Fabian Denner

Fully-Coupled Balanced-Force VOF Framework for Arbitrary Meshes with Least-Squares Curvature Evaluation from Volume Fractions

A fully-coupled balanced-force numerical framework for two-phase flows with surface tension on arbitrary collocated meshes is presented, including a novel method to evaluate the curvature from volume fractions. The presented framework reduces the imbalances at the interface to solver tolerance and provides stable and reliable results for density ratios of 106 and larger. The new method to evaluate the curvature is based on a least-squares fit, providing better or equal accuracy compared to implementations found in the literature, which are generally limited to structured meshes, and the accuracy on Cartesian and tetrahedral meshes is shown to be comparable.

Numerical time-step restrictions as a result of capillary waves

The propagation of capillary waves on material interfaces between two fluids imposes a strict constraint on the numerical time-step applied to solve the equations governing this problem and is directly associated with the stability of interfacial flow simulations. The explicit implementation of surface tension is the generally accepted reason for the restrictions on the temporal resolution caused by capillary waves. In this article, a fully-coupled numerical framework with an implicit treatment of surface tension is proposed and applied, demonstrating that the capillary time-step constraint is in fact a constraint imposed by the temporal sampling of capillary waves, irrespective of the type of implementation. The presented results show that the capillary time-step constraint can be exceeded by several orders of magnitude, with the explicit as well as the implicit treatment of surface tension, if capillary waves are absent. Furthermore, a revised capillary time-step constraint is derived by studying the temporal resolution of capillary waves based on numerical stability and signal processing theory, including the Doppler shift caused by an underlying fluid motion. The revised capillary time-step constraint assures a robust, aliasing-free result, as demonstrated by representative numerical experiments, and is in the static case less restrictive than previously proposed time-step limits associated with capillary waves.

Compressive VOF method with skewness correction to capture sharp interfaces on arbitrary meshes

The accurate and efficient modelling of two-phase flows is at present mostly limited to structured, unskewed meshes, due to the additional topological and numerical complexity of arbitrary, unstructured meshes. Compressive VOF methods which discretize the interface advection with algebraic differencing schemes are computationally efficient and inherently applicable to arbitrary meshes. However, compressive VOF methods evidently suffer severely from numerical diffusion on meshes with topological skewness. In this paper we present a compressive VOF method using a state-of-the-art donor-acceptor advection scheme which includes novel modifications to substantially reduce numerical diffusion on arbitrary meshes without adding computational complexity. The new methodology accurately captures evolving interfaces on any arbitrary, non-overlapping mesh and conserves mass within the limits of the applied solver tolerance. A thorough validation of the presented methods is conducted, examining the pure advection of the interface indicator function as well as the application to evolving interfaces with surface tension. Crucially, the results on equidistant Cartesian and arbitrary tetrahedral meshes are shown to be comparable and accurate.

TVD differencing on three-dimensional unstructured meshes with monotonicity-preserving correction of mesh skewness

Total variation diminishing (TVD) schemes are a widely applied group of monotonicity-preserving advection differencing schemes for partial differential equations in numerical heat transfer and computational fluid dynamics. These schemes are typically designed for one-dimensional problems or multidimensional problems on structured equidistant quadrilateral meshes. Practical applications, however, often involve complex geometries that cannot be represented by Cartesian meshes and, therefore, necessitate the application of unstructured meshes, which require a more sophisticated discretisation to account for their additional topological complexity. In principle, TVD schemes are applicable to unstructured meshes, however, not all the data required for TVD differencing is readily available on unstructured meshes, and the solution suffers from considerable numerical diffusion as a result of mesh skewness. In this article we analyse TVD differencing on unstructured three-dimensional meshes, focusing on the non-linearity of TVD differencing and the extrapolation of the virtual upwind node. Furthermore, we propose a novel monotonicity-preserving correction method for TVD schemes that significantly reduces numerical diffusion caused by mesh skewness. The presented numerical experiments demonstrate the importance of accounting for the non-linearity introduced by TVD differencing and of imposing carefully chosen limits on the extrapolated virtual upwind node, as well as the efficacy of the proposed method to correct mesh skewness.

Frequency dispersion of small-amplitude capillary waves in viscous fluids

This work presents a detailed study of the dispersion of capillary waves with small amplitude in viscous fluids using an analytically derived solution to the initial value problem of a small-amplitude capillary wave as well as direct numerical simulation. A rational parametrization for the dispersion of capillary waves in the underdamped regime is proposed, including predictions for the wave number of critical damping based on a harmonic-oscillator model. The scaling resulting from this parametrization leads to a self-similar solution of the frequency dispersion of capillary waves that covers the entire underdamped regime, which allows an accurate evaluation of the frequency at a given wave number, irrespective of the fluid properties. This similarity also reveals characteristic features of capillary waves, for instance that critical damping occurs when the characteristic time scales of dispersive and dissipative mechanisms are balanced. In addition, the presented results suggest that the widely adopted hydrodynamic theory for damped capillary waves does not accurately predict the dispersion when viscous damping is significant, and an alternative definition of the damping rate, which provides consistent accuracy in the underdamped regime, is presented.

Fully-coupled pressure-based finite-volume framework for the simulation of fluid flows at all speeds in complex geometries

Abstract A generalized finite-volume framework for the solution of fluid flows at all speeds in complex geometries and on unstructured meshes is presented. Starting from an existing pressure-based and fully-coupled formulation for the solution of incompressible flow equations, the additional implementation of pressure–density–energy coupling as well as shock-capturing leads to a novel solver framework which is capable of handling flows at all speeds, including quasi-incompressible, subsonic, transonic and supersonic flows. The proposed numerical framework features an implicit coupling of pressure and velocity, which improves the numerical stability in the presence of complex sources and/or equations of state, as well as an energy equation discretized in conservative form that ensures an accurate prediction of temperature and Mach number across strong shocks. The framework is verified and validated by a large number of test cases, demonstrating the accurate and robust prediction of steady-state and transient flows in the quasi-incompressible as well as subsonic, transonic and supersonic speed regimes on structured and unstructured meshes as well as in complex domains.

Estimation of curvature from volume fractions using parabolic reconstruction on two-dimensional unstructured meshes

Abstract This paper proposes a method to estimate the curvature of an interface represented implicitly by discrete volume fractions on an unstructured two-dimensional mesh. The method relies on the computation of local parabolic reconstructions of the interface. The parabolic reconstruction of the interface in a given computational cell is obtained by solving a local non-linear minimisation problem, and only requires additional information from two neighbouring cells. This compactness ensures a robust behaviour on poorly-resolved interfaces. The proposed method is proven to be analogous to the height-function method for Cartesian configurations with consistent heights, and can be interpreted as a generalisation of the height-function method to meshes of any type. Tests are conducted on a range of interfaces with known curvature. The method is shown to converge with mesh refinement with the same order of accuracy as the height-function method for all three types of meshes tested, i.e. Cartesian, triangular, and polygonal.

Dispersion and viscous attenuation of capillary waves with finite amplitude

Abstract We present a comprehensive study of the dispersion of capillary waves with finite amplitude, based on direct numerical simulations. The presented results show an increase of viscous attenuation and, consequently, a smaller frequency of capillary waves with increasing initial wave amplitude. Interestingly, however, the critical wavenumber as well as the wavenumber at which the maximum frequency is observed remain the same for a given two-phase system, irrespective of the wave amplitude. By devising an empirical correlation that describes the effect of the wave amplitude on the viscous attenuation, the dispersion of capillary waves with finite initial amplitude is shown to be, in very good approximation, self-similar throughout the entire underdamped regime and independent of the fluid properties. The results also shown that analytical solutions for capillary waves with infinitesimal amplitude are applicable with reasonable accuracy for capillary waves with moderate amplitude.

Self-similarity of solitary waves on inertia-dominated falling liquid films.

We propose consistent scaling of solitary waves on inertia-dominated falling liquid films, which accurately accounts for the driving physical mechanisms and leads to a self-similar characterization of solitary waves. Direct numerical simulations of the entire two-phase system are conducted using a state-of-the-art finite volume framework for interfacial flows in an open domain that was previously validated against experimental film-flow data with excellent agreement. We present a detailed analysis of the wave shape and the dispersion of solitary waves on 34 different water films with Reynolds numbers Re=20-120 and surface tension coefficients σ=0.0512-0.072 N m(-1) on substrates with inclination angles β=19°-90°. Following a detailed analysis of these cases we formulate a consistent characterization of the shape and dispersion of solitary waves, based on a newly proposed scaling derived from the Nusselt flat film solution, that unveils a self-similarity as well as the driving mechanism of solitary waves on gravity-driven liquid films. Our results demonstrate that the shape of solitary waves, i.e., height and asymmetry of the wave, is predominantly influenced by the balance of inertia and surface tension. Furthermore, we find that the dispersion of solitary waves on the inertia-dominated falling liquid films considered in this study is governed by nonlinear effects and only driven by inertia, with surface tension and gravity having a negligible influence.

Pressure-based algorithm for compressible interfacial flows with acoustically-conservative interface discretisation

Abstract A pressure-based algorithm for the simulation of compressible interfacial flows is presented. The algorithm is based on a fully-coupled finite-volume framework for unstructured meshes with collocated variable arrangement, in which the governing conservation laws are discretised in conservative form and solved in a single linear system of equations for velocity, pressure and specific total enthalpy, with the density evaluated by an equation of state. The bulk phases are distinguished using the Volume-of-Fluid (VOF) method and the motion of the fluid interface is captured by a state-of-the-art compressive VOF method. A new interface discretisation method is proposed, derived from an analogy with a contact discontinuity, that performs local changes to the discrete values of density and total enthalpy based on the assumption of thermodynamic equilibrium, and does not require a Riemann solver. This interface discretisation method yields a consistent definition of the fluid properties in the interface region, including a unique definition of the speed of sound and the Rankine–Hugoniot relations, and conserves the acoustic features of the flow, i.e. compression and expansion waves. A variety of representative test cases of gas–gas and gas–liquid flows, ranging from acoustic waves and shock tubes to shock-interface interactions in one-, two- and three-dimensional domains, is used to demonstrate the capabilities and versatility of the presented algorithm in all Mach number regimes. The propagation, reflection and transmission of acoustic waves, shock waves and rarefaction fans in interfacial flows are predicted accurately, even for difficult cases that feature fluids with shock impedance matching, transonic shock tubes or strong shocks in gas–liquid flows, as well as on unstructured meshes.