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A two‐dimensional mathematical model of flame spread and solid burning is presented. For the gas phase, it consists of variable density, fully elliptic Navier‐Stokes momentum, energy and chemical species mass equations. Combustion processes are treated according to a one‐step, finite‐rate, reaction. The solid phase model describes a porous cellulosic fuel for a range of thicknesses from the thermally thin to the thermally thick limit. Conductive and convective heat transfer takes place as the solid degrades, by two first order Arrhenius reactions, to volatiles and chars. Variations of solid phase densities account for fuel burn‐out. Effects of gas phase and surface radiation are also included. A steady formulation of gas phase equations with respect to the unsteady solid phase mathematical model is proposed, gas phase characteristic times being much shorter than those of the solid phase. The non‐constant density Navier‐Stokes equations are formulated in terms of vorticity and stream function, avoiding the pressure‐velocity coupling and, at the same time, the adoption of a sample‐fixed coordinate system allows unsteady flame spread processes to be simulated. The solution is computed numerically by means of an iterative, operator‐splitting method based on implicit finite‐difference approximations. Numerical simulations of the dynamics of flame spread over cellulosic solids are presented and extinction limits as a consequence of reduced rates of fuel generation are determined.

This paper aims to describe the 2D meshless local boundary integral equation (LBIE) method for solving the Navier–Stokes equations.

The velocity–vorticity formulation is selected to eliminate the pressure gradient of the equations. The local integral representations of flow kinematics and transport kinetics are derived. The integral equations are discretized using the local RBF interpolation of velocities and vorticities, while the unknown fluxes are kept as independent variables. The resulting volume integrals are computed using the general radial transformation algorithm.

The efficiency and accuracy of the method are illustrated with several examples chosen from reference problems in computational fluid dynamics.

The meshless LBIE method is applied to the 2D Navier–Stokes equations. No derivatives of interpolation functions are used in the formulation, rendering the present method a robust numerical scheme for the solution of fluid flow problems.

The flow field around an oscillating airfoil is evaluated numerically, using the stream function‐vorticity formulation of the Navier‐Stokes equations. An algebraic turbulence model, adapted from the Baldwin‐Lomax model, is included in solving the time‐averaged Reynolds equations. Computed pressure distribution for turbulent flow past a stationary airfoil is compared with measurements. Finally, for the oscillating airfoil cases, the computations are performed in order to determine the history of pressure distribution and to identify the nature of the vortex initiation on the suction surface for laminar and turbulent flow. Our results for laminar flow show that minute circular shaped vortices are formed on the surface prior to the dominant vortex formation. Flattened vortices are formed on the surface in turbulent flow, prior to the formation of the dominant large vortex structure.

The purpose of the paper is to study the steady and periodic solution of a lid‐driven cavity flow problem with the gradual increase of Reynolds number (Re) up to 10,000.

The problem is solved by unsteady stream function‐vorticity formulation using the clustered grids. The alternating direction implicit (ADI) method and the central difference scheme have been used for discretization of the governing equations. Total vorticity error and the total kinetic energy have been considered for ensuring the state of flow condition. The midplane velocity distribution and the top wall vortex distribution are compared with the results of other authors and found to show good agreement.

Kinetic energy variation with time is studied for large time computation. Below 7,500, it becomes constant signifying the flow to be in steady‐state. At Re=10,000, the fluid flow has an oscillating nature. The dimensionless period of oscillation is found to be 1.63. It is demonstrated that the present computation is able to capture the periodic solution after the bifurcation very accurately.

The findings will be useful in conducting a steady and periodic solution of variety of fluid flows or thermally‐driven fluid flows.

This paper presents a spectral multidomain method for solving the Navier‐Stokes equations in the vorticity‐stream function formulation. The algorithm is based on an extensive use of the influence matrix technique and so leads to a direct method without any iterative process. Numerical results concerning the Czochralski melt configuration are reported and compared with spectral monodomain solutions to show the advantage of the domain decomposition for such a problem which solution presents a singular behaviour.

This paper aims to develop a multidomain boundary element method (BEM) for modeling 2D complex turbulent thermal flow using low Reynolds two‐equation turbulence models.

The integral boundary domain equations are discretised using mixed boundary elements and a multidomain method also known as a subdomain technique. The resulting system matrix is an overdetermined, sparse block banded and solved using a fast iterative linear least squares solver.

The simulation of a turbulent flow over a backward step is in excellent agreement with the finite volume method using the same turbulent model. A grid consisting of over 100,000 elements could be solved in the order of a few minutes using a 3.0 Ghz P4 and 1 GB memory indicating good efficiency.

The paper shows, for the first time, that the BEM is applicable to thermal flows using k‐ε.

The Rayleigh‐Benard‐Marangoni problem for natural convection in a rectangular cavity with thermocapillary forces on a free surface is investigated using a stream function‐vorticity formulation. The nonlinear system is iteratively decoupled and high‐degree p finite elements are used for the discretization of the physical domain. The linear systems arising from the discretization at each iteration are solved using a spectral multilevel scheme, which is a natural preconditioner for high‐p (spectral) elements. The spectral multilevel solver lends itself to parallelization in an element‐by‐element (EBE) framework. Simulation results are presented and compared to previously published results. The multilevel efficiency is compared to previous results for the driven cavity problem. Parallel performance studies are presented for the Cray T3E distributed memory architecture.

Modelling accurately the transient behaviour of natural convection flow in enclosures been a challenging task because of a variety of numerical errors which have limited achieving the higher order temporal accuracy. A fourth-order accurate finite difference method in both space and time is proposed to overcome these numerical errors and accurately model the transient behaviour of natural convection flow in enclosures using vorticity–streamfunction formulation.

Fourth-order wide stencil formula with appropriate one-sided difference extrapolation technique near the boundary is used for spatial discretisation, and classical fourth-order Runge–Kutta scheme is applied for transient term discretisation. The proposed method is applied on two transient case studies, i.e. convection–diffusion of a Gaussian Pulse and Taylor Vortex flow having analytical solution.

Error magnitude comparison and rate of convergence analysis of the proposed method with these analytical solutions establish fourth-order accuracy and prove the ability of the proposed method to truly capture the transient behaviour of incompressible flow. Also, to test the transient natural convection flow behaviour, the algorithm is tested on differentially heated square cavity at high Rayleigh number in the range of 10³-10⁸, followed by studying the transient periodic behaviour in a differentially heated vertical cavity of aspect ratio 8:1. An excellent comparison is obtained with standard benchmark results.

The developed method is applied on 2D enclosures; however, the present methodology can be extended to 3D enclosures using velocity–vorticity formulations which shall be explored in future.

The proposed methodology to achieve fourth-order accurate transient simulation of natural convection flows is novel, to the best of the authors’ knowledge. Stable fourth-order vorticity boundary conditions are derived for boundary and external boundary regions. The selected case studies for comparison demonstrate not only the fourth-order accuracy but also the considerable reduction in error magnitude by increasing the temporal accuracy. Also, this study provides novel benchmark results at five different locations within the differentially heated vertical cavity of aspect ratio 8:1 for future comparison studies.

To conduct a numerical study on two‐dimensional, transient, buoyant flow inside an air‐filled 45°‐inclined enclosure, heated and cooled on adjacent walls.

The governing equations obtained through the stream function‐vorticity formulation are solved using finite differences. Flow characteristics have been investigated for an aspect ratio of 1. Calculations are carried out for the Rayleigh numbers in the range of 10³≤Ra≤5×10⁷.

With the increasing Rayleigh number, four distinct flow regimes were identified based on the time variations of the mid‐point stream function and the mean Nusselt number at the heated wall as well as those of flow and temperature fields: steady flow with symmetric two cells at low Ra; steady flow with asymmetric two cells at lower moderate Ra; oscillatory flow with the periodic nature at upper moderate Ra; and oscillatory flow in chaotic nature at higher Ra range.

The distinct flow regimes are observed only at ϕ=45°; a small deviation of the tilting angle from ϕ=45° results in the disappearance of the distinction.

A finite difference solution for steady natural convective flow within the human eye, modelled as a sphere with a specified temperature distribution over its surface, has been obtained. The stream function—vorticity formulation of the equations of motion for the unsteady axisymmetric flow is used; interest lying in the final steady solution. Forward differences are used for the time derivatives and second‐order central differences for the space derivatives. The alternating direction implicit method is used for solution of the discretization equations. Local one‐dimensional grid adaptation is used to resolve the steep gradients in some regions of the flow at large Rayleigh numbers. The break‐up into multi‐cellular flow is found at high Rayleigh numbers. Results identify regions of stagnant fluid in locations similar to those of blind spots in the eye.