A 3D sharp and conservative VOF method for modeling the contact line dynamics with hysteresis on complex boundaries

Abstract

We propose a sharp and conservative 3D numerical method for simulating moving contact lines on complex geometries, developed within a coupled geometric Volume-of-Fluid (VOF) and embedded boundary framework. The first major contribution is a modified VOF advection and reconstruction scheme specifically designed for mixed cells containing liquid, gas, and solid phases. This formulation ensures strict local mass conservation in the presence of arbitrarily shaped embedded boundaries. To overcome the severe time-step limitation caused by small cut cells, a redistribution advection strategy is introduced, which completely removes the CFL constraint while preserving both local and global volume conservation. The second key contribution is a novel 3D contact angle imposition technique built upon the height function framework. By incorporating a pre-fitting paraboloid procedure, the method achieves robust curvature estimation and accurate enforcement of contact angle conditions on irregular solid surfaces. In addition, contact angle hysteresis is modeled to capture more realistic wetting dynamics. A series of challenging benchmark tests have been conducted to demonstrate the accuracy, robustness, and superiority of the proposed method compared with existing sharp-interface approaches. This study, for the first time, establishes a fully geometric and conservative VOF-based scheme capable of accurately resolving contact line dynamics on arbitrarily complex 3D surfaces.

Figures (33)

• Figure 1: Schematic illustration of the moving contact line problem on (a) a flat solid substrate (regular boundary) and (b) an irregular solid substrate (complex geometric boundary). $\Omega_g$, $\Omega_l$, and $\Omega_s$ denote the gas, liquid, and solid phases, respectively. $\Gamma_l$ and $\Gamma_s$ indicate the liquid/gas interface and the solid surface, with $\textbf{n}_l$ and $\textbf{n}_s$ representing their corresponding normal vectors. The symbol $\theta$ denotes the contact angle.
• Figure 2: (a) Schematic of the geometric VOF advection method on regular boundaries. (b) Definition of the height function in 3D on regular boundaries, including its evaluation in ghost cells. (c) Contact line contour on the solid boundary.
• Figure 3: Schematic of the liquid/gas interface reconstruction within a mixed cell. $\gamma$ denotes the signed distance from the cell center to the interface. $V_i$ represents the volume of each subdivided region. $D_l$ and $D_u$ are the heights from the lowest vertex to the lower and upper planes of the sub-region, respectively. The $M$-region contains the liquid/gas interface. $A_l$ and $A_u$ are the polygonal cross-sectional areas at the lower and upper planes, and $\lambda$ denotes the interface height within the sub-region.
• Figure 4: Illustration of the advected liquid volume $V$ within a mixed cell. The advection interval has a width of $un = u_f \cdot dt$.
• Figure 5: Sketch of the geometric VOF advection scheme in mixed cells. (a) $s_f$ denotes the fluid fraction of the face, and $\Delta$ is the mesh size. $I$ indicates the face intersected by the embedded solid boundary. $V_1$ and $V_2$ represent the volumes of the dark blue and dark grey regions, respectively. (b) ${un}^*$ denotes the corrected width of the advection interval, and $V_3$ is the green region, numerically equal to $V_2$.