TY - JOUR
T1 - An energy-conservative DG-FEM approach for solid–liquid phase change
AU - Kaaks, Bouke Johannes
AU - Rohde, Martin
AU - Kloosterman, Jan Leen
AU - Lathouwers, Danny
PY - 2023
Y1 - 2023
N2 - We present a discontinuous Galerkin method for melting/solidification problems based on the “linearized enthalpy approach,” which is derived from the conservative form of the energy transport equation and does not depend on the use of a so-called mushy zone. We use the symmetric interior penalty method and the Lax–Friedrichs flux to discretize diffusive and convective terms, respectively. Time is discretized with a second-order implicit backward differentiation formula, and two outer iterations with second-order extrapolation predictors are used for the coupling of the momentum and energy. The numerical method was validated with three different benchmark cases, i.e., the one-dimensional Stefan problem, octadecane melting in a square cavity and gallium melting in a rectangular cavity. The performance of the method was quantified based on the L 2 norm error and the number of iterations needed to convergence the energy equation at each time step. For all three validation cases, a mesh convergence rate of approximately O(h) was obtained, which is below the expected accuracy of the numerical method. Only for the gallium melting case, the use of a higher-order method proved to be beneficial. The results from the present numerical campaign demonstrate the promise of the discontinuous Galerkin finite element method for modeling certain solid–liquid phase change problems where large gradients in the flow field are present or the phase change is highly localized, however, further enhancement of the method is needed to fully benefit from the use of a higher-order numerical method when solving solid–liquid phase change problems.
AB - We present a discontinuous Galerkin method for melting/solidification problems based on the “linearized enthalpy approach,” which is derived from the conservative form of the energy transport equation and does not depend on the use of a so-called mushy zone. We use the symmetric interior penalty method and the Lax–Friedrichs flux to discretize diffusive and convective terms, respectively. Time is discretized with a second-order implicit backward differentiation formula, and two outer iterations with second-order extrapolation predictors are used for the coupling of the momentum and energy. The numerical method was validated with three different benchmark cases, i.e., the one-dimensional Stefan problem, octadecane melting in a square cavity and gallium melting in a rectangular cavity. The performance of the method was quantified based on the L 2 norm error and the number of iterations needed to convergence the energy equation at each time step. For all three validation cases, a mesh convergence rate of approximately O(h) was obtained, which is below the expected accuracy of the numerical method. Only for the gallium melting case, the use of a higher-order method proved to be beneficial. The results from the present numerical campaign demonstrate the promise of the discontinuous Galerkin finite element method for modeling certain solid–liquid phase change problems where large gradients in the flow field are present or the phase change is highly localized, however, further enhancement of the method is needed to fully benefit from the use of a higher-order numerical method when solving solid–liquid phase change problems.
KW - Energy-conservative discontinuous Galerkin method
KW - linearized enthalpy approach
KW - solid–liquid phase change
UR - http://www.scopus.com/inward/record.url?scp=85159721782&partnerID=8YFLogxK
U2 - 10.1080/10407790.2023.2211231
DO - 10.1080/10407790.2023.2211231
M3 - Article
AN - SCOPUS:85159721782
SN - 1040-7790
VL - 84
SP - 487
EP - 513
JO - Numerical Heat Transfer, Part B: Fundamentals
JF - Numerical Heat Transfer, Part B: Fundamentals
IS - 5
ER -