Analysis of unsteady gas–liquid flows in a rectangular tank: Comparison of Euler–Eulerian and Euler–Lagrangian simulations

We study the dynamics of gas–liquid flows experimentally and computationally in a rectangular bubble column where the gas source is introduced at the corner. The flow in this reactor is complex and inherently unsteady in nature. The two-dimensional liquid phase velocity field is calculated by an Eulerian approach solving the unsteady Reynolds Averaged Navier Stokes equations. The conservation equations are closed using a two parameter turbulence model. The two-way coupling was accounted for by adding source terms in the conservation equations of the continuous phase to take into account the interaction with the dispersed phase. Bubble tracking is achieved through a Lagrangian approach. Here the equations of motion are solved taking into account the drag, pressure, buoyancy and gravity forces. The time-averaged flows along with the variables which characterize turbulence are analyzed for a wide range of gas flow-rates using Euler–Lagrangian simulations. These simulation predictions are validated with Euler–Eulerian simulations where the gas-phase distribution is captured as a void fraction and PIV experiments. The motion of bubbles induces turbulence in the flow. The applicability of two parameter models for turbulence like the standard k–ε model on time-averaged flow properties is addressed. From the results of the time averaged velocity field, turbulence intensity, turbulent viscosity and gas hold-up profiles, it is concluded that the Euler–Lagrangian model is applicable at lower gas flow-rates. The Euler–Eulerian approach was found to be valid at lower as well as higher gas flow-rates.     

Numerical modelling of 3D stratified free surface flows: a case study of sediment dumping

A three‐dimensional numerical model has been developed to simulate stratified flows with free surfaces. The model is based on the Reynolds‐averaged Navier–Stokes (RANS) equations with variable fluid density. The equations are solved in a transformed σ‐coordinate system with the use of operator‐splitting method (Int. J. Numer. Meth. Fluids 2002; 38 :1045–1068). The numerical model is validated against the one‐dimensional diffusion problem and the two‐dimensional density‐gradient flow. Excellent agreements are obtained between numerical results and analytical solutions. The model is then used to study transport phenomena of dumped sediments into a water body, which has been modelled as a strongly stratified flow. For the two‐dimensional problem, the numerical results compare well with experimental data in terms of mean particle falling velocity and spreading rate of the sediment cloud for both coarse and medium‐size sediments. The model is also employed to study the dumping of sediments in a three‐dimensional environment with the presence of free surface. It is found that during the descending process an annulus‐like cloud is formed for fine sediments whereas a plate‐like cloud for medium‐size sediments. The model is proven to be a good tool to simulate strongly stratified free surface flows. Copyright © 2005 John Wiley & Sons, Ltd.     

Simple,coupled algorithms for solving creeping flows and their application to electro‐osmotic flows

In this study we developed simple, coupled algorithms for solving low‐Reynolds‐number flows applicable to micro‐scale flows such as electro‐osmotic flows. The most popular scheme, i.e. the projection method, is not suitable for such flows because of its undesirable slip effect on boundaries at low‐Reynolds‐numbers. In our method, the velocity and pressure are strongly coupled, and the momentum and pressure equations are solved iteratively by using the successive over relaxation (SOR) method while exchanging the unknown variables as soon as they have been updated. The developed methods are applied to a model flow for evaluating their performance. It was found that the coupled schemes are indeed superior to a projection method, i.e. the fractional‐step method, in both numerical accuracy and CPU time. The code is then applied to a dc electro‐osmotic flow within a cavity driven by electrical force acting on the ions spread in the fluid. In this application, the system of equations for the fluid flow and that for the ion transport are solved in a decoupled way, but each system is solved by using fully implicit schemes. From the simulations and by introducing the concept of vorticity source, we can identify two roles of the body force, one contributing to build‐up of the osmotic pressure and the other to the fluid flow. The interesting reverse flow occurring after the external potentials applied on the electrodes have been shut off is also investigated in terms of the vorticity source. Copyright © 2010 John Wiley & Sons, Ltd.     

Theoretical and numerical studies on the flow multiplicity phenomenon for gas–solids two-phase flows in CFB risers

The dependence of the fully-developed flow profiles on the inlet flow conditions for gas–solids two-phase flows, i.e. the flow multiplicity phenomenon, in circulating fluidized bed (CFB) risers was proposed and discussed in this article. The flow multiplicity phenomenon for gas–solids two-phase flows was first proved mathematically based on the conservation equations of mass and momentum. Then the CFD model using Eulerian–Eulerian approach with k–ε turbulence model for each phase was further adopted to analyze the details of this flow multiplicity phenomenon. It is theoretically and numerically revealed that for gas–solids two-phase flows, the flow profiles in the fully-developed region are always dominated by the flow profiles at the inlet. The solids concentration profile is closely coupled with the velocity profile, and the inlet solids concentration and velocity profiles can largely influence the fully-developed concentration and velocity profiles.     

Drag law for monodisperse gas–solid systems using particle-resolved direct numerical simulation of flow past fixed assemblies of spheres

Gas–solid momentum transfer is a fundamental problem that is characterized by the dependence of normalized average fluid–particle force F on solid volume fraction ? and the Reynolds number based on the mean slip velocity Re_m. In this work we report particle-resolved direct numerical simulation (DNS) results of interphase momentum transfer in flow past fixed random assemblies of monodisperse spheres with finite fluid inertia using a continuum Navier–Stokes solver. This solver is based on a new formulation we refer to as the Particle-resolved Uncontaminated-fluid Reconcilable Immersed Boundary Method (PUReIBM). The principal advantage of this formulation is that the fluid stress at the particle surface is calculated directly from the flow solution (velocity and pressure fields), which when integrated over the surfaces of all particles yields the average fluid–particle force. We demonstrate that PUReIBM is a consistent numerical method to study gas–solid flow because it results in a force density on particle surfaces that is reconcilable with the averaged two-fluid theory. The numerical convergence and accuracy of PUReIBM are established through a comprehensive suite of validation tests. The normalized average fluid–particle force F is obtained as a function of solid volume fraction ? (0.1 ? ? ? 0.5) and mean flow Reynolds number Re_m (0.01 ? Re_m ? 300) for random assemblies of monodisperse spheres. These results extend previously reported results of  and  to a wider range of ?, Re_m, and are more accurate than those reported by Beetstra et al. (2007). Differences between the drag values obtained from PUReIBM and the drag correlation of Beetstra et al. (2007) are as high as 30% for Re_m in the range 100–300. We take advantage of PUReIBM’s ability to directly calculate the relative contributions of pressure and viscous stress to the total fluid–particle force, which is useful in developing drag correlations. Using a scaling argument, Hill et al. (2001b) proposed that the viscous contribution is independent of Re_m but the pressure contribution is linear in Re_m (for Re_m > 50). However, from PUReIBM simulations we find that the viscous contribution is not independent of the mean flow Reynolds number, although the pressure contribution does indeed vary linearly with Re_m in accord with the analysis of Hill et al. (2001b). An improved correlation for F in terms of ? and Re_m is proposed that corrects the existing correlations in Re_m range 100–300. Since this drag correlation has been inferred from simulations of fixed particle assemblies, it does not include the effect of mobility of the particles. However, the fixed-bed simulation approach is a good approximation for high Stokes number particles, which are encountered in most gas–solid flows. This improved drag correlation can be used in CFD simulations of fluidized beds that solve the average two-fluid equations where the accuracy of the drag law affects the prediction of overall flow behavior.     

基于两相紊流模型的非平衡输沙研究

基于雷诺平均的水沙两相流方程, 建立了一个非平衡全沙输移二维数学模型. 模型考虑相间相对运动以及多颗粒之间的相互影响, 通过相间作用力进行两相耦合.和传统的单相流模型以及低浓度两相流模型相比, 该模型摆脱了依赖经验公式给定床面边界条件的局限性. 针对明渠净冲刷问题, 在合理给定水相和泥沙相边界条件的前提下计算了泥沙浓度分布的沿程变化, 并利用物理模型实验的结果和理论解验证了数学模型的正确性,同时也分析了明渠净冲刷问题中紊动扩散和重力沉降现象的特征.      

Modeling of three-phase heavy oil–water–gas bubbly flow in upward vertical pipes

A bubbly gas–bubbly oil flow pattern may occur when water, heavy oil and gas flow simultaneously in vertical pipes in such a way that water is the continuous phase. In this work, a one-dimensional, thermal, transient two-fluid mathematical model, for such flow, is presented. The model consists of mass, momentum and energy conservation equations for every phase whose numerical solution is based on the finite difference technique in the implicit scheme. The model is able to predict pressure, temperature, volumetric fraction and velocity profiles. For accurate modeling of multiphase flows, the key issue is to specify the adequate closure relationships, thus drag and virtual mass forces for the gas and oil phases were taken into account and special attention was paid on the gas–oil drag force. When this force was included into the model it was found that: (1) such force had the same order of magnitude than the oil drag force and both forces were smaller than the gas drag force, (2) the pressure, gas and oil velocities and gas and oil volume fraction profiles were affected, (3) the numerical stability was increased. The model predictions are in agreement with experimental data reported in literature.     

Numerical study of breaking waves by a two‐phase flow model

A two‐phase flow model, which solves the flow in the air and water simultaneously, is presented for modelling breaking waves in deep and shallow water, including wave pre‐breaking, overturning and post‐breaking processes. The model is based on the Reynolds‐averaged Navier–Stokes equations with the k ?ε turbulence model. The governing equations are solved by the finite volume method in a Cartesian staggered grid and the partial cell treatment is implemented to deal with complex geometries. The SIMPLE algorithm is utilised for the pressure‐velocity coupling and the air‐water interface is modelled by the interface capturing method via a high resolution volume of fluid scheme. The numerical model is validated by simulating overturning waves on a sloping beach and over a reef, and deep‐water breaking waves in a periodic domain, in which good agreement between numerical results and available experimental measurements for the water surface profiles during wave overturning is obtained. The overturning jet, air entrainment and splash‐up during wave breaking have been captured by the two‐phase flow model, which demonstrates the capability of the model to simulate free surface flow and wave breaking problems.Copyright © 2011 John Wiley & Sons, Ltd.     

A comparative study of central and upwind difference schemes using the primitive variables

The use of the velocity-pressure formulation of the Navier-Stokes equations for the numerical solution of fluid flow problems is favoured for free-surface problems, more involved flow configurations, and three-dimensional flows. Many engineering problems involve such features in addition to strong inertial effects. The computational instabilities arising from central-difference schemes for the convective terms of the governing equations impose serious limitations on the range of Reynolds numbers that can be investigated by the numerical method. Solutions for higher Reynolds numbers Re > 1000 could be reached using upwind-difference schemes. A comparative study of both schemes using a method based on the primitive variables is presented. The comparison is made for the model problem of the driven flow in a square cavity. Using a central scheme stable solutions of the pressure and velocity fields were obtained for Reynolds numbers up to 5000. The streamfunction and vorticity fields were calculated from the resulting velocity field and compared with previous solutions. It is concluded that total upwind differencing results in a considerable change in the flow pattern due to the false diffusion. For practical calculations, by a proper choice of a small amount of partial upwind differencing the vorticity diffusion near the walls and the global features of the solutions are not sigificantly altered.     

On stagnation pressure increases in calorically perfect,ideal gases

When stagnation pressure rises in a natural or numerically simulated flow it is frequently a cause for concern, as one usually expects viscosity and turbulence to cause stagnation pressure to decrease. In fact, if stagnation pressure increases, one may suspect measurement or numerical errors. However, this need not be the case, as the laws of nature do not require that stagnation pressure continually decreases. In order to help clarify matters, the objective of this work is to understand the conditions under which stagnation pressure will rise in the unsteady/steady flows of compressible, viscous, calorically perfect, ideal gases. Furthermore, at a more practical level, the goal is to understand the conditions under which stagnation pressure will increase in flows simulated with the Reynolds averaged Navier–Stokes equations and eddy-viscosity turbulence models. In order to provide an improved understanding of increases in stagnation pressure for both these scenarios, transport equations are derived that govern its behavior in the unaveraged and Reynolds averaged settings. These equations are utilized to precisely determine the relationship between changes in stagnation pressure and zeroth, first, and second derivatives of fundamental flow quantities. Furthermore, these equations are utilized to demonstrate the relationship between changes in stagnation pressure and fundamental non-dimensional quantities that govern the conductivity, viscosity, and compressibility of the flow. In addition, based on an analysis of the Reynolds averaged equation (for eddy-viscosity turbulence models), it is shown that stagnation pressure is particularly likely to experience a spurious rise at the outer edges of shear layers that are undergoing convex curvature. Thereafter, numerical experiments are performed which confirm the primary aspects of the theoretical analysis.     

Unsteady RANS simulation of oscillating mould flows

Mould flow oscillations are of major importance for the performance of the continuous casting process. They are suspected to promote entrainment of slag and other unwanted secondary phases into the melt pool. These oscillating turbulent flows are investigated by means of numerical simulations. The numerical model is based on the equation of continuity and the unsteady Reynolds averaged Navier–Stokes equations. The system of flow equations is closed by a Reynolds stress turbulence model in combination with non‐equilibrium wall functions. The unsteady simulation resolves low‐frequency oscillations of the flow field. These frequencies and numerically resolved mean values are in agreement with results of corresponding model experiments. The proposed model should be advantageous in order to investigate the mechanisms of the oscillations and the process of slag entrainment in more detail. Copyright © 2006 John Wiley & Sons, Ltd.     

Control of the aerodynamic characteristics of a profile oscillating with respect to the incidence angle in subsonic viscous flow

The results of a numerical simulation of the unsteady subsonic viscous gas flow around a two-dimensional profile oscillating with respect to the incidence angle are presented and the possibility of controlling the nonstationary aerodynamic characteristics is considered. The hysteresis phenomena typical of oscillatory profile motions are investigated, the dependence of the lift force and drag is found for various laws of periodic variation of the incidence angle with time, and the effect of the frequency and amplitude of the angular profile oscillations on the shape of the hysteresis curves is studied. The calculations were based on the numerical solution of the nonstationary Navier-Stokes equations averaged in the Reynolds sense (Reynolds equations) which were closed using the k-ω turbulence model with modeling of the laminar/turbulent transition.     

Ensemble phase averaged equations for multiphase flows in porous media. Part 1: The bundle-of-tubes model

A bundle-of-tubes construct is used as a model system to study ensemble averaged equations for multiphase flow in a porous material. Momentum equations for the fluid phases obtained from the method are similar to Darcy’s law, but with additional terms. We study properties of the additional terms, and the conditions under which the averaged equations can be approximated by the diffusion model or the extended Darcy’s law as often used in models for multiphase flows in porous media. Although the bundle-of-tubes model is perhaps the simplest model for a porous material, the ensemble averaged equation technique developed in this paper assumes the very same form in more general treatments described in Part 2 of the present work (Zhang, D.Z., 2009. Ensemble Phase Averaged Equations for Multiphase Flows in Porous Media, Part 2: A General Theory. Int. J. Multiphase Flow 35, 640–649). Any model equation system intended for the more general cases must be understood and tested first using simple models. The concept of ensemble phase averaging is dissected here in physical terms, without involved mathematics through its application to the idealized bundle-of-tubes model for multiphase flow in porous media.     

Statistical models for predicting the effect of bidisperse particle collisions on particle velocities and stresses in homogeneous anisotropic turbulent flows

The purpose of this paper is to present and compare two statistical models for predicting the effect of collisions on particle velocities and stresses in bidisperse turbulent flows. These models start from a kinetic equation for the probability density function (PDF) of the particle velocity distribution in a homogeneous anisotropic turbulent flow. The kinetic equation describes simultaneously particle–turbulence and particle–particle interactions. The paper is focused on deriving the collision terms in the governing equations of the PDF moments. One of the collision models is based on a Grad-like expansion for the PDF of the velocity distributions of two particles. The other model stems from a Grad-like expansion for the joint fluid–particle PDF. The validity of these models is explored by comparing with Lagrangian simulations of particle tracking in uniformly sheared and isotropic turbulent flows generated by LES. Notwithstanding the fact that the fluid turbulence may be isotropic, the particle velocity fluctuations are anisotropic due to the impact of gravitational settling. Comparisons of the model predictions and the numerical simulations show encouraging agreement.     

Numerical simulation of turbulent flows in trapezoidal meandering compound open channels

Three‐dimensional (3D) numerical study is presented to investigate the turbulent flow in meandering compound open channels with trapezoidal cross‐sections. The flow simulation is carried out by solving the 3D Reynolds‐averaged continuity and Navier–Stokes equations with Reynolds stress equation model (RSM) for steady‐state flow. Finite volume method (FVM) is applied to numerically solve the governing equations of fluid flow. The velocity magnitude, tangential velocity, transverse velocity and Reynolds stresses are calculated for various flow conditions. Good agreement between the simulated and available laboratory measurements was obtained, indicating that the RSM can accurately predict the complicated flow phenomenon. Comparison of the calculated secondary currents of four cases (one being inbank flow and other three being overbank flow) with different water depths reveals that (i) the inbank flow exhibits different flow behaviors from that of the overbank flow does and (ii) the water depth has significant effects on the magnitude and direction of secondary currents. Copyright © 2010 John Wiley & Sons, Ltd.     

Assessment of a high-order accurate Discontinuous Galerkin method for turbomachinery flows

In this work the capabilities of a high-order Discontinuous Galerkin (DG) method applied to the computation of turbomachinery flows are investigated. The Reynolds averaged Navier–Stokes equations coupled with the two equations k-ω turbulence model are solved to predict the flow features, either in a fixed or rotating reference frame, to simulate the fluid flow around bodies that operate under an imposed steady rotation. To ensure, by design, the positivity of all thermodynamic variables at a discrete level, a set of primitive variables based on pressure and temperature logarithms is used. The flow fields through the MTU T106A low-pressure turbine cascade and the NASA Rotor 37 axial compressor have been computed up to fourth-order of accuracy and compared to the experimental and numerical data available in the literature.     

3D free‐surface flow computation using a RANSE/Fourier–Kochin coupling

A coupling method for numerical calculations of steady free‐surface flows around a body is presented. The fluid domain in the neighbourhood of the hull is divided into two overlapping zones. Viscous effects are taken in account near the hull using Reynolds‐averaged Navier–Stokes equations (RANSE), whereas potential flow provides the flow away from the hull. In the internal domain, RANSE are solved by a fully coupled velocity, pressure and free‐surface elevation method. In the external domain, potential‐flow theory with linearized free‐surface condition is used to provide boundary conditions to the RANSE solver. The Fourier–Kochin method based on the Fourier–Kochin formulation, which defines the velocity field in a potential‐flow region in terms of the velocity distribution at a boundary surface, is used for that purpose. Moreover, the free‐surface Green function satisfying this linearized free‐surface condition is used. Calculations have been successfully performed for steady ship‐waves past a serie 60 and then have demonstrated abilities of the present coupling algorithm. Copyright © 2003 John Wiley & Sons, Ltd.