The progress of the two-phase flow research has been divided into four stages in this series of the report. These periods are (I) 1948-1959, (II) 1960-1970, (III) 1971-1979, (IV) 1980-1988. In Chap. 8 in this report, various methods of the theoretical analysis of water circulation in boilers in the time before the period (I), when analog and digital computers did not exist, have been introduced. And short review of development of theoretical analysis of flow instabilities in two-phase flow in the periods (II)-(IV) has been described based on the history of publication of books, reviews, and dissertations concerning two-phase flow, as the introdution of Chap. 9. The main parts of Chap. 9 will be presented in the next report.

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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.

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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.

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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.

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The high speed mixing granulator, the fluidized bed granulator/dryer, the tumbling fluidized bed to which a rotor is added, and the rotating drum type tablet coating equipment are widely used for solid dosage manufacture. Rationalization of operating conditions according to the physical properties of various materials has been a big subject. Therefore, for these equipments, the phenomenon inside these equipment are analyzed especially on particle motion by DEM analysis. Some example data examined about the features of these equipments performance are introduced, and the application to future development of equipments is explored.

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Bubbles never attach to the surface of a solid body of good wettability, while bubble attachment is usually observed when the wettability of the surface of a solid body is poor. This phenomenon was used to separate gas and liquid in gas-liquid two-phase flows in a vertical pipe. Air and water ascending in the vertical pipe was satisfactorily separated by using a T-junction. The dispersed gas phase, i.e., bubbles and slugs, passed preferably through the poorly wetted side of the T-junction. The separation efficiency increased with a decrease in the air flow rate.

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The motion of single bubbles in turbulent flow field in vertical ducts was calculated using a two-way bubble tracking method, in which Eulerian and Lagrangian descriptions are used for continuous liquid phase and individual bubbles, respectively. It is considered in the present flow condition that the oscillatory motion of bubbles is caused by the interaction between the bubble and shear-induced turbulent eddy produced in the surrounding liquid; furthermore, it has been experimentally observed even in stagnant liquid that the oscillation of bubble rise path is induced by the fluctuation of wake structure formed behind the bubble. For these reasons, in the calculations, the fluctuating components associated with the shear-induced turbulence and the wake structure behind the rising bubble were added to the mean velocity of liquid phase in a stochastic manner, which enabled to express the oscillatory motion of bubbles. Since these fluctuating components are associated with different mechanisms, their amplitudes and time scales were estimated separately. As a result of several numerical calculations, it was demonstrated that the statistical features of the motion of single bubbles observed in several experiments are to be reasonably predicted with the present method.

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A new type of solid-liquid separator, called Tapered Drum Rotating Separator, was proposed by the authors. This system has two operation modes; one is a separating mode and another a discharging one. As a fundamental study on the separating mode, the numerical simulation of the motion of a solid particle thrown into the rotating cylindrical container was carried out by the Runge-Kutta numerical integration method. In was clarified that, while at the beginning of the motion, the smaller the diameter of the particle becomes, the larger the radial deflection, because of its very quick tangential acceleration. After a certain duration, the larger the diameter of the particle becomes, the larger the radial deflection. Then there exists the minimum reaching time at the wall for each particle size and the drum diameter. The numerical simulations were also carried out for different density of particle and show same characteristics of motion as those of different sizes qualitatively. The numerical results were compared with the experimental one obtained by the flow visualization technique. The numerical value, for example, the reaching time at the wall agrees well with the experimental one, except for very small particle and low drum revolution.

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Characteristics of velocities of large bubble and of large solid particle of gas-liquid-solid three-phase slug flow with large particles are experimentally studied in a vertical pipe whose blockage ratio is larger than 0.8. Strong interactions between large bubble and large particle such as the destruction of a large bubble into small and/or medium bubbles by a large particle, the recovery of fragments of bubbles into their corresponding large bubble, the deceleration of large particle during its passing through a large bubble, and the acceleration of a large particle in the top part of liquid slug are observed. Three kinds of velocities are defined for large bubble velocity and for large particle velocity, respectively, in this turbulent and violent three-phase slug flow. The characteristics of these velocities are discussed against volumetric fluxes of gas phase, liquid phase, solid phase and total. Changes of each velocity from corresponding velocities observed in fundamental two-phase flows, namely gas-liquid two-phase flow and liquid-solid two-phase flow with large particles are explained.

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