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The accurate reconstruction of immiscible fluid-fluid interfaces from the volume fraction field is a critical component of geometric Volume of Fluid (VOF) methods. A common strategy is the Piecewise Linear Interface Calculation (PLIC), which fits a plane in each mixed-phase computational cell. However, recent work goes beyond PLIC by using two planes or even a paraboloid. To select such planes or paraboloids, complex optimization algorithms as well as carefully crafted heuristics are necessary. Yet, the potential exists for a well-trained machine learning model to efficiently provide broadly applicable solutions to the interface reconstruction problem at lower costs. In this work, the viability of a machine learning approach is demonstrated in the context of a single plane reconstruction. A feed-forward deep neural network is used to predict the normal vector of a PLIC plane given volume fraction and phasic barycenter data in a $3\times3\times3$ stencil. The PLIC plane is then translated in its cell to ensure exact volume conservation. Our proposed neural network PLIC reconstruction (PLIC-Net) is equivariant to reflections about the Cartesian planes. Training data is analytically generated with $\mathcal{O}(10^6)$ randomized paraboloid surfaces, which allows for sampling a wide range of interface shapes. PLIC-Net is tested in multiphase flow simulations where it is compared to standard (E)LVIRA reconstruction algorithms, and the impact of training data statistics on PLIC-Net's performance is also explored. It is found that PLIC-Net greatly limits the formation of spurious planes and generates cleaner numerical break-up of the interface. Additionally, the computational cost of PLIC-Net is lower than that of (E)LVIRA. These results establish that machine learning is a viable approach to VOF interface reconstruction and is superior to current reconstruction algorithms for some cases.

The mesh-dependency of the breakup of liquid films, including their breakup length scales and resulting drop size distributions, has long been an obstacle inhibiting the computational modeling of large-scale spray systems. With the aim of overcoming this barrier, this work presents a framework for the prediction and modeling of subgrid-thickness liquid film formation and breakup within two-phase simulations using the volume of fluid method. A two-plane interface reconstruction is used to capture the development of liquid films as their thickness decreases below the mesh size. The breakup of the film is predicted with a semi-analytical model that incorporates the film geometry captured through the two-plane reconstruction. The framework is validated against experiments of the bag breakup of a liquid drop at $\text{We}=13.8$ through the comparison of the resulting drop size and velocity distributions. The generated distributions show good agreement with experimental results for drop resolutions as low as 25.6 cells across the initial diameter. The presented framework enables these drop breakup simulations to be performed at a computational cost three orders of magnitude lower than the cost of simulations utilizing adaptive mesh refinement.

A novel interface reconstruction strategy for volume of fluid (VOF) methods is introduced that represents the liquid-gas interface as two planes that co-exist within a single computational cell. In comparison to the piecewise linear interface calculation (PLIC), this new algorithm greatly improves the accuracy of the reconstruction, in particular when dealing with thin structures such as films. The placement of the two planes requires the solution of a non-linear optimization problem in six dimensions, which has the potential to be overly expensive. An efficient solution to this optimization problem is presented here that exploits two key ideas: an algorithm for extracting multiple plane orientations from transported surface data, and an efficient and mass-conserving distance-finding algorithm that accounts for two planes with arbitrary orientation. Additionally, a simple and robust strategy is presented to accurately represent the surface tension forces produced at the interface of subgrid-thickness films. The performance of this new VOF reconstruction is demonstrated on several test cases that illustrate the capability to handle arbitrarily thin films.

Over the past decades, the volume-of-fluid (VOF) method has been the method of choice for simulating atomization processes, owing to its unique ability to discretely conserve mass. Current state-of-the-art VOF methods, however, rely on the piecewise-linear interface calculation (PLIC) to represent the interface used when calculating advection fluxes. This renders the estimated curvature of the transported interface zeroth-order accurate at best, adversely impacting the simulation of surface-tension-driven flows. In the past few years, there have been several attempts at using piecewise-parabolic interface approximations instead of piecewise-linear ones for computing advection fluxes, albeit all limited to two-dimensional cases or not inherently mass conservative. In this contribution, we present our most recent work on three-dimensional piecewise-parabolic interface reconstruction and apply it in the context of the VOF method. As a result of increasing the order of the interface representation, the reconstruction of the interface and the estimation of its curvature now become a single step instead of two separate ones. The performance of this new approach is assessed both in terms of accuracy and stability and compared to the classical PLIC-VOF approach on a range of canonical test-cases and cases of surface-tension-driven instabilities.

In liquid-fueled combustion systems, optimization of the fuel atomization process is critical to reducing fuel consumption and lowering pollutant emissions. Accurate and efficient computational modeling of liquid atomization can open the door to spray optimization, however it presents a significant challenge to modelers due to the extremely complex flow field and wide range of length and time scales involved. In this work, a multi-scale and multi-domain simulation strategy is used to model end-to-end the turbulent spray produced by a swirled two-fluid coaxial atomizer, a device that utilize a high-speed swirled gas stream to destabilize a co-flowing low-speed liquid, widely used in systems such as fuel injectors. Our computational method relies on sub-grid scale modeling; in particular, we will introduce a thin structure break-up model to account for topology changes, converting thin liquid structures into spherical Lagrangian particles. With such simulations, the impact of swirl on the break-up process can be analyzed by varying the swirl ratio, and we aim to quantitatively validate our simulations against experiments at identical operating conditions, including drop size statistics., Comment: 8 pages, 12 figures. Conference: 33rd Annual Conference on Liquid Atomization and Spray Systems (ILASS-Americas 2023)

This paper evaluates and compares the accuracy and robustness of curvature estimation methods for three-dimensional interfaces represented implicitly by discrete volume fractions on a Cartesian mesh. The height function (HF) method is compared to three paraboloid fitting methods: fitting to the piecewise linear interface reconstruction centroids (PC), fitting to the piecewise linear interface reconstruction volumetrically (PV), and volumetrically fitting (VF) the paraboloid directly to the volume fraction field. The numerical studies presented in this work find that while the curvature error from the VF method converges with second-order accuracy as with the HF method for static interfaces represented by exact volume fractions, the PV method best balances low curvature errors with low computational cost for dynamic interfaces when the interface reconstruction and advection are coupled to a two-phase Navier-Stokes solver.

We provide closed-form expressions for the first moments (i.e., the volume and volume-weighted centroid) of a polyhedron clipped by a paraboloid, that is, of a polyhedron intersected with the subset of the three-dimensional real space located on one side of a paraboloid. These closed-form expressions are derived following successive applications of the divergence theorem and the judicious parametrization of the intersection of the polyhedron's faces with the paraboloid. We provide means for identifying ambiguous discrete intersection topologies, and propose a corrective procedure for preventing their occurence. Finally, we put our proposed closed-form expressions and numerical approach to the test with millions of random and manually engineered polyhedron/paraboloid intersection configurations. The results of these tests show that we are able to provide robust machine-accurate estimates of the first moments at a computational cost that is within one order of magnitude of that of state-of-the-art half-space clipping algorithms.

In this paper a numerical methodology for surface tension modeling is presented, with an emphasis on the implementation in the OpenFOAM framework. The methodology relies on a combination of (i) a well-balanced approach based on the Ghost Fluid Method (GFM), including the jump of density and pressure directly in the numerical discretization of the pressure equation, and (ii) Height Functions to evaluate the interface curvature, implemented, to the authors' knowledge, for the first time in OpenFOAM. The method is able to significantly reduce spurious currents (almost to machine accuracy) for a stationary droplet, showing second order convergence both for the curvature and the interface shape. Accurate results are also obtained for additional test cases such as translating droplets, capillary oscillations and rising bubbles, for which numerical results are comparable to what obtained by other numerical codes in the same conditions. Finally, the Height Functions method is extended to include the treatment of contact angles, both for sessile droplets and droplets suspended under the effect of gravity, showing a very good agreement with the theoretical prediction. The code works in parallel mode and details on the actual implementation in OpenFOAM are included to facilitate the reproducibility of the results., Comment: The name of the journal to which the draft has been submitted has been replaced with the Editor

This work demonstrates a computational framework for simulating vaporizing, liquid-gas flows. It is developed for the general vaporization problem which solves the vaporization rate based as from the local thermodynamic equilibrium of the liquid-gas system. This includes the commonly studied vaporization regimes of film boiling and isothermal evaporation. The framework is built upon a Cartesian grid solver for low-Mach, turbulent flows which has been modified to handle multiphase flows with large density ratios. Interface transport is performed using an unsplit volume of fluid solver. A novel, divergence-free extrapolation technique is used to create a velocity field that is suitable for interface transport. Sharp treatments are used for the vapor mass fractions and temperature fields. The pressure Poisson equation is treated using the Ghost Fluid Method. Interface equilibrium at the interface is computed using the Clausius-Clapeyron relation, and is coupled to the flow solver using a monotone, unconditionally stable scheme. It will be shown that correct prediction of the interface properties is fundamental to accurate simulations of the vaporization process. The convergence and accuracy of the proposed numerical framework is verified against solutions in one, two, and three dimensions. The simulations recover first order convergence under temporal and spatial refinement for the general vaporization problem. The work is concluded with a demonstration of unsteady vaporization of a droplet at intermediate Reynolds number.

This note introduces a regression technique for finding a class of nonlinear integro-differential operators from data. The method parametrizes the spatial operator with neural networks and Fourier transforms such that it can fit a class of nonlinear operators without needing a library of a priori selected operators. We verify that this method can recover the spatial operators in the fractional heat equation and the Kuramoto-Sivashinsky equation from numerical solutions of the equations., Comment: 5 pages, 3 figures, preprint submitted to the Journal of Computational Physics

A volume-filtered Euler-Lagrange large eddy simulation methodology is used to predict the physics of turbulent liquid-solid slurry flow through a horizontal pipe. A dynamic Smagorinsky model based on Lagrangian averaging is employed to account for the sub-filter scale effects in the liquid phase. A fully conservative immersed boundary method is used to account for the pipe geometry on a uniform cartesian grid. The liquid and solid phases are coupled through volume fraction and momentum exchange terms. Particle-particle and particle-wall collisions are modeled using a soft-sphere approach. A series of simulations have been performed by varying the superficial liquid velocity to be consistent with the experimental data by Dahl et al. (2003). Depending on the liquid flow rate, a particle bed can form and develop different patterns, which are discussed in the light of regime diagrams proposed in the literature. The fluctuation in the height of the liquid-bed interface is characterized to understand the space and time evolution of these patterns. Statistics of engineering interest such as mean velocity, mean concentration, and mean streamwise pressure gradient driving the flow are extracted from the numerical simulations and presented. Sand hold-up calculated from the simulation results suggest that this computational strategy is capable of accurately predicting critical deposition velocity.

A conjugate heat transfer (CHT) immersed boundary (IB and CHTIB) method is developed for use with laminar and turbulent flows with low to moderate Reynolds numbers. The method is validated with the canonical flow of two co-annular rotating cylinders at $Re=50$ which shows second order accuracy of the $L_{2}$ and $L_{\infty}$ error norms of the temperature field over a wide rage of solid to fluid thermal conductivities, $\kappa_{s}/\kappa_{f} = \left(9-100\right)$. To evaluate the CHTIBM with turbulent flow a fully developed, heated, turbulent channel $\left(Re_{u_{\tau}}=150\text{ and } \kappa_{s}/\kappa_{f}=4 \right)$ is used which shows near perfect correlation to previous direct numerical simulation (DNS) results. The CHTIB method is paired with a momentum IB method (IBM), both of which use a level set field to define the wetted boundaries of the fluid/solid interfaces and are applied to the flow solver implicitly with rescaling of the difference operators of the finite volume (FV) method (FVM).