Integral transform solution of natural convection in a square cavity with volumetric heat generation

Abstract

-The generalized integral transform technique (GITT)is employed to obtain a hybrid numericalanalytical solution of natural convection in a cavity with volumetric heat generation. The hybrid nature of this approach allows for the establishment of benchmark results in the solution of non-linear partial differential equation systems, including the coupled set of heat and fluid flow equations that govern the steady natural convection problem under consideration. Through performing the GITT, the resulting transformed ODE system is then numerically solved by making use of the subroutine DBVPFD from the IMSL Library. Therefore, numerical results under user prescribed accuracy are obtained for different values of Rayleigh numbers, and the convergence behavior of the proposed eigenfunction expansions is illustrated. Critical comparisons against solutions produced by ANSYS CFX 12.0 are then conducted, which demonstrate excellent agreement. Several sets of reference results for natural convection with volumetric heat generation in a bi-dimensional square cavity are also provided for future verification of numerical results obtained by other researchers.

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

“…The GITT has been applied to the solution of heat and fluid flow problems under the Navier–Stokes formulation in both primitive variables (Lima et al , 2007; Curi et al , 2014; Souza et al , 2016) and streamfunction-only (Perez-Guerrero and Cotta, 1992; Perez-Guerrero et al , 2000; Silva et al , 2010; An et al , 2013; Matt et al , 2017; Pontes et al , 2018; Fu et al , 2018) formulations. The latter approach has been preferred due to the enhanced convergence, the elimination of the pressure field, and automatic satisfaction of the continuity equation granted by the definition of the streamfunction, though restricted to two-dimensional fluid flow problems.…”

Integral transform solution of natural convection in a cylinder cavity with uniform internal heat generation

“…In the last decades, the hybrid analytical-numerical computational methodology (GITT) has been recognised as a powerful computational tool to explain the heat transfer and diffusion-convection phenomena (Nogueira and Cotta, 1990;Pontedeiro et al, 2008;Monteiro et al, 2009;Knupp et al, 2014Knupp et al, , 2015aKnupp et al, , 2015bCotta et al, 2016) and for the solutions of Navier-Stokes equations (Lima et al, 1997;Pereira et al, 1998Pereira et al, , 2000de Lima et al, 2007;Paz et al, 2007;Silva et al, 2010). In the convection behaviour analysis, the GITT technique has been approved as an accurate hybrid analytical-numerical methodology (Leal et al, 1999(Leal et al, , 2000Alves et al, 2002;Neto et al, 2006;Monteiro et al, 2010;An et al, 2013). Leal et al (1999Leal et al ( , 2000 used the GITT methodology to investigate the steady and transient convection behaviours in a steady and transient laminar flow in a square cavity, respectively.…”

“…Later, Neto et al (2006) performed a study to investigate the effect of the aspect ratio on the natural convection using a three-dimensional model. Recently, the natural convection in enclosure with the uniformly distributed heat generation was investigated using the GITT technique (An et al, 2013).…”

Linear Stability Analysis and CFD Simulation of Double-Layer Rayleigh-Bénard Convection

“…Even though it has been investigated for more than a century, the RB problem has been the subject of a large number of studies in recent years, involving, in particular, modifications of the classical RB system. For example, the problem of natural and mixed convection in enclosures is very important in the study of the cooling of electronic devices and thermal comfort (Fontana et al, 2015;An et al, 2013;Mariani and Coelho, 2007).…”