The Navier-Stokes equations represent the fundamental governing equations for fluid flow. The incompressible Navier-Stokes equations in conservation form are

$\frac{\partial U_i}{\partial x_i}=0$

$\frac{\partial U_i}{\partial t}+U_j\frac{\partial U_i}{\partial x_j}=-\frac{1}{\rho }\frac{\partial P}{\partial x_i}+\frac{\partial }{\partial x_j}\left[v\left(\frac{\partial U_i}{\partial x_j}+\frac{\partial U_j}{\partial x_i}\right)+{\tau }_{ij}\right]$

The quantity ${\tau }_{ij}=-\overline{{u^{\prime }}_i{u^{\prime }}_j}=v_t\left(\frac{\partial U_i}{\partial x_j}+\frac{\partial U_j}{\partial x_i}\right)-\frac{2}{3}K{\delta }_{ij}$ is known as the Reynolds stress tensor, which is symmetric and $v_t$ is the kinetic eddy viscosity assumed as an isotropic scalar quantity.

$\frac{\partial U_i}{\partial t}+U_j\frac{\partial U_i}{\partial x_j}=-\frac{1}{\rho }\frac{\partial P}{\partial x_i}+\frac{\partial }{\partial x_j}\left[v\left(\frac{\partial U_i}{\partial x_j}+\frac{\partial U_j}{\partial x_i}\right)+v_t\left(\frac{\partial U_i}{\partial x_j}+\frac{\partial U_j}{\partial x_i}\right)-\frac{2}{3}K{\delta }_{ij}\right]$

In radiation problems, the conservation of energy is given by

$\frac{\partial T}{\partial t}+U_j\frac{\partial }{\partial x_j}\left(T\right)=\frac{k}{\rho C_p}\frac{{\partial }^2}{\partial x_i^2}\left(T\right)+\frac{1}{\rho C_p}{\stackrel{\dot{}}{Q}}_i$

where $T$ is the temperature of the fluid, $k$ is the thermal conductivity, $C_p$ is the specific heat of the fluid at constant pressure and ${\stackrel{\dot{}}{Q}}_i$ is the net heat transfer from surface $i$.

Radiation heat transfer between surfaces depends on the orientation of the surfaces relative to each other as well as their radiation properties and temperatures. Consider an enclosure consisting of $N$ black surfaces maintained at specified temperatures. For each surface $i$, we can write

${\stackrel{\dot{}}{Q}}_i=\underset{j=1}{\overset{N}{{\displaystyle \sum }}}{\stackrel{\dot{}}{Q}}_{ij}=\underset{j=1}{\overset{N}{{\displaystyle \sum }}}{A}_i{F}_{ij}\sigma \left(T_i^4-T_j^4\right)$

where $A_i$ is surface area and $F_{ij}$ indicates the fraction of the radiation leaving surface $i$ that strikes surface $j$ directly.

In this study, non-black surfaces are considered. In an N-surface enclosure, the conservation of energy principle requires that the net heat transfer from surface i to be equal to the sum of the net heat transfers from i to each of the N surfaces of the enclosure.

${\stackrel{\dot{}}{Q}}_i=\underset{j=1}{\overset{N}{{\displaystyle \sum }}}{\stackrel{\dot{}}{Q}}_{ij}=\underset{j=1}{\overset{N}{{\displaystyle \sum }}}\frac{J_i-J_j}{R_{ij}}$

where $R_{ij}=\frac{1}{A_i{F}_{ij}}$ is called the space resistance to radiation and $J_i$ and $J_j$ are the radiosity for the surface $i$ to $j$. The standard k-ε turbulence model was used in this study because the flow is fully turbulent and the effects of molecular viscosity are negligible [7] . The turbulent kinetic energy, $k$, and its rate of dissipation, $\epsilon$, in the flow field are calculated from two additional transport equations. The model transport equation for $k$ is derived from the exact equation, while the model transport equation for $\epsilon$ was obtained using physical reasoning and bears little resemblance to its mathematically exact counterpart. The k-ε transport equations [7] are:

$\frac{\partial k}{\partial t}+U_j\frac{\partial k}{\partial x_j}=\frac{\partial }{\partial x_j}\left[\left(\nu +\frac{{\nu }_t}{{\sigma }_k}\right)\frac{\partial k}{\partial x_j}\right]+\frac{1}{\rho }{P}_k-\epsilon$

$\frac{\partial \epsilon }{\partial t}+U_j\frac{\partial \epsilon }{\partial x_j}=\frac{\partial }{\partial x_j}\left[\left(\nu +\frac{{\nu }_t}{{\sigma }_{\epsilon }}\right)\frac{\partial \epsilon }{\partial x_j}\right]+C_{\epsilon 1}\frac{\epsilon }{k\rho }{P}_k-C_{\epsilon 2}\frac{{\epsilon }^2}{k}$

where $P_k$ is the production rate of turbulent kinetic energy, which depends on the turbulent viscosity and the velocity distribution. In Table 1 , the values of all the parameters used by COMSOL Multiphysics are provided.

In this study, the CFD software based on the finite element method (Galerkin weighted residual method) has been used to solve the set of equations provided by k-ε models. In the case of vertical wind direction, the computational domain was constructed that had a height of $4H$, width of $9H$ and length of $13H$ where $H=0.25\text{m}$. The ability to accurately model outdoor airflow around buildings is necessary to provide the correct boundary conditions for the indoor building environment that is naturally ventilated. As far as the boundary conditions are concerned, at the inlet of upwind boundary, a logarithmic profile of the stream wise velocity component $U$ has been applied: $U\left(h\right)=\frac{U_0}{K}\ln \left(\frac{h}{h_0}\right)$; where $h$ is the distance from the ground, $K=0.41$ is the Von Karman’s constant, $U_0$ and $h_0$ have been determined through the best fit of the available data and experimental data. The velocity components along with vertical and span wise direction were equated to zero. In addition, as the distribution of $k$ and $\epsilon$ on the inlet boundary was unknown, the following relations have used: $k=\frac{3}{2}\left(U_{avg}\cdot T_i{}^2\right)$; $\epsilon =c_{\mu }^{3/4}\frac{k^{3/2}}{l_t}$; where $U_{avg}$ is the average flow velocity, $T_i$ is the turbulence intensity and $l_t$ is the turbulence length scale. To solve turbulence conditions, $T_i=1\text{\%}$ [25] and $l_t=0.4\text{m}$ [7] have been used in this research. A constant pressure is assumed at the outlet of the computational domain, while the gradients of all the dependent variables are assumed to vanish. The ambient conditions at the domain boundary were represented by specifying ambient pressure of 1 atm and temperature of 33˚C. For all cases, 33˚C heated air entered into the region through the inlet. Standard wall functions were used and the no-slip velocity boundary condition was applied for the fluid-wall interaction. The wall of the room is modelled as heat-transmitting concrete material with emission $\epsilon =0.85$. All objects except the transparent window are modelled as grey bodies. Owing to the ellipticity of the earth’s orbit, the actual solar constant changes throughout the year within ±3.4%. As a result, the solar radiation reaching the earth’s surface is about 950 W/m² on a clear day and much less on a cloudy day, in the wavelength band 0.3 to 2.5 µm. The thermal expansion coefficient of the air is based on the ambient temperature. The commercial software COMSOL Multiphysics 5.2 based on finite element method is used for computation.

Physics-controlled mesh is prepared using COMSOL Multiphysics Meshing. The element sizes for this meshing are different, such as fine, extra fine, extremely fine, finer, coarse, coarser, extra coarse, extremely coarse, and normal. The number of mesh depends on different mesh types and element sizes. In 2D elements, triangular and rectangular are used and tetrahedral, Pyramidal, Prismatic and hexahedral are used for 3D elements. Table 2 shows that extremely coarse is suitable to use in this study. Total number of mesh for 3D cases is around 35,000 - 40,000, whereas 1800 - 2300 for 2D cases. In Figure 7 , the computational domain is meshing with total number of elements around 35,000 to 40,000.

Wind-driven single-sided ventilation experiments were performed in the present study. In this validation study, CFD has been used to analyze single-sided ventilation by studying wind-driven windward flow through double opening. A single-sided ventilation experiment was used to validate the CFD model [11] . For the CFD simulation, the building model was set up and placed within a larger computational domain. After setting up the CFD model according to the experimental setup, the simulation was performed. In Figure 8 , three vertical lines in the middle section of the domain A, B, and C are identified. Locations A and C have been chosen in order to describe the flow pattern close to the openings.

The velocity components have been determined, along streamwise and vertical direction, respectively. The third velocity component in x-axis has been neglected as it is supposed to be zero, due to the symmetry of domain. Figure 9(a) and Figure 9(b) show the velocity field results for the experiment model and CFD modeling results respectively, at every section of the building. The plots of Figure 9 show the vertical distribution of x-direction velocity U for ventilation patterns evaluated. Table 3 shows the mean absolute error analysis of velocity distribution of cubic room. From these validation cases, it is clear that the agreement between experimental results and CFD modeling for the interaction between outdoor and indoor flow in single-sided ventilation is very good.

The aim of this study is to examine the wind profiles inside and outsides of the buildings through single/double opening(s). In this study, four physical models have been used in context of weather of Dhaka city. To compare with others results, three vertical lines are considered (from A to C), shown in Figure 8 . Meteorological wind speeds are varying from 2 - 4 m/s in Dhaka city. Initially, at a low 2 m/s wind speed, buoyancy effects were dominant and consequently drove air in through the lower opening and out through the upper opening. At a wind speed of 4 m/s, this buoyancy effect diminished as the wind-driven force through the upper opening increased. The buoyancy and wind forces were approximately equal at this wind speed, resulting in the minimum ventilation rate over the range of wind speed.

In this analysis, the flow pattern inside the building as well as computational domain for all cases are considered. To show the flow pattern, air flow distribution and velocity distribution are essential. Figures 10-14 show the air flow distribution and velocity distribution in computational domain and cubic building for all cases at 3^rd second. It was observed that the velocities change from time to time. After a few seconds, the changes are going to come down, which is approximately 3 seconds at that time-dependent results bear a physical resemblance to steady. Air flow distribution and stream lines are almost same. Since, after 3^rd second, the changes are not worth watching, so the vertical velocity is calculated for 3^rd second. Section-based comparison of streamwise velocity and vertical velocity for 3^rd second is shown in Figure 10 and Figure 11 respectively. In computational domain, the behavior of streamwise velocity distribution for all cases is shown in Figure 12 . The difference is evident in the regions close to horizontal surfaces. Figure 13 shows the velocity distribution for all cases in xy-plane at time (t) = 3 sec. It is interesting to notice that leeward ventilation seems to produce an air movement inside the enclosed space more significant than windward ventilation, especially in the region close to the opening. In the windward single opening (Case 1), the air enters into the upper side of the opening and comes out through the lower side of that opening, but for the leeward single opening (Case 2), the air enters into the lower side of the opening and comes out through the upper side of that opening. Also, in the windward double openings (Case 3), the air enters into the upper opening

and comes out through the lower opening, but for the leeward double openings (Case 4), the air enters into the building through the lower opening and comes out through the upper opening. Figure 14 shows the air flow distribution of the cubic room.

Above the roof of the building (height > 0.25), the velocity components U and V for all cases are similar in every section but in the cubic building, directions and velocity components are some disagreements in few sections. To choose the location of the opening(s) and description of the velocity profile inside the building and near the openings is the main purpose of this study. All the numerical results have been obtained by k-ε model.

Temperature is the main component for thermal comfort. The information about temperature profile in cubic buildings, as well as computational domain, is provided in this section. Ambient temperature is considered to calculate the temperature distribution. The graphs in Figure 15 display the temperature distribution profile for all (four) cases in Section A, Section B and Section C respectively at

time (t) = 3 sec. An interesting observation is that air temperature is fluctuating so much for windward opening(s) (Case 1 and Case 3) in Section A because hot air that exists outside the room is entered easily into the room but the room temperature is comparatively cool. This cold temperature influences the outside temperature when air comes out of the room. From the figure, it is clear that the air temperature fluctuation is decreasing in Section A when time is increasing gradually. For every case, the outside temperature of the room is around 306 K where the minimum and the maximum room temperature is approximately 293 K and 300 K respectively. In all cases, the low temperature is existing in the floor. When time is increasing, the roof of the cubic building is heated gradually for a certain period, which gives rise to the room temperature. However, the ASHRAE Standard 55 [26] states that occupants may feel uncomfortable due to contact with floor surfaces that are too warm or too cool, giving an allowable range of floor temperatures between 292 K and 303 K. In this study, only solar radiation effect is regarded but no other heat source or mechanical system is accounted here. Since room temperature is a vital factor so average volume temperature is needed. Table 4 shows the details of the average volume temperature for all cases at different times. The average volume temperature for double windward openings and double leeward openings is much closer. At different times, the average volume temperature for double leeward opening is comparatively less than the double windward openings.

Bangladesh is situated in both the eastern and northern hemispheres. So, wind speed and wind direction is an important part for room temperature. Since this research is time-dependent and also related to solar radiation, so sunlight is necessary. In this study, a clear sunny day at 12:00 pm is considered to calculate ventilation rate for a cubic room that is filled with air. The air flow rate for all cases is shown in Table 5 . It is observed that maximum and minimum ventilation rates occur in double windward openings and single leeward opening respectively. From this table, it is clear that double windward openings allow more air to enter into the room than other openings and that’s why more ventilation rate is accounted.

Air-Change-per-Hour (ACH) value is calculated for each case so that the ventilation findings can be compared with indoor air-quality standards. The ACH value for single windward, single leeward, double windward and double leeward are 6.451, 3.072, 10.368 and 5.606 respectively.

Radiosity is the summation of the reflected and the emitted radiation. Since all objects except the transparent window are modelled as grey bodies, so radiosity is another parameter which is defined as the rate at which radiation leaves a given surface per unit area. In this study, the lower surface of the computational domain and building are considered as sand and concrete materials, respectively. Figure 16 shows the surface radiosity for all cases of computational domain and a cubic room, respectively. Figures show that the range of the radiosity is 353 to 405 for the room, as well as computational domain at 12:00 pm. In a cubic room, two walls and the floor are free from surface radiosity because these walls are situated in opposite side of the sunlight.

The thermal comfort of the occupant within the indoor environment is an important for indoor air quality. Thermal comfort essentially means the condition of mental satisfaction with the thermal environment. In practice, it is not necessary to provide thermal comfort that satisfies each person. Instead, a level of thermal comfort that can satisfy the majority of occupants should be provided. The indoor thermal environment also influences the perception of air quality and can directly or indirectly affect occupant health. Thus, the strategies and methods for maintaining suitable indoor environment conditions, such as indoor air quality and thermal comfort, vary for different locations as well. After the effect of solar radiation, it is clear that in all cases, air temperature inside the room maintains the ASHRAE [26] standard in the context of the weather in Dhaka city.