Full numerical simulations of multifluid flows

To fully understand the behavior of multifluid systems, one must have good insight into the basic micromechanisms that govern the evolution of a single ‘‘structure’’ (e.g., a bubble or a drop) and the interactions of a few such basic entities. In addition to the usual questions about the relative magnitude of the various physical effects (inertia, viscosity, and surface tension) for multifluid systems, the effects of surface phenomena such as contaminants must be addressed. Full numerical simulations are, in principle, ideally suited to provide this information. Not only are all the quantitative data readily available, but various physical processes can be turned on and off at will. In practice, however, simulations of multifluid problems are one of the difficult areas of computational fluid dynamics. Almost all current studies of multifluid problems make a number of simplifications, such as inviscidness, Stokes flow, two‐dimensionality, or axisymmetry. Although such models capture some of the important behavior, they often put severe constraints on the problems that can be investigated.Many of the fundamental processes in multifluid flow involve fully three‐dimensional flows, where both inertia and viscous effects must be accounted for. To address these effects, we have recently developed a front‐tracking method for multifluid, incompressible flows that appears to be both accurate and robust. The method has been implemented for both two‐ as well as fully three‐dimensional situations. In this paper, we will discuss two problems that we are currently investigating using this numerical method: the Rayleigh–Taylor instability and the motion of bubbles and drops. For fluid mixing induced by unstable stratification, the Rayleigh–Taylor instability where a heavy fluid falls into a lighter underlying fluid, is the prototypical example. Indeed, for such flows its importance is similar to that of the Kelvin–Helmholtz instability for fluid mixing induced by a shear flow. For small density stratification, we show that three‐dimensionality can lead to a large amplitude vortex structure that differs considerably from what two‐dimensional simulations predict. The different vortical configuration leads to more rapid nonlinear growth for the fully three‐dimensional case, even though the linear growth rate is the same. We also show how viscosity stratification modifies the evolution. For the weakly stratified case, where inviscid calculations predict symmetric evolution with respect to the heavy and the light fluid, viscosity stratification leads to considerable asymmetry, with the more viscous fluid forming big round bubbles and the less viscous one being confined to narrow fingers. The effect of density stratification for viscous three‐dimensional motion will also be discussed. For many mixing problems, the long‐time state consists of a dispersed phase that forms drops or bubbles in another phase. We will discuss preliminary investigations of such flows. Calculations of rising bubbles for various values of surface tension and viscosity (both in two and three dimensions) appear to correlate well with experimental observations and steady‐state calculations in the literature. The interactions of bubbles with each other, density interfaces, and vortices will also be discussed.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/69841/2/PFADEB-3-5-1455-1.pd

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

Computations of multi-fluid flows

Full numerical simulations of three-dimensional flows of two or more immiscible fluids of different densities and viscosities separated by a sharp interface with finite surface tension are discussed. The method used is based on a finite difference approximation of the full Navier-Stokes equations and explicit tracking of the interface between the fluids. Preliminary simulations of the Rayleigh-Taylor instability and the motion of bubbles are shown.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/29741/1/0000078.pd

Effects of small surface tension in Hele-Shaw multifinger dynamics: an analytical and numerical study

We study the singular effects of vanishingly small surface tension on the dynamics of finger competition in the Saffman-Taylor problem, using the asymptotic techniques described in [S. Tanveer, Phil. Trans. R. Soc. Lond. A 343, 155 (1993)]and [M. Siegel, and S. Tanveer, Phys. Rev. Lett. 76, 419 (1996)] as well as direct numerical computation, following the numerical scheme of [T. Hou, J. Lowengrub, and M. Shelley,J. Comp. Phys. 114, 312 (1994)]. We demonstrate the dramatic effects of small surface tension on the late time evolution of two-finger configurations with respect to exact (non-singular) zero surface tension solutions. The effect is present even when the relevant zero surface tension solution has asymptotic behavior consistent with selection theory.Such singular effects therefore cannot be traced back to steady state selection theory, and imply a drastic global change in the structure of phase-space flow. They can be interpreted in the framework of a recently introduced dynamical solvability scenario according to which surface tension unfolds the structually unstable flow, restoring the hyperbolicity of multifinger fixed points.Comment: 16 pages, 15 figures, submitted to Phys. Rev

A New Class of Nonsingular Exact Solutions for Laplacian Pattern Formation

We present a new class of exact solutions for the so-called {\it Laplacian Growth Equation} describing the zero-surface-tension limit of a variety of 2D pattern formation problems. Contrary to common belief, we prove that these solutions are free of finite-time singularities (cusps) for quite general initial conditions and may well describe real fingering instabilities. At long times the interface consists of N separated moving Saffman-Taylor fingers, with ``stagnation points'' in between, in agreement with numerous observations. This evolution resembles the N-soliton solution of classical integrable PDE's.Comment: LaTeX, uuencoded postscript file

Phase-field model for Hele-Shaw flows with arbitrary viscosity contrast. II. Numerical study

We implement a phase-field simulation of the dynamics of two fluids with arbitrary viscosity contrast in a rectangular Hele-Shaw cell. We demonstrate the use of this technique in different situations including the linear regime, the stationary Saffman-Taylor fingers and the multifinger competition dynamics, for different viscosity contrasts. The method is quantitatively tested against analytical predictions and other numerical results. A detailed analysis of convergence to the sharp interface limit is performed for the linear dispersion results. We show that the method may be a useful alternative to more traditional methods.Comment: 13 pages in revtex, 5 PostScript figures. changes: 1 reference added, figs. 4 and 5 rearrange

Phase-field model for Hele-Shaw flows with arbitrary viscosity contrast. I. Theoretical approach

We present a phase-field model for the dynamics of the interface between two inmiscible fluids with arbitrary viscosity contrast in a rectangular Hele-Shaw cell. With asymptotic matching techniques we check the model to yield the right Hele-Shaw equations in the sharp-interface limit and compute the corrections to these equations to first order in the interface thickness. We also compute the effect of such corrections on the linear dispersion relation of the planar interface. We discuss in detail the conditions on the interface thickness to control the accuracy and convergence of the phase-field model to the limiting Hele-Shaw dynamics. In particular, the convergence appears to be slower for high viscosity contrasts.Comment: 17 pages in revtex. changes: 1 reference adde

Comparing resonant photon tunneling via cavity modes and Tamm plasmon polariton modes in metal-coated Bragg mirrors

Resonant photon tunneling was investigated experimentally in multilayer structures containing a high-contrast (TiO2/SiO2) Bragg mirror capped with a semitransparent gold film. Transmission via a fundamental cavity resonance was compared with transmission via the Tamm plasmon polariton resonance that appears at the interface between a metal film and a one-dimensional photonic bandgap structure. The Tamm-plasmon-mediated transmission exhibits a smaller dependence on the angle and polarization of the incident light for similar values of peak transmission, resonance wavelength, and finesse. Implications for transparent electrical contacts based on resonant tunneling structures are discussed

Gravity and elevation changes at Askja, Iceland

Ground tilt measurements demonstrate that Askja is in a state of unrest, and that in the period 1988 - 1991 a maximum 48 +/- 3 µrad tilt occurred down towards the centre of the caldera. This is consistent with 126 mm of deflation at the centre of the caldera with a 2.5 - 3.0 km depth to the source of deformation. The volume of the subsidence bowl is 6.2 x 106 m3. When combined with high precision microgravity measurements, the overall change in sub-surface mass may be quantified. After correction for the observed elevation change using the free air gradient of gravity measured for each station, the total change in mass is estimated to be less than 109 kg. A small residual ground inflation and net gravity increase in the eastern part of the caldera may be caused by dyke intrusion in this region. The minimum dimensions of such an intrusion or complex of intrusions are 1m width, up to 100m deep and up to several hundred metres thick

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