We consider two-fluid flow problems in an Arbitrary Lagrangian Eulerian (ALE) framework. The purpose of this work is twofold. First, we address the problem of the moving contact line, namely the line common to the two fluids and the wall. Second, we perform a stability analysis in the energy norm for various numerical schemes, taking into account the gravity and surface tension effects. The problem of the moving contact line is treated with the so-called Generalized Navier Boundary Conditions. Owing to these boundary conditions, it is possible to circumvent the incompatibility between the classical no-slip boundary condition and the fact that the contact line of the interface on the wall is actually moving. The energy stability analysis is based in particular on an extension of the Geometry Conservation Law (GCL) concept to the case of moving surfaces. This extension is useful to study the contribution of the surface tension. The theoretical and computational results presented in this paper allow us to propose a strategy which offers a good compromise between efficiency, stability and artificial diffusion.
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 methodolog…
We propose, study, and compute solutions to a class of optimal control problems for hyperbolic systems of conservation laws and their viscous regularization. We take barotropic compressible Navier--Stokes equations (BNS) as a canonical example. We first apply the entropy--entropy flux--metric condition for BNS. We select an entropy function and rewrite BNS to a summation of flux and metric gradient of entropy. We then develop a metric variational problem for BNS, whose critical points form a primal-dual BNS system. We design a finite difference scheme for the variational system. The numerical …
Projection-based model order reduction of an ordinary differential equation (ODE) results in a projected ODE. Based on this ODE, an existing reduced-order model (ROM) for finite volume discretizations satisfies the underlying conservation law over arbitrarily chosen subdomains. However, this ROM does not satisfy the projected ODE exactly but introduces an additional perturbation term. In this work, we propose a novel ROM with the same subdomain conservation properties which indeed satisfies the projected ODE exactly. We apply this ROM to the incompressible Navier-Stokes equations and show with…
An energy--stable ALE--FEM with compensated tangential velocity for Navier--Stokes free--boundary flows
The Hong Kong Polytechnic University（香港理工大学）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；University of Chinese Academy of Sciences（中国科学院大学）
The framework of inner product norm preserving relaxation Runge-Kutta methods (David I. Ketcheson, \emph{Relaxation Runge-Kutta Methods: Conservation and Stability for Inner-Product Norms}, SIAM Journal on Numerical Analysis, 2019) is extended to general convex quantities. Conservation, dissipation, or other solution properties with respect to any convex functional are enforced by the addition of a {\em relaxation parameter} that multiplies the Runge-Kutta update at each step. Moreover, other desirable stability (such as strong stability preservation) and efficiency (such as low storage requir…
The entropy conservative, curvilinear, nonconforming, p-refinement algorithm for hyperbolic conservation laws of Del Rey Fernandez et al. (2019), is extended from the compressible Euler equations to the compressible Navier-Stokes equations. A simple and flexible coupling procedure with planar interpolation operators between adjoining nonconforming elements is used. Curvilinear volume metric terms are numerically approximated via a minimization procedure and satisfy the discrete geometric conservation law conditions. Distinct curvilinear surface metrics are used on the adjoining interfaces to c…
The resolution of the incompressible Navier-Stokes equations is tricky, and it is well known that one of the major issue is to compute a divergence free velocity. The non-conforming Crouzeix-Raviart finite element are convenient since they induce local mass conservation. Moreover they are such that the stability constant of the Fortin operator is equal to 1. This implies that they can easily handle anisotropic mesh [1, 2]. However spurious velocities may appear and damage the approximation. We propose a scheme here that allows to reduce the spurious velocities. It is based on a new discretisat…
This paper introduces a new immersed boundary (IB) method for viscous incompressible flow, based on a Fourier spectral method for the fluid solver and on the nonuniform fast Fourier transform (NUFFT) algorithm for coupling the fluid with the immersed boundary. The new Fourier spectral immersed boundary (FSIB) method gives improved boundary resolution in comparison to the standard IB method. The interpolated velocity field, in which the boundary moves, is analytically divergence-free. The FSIB method is gridless and has the meritorious properties of volume conservation, exact translation invari…
The paper is devoted to the simulation of maritime two-phase flows of air and water. Emphasis is put on an extension of the classical Volume-of-Fluid (VoF) method by a diffusive contribution derived from a Cahn-Hilliard (CH) model and its benefits for simulating immiscible, incompressible two-phase flows. Such flows are predominantly simulated with implicit VoF schemes, which mostly employ heuristic downwind-biased approximations for the concentration transport to mimic a sharp interface. This strategy introduces a severe time step restriction and requires pseudo time-stepping of steady flows.…
