Journal ArticleDOI
A fourth‐order compact finite difference scheme for the steady stream function–vorticity formulation of the Navier–Stokes/Boussinesq equations
TLDR
In this paper, a fourth-order compact finite difference scheme on the nine-point 2D stencil is formulated for solving the steady-state Navier-Stokes/Boussinesq equations for two-dimensional, incompressible fluid flow and heat transfer using the stream function-vorticity formulation.Abstract:
A fourth-order compact finite difference scheme on the nine-point 2D stencil is formulated for solving the steady-state Navier-Stokes/Boussinesq equations for two-dimensional, incompressible fluid flow and heat transfer using the stream function-vorticity formulation. The main feature of the new fourth-order compact scheme is that it allows point-successive overrelaxation (SOR) or point-successive under-relaxation iteration for all Rayleigh numbers Ra of physical interest and all Prandtl numbers Pr attempted. Numerical solutions are obtained for the model problem of natural convection in a square cavity with benchmark solutions and compared with some of the accurate results available in the literatureread more
Citations
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Journal ArticleDOI
A fourth-order compact ADI method for solving two-dimensional unsteady convection-diffusion problems
Z. F. Tian,Y. B. Ge +1 more
TL;DR: In this paper, an exponential high-order compact (EHOC) alternating direction implicit (ADI) method, in which the Crank-Nicolson scheme is used for the time discretization and an exponential fourthorder compact difference formula for the steady-state 1D convection-diffusion problem is used to solve the problem, is presented for the solution of the unsteady convection--diffusion problems, which requires only a regular fivepoint 2D stencil similar to that in the standard second-order methods.
Journal ArticleDOI
High-order compact exponential finite difference methods for convection-diffusion type problems
Z. F. Tian,S. Q. Dai +1 more
TL;DR: The method developed in this article has been applied to obtain the numerical solutions of the lid driven cavity flow problem governed by the 2D incompressible Navier-Stokes equations using the stream function-vorticity formulation.
Journal ArticleDOI
Lattice-Boltzmann simulations of the thermally driven 2D square cavity at high Rayleigh numbers
TL;DR: The D2Q9+D2Q5 MRT-TLBE is shown to be second-order accurate and to be capable of yielding results of benchmark quality, including various Nusselt numbers and local hydrodynamic intensities, which agree well with existing benchmark data obtained by other methods.
Journal ArticleDOI
A higher order compact finite difference algorithm for solving the incompressible Navier-Stokes equations
TL;DR: In this article, a higher order compact finite difference algorithm is developed for solving the 2D unsteady incompressible Navier-Stokes equations in primitive variable, which is established on a staggered grid system and is at least third order accurate in space.
Journal ArticleDOI
Multigrid method and fourth-order compact difference discretization scheme with unequal meshsizes for 3D poisson equation
TL;DR: A fourth-order compact difference discretization scheme with unequal meshsizes in different coordinate directions is employed to solve a three-dimensional (3D) Poisson equation on a cubic domain and two multgrid methods are developed to solve the resulting sparse linear systems.
References
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Journal ArticleDOI
Natural convection of air in a square cavity: A bench mark numerical solution
TL;DR: In this paper, the authors used mesh refnement and extrapolation to obtain an accurate solution of the equations describing two-dimensional natural convection in a square cavity with differentially heated side walls.
Journal ArticleDOI
Natural convection in a square cavity: A comparison exercise
G. de Vahl Davis,I. P. Jones +1 more
TL;DR: In this article, a number of contributed solutions to the problem of laminar natural convection in a square cavity have been compared with what is regarded as a solution of high accuracy, and the purposes of this exercise have been to confirm the accuracy of the bench mark solution and to provide a basis for the assessment of the various methods and computer codes used to obtain the contributed solutions.
Book
Numerical Solution of Convection-Diffusion Problems
TL;DR: In this paper, the authors presented results from mathematical analysis on different schemes for steady problems. But they did not discuss the effect of finite element methods and Petrov-Galerkin methods on the performance of steady problems and unsteady problems.
Journal ArticleDOI
Higher order accurate difference solutions of fluid mechanics problems by a compact differencing technique
TL;DR: In this paper, a general fourth order differencing scheme was developed and applied to three viscous test problems to verify the accuracy and applicability of the technique, which is atypical since only three nodes are necessary to obtain the desired fourth order accuracy.
Journal ArticleDOI
Finite volume multigrid prediction of laminar natural convection: Bench-mark solutions
TL;DR: In this article, a finite volume multigrid procedure for the prediction of laminar natural convection flows is presented, enabling efficient and accurate calculations on very fine grids, which is fully conservative and uses second-order central differencing for convection and diffusion fluxes.
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