Conference Agenda

Flows with both compressible and (nearly) incompressible species appear in many engineering applications, such as cavitation near propellers or oil-dragging action by blow-by gases in internal combustion engine piston sealing rings[1].

Simulation of these systems can offer insight into the behavior of these systems in the early design stages of products or when investigating a large design space, where building prototypes is more expensive than simulating them.

Nevertheless, simulation requires accurate and efficient models for these complex flow phenomena.

The level-set method has been used for incompressible two-phase flow with success[2].

It offers intuitive computation of surface curvature based on the signed distance field.

For systems where one of the phases is compressible while the other is incompressible, material models have to be used that capture the behavior in both cases.

Also, the level-set method does not guarantee mass conservation, requiring special treatment of advection and re-initialization[3].

The models and numerical methods used for this will be presented, as well as

results for test and simple application cases.

[1] Benoist Thirouard and Tian Tian, “Oil transport in the piston ring pack (Part I): identification and characterization of the main oil transport routes and mechanisms”, Technical report, SAE Technical Paper, 2003

[2] Violeta Karyofylli, Loïc Wendling, Michel Make, Norbert Hosters, and Marek Behr, “Simplex space-time meshes in thermally coupled two-phase flow simulations of mold filling”, Computers & Fluids, 192 (2019) 104261.

[3] Elin Olsson and Gunilla Kreiss, “A conservative level set method for two-phase flow”, Journal of computational physics, 210 (2005) 225–246.

Facing phase-change systems ubiquitous in engineering and geophysical applications, we today leverage a range of problem-tailored numerical techniques. Monolithically formulated mathematical models, also referred to as fixed grid techniques [1], consider a single domain containing two or more phases with distinct material parameters, but impose the same governing equations everywhere. It is well known that numerical difficulties arise from the steeply varying material behaviour and the resulting strong non-linearities at the phase-change interface (PCI) [2].

At sufficiently large spatial scales, mixture solidification problems, such as the solidification of alloys or saltwater systems, feature a continuous mushy-layer transition from pure solid to pure liquid. At thermodynamic equilibrium, fixed grid techniques considering just one energy conservation equation for all phases are the de facto standard. While these models theoretically reduce to the classical Stefan problem as the mixture impurity tends to zero, they are often challenging to solve numerically. This is because the implicitly assumed continuity of the mixture PCI, acting as a regularization of the interface non-linearity, is lost in the limit of a sharp Stefan-problem type interface.

In this contribution, we contextualize numerical approaches to single-equation models with applications to solidification of pure substances and mixtures. In particular, we discuss the choice of primary variables in the energy equation, providing a comparative study of enthalpy-based and temperature-based formulations [3]. We examine the two approaches in terms of ease of implementation, accuracy and computational effort and provide a reference simulation based on a level-set method [4]. The methods will be applied both to a solidifying pure substance, where the interface propagation can be interpreted as a Stefan problem, and a binary mixture with a continuous mushy type phase-transition region. Based on our numerical experiments, we will conclude on guidelines to picking the most efficient formulation for the considered problem.

[1] Voller et al., "Fixed grid techniques for phase change problems: A review", 1990

[2] Krabbenhoft et al., "An implicit mixed enthalpy–temperature method for phase-change problems", 2007

[3] Terschanski et al.,: "Reactive Transport Models for Ice-Ocean Interfaces", Poster presented at SIAM CSE Amsterdam, Feb. 28, 2023

[4] Boledi et al., "A level-set based space-time finite element approach to the modelling of solidification and melting processes", 2022

Phase transition problems in the setting of non-equilibrium thermodynamics appear in various industrial and academical problems. More specifically, sublimation and deposition phenomena involving rarefied gases are important processes in freeze drying in the pharmaceutical area or in the behavior of planetary atmospheres. In these cases, the low-pressure regime is the reason for the rarefaction and consequently for the non-equilibrium behavior.

Modeling non-equilibrium behavior of rarefied gases is challenging as default continuum models can no longer accurately describe them. Instead, the kinetic theory is consulted to derive suitable descriptions.

The Stefan problem is a classical model for phase change problems, originally designed for solid-liquid interactions. We are generalizing this model for solid-gas interaction, so for sublimation and deposition problems. The gas phase is treated rarefied, where the non-equilibrium effects are introduced. A dependency on the well-studied Knudsen and Mach numbers, defined at the phase transition interface, is established. Depending on the level of the rarefaction, the resulting differences from the classical model are significant.

Motivated by the vast number of applications of higher-order partial differential equations representing gradient flow like the Cahn-Hilliard equation, the Allen-Cahn equation, the phase-field crystal equation, and the Swift–Hohenberg equation, among others, in this contribution we propose an efficient combination of the Preconditioned Conjugated Gradient (PCG) solver and the recently proposed Iterative Sherman-Morrison Inversion (ISMI) to solve the systems of linear equations that arise during an implicit-time integration of these equations if Fourier-spectral methods are employed for the spatial discretization. PCG is computationally expensive when compared to ISMI, which has a superior convergence, especially during the first few iterations but its computational edge over PCG is lost if too many solver iterations are carried out due to a higher storage demand. Therefore, in this work we propose to selectively combine PCG and ISMI solvers such that the advantages of both solvers are exploited at different stages of the solution scheme which improves the convergence of the residual error of the linear system and thereby the computational efficiency considerably in comparison to standalone versions of the solvers. Some numerical examples are presented in the context of all the aforementioned types of gradient flow in 2D and 3D to demonstrate the benefits of the selective combination in the context of different phase distributions.