A study on the numerical stability of the two-fluid model near ill-posedness

The two-fluid model is widely used in studying gas–liquid flow inside pipelines because it can qualitatively predict the flow field at low computational cost. However, the two-fluid model becomes ill-posed when the slip velocity exceeds a critical value, and computations can be quite unstable before the flow reaches the ill-posed condition. In this work, computational stability of various convection schemes together with the Euler implicit method for the time derivatives in conjunction with the two-fluid model is analyzed. A pressure correction algorithm for the two-fluid model is carefully implemented to minimize its effect on numerical stability. von Neumann stability analysis shows that the central difference scheme is more accurate and more stable than the 1st-order upwind, 2nd-order upwind, and QUICK schemes. The 2nd-order upwind scheme is much more susceptible to instability than the 1st-order upwind scheme and is inaccurate for short waves. Excellent agreement is obtained between the predicted and computed growth rates of harmonic disturbances. The instability associated with the two-fluid model discretized system of equations is related to but quantitatively different from the instability associated with ill-posedness of the two-fluid model. When the computation becomes unstable due to the ill-posedness, the machine roundoff errors from a selected range of short wavelengths, which scale with the grid size, are amplified rapidly to render the computation of any targeted long wavelength variation useless. For the viscous two-fluid model with wall friction and interfacial drag, a small-amplitude long wavelength disturbance grows due to viscous Kelvin–Helmholtz instability without triggering the grid scale short waves when the system remains well posed. Under such a condition, central difference is found to be the most accurate discretization scheme among those investigated.     

有限深水中骑行波的显式Hamilton描述

小尺度波（扰动波）迭加在大尺度波（未受扰动波）上形成的波动一般之为“骑行波”。研究了有限可变深度的理想不可压缩流体中的骑行波的显式Hamliltn表示，考虑了自由面上流体与空气之间的表面张力。采用自由面高度和自由面上速度势构成的Hamilton正则变量表示骑行波的动能密度，并在未受扰动波的自由面上作一阶展开。运用复变函数论方法处理了二维流动。先用保角变换将物理平面上的流动区域变换到复势平面上的无限长带形区域，然后在复势平面上用Fourier变换解出Laplace方程，最后经Fourier逆变换求出了扰动波速度热所满足的积分方程。作为特例考虑了平坦底部的流动，导出了动能密度的显式表达式。这里给出的积分方程可以替代相当难解的Hamilton正则方程。通过求解积分方程可得出agrange密度的显式表达式。本文提出的方法约研究骑行波的Hamilton描述以及波的相互作用问题提供了新的途径，有助于了解海面的小尺度波的精细结构。     

Inviscid instability of two-fluid free surface flow down an incline

The inviscid temporal stability analysis of two-fluid parallel shear flow with a free surface, down an incline, is studied. The velocity profiles are chosen as piecewise-linear with two limbs. The analysis reveals the existence of unstable inviscid modes, arising due to wave interaction between the free surface and the shear-jump interface. Surface tension decreases the maximum growth rate of the dominant disturbance. Interestingly, in some limits, surface tension destabilises extremely short waves in this flow. This can happen because of the interaction with the shear-jump interface. This flow may be compared with a corresponding viscous two-fluid flow. Though viscosity modifies the stability properties of the flow system both qualitatively and quantitatively, there is qualitative agreement between the viscous and inviscid stability analysis when the less viscous fluid is closer to the free surface.     

Global volume conservation in unsteady free surface flows with energy absorbing far‐end boundaries

A wave absorption filter for the far‐end boundary of semi‐infinite large reservoirs is developed for numerical simulation of unsteady free surface flows. Mathematical model is based on finite volume solution of the Navier–Stokes equations and depth‐integrated continuity equation to track the free surface. The Sommerfeld boundary condition is applied at the far‐end of the truncated computational domain. A dissipation zone is formed by applying artificial pressure on water surface to dissipate the kinetic energy of the outgoing waves. The computational scheme is tested to verify the conservation of total fluid volume in the domain for long simulation durations. Combination of the Sommerfeld boundary and dissipation zone can effectively minimize reflections and prevent cumulative changes in total fluid volume in the domain. Solitary wave, nonlinear periodic waves and irregular waves are simulated to illustrate the numerical developments. Earthquake excited surface waves and nonlinear hydrodynamic pressures in a dam–reservoir are computed. Copyright © 2009 John Wiley & Sons, Ltd.     

粗糙颗粒动理学及稠密气固两相流动的数值模拟

考虑颗粒碰撞过程中摩擦作用,给出了粗糙颗粒碰撞动力学.引入颗粒相拟总温来表征颗粒平动和转动脉动能量的特征.基于气体分子运动论,建立颗粒碰撞中平动和旋转共同作用的粗糙颗粒动理学,给出了颗粒相压力和黏度等输运参数计算模型.运用基于颗粒动理学的欧拉-欧拉气固两相流模型,数值模拟了流化床内气体颗粒两相流动特性,分析了颗粒旋转流动对颗粒碰撞能量交换和耗散的影响.模拟得到的流化床内径向颗粒浓度和提升管内颗粒轴向速度与他人实验结果相吻合.模拟结果表明随着颗粒浓度的增加,颗粒相压力和能量耗散逐渐增加,而颗粒拟总温先增加后下降.随着颗粒粗糙度系数的增加,床内平均颗粒相拟总温和能量耗散增加,表明颗粒旋转产生的摩擦将导致颗粒旋转脉动能量的改变,影响床内气体-颗粒两相宏观流动特性.      

Stability analysis of annular flow structure, using a discrete form of the “two-fluid model”

The differential form of the “two-fluid model” for annular flow, neglecting surface tension, is ill-posed, and it is not suited for examining the stability of the steady-state solutions with respect to the average film thickness. It is shown here that a discrete (difference) representation of the two-fluid model may lead to an appropriate criterion for the stability of the steady-state solutions. Exactly the same criterion is obtained from the requirement that the kinematic waves will propagate in the downstream direction. The suggested discrete form of the “two-fluid model” is used to perform transient simulation and for examining the system response to finite disturbances.     

Numerical modeling of slug flow initiation in a horizontal channels using a two-fluid model

This paper presents a methodology for modeling slug initiation and growth in horizontal ducts. Transient two-fluid equations are solved numerically using a class of high-resolution shock capturing methods. The advantage of this method is that slug formation and growth in a stratified regime can be calculated directly from the solutions to the flow field differential equations. In addition, by using high-resolution shock capturing methods that do not contain numerical diffusion, the discontinuity generated by slugging in the flow field can be modeled with good accuracy. The two-fluid model is shown to be well-posed mathematically only under certain conditions. Under these circumstances, the two-fluid model is capable of correctly predicting and modeling the flow physics. When ill-posed, an unbounded instability occurs in the flow field solution, and the instability amplitude increases exponentially with decreasing mesh sizes. This work shows that there are three zones associated with slug formation. In addition, long wavelength slugs are shown to initiate from short wavelength waves. These short waves are generated at the interface of the two phases by the Kelvin-Helmholtz hydrodynamic instability. The results obtained through numerical modeling show good agreement with experimental results.     

Shock wave interaction in a dusty gas and the appearance of fully dispersed waves

A problem of regular (symmetric and asymmetric) interaction of plane shock waves in a steady-state dusty-gas flow is considered. The possibility of the formation of wave structures is revealed, in which either all or some of the incident or reflected waves degenerate into fully dispersed waves, i.e. zones in which the parameters of both phases vary continuously. Using the Rankine-Hugoniot relations for a one-velocity “effective-gas” model, the ranges of nondimensional governing parameters (the Mach number, the angles between the incident waves and the free stream, the phase specific-heat ratio, and the particle mass concentration) are found, which correspond to different wave configurations. In the framework of a two-fluid dusty-gas model, the flow structure in the region of symmetric interaction of the shocks is calculated numerically for typical configurations containing fully dispersed waves. The flow in the region of a normal fully dispersed wave is also calculated. Good agreement between the calculated wave structure and the data known in the literature is obtained. A range of governing parameters in which the carrier-phase temperature has a local maximum inside the wave structure is found.     

Head-on collision between two mKdV solitary waves in a two-layer fluid system

In this paper, based on the equations presented in [2], the head-on collision between two solitary waves described by the modified KdV equation (the mKdV equation, for short) is investigated by using the reductive perturbation method combined with the PLK method. These waves propagate at the interface of a two-fluid system, in which the density ratio of the two fluids equals the square of the depth ratio of the fluids. The second order perturbation solution is obtained. It is found that in the case of disregarding the nonuniform phase shift, the solitary waves preserve their original profiles after collision, which agrees with Fornberg and Whitham's numerical result of overtaking collision161 whereas after considering the nonuniform phase shift, the wave profiles may deform after collision.     

Kinetic models for dilute solutions of dumbbells in non-homogeneous flows revisited

We propose a two-fluid theory to model a dilute polymer solution assuming that it consists of two phases, polymer and solvent, with two distinct macroscopic velocities. The solvent phase velocity is governed by the macroscopic Navier–Stokes equations with the addition of a force term describing the interaction between the two phases. The polymer phase is described on the mesoscopic level using a dumbbell model and its macroscopic velocity is obtained through averaging. We start by writing down the full phase-space distribution function for the dumbbells and then obtain the inertialess limits for the Fokker–Planck equation and for the averaged friction force acting between the phases from a rigorous asymptotic analysis. The resulting equations are relevant to the modelling of strongly non-homogeneous flows, while the standard kinetic model is recovered in the locally homogeneous case.     

Simulation of large amplitude waves in a slug tracking scheme compared to roll wave experiments at high pressure

Due to the similarities between large amplitude roll waves and slug flow in two-phase gas–liquid pipe flow, a slug tracking scheme is presented with the addition of a simplified model for roll waves. The waves are treated in a similar way to slugs, modelled as objects moving at the wave velocity and with a pressure variation across them. The two-fluid model is solved on a stationary staggered grid in stratified sections between moving waves and slugs. The model is dynamic meaning that the growth and decay of waves and slugs can be simulated. The wave model implementation within the tracking scheme is discussed and demonstrated in comparison to existing experimental data on wave velocities and averaged pressure drops. The results from the tracking scheme compared well to the experiments when waves were initiated with the experimental frequency. Wave initiation remains as a modelling challenge.     

A CORRELATION EQUATION FOR CALCULATING INCLINED JET PENETRATION LENGTH IN A GAS-SOLID FLUIDIZED BED

Numerical simulation of gas-solid flow in a two-dimensional fluidized bed with an inclined jet was performed. The numerical model is based on the two-fluid model of gas and solids phase in which the solids constitutive equations are based on the kinetic theory of granular flow. The improved ICE algorithm, which can be used for both low and high-velocity fluid flow, were used to solve the model equations. The mechanism of jet formation was analyzed using both numerical simulations and experiments. The emergence and movement of gas bubbles were captured numerically and experimentally. The influences of jet velocity, nozzle diameter, nozzle inclination and jet position on jet penetration length were obtained. A semi-empirical expression was derived and the parameters were correlated from experimental data. The correlation equation, which can be easily used to obtain the inclined jet penetration length, was compared with our experimental data and published correlation equations.     

Numerical computations of two-dimensional solitary waves generated by moving disturbances

Two-dimensional solitary waves generated by disturbances moving near the critical speed in shallow water are computed by a time-stepping procedure combined with a desingularized boundary integral method for irrotational flow. The fully non-linear kinematic and dynamic free-surface boundary conditions and the exact rigid body surface condition are employed. Three types of moving disturbances are considered: a pressure on the free surface, a change in bottom topography and a submerged cylinder. The results for the free surface pressure are compared to the results computed using a lower-dimensional model, i.e. the forced Korteweg–de Vries (fKdV) equation. The fully non-linear model predicts the upstream runaway solitons for all three types of disturbances moving near the critical speed. The predictions agree with those by the fKdV equation for a weak pressure disturbance. For a strong disturbance, the fully non-linear model predicts larger solitons than the fKdV equation. The fully non-linear calculations show that a free surface pressure generates significantly larger waves than that for a bottom bump with an identical non-dimensional forcing function in the fKdV equation. These waves can be very steep and break either upstream or downstream of the disturbance.     

Nonlinear fourth sound

The two-fluid model for He II is used to derive the Mach number dependence on shock strength for weak fourth sound shock waves. The shock structure profile is found to be a hyperbolic tangent with shock thickness proportional to shock strength. The steepening coefficient is evaluated, and methods for producing nonlinear fourth sound are discussed.     

Characteristic Properties and Exact Solutions of the Kinetic Equation of Bubbly Liquid

The paper considers a kinetic model for the motion of incompressible bubbles in an ideal liquid that takes into account their collective interaction in the case of one spatial variable. Generalized characteristics and a characteristic form of the equations are found. Necessary and sufficient hyperbolicity conditions of the integrodifferential model of rarefied bubbly flow are formulated. Exact solutions of the kinetic equation for the class of traveling waves are derived. A solution of the linearized equation is obtained.     

变深度浅水域中非定常船波

以Green—Naghdi(G—N)方程为基础，采用波动方程／有限元法计算船舶经过变深度浅水域时非定常波浪特性．把运动船舶对水面的扰动作为移动压强直接加在Green-Naghdi方程里，以描述运动船体和水面的相互作用．以Series60 CB＝0．6船为算例，给出自由面坡高，波浪阻力在船舶经过一个水下凸包时变化规律，并与浅水方程的结果进行了比较．计算结果表明，当船舶经过凸包时，波浪阻力先增加，后减少，并逐渐趋于正常．同时发现，当船速小于临界速度时(Fr＝√gh＜1．0)，G—N方程给出的船后尾波比浅水方程的结果明显，波浪阻力也比浅水方程的结果有所提高，频率散射必须考虑．当船速大于临界速度时(Fr＝√gh＞1．0)，G—N方程的计算结果与浅水方程差别不大，频率散射的影响可以忽略．     

Second-order cnoidal waves at the free surface and interface of a two-fluid system

In this paper, using the PLK method and reductive perturbation method, we obtained the second approximation to cnoidal waves at the free surface and interface for the two-fluid system considered in [1]. The corresponding results in [3] and [4] may be obtained as special cases in this paper.Projects Supported by the Science Fund of the Chinese Academy of Sciences.     

Two-equation and multi-fluid turbulence models for Rayleigh–Taylor mixing

This paper presents a new, improved version of the K–L model, as well as a detailed investigation of K–L and multi-fluid models with reference to high-resolution implicit large eddy simulations of compressible Rayleigh–Taylor mixing. The accuracy of the models is examined for different interface pressures and specific heat ratios for Rayleigh–Taylor flows at initial density ratios 3:1 and 20:1. It is shown that the original version of the K–L model requires modifications in order to provide comparable results to the multi-fluid model. The modifications concern the addition of an enthalpy diffusion term to the energy equation; the formulation of the turbulent kinetic energy (source) term in the K equation; and the calculation of the local Atwood number. The proposed modifications significantly improve the results of the K–L model, which are found in good agreement with the multi-fluid model and implicit large eddy simulations with respect to the self-similar mixing width; peak turbulent kinetic energy growth rate, as well as volume fraction and turbulent kinetic energy profiles. However, a key advantage of the two-fluid model is that it can represent the degree of molecular mixing in a direct way, by transferring mass between the two phases. The limitations of the single-fluid K–L model as well as the merits of more advanced Reynolds-averaged Navier–Stokes models are also discussed throughout the paper.     

Propagation of nonlinear perturbations in a quasineutral collisionless plasma

For the nonlinear kinetic equation describing the one-dimensional motion of a quasineutral collisionless plasma, perturbation velocities are determined and conditions of generalized hyperbolicity are formulated. Exact (in particular, periodical) solutions of the model are constructed and interpreted physically for the class of traveling waves. Differential conservation laws approximating the basic integrodifferential equation are proposed. These laws are used to perform numerical calculations of wave propagation, which show the possibility of turnover of the kinetic distribution function.     

Investigation on effect of drag models on flow behavior of power-law fluid–solid two-phase flow in fluidized bed

In this study, a Eulerian-Eulerian two-fluid model combined with the kinetic theory of granular flow is adopted to simulate power-law fluid–solid two-phase flow in the fluidized bed. Two new power-law liquid–solid drag models are proposed based on the rheological equation of power-law fluid and pressure drop. One called model A is a modified drag model considering tortuosity of flow channel and ratio of the throat to pore, and the other called model B is a blending drag model combining drag coefficients of high and low particle concentrations. Predictions are compared with experimental data measured by Lali et al., where the computed porosities from model B are closer to the measured data than other models. Furthermore, the predicted pressure drop rises as liquid velocity increases, while it decreases with the increase of particle size. Simulation results indicate that the increases of consistency coefficient and flow behavior index lead to the decrease of drag coefficient, and particle concentration, granular temperature, granular pressure, and granular viscosity go down accordingly.