Results

Boettinger, W.J., Warren, J.A., Beckermann, C., and Karma, A., "Phase-Field Simulation of Solidification," Annual Review of Materials Research, Vol. 32, pp. 163-194, 2002.

2-D simulations of dendritic growth of a pure substance with flow in the melt

 by Xinglin Tong

Dendritic growth image
Still image
Animated gif of flow
Full movie
Gif of flow without the crystal growing
The flow without the crystal growing
Gif of the dendrite rotated
The dendrite rotated 45 degrees with respect to the flow

Parameters: Dimensionless supercooling = 0.55; anisotropy strength: 0.03, Prandtl number = 23.1; D = 2; uniform inlet velocity = 1.0 (in dimensionless form); others.

Computational details: Grid: 1024x2048; the entire transient simulation took about 200 CPU hours on an alpha workstation - corresponding simulations without flow take about 2 hours (the no-flow version of the code is from A. Karma).

We would like to offer this and other simulations as a computational benchmark. We can provide quantitative information on the tip speeds, curvatures, velocities, etc.

3-D simulations of dendritic growth of a pure substance with flow in the melt

by Yili Lu

3D parallel image
3d parallel image
3D upstream image
3D upstream image
3D downstream image
3D downstream image
3D parallel animation
3D parallel animation
3D upstream animation
3D upstream animation
3D downstream animation
3D downstream animation
Animation of flow without crystal growing
Animation of the flow without the crystal growing

2-D simulations of free dendritic growth of a dilute binary alloy with coupled heat and solute diffusion

 by Juan Ramirez

View of entire dendrite
View of entire dendrite
Close-up of center region of dendrite
Close-up of center region

The above simulations are described in detail in the following publication: Ramirez, J.C., and Beckermann, C., "Examination of Binary Alloy Free Dendritic Growth Theories with a Phase-Field Model," Acta Materialia, Vol. 53, pp. 1721-1736, 2005.

Parameters: dimensionless supercooling = 0.55, dimensionless composition = 0.04, anisotropy strength = 0.02, Lewis number = 100, partition coefficient = 0.15, no solute diffusion in solid, no kinetics, no solute trapping.

The left movie shows how the grid is adapted in the simulation: a fine grid is used only in the inner quadrant where the dendrite grows and where there are strong species concentration gradients (U field); a grid that is four times coarser is used for the outer quadrants in order to accomodate the long tails of the thermal boundary layer (Theta field) in front of the growing dendrite; the total domain size is constant; as the dendrite grows, the grid is adapted periodically (five times in the movie) by enlarging the fine grid region at the expense of the course grid region. This saves a lot of computer time.

The simulation was performed using the phase-field model published in: J.C. Ramirez, C. Beckermann, A. Karma, and H.J. Diepers, "Phase-field modeling of binary alloy solidification with coupled heat and solute diffusion," Physical Review E, Vol. 69, 051607 (16 pages), 2004.

2-D simulations of coupled columnar and equiaxed dendritic growth in directional solidification of a binary alloy

by Arnoldo Badillo

Image showing imposed temperature gradient
Image showing imposed temperature gradient
Image showing imposed temperature gradient

Images are for increasing imposed temperature gradient from left to right.

Relevant Publications

  • Sun, Y., and Beckermann, C., "Phase-Field Modeling of Bubble Growth and Flow in a Hele-Shaw Cell, " Int. J. Heat Mass Transfer, Vol. 53, 2010, pp. 2969-2978.
  • Sun, Y., and Beckermann, C., "Effect of Solid-Liquid Density Change on Dendrite Tip Velocity and Shape Selection," J. Crystal Growth, Vol. 311, 2009, pp. 4447-4453.
  • Sun, Y., and Beckermann, C., "A Two-Phase Diffuse-Interface Model for Hele-Shaw Flows with Large Property Contrasts," Physica D, Vol. 237, 2008, pp. 3089-3098, 2008.
  • Sun, Y., and Beckermann, C., "Sharp Interface Tracking Using the Phase-Field Equation," J. Computational Physics, Vol. 220, 2007, pp. 626-653.
  • Badillo, A., and Beckermann, C., "Phase-Field Simulation of the Columnar-to-Equiaxed Transition in Alloy Solidification," Acta Materialia, Vol. 54, 2006, pp. 2015-2026.
  • Sun, Y., and Beckermann, C., "Phase-Field Simulation of Two-Phase Micro-Flows in a Hele-Shaw Cell," in Computational Methods in Multiphase Flow III, eds. A.A. Mammoli and C.A. Brebbia, WIT Press, Southampton, UK, 2005, pp. 147-157.
  • Lu, Y., Beckermann, C., and Ramirez, J.C., "Three-Dimensional Phase-Field Simulations of the Effect of Convection on Free Dendritic Growth," J. Crystal Growth, Vol. 280, pp. 320-334, 2005.
  • Ramirez, J.C., and Beckermann, C., "Examination of Binary Alloy Free Dendritic Growth Theories with a Phase-Field Model," Acta Materialia, Vol. 53, pp. 1721-1736, 2005.
  • Sun, Y., and Beckermann, C., "Diffuse Interface Modeling of Two-Phase Flows Based on Averaging: Mass and Momentum Equations," Physica D, Vol. 198, pp. 281-308, 2004.
  • Ramirez, J.C., Beckermann, C., Karma, A., and Diepers, H.-J., "Phase-Field Modeling of Binary Alloy Solidification with Coupled Heat and Solute Diffusion," Physical Review E, Vol. 69, 051607 (16 pages), 2004.
  • Sun, Y., and Beckermann, C., "A Diffuse Interface Model for Two-Phase Flows Based on Averaging," in Multiphase Phenomena and CFD Modeling and Simulation in Materials Processes, eds. L. Nastac and B.Q. Li, TMS, Warrendale, PA, 2004, pp. 109-118.
  • Ramirez, J.C., and Beckermann, C., "Examination of Binary Alloy Free Dendritic Growth Theories with a Phase-Field Model," in Solidification Processes and Microstructures - A Symposium in Honor of Wilfried Kurz, eds. M. Rappaz, C. Beckermann, and R. Trivedi, TMS, Warrendale, PA, 2004, pp. 373-378.
  • Lu, Y., Beckermann, C. and Karma, A., "Convection Effects in Three-Dimensional Dendritic Growth," Proceedings of ASME IMECE2002, Paper No. IMECE2002-32838, Nov. 2002.
  • Boettinger, W.J., Warren, J.A., Beckermann, C., and Karma, A., "Phase-Field Simulation of Solidification," Annual Review of Materials Research, Vol. 32, pp. 163-194, 2002.
  • Lu, Y., Beckermann, C. and Karma, A., "Convection Effects in Three-Dimensional Dendritic Growth," Proceedings of the 2001 Fall MRS Meeting in Boston, MA, 2001 (invited paper).
  • Tong, X., Beckermann, C., Karma, A. and Li, Q., "Phase-Field Simulations of Dendritic Crystal Growth in a Forced Flow," Physical Review E, Vol. 63, 061601 (16 pages), 2001.
  • Tong, X., Beckermann, C., and Karma, A., "Velocity and Shape Selection of Dendritic Crystals in a Forced Flow," Physical Review E, Vol. 61, pp. R49-R52, 2000.
  • Beckermann, C., Diepers, H.J., Steinbach, I., Karma, A., and Tong, X., "Modeling Melt Convection in Phase-Field Simulations of Solidification," J. Computational Physics, Vol. 154, pp. 468-496, 1999.
  • Diepers, H.J., Beckermann, C., and Steinbach, I., "Simulation of Convection and Ripening in a Binary Alloy Mush Using the Phase-Field Method," Acta Materialia, Vol. 47, pp. 3663-3678, 1999.
  • Diepers, H.J., Beckermann, C., and Steinbach, I., "A Phase-Field Method for Alloy Solidification with Convection," in Solidification Processing 1997, eds. J. Beech and H. Jones, Dept. Engineering Materials, The University of Sheffield, Sheffield, UK, 1997, pp. 426-430.
  • Tong, X., Beckermann, C., and Karma, A., "Phase-Field Simulation of Dendritic Growth with Convection," in Modeling of Casting, Welding and Advanced Solidification Processes VIII, eds. B.G. Thomas and C. Beckermann, TMS, Warrendale, PA, 1998, pp. 613-620.
  • Diepers, H.J., Beckermann, C., and Steinbach, I., "Modeling of Convection-Influenced Coarsening of a Binary Alloy Mush Using the Phase-Field Method," in Modeling of Casting, Welding and Advanced Solidification Processes VIII, eds. B.G. Thomas and C. Beckermann, TMS, Warrendale, PA, 1998, pp. 565-572.