数値計算法

In recent years, the two-fluid model has often been used in thermalhydraulic analyses. This model is considered to be theoretically more strict and to be more relevant in treating such two-phase flow phenomena as these that are affected by thermal or hydrodynamic non-equilibrium, because the model describes the conservation of mass, momentum and energy for both gas and liquid phases, separately. In order for the model to display its real worth, however, it is important that the constitutive equations be used properly. This review paper, therefore, presents the constitutive equations for flow-regime transitions, interfacial area concentration, behavior of liquid droplets, interfacial friction, interfacial heat transfer, wall friction and wall heat transfer that are actually used in computer codes for thermal-hydraulic analyses.

Recent development on the subgrid scale modeling for the large eddy simulation (LES) of turbulent flows is reviewed. Special attention is focused on the dynamic models which use properties of the smallest resolved eddies to estimate the subgrid scale stresses. Taking into account the multiple scale structure of multiphase flows, the extension of subgrid scale model for turbulent multiphase flows is surveyed.

To realize LES for a multiphase turbulent flow with a quite large number of particles, the authors have recently developed a new multiphase LES, named “GAL-LES model”. The GAL (Grid- Averaged Lagrangian) model is based on a grid-averaging operation for a Lagrangian-type equation of a particle motion and reallocation procedure of a particle cloud with its centroid movement to each grid. The GAL-LES model has been applied to a particle plume and a sheet-flow-type sediment transport, showing that the present model is found to simulate the essential features of the large-eddy structure and gives good agreements with the existing experimental data.

The development process of the computational mechanics of sediment transport is reviewed briefly. First, the importance of the description of the irregular motion of the sediment particle, which consists the essential part of the non-equilibrium sediment transport process, is explained. Secondary, the multiphase flow model for the modeling of the flow/sediment interaction is reviewed. As the increase of the sediment concentration, not only the flow/sediment interaction but also the particle/particle interaction, or interparticle collision, play the significant role. The application of the granular material model of the interparticle collision to the sediment transport process is also reviewed.Finally the unresolved problems and the prospects of the computational mechanics of sediment transport are discussed briefly.

The methods to simulate gas-liquid two-phase flow have developed due to the progress of the computer technology and computational fluid dynamics. Then, the simulation methods lead the useful information in the industrial field and the detailed results that could not be obtained by the experimental methods. In this review, these methods are classified into the Eulerian method and the Lagrangian method. This report focuses on the interface capturing technique of each method. Detailed capturing procedure, the typical results and advantage/disadvantage of each method are disscussed.

This paper describes a review of the numerical simulations of multiphase fluid flow by the lattice Boltzmann method (LBM), in which a macroscopic fluid consists of mesoscopic particles repeating collisions and translations. One of the advantages is that an interface can be reproduced in a self-organizing way by repulsive interaction between particles. Two- and three-dimensional LBM simulation results are presented, spontaneous phase separation, deformation of bubble due to shear stress, and single- and two-bubble motions in a tube under gravity.

We survey some numerical simulation methods for dispersed flows and point out that the vorticity generation mechanism due to the horizontal gradient of the void fraction is regarded in the conventional bubbly flow simulation based on the averaged equation, while one due to the boundary condition on the bubble-liquid interface is disregarded. In order to make clear the local structure of the pseudo turbulence in the bubbly flow, the direct numerical simulation of the Multi-bubble system is carried out. The simulation based on the averaged equation is also carried out. Constitutive equations, where not only SGS stresses but also boundary conditions of the pressure and the vorticity on the interface are taken into account, are derived for the averaged equations.

A heterogeneous nucleation of liquid droplets on a solid surface from vapor and a heterogeneous nucleation of vapor bubbles on a solid surface from liquid were simulated by the molecular dynamics method. Argon liquid or vapor was represented by Lennard-Jones molecules and a solid surface was represented by harmonic molecules with the constant temperature heat bath model using the phantom molecules outside of harmonic molecules. Nucleation of liquid droplet was realized by suddenly cooling a solid surface in contact with dense argon vapor. From the number distribution of clusters appeared on solid surface, heterogeneous nucleation rate and free energy of clusters dependent on the cluster size were measured. On the other hand, liquid argon between parallel solid surfaces was expanded in order to simulate the vapor bubble nucleation on the solid surface. With visualizations of the void patterns, molecular-level nucleation dynamics were explored for slowly and rapidly expanding systems. For both systems, the heterogeneous nucleation rate and the critical radius were not far from the prediction of the classical theory.

Mathematical models are necessary more or less in the simulation. In general, particle-fluid interaction and particle-particleinteraction are modeled in discrete particle simulation. The more universal the models, the more bearable for various purposes. Some examples of simulations which are regarded as earliest works in this field are shown.

In this paper the turbulent models for dilute gas-particle flows are briefly reviewed and the problems encountered in the turbulent simulations are discussed. Then we describe the two way coupling Large Eddy Simulations in which the effect of particle existence on subgrid-scale flows have been taken into account. We survey these models based on the comparison of the calculated results of three-dimensional Navier-Stokes equations and the Lagrangian particle motion equations and the experimental data.

Vortex methods have been thus far applied to simulate gas-particle free turbulent flows, such as mixing layer, wake and jet, to investigate the particle diffusion due to the vortex structure of the gas-phase. The simulations are performed only under low mass loading ratio, because the vortex methods are based on a one-way coupling between the two phases. This article describes a two-way vortex method proposed by the author. The gas flow and the particle motion are simultaneously calculated by the Lagrangian method. The applicability of the method is also demonstrated by presenting the numerical simulations of a particle-laden mixing layer and a slit nozzle gas-particle two-phase jet.

We review the recent progress and successful applications of lattice Boltzmann method (LBM) to computational fluid dynamics. To clarify the important issue in the LBM simulation, this report shows the recent progress in the LBM, and summarizes both the advantages and disadvantages of the LBM. We also discuss the immersed boundary-lattice Boltzmann method (IB-LBM) that has received much attention in recent years. Due to the common feature of using the Cartesian mesh, the IB-LBM successfully calculates the rigid particle motions in a viscous fluid. We present one of key issues in the IB-LBM, and examine the applicability of the Immersed Boundary Method to the lattice kinetic scheme (LKS) for particulate flow.

Treating the moving contact lines (fluid-fluid interfaces on solid surface) in numerical simulations are overviewed. At first, mathematical failure in analytical solution for flow between moving wall and static fluid-fluid interface is introduced. Analytical models for the relation between the dynamic contact angle and the contact line moving velocity are also introduced. Defect of numerical simulations based on implicit interface representations are discussed. Then, a front-tracking method combined with generalized Navier boundary condition is presented. Furthermore, it is shown that an analytical asymptotic relation between macroscopic and microscopic behaviors is very useful for combining the microscopic model into macroscopic simulation method.

Dendrite morphology is drastically changed due to the liquid flow, such as forced and natural convections, during the solidification of metallic alloys. We have developed a phase-field lattice Boltzmann method to simulate the dendrite growth during the alloy solidification in the presence of the liquid flow. Here, we have employed the phase-field and lattice Boltzmann methods to express the dendrite growth and liquid flow, respectively. In addition, a parallel computation using multiple graphical processing units in a supercomputer has been enabled to accelerate the phase-field lattice Boltzmann simulation. In this report, the phase-field lattice Boltzmann model is briefly explained, and some dendrite growth simulations using the model are introduced.

I introduced two examples of molecular dynamics (MD) simulation of nano-scale bubble with phase change corresponding to cavitation. One is a nucleation-growth process starting from homogeneous bubble nucleation, and a characteristic length reflecting the time change of the size distribution was extracted and compared with a macroscopic theory for Ostwald ripening. The other is a simulation of nano-scale bubble collapsing. In the latter simulation, we evaluated the time change of bubble radius and compared it with some bubble dynamics models derived by fluid mechanics assuming continuity of fluid.

Incompressible flows have much lower flow speeds to compare with the sound speed and called low Mach number flows. So far, we have simulated incompressible flows by using semi-implicit solvers, in which we iteratively compute the linear equation derived from the pressure Poisson equation. In the case of incompressible gas-liquid two-phase flows, the sparse matrix would be very stiff with non-zero elements changing largely from the gas density to the liquid density at the interface. Even if we use a sophisticated Krylov sub-space iteration solver with the multi-grid preconditioner, the convergence becomes poor for large-scale problems. We have developed a weakly compressible gas-liquid two-phase solver based on fully-explicit time integration of compressible Navier-Stokes equation. For several benchmark problems, the results of the weakly compressible solver are in good agreement with those of the semi-implicit incompressible solver. Since the weakly compressible solver uses a local stencil access, it is possible to implement AMR (Adaptive Mesh Refinement) method with less programming cost. Fine meshes are assigned near the gas-liquid interface and we efficiently compute a liquid film generated by a spoon and a growing soap bubble with the AMR two-phase flow solver.

Capsule refers to a particle consisting of a liquid droplet enclosed by a thin-membrane. Analyzing the behavior of capsules in flow is needed to predict the transport and breakup of the capsules, and to understand the rheological property of the capsule suspension. We have developed a graphics processing unit (GPU) accelerated boundary element method (BEM) for the fluid-structure interaction analysis of capsule suspensions at the Stokes flow regime, where BEM for fluid mechanics is coupled with the finite element method (FEM) for membrane mechanics. Here, we provide an overview of the GPU-accelerated BEM and its application to the rheological analysis of capsules suspensions.

The numerical analysis of flow-induced structures of complex fluids is one of effective approaches for elucidating the mechanism of their complicated flow behavior. In this article, three examples of the numerical simulation of the flow-induced structure using the Brownian dynamics (BD) method and the multi-particle collision dynamics (MPCD) method are presented: (a) BD simulation of shear flows of polymer/clay suspensions, (b) MPCD simulation of shear flows of linear polymer solutions, and (c) MPCD simulation of bio-convection of phototactic microalgae in water. In these simulations, complex fluids are modeled by dispersion systems of basic elements of the fluid inner structure such as particles and model polymers, and their behavior is analyzed for the investigation of the flow-induced structure.

A method for calculating fluid motion around an object placed in a fluid containing a gas-liquid interface using the Immersed Boundary Method is introduced. Instead of cumbersome algorithms such as the existing the Volume Of Fluid method, the level set method and so on, a simplified algorithm is proposed by directly solving a variable in terms of pressure divided by the fluid density. Some calculated results of the application of the method with k-ω turbulence model are shown.