^{1, 2}

^{1}

^{1}

^{2}

The effects of various parameters, Rayleigh number (Ra), Darcy number (Da), and
wave amplitude (^{4}, 10^{5}, and 10^{6}, Da = 10^{−5}, 10^{−4}, and 10^{−3}, and

The study of natural convection in an enclosure has received significant attention because the convection motion plays an important role in thermal characteristic and can be found in many applications. Due to a large number of technical applications, for example, geophysics, geothermal reservoirs, insulation of building, crude oil production, separation processes in industries, and so on, many researchers have studied natural convection inside the different shapes of enclosures, triangle, rectangle, nonrectangle, or irregular enclosure like wavy surface, with various boundary conditions to analyze the fluid flow and thermal behaviors. Triangular enclosure can be used in the application of roof structure in order to study the natural convection under summer and winter day boundary conditions which are found in Asan and Namli [

For a complicated enclosure with wavy surface, Das and Mahmud [

The objective of this study is to investigate the flow field and temperature distribution due to natural convection inside the trapezoidal enclosure with wavy top surface where the enclosure is heated from bottom and discrete heat sources are located on inclined boundaries by using finite element method. The physical model and mathematical formulation are described in Section

The following notations are applied throughout the paper.

Da: Darcy number

Pr: Prandtl number

Ra: Rayleigh number

Figure

Physical model and boundary conditions of the enclosure with top wavy surface.

For the boundary conditions, it is assumed that there is no slip on boundary. Thus, the velocities in

on the left,

To analyze the effects on the flow and temperature characteristic, the governing equations mentioned previously are coded into FlexPDE which is a software package performing the operations necessary to turn a description of a partial differential equations system into a finite element model. Then, it solves the system and presents graphical and tabular outputs of the results. The results obtained from varying values of interested parameters are displayed by streamlines and isotherms in which graphical outputs from FlexPDE are modified to show numerical values.

Figures

Streamlines (left) and isotherms (right) for

Streamlines (left) and isotherms (right) for

The effects of increasing Ra on temperature distribution are illustrated by isotherms as shown in the second column of Figure

Streamlines and isotherms for different Da with constant

The results obtained from changing wave amplitude are shown in Figure

Streamlines (left) and isotherms (right) for

The effects of interested parameters, Rayleigh number, Darcy number, and wave amplitude on natural convection inside the trapezoidal enclosure with wavy top boundary are investigated. The values of parameters have been chosen based on wide range of applicability,

the magnitudes of the flow and temperature distribution are small for

at

the result of a variation in Ra shows similar trend with a variation in Da; that is, the fluid circulation at constant Da (

fluid circulation is not found at the top corners of the enclosure for all cases,

the increase of wave amplitude from 0.9 to 1.1 has no effect on flow intensity but it affects the flow pattern due to the inner wave,

the wavy top surface has small influence on temperature distribution compared to the influence of Ra and Da.

This research is (partially) supported by Centre of Excellence in Mathematics, the Commission on Higher Education, Thailand. The authors would like to thank Department of Mathematics, Faculty of Science, Khon Kaen University (Thailand), for computational resources necessary for this work.