A MULTI-SCALE/MULTI-PHYSICS MODELING FRAMEWORK FOR SOLIDIFICATION SYSTEMS Vaughan R Voller Saint Anthony Falls Lab University of Minnesota Acknowledgments National Science Foundation who through have supported some of the multi-scale work National Center For Earthsurface Dynamics For the Aditya Birla visiting Chair in the department of Mechanical Engineering

CFD- Model Experiment Plant/field Data e.g. optimization of wheel casting Cleary et al, Fluent 1. Process Optimization By blending models, plant/ field and experiment Multi-physics –Multi-Scale KEY AREA OF INTEREST IN CFD 3 Flow +  LIDAR (1mm) measurement Of bed topography 2. Dovetailing Modeling and High Resolution Distributed Measurement Techniques

Multi-physics –Multi-Scale Framework defines the domain of the problem Three Scales where nodal values of process variables are defined and stored phenomena at scales below the grid resolution are incorporated into the analysis via the use of volume averaging and the development of constitutive relationships. Solidification Phenomena (The Science) occur at the local scale of the Solid Liquid Interface (~ microns) Solidification Process (The Engineering) occur at the global scale of a product (~meters) To make progress in the “science” and “engineering” we need to bridge between these scales

Example: Macro-Segregation Scheil vs lever Distribution Of solute at scale of process

Growth of Equiaxed Crystal In under-cooled melt A microstructure model Phase change temperature depends on interface curvature, speed and concentration Sub-grid models Account for Crystal anisotropy and “smoothing” of interface jumps

In Detail Four fold symmetry Sub grid constitutive If f= 0 or f = 1If 0 < f < 1 curvature Local direction Capillary length m in Al alloys Easy and Direct ENTHALPY

seed Typical grid Size 200x200 ¼ geometry At end of time step if solidification Completes in cell i Force solidification in ALL fully liquid neighboring cells. Physical domain ~ 2-10 microns Initially insulated cavity contains liquid metal with bulk undercooling T 0 < 0. Solidification induced by placing solid seed at center. Some Results

4d o (blue) 3.25d o (black) 2.5d o (red) Dendrite shape with 3 grid sizes shows reasonable independence  = 0.05, T 0 = Dimensionless time  = 6000  = 0.25,  = 0.75

at t =37,000, T 0 = -0.55,  = 0.05,  = 2.5d o ( ¼ box size 800x800) Tip Velocity Approaches Theoretical Limit

Verification 1 Looks Right!!  = 0.05, T 0 = -0.65,  x = 3.333d 0 Enthalpy Calculation Dimensionless time  = 0 (1000) 6000  = 0.05, T 0 = -0.55,  x = d 0 Level Set Kim, Goldenfeld and Dantzig Dimensionless time  = 37,600 Red my calculation for these parameters With grid size

The Solid color is solved with a 45 deg twist on the anisotropy and then twisted back —the white line is with the normal anisotropy Note: Different “smear” parameters are used in 0 0 and 45 0 case Tip position with time Dimensionless time  = 6000  = 0.05, T 0 = Not perfect: In 45 0 case the tip velocity at time 6000 (slope of line) is below the theoretical limit. Low Grid Anisotropy

 = 0.05, T 0 = time  = 6000  = 0.25,  = 0.75,  x =4d 0 FAST-CPU This On This In 60 seconds !

Playing Around A Problem with Noise Multiple Grains-multiple orientations Grains in A Flow Field Thses calculations were performed by Andrew Kao, University of Greenwich, London Under supervision of Prof Koulis Pericleous and Dr. Georgi Djambazov.

Together they cover a vast range of scales Have Presented Two examples of multi-scale multi-physics framework Model 1 Model 2 Can we Eliminate The middle Man An model Across Scales in One model? “Direct Microscale modeling” Do we really want to do it?

3 Largest grid sizes (nodes/elements) reported in 11 MCWASP proceedings Dating back to 1980 and ending in 2006 Update of Voller and Porte-Agel JCP 2003

Have Presented Two examples of multi-scale multi-physics framework In The mean time the Framework of Process-Grid-Sub-grid Is an adequate bridge Can produce insightful results And in some ways may provide more insight in to the process as opposed to a direct simulation!