Transition in a numerical model of contact line dynamics and forced dewetting

We investigate the transition to a Landau-Levich-Derjaguin film in forced dewetting using a quadtree adaptive solution to the Navier-Stokes equations with surface tension. We use a discretization of the capillary forces near the receding contact line that yields an equilibrium for a specified contact angle $\theta_\Delta$ called the numerical contact angle. Despite the well-known contact line singularity, dynamic simulations can proceed without any explicit additional numerical procedure. We investigate angles from $15^\circ$ to $110^\circ$ and capillary numbers from $0.00085$ to $0.2$ where the mesh size $\Delta$ is varied in the range of $0.0035$ to $0.06$ of the capillary length $l_c$. To interpret the results, we use Cox's theory which involves a microscopic distance $r_m$ and a microscopic angle $\theta_e$. In the numerical case, the equivalent of $\theta_e$ is the angle $\theta_\Delta$ and we find that Cox's theory also applies. We introduce the scaling factor or gauge function $\phi$ so that $r_m = \Delta/\phi$ and estimate this gauge function by comparing our numerics to Cox's theory. The comparison provides a direct assessment of the agreement of the numerics with Cox's theory and reveals a critical feature of the numerical treatment of contact line dynamics: agreement is poor at small angles while it is better at large angles. This scaling factor is shown to depend only on $\theta_\Delta$ and the viscosity ratio $q$. In the case of small $\theta_e$, we use the prediction by Eggers [Phys. Rev. Lett., vol. 93, pp 094502, 2004] of the critical capillary number for the Landau-Levich-Derjaguin forced dewetting transition. We generalize this prediction to large $\theta_e$ and arbitrary $q$ and express the critical capillary number as a function of $\theta_e$ and $r_m$. An analogy can be drawn between $r_m$ and the numerical slip length.Comment: This version of the paper includes the corrections indicated in Ref. [1

A momentum-conserving, consistent, Volume-of-Fluid method for incompressible flow on staggered grids

The computation of flows with large density contrasts is notoriously difficult. To alleviate the difficulty we consider a consistent mass and momentum-conserving discretization of the Navier-Stokes equation. Incompressible flow with capillary forces is modelled and the discretization is performed on a staggered grid of Marker and Cell type. The Volume-of-Fluid method is used to track the interface and a Height-Function method is used to compute surface tension. The advection of the volume fraction is performed using either the Lagrangian-Explicit / CIAM (Calcul d'Interface Affine par Morceaux) method or the Weymouth and Yue (WY) Eulerian-Implicit method. The WY method conserves fluid mass to machine accuracy provided incompressiblity is satisfied which leads to a method that is both momentum and mass-conserving. To improve the stability of these methods momentum fluxes are advected in a manner "consistent" with the volume-fraction fluxes, that is a discontinuity of the momentum is advected at the same speed as a discontinuity of the density. To find the density on the staggered cells on which the velocity is centered, an auxiliary reconstruction of the density is performed. The method is tested for a droplet without surface tension in uniform flow, for a droplet suddenly accelerated in a carrying gas at rest at very large density ratio without viscosity or surface tension, for the Kelvin-Helmholtz instability, for a falling raindrop and for an atomizing flow in air-water conditions

Numerical validation of a κ-ω-κ θ -ω θ heat transfer turbulence model for heavy liquid metals

The correct prediction of heat transfer in turbulent flows is relevant in almost all industrial applications but many of the heat transfer models available in literature are validated only for ordinary fluids with Pr ≃ 1. In commercial Computational Fluid Dynamics codes only turbulence models with a constant turbulent Prandtl number of 0.85 — 0.9 are usually implemented but in heavy liquid metals with low Prandtl numbers it is well known that these models fail to reproduce correlations based on experimental data. In these fluids heat transfer is mainly due to molecular diffusion and the time scales of temperature and velocity fields are rather different, so simple turbulence models based on similarity between temperature and velocity cannot reproduce experimental correlations. In order to reproduce experimental results and Direct Numerical Simulation data obtained for fluids with Pr ≃ 0.025 we introduce a κ-ε-κ θ -ε θ turbulence model. This model, however, shows some numerical instabilities mainly due to the strong coupling between κ and ε on the walls. In order to fix this problem we reformulate the model into a new four parameter κ-ω-κ θ -ω θ where the dissipation rate on the wall is completely independent on the fluctuations. The model improves numerical stability and convergence. Numerical simulations in plane and channel geometries are reported and compared with experimental, Direct Numerical Simulation results and with results obtained with the κ-ε formulation, in order to show the model capabilities and validate the improved κ-ω model

Numerical analysis and simulation of the dynamics of mountain glaciers

In this chapter, we analyze and approximate a nonlinear stationary Stokes problem that describes the motion of glacier ice. The existence and uniqueness of solutions are proved and an a priori error estimate for the finite element approximation is found. In a second time, we combine the Stokes problem with a transport equation for the volume fraction of ice, which describes the time evolution of a glacier. The accumulation due to snow precipitation and melting are accounted for in the source term of the transport equation. A decoupling algorithm allows the diffusion and the advection problems to be solved using a two-grids method. As an illustration, we simulate the evolution of Aletsch glacier, Switzerland, over the 21st century by using realistic climatic conditions

A Lattice Boltzmann method for simulations of liquid-vapor thermal flows

We present a novel lattice Boltzmann method that has a capability of simulating thermodynamic multiphase flows. This approach is fully thermodynamically consistent at the macroscopic level. Using this new method, a liquid-vapor boiling process, including liquid-vapor formation and coalescence together with a full coupling of temperature, is simulated for the first time.Comment: one gzipped tar file, 19 pages, 4 figure

PArallel, Robust, Interface Simulator (PARIS)

Paris (PArallel, Robust, Interface Simulator) is a finite volume code for simulations of immiscible multifluid or multiphase flows. It is based on the "one-fluid" formulation of the Navier-Stokes equations where different fluids are treated as one material with variable properties, and surface tension is added as a singular interface force. The fluid equations are solved on a regular structured staggered grid using an explicit projection method with a first-order or second-order time integration scheme. The interface separating the different fluids is tracked by a Front-Tracking (FT) method, where the interface is represented by connected marker points, or by a Volume-of-Fluid (VOF) method, where the marker function is advected directly on the fixed grid. Paris is written in Fortran95/2002 and parallelized using MPI and domain decomposition. It is based on several earlier FT or VOF codes such as Ftc3D, Surfer or Gerris. These codes and similar ones, as well as Paris, have been used to simulate a wide range of multifluid and multiphase flows

Finite element simulation of three-dimensional free-surface flow problems

An adaptive finite element algorithm is described for the stable solution of three-dimensional free-surface-flow problems based primarily on the use of node movement. The algorithm also includes a discrete remeshing procedure which enhances its accuracy and robustness. The spatial discretisation allows an isoparametric piecewise-quadratic approximation of the domain geometry for accurate resolution of the curved free surface. The technique is illustrated through an implementation for surface-tension-dominated viscous flows modelled in terms of the Stokes equations with suitable boundary conditions on the deforming free surface. Two three-dimensional test problems are used to demonstrate the performance of the method: a liquid bridge problem and the formation of a fluid droplet

Fluctuations of elastic interfaces in fluids: Theory and simulation

We study the dynamics of elastic interfaces-membranes-immersed in thermally excited fluids. The work contains three components: the development of a numerical method, a purely theoretical approach, and numerical simulation. In developing a numerical method, we first discuss the dynamical coupling between the interface and the surrounding fluids. An argument is then presented that generalizes the single-relaxation time lattice-Boltzmann method for the simulation of hydrodynamic interfaces to include the elastic properties of the boundary. The implementation of the new method is outlined and it is tested by simulating the static behavior of spherical bubbles and the dynamics of bending waves. By means of the fluctuation-dissipation theorem we recover analytically the equilibrium frequency power spectrum of thermally fluctuating membranes and the correlation function of the excitations. Also, the non-equilibrium scaling properties of the membrane roughening are deduced, leading us to formulate a scaling law describing the interface growth, W^2(L,T)=L^3 g[t/L^(5/2)], where W, L and T are the width of the interface, the linear size of the system and the temperature respectively, and g is a scaling function. Finally, the phenomenology of thermally fluctuating membranes is simulated and the frequency power spectrum is recovered, confirming the decay of the correlation function of the fluctuations. As a further numerical study of fluctuating elastic interfaces, the non-equilibrium regime is reproduced by initializing the system as an interface immersed in thermally pre-excited fluids.Comment: 15 pages, 11 figure

On the reliability of computed chaotic solutions of nonlinear differential equations

In this paper a new concept, namely the critical predictable time $T_c$, is introduced to give a more precise description of computed chaotic solutions of nonlinear differential equations: it is suggested that computed chaotic solutions are unreliable and doubtable when $t > T_c$. This provides us a strategy to detect reliable solution from a given computed result. In this way, the computational phenomena, such as computational chaos (CC), computational periodicity (CP) and computational prediction uncertainty, which are mainly based on long-term properties of computed time series, can be completely avoided. So, this concept also provides us a time-scale to determine whether or not a particular time is long enough for a given nonlinear dynamic system. Besides, the influence of data inaccuracy and various numerical schemes on the critical predictable time is investigated in details by using symbolic computation software as a tool. A reliable chaotic solution of Lorenz equation in a rather large interval $0 \leq t < 1200$ non-dimensional Lorenz time units is obtained for the first time. It is found that the precision of initial condition and computed data at each time-step, which is mathematically necessary to get such a reliable chaotic solution in such a long time, is so high that it is physically impossible due to the Heisenberg uncertainty principle in quantum physics. This however provides us a so-called "precision paradox of chaos", which suggests that the prediction uncertainty of chaos is physically unavoidable, and that even the macroscopical phenomena might be essentially stochastic and thus could be described by probability more economically.Comment: 29 pages, 13 figures, 1 tabl