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An Enthalpy Based Scheme for Simulating Dendritic Growth V.R. Voller

Solvabillity. (kim et al). Concentration. k = 0.15, Mc = 0.1, T 0 = -.55, Le = 20.0. e = 0.02, D x = 2.5d 0. Profile along dashed line. Concentration field at time t = 30,000. An Enthalpy Based Scheme for Simulating Dendritic Growth V.R. Voller.

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An Enthalpy Based Scheme for Simulating Dendritic Growth V.R. Voller

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  1. Solvabillity (kim et al) Concentration k = 0.15, Mc = 0.1, T0 = -.55, Le = 20.0 e = 0.02, Dx = 2.5d0 Profile along dashed line Concentration field at time t = 30,000 An Enthalpy Based Scheme for Simulating Dendritic Growth V.R. Voller This work is based on an original idea of Tacke 1988— modified here to allow for anisotropy and solute Problem and Solution Problem Growing a solid seed in an undercooeld binary alloy melt T0 < 0 Governing equations Solution Use FIXED finite difference grid Enthalpy Solute Conservation Solve for H and C Explicitly Surface of seed is undercooled If f = 0 or f = 1 set crystal anisotropy solute undercooling curvature BUT If 0 < f < 1 THEN Iterate until potential: continuous at interface Calculate k and q from current f filed f = liqu. frac. Calculate Undercooling Ti resulting in crystal growth Update Tricks and Devices Numerical Considerations Remedial scheme: smear out f value, e.g., Remedial Scheme: Use nine volume stencil to calculate derivatives When cell first reaches f = 0 “infect” every fully liquid neighboring cell (f =1) with a small solid “seed” Like a CA RULE Problem: range of cells with 0 < f < 1 restricted to width of one cell Accuracy in curvature calc? Initial condition— rectangle of side 3 Dx Testing Comparison with other Methods Grid Dependence-Tip velocity Black-- Phase field and Level set from Kim, Goldenfeld and Dantzig Red-- Current work Dendrite shape with 3 grid sizes shows reasonable independence Long term tip dynamics approaches theory Dimensionless time t = 37,600 Tip velocity 3.25do (black) 4do (blue) 2.5do (red) e = 0.05, T0 = -0.55 e = 0.05, T0 = -0.65 Dimensionless time t = 6000 BUT results begin to deteriorate if grid is made smaller !! a = 0.25, b = 0.75 Low Grid Anisotropy Comparison with one-d Analytical Solution 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 Dimensionless time t = 6000 Constant Ti, Ci k = 0.1, Mc = 0.1, T0 = -.5, Le = 1.0 Symbol-numeric sol. Tip position with time Red-line Numeric sol. Covers analytical e = 0.05, T0 = -0.65 Concentration and Temperature at dimensionless time t =100 Front Movement Not perfect: In 450 case the tip velocity at time 6000 (slope of line) is below the theoretical limit. Note: Different “smear” parameters are used in 00 and 450 case Results FAST-CPU Effect of Lewis Number This k = 0.15, Mc = 0.1, T0 = -.65 • = 0.05, • Dx =3.333d0 On This In 60 seconds ! time t = 6000 e = 0.05, T0 = -0.65 All predictions at time t =6000 a = 0.25, b = 0.75, Dx =4d0 Comments The proposed enthalpy method is EASY to program and can RAPIDLY produce ACCURATE RESULTS consistent with known growth characteristics. The approach can account for growth in undercooled ALLOYS A DRAWBACK GRID dependence can be limited BUT choice of “smearing” parameters is ad-hoc. Talking Point: Method has features of both CA, Level Set and Phase Fields. How is the enthalpy method related to these alternative methods ?

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