Modelling binary alloy solidification with adaptive mesh refinement
Abstract
The solidification of a binary alloy results in the formation of a porous mushy layer, within which spontaneous localisation of fluid flow can lead to the emergence of features over a range of spatial scales. We describe a finite volume method for simulating binary alloy solidification in two dimensions with local mesh refinement in space and time. The coupled heat, solute, and mass transport is described using an enthalpy method with flow described by a Darcy-Brinkman equation for flow across porous and liquid regions. The resulting equations are solved on a hierarchy of block-structured adaptive grids. A projection method is used to compute the fluid velocity, whilst the viscous and nonlinear diffusive terms are calculated using a semi-implicit scheme. A series of synchronization steps ensure that the scheme is flux-conservative and correct for errors that arise at the boundaries between different levels of refinement. We also develop a corresponding method using Darcy's law for flow in a porous medium/narrow Hele-Shaw cell. We demonstrate the accuracy and efficiency of our method using established benchmarks for solidification without flow and convection in a fixed porous medium, along with convergence tests for the fully coupled code. Finally, we demonstrate the ability of ourmore »
- Authors:
- Publication Date:
- Research Org.:
- Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)
- Sponsoring Org.:
- USDOE Office of Science (SC)
- OSTI Identifier:
- 1577402
- Alternate Identifier(s):
- OSTI ID: 1603566
- Grant/Contract Number:
- AC02-05CH11231; IE141071
- Resource Type:
- Published Article
- Journal Name:
- Journal of Computational Physics: X
- Additional Journal Information:
- Journal Name: Journal of Computational Physics: X Journal Volume: 5 Journal Issue: C; Journal ID: ISSN 2590-0552
- Publisher:
- Elsevier
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 36 MATERIALS SCIENCE; binary alloy solidification; adaptive mesh refinement; mushy layer convection; finite volume
Citation Formats
Parkinson, James R. G., Martin, Daniel F., Wells, Andrew J., and Katz, Richard F. Modelling binary alloy solidification with adaptive mesh refinement. United States: N. p., 2020.
Web. doi:10.1016/j.jcpx.2019.100043.
Parkinson, James R. G., Martin, Daniel F., Wells, Andrew J., & Katz, Richard F. Modelling binary alloy solidification with adaptive mesh refinement. United States. https://doi.org/10.1016/j.jcpx.2019.100043
Parkinson, James R. G., Martin, Daniel F., Wells, Andrew J., and Katz, Richard F. Wed .
"Modelling binary alloy solidification with adaptive mesh refinement". United States. https://doi.org/10.1016/j.jcpx.2019.100043.
@article{osti_1577402,
title = {Modelling binary alloy solidification with adaptive mesh refinement},
author = {Parkinson, James R. G. and Martin, Daniel F. and Wells, Andrew J. and Katz, Richard F.},
abstractNote = {The solidification of a binary alloy results in the formation of a porous mushy layer, within which spontaneous localisation of fluid flow can lead to the emergence of features over a range of spatial scales. We describe a finite volume method for simulating binary alloy solidification in two dimensions with local mesh refinement in space and time. The coupled heat, solute, and mass transport is described using an enthalpy method with flow described by a Darcy-Brinkman equation for flow across porous and liquid regions. The resulting equations are solved on a hierarchy of block-structured adaptive grids. A projection method is used to compute the fluid velocity, whilst the viscous and nonlinear diffusive terms are calculated using a semi-implicit scheme. A series of synchronization steps ensure that the scheme is flux-conservative and correct for errors that arise at the boundaries between different levels of refinement. We also develop a corresponding method using Darcy's law for flow in a porous medium/narrow Hele-Shaw cell. We demonstrate the accuracy and efficiency of our method using established benchmarks for solidification without flow and convection in a fixed porous medium, along with convergence tests for the fully coupled code. Finally, we demonstrate the ability of our method to simulate transient mushy layer growth with narrow liquid channels which evolve over time.},
doi = {10.1016/j.jcpx.2019.100043},
journal = {Journal of Computational Physics: X},
number = C,
volume = 5,
place = {United States},
year = {2020},
month = {1}
}
https://doi.org/10.1016/j.jcpx.2019.100043
Works referenced in this record:
A Cartesian Grid Embedded Boundary Method for Poisson's Equation on Irregular Domains
journal, November 1998
- Johansen, Hans; Colella, Phillip
- Journal of Computational Physics, Vol. 147, Issue 1
Instabilities of the liquid and mushy regions during solidification of alloys
journal, April 1992
- Worster, M. Grae
- Journal of Fluid Mechanics, Vol. 237
The origin of freckles in unidirectionally solidified castings
journal, August 1970
- Copley, S. M.; Giamei, A. F.; Johnson, S. M.
- Metallurgical Transactions, Vol. 1, Issue 8
Adaptive mesh refinement for hyperbolic partial differential equations
journal, March 1984
- Berger, Marsha J.; Oliger, Joseph
- Journal of Computational Physics, Vol. 53, Issue 3
Modeling Melt Convection in Phase-Field Simulations of Solidification
journal, September 1999
- Beckermann, C.; Diepers, H. -J; Steinbach, I.
- Journal of Computational Physics, Vol. 154, Issue 2
Nonlinear analysis of buoyant convection in binary solidification with application to channel formation
journal, July 1993
- Amberg, Gustav; Homsy, G. M.
- Journal of Fluid Mechanics, Vol. 252
Local adaptive mesh refinement for shock hydrodynamics
journal, May 1989
- Berger, M. J.; Colella, P.
- Journal of Computational Physics, Vol. 82, Issue 1
The Formation of Freckles in Binary Alloys
journal, January 1985
- Fowler, A. C.
- IMA Journal of Applied Mathematics, Vol. 35, Issue 2
A Cell-Centered Adaptive Projection Method for the Incompressible Euler Equations
journal, September 2000
- Martin, Daniel F.; Colella, Phillip
- Journal of Computational Physics, Vol. 163, Issue 2
Chimneys on the Earth's inner-outer core boundary?
journal, March 1994
- Bergman, Michael I.; Fearn, David R.
- Geophysical Research Letters, Vol. 21, Issue 6
Efficient Computation of Dendritic Microstructures Using Adaptive Mesh Refinement
journal, April 1998
- Provatas, Nikolas; Goldenfeld, Nigel; Dantzig, Jonathan
- Physical Review Letters, Vol. 80, Issue 15
Solidification of an alloy from a cooled boundary
journal, June 1986
- Worster, M. Grae
- Journal of Fluid Mechanics, Vol. 167, Issue -1
Seaweed to Dendrite Transition in Directional Solidification
journal, October 2003
- Provatas, Nikolas; Wang, Quanyong; Haataja, Mikko
- Physical Review Letters, Vol. 91, Issue 15
Natural convective heat transfer in a fluid saturated variable porosity medium
journal, October 1997
- Nithiarasu, P.; Seetharamu, K. N.; Sundararajan, T.
- International Journal of Heat and Mass Transfer, Vol. 40, Issue 16
Convection in Mushy Layers
journal, January 1997
- Worster, M. Grae
- Annual Review of Fluid Mechanics, Vol. 29, Issue 1
A cell-centered adaptive projection method for the incompressible Navier–Stokes equations in three dimensions
journal, January 2008
- Martin, Daniel F.; Colella, Phillip; Graves, Daniel
- Journal of Computational Physics, Vol. 227, Issue 3
Compositional convection in a reactive crystalline mush and melt differentiation
journal, January 1992
- Tait, Stephen; Jaupart, Claude
- Journal of Geophysical Research, Vol. 97, Issue B5
Visualizing brine channel development and convective processes during artificial sea-ice growth using Schlieren optical methods
journal, February 2016
- Middleton, C. A.; Thomas, C.; De Wit, A.
- Journal of Glaciology, Vol. 62, Issue 231
Natural convection in a mushy layer
journal, March 1991
- Worster, M. Grae
- Journal of Fluid Mechanics, Vol. 224
Steady-state mushy layers: experiments and theory
journal, January 2007
- Peppin, S. S. L.; Aussillous, P.; Huppert, Herbert E.
- Journal of Fluid Mechanics, Vol. 570
Measurements of the temperature field of mushy and liquid regions during solidification of aqueous ammonium chloride
journal, March 1998
- Solomon, T. H.; Hartley, R. R.
- Journal of Fluid Mechanics, Vol. 358
Finite element analysis of directional solidification of multicomponent alloys
journal, January 1998
- Felicelli, S. D.; Heinrich, J. C.; Poirier, D. R.
- International Journal for Numerical Methods in Fluids, Vol. 27, Issue 1-4
An Unsplit 3D Upwind Method for Hyperbolic Conservation Laws
journal, November 1994
- Saltzman, Jeff
- Journal of Computational Physics, Vol. 115, Issue 1
Modeling Freckle Segregation with Mesh Adaptation
journal, September 2011
- Sajja, Udaya K.; Felicelli, Sergio D.
- Metallurgical and Materials Transactions B, Vol. 42, Issue 6
The convergence rate for difference approximations to mixed initial boundary value problems
journal, May 1975
- Gustafsson, Bertil
- Mathematics of Computation, Vol. 29, Issue 130
A Thermodynamically Consistent Model of a Mushy zone
journal, January 1983
- Hills, R. N.; Loper, D. E.; Roberts, P. H.
- The Quarterly Journal of Mechanics and Applied Mathematics, Vol. 36, Issue 4
Simulation of directional solidification, thermochemical convection, and chimney formation in a Hele-Shaw cell
journal, December 2008
- Katz, Richard F.; Worster, M. Grae
- Journal of Computational Physics, Vol. 227, Issue 23
Natural convection solid/liquid phase change in porous media
journal, January 1988
- Beckermann, C.; Viskanta, R.
- International Journal of Heat and Mass Transfer, Vol. 31, Issue 1
Nonlinear mushy-layer convection with chimneys: stability and optimal solute fluxes
journal, January 2013
- Wells, Andrew J.; Wettlaufer, J. S.; Orszag, Steven A.
- Journal of Fluid Mechanics, Vol. 716
Solidification of a binary alloy: Finite-element, single-domain simulation and new benchmark solutions
journal, July 2006
- Le Bars, Michael; Worster, M. Grae
- Journal of Computational Physics, Vol. 216, Issue 1
Weak convection, liquid inclusions and the formation of chimneys in mushy layers
journal, June 1999
- Schulze, T. P.; Grae Worster, M.
- Journal of Fluid Mechanics, Vol. 388
the phase evolution of Young Sea Ice
journal, May 1997
- Wettlaufer, J. S.; Worster, M. Grae; Huppert, Herbert E.
- Geophysical Research Letters, Vol. 24, Issue 10
A Conservative Adaptive Projection Method for the Variable Density Incompressible Navier–Stokes Equations
journal, May 1998
- Almgren, Ann S.; Bell, John B.; Colella, Phillip
- Journal of Computational Physics, Vol. 142, Issue 1
Stability of a Planar Interface During Solidification of a Dilute Binary Alloy
journal, February 1964
- Mullins, W. W.; Sekerka, R. F.
- Journal of Applied Physics, Vol. 35, Issue 2
A numerical investigation of steady convection in mushy layers during the directional solidification of binary alloys
journal, February 1998
- Schulze, T. P.; Worster, M. Grae
- Journal of Fluid Mechanics, Vol. 356
An adaptive finite element scheme for transient problems in CFD
journal, April 1987
- Löhner, Rainald
- Computer Methods in Applied Mechanics and Engineering, Vol. 61, Issue 3
Modelling of macrosegregation during solidification processes using an adaptive domain decomposition method
journal, June 2003
- Kaempfer, Th U.; Rappaz, M.
- Modelling and Simulation in Materials Science and Engineering, Vol. 11, Issue 4
Convective flow and heat transfer in a tall porous cavity side-cooled with temperature profile
journal, December 2009
- Yedder, Ridha Ben; Erchiqui, Fouad
- International Journal of Heat and Mass Transfer, Vol. 52, Issue 25-26
Weakly nonlinear analysis of convection in mushy layers during the solidification of binary alloys
journal, November 1995
- Anderson, D. M.; Worster, M. Grae
- Journal of Fluid Mechanics, Vol. 302
Conditions for defect-free solidification of aqueous ammonium chloride in a quasi two-dimensional directional solidification facility
journal, July 2008
- Whiteoak, S. H.; Huppert, Herbert E.; Worster, M. Grae
- Journal of Crystal Growth, Vol. 310, Issue 15
On weakly nonlinear convection in mushy layers during solidification of alloys
journal, January 2008
- Okhuysen, B. S.; Riahi, D. N.
- Journal of Fluid Mechanics, Vol. 596
2-Dimensional FEM modeling of macrosegregation in the directional solidification with mesh adaptation
journal, June 2009
- Liu, Weitao; Xie, Changchuan; Bellet, Michel
- Acta Metallurgica Sinica (English Letters), Vol. 22, Issue 3
Multidimensional upwind methods for hyperbolic conservation laws
journal, March 1990
- Colella, Phillip
- Journal of Computational Physics, Vol. 87, Issue 1
Steady-state chimneys in a mushy layer
journal, March 2002
- Chung, C. A.; Worster, M. Grae
- Journal of Fluid Mechanics, Vol. 455
Growth of and brine drainage from NaCl-H 2 O freezing: A simulation of young sea ice
journal, January 2004
- Oertling, Annette B.
- Journal of Geophysical Research, Vol. 109, Issue C4
Experimental study of convection in a mushy layer during directional solidification
journal, June 1995
- Chen, C. F.
- Journal of Fluid Mechanics, Vol. 293
Adaptive mesh refinement for coupled elliptic-hyperbolic systems
journal, October 2006
- Pretorius, Frans; Choptuik, Matthew W.
- Journal of Computational Physics, Vol. 218, Issue 1