两相临界流

由于两相临界流在压水反应堆安全分析中的重要性，在过去的４０年内，两相临界流一直是一个活跃的分支学科。本文对压水堆失水事故常用的两相临界流均匀平衡模型、滑移模型、考虑相间热力学非平衡效应的简化模型及两流体模型做了综述。结论是到目前为止，没有一个数学模型能准确预测各种参数范围、各种通道形状的临界流量。特定的模型只能适用于特定的情况。本文对非平衡两相流中汽泡成核、汽泡生长、相界面间质量、动量、能量交换的机理给予了更多的关注。     

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.     

Numerical analysis of turbulent flow in a rotating U-bend with roughened walls

A numerical analysis has been performed for a developing turbulent flow in a rotating U-bend of strong curvature with rib-roughened walls using an anisotropic turbulent model. In this calculation, an algebraic Reynolds stress model is used to precisely predict Reynolds stresses, and a boundary-fitted coordinate system is introduced as a method of coordinate transformation to set the exact boundary conditions along the complicated shape of U-bend with rib-roughened walls. Calculated results for mean velocity and Reynolds stresses are compared to the experimental data in order to validate the proposed numerical method and the algebraic Reynolds stress model. Although agreement is certainly not perfect in all details, the present method can predict characteristic velocity profiles and reproduce the separated flow generated near the outer wall, which is located just downstream of the curved duct. The Reynolds stresses predicted by the proposed turbulent model agree well with the experimental data, except in regions of flow separation.     

Parametric study of a model for determining the liquid flow-rates from the pressure drop and water hold-up in oil–water flows

A flow-pattern-dependent model, traditionally used for calculation of pressure drop and water hold-up, is accustomed for calculation of the liquid production rates in oil–water horizontal flow, based on the known pressure drop and water hold-up. The area-averaged steady-state one-dimensional two-fluid model is used for stratified flow, while the homogeneous model is employed for dispersed flow. The prediction errors appear to be larger when the production rates are calculated instead of pressure drop and water hold-up. The difference in the calculation accuracies between the direct and inverse calculation is most probably caused by the different uncertainties in the measured values of the input variables and a high sensitivity of the calculated phase flow-rates on even small change of the water hold-up for certain flow regimes. In order to locate the source of error in the standard two-fluid model formulation, several parametric studies are performed. In the first parametric study, we investigate under which conditions the momentum equations are satisfied when the measured pressure drop and water hold-up are imposed. The second and third parametric studies address the influence of the interfacial waves and drop entrainment on the model accuracy, respectively. These studies show that both interfacial waves and drop entrainment can be responsible for the augmentation of the wall-shear stress in oil–water flow. In addition, consideration of the interfacial waves offers an explanation for some important phenomena of the oil–water flow, such as the wall-shear stress reduction.     

Two-fluid non-linear model for flow in catheterized blood vessels

Blood flow through a catheterized artery is analyzed, assuming the flow is steady and blood is treated as a two-fluid model with the suspension of all the erythrocytes in the core region as a Casson fluid and the plasma in the peripheral region as a Newtonian fluid. The expressions for velocity, flow rate, wall shear stress and frictional resistance are obtained. The variations of these flow quantities with yield stress, catheter radius ratio and peripheral layer thickness are discussed. It is noticed that the velocity and flow rate decrease while the wall shear stress and resistance to flow increase when the yield stress or the catheter radius ratio increases while all the other parameters were held fixed. It is found that the velocity and flow rate increase while the wall shear stress and frictional resistance decrease with the increase of the peripheral layer thickness. The estimates of the increase in the frictional resistance are significantly very small for the present two-fluid model than those of the single-fluid Casson model.     

Closure relations effects on the prediction of the stratified two-phase flow stability via the two-fluid model

The possibility of predicting the exact long wave linear stability boundary via the two-fluid (TF) model for horizontal and inclined stratified two-phase flow is examined. The application of the TF model requires the introduction of empirical closure relations for the velocity profile shape factors and for the wave induced wall and interfacial shear stresses. The latter are recognized as the problematic closure laws. In order to explore the closure relations effects and to suggest the necessary modifications that can improve the stability predictions of the TF model, the results are compared with the exact long wave solution of the Orr–Sommerfeld equations for the two-plate geometry. It is demonstrated that with the shape factors corrections and the inclusion of wave induced stresses effects, the TF model is able to fully reproduce the exact long wave neutral stability curves. The wave induced shear stresses in phase with the wave slope, which give rise to the so called “sheltering force”, were found to have a remarkable destabilizing effect in many cases of horizontal and inclined flows. In such cases, the sheltering effects must be included in the TF model, otherwise the region of smooth stratified flow would be significantly over predicted. Based on the results of the exact analysis, a simple closure relation for the sheltering term in the TF model is provided.     

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.     

Three-phase upward flow in a vertical pipe

In this paper we develop an approach to design a three-phase, gas–solid–liquid flow system that transports pneumatically scarified solid particles, including sticky ones, through a vertical pipe. The proposed system permits the introduction and maintenance of a liquid film that coats the pipe’s inner wall and acts as a lubricant that ensures sticky particles continue to move upward without permanently adhering to the pipe wall. The system’s operating conditions fall within the boundaries of the annular dispersed region on a typical flow pattern map of vertical flow of a gas–liquid mixture. High gas superficial velocities combined with low liquid superficial velocities characterize such a region. A combination of a modified one-dimensional, two-fluid annular dispersed flow model and a one-dimensional pneumatic conveying model is shown to describe this transport process satisfactorily. Solution of the combined models produces all the necessary design parameters including power requirements and superficial velocities of the two-fluid media needed to transport a given amount of solid particles. Results of model calculations are compared with rare three-phase flow data obtained prior to the development of the present model, by an independent experimental team that used the physical conditions of the present approach. Reasonable agreement justifies the use of the combined model for engineering design purposes.     

Numerical modelling of cavities on axisymmetric bodies at zero and non-zero angle of attack

The flow about submerged, fully cavitating axisymmetric bodies at both zero and non-zero angle of attack is considered in this paper. A cavity closure model that relates the point of detachment, the angle that the separating streamline makes with the body and the cavity length is described. The direct boundary element method is used to solve the potential flow problem and to determine the cavity shape. A momentum integral boundary layer solver is included in the formulation so that shear stresses can be incorporated into the drag calculations. The numerical predictions based on the proposed closure model are compared with water tunnel measurements and photographs.     

On the feasibility of heat-exchange processes in uninsulated equipment

Strictly speaking, the common two-fluid heat exchanger is a special version of a three-fluid exchanger because the environment also participates in the energy exchange. In a number of situations this effect is significant and leads to either reduced or increased area requirements, depending on the purpose of the exchange operation. With certain combinations of the system parameters it is even impossible to achieve an envisaged temperature change. For some typical flow arrangements these combinations are established in an analytic form so that one can rapidly assess the feasibility of a desired process without actually performing the rather tedious design calculations.     

A two-fluid model for pulsatile flow in catheterized blood vessels

The pulsatile flow of blood through a catheterized artery is analyzed, assuming the blood as a two-fluid model with the suspension of all the erythrocytes in the core region as a Casson fluid and the peripheral region of plasma as a Newtonian fluid. The resulting non-linear implicit system of partial differential equations is solved using perturbation method. The expressions for shear stress, velocity, flow rate, wall shear stress and longitudinal impedance are obtained. The variations of these flow quantities with yield stress, catheter radius ratio, amplitude, pulsatile Reynolds number ratio and peripheral layer thickness are discussed. It is observed that the velocity distribution and flow rate decrease, while, the wall shear, width of the plug flow region and longitudinal impedance increase when the yield stress increases. It is also found that the velocity increases, but, the longitudinal impedance decreases when the thickness of the peripheral layer increases. The wall shear stress decreases non-linearly, while, the longitudinal impedance increases non-linearly when the catheter radius ratio increases. The estimates of the increase in the longitudinal impedance are considerably lower for the present two-fluid model than those of the single-fluid model.     

Numerical analysis of developing turbulent flow in a 180° bend tube by an algebraic Reynolds stress model

A numerical analysis has been performed for three‐dimensional developing turbulent flow in a 180° bend tube with straight inlet and outlet section used by an algebraic Reynolds stress model. To our knowledge, numerical investigations, which show the detailed comparison between calculated results and experimental data including distributions of Reynolds stresses, are few and far between. From this point of view, an algebraic Reynolds stress model in conjunction with boundary‐fitted co‐ordinate system is applied to a 180° bend tube in order to predict the anisotropic turbulent structure precisely. Calculated results are compared with the experimental data including distributions of Reynolds stresses. As a result of this analysis, it has been found that the calculated results show a comparatively good agreement with the experimental data of the time‐averaged velocity and the secondary vectors in both the bent tube and straight outlet sections. For example, the location of the maximum streamwise velocity, which appears near the top or bottom wall in the bent tube, is predicted correctly by the present method. As for the comparison of Reynolds stresses, the present method has been found to simulate many characteristic features of streamwise normal stress and shear stresses in the bent tube qualitatively and has a tendency to under‐predict its value quantitatively. Judging from the comparison between the calculated and the experimental results, the algebraic Reynolds stress model is applicable to the developing turbulent flow in a bent tube that is known as a flow with a strong convective effect. Copyright © 2005 John Wiley & Sons, Ltd.     

A comparative study of poiseuille flow of a polar fluid under various boundary conditions with applications to blood flow

In this paper, Poiseuille flow of a polar fluid (model of a red blood cell suspension) under various boundary conditions at the wall, viz., slip or no-slip in the axial velocity and couple stresses zero or non-zero at the boundary, is considered from the point of view of its applications to blood flow. Analytic expressions for axial and rotational velocities, flow rate, effective viscosity and stresses are obtained. The magnitudes of the length ratioL and the coupling number N are determined in accordance with concentration and tube radius (in the existing literature, values ofL andN are chosen arbitrarily). Velocity profiles (both axial and rotational) and the variation of the effective viscosity with concentration, tube radius and for various values of the boundary condition parameters are shown graphically. The analytic results obtained are compared with experimental results (for blood flow). It is found that they are in a reasonably good agreement. The effective viscosity exhibits the Inverse Fahraeus-Lindquist Effect in all the cases (including the slip or no-slip in the velocity fields). A method is given for determining the non-zero couple stress boundary condition for a given concentration. Applications of this theory to blood flow are briefly discussed.     

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.     

Prediction of liquid level distribution in horizontal gas-liquid stratified flows with interfacial level gradient

A one-dimensional momentum equation has been derived based on a two-fluid model and used to predict the axial distribution of liquid level or void fraction in steady, cocurrent, gas-liquid stratified flows in horizontal circular pipes and rectangular channels. The equation is carefully formulated to incorporate the effect of interfacial level gradient. Two different critical liquid levels are found from the momentum equation and are adopted as a boundary condition to calculate the liquid level or void fraction distribution upstream of the channel exit. The predicted void fraction distributions are compared with the experimental data obtained in a rectangular channel in this work and other data reported for large-diameter pipes. Good agreement is shown for air-kerosene, air-water and stream-water stratified flows with a smooth gas-liquid interface.     

On the numerical study of bubbly flow created by ventilated cavity in vertical pipe

The dispersion of bubbles into a down-liquid flow in a vertical pipe is investigated. At low flow rates, the intended design of a swarm of discrete bubbles is achieved. At high flow rates, a ventilated cavity is nonetheless formed, which is attached close to the gas sparger. Behind this ventilated cavity, three different flow regimes characterize the complex bubbly flow field downstream of the down-liquid flow: vortex region with high void fraction, transitional region and pipe flow region. In this study, a numerical model that solved the entire development of the gas–liquid flow including the extended single-phase liquid region upstream to the wall-jet and recirculating-vortex zones in order to allow a more realistic determination of the boundary conditions of the down-liquid flow was adopted. Coupling with the Eulerian–Eulerian two-fluid model to solve the respective gas and liquid phases, a population balance model was also applied to predict the bubble size distribution in the wake right below the cavity base as well as further downstream in the transitional and fully-developed pipe flow regions. The numerical model was evaluated by comparing the numerical results against the data derived from theoretical, numerical and experimental approaches. Prediction of the Sauter mean bubble diameter distributions by the population balance approach at different axial locations confirmed the dominance of breakage due to the high turbulent intensity below the ventilated cavity which led to the generation of small gas bubbles at high void fraction. Further downstream, the coalescence effect dominated leading to merging of the small bubbles to form bigger bubbles.     

Numerical analysis of turbulent structure in compound meandering open channel by algebraic Reynolds stress model

The turbulent flow in a compound meandering channel with a rectangular cross section is one of the most complicated turbulent flows, because the flow behaviour is influenced by several kinds of forces, including centrifugal forces, pressure‐driven forces and shear stresses generated by momentum transfer between the main channel and the flood plain. Numerical analysis has been performed for the fully developed turbulent flow in a compound meandering open‐channel flow using an algebraic Reynolds stress model. The boundary‐fitted coordinate system is introduced as a method for coordinate transformation in order to set the boundary conditions along the complicated shape of the meandering open channel. The turbulence model consists of transport equations for turbulent energy and dissipation, in conjunction with an algebraic stress model based on the Reynolds stress transport equations. With reference to the pressure–strain term, we have made use of a modified pressure–strain term. The boundary condition of the fluctuating vertical velocity is set to zero not only for the free surface, but also for computational grid points next to the free surface, because experimental results have shown that the fluctuating vertical velocity approaches zero near the free surface. In order to examine the validity of the present numerical method and the turbulent model, the calculated results are compared with experimental data measured by laser Doppler anemometer. In addition, the compound meandering open channel is clarified somewhat based on the calculated results. As a result of the analysis, the present algebraic Reynolds stress model is shown to be able to reasonably predict the turbulent flow in a compound meandering open channel. Copyright © 2005 John Wiley & Sons, Ltd.     

Flow pattern transition,pressure gradient,hold-up predictions in gas/non-Newtonian power-law fluid stratified flow

This work focuses on gas/non-Newtonian power-law fluid stratified pipe flow. Two different theoretical approaches to obtain pressure gradient and hold-up predictions are presented: the steady fully developed two-fluid model and the pre-integrated model. The theoretical predictions are compared with experimental data available for horizontal and for slightly downward inclined air/shear thinning fluid stratified flow taken from literature. The predictions of the pre-integrated model are validated showing a good agreement when compared with experimental data. The criteria for the transition from the stratified flow pattern are applied to gas/non-Newtonian stratified flow. The neutral stability analysis (smooth/wavy stratified flow) and the well-posedness (existence region of stratified flow) of governing equations are carry out. The predicted transition boundaries are obtained using the steady fully developed two-fluid model and the pre-integrated model, where the shape factors and their derivatives are accounted for. A comparison between the predicted boundaries and experimental flow pattern maps is presented and shows a good agreement. A comment on the shear stress modeling by the pre-integrated model is provided.