Integral transform solutions of transient natural convection in enclosures with variable fluid properties

Abstract

This paper is aimed at the application of the Generalized Integral Transform Technique to the transient version of the classical differentially heated square cavity problem, considering both constant and variable fluid properties. The streamfunction-only formulation of the flow equations and the associated energy equation under laminar flow regime are employed in seeking a hybrid numerical–analytical solution to this natural convection problem. The computational procedure is carefully validated and a thorough convergence analysis is undertaken, yielding sets of reference results. The computed transient behavior of the coupled heat and fluid flow phenomena is compared to some previously reported results. The solution for variable fluid properties with partial Boussinesq approximation (density variation in the body force term only) is presented and compared with the constant properties results. Both models are investigated for different values of the Rayleigh number, from 10³ to 10⁵, and Prandtl number equal to 0.71.

Introduction

Natural convection inside cavities offers challenging test cases for the covalidation of numerical methods devised for the solution of coupled heat and fluid flow phenomena, governed by the continuity, the Navier–Stokes and the energy equations. The establishment of reliable benchmark results in both steady- and transient-state then becomes of major interest in allowing for critical comparisons among different scheme variants and computational implementation strategies.

In recent years, the so-called Generalized Integral Transform Technique (GITT) [1] has been successfully employed in the hybrid numerical–analytical solution of several classes of problems in diffusion and convection–diffusion, while being extended to the solution of the Navier–Stokes equations, either in isothermal flows or coupled to the energy equation formulated for the fluid motion [2], [3], [4], [5], [6].

The integral transform method, due to its hybrid numerical–analytical structure, offers the attractive feature of automatically controlling the global error in the computation, in a way similar to a purely analytical approach. This aspect enables the essential confidence on the final converged results accuracy, making this type of approach particularly suitable for the confirmation and/or generation of benchmark results for different classes of problems in heat and fluid flow. Among various other contributions, of specific relevance to the application here considered, it is worth mentioning the integral transform solutions of the Navier–Stokes equations under the streamfunction-only formulation, for incompressible flow within cavities and channels [2], [3] and natural convection under the Boussinesq approximation inside rectangular enclosures for both steady and transient states [4], [5]. In addition, natural convection within porous rectangular enclosures was accurately solved through the same integral transform approach [6].

In 1983, De Vahl Davis [7] provided the first set of benchmark solutions for the steady natural convection in a enclosed square cavity with differentially heated vertical walls and insulated top and bottom walls, utilizing a second-order finite differences method and the Richardson extrapolation scheme. Following this pioneering work, different solution strategies have been reported in the literature. Among the most relevant works to our present objectives, one can cite Saitoh and Hirose [8], who utilized a non-conservative fourth-order finite differences approach in 1989, Hortmann et al. [9], who have made use of the finite volume method in 1990, and Lé Queré [10], who employed a pseudo-spectral Chebyshev algorithm to provide accurate solutions to values of Rayleigh number from 10⁶ to 10⁸, in 1991.

In contrast with steady natural convection, transient analysis of natural convection in a cavity has received much less attention in the literature, despite its scientific and technological relevance. A brief literature review includes the contributions of Wilkes and Churchill [11] in 1966, who provided a pioneering transient analysis using an implicit alternating direction (ADI) finite difference method, Patterson and Imberger [12], in a classical work, who used a simple scale analysis to give some insight into the possible transient behavior and obtained a number of numerical solutions using a modified version of the finite difference method proposed by Chorin [13]. More recently, Sai et al. [14] presented solutions for the transient problem in the Rayleigh number range of 10³–10⁶ by the application of the finite element method based on the first-order projection scheme, which is an extension of Chorin's algorithm. Ramaswamy et al. [15] applied a semi-implicit projection-type finite element method to perform two numerical tests, oscillatory cavity flow with heat transfer and transient buoyancy-driven flow in a square cavity.

Parallel to these research efforts, and also in recent years, the influence of the fluid properties variation with temperature has appeared as an important aspect to be analyzed in this class of problems. The well-known Boussinesq approximation has been extensively employed, but very little research has been undertaken to inspect the influence of variable thermophysical properties in the flow structure, with or without the Boussinesq simplification. Among other researches of relevance to our present purposes, Bergles [16] presented correlation formulae to compute the influence of each property (viscosity and conductivity) for forced convection in tubes, considering incompressible flow. Gray and Giorgini [17] studied the limit of application of the Boussinesq approximation to external flows of water and air, using two orders of approximation: strict and extended. They presented also graphics to indicate accurate limits to those hypothesis. The stability and the limits of the Boussinesq approximation were also the subject in the works of Graham [18] and Spradley and Churchill [19], both using the finite difference method to compute the lid-driven cavity problem for a compressible fluid with variable properties. Suslov and Paollucci [20] reproduced and extended the results of these works, aimed at finding the critical Rayleigh number, and at showing the presence of two regimen of instability, one of them due to the non-Boussinesq effects. Yu et al. [21] tried to present a benchmark for the compressible problem (the lid-driven cavity), using finite element analysis, and handling the common limitations of this method when applied to low-Mach number compressible flows. Finally, Zhong et al. [22] revised the work of Graham [18], centering their study in the validity of the Boussinesq approximation. They found a more strict limit than the one presented in [17], despite of the good agreement achieved for Nusselt number calculations.

Here, the results obtained for variable properties will be inspected within and outside the limits predicted in [22], considering variations of dynamic viscosity, thermal conductivity, specific heat and density in the body force term only. Thus, maintaining the Boussinesq approximation, the temperature variation effects, on the flow structure, of all other thermophysical properties will be analyzed. We first reproduce through integral transformation, the transient natural convection solutions under the Boussinesq approximation and constant properties to reconfirm some previously reported reference results, for which there is an evident lack of formal confidence on their global accuracy control. The method is then validated and the solutions are compared with those obtained for the variable properties situation, when the very desirable hybrid characteristics of the GITT are explored in a situation of marked non-linear effects, in automatically controlling the global error in the simulation.