The paper describes the development of a computer program for the prediction of the flow,heat-transfer, and combustion processes in a three-dimensional furnace. The mathematical model involves the solution of the differential equations for momentum, continuity, chemical-species concentrations, stagnation enthalpy, radiation fluxes, turbulence quantities, and concentration of particles in various size ranges. The mathematical framework is outlined in some detail, and the results of some recent computations are presented. The conclusions are: that substantial progress has already been made toward the developmentof a computer model for the transfer processes in a furnace; and that further work is needed to validate the computer model and to refine the physical inputs.

Two procedures are described for solving the Navier-Stokes equations for steady, fully three-dimensional flows: both are extensions of earlier methods devised for three-dimensional boundary layers, and have the following common features: (i) the main dependent variables are the velocities and pressure; (ii) the latter are computed on a number of staggered, interlacing grids, each of which is associated with a particular variable; (iii) a hybrid central-upwind difference scheme is employed; and (iv) the solution algorithms are sufficiently implicit to obviate the need to approach the steady state via the time evolution of the flow, as is required by wholly explicit methods.

A general, numerical, marching procedure is presented for the calculation of the transport processes in three-dimensional flows characterised by the presence of one coordinate in which physical influences are exerted in only one direction. Such flows give rise to parabolic differential equations and so can be called three-dimensional parabolic flows. The procedure can be regarded as a boundary-layer method, provided it is recognised that, unlike earlier published methods with this name, it takes full account of the cross-stream diffusion of momentum, etc., and of the pressure variation in the cross-stream plane. The pressure field is determined by: first calculating an intermediate velocity field based on an estimated pressure field; and then obtaining appropriate correction so as to satisfy the continuity equation. To illustrate the procedure, calculations are presented for the developing laminar flow and heat transfer in a square duct with a laterally-moving wall.

A general, implicit, numerical, marching procedure is presented for the solution of parabolic partial differential equations, with special reference to those of the boundary layer. The main novelty lies in the choice of a grid which adjusts its width so as to conform to the thickness of the layer in which significant property gradients are present. The non-dimensional stream function is employed as the independent variable across the layer.