Gretar Tryggvason

A front-tracking method for viscous, incompressible, multi-fluid flows

Abstract A method to simulate unsteady multi-fluid flows in which a sharp interface or a front separates incompressible fluids of different density and viscosity is described. The flow field is discretized by a conservative finite difference approximation on a stationary grid, and the interface is explicitly represented by a separate, unstructured grid that moves through the stationary grid. Since the interface deforms continuously, it is necessary to restructure its grid as the calculations proceed. In addition to keeping the density and viscosity stratification sharp, the tracked interface provides a natural way to include surface tension effects. Both two- and three-dimensional, full numerical simulations of bubble motion are presented.

Computational methods for multiphase flow

Preface 1. Introduction: a computational approach to multiphase flow A. Prosperetti and G. Tryggvason 2. Direct numerical simulations of finite Reynolds number flows G. Tryggvason and S. Balachandar 3. Immersed boundary methods for fluid interfaces G. Tryggvason, M. Sussman and M. Y. Hussaini 4. Structured grid methods for solid particles S. Balachandar 5. Finite element methods for particulate flows H. Hu 6. Lattice Boltzmann methods for multiphase flows S. Chen, X. He and L. S. Luo 7. Boundary integral methods for Stokes flows J. Blawzdziewic 8. Averaged equations for multiphase flows A. Prosperetti 9. Point particle methods for disperse flows K. Squires 10. Segregated methods for two-fluid models A. Prosperetti, S. Sundaresan, S. Pannala and D. Z. Zhang 11. Coupled methods for multi-fluid models A. Prosperetti References Index.

Computations of boiling flows

Abstract A numerical method to simulate liquid–vapor phase change is presented. The method is based on the so-called single field formulation where one set of equations for conservation of mass, momentum and energy are written for the entire flow field. Interfacial source terms for surface tension, interphase mass transfer and latent heat are added as delta functions that are non-zero only at the phase boundary. The equations are discretized by a finite difference method on a regular grid and the phase boundary is explicitly tracked by a moving front. A comparison of numerical results to the exact solution of a one-dimensional test problem shows excellent agreement. The method is applied to film boiling, where vapor bubbles are generated from a thin film next to a hot wall. Although the film boiling simulations presented here are two-dimensional, the resulting heat transfer rate and wall temperatures are found to be in good agreement with experimental observations.

Direct numerical simulations of bubbly flows. Part 1. Low Reynolds number arrays

Direct numerical simulations of the motion of two- and three-dimensional finite Reynolds number buoyant bubbles in a periodic domain are presented. The full Navier–Stokes equations are solved by a finite difference/front tracking method that allows a fully deformable interface between the bubbles and the ambient fluid and the inclusion of surface tension. The rise Reynolds numbers are around 20–30 for the lowest volume fraction, but decrease as the volume fraction is increased. The rise of a regular array of bubbles, where the relative positions of the bubbles are fixed, is compared with the evolution of a freely evolving array. Generally, the freely evolving array rises slower than the regular one, in contrast to what has been found earlier for low Reynolds number arrays. The structure of the bubble distribution is examined and it is found that while the three-dimensional bubbles show a tendency to line up horizontally, the two-dimensional bubbles are nearly randomly distributed. The effect of the number of bubbles in each period is examined for the two-dimensional system and it is found that although the rise Reynolds number is nearly independent of the number of bubbles, the velocity fluctuations in the liquid (the Reynolds stresses) increase with the size of the system. While some aspects of the fully three-dimensional flows, such as the reduction in the rise velocity, are predicted by results for two-dimensional bubbles, the structure of the bubble distribution is not. The magnitude of the Reynolds stresses is also greatly over-predicted by the two-dimensional results.

Numerical simulations of the Rayleigh-Taylor instability

Two Lagrangian-Eulerian vortex methods to simulate the motion of an interface between inviscid fluids of different densities are presented. The representation of the interface as a vortex sheet eliminates numerical diffusion, and by coupling the tracked interface with z stationary grid (using the well-known vortex-in-ceil method) the high cost associated with traditional vortex m&hods is reduced. These methods are applied to the Rayleigh-Taylor instability. For finite density ratios the appearance of rolled up vortices and a possibie singularity formation has limited simulations in the past. By providing proper regularization our methods ovcrcomc some of these difficulties. (T I’M Acndcmic Press. Inc. 1. INTROD~JCTION One of the classic examples of hydrodynamic instability is the mixing of two fluids that takes place if a heavy fluid initially lies above a lighter one in a gravitational field. The first to investigate this problem was Lord Rayleigh [43], who late in the last century considered the linear stability problem for inviscid fluids with various stratification profiles, including sharply stratified fluids. In the early fifties 6.1. Taylor [Sl] renewed interest in this problem by pointing out that it is formally identical to the problem of an interface accelerated toward the heavy fluid. This observation made controlled laboratory experiments possible, and at Taylor’s suggestion D. J. Lewis [29] performed several such experiments, confirming Taylor’s and Rayleigh’s theoretical predictions for the linear stage and shedding some light on the large amplitude evolution. Several investigators have since repeated Lewis’ experiments using devices ranging from rubber bands (Emmons, Chang, and Watson [IX]) to rocket motors (Read [742]) to produce rhe desired acceleration. The stability analysis of Rayleigh and Taylor has been extended to take into account other physical effects such as surface tension and viscosity (Bellman and Pennington [S], Chandrasekhar [ 13]), compressibility (Mitchner and Landshoff [36]), and weakly nonlinear amplitudes (e.g., Emmons cl al. [ 18]i. Although these analytical investigations have revealed several interesting properties, most have not addressed the large amplitude behavior of a single bubble, or the interactions of these bubbles at large amplitudes. For a single bubble of negligible density penetrating the heavy fluid the asymptotic shape at large amplitudes has keen

A front-tracking method for computation of interfacial flows with soluble surfactants

A finite-difference/front-tracking method is developed for computations of interfacial flows with soluble surfactants. The method is designed to solve the evolution equations of the interfacial and bulk surfactant concentrations together with the incompressible Navier-Stokes equations using a non-linear equation of state that relates interfacial surface tension to surfactant concentration at the interface. The method is validated for simple test cases and the computational results are found to be in a good agreement with the analytical solutions. The method is then applied to study the cleavage of drop by surfactant-a problem proposed as a model for cytokinesis [H.P. Greenspan, On the dynamics of cell cleavage, J. Theor. Biol. 65(1) (1977) 79; H.P. Greenspan, On fluid-mechanical simulations of cell division and movement, J. Theor. Biol., 70(1) (1978) 125]. Finally the method is used to model the effects of soluble surfactants on the motion of buoyancy-driven bubbles in a circular tube and the results are found to be in a good agreement with available experimental data.

Computations of multi-fluid flows

Abstract 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.

Head-on collision of drops—A numerical investigation

The head‐on collision of equal sized drops is studied by full numerical simulations. The Navier–Stokes equations are solved for the fluid motion both inside and outside the drops using a front tracking/finite difference technique. The drops are accelerated toward each other by a body force that is turned off before the drops collide. When the drops collide, the fluid between them is pushed outward leaving a thin layer bounded by the drop surface. This layer gets progressively thinner as the drops continue to deform, and in several of our calculations we artificially remove this double layer at prescribed times, thus modeling rupture. If no rupture takes place, the drops always rebound, but if the film is ruptured the drops may coalesce permanently or coalesce temporarily and then split again. Although the numerically predicted boundaries between permanent and temporary coalescence are found to be consistent with experimental observations, the exact location of these boundaries in parameter space is found ...

Dynamics of homogeneous bubbly flows Part 1. Rise velocity and microstructure of the bubbles

Direct numerical simulations of the motion of up to 216 three-dimensional buoyant bubbles in periodic domains are presented. The full Navier–Stokes equations are solved by a parallelized finite-difference/front-tracking method that allows a deformable interface between the bubbles and the suspending fluid and the inclusion of surface tension. The governing parameters are selected such that the average rise Reynolds number is about 12–30, depending on the void fraction; deformations of the bubbles are small. Although the motion of the individual bubbles is unsteady, the simulations are carried out for a sufficient time that the average behaviour of the system is well defined. Simulations with different numbers of bubbles are used to explore the dependence of the statistical quantities on the size of the system. Examination of the microstructure of the bubbles reveals that the bubbles are dispersed approximately homogeneously through the flow field and that pairs of bubbles tend to align horizontally. The dependence of the statistical properties of the flow on the void fraction is analysed. The dispersion of the bubbles and the fluctuation characteristics, or ‘pseudo-turbulence’, of the liquid phase are examined in Part 2.

Numerical experiments on Hele Shaw flow with a sharp interface

The fingering instability of an interface between two immiscible fluids in a Hele Shaw cell is simulated numerically. The algorithm used is based on a transcription of the equations of motion for the interface in which it formally becomes a generalized vortex sheet. The evolution of this sheet is computed using a variant of the vortex-in-cell method. The resulting scheme and code make it possible to follow the collective behaviour of many competing and interacting fingers well into the nonlinear, large-amplitude regime. It is shown that in this regime the evolution is controlled essentially by just one dimensionless parameter, the ratio of fluid viscosities. The effects of varying this parameter are studied and the results compared with experimental investigations. Scaling properties of the average density profile across the evolving mixed layer between the two homogeneous fluid phases are investigated. Many phenomena are observed that must be characterized as collective interactions and thus cannot be understood in terms of flows with just a single finger.