PCIC: Cylindrical Volume Moment Calculation and Interface Reconstruction for Sub-Grid Modeling in Volume of Fluid Methods

Abstract

The accurate modeling of topology changes remains a significant challenge in geometric Volume of Fluid (VOF) simulations. When using traditional single-plane reconstruction (PLIC), fluid structures smaller than the mesh size cannot be resolved and spurious numerical breakup is triggered; this impacts important flow statistics such as drop size distributions. Recent advances have introduced paraboloid and two-plane reconstructions, which have improved high-curvature performance and enabled sub-grid film reconstructions, respectively. However, sub-grid ligament reconstructions have remained elusive. In this work, a novel cylindrical interface reconstruction strategy called PCIC is introduced for sub-grid ligament modeling. PCIC is facilitated by deriving the analytical volume moments of quadratic cylinders clipping polyhedra; this allows for exact mass conservation during volume moment transport. From the transported moments, a straight circular cylinder can be defined in the center cell of a 5x5x5 stencil. First, a quadratic principal curve is fitted to the normalized first-order moments in the stencil (the liquid barycenters), from which the cylinder's orientation and origin are approximated. The cylinder radius is then chosen to conserve volume. On-the-fly ligament detection is achieved using connected-component labeling and moments of inertia criteria, which allows for simulations to automatically choose between PLIC and PCIC in each interface cell at runtime. PCIC is demonstrated in multiphase flow test cases, where it exhibits robust reconstruction of sub-grid ligaments. This allows for relatively low-resolution PCIC simulations to provide comparable results to traditional high-resolution simulations.

Figures (20)

• Figure 1: Notation summary: $\Omega$ is a subset of $\mathbb{R}^3$, here illustrated as a cube. $\mathcal{P} \subset \Omega$ is a polyhedron, here illustrated as a regular dodecahedron, and $\mathcal{Q} \subset \Omega$ is the clipping region bounded inside $\Omega$ by the quadratic surface $\mathcal{S}$, here illustrated as a hyperbolic paraboloid. Inside $\Omega$, the surface $\mathcal{S}$ can be parameterized by a single-valued function of the coordinates $x$ and $y$. The portion of $\mathcal{S}$ inside $\mathcal{P}$ is denoted as $\tilde{\mathcal{S}} \equiv \mathcal{S} \cap \mathcal{P}$. The intersection of $\mathcal{P}$ with $\mathcal{Q}$ is denoted as $\hat{\mathcal{P}} \equiv \mathcal{P} \cap \mathcal{Q}$.
• Figure 2: Integration domains for the contribution of face $\hat{\mathcal{F}}_i$ to $\boldsymbol{\mathcal{M}}^{\hat{\mathcal{P}}_1}$, $\boldsymbol{\mathcal{M}}^{\hat{\mathcal{P}}_2}$ and $\boldsymbol{\mathcal{M}}^{\hat{\mathcal{P}}_3}$
• Figure 3: When a polyhedron that intersects the $xy$-plane is to be clipped by a quadratic cylinder centered around $\mathbf{e}_x$, the polyhedron is first clipped by the $xy$-plane and then each half is considered independently according to Section \ref{['def:halfspace']}.
• Figure 4: Unit cube centered at ${1/21/23/2 - k/2^\intercal}$ clipped by the elliptic cylinder parametrically defined as $y^2 + z^2 = 1$.
• Figure 5: Moments of the unit cube centered at ${1/21/23/2 - k/2^\intercal}$ clipped by the elliptic cylinder defined as $y^2 + z^2 = 1$ (left); Errors in the estimation of the moments in $64$-bit floating-point format (right). The geometric moments and the corresponding estimation errors are scaled by their maximum value. The moments are computed using the closed-form expressions derived in Section \ref{['sec:cylinder_case']}. The $64$-bit machine epsilon $\epsilon_{64} = 2^{-52}$ is shown as the dashed red line.