arXiv subjects

M. ten Eikelder

Publications and source records attributed to M. ten Eikelder.

11 recordsLinked to original sources

arXiv2023

The divergence-free velocity formulation of the consistent Navier-Stokes Cahn-Hilliard model with non-matching densities, divergence-conforming discretization, and benchmarks

The prototypical diffuse-interface model that describes multi-component flows is the Navier-Stokes Cahn-Hilliard model (NSCH). Over the last decades many NSCH models have appeared that claim to describe the same physical phenomena, yet are distinct from one another. In a recent article [M.F.P. ten Eikelder, K.G. van der Zee, I. Akkerman, and D. Schillinger, Math. Mod. Meth. Appl. S. 33, pp 175-221, 2023.] we have established a unified framework of virtually all NSCH models. The framework reveals that there is only a single consistent NSCH model that naturally emanates from the underlying mixture theory. In the current article we present, verify and validate this novel consistent NSCH model by means of numerical simulation. To this purpose we discretize a divergence-free velocity formulation of the NSCH model using divergence-conforming isogeometric spaces. We compare computations of our consistent model to results of existing models from literature. The predictive capability of the numerical methodology is demonstrated via three-dimensional computations of a rising bubble and the contraction of a liquid filament that compare well with experimental data.

physics.flu-dyn↗

arXiv2023

Constraints for eliminating the Gibbs phenomenon in finite element approximation spaces

One of the major challenges in finite element methods is the mitigation of spurious oscillations near sharp layers and discontinuities known as the Gibbs phenomenon. In this article, we propose a set of functionals to identify spurious oscillations in best approximation problems in finite element spaces. Subsequently, we adopt these functionals in the formulation of constraints in an effort to eliminate the Gibbs phenomenon. By enforcing these constraints in best approximation problems, we can entirely eliminate over- and undershoot in one dimensional continuous approximations, and significantly suppress them in one- and higher-dimensional discontinuous approximations.

math.NA↗

arXiv2023

Thermodynamically consistent diffuse-interface mixture models of incompressible multicomponent fluids

In this paper we derive a class of thermodynamically consistent diffuse-interface mixture models of incompressible multicomponent fluids. The class of mixture models is fully compatible with the continuum theory of mixtures. The resulting mixture models may be formulated either in constituent or in mixture quantities. This permits a direct comparison with the Navier-Stokes Cahn-Hilliard model with non-matching densities, which reveals the key modeling simplifications of the latter.

physics.flu-dyn↗

arXiv2023

A unified framework for Navier-Stokes Cahn-Hilliard models with non-matching densities

Over the last decades, many diffuse-interface Navier-Stokes Cahn-Hilliard models with non-matching densities have appeared in the literature. These models claim to describe the same physical phenomena, yet they are distinct from one another. The overarching objective of this work is to bring all of these models together by laying down a unified framework of Navier-Stokes Cahn-Hilliard models with non-zero mass fluxes. Our development is based on three unifying principles: (1) there is only one system of balance laws based on continuum mixture theory that describes the physical model, (2) there is only one natural energy-dissipation law that leads to quasi-incompressible Navier-Stokes Cahn-Hilliard models, (3) variations between the models only appear in the constitutive choices. The framework presented in this work now completes the fundamental exploration of alternate non-matching density Navier-Stokes Cahn-Hilliard models that utilize a single momentum equation for the mixture velocity, but leaves open room for further sophistication in the energy functional and constitutive dependence.

math.AP↗

arXiv2022

A space-time framework for periodic flows with applications to hydrofoils

In this paper we propose a space-time framework for the computation of periodic flows. We employ the isogeometric analysis framework to achieve higher-order smoothness in both space and time. The discretization is performed using residual-based variational multiscale modelling and weak boundary conditions are adopted to enhance the accuracy near the moving boundaries of the computational domain. We show conservation properties and present a conservative method for force extraction. We apply our framework to the computation of a heaving and pitching hydrofoil. Numerical results display very accurate results on course meshes.

math.NA↗

arXiv2020

An energy-dissipative level-set method for the incompressible two-phase Navier-Stokes equations with surface tension using functional entropy variables

This paper presents the first energy-dissipative level-set method for the incompressible Navier-Stokes equations with surface tension. The methodology relies on the recently proposed concept of functional entropy variables. Discretization in space is performed with isogeometric analysis. Temporal-integration is performed with a new perturbed midpoint scheme. The fully-discrete scheme is unconditionally energy-dissipative, pointwise divergence-free and satisfies the maximum principle for the density. Numerical examples in two and three dimensions verify the energetic-stability of the methodology.

math.NA↗

arXiv2019

Variation entropy: a continuous local generalization of the TVD property using entropy principles

This paper presents the notion of a variation entropy. This concept is an entropy framework for the gradient of the solution of a conservation law instead of on the solution itself. It appears that all semi-norms are admissible variation entropies. This provides insight into the total variation diminishing property and justifies it from entropy principles. The framework allows to derive new local variation diminishing properties in the continuous form. This can facilitate the design of new numerical methods for problems that contain discontinuities.

math.NA↗

arXiv2019

Isogeometric analysis of linear free-surface potential flow

This paper presents a novel variational formulation to simulate linear free-surface flow. The variational formulation is suitable for higher-order finite elements and higher-order and higher-continuity shape functions as employed in Isogeometric Analysis (IGA). The novel formulation combines the interior and free-surface problems in one monolithic formulation. This leads to exact energy conservation and superior performance in terms of accuracy when compared to a traditional segregated formulation. This is confirmed by the numerical computation of traveling waves in a periodic domain and a three-dimensional sloshing problem. The isogeometric approach shows significant improved performance compared to traditional finite elements. Even on very coarse quadratic NURBS meshes the dispersion error is virtually absent.

math.NA↗

arXiv2019

An acoustic-convective splitting-based approach for the Kapila two-phase flow model

In this paper we propose a new acoustic-convective splitting-based numerical scheme for the Kapila five-equation two-phase flow model. The splitting operator decouples the acoustic waves and convective waves. The resulting two submodels are alternately numerically solved to approximate the solution of the entire model. The Lagrangian form of the acoustic submodel is numerically solved using an HLLC-type Riemann solver whereas the convective part is approximated with an upwind scheme. The result is a simple method which allows for a general equation of state. Numerical computations are performed for standard two-phase shock tube problems. A comparison is made with a non-splitting approach. The results are in good agreement with reference results and exact solutions.

math.NA↗

arXiv2019

Toward free-surface flow simulations with correct energy evolution: an isogeometric level-set approach with monolithic time-integration

This paper presents a new monolithic free-surface formulation that exhibits correct kinetic and potential energy behavior. We focus in particular on the temporal energy behavior of two-fluids flow with varying densities. Correct energy behavior here means that the actual energy evolution of the numerical solution matches the evolution as predicted by the discrete two-fluid equations. We adopt the level-set method to describe the two-fluid surface. To ensure the correct energy behavior we augment the interface convection equation with kinetic and potential energy constraints. We solve the resulting formulation consisting of the fluid and interface equations in a monolithic fashion using a recently proposed level-set method [I. Akkerman. Monotone level-sets on arbitrary meshes without redistancing. Computers & Fluids, 146:74-85, 2017.]. For the spatial discretization divergence-conforming NURBS are adopted. The resulting discrete equations are solved with a quasi-newton method which partially decouples the constraints from the rest of the problem. The simulation of a dambreak problem numerically supports the correct energy behavior of the proposed methodology. This improves the solution quality significantly upon a more traditional approach. Due to the excellent accuracy per degree of freedom one can suffice with a much lower resolution.

math.NA↗

arXiv2018

Correct energy evolution of stabilized formulations: The relation between VMS, SUPG and GLS via dynamic orthogonal small-scales and isogeometric analysis. I: The convective-diffusive context

This paper presents the construction of novel stabilized finite element methods in the convective-diffusive context that exhibit correct-energy behavior. Classical stabilized formulations can create unwanted artificial energy. Our contribution corrects this undesired property by employing the concepts of dynamic as well as orthogonal small-scales within the variational multiscale framework (VMS). The desire for correct energy indicates that the large- and small-scales should be $H_0^1$-orthogonal. Using this orthogonality the VMS method can be converted into the streamline-upwind Petrov-Galerkin (SUPG) or the Galerkin/least-squares (GLS) method. Incorporating both large- and small-scales in the energy definition asks for dynamic behavior of the small-scales. Therefore, the large- and small-scales are treated as separate equations. Two consistent variational formulations which depict correct-energy behavior are proposed: (i) the Galerkin/least-squares method with dynamic small-scales (GLSD) and (ii) the dynamic orthogonal formulation (DO). The methods are presented in combination with an energy-decaying generalized-$α$ time-integrator. Numerical verification shows that dissipation due to the small-scales in classical stabilized methods can become negative, both on a local and global scale. The results show that without loss of accuracy the correct-energy behavior can be recovered by the proposed methods. The computations employ NURBS-based isogeometric analysis for the spatial discretization.

math.NA↗