Mathematical Problems in Engineering

Volume 2017, Article ID 3834170, 11 pages

https://doi.org/10.1155/2017/3834170

## Lattice Boltzmann Simulation of Natural Convection in an Annulus between a Hexagonal Cylinder and a Square Enclosure

Faculty of Sciences Semlalia, Cadi Ayyad University, LMFE, BP 2390, 40000 Marrakech, Morocco

Correspondence should be addressed to Z. Zrikem; am.ca.acu@mekirz

Received 1 February 2017; Accepted 6 April 2017; Published 18 June 2017

Academic Editor: Arturo Pagano

Copyright © 2017 L. El Moutaouakil et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Laminar natural convection in a water filled square enclosure containing at its center a horizontal hexagonal cylinder is studied by the lattice Boltzmann method. The hexagonal cylinder is heated while the walls of the cavity are maintained at the same cold temperature. Two orientations are treated, corresponding to two opposite sides of the hexagonal cross-section which are horizontal (case I) or vertical (case II). For each case, the results are presented in terms of streamlines, isotherms, local and average convective heat transfers as a function of the dimensionless size of the hexagonal cylinder cross-section (), and the Rayleigh number ().

#### 1. Introduction

The numerical simulation of natural convection of various fluids in different geometries has become the most widely used approach because of its low cost and the accuracy of its predictions. This is justified by the availability of reliable mathematical models and robust numerical methods that predict the fluid flow and heat transfer characteristics in practically all situations encountered. The lattice Boltzmann method is relatively new compared to the classical approaches based on the Navier-Stokes equations. This technique has been used with a great success to simulate different physical behaviors of magnetohydrodynamic fluids [1, 2], inhomogeneous mediums [3, 4], phase change materials [5, 6], flows with chemical reactions [7], and porous media [8, 9].

The lattice Boltzmann method is easier to implement for the numerical simulation of laminar natural flows in the complex geometries. Thus, it has been successfully implemented to predict the flow structures and heat transfers in enclosures of different shapes [10–12]. But its performances are even more remarkable in the simulation of natural convection in the annulus between a rectangular enclosure and a cylinder of circular cross-section [13–20], square cross-section [21–23], or elliptical cross-section [24]. However, one finds in the literature many other numerical works that treated the natural convection in such geometries by using the Navier-Stokes equations [25–28]. Generally, the system is heated by the cylinder and cooled by the cavity walls which are maintained at the same temperature.

In these studies, the effects of the shape, size, and position of the heating body on the dynamic and thermal fields in the cavity are analyzed. Thus, Hussain and Hussein [27] studied the effect of the position of a circular cylinder inside a square enclosure. These authors have shown that the position of the horizontal cylinder has an influence on the convective heat transfer which is proportional to the Rayleigh number. Concerning the size effect, it has been studied by Moukalled and Acharya [25] which showed that the average heat transfer is proportional to the cross-section size of the cylinder placed in the center of the cavity. The simultaneous effects of the size and position of a circular cylinder inside a square cavity were analyzed by Ding et al. [13]. The results found indicate that these two parameters have a great influence on the dynamic and thermal fields in the annulus between the circular cross-section and the cavity walls. The effect of the cross-section geometry of the heating cylinder on natural convection within a square cavity is studied by Ravnik and Škerget [28]. These authors found that a cylinder of elliptical cross-section slightly improves the heat transfer through the walls of the cavity.

The literature review showed that the case of a horizontal hexagonal cylinder inserted in a square enclosure has not been studied. For this purpose, the present work is dedicated to the lattice Boltzmann simulation of two-dimensional natural convection around a horizontal hexagonal cylinder placed in a square cavity filled with water. The heating cylinder is at a hot temperature , while the cavity walls are kept at a cold temperature . The effects of the Rayleigh number, the size and orientation of the hexagonal cylinder on the streamlines, isotherms, and local and average heat transfers are presented and discussed below.

#### 2. Mathematical Formulation

The system under consideration is a square cavity (), filled with water (), and having four walls kept at a cold temperature . A cylinder of hexagonal cross-section of side is placed in the center of the cavity and maintained at a hot temperature . The hexagon dimensions are such that . Two orientations are considered, corresponding to two opposite sides of the hexagonal cross-section which are horizontal (case I) or vertical (case II) (Figure 1). The dimensions of the cavity and the temperature difference () are varied so that .