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

A visualization method is used to obtain the main features of the hydrodynamic field for flow past a circular cylinder moving at a uniform speed in a direction perpendicular to its generating lines in a tank filled with a viscous liquid. Photographs are presented to show the particular fineness of the experimental technique. More especially, the closed wake and the velocity distribution behind the obstacle are investigated; the changes in the geometrical parameters describing the eddies with Reynolds number (5 < Re < 40) and with the ratio λ between the diameters of the cylinder and tank are given. A comparison with existing numerical and experimental results is presented and some remarks are made about the calculation techniques proposed up to the present. The limits of the Reynolds-number range for which the twin vortices exist and adhere stably to the cylinder are determined.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112077000147

Experimental determination of the main features of the viscous flow in the wake of a circular cylinder in uniform translation. Part 2. Unsteady 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

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

[1] Natural terrestrial radioactivity and cosmic ray ionization lead to the formation of air ions and charged aerosol particles even away from regions of active charge separation, such as in thunderstorms. The natural electrified state of the atmosphere has been studied for over a century; however, the effect of ionization on the physical properties of aerosols and clouds has rarely been studied in its own right except in thunderstorms. Here we review the status of our understanding of atmospheric charged particles and their influence on aerosol and cloud microphysical processes. We also review mechanisms that have been recently proposed to connect variations in the atmospheric ionization rate with variations in global cloudiness and weather systems. We conclude that a mechanism linking cosmic ray ionization and cloud properties cannot be excluded and that there are established electrical effects on aerosol and cloud microphysics. Necessary further work includes measurements of cloud, droplet, and aerosol charging and ion-aerosol conversion, together with modeling of the electrical aspects of nonthunderstorm cloud microphysics.

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/2002RG000114

Ion-aerosol-cloud processes in the lower atmosphere

Accurate solutions of the steady-state Navier-Stokes equations of motion have been obtained by means of a numerical method to determine the hydrodynamic drag on a rigid sphere falling at its terminal velocity in an unbounded fluid. The calculations were carried out for Reynolds numbers between 0.01 and 400. The numerical solutions were compared with the theoretical results of Stokes, Oseen, Goldstein, Proudman and Pearson, Jenson, Rimon and Cheng, and Carrier, and with the recent experimental data of Maxworthy, and Pruppacher and Steinberger. At the lowest Reynolds numbers the numerical solutions show closest agreement with the theory of Proudman and Pearson and at intermediate Reynolds numbers with the semi-theoretical relationship proposed by Carrier. At higher Reynolds numbers our present results agree well with the calculations of Hamielec et al. for Reynolds numbers of 40 and 100 and with the numerical results of Rimon and Cheng; they depart, however, significantly from the results of Jenson...

A Numerical Study of the Drag on a Sphere at Low and Intermediate Reynolds Numbers

The results of a numerical evaluation of the Navier-Stokes equations of motion for the case of a viscous fluid streaming past a sphere are presented in terms of the length of the standing eddy behind the sphere and in terms of the angle of flow separation at the sphere. Emphasis was placed on calculating these quantities at Reynolds numbers between 20 and 40 where no reliable theoretical or experimental values are available. In support of these calculations, it is shown that the values for the drag on a sphere previously calculated by us from the Navier-Stokes equations of motion by the same numerical technique as that used for calculating the eddy length and angle of flow separation agree well with our recent, extensive drag measurements for a wide Reynolds number interval. Our results are used to make a comparison between drag and flow field as predicted by analytical solutions and numerical solutions to the Navier-Stokes equations of motion. Some limitations of the analytical solutions to predict correct values for the drag, and to describe the correct nature of the flow field, are pointed out. It is shown further that a plot of [(D/Ds) – 1] versus log NRe, where D is the actual drag on a sphere, Ds is the Stokes drag, and NRe is the Reynolds number, reveals that the variation of the drag on a sphere with Reynolds number follows well defined regimes, which correlate well with the regimes of the flow field around a sphere. A similar relationship between ‘drag-regime’ and flow field pattern is discussed for the case of viscous flow past a cylinder.

B. P. Le Clair

Papers

A Numerical Study of the Drag on a Sphere at Low and Intermediate Reynolds Numbers

Some relations between drag and flow pattern of viscous flow past a sphere and a cylinder at low and intermediate Reynolds numbers

Viscous Flow through Particle Assemblages at Intermediate Reynolds Numbers