Natural convection inside two-dimensional cavities: the integral transform method

Abstract

The Integral Transform Method is employed in the hybrid numerical‐analytical solution of laminar natural convection inside rectangular enclosures. The hybrid nature of this approach allows for the establishment of benchmark results in the solution of non‐linear partial differential systems, including the coupled set of heat and fluid flow equations that govern the steady natural convection problem under consideration. Therefore, numerical results under user prescribed accuracy are obtained for different values of the governing parameters, and the convergence behaviour of the proposed eigenfunction expansions is illustrated. Critical comparisons against previously reported benchmark solutions are then performed. Copyright © 1999 John Wiley & Sons, Ltd.

Vector eigenfunction expansion in the integral transform solution of transient natural convection

“…The initial and boundary conditions are chosen in accordance with the ones used in previous works (Leal et al , 1999; Leal et al , 2000). No-slip and impermeable wall conditions are imposed on all four boundaries of the cavity.…”

“…Results extending the benchmark to higher Rayleigh numbers (Saitoh and Hirose, 1989; Le Quéré, 1991) and transient regime (Ramaswamy et al , 1992; Sai et al , 1994) have also been provided. The GITT itself, under the streamfunction-only formulation, was also used to solve natural convection inside a rectangular cavity in both steady and transient regimes (Cotta and Leal, 1998; Leal et al , 1999; Leal et al , 2000), providing an independent confirmation of the available benchmark results.…”

A fourth-order accurate finite difference method to evaluate the true transient behaviour of natural convection flow in enclosures

“…These numerical errors have essentially restricted the accuracy of implicit methods to at most second-order in time and have not been able to truly capture the transient behaviour of the incompressible flows unless a very small timestep is used. Unconventional numerical methods have been developed to avoid these numerical errors and achieve higher order accuracy in time, for example, the time-space method of Wakashima and Saitoh (2004), discrete singular convolution method for numerical simulation of convective heat transfer problems by Wan et al (2001) and integral transform technique by Silva et al (2011) and Leal et al (1999), Leal et al (2000). (Arpino et al, 2014), proposed AC-CBS algorithm based on primitive variable approach using third-order Backward difference formula for time derivative discretisation.…”

An efficient Legendre‐Galerkin spectral method for the natural convection in two‐dimensional cavities

“…The thermal cavity flow can be applied to a wide range of practical technological problems such as ventilation, crystal growth in liquids, nuclear reactor safety, and the design of high-powered laser systems. This problem was numerical studied by lots of approximated methods, for example, Fourier spectral method with periodic boundary conditions, 1 finite-difference method, 2,3 finite-element method, 4 finite-volume method, 5,6 Chebyshev pseudospectral method, 7-9 spectral element method, 10 integral transform method, 11 Lattice Boltzmann simulations, 12,13 and so on. Most of them listed some tabulated solutions with four Rayleigh number (Ra), namely 10 3 , 10 4 , 10 5 , and 10 6 and the Prandtl number (Pr) with 0.7,1 which corresponds to a cavity filled with air.…”