Boiling heat transfer by phase-field method

Abstract In this work, we propose and test the validity of a phase-field method tailored specifically for modeling boiling phenomena. The method relies on numerical solutions of the Navier–Stokes equations coupled with a phase-field method and the energy equation. The continuity and Navier–Stokes equations have been modified introducing a source term that accounts for phase change. Likewise, in the conservative Allen–Cahn equation (phase-field method) a source term that accounts for the volume is introduced. The system of governing equations is solved using a projection-correction method and equations are discretized using a second-order finite difference approach. Thanks to the numerical discretization employed, a constant coefficient Poisson equation for pressure is obtained, which can be efficiently solved using FFT-based direct solvers. The proposed method is validated against several benchmarks: an interface undergoing vaporization at a constant rate, the Stefan problem, the adsorption problem, and the growth of a 2D vapor bubble. For all the benchmarks, the present method well matches with analytical and archival literature results for a wide range of vapor-to-liquid density ratios, from $$\rho _v/\rho _l = 1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>ρ</mml:mi> <mml:mi>v</mml:mi> </mml:msub> <mml:mo>/</mml:mo> <mml:msub> <mml:mi>ρ</mml:mi> <mml:mi>l</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> down to $$\rho _v/\rho _l \simeq 5 \times 10^{-4}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>ρ</mml:mi> <mml:mi>v</mml:mi> </mml:msub> <mml:mo>/</mml:mo> <mml:msub> <mml:mi>ρ</mml:mi> <mml:mi>l</mml:mi> </mml:msub> <mml:mo>≃</mml:mo> <mml:mn>5</mml:mn> <mml:mo>×</mml:mo> <mml:msup> <mml:mn>10</mml:mn> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> (where $$\rho _v$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ρ</mml:mi> <mml:mi>v</mml:mi> </mml:msub> </mml:math> identifies the vapor density and $$\rho _l$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ρ</mml:mi> <mml:mi>l</mml:mi> </mml:msub> </mml:math> the liquid density).