In this work, we present an extended Discontinuous Galerkin method for simulating transient, incompressible two-phase flows, which include heat transfer and thermally driven evaporation at the interface of single-component systems. This expands our previous work to include the consideration of non-material interfaces and a coupling between velocities and temperature gradients at the interface. The phase boundary is represented by the zero-set of a level set function, while effects due to surface tension are treated by the Laplace-Beltrami formulation. This sharp interface model allows for a sub-grid accurate representation of the solution fields. By using compactly supported polynomial solutions, discontinuities at the interface can be sharply represented without employing additional reconstruction schemes. The approach is validated through well-known evaporation test cases. This includes two 1D test cases, known as Stefan and Sucking problem, a 2D film boiling and finally the 3D growth of a vapor bubble, known as Scriven test case.

Nonlinear axisymmetric shape oscillations of a Newtonian drop in a vacuum are investigated using two different theoretical methods, for fundamental interest and for the significance of the oscillations in transport processes across the drop surface. The extended discontinuous Galerkin method is contrasted to the weakly nonlinear theory. While the former allows large drop surface deformation amplitudes to be analyzed with high precision and drop volume errors below 0.11% even at the largest deformations, the latter provides analytical insight into the origin of quasiperiodic time behavior of the oscillations and reveals the oscillation modes coupled in the nonlinear motion. Results from both methods for moderate initial deformation amplitudes at modes of initial drop deformation m=2, 3, and 4 are in excellent agreement, showing the time asymmetry of the oscillation and the decrease of the oscillation frequency with increasing deformation amplitude. The Fourier power spectra for the first oscillation period exhibit decreased dominant frequencies as compared to the linear results as well as the mode coupling as nonlinear effects. The numerical method is used to compute the oscillatory and damping behavior of viscous drops, as well as the interconversion of kinetic and surface energies during the oscillations at strong initial deformations. Published by the American Physical Society 2024

In this work, an extended discontinuous Galerkin (extended DG/XDG also called unfitted DG) solver for two‐dimensional flow problems exhibiting moving contact lines is presented. The generalized Navier boundary condition is employed within the XDG discretization for the handling of the moving contact lines. The spatial discretization is based on a symmetric interior penalty method and the numerical treatment of the surface tension force is done via the Laplace–Beltrami formulation. The XDG method adapts the approximation space conformal to the position of the interface and allows a sub‐cell accurate representation within the sharp interface formulation. The interface is described as the zero set of a signed‐distance level‐set function and discretized by a standard DG method. No adaption of the level‐set evolution algorithm is needed for the extension to moving contact line problems. The developed solver is validated against typical two‐dimensional contact line driven flow phenomena including droplet simulations on a wall and the two‐phase Couette flow.

In this paper the one-dimensional two-phase Stefan problem is studied analytically leading to a system of non-linear Volterra-integral-equations describing the heat distribution in each phase. For this the unified transform method has been employed which provides a method via a global relation, by which these problems can be solved using integral representations. To do this, the underlying partial differential equation is rewritten into a certain divergence form, which enables to treat the boundary values as part of the integrals. Classical analytical methods fail in the case of the Stefan problem due to the moving interface. From the resulting non-linear integro-differential equations the one for the position of the phase change can be solved in a first step. This is done numerically using a fix-point iteration and spline interpolation. Once obtained, the temperature distribution in both phases is generated from their integral representation.

The software package BoSSS serves the discretization of (steady-state or time-dependent) partial differential equations with discontinuous coefficients and/or time-dependent domains by means of an eXtended Discontinuous Galerkin (XDG, resp. DG) method, aka. cut-cell DG, aka. unfitted DG. This work consists of two major parts: First, the XDG method is introduced and a formal notation is developed, which captures important numerical details such as cell-agglomeration and a multigrid framework. In the second part, iterative solvers for extended DG systems are presented and their performance is evaluated.

Presently, the oscillation of a liquid droplet in a dynamically negligible outer medium subject to surface tension and small viscosity is investigated. By using the potential flow assumption, the unified transform method by Fokas is employed to reduce the corresponding free boundary problem formulated on a time-dependent domain into a nonlinear system of integro-differential equations (IDEs). This new system depends on one less spatial variable and is now defined on a time-independent domain. Most importantly, the resulting set of equations governs the general droplet oscillation with arbitrarily large deviations from the spherical shape. As the nonlinearity of the above IDE system up to now prevented an analytical solution, the Poincaré expansion technique is employed, retaining terms up to the second order. By decomposing the unknowns into normal modes, these equations are uncoupled and the resulting ordinary differential equations for the mode amplitudes are solved, and the results are compared to those of previous works. It should be stressed that the present analysis is limited to small viscosity, or, in other words, for small Ohnesorge numbers. The reason for this is that, inside of the droplet, a potential flow is assumed and the viscous effect is taken into account only at the droplet surface by the jump condition of momentum. This is only reasonable for a small viscosity and a short time. Otherwise, vorticity is generated at the interface and diffuses toward the inside of the droplet.

The rise of liquid in capillaries, or between two parallel plates as the 2D variant thereof, represents a challenging test case for two-phase flow solvers without a full analytic solution. Four different numerical approaches are compared for the rise of liquid, also providing reference data being of high relevance for capillarity-dominated wetting processes. The used methods are an Arbitrary Lagrangian-Eulerian method (OpenFOAM solver interTrackFoam), a geometric Volume of Fluid code (FS3D), an algebraic Volume of Fluid method (OpenFOAM solver interFoam), and a level-set based extended discontinuous Galerkin discretization (BoSSS). While the transient rise height shows excellent agreement between the different implementations, the velocity fields at the interface demonstrate a different level of local accuracy of the available approaches. Reducing the slip length reduces the overall dynamics of the system, thus yielding a qualitative change in the rise behavior - a behavior that is not covered by simplified ODE models. The obtained rise height results are vailable online: http://dx.doi.org/10.25534/tudatalib-173 (C) 2020 The Author(s). Published by Elsevier Inc.

The two-phase Couette flow with transpiration through both walls is considered, where there is a constant blowing v0 at the lower wall and a corresponding suction at the upper wall. The interface between both fluids is initially flat and, hence, stays flat as it moves upward at the constant speed of the transpiration velocity v0. The corresponding initial value problem is subject to three dimensionless numbers consisting of the Reynolds number Re and the viscosity and density ratios, ϵ and γ. The solution is obtained by the unified transform method (Fokas method) in the form of an integral representation depending on initial and all boundary values including the Dirichlet and Neumann values at the interface. The unknown values at the moving interface are determined by a system of linear Volterra integral equations (VIEs). The VIEs are of the second kind with continuous and bounded kernels. Hence, the entire two-phase spatiotemporal 1 + 1 system has dimensionally reduced. The system of VIEs is solved via a standard marching method. For the numerical computation of the complex integral contours, a parameterized hyperbola is used. The influence of the dimensionless numbers Re, γ, and ϵ is studied exemplarily. The most notable effect results from ϵ that gives rise to a kink in the velocity at the moving interface. Both ratios, ϵ and γ, allow for very different flow regimes in each fluid phase such as nearly pure Couette flows and transpiration dominated flows with strongly curved velocity profiles. Those regimes are mainly determined by the effective Reynolds number in the respective phases.

Four different numerical approaches are compared for the rise of liquid between two parallel plates. These are an Arbitrary Lagrangian-Eulerian method (OpenFOAM solver interTrackFoam), a geometric volume of fluid code (FS3D), an algebraic volume of fluid method (OpenFOAM solver interFoam), and a level set approach (BoSSS). The first three approaches discretize the bulk equation using a finite volume method while the last one employs an extended discontinuous Galerkin discretization. The results are compared to ODE models which are the classical rise model and an extended model that incorporates a Navier slip boundary condition on the capillary walls and levels at a corrected stationary rise height. All physical parameters are based on common requirements for the initial conditions, short simulation time, and a non-dimensional parameter study. The comparison shows excellent agreement between the different implementations with minor quantitative deviations for the adapted interFoam implementation. While the qualitative agreement between the full solutions of the continuum mechanical approach and the reference model is good, the quantitative comparison is only reasonable, especially for cases with increasing oscillations. Furthermore, reducing the slip length changes the solution qualitatively as oscillations are completely damped in contrast to the solution of the ODE models. To provide reference data for a full continuum simulation of the capillary rise problem, all results are made available online.