The present paper is devoted to the computation of two-phase flows using the two-fluid approach. The overall model is hyperbolic and has no conservative form. No instantaneous local equilibrium between phases is assumed, which results in a two-velocity two-pressure model. Original closure laws for interfacial velocity and interfacial pressure are proposed. These closures allow to deal with discontinuous solutions such as shock waves and contact discontinuities without ambiguity with the definition of Rankine–Hugoniot jump relations. Each field of the convective system is investigated, providing maximum principle for the volume fraction and the positivity of densities and internal energies are ensured when focusing on the Riemann problem. Two-finite volume methods are presented, based on the Rusanov scheme and on an approximate Godunov scheme. Relaxation terms are taken into account using a fractional step method. Eventually, numerical tests illustrate the ability of both methods to compute two-phase flows.

We study the construction of subgrid-scale models for large-eddy simulation of incompressible turbulent flows. In particular, we aim to consolidate a systematic approach of constructing subgrid-scale models, based on the idea that it is desirable that subgrid-scale models are consistent with the mathematical and physical properties of the Navier-Stokes equations and the turbulent stresses.To that end, we first discuss in detail the symmetries of the Navier-Stokes equations, and the near-wall scaling behavior, realizability and dissipation properties of the turbulent stresses. We furthermore summarize the requirements that subgrid-scale models have to satisfy in order to preserve these important mathematical and physical properties. In this fashion, a framework of model constraints arises that we apply to analyze the behavior of a number of existing subgrid-scale models that are based on the local velocity gradient. We show that these subgrid-scale models do not satisfy all the desired properties, after which we explain that this is partly due to incompatibilities between model constraints and limitations of velocity-gradient-based subgrid-scale models. However, we also reason that the current framework shows that there is room for improvement in the properties and, hence, the behavior of existing subgrid-scale models. We furthermore show how compatible model constraints can be combined to construct new subgrid-scale models that have desirable properties built into them. We provide a few examples of such new models, of which a new model of eddy viscosity type, that is based on the vortex stretching magnitude, is successfully tested in large-eddy simulations of decaying homogeneous isotropic turbulence and turbulent plane-channel flow.

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When simulating high Reynolds number two-phase flow, boundary layers develop at the interface, which are much thinner compared to the capillary length-scales that are of interest. Resolving such an interface layer is expensive and therefore it is often not resolved in a simulation. Numerically such an underresolved interface layer results in a velocity discontinuity tangential to the interface. We propose to include such tangential velocity discontinuities in our numerical model. This results in a sharp two-fluid model for two-phase flow, where only the interface-normal component of the velocity field is smooth. This condition is implicitly enforced via a new jump condition on the pressure gradient, which we discretize using a multidimensional variant of the ghost fluid method [6]. Results are shown of breaking waves [2] as well as (breaking) waves impacting a solid wall [3] where we compare to state-of-the-art methods [3, 4]. We show that our proposed method is able to accurately simulate high Reynolds number two-phase flow without the need for resolving, or artificially thickening, of the interface layer.

We focus on subgrid-scale modeling for large-eddy simulation of incompressible turbulent flows. In particular, we follow a systematic approach that is based on the idea that subgrid-scale models should preserve fundamental properties of the Navier-Stokes equations and turbulent stresses. To that end, we discuss the symmetries and conservation laws of the Navier-Stokes equations, as well as the near-wall scaling, realizability and dissipation behavior of the turbulent stresses. Regarding each of these properties as a model constraint, we obtain a framework that can be used to assess existing and create new subgrid-scale models. We show that several commonly used velocity-gradient-based subgrid-scale models do not exhibit all the desired properties. Although this can partly be explained by incompatibilities between model constraints, we believe there is room for improvement in the properties of subgrid-scale models. As an example, we provide a new eddy viscosity model, based on the vortex stretching magnitude, that is successfully tested in large-eddy simulations of turbulent plane-channel flow.

The numerical modelling of convection dominated high density ratio two-phase flow poses several challenges, amongst which is resolving the relatively thin shear layer at the interface. To this end we propose a sharp discretisation of the two-velocity model of the two-phase Navier-Stokes equations. This results in the ability to model the shear layer, rather than resolving it, by allowing for a velocity discontinuity in the direction(s) tangential to the interface.In a previous paper (Remmerswaal and Veldman (2022), arXiv:2209.14934) we have discussed the transport of mass and momentum, where the two fluids were not yet coupled. In this paper an implicit coupling of the two fluids is proposed, which imposes continuity of the velocity field in the interface normal direction. The coupling is included in the pressure Poisson problem, and is discretised using a multidimensional generalisation of the ghost fluid method. Furthermore, a discretisation of the diffusive forces is proposed, which leads to recovering the continuous one-velocity solution as the interface shear layer is resolved.The proposed two-velocity formulation is validated and compared to our one-velocity formulation, where we consider a multitude of two-phase flow problems. It is demonstrated that the proposed two-velocity model is able to consistently, and sharply, approximate solutions to the inviscid Euler equations, where velocity discontinuities appear analytically as well. Furthermore, the proposed two-velocity model is shown to accurately model the interface shear layer in viscous problems, and it is successfully applied to the simulation of breaking waves where the model was used to sharply capture free surface instabilities.

Local pressure measurements during liquid impacts in LNG tanks show a large variability, due to the free-surface roughness and the liquid fragmentation resulting from free-surface instabilities initiated by the gas escaping redfrom the closing gas pocket. It also makes the experimental discrimination of the effect of any multiphase parameter (density ratio, compressibility, phase transition) on impact loads more complex. The main challenges are (i) to capture both experimentally and numerically the development of the free surface instabilities; and (ii) to quantify their effect on the variability of local pressures. Numerical modeling of the initiation of the free surface instabilities requires an accurate VOF-variant. Surface tension forces are modelled via a Young-Laplace jump condition, where the interface curvature is computed using a local height function. A parabolic interface reconstruction is required to ensure convergence of the curvature in space and time. Advection is performed with a modified EMPFA method. Spurious velocities are avoided by careful treatment of the (almost) discontinuities of the flow field. The model will be demonstrated on various simulations of breaking waves.

For capillary driven flow the interface curvature is essential in the modelling of surface tension via the imposition of the Young-Laplace jump condition. We show that traditional geometric volume of fluid (VoF) methods, that are based on a piecewise linear approximation of the interface, do not lead to an interface curvature which is convergent under mesh refinement in time-dependent problems.Instead, we propose to use a piecewise parabolic approximation of the interface, resulting in a class of piecewise parabolic interface calculation (PPIC) methods. In particular, we introduce the parabolic LVIRA and MoF methods, PLVIRA and PMoF, respectively. We show that a Lagrangian remapping method is sufficiently accurate for the advection of such a parabolic interface.It is numerically demonstrated that the newly proposed PPIC methods result in an increase of reconstruction accuracy by one order, convergence of the interface curvature in time-dependent advection problems and Weber number independent convergence of a droplet translation problem, where the advection method is coupled to a two-phase Navier-Stokes solver.