This study describes the development and validation of Computational Fluid Dynamics (CFD) methodology for the simulation of dispersed two-phase flows. A two-fluid (Euler-Euler) methodology previously developed at Imperial College is adapted to high phase fractions. It employs averaged mass and momentum conservation equations to describe the time-dependent motion of both phases and, due to the averaging process, requires additional models for the inter-phase momentum transfer and turbulence for closure. The continuous phase turbulence is represented using a two-equation k − ε−turbulence model which contains additional terms to account for the effects of the dispersed on the continuous phase turbulence. The Reynolds stresses of the dispersed phase are calculated by relating them to those of the continuous phase through a turbulence response function. The inter-phase momentum transfer is determined from the instantaneous forces acting on the dispersed phase, comprising drag, lift and virtual mass. These forces are phase fraction dependent and in this work revised modelling is put forward in order to capture the phase fraction dependency of drag and lift. Furthermore, a correlation for the effect of the phase fraction on the turbulence response function is proposed. The revised modelling is based on an extensive survey of the existing literature. The conservation equations are discretised using the finite-volume method and solved in a solution procedure, which is loosely based on the PISO algorithm, adapted to the solution of the two-fluid model. Special techniques are employed to ensure the stability of the procedure when the phase fraction is high or changing rapidely. Finally, assessment of the methodology is made with reference to experimental data for gas-liquid bubbly flow in a sudden enlargement of a circular pipe and in a plane mixing layer. Additionally, Direct Numerical Simulations (DNS) are performed using an interface-capturing methodology in order to gain insight into the dynamics of free rising bubbles, with a view towards use in the longer term as an aid in the development of inter-phase momentum transfer models for the two-fluid methodology. The direct numerical simulation employs the mass and momentum conservation equations in their unaveraged form and the topology of the interface between the two phases is determined as part of the solution. A novel solution procedure, similar to that used for the two-fluid model, is used for the interface-capturing methodology, which allows calculation of air bubbles in water. Two situations are investigated: bubbles rising in a stagnant liquid and in a shear flow. Again, experimental data are used to verify the computational results.

Computational fluid dynamics of dispersed two-phase flows at high phase fractions

Preface. 1 Introduction. 1.1 Importance of Heat Transfer. 1.2 Heat Transfer Modes. 1.3 The Laws of Heat Transfer. 1.4 Formulation of Heat Transfer Problems. 1.4.1 Heat transfer from a plate exposed to solar heat flux. 1.4.2 Incandescent lamp. 1.4.3 Systems with a relative motion and internal heat generation. 1.5 Heat Conduction Equation. 1.6 Boundary and Initial Conditions. 1.7 Solution Methodology. 1.8 Summary. 1.9 Exercise. Bibliography. 2 Some Basic Discrete Systems. 2.1 Introduction. 2.2 Steady State Problems. 2.2.1 Heat flow in a composite slab. 2.2.2 Fluid flow network. 2.2.3 Heat transfer in heat sinks (combined conduction-convection). 2.2.4 Analysis of a heat exchanger. 2.3 Transient Heat Transfer Problem (Propagation Problem). 2.4 Summary. 2.5 Exercise. Bibliography. 3 The Finite Elemen t Method. 3.1 Introduction. 3.2 Elements and Shape Functions. 3.2.1 One-dimensional linear element. 3.2.2 One-dimensional quadratic element. 3.2.3 Two-dimensional linear triangular elements. 3.2.4 Area coordinates. 3.2.5 Quadratic triangular elements. 3.2.6 Two-dimensional quadrilateral elements. 3.2.7 Isoparametric elements. 3.2.8 Three-dimensional elements. 3.3 Formulation (Element Characteristics). 3.3.1 Ritz method (Heat balance integral method-Goodman's method). 3.3.2 Rayleigh-Ritz method (Variational method). 3.3.3 The method of weighted residuals. 3.3.4 Galerkin finite element method. 3.4 Formulation for the Heat Conduction Equation. 3.4.1 Variational approach. 3.4.2 The Galerkin method. 3.5 Requirements for Interpolation Functions. 3.6 Summary. 3.7 Exercise. Bibliography. 4 Steady State Heat Conduction in One Dimension. 4.1 Introduction. 4.2 Plane Walls. 4.2.1 Homogeneous wall. 4.2.2 Composite wall. 4.2.3 Finite element discretization. 4.2.4 Wall with varying cross-sectional area. 4.2.5 Plane wall with a heat source: solution by linear elements. 4.2.6 Plane wall with a heat source: solution by quadratic elements. 4.2.7 Plane wall with a heat source: solution by modified quadratic equations (static condensation). 4.3 Radial Heat Flow in a Cylinder. 4.3.1 Cylinder with heat source. 4.4 Conduction-Convection Systems. 4.5 Summary. 4.6 Exercise. Bibliography. 5 Steady State Heat Conduction in Multi-dimensions. 5.1 Introduction. 5.2 Two-dimensional Plane Problems. 5.2.1 Triangular elements. 5.3 Rectangular Elements. 5.4 Plate with Variable Thickness. 5.5 Three-dimensional Problems. 5.6 Axisymmetric Problems. 5.6.1 Galerkin's method for linear triangular axisymmetric elements. 5.7 Summary. 5.8 Exercise. Bibliography. 6 Transient Heat Conduction Analysis. 6.1 Introduction. 6.2 Lumped Heat Capacity System. 6.3 Numerical Solution. 6.3.1 Transient governing equations and boundary and initial conditions. 6.3.2 The Galerkin method. 6.4 One-dimensional Transient State Problem. 6.4.1 Time discretization using the Finite Difference Method (FDM). 6.4.2 Time discretization using the Finite Element Method (FEM). 6.5 Stability. 6.6 Multi-dimensional Transient Heat Conduction. 6.7 Phase Change Problems-Solidification and Melting. 6.7.1 The governing equations. 6.7.2 Enthalpy formulation. 6.8 Inverse Heat Conduction Problems. 6.8.1 One-dimensional heat conduction. 6.9 Summary. 6.10 Exercise. Bibliography. 7 Convection Heat Transfer 173 7.1 Introduction. 7.1.1 Types of fluid-motion-assisted heat transport. 7.2 Navier-Stokes Equations. 7.2.1 Conservation of mass or continuity equation. 7.2.2 Conservation of momentum. 7.2.3 Energy equation. 7.3 Non-dimensional Form of the Governing Equations. 7.3.1 Forced convection. 7.3.2 Natural convection (Buoyancy-driven convection). 7.3.3 Mixed convection. 7.4 The Transient Convection-diffusion Problem. 7.4.1 Finite element solution to convection-diffusion equation. 7.4.2 Extension to multi-dimensions. 7.5 Stability Conditions. 7.6 Characteristic-based Split (CBS) Scheme. 7.6.1 Spatial discretization. 7.6.2 Time-step calculation. 7.6.3 Boundary and initial conditions. 7.6.4 Steady and transient solution methods. 7.7 Artificial Compressibility Scheme. 7.8 Nusselt Number, Drag and Stream Function. 7.8.1 Nusselt number. 7.8.2 Drag calculation. 7.8.3 Stream function. 7.9 Mesh Convergence. 7.10 Laminar Isothermal Flow. 7.10.1 Geometry, boundary and initial conditions. 7.10.2 Solution. 7.11 Laminar Non-isothermal Flow. 7.11.1 Forced convection heat transfer. 7.11.2 Buoyancy-driven convection heat transfer. 7.11.3 Mixed convection heat transfer. 7.12 Introduction to Turbulent Flow. 7.12.1 Solution procedure and result. 7.13 Extension to Axisymmetric Problems. 7.14 Summary. 7.15 Exercise. Bibliography. 8 Convection in Porous Media. 8.1 Introduction. 8.2 Generalized Porous Medium Flow Approach. 8.2.1 Non-dimensional scales. 8.2.2 Limiting cases. 8.3 Discretization Procedure. 8.3.1 Temporal discretization. 8.3.2 Spatial discretization. 8.3.3 Semi- and quasi-implicit forms. 8.4 Non-isothermal Flows. 8.5 Forced Convection. 8.6 Natural Convection. 8.6.1 Constant porosity medium. 8.7 Summary. 8.8 Exercise. Bibliography. 9 Some Examples of Fluid Flow and Heat Transfer Problems. 9.1 Introduction. 9.2 Isothermal Flow Problems. 9.2.1 Steady state problems. 9.2.2 Transient flow. 9.3 Non-isothermal Benchmark Flow Problem. 9.3.1 Backward-facing step. 9.4 Thermal Conduction in an Electronic Package. 9.5 Forced Convection Heat Transfer From Heat Sources. 9.6 Summary. 9.7 Exercise. Bibliography. 10 Implementation of Computer Code. 10.1 Introduction. 10.2 Preprocessing. 10.2.1 Mesh generation. 10.2.2 Linear triangular element data. 10.2.3 Element size calculation. 10.2.4 Shape functions and their derivatives. 10.2.5 Boundary normal calculation. 10.2.6 Mass matrix and mass lumping. 10.2.7 Implicit pressure or heat conduction matrix. 10.3 Main Unit. 10.3.1 Time-step calculation. 10.3.2 Element loop and assembly. 10.3.3 Updating solution. 10.3.4 Boundary conditions. 10.3.5 Monitoring steady state. 10.4 Postprocessing. 10.4.1 Interpolation of data. 10.5 Summary. Bibliography. A Green's Lemma. B Integration Formulae. B.1 Linear Triangles. B.2 Linear Tetrahedron. C Finite Element Assembly Procedure. D Simplified Form of the Navier-Stokes Equations. Index.

Fundamentals of the Finite Element Method for Heat and Fluid Flow

Expressions for the probability of collision (Pc) and adhesion (Pa) have been derived for fine particle flotation by calculating the trajectory of particles as they flow past a bubble in streamline How. Three different flow regimes have been considered in the present work, i.e., Stokes, potential and intermediate. For the intermediate flow conditions in which most flotation operations are carried out, the particle trajectories have been determined using an empirical stream function derived in the present work. For the case of a very hydrophobic coal sample, the values of the probability of collection (P) determined experimentally have been found to be in close agreement with the theoretically predicted Pc values over a range of bubble and particle sizes The expression for Pa has been derived by determining the time it takes for a particle to slide along the surface of a bubble after collision. It has been assumed that the bubble-particte adhesion occurs when the sliding time is equal to or exceed...

The Effect of Bubble Size on Fine Particle Flotation

Sphere drag data from throughout the twentieth century are available in tabular form. However, much of the data arose from experiments in small diameter cylindrical vessels, where the results might have been influenced by the wall effect. Wall effect corrections developed by others were applied to 178 of the 480 data points collected. This corrected data set is believed to be free of the influence of wall effects. Existing drag and settling velocity correlations were compared to this data set. In addition, new correlations of the same forms were developed using the corrected data. Two new correlations of sphere terminal velocity are proposed, one applicable for all Reynolds numbers less than 2310 5 , and the other designed to predict settling velocities with exceptional accuracy for terminal Reynolds numbers less than 4,000, a region that contains almost all applications of interest in environmental engineering. The trial and error solution for settling velocity using the Fair and Geyer equation for drag should be retired in favor of the direct calculation available from these new correlations.

Sphere Drag and Settling Velocity Revisited

This work reports the first part of a series of numerical simulations carried out in order to improve knowledge of the forces acting on a sphere embedded in accelerated flows at finite Reynolds number, Re. Among these forces added mass and history effects are particularly important in order to determine accurately particle and bubble trajectories in real flows. To compute these hydrodynamic forces and more generally to study spatially or temporally accelerated flows around a sphere, the full Navier–Stokes equations expressed in velocity–pressure variables are solved by using a finite-volume approach. Computations are carried out over the range 0.1 ≤ Re ≤ 300 for flows around both a rigid sphere and an inviscid spherical bubble, and a systematic comparison of the flows around these two kinds of bodies is presented.Steady uniform flow is first considered in order to test the accuracy of the simulations and to serve as a reference case for comparing with accelerated situations. Axisymmetric straining flow which constitutes the simplest spatially accelerated flow in which a sphere can be embedded is then studied. It is shown that owing to the viscous boundary condition on the body as well as to vorticity transport properties, the presence of the strain modifies deeply the distribution of vorticity around the sphere. This modification has spectacular consequences in the case of a rigid sphere because it influences strongly the conditions under which separation occurs as well as the characteristics of the separated region. Another very original feature of the axisymmetric straining flow lies in the vortex-stretching mechanism existing in this situation. In a converging flow this mechanism acts to reduce vorticity in the wake of the sphere. In contrast when the flow is divergent, vorticity produced at the surface of the sphere tends to grow indefinitely as it is transported downstream. It is shown that in the case where such a diverging flow extends to infinity a Kelvin–Helmholtz instability may occur in the wake.Computations of the hydrodynamic force show that the effects of the strain increase rapidly with the Reynolds number. At high Reynolds numbers the total drag is dramatically modified and the evaluation of the pressure contribution shows that the sphere undergoes an added mass force whose coefficient remains the same as in inviscid flow or in creeping flow, i.e. CM = ½, whatever the Reynolds number. Changes found in vorticity distribution around the rigid sphere also affect the viscous drag, which is markedly increased (resp. decreased) in converging (resp. diverging) flows at high Reynolds numbers.

Accelerated flows past a rigid sphere or a spherical bubble. Part 1. Steady straining flow

Numerical solutions of the transient uniform flow around a sphere are obtained. The transition takes place between an initial potential flow and a fully developed viscous field. The fluid is incompressible, homogeneous, and its flow is governed by the complete Navier‐Stokes equations. The range of Reynolds number studied is Re = 1–1000 where a recirculatory wake appears and the nonlinear terms are essential, that is, they cannot be neglected or approximated. The flow is assumed to be axisymmetric throughout this range. A time‐dependent stream function‐vorticity formulation is adopted. The solution is obtained by constructing a finite difference approximation to the vorticity transport equation on an expanding spherical polar grid system. Central differencing of second‐order accuracy both in time (Dufort‐Frankel) and space is utilized. Experiments with numerical stability show an appreciable deviation from linearized stability analysis due to the large gradients of vorticity in the field. Quantitative physical results are obtained. The geometrical parameters characterizing the recirculatory wake compare favorably with those recorded in physical experiments. The detailed distribution of the vorticity on the sphere agrees with results obtained via the steady‐state approach at Re = 10, 40, and 100. The computed drag coefficient CD agrees well with the standard drag curve over the range of Reynolds numbers investigated.

Y. Rimon

Papers

Numerical Solution of a Uniform Flow over a Sphere at Intermediate Reynolds Numbers