Over the last decades, many diffuse-interface Navier-Stokes Cahn-Hilliard (NSCH) 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 NSCH 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 NSCH 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 NSCH 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.

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Two well-established classes of the interface capturing models are the level-set and phase-field models. Level-set formulations satisfy the maximum principle for the density but are not energy-stable. On the other hand, phase-field models do satisfy the second law of thermodynamics but lack the maximum principle for the density. In this paper we derive a novel model for incompressible immiscible two-phase flow with non-matching densities and surface tension that is both energetically-stable and satisfies the maximum principle for the density. The model finds its place at the intersection of level-set and phase-field models. Its derivation is based on a diffusification of the incompressible two-phase Navier–Stokes equations with non-matching densities and surface tension and involves functional entropy variables. Additionally, we present an associated fully-discrete energy-stable method. Isogeometric analysis is used for the spatial discretization and the temporal-integration is performed with a new time-integration scheme that is a perturbation of the second-order 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.

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We show that in the variational multiscale framework, the weak enforcement of essential boundary conditions via Nitsche's method corresponds directly to a particular choice of projection operator. The consistency, symmetry and penalty terms of Nitsche's method all originate from the fine-scale closure dictated by the corresponding scale decomposition. As a result of this formalism, we are able to determine the exact fine-scale contributions in Nitsche-type formulations. In the context of the advection–diffusion equation, we develop a residual-based model that incorporates the non-vanishing fine scales at the Dirichlet boundaries. This results in an additional boundary term with a new model parameter. We then propose a parameter estimation strategy for all parameters involved that is also consistent for higher-order basis functions. We illustrate with numerical experiments that our new augmented model mitigates the overly diffusive behavior that the classical residual-based fine-scale model exhibits in boundary layers at boundaries with weakly enforced essential conditions.

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In this paper we show that the variational multiscale method together with the variation entropy concept form the underlying theoretical framework of discontinuity capturing. The variation entropy [M.F.P. ten Eikelder and I. Akkerman, Comput. Methods Appl. Mech. Engrg. 355 (2019) 261-283] is the recently introduced concept that equips total variation diminishing solutions with an entropy foundation. This is the missing ingredient in order to show that the variational multiscale method can capture sharp layers. The novel framework naturally equips the variational multiscale method with a class of discontinuity capturing operators. This class includes the popular YZβ method and methods based on the residual of the variation-entropy. The discontinuity capturing mechanisms do not contain ad hoc devices and appropriate length scales are derived. Numerical results obtained with quadratic NURBS are virtually oscillation-free and show sharp layers, which confirms the viability of the methodology.

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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.

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Numerical procedures and simulation techniques in science and engineering have progressed significantly during the last decades. The finite element method plays an important role in this development and has gained popularity in many fields including fluid mechanics. A recent finite element solution strategy is isogeometric analysis. Isogeometric analysis replaces the usual finite element basis functions by higher-order splines. This leads to significantly more accurate results and equips the numerical method with several desirable properties. By naively applying the finite element isogeometric method one may obtain solutions that are seriously perturbed and are as such not physically relevant. The reason is often linked to the stability of the method; a finite element method is not a priori stable. The overall objective of this thesis is centered around this point. The aim is to develop numerical techniques that inherit the stability properties of the underlying physical system. In particular we are interested in finite element techniques that can be applied to free-surface flow simulations. Stability issues in free-surface flow computations may already appear in single-fluid flow problems. Other causes of instabilities are steep layers or discontinuities and instabilities arising from the numerical treatment of the interface that separates the fluids. This thesis addresses each of these topics. Several stabilized finite element methods @en

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.@en

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 [26]. 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. As we focus on the energy behavior of time integration in case of varying densities, we restrict ourselves to low-Reynolds-number flow allowing simple Galerkin discretizations. High-Reynolds-number two-fluid flows that require stabilization are beyond the scope of the current paper. The simulation of a dambreak problem numerically supports the correct energy behavior of the proposed methodology. The proposed methodology 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.@en

This paper presents the construction of a correct-energy stabilized finite element method for the incompressible Navier–Stokes equations. The framework of the methodology and the correct-energy concept have been developed in the convective–diffusive context in the preceding paper [M.F.P. ten Eikelder, I. Akkerman, 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, Comput. Methods Appl. Mech. Engrg. 331 (2018) 259–280]. The current work extends ideas of the preceding paper to build a stabilized method within the variational multiscale (VMS) setting which displays correct-energy behavior. Similar to the convection–diffusion case, a key ingredient is the proper dynamic and orthogonal behavior of the small-scales. This is demanded for correct energy behavior and links the VMS framework to the streamline-upwind Petrov–Galerkin (SUPG) and the Galerkin/least-squares method (GLS). The presented method is a Galerkin/least-squares formulation with dynamic divergence-free small-scales (GLSDD). It is locally mass-conservative for both the large- and small-scales separately. In addition, it locally conserves linear and angular momentum. The computations require and employ NURBS-based isogeometric analysis for the spatial discretization. The resulting formulation numerically shows improved energy behavior for turbulent flows comparing with the original VMS method.

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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 H₀ 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, on both a local and a 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.

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