G. D. Raithby

ENHANCEMENTS OF THE SIMPLE METHOD FOR PREDICTING INCOMPRESSIBLE FLUID FLOWS

Variations of the SIMPLE method of Patankar and Spalding have been widely used over the past decade to obtain numerical solutions to problems involving incompressible flows. The present paper shows several modifications to the method which both simplify its implementation and reduce solution costs. The performances of SIMPLE, SIMPLER, and SIMPLEC (the present method) are compared for two recirculating flow problems. The paper is addressed to readers who already have experience with SIMPLE or its variants.

COMPUTATION OF RADIANT HEAT TRANSFER ON A NONORTHOGONAL MESH USING THE FINITE-VOLUME METHOD

Abstract The finite-volume method has been shown to effectively predict radiant exchange in geometrically simple enclosures where the medium is gray, absorbing, emitting, and scattering. Cartesian and circular cylindrical meshes have always been used. The present article shows that the method applies equally well to geometrically complex enclosures where nonorthogonal, boundary-fitted meshes are used. This development permits radiant heat transfer to be computed on the same mesh employed to solve the equations of fluid motion.

NUMERICAL SOLUTION OF PROBLEMS IN INCOMPRESSIBLE FLUID FLOW: TREATMENT OF THE VELOCITY-PRESSURE COUPLING

Coupling between the momentum and mass conservation equations for “ incompressible” flows is often the major cause of the slow convergence of iterative solution techniques. Several methods of handling this coupling, some of which are novel, are examined, and results of their application to a test problem are compared. The application is made in a manner that completely isolates the effect of the coupling and leads to a clearer understanding of how the methods perform. Several recommendations are made for potential users.

The Segregated Approach to Predicting Viscous Compressible Fluid Flows

Extension des methodes de Patankar et Spalding a la resolution des equations de mouvement des ecoulements de fluide compressible

IMPROVED FINITE-DIFFERENCE METHODS BASED ON A CRITICAL EVALUATION OF THE APPROXIMATION ERRORS

When finite-difference methods are used to solve the benchmark problem of natural convection in a square cavity, a very fine grid is required to obtain predictions that are accurate to 1-2%. The derivation of the finite-difference equations requires the introduction of many approximations; this study systematically evaluates these approximations to establish which are mainly responsible for the fine-grid requirement. The poorest approximations are then improved one by one, resulting in a scheme that yields highly accurate predictions using a relatively coarse grid. The method of evaluating the accuracy of the approximations, the improved approximations themselves, and the solution method used all contain novel features. Storage and computing time requirements for a new sparse matrix solver, which was used in the current study to simultaneously solve for stream function and vorticity, are presented.

APPLICATION OF ADDITIVE CORRECTION MULTIGRID TO THE COUPLED FLUID FLOW EQUATIONS

Hutchinson and Raithby [10] proposed a simple additive correction multigrid (ACM) method that has proved effective in reducing the cost of solving an algebraic equation set for a single variable. The equations of fluid motion and heat transfer lead to coupled sets of equations. The present paper outlines how ACM can be extended to solve these equations and reports results for a fluid dynamics and a natural convection problem (where the temperature-velocity coupling is important). The application of ACM to the coupled equations proves to be extremely attractive.

FINITE-ELEMENT SOLUTION PROCEDURES FOR SOLVING THE INCOMPRESSIBLE, NAVIER-STOKES EQUATIONS USING EQUAL ORDER VARIABLE INTERPOLATION

The conventional finite-element formulation of the equations of motion (written in pressure-velocity variables) requires that the order of interpolation for pressure be one less than that used for the velocity components. This constraint is inconvenient and can be argued to be physically inconsistent when inertial effects are dominant. The origins of the constraint are discussed and three new finite-element formulations are advanced that permit equal order representation of pressure and velocity. Of these, the velocity correction scheme, similar to that commonly used in finite-difference procedures, offers superior performance for the examples examined in this paper.

Prediction of heat and fluid flow in complex geometries using general orthogonal coordinates

A method is presented for solving the equations of motion in orthogonal curvilinear coordinates using a computational grid that conforms to the boundaries and is orthogonal in the interior. Details of the derivation are provided. The particular formulation proposed maintains simplicity, clarity, and flexibility by special treatment of the stresses and diffusive fluxes. Because the equations are simple generalizations of those that arise when specific analytic coordinates are used, standard solution methods are applicable. The method is demonstrated for two problems, one of which involves both heat and fluid flow.

HEAT TRANSFER BY NATURAL CONVECTION ACROSS VERTICAL AIR LAYERS

Finite-difference predictions of natural convection in vertical air layers are reported for a wide range of Rayleigh numbers and aspect ratios. Calculations were carried out for both perfectly conducting and adiabatic boundaries at the top and bottom ends of the layer. Comparisons between the predictions and measurements show that the calculated heat transfer rates are valid only over a limited range of parameters, and that the points of departure correlate closely with predicted points of instability.

A transient two-fluid model for the simulation of slug flow in pipelines—I. Theory

A one-dimensional transient two-fluid model is developed to predict transient slug flow in pipelines. To account for the interphase interactions, new constitutive relations for the drag coefficient and the virtual mass force for the slug flow regime are derived by applying the conservation equations to a geometrically simplified slug unit. New coefficients in the pressure gradient term in the two-fluid momentum conservation equations are also obtained to account for the non-uniform distribution of the phases and of the pressure drop along a slug unit. The new relations yield a more accurate treatment of the hydrodynamics of slug flow than traditional two-fluid models. Constitutive relations for other flow regimes can also be incorporated into the model, allowing the analysis of general transient two-phase flows in pipelines.