A conservation-based method for simulating the inspiral of binary black holes

We present a new approach to studying the evolution of massive black hole binaries in a stellar environment. By imposing conservation of total energy and angular momentum in scattering experiments, we find the dissipation forces that are exerted on the black holes by the stars, and thus obtain the decaying path of the binary from the classical dynamical friction regime down to subparsec scales. Our scheme lies between scattering experiments and N-body simulations. While still resolving collisions between stars and black holes, it is fast enough and allows to use a large enough number of particles to reach a smooth and convergent result. We studied both an equal mass and a 10:1 mass ratio binaries under various initial conditions. We show that while an equal mass binary stalls at a nearly circular orbit, a runaway growth of eccentricity occurs in the unequal mass case. This effect reduces the timescale for black hole coalescence through gravitational radiation to well below the Hubble time, even in spherical and gasless systems formed by dry mergers.Comment: 11 pages, 9 figure

Numerical methods for the simulation of an aggregation-driven droplet size distribution

A droplet size distribution in a turbulent flow field is considered and modeled by means of a population balance system. This paper studies different numerical methods for the 4D population balance equation and their impact on an output of interest, the time-space-averaged droplet size distribution at the outlet which is known from experiments. These methods include different interpolations of the experimental data at the inlet, various discretizations in time and space, and different schemes for computing the aggregation integrals. It will be shown that notable changes in the output of interest might occur. In addition, the efficiency of the studied methods is discussed

Numerical methods for the simulation of an aggregation-driven droplet size distribution

A droplet size distribution in a turbulent flow field is considered and modeled by means of a population balance system. This paper studies different numerical methods for the 4D population balance equation and their impact on an output of interest, the time-space-averaged droplet size distribution at the outlet which is known from experiments. These methods include different interpolations of the experimental data at the inlet, various discretizations in time and space, and different schemes for computing the aggregation integrals. It will be shown that notable changes in the output of interest might occur. In addition, the efficiency of the studied methods is discussed

Numerical simulation of spray coalescence in an eulerian framework : direct quadrature method of moments and multi-fluid method

The scope of the present study is Eulerian modeling and simulation of polydisperse liquid sprays undergoing droplet coalescence and evaporation. The fundamental mathematical description is the Williams spray equation governing the joint number density function f(v, u; x, t) of droplet volume and velocity. Eulerian multi-fluid models have already been rigorously derived from this equation in Laurent et al. (2004). The first key feature of the paper is the application of direct quadrature method of moments (DQMOM) introduced by Marchisio and Fox (2005) to the Williams spray equation. Both the multi-fluid method and DQMOM yield systems of Eulerian conservation equations with complicated interaction terms representing coalescence. In order to validate and compare these approaches, the chosen configuration is a self-similar 2D axisymmetrical decelerating nozzle with sprays having various size distributions, ranging from smooth ones up to Dirac delta functions. The second key feature of the paper is a thorough comparison of the two approaches for various test-cases to a reference solution obtained through a classical stochastic Lagrangian solver. Both Eulerian models prove to describe adequately spray coalescence and yield a very interesting alternative to the Lagrangian solver

ON THE IMPLEMENTATION OF MOMENT TRANSPORT EQUATIONS IN OPENFOAM TO PRESERVE CONSERVATION, BOUNDEDNESS AND REALIZABILITY

Different industrial scale multiphase systems can be successfully described by considering their polydispersity (e.g. particle/droplet/bubble size and velocity distributions) and phase coupling issues are properly overcome only by considering the evolution in space and time of such distributions, dictated by the so-called Generalized Population Balance Equation (GPBE). A computationally efficient approach for solving the GPBE is represented by the quadrature-based moment methods, where the evolution of the entire particle/droplet/bubble population is recovered by tracking some specific moments of the distribution and the quadrature approximation is used to solve the "closure problem" typical of moment-based methods. In this contribution some crucial computational and numerical details concerning the implementation of these methods into the opensource Computational Fluid Dynamics (CFD) code OpenFOAM are discussed. These aspects are in fact very often overlooked, resulting in implementations that do not satisfy the properties of conservation, realizability and boundedness. These constraints have to be satisfied in a consistent way, with respect to what done with the other conserved transported variables (e.g. volume fraction of the disperse phase) also when higher-order discretization schemes are used. These issues are illustrated on examples taken on our work on the simulation of fluid-fluid multiphase system

Quadrature-based models for multiphase and turbulent reacting flows

The simulation of physical systems requires accurate and robust methods with relatively low cost and it is still the challenge in many applications of engineering processes, specifically in multiphase flow systems. Soot formation, distribution of the aerosols in the atmosphere, reactive precipitation, and combustion modeling are some examples of these processes. Computer simulations of theses systems require a model that can be adapted to that reality. In this study, a quadrature based method of moments (QBMM) is used to address the problems related to the reactive multiphase flow systems. First, the log-normal kernel density function is implemented into the extended quadrature method of moments (Ln-EQMOM). Ln-EQMOM is verified reconstructing the NDF and calculating the moments of a distribution obtained by the linear combination of two log-normal distributions. Later, this numerical procedure is used for problems of aggregation and breakup of fine particles to solve the population balance equation (PBE). The results are compared to the rigorous solutions reported for the cases under consideration \citep{vanni2000}. Finally, the method is verified using two analytically known problems (\textit{e.g.} coalescence and condensation). In comparison to EQMOM with $\Gamma$ kernel density function \citep{yuan2012}, Ln-EQMOM is faster in terms of computations and it preserves the moments more accurately. Then EQMOM with $\beta$ kernel density function is implemented to approximate the solution of the transport equation for the composition probability density function (PDF) of a passive scalar using the Fokker-Planck model to treat the molecular mixing term. The results then compared in a similar condition to those obtained with direct numerical simulation (DNS). The $L_2$ norm of the PDF is reported for two test cases that have been considered. Later the new approach is introduced to address the problems includes the mixing and reaction. Conditional quadrature method of moments (CQMOM) and using the joint composition PDF for the mixture fraction and progress variables, it is possible to address the problems with two consecutive competitive reactions, one reaction and fast reaction, all including the mixing of reactants. direct quadrature method of moments (DQMOM) also expressed for the joint composition PDF. Results obtained with CQMOM and DQMOM are compared with each other. Finally, the CQMOM approach for mixing problems was tested considering two consecutive competitive reactions to verify the implementation and validate the proposed approach. Coupled mixing-PBE approach was then used to investigate polymer aggregation in a multi-inlet vortex reactor (MIVR), typically used to perform flash nanoprecipitation for the production of nanoparticles used in pharmaceutical applications

A numerical method for the simulation of an aggregation-driven population balance system

A population balance system which models the synthesis of urea is studied in this paper. The equations for the flow field, the mass and the energy balances are given in a three-dimensional domain and the equation for the particle size distribution (PSD) in a four-dimensional domain. This problem is convection-dominated and aggregation-driven. Both features require the application of appropriate numerical methods. This paper presents a numerical approach for simulating the population balance system which is based on finite element schemes, a finite difference method and a modern method to evaluate convolution integrals that appear in the aggregation term. Two experiments are considered and the numerical results are compared with experimental data. Unknown parameters in the aggregation kernel have to be calibrated. For appropriately chosen parameters, good agreements are achieved of the experimental data and the numerical results computed with the proposed method. A detailed study of the computational results reveals the influence of different parts of the aggregation kernel

A contribution to the understanding of crystallization

Thermodynamic and kinetic studies of the vapor/liquid phase transition of a Lennard-Jones model fluid are presented which challenge the conventional interpretation of the mechanism of the condensation process. We have lifted the usual approximations (incompressible phases, constant surface tension, and non-depleting vapor phase) applied in classical nucleation theory to investigate the nature of the barrier to condensation. Based on the 1[superscript]st Yvon-Born-Green integro-differential equation we have developed a thermodynamically consistent molecular theory which accurately predicts the radially dependent surface tension and the location of the surface of tension of microscopically small droplets. The droplet size dependence of the inter-facial free energy is sufficiently strong that the free energy barrier to the nucleation is absent;Computer simulations of ionic and neutral fluids have been performed to study the dynamical behavior of the fluid during the phase separation. We find that phase transitions in the metastable region for both systems are characterized by the instantaneous formation of concentration fluctuations. In the vapor phase of the Lennard-Jones fluid nearly spherical, disjoint, high density regions are formed, whereas in the ionic vapor a network of charged chains is observed. The connectivity between the clusters and their linearity diminishes with charge asymmetry. The short induction time where small clusters are spontaneously formed is followed by two rate determining regimes. First, the clusters absorb surrounding atoms and smaller clusters. During this regime, the evolution of the number of atoms in the cluster is linear in time and can be described by a modified Lifshitz-Slyozov theory. Second, the clusters undergo Brownian motion and further growth is mainly driven by coalescence. The Brownian character of this motion is due to unsymmetric internal motion near the surface. This is in contrast to the usual interpretation of the origin of Brownian motion as environmental noise;These results support our earlier conclusions that the free energy of formation of a spherical droplet is irrelevant to a description of vapor condensation, but require an alternative mechanism for the persistence of metastability. We have constructed a new model for the vapor/liquid phase transition, which regards the phase separation as a cascade of Brownian walkers whose mass grows linearly in time. The nucleation rate is then simply determined by the sum over the first passage times required for the binary coalescence event. These intervals scale like time to the 11/6 and supersaturation to the -2/3 providing an explanation of the sensitivity of nucleation rate to supersaturation