Weak convection, liquid inclusions and the formation of chimneys in mushy layers

1999 ◽  
Vol 388 ◽  
pp. 197-215 ◽  
Author(s):  
T. P. SCHULZE ◽  
M. GRAE WORSTER
Keyword(s):  
Solid Fraction ◽  
Stokes Equations ◽  
Numerical Study ◽  
Solidification Rate ◽  
Mushy Region ◽  
Mushy Layer ◽  
Liquid Region ◽  
Interfacial Conditions ◽  
Buoyancy Driven Convection ◽  
Steady Convection

We present a numerical study of steady convection in a two-dimensional mushy layer during solidification of a binary mixture at a constant speed V. The mushy layer is modelled as a reactive porous medium whose permeability is a function of the local solid fraction. The flow in the liquid region above the mushy layer is modelled using the Stokes equations (i.e. the Prandtl number is taken to be infinite). The calculations follow the development of buoyancy-driven convection as the flow amplitude is increased to the level where the solid fraction is driven to zero at some point within the mushy region. We show that this event cannot occur before the local buoyancy-driven volume flux exceeds the solidification rate V. Further increases in the flow amplitude lead to the formation of a region with negative solid fraction, indicating the need to switch from the Darcy approximation to the Stokes flow approximation. These regions ultimately become what are known as chimneys. We exhibit solutions which give the detailed structure of the temperature, solute, flow and solid fraction fields within the mushy layer. A key finding of the numerics is that these fledgling chimneys emerge from the interior of the mushy layer, rather than eating their way down from the top of the layer, as the amplitude of the steady convection is increased. We discuss some qualitative features of the resulting liquid inclusions and, in the light of these, reassess the interfacial conditions between mushy and liquid regions.

A numerical investigation of steady convection in mushy layers during the directional solidification of binary alloys

1998 ◽  
Vol 356 ◽  
pp. 199-220 ◽  
Author(s):  
T. P. SCHULZE ◽  
M. GRAE WORSTER
Keyword(s):  
Directional Solidification ◽  
Solid Fraction ◽  
Stokes Equations ◽  
Numerical Study ◽  
Vertical Channel ◽  
Navier Stokes ◽  
Mushy Layer ◽  
Navier Stokes Equations ◽  
Strongly Nonlinear ◽  
Steady Convection

We present a numerical study of steady convection in a two-dimensional mushy layer during the directional solidification of a binary mixture. The calculations reveal the internal structure of strongly nonlinear states featuring upflow which has been focused into solid-free regions known as chimneys. The mushy layer is modelled as a porous medium whose permeability is a function of the local solid fraction. The mushy layer is coupled to a chimney that is modelled as a narrow vertical channel where lubrication scalings are used to simplify the Navier–Stokes equations. We use these methods to exhibit solutions which give the detailed structure of the temperature, solute, flow and solid-fraction fields within the mushy layer. A key finding of the numerics is that there are two distinct chimney solutions at low Rayleigh numbers, presumably corresponding to stable and unstable portions of a subcritical solution branch. We also explore the relationship between convective solutions with and without chimneys.

Experimental study of convection in a mushy layer during directional solidification

1995 ◽  
Vol 293 ◽  
pp. 81-98 ◽  
Author(s):  
C. F. Chen
Keyword(s):  
Directional Solidification ◽  
Solid Fraction ◽  
Convective Motion ◽  
Mushy Layer ◽  
Liquid Region ◽  
Dye Tracing ◽  
X Ray ◽  
Local Permeability ◽  
Before And After ◽  
Theoretical Predictions

Results of experiments using a number of techniques to study the nature of convection in a mushy layer generated by directional solidification of aqueous ammonium chloride solutions are reported. The techniques include flow visualization using a dye tracing method to study convection within the mushy layer before and after the onset of plume convection, and X-ray tomography to measure the solid fraction of a growing mush. The principal results are as follows. (i) Prior to the onset of chimneys, there is no convective motion in the mush, in spite of the vigorous finger convection at the mush-liquid interface. (ii) When the plume convection is fully developed, the flow of fluid in the mush consists of a nearly uniform downward motion toward the bottom of the tank, horizontal motion along the bottom toward the chimneys, then upward plume motion through the chimneys in the liquid region above the mush. (iii) The solid fraction of a growing mush as determined by X-ray tomography shows a significant decrease toward the bottom of the tank after the chimneys are fully developed. As a result, the concomitant increase in the local permeability can be as much as 50%. Some of the results reported herein confirm theoretical predictions of Worster (1992) and Amberg & Homsey (1993). Others reveal phenomena not observed heretofore.

scholarly journals On the thermodynamic boundary conditions of a solidifying mushy layer with outflow

2014 ◽  
Vol 762 ◽  
Author(s):  
David W. Rees Jones ◽  
M. Grae Worster
Keyword(s):  
Transition Region ◽  
Stokes Equations ◽  
Fluid Velocity ◽  
Darcy Number ◽  
Forced Flow ◽  
Mushy Layer ◽  
Liquid Region ◽  
Precise Knowledge ◽  
Steady Solutions ◽  
Velocity Changes

AbstractThe free-boundary problem between a liquid region and a mushy layer (a reactive porous medium) must respect both thermodynamic and fluid dynamical considerations. We develop a steady two-dimensional forced-flow configuration to investigate the thermodynamic condition of marginal equilibrium that applies to a solidifying mushy layer with outflow and requires that streamlines are tangent to isotherms at the interface. We show that a ‘two-domain’ approach in which the mushy layer and liquid region are distinct domains is consistent with marginal equilibrium by extending the Stokes equations in a narrow transition region within the mushy layer. We show that the tangential fluid velocity changes rapidly in the transition region to satisfy marginal equilibrium. In convecting mushy layers with liquid channels, a buoyancy gradient can drive this tangential flow. We use asymptotic analysis in the limit of small Darcy number to derive a regime diagram for the existence of steady solutions. Thus we show that marginal equilibrium is a robust boundary condition and can be used without precise knowledge of the fluid flow near the interface.

scholarly journals Convection in a mushy layer along a vertical heated wall

2021 ◽  
Vol 926 ◽  
Author(s):  
S. Boury ◽  
C.R. Meyer ◽  
G.M. Vasil ◽  
A.J. Wells
Keyword(s):  
Boundary Layer ◽  
Numerical Study ◽  
Heated Wall ◽  
Solid Matrix ◽  
Mushy Region ◽  
Mushy Layer ◽  
Icy Moons ◽  
Vertical Boundary ◽  
Theoretical Predictions ◽  
Region Ii

Motivated by the mushy zones of sea ice, volcanoes and icy moons of the outer solar system, we perform a theoretical and numerical study of boundary-layer convection along a vertical heated wall in a bounded ideal mushy region. The mush is comprised of a porous and reactive binary alloy with a mixture of saline liquid in a solid matrix, and is studied in the near-eutectic approximation. Here, we demonstrate the existence of four regions and study their behaviour asymptotically. Starting from the bottom of the wall, the four regions are (i) an isotropic corner region; (ii) a buoyancy dominated vertical boundary layer; (iii) an isotropic connection region; and (iv) a horizontal boundary layer at the top boundary with strong gradients of pressure and buoyancy. Scalings from numerical simulations are consistent with the theoretical predictions. Close to the heated wall, the convection in the mushy layer is similar to a rising buoyant plume abruptly stopped at the top, leading to increased pressure and temperature in the upper region, whose impact is discussed as an efficient melting mechanism.

scholarly journals Dynamics of Single Droplet Splashing on Liquid Film by Coupling FVM with VOF

Processes ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 841
Author(s):  
Yuzhen Jin ◽  
Huang Zhou ◽  
Linhang Zhu ◽  
Zeqing Li
Keyword(s):  
Liquid Film ◽  
Weber Number ◽  
Stokes Equations ◽  
Numerical Study ◽  
Three Dimensional ◽  
Navier Stokes ◽  
Single Droplet ◽  
Navier Stokes Equations ◽  
Volume Method ◽  
The Relationship

A three-dimensional numerical study of a single droplet splashing vertically on a liquid film is presented. The numerical method is based on the finite volume method (FVM) of Navier–Stokes equations coupled with the volume of fluid (VOF) method, and the adaptive local mesh refinement technology is adopted. It enables the liquid–gas interface to be tracked more accurately, and to be less computationally expensive. The relationship between the diameter of the free rim, the height of the crown with different numbers of collision Weber, and the thickness of the liquid film is explored. The results indicate that the crown height increases as the Weber number increases, and the diameter of the crown rim is inversely proportional to the collision Weber number. It can also be concluded that the dimensionless height of the crown decreases with the increase in the thickness of the dimensionless liquid film, which has little effect on the diameter of the crown rim during its growth.

scholarly journals Experimental and Numerical Study on Water Filling and Air Expelling Process in a Pipe with Multiple Air Valves under Water Slow Filling Condition

Water ◽  
2019 ◽  
Vol 11 (12) ◽  
pp. 2511
Author(s):  
Jintao Liu ◽  
Di Xu ◽  
Shaohui Zhang ◽  
Meijian Bai
Keyword(s):  
Numerical Model ◽  
Stokes Equations ◽  
Numerical Study ◽  
Practical Method ◽  
Navier Stokes ◽  
Physical Processes ◽  
Two Phase ◽  
Saint Venant Equations ◽  
Water Filling ◽  
Air Valve

This paper investigates the physical processes involved in the water filling and air expelling process of a pipe with multiple air valves under water slow filling condition, and develops a fully coupledwater–air two-phase stratified numerical model for simulating the process. In this model, the Saint-Venant equations and the Vertical Average Navier–Stokes equations (VANS) are respectively applied to describe the water and air in pipe, and the air valve model is introduced into the VANS equations of air as the source term. The finite-volume method and implicit dual time-stepping method (IDTS) with two-order accuracy are simultaneously used to solve this numerical model to realize the full coupling between water and air movement. Then, the model is validated by using the experimental data of the pressure evolution in pipe and the air velocity evolution of air valves, which respectively characterize the water filling and air expelling process. The results show that the model performs well in capturing the physical processes, and a reasonable agreement is obtained between numerical and experimental results. This agreement demonstrates that the proposed model in this paper offers a practical method for simulating water filling and air expelling process in a pipe with multiple air valves under water slow filling condition.

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.

A numerical study of the interaction between unsteay free-stream disturbances and localized variations in surface geometry

1989 ◽  
Vol 209 ◽  
pp. 285-308 ◽  
Author(s):  
R. J. Bodonyi ◽  
W. J. C. Welch ◽  
P. W. Duck ◽  
M. Tadjfar
Keyword(s):  
Numerical Solutions ◽  
Free Stream ◽  
Stokes Equations ◽  
Numerical Study ◽  
Navier Stokes ◽  
Surface Geometry ◽  
Navier Stokes Equations ◽  
S Waves ◽  
Tollmien Schlichting

A numerical study of the generation of Tollmien-Schlichting (T–S) waves due to the interaction between a small free-stream disturbance and a small localized variation of the surface geometry has been carried out using both finite–difference and spectral methods. The nonlinear steady flow is of the viscous–inviscid interactive type while the unsteady disturbed flow is assumed to be governed by the Navier–Stokes equations linearized about this flow. Numerical solutions illustrate the growth or decay of the T–S waves generated by the interaction between the free-stream disturbance and the surface distortion, depending on the value of the scaled Strouhal number. An important result of this receptivity problem is the numerical determination of the amplitude of the T–S waves.

A Numerical Study of Unsteady Natural Convection in a Rectangular Enclosure: The Effect of Compressibility

2004 ◽  
Author(s):  
K. M. Akyuzlu ◽  
Y. Pavri ◽  
A. Antoniou
Keyword(s):  
Mathematical Model ◽  
Stokes Equations ◽  
Numerical Study ◽  
Vertical Wall ◽  
Ideal Gas ◽  
Working Fluid ◽  
Two Dimensional ◽  
Algebraic Equations ◽  
Rectangular Enclosure ◽  
Circulation Patterns

A two-dimensional, mathematical model is adopted to investigate the development of buoyancy driven circulation patterns and temperature contours inside a rectangular enclosure filled with a compressible fluid (Pr=1.0). One of the vertical walls of the enclosure is kept at a higher temperature then the opposing vertical wall. The top and the bottom of the enclosure are assumed insulated. The physics based mathematical model for this problem consists of conservation of mass, momentum (two-dimensional Navier-Stokes equations) and energy equations for the enclosed fluid subjected to appropriate boundary conditions. The working fluid is assumed to be compressible through a simple ideal gas relation. The governing equations are discretized using second order accurate central differencing for spatial derivatives and first order forward finite differencing for time derivatives where the computation domain is represented by a uniform orthogonal mesh. The resulting nonlinear equations are then linearized using Newton’s linearization method. The set of algebraic equations that result from this process are then put into a matrix form and solved using a Coupled Modified Strongly Implicit Procedure (CMSIP) for the unknowns (primitive variables) of the problem. A numerical experiment is carried out for a benchmark case (driven cavity flow) to verify the accuracy of the proposed solution procedure. Numerical experiments are then carried out using the proposed compressible flow model to simulate the development of the buoyancy driven circulation patterns for Rayleigh numbers between 103 and 105. Finally, an attempt is made to determine the effect of compressibility of the working fluid by comparing the results of the proposed model to that of models that use incompressible flow assumptions together with Boussinesq approximation.

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