this model to the case in which the system is in a state of uniform rotation about the vertical. Of particular interest is to determine how the rotation of the system controls the bifurcating convection with both the oblique-roll planform and the planform of hexagonal symmetry. We find that two-dimensional oblique rolls can be either subcritically or supercritically bifurcating, depending on a pair of parameters (K1/CS, [Tscr ]), where K1 measures how the permeability linearly varies with the local solid fraction, CS relates the compositional difference between the liquid and solid phases to the variation of composition throughout the mushy layer, and the Taylor number [Tscr ] gives a measure of the local Coriolis acceleration relative to the viscous dissipation in a porous medium. The three-dimensional convection with hexagonal symmetry is found to be transcritical. Furthermore, distorted hexagons with upflow at the centres can be either subcritical or supercritical, depending on the value of the Taylor number [Tscr ].

Interactions between steady and oscillatory convection in mushy layers

2010 ◽  
Vol 645 ◽  
pp. 411-434 ◽  
Author(s):  
PETER GUBA ◽  
M. GRAE WORSTER
Keyword(s):  
Standing Waves ◽  
Mushy Layer ◽  
Two Dimensional ◽  
Oscillatory Convection ◽  
Mushy Layers ◽  
Theoretical Predictions ◽  
Convection Patterns ◽  
The Stability ◽  
Nonlinear Solutions ◽  
Steady Convection

We study nonlinear, two-dimensional convection in a mushy layer during solidification of a binary mixture. We consider a particular limit in which the onset of oscillatory convection just precedes the onset of steady overturning convection, at a prescribed aspect ratio of convection patterns. This asymptotic limit allows us to determine nonlinear solutions analytically. The results provide a complete description of the stability of and transitions between steady and oscillatory convection as functions of the Rayleigh number and the compositional ratio. Of particular focus are the effects of the basic-state asymmetries and non-uniformity in the permeability of the mushy layer, which give rise to abrupt (hysteretic) transitions in the system. We find that the transition between travelling and standing waves, as well as that between standing waves and steady convection, can be hysteretic. The relevance of our theoretical predictions to recent experiments on directionally solidifying mushy layers is also discussed.

Segregation Patterns and the Phase-Field 3D-Model for Adiabatic Phase Transition in Al-Cu Alloys

2013 ◽  
Vol 668 ◽  
pp. 870-874
Author(s):  
Heng Min Ding ◽  
Tie Qiao Zhang ◽  
Lv Chun Pu
Keyword(s):  
Dendritic Growth ◽  
3D Model ◽  
Three Dimensional ◽  
Cartesian Grid ◽  
Solute Diffusion ◽  
Solute Redistribution ◽  
Cu Alloys ◽  
Solid Phases ◽  
Solid Liquid ◽  
Liquid And Solid Phases

In the paper, a model basing on solute conservative in every unit is developed for solving the solute diffusion equation during solidification. The model includes time-dependent calculations for temperature distribution, solute redistribution in the liquid and solid phases. Three-dimensional computations are performed for Al-Cu dendritic growth into an adiabatic and highly supersaturated liquid phase. A numerical algorithm was developed to explicitly track the sharp solid/liquid (S/L) interface on a fixed Cartesian grid. Three-dimensional mesoscopic calculations were performed to simulate the evolution of equiaxed dendritic morphologies.

Localisation of convection in mushy layers by weak background flow

2011 ◽  
Vol 675 ◽  
pp. 518-528 ◽  
Author(s):  
S. M. ROPER ◽  
S. H. DAVIS ◽  
P. W. VOORHEES
Keyword(s):  
Three Dimensional ◽  
Mushy Layer ◽  
Thermal Gradients ◽  
Two Dimensional ◽  
Amplitude Equations ◽  
Directionally Solidified ◽  
Background Flow ◽  
Weakly Nonlinear ◽  
Weakly Nonlinear Analysis ◽  
Mushy Layers

It is known that freckles form at the sidewalls of directionally solidified materials. We present a weakly nonlinear analysis of the effects of a weak and slowly varying background flow formed by non-axial thermal gradients on convection near onset in a mushy layer. We find that in the two-dimensional case, the onset of mush convection occurs away from the walls. However if three-dimensional disturbances are allowed, the onset occurs near the walls of the container confining the mush. We derive amplitude equations governing this behaviour and simulate their evolution numerically.

A nonlinear stability analysis of the freezing of a dilute binary alloy

1970 ◽  
Vol 268 (1191) ◽  
pp. 351-380 ◽  
Keyword(s):  
Binary Alloy ◽  
Experimental Situation ◽  
Solute Concentration ◽  
Finite Amplitude ◽  
Linear Stability Theory ◽  
Nonlinear Stability Analysis ◽  
Solid Phases ◽  
The Stability ◽  
Parameter Values ◽  
Liquid And Solid Phases

An investigation is made of the stability of the shape of a moving planar interface between the liquid and solid phases in the freezing of a dilute binary alloy. A nonlinear model is used to describe an experimental situation in which solidification is controlled so that the mean position of the interface moves with constant speed. The model postulates two-dimensional diffusion of solute and heat such that: 1. Convection in the liquid is negligible. 2. Diffusion of the solute in the solid is negligible. 3. Solute concentration in the liquid is small. 4. The effects of interface attachment kinetics are negligible. 5. The extent of the liquid and solid phases is infinite. 6. c8=c1 where c8(c1) is the specific heat per unit volume of the solid (liquid). 7. (D/Dth) 1, , where D is the diffusion coefficient of the solute in the liquid and Dth is the thermal diffusivity in the liquid. 8. G= G where G is the imposed temperature gradient in the liquid and Gcis the critical value of G at which linear theory predicts the onset of instability. The analysis is expected to be asymptotically valid as G-Gc. It is found that the interface can be unstable to finite amplitude disturbances even when linear stability theory predicts stability to infinitesimal disturbances. Further, cellular structure can be anticipated for certain ranges of parameter values. These results are in accord with relevant experimental evidence.

Stability of convection rolls in a layer with stress-free boundaries

1985 ◽  
Vol 150 ◽  
pp. 487-498 ◽  
Author(s):  
E. W. Bolton ◽  
F. H. Busse
Keyword(s):  
Three Dimensional ◽  
Finite Amplitude ◽  
Two Dimensional ◽  
Free Boundaries ◽  
Stability Boundaries ◽  
Prandtl Numbers ◽  
Steady Solutions ◽  
The Stability ◽  
Convection Rolls ◽  
Steady Convection

Steady finite-amplitude solutions for two-dimensional convection in a layer heated from below with stress-free boundaries are obtained numerically by a Galerkin method. The stability of the steady convection rolls with respect to arbitrary three-dimensional infinitesimal disturbances is investigated. Stability is found only in a small fraction of the Rayleigh-number-wavenumber space where steady solutions exist. The cross-roll instability and the oscillatory and monotonic skewed varicose instabilities are most important in limiting the stability of steady convection rolls. The Prandtlnumbers P = 0.71, 7, 104 areemphasized, but the stability boundaries are sufficiently smoothly dependent on the parameters of the problem to permit qualitative extrapolations to other Prandtl numbers.

Complex investigation of the thermal and electrophysical properties of refractory oxides in the liquid and solid phases

1980 ◽  
Vol 38 (4) ◽  
pp. 333-337
Author(s):  
�. �. Shpil'rain ◽  
D. N. Kagan ◽  
L. S. Barkhatov ◽  
L. I. Zhmakin ◽  
V. V. Koroleva
Keyword(s):  
Electrophysical Properties ◽  
Complex Investigation ◽  
Refractory Oxides ◽  
Solid Phases ◽  
Liquid And Solid Phases

scholarly journals Numerical simulation of Faraday waves

2009 ◽  
Vol 635 ◽  
pp. 1-26 ◽  
Author(s):  
NICOLAS PÉRINET ◽  
DAMIR JURIC ◽  
LAURETTE S. TUCKERMAN
Keyword(s):  
Stokes Equations ◽  
Three Dimensional ◽  
Oscillation Period ◽  
Finite Amplitude ◽  
Three Dimensions ◽  
Navier Stokes ◽  
Faraday Waves ◽  
Temporal Behaviour ◽  
Two Fluids ◽  
Front Tracking Method

We simulate numerically the full dynamics of Faraday waves in three dimensions for two incompressible and immiscible viscous fluids. The Navier–Stokes equations are solved using a finite-difference projection method coupled with a front-tracking method for the interface between the two fluids. The critical accelerations and wavenumbers, as well as the temporal behaviour at onset are compared with the results of the linear Floquet analysis of Kumar & Tuckerman (J. Fluid Mech., vol. 279, 1994, p. 49). The finite-amplitude results are compared with the experiments of Kityk et al (Phys. Rev. E, vol. 72, 2005, p. 036209). In particular, we reproduce the detailed spatio-temporal spectrum of both square and hexagonal patterns within experimental uncertainty. We present the first calculations of a three-dimensional velocity field arising from the Faraday instability for a hexagonal pattern as it varies over its oscillation period.

Thermocapillary Convection With Undeformable Curved Surfaces in Open Cylinders

2001 ◽  
Author(s):  
Bok-Cheol Sim ◽  
Abdelfattah Zebib
Keyword(s):  
Free Surface ◽  
Critical Frequency ◽  
Thermocapillary Convection ◽  
Three Dimensional ◽  
Quantitative Agreement ◽  
Fluid Volume ◽  
Uniform Heat Flux ◽  
Free Surfaces ◽  
Space Experiments ◽  
Steady Convection

Abstract Thermocapillary convection driven by a uniform heat flux in an open cylindrical container of unit aspect ratio is investigated by two- and three-dimensional numerical simulations. The undeformable free surface is either flat or curved as determined by the fluid volume (V ≤ 1) and the Young-Laplace equation. Convection is steady and axisymmetric at sufficiently low values of the Reynolds number (Re) with either flat or curved interfaces. Only steady convection is possible in strictly axisymmetric computations. Transition to oscillatory three-dimensional motions occurs as Re increases beyond a critical value dependent on Pr and V. With a flat free surface (V = 1), two-lobed pulsating waves are found on the free surface and prevail with increasing Re. While the critical Re increases with increasing Pr, the critical frequency decreases. In the case of a concave surface, four azimuthal waves are found rotating clockwise on the surface. The critical Re decreases with increasing fluid volume, and the critical frequency is found to increase. The numerical results with either flat or curved free surfaces are in good quantitative agreement with space experiments.

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