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.

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

“…An optimum grid resolution is the highest grid resolution evaluated at each Rayleigh number as mentioned in Table IV. In the successive Tables V-X we compare the present steady state solutions with benchmark solutions obtained by de Vahl Davis (Ra = 10 3 -10 6 ) (De Vahl Davis, 1983) and Le quere (Ra = 10 6 -10 8 ) (Le Quéré, 1991) and some well-established numerical results available in the literature (Wan et al, 2001;Leal et al, 1999;Dennis and Hudson, 1989;Tian and Ge, 2003;Kalita et al, 2001;Chen et al, 1992;Mayne et al, 2000;Yapici and Obut, 2015;Syrjälä, 1996;Hortmann et al, 1990;Kondo, 1994). The results obtained are in excellent agreement with these benchmark results.…”

“…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.…”

Hybrid methods: the generalizaed integral transform technique

“…and L is the enclosure height and length, while Th and Te are the hot and cold wall temperatures, respectively. compared against previously reported purely numerical solutions, as detailed in [35]. The agreement is indeed excellent, reconfirming some of the most recent and careful benchmarking efforts available in the literature, even for the slower converging Nusselt number expansions.…”

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

“…The results obtained in the present work compares well with the benchmarks obtained with both transient and steady formulations. Relative differences against the benchmark results of Leal et al (1999) are evaluated to be approximately 0.3 per cent in the worst case, thus showing the adequacy of the methodology here proposed.…”

“…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.…”