Use of lagrangian methods to describe drop deposition and distribution in horizontal gas—liquid annular flows

Drop distribution and deposition in horizontal gas—liquid annular flow is described by a diffusion model, which views the concentration field as the result of dispersion from a distribution of sources. Drops originating from a wall source are considered to diffuse in a field of homogeneous turbulence, while simultaneously being swept downward by the gravitational field. Deposition is assumed to be controlled by two mechanisms operating in parallel, and boundary conditions are derived which correctly satisfy conservation of mass. This analysis for an instantaneous source is shown to be equivalent to considering diffusion in a coordinate system moving with the settling velocity of the particles. The results are found to be useful for understanding droplet distribution and deposition.     

A DNS study of laminar bubbly flows in a vertical channel

Direct numerical simulations are used to examine laminar bubbly flows in vertical channels. For equal size nearly spherical bubbles the results show that at steady state the number density of bubbles in the center of the channel is always such that the fluid mixture there is in hydrostatic equilibrium. For upflow, excess bubbles are pushed to the walls, forming a bubble rich wall-layer, one bubble diameter thick. For downflow, bubbles are drawn into the channel center, leading to a wall-layer devoid of bubbles, of a thickness determined by how much the void fraction in the center of the channel must be increased to reach hydrostatic equilibrium. The void fraction profile can be predicted analytically using a very simple model and the model also gives the velocity profile for the downflow case. For the upflow, however, the velocity increase across the wall-layer must be obtained from the simulations. The slip velocity of the bubbles in the channel core and the velocity fluctuations are predicted reasonably well by results for homogeneous flows.     

Application of the ultrasonic technique and high-speed filming for the study of the structure of air–water bubbly flows

Multiphase flows are very common in industry, oftentimes involving very harsh environments and fluids. Accordingly, there is a need to determine the dispersed phase holdup using noninvasive fast responding techniques; besides, knowledge of the flow structure is essential for the assessment of the transport processes involved. The ultrasonic technique fulfills these requirements and could have the capability to provide the information required. In this paper, the potential of the ultrasonic technique for application to two-phase flows was investigated by checking acoustic attenuation data against experimental data on the void fraction and flow topology of vertical, upward, air–water bubbly flows in the zero to 15% void fraction range. The ultrasonic apparatus consisted of one emitter/receiver transducer and three other receivers at different positions along the pipe circumference; simultaneous high-speed motion pictures of the flow patterns were made at 250 and 1000 fps. The attenuation data for all sensors exhibited a systematic interrelated behavior with void fraction, thereby testifying to the capability of the ultrasonic technique to measure the dispersed phase holdup. From the motion pictures, basic gas phase structures and different flows patterns were identified that corroborated several features of the acoustic attenuation data. Finally, the acoustic wave transit time was also investigated as a function of void fraction.     

Roe‐type Riemann solver for gas–liquid flows using drift‐flux model with an approximate form of the Jacobian matrix

This work presents an approximate Riemann solver to the transient isothermal drift ‐ flux model. The set of equations constitutes a non‐linear hyperbolic system of conservation laws in one space dimension. The elements of the Jacobian matrix A are expressed through exact analytical expressions. It is also proposed a simplified form of A considering the square of the gas to liquid sound velocity ratio much lower than one. This approximation aims to express the eigenvalues through simpler algebraic expressions. A numerical method based on the Gudunov's fluxes is proposed employing an upwind and a high order scheme. The Roe linearization is applied to the simplified form of A . The proposed solver is validated against three benchmark solutions and two experimental pipe flow data. Copyright © 2015 John Wiley & Sons, Ltd.     

Computation of particle dispersion in turbulent liquid flows using an efficient Lagrangian trajectory model

The dispersion of solid particles in a turbulent liquid flow impinging on a centrebody through an axisymmetric sudden expansion was investigated numerically using a Eulerian–Lagrangian model. Detailed experimental measurements at the inlet were used to specify the inlet conditions for two-phase flow computations. The anisotropy of liquid turbulence was accounted for using a second-moment Reynold stress transport model. A recently developed stochastic–probabilistic model was used to enhance the computational efficiency of Lagrangian trajectory computations. Numerical results of the stochastic–probabilistic model using 650 particle trajectories were compared with those of the conventional stochastic discrete-delta-function model using 18 000 particle trajectories. In addition, results of the two models were compared with experimental measurements. © 1998 John Wiley & Sons, Ltd.     

Two-way coupling of Eulerian–Lagrangian model for dispersed multiphase flows using filtering functions

Eulerian–Lagrangian approaches for dispersed multiphase flows can simulate detailed flow structures with a much higher spatial resolution than the Eulerian–Eulerian approaches. However, there are still unsolved problems regarding the calculation method for accurate two-way interaction, especially on the numerical instability due to the dispersion migration through discrete computational grids. Inadequate solvers sometimes produce false velocity fluctuation which makes the simulation unstable. In this paper, a new calculation method for dispersion-to-continuous phase interaction, which is accompanied by spherical dispersion migration, is proposed. The basic principle of the method is the introduction of Lagrangian filtering functions which convert discrete dispersion volume fractions to a spatially differentiable distribution. The performance of linear, Gaussian and sinewave filtering functions is examined by simple benchmark tests and applied to the simulation of dispersion-generated fluctuation. Using the present method, three-dimensional continuous phase flow structures induced by rising spherical bubbles and/or settling solid particles are demonstrated.     

Simulations of free surface flows with implementation of surface tension and interface sharpening in the two-fluid model

Two-fluid model used for free surface flows with large characteristic scales is improved; the smeared interface is sharpened with conservative level set method and the surface tension force with wetting angle is implemented. Surface tension force is split between two phases with several models. Detailed analysis showed the splitting of surface tension force with volume averaging as the most appropriate. The improved two-fluid model with interface sharpening and implemented surface tension is validated on several test cases. The pressure jump over a droplet interface test case showed that the pressure jump in simulation converges with grid refinement to the analytical one. The parasitic currents in simulation are one order of magnitude larger than in simulation with volume of fluid model. In the oscillating droplet test case the time period of oscillating droplet with initially ellipsoid or square shape is similar to the analytical time period. In the rising bubble test case, the rising bubble position, terminal velocity, and circularity are similar to the one observed in simulations with level set model. The wetting angle is implemented in the two-fluid model with interface sharpening and surface tension force. Model is tested in the simulation of droplet in contact with wall with different wetting angles.     

Two-fluid model for transient analysis of slug flow in oil wells

In this work it is presented a transient, one-dimensional, adiabatic model for slug flow simulation, which appears when liquid (mixture of oil and water) and gas flow simultaneously through pipes. The model is formed by space and time averaged conservation equations for mass, momentum and energy for each phase, the numerical solution is based on the finite difference technique in the implicit scheme. Velocity, pressure, volumetric fraction and temperature profiles for both phases were predicted for inclination angles from the horizontal to the vertical position (unified model) and ascendant flow. Predictions from the model were validated using field data and ten correlations commonly used in the oil industry. The effects of gas heating or cooling, due to compression and expansion processes, on the predictions and numerical stability, were studied. It was found that when these effects are taken into account, a good behavior of temperature predictions and numerical stability are obtained. The model presents deviations lower than 14% regarding field data and it presents better predictions than most of the correlations.     

A low cost implementation of the Phillips model in solid–liquid flow modeling

The interpolation requirements for the loosely coupled finite element solution of the Navier–Stokes equations and Phillips shear‐induced particle diffusion model are discussed. It is shown that a second‐order approximation of the fluid velocity field is required to adequately capture the spatial derivatives of the rate‐of‐strain tensor. To circumvent this limitation, a shear‐rate smoothing procedure is introduced, thereby allowing the use of lower‐order approximations for the fluid phase. Numerical experiments comparing the convergence and CPU cost of the different tetrahedral interpolation bases are presented. Copyright © 2009 John Wiley & Sons, Ltd.     

Two-phase flow distribution of air–water annular flow in a parallel flow heat exchanger

The air and water flow distribution are experimentally studied for a round header – flat tube geometry simulating a parallel flow heat exchanger. The number of branch flat tube is 30. The effects of tube outlet direction, tube protrusion depth as well as mass flux, and quality are investigated. The flow at the header inlet is identified as annular. For the downward flow configuration, the water flow distribution is significantly affected by the tube protrusion depth. For flush-mounted configuration, most of the water flows through frontal part of the header. As the protrusion depth increases, more water is forced to the rear part of the header. The effect of mass flux or quality is qualitatively the same as that of the protrusion depth. Increase of the mass flux or quality forces the water to rear part of the header. For the upward flow configuration, however, most of the water flows through rear part of the header. The protrusion depth, mass flux, or quality does not significantly alter the flow pattern. Possible explanations are provided based on the flow visualization results. Negligible difference on the water flow distribution was observed between the parallel and the reverse flow configuration.     

A non‐linear k–ε model with realizability for prediction of flows around bluff bodies

The incompressible flow around bluff bodies (a square cylinder and a cube) is investigated numerically using turbulence models. A non‐linear k–ε model, which can take into account the anisotropy of turbulence with less CPU time and computer memory then RSM or LES, is adopted as a turbulence model. In tuning of the model coefficients of the non‐linear terms are adjusted through the examination of previous experimental studies in simple shear flows. For the tuning of the coefficient in the eddy viscosity (=C_μ), the realizability constraints are derived in three types of basic 2D flow patterns, namely, a simple shear flow, flow around a saddle and a focal point. C_μ is then determined as a function of the strain and rotation parameters to satisfy the realizability. The turbulence model is first applied to a 2D flow around a square cylinder and the model performance for unsteady flows is examined focussing on the period and the amplitude of the flow oscillation induced by Karman vortex shedding. The applicability of the model to 3D flows is examined through the computation of the flow around a surface‐mounted cubic obstacle. The numerical results show that the present model performs satisfactorily to reproduce complex turbulent flows around bluff bodies. Copyright © 2003 John Wiley & Sons, Ltd.     

Velocity–pressure coupling in finite difference formulations for the Navier–Stokes equations

A new numerical procedure for solving the two‐dimensional, steady, incompressible, viscous flow equations on a staggered Cartesian grid is presented in this paper. The proposed methodology is finite difference based, but essentially takes advantage of the best features of two well‐established numerical formulations, the finite difference and finite volume methods. Some weaknesses of the finite difference approach are removed by exploiting the strengths of the finite volume method. In particular, the issue of velocity–pressure coupling is dealt with in the proposed finite difference formulation by developing a pressure correction equation using the SIMPLE approach commonly used in finite volume formulations. However, since this is purely a finite difference formulation, numerical approximation of fluxes is not required. Results presented in this paper are based on first‐ and second‐order upwind schemes for the convective terms. This new formulation is validated against experimental and other numerical data for well‐known benchmark problems, namely developing laminar flow in a straight duct, flow over a backward‐facing step, and lid‐driven cavity flow. Copyright © 2010 John Wiley & Sons, Ltd.     

Heat transfer measurements and correlations for air–water flow of different flow patterns in a horizontal pipe

Heat transfer coefficients were measured and new correlations were developed for two-phase, two-component (air and water) heat transfer in a horizontal pipe for different flow patterns. Flow patterns were observed in a transparent circular pipe using an air–water mixture. Visual identification of the flow patterns was supplemented with photographic data, and the results were plotted on the flow regime map proposed by Taitel and Dukler and agreed quite well with each other. A two-phase heat transfer experimental setup was built for this study and a total of 150 two-phase heat transfer data with different flow patterns were obtained under a uniform wall heat flux boundary condition. For these data, the superficial Reynolds number ranged from 640 to 35,500 for the liquid and from 540 to 21,200 for the gas. Our previously developed robust two-phase heat transfer correlation for a vertical pipe with modified constants predicted the horizontal pipe air–water heat transfer experimental data with very good accuracy. Overall the proposed correlations predicted the data with a mean deviation of 1.0% and an rms deviation of 12%.     

Görtler vortices in Falkner–Skan flows with suction and blowing

In this paper, we use nonlinear calculations to study curved boundary‐layer flows with pressure gradients and self‐similar suction or blowing. For an accelerated outer flow, stabilization occurs in the linear region while the saturation amplitude of vortices is larger than for flows with a decelerating outer flow. The combined effects of boundary‐layer suction and a favourable pressure gradient can give a significant stabilization of the flow. Streamwise vortices can be amplified on both concave and convex walls for decelerated Falkner–Skan flow with an overshoot in the velocity profile. The disturbance amplitude is generally lower far downstream compared with profiles without overshoot. Copyright © 2007 John Wiley & Sons, Ltd.     

Finite element simulation of turbulent Couette–Poiseuille flows using a low Reynolds number k–ϵ model

Developing Couette–Poiseuille flows at Re=5000 are studied using a low Reynolds number k–ϵ two‐equation model and a finite element formulation. Mesh‐independent solutions are obtained using a standard Galerkin formulation and a Galerkin/least‐squares stabilized method. The predictions for the velocity and turbulent kinetic energy are compared with available experimental results and to the DNS data. Second moment closure's solutions are also compared with those of the k–ϵ model. The deficiency of eddy viscosity models to predict dissymmetric low Reynolds number channel flows has been demonstrated. Copyright © 1999 John Wiley & Sons, Ltd.     

Simulation of gas–solid particle flows over a wide range of concentration

A two-fluid model of gas–solid particle flows that is valid for a wide range of the solid-phase volume concentration (dilute to dense) is presented. The governing equations of the fluid phase are obtained by volume averaging the Navier–Stokes equations for an incompressible fluid. The solid-phase macroscopic equations are derived using an approach that is based on the kinetic theory of dense gases. This approach accounts for particle–particle collisions. The model is implemented in a control-volume finite element method for simulations of the flows of interest in two-dimensional, planar or axisymmetric, domains. The chosen mathematical model and the proposed numerical method are applied to three test problems and one demonstration problem. © 1998 John Wiley & Sons, Ltd.     

A simplified v2–f model for near‐wall turbulence

A simplified version of the v²–f model is proposed that accounts for the distinct effects of low‐Reynolds number and near‐wall turbulence. It incorporates modified C_ε(1,2) coefficients to amplify the level of dissipation in non‐equilibrium flow regions, thus reducing the kinetic energy and length scale magnitudes to improve prediction of adverse pressure gradient flows, involving flow separation and reattachment. Unlike the conventional v²–f, it requires one additional equation (i.e. the elliptic equation for the elliptic relaxation parameter f_µ) to be solved in conjunction with the k–ε model. The scaling is evaluated from k in collaboration with an anisotropic coefficient C_v and f_µ. Consequently, the model needs no boundary condition on and avoids free stream sensitivity. The model is validated against a few flow cases, yielding predictions in good agreement with the direct numerical simulation (DNS) and experimental data. Copyright © 2007 John Wiley & Sons, Ltd.