# Numerical Study of the Effect of a Heated Cylinder on Natural Convection in a Square Cavity in the Presence of a Magnetic Field

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## Abstract

**:**

^{3}–10

^{5}), the Hartmann number (Ha = 0–200) at Pr = 0.71 on the flow field, thermal pattern and the variation of heat inside the enclosure. The clarifications of the numerical result were exhibited in the form of streamlines, isotherms, velocity profiles and temperature profiles, local and mean Nusselt number, along with heated cylinder configurations. From the obtained outcomes, it was observed that the rate of heat transport, as well as the local Nusselt number, decreased for the LBC and LTC configurations, but increased for the RBC and RTC configurations with the increase of the Hartmann number within the square cavity. In addition, the mean Nusselt number for the LBC, RBC, LTC and RTC configurations increased when the Hartmann number was absent, but decreased when the Hartmann number increased in the cavity. The computational results were verified in relation to a published work and were found to be in good agreement.

## 1. Introduction

## 2. Problem Definition

_{0}) with the y-axis, was used in the present model. The dimension of the cavity was defined by its height (H) and length (L). The gravitational force (g) always worked in the vertically downward direction. The left and right walls of the cavity were thermally insulated (T

_{i}). The base wall of the cavity was considered to be at a uniform hot temperature (T

_{h}) and the top wall was maintained at a cold temperature (T

_{c}), where T

_{h}> T

_{c}. Furthermore, a heated cylinder of a diameter D was placed in various positions within the square cavity. The diameter of the cylinder was made to be one third of the cavity’s height. The electrically conductive fluid with Pr = 0.71 [10] was placed in the square cavity and the flow of fluid was thought to be Newtonian and laminar. In addition, stable fluid properties were seen, and the boundary walls of the cavity were no-slip.

## 3. Mathematical Modeling

_{p}, g, and ΔT.

_{local}) and mean Nusselt number (Nu

_{av}) on the heated part of the cavity, were determined as follows:

## 4. Numerical Details

_{x}, S

_{y}) alongside the outflow boundary S

_{0}and (11) specifies the components of velocity and heat flux (q

_{w}), which flows into or out from field alongside S

_{w}.

_{0}and S

_{w}as:

^{5}, which had been solved for streamlines (stream function) and isotherms for 2D magneto-hydrodynamic free convection flow through the square cavity. The result was checked for streamlines and isotherms and then the present work was compared with the reported reference of Jani et al. [10] and presented in Figure 2. From the above comparisons of the figures, we found a good agreement between the present work and Jani et al. [10], which is displayed in Figure 2. Furthermore, mesh configuration is a technique in which a large domain is subdivided into a set of sub domains called finite elements, control volume and so on. A lot of boundary value problems of several engineering fields have been solved with the aid of irregular geometry via a set of finite elements. The answer for the current geometry for the specific non-dimensional parameters was computed at discrete locations called numerical grids. The mesh structure for the current problem is provided in Figure 3.

^{5}, considering assorted size of mesh. The manifest meshing is shown in Table 1 and Figure 4, where the average Nusselt number is calculated. It was found that further increments of Nu

_{av}have insignificant transform. Throughout the study, for 23,780 nodes and 3568 elements, the mesh configuration was chosen for accurate simulation to find the optimized, desired result in the present study.

## 5. Results and Discussion

#### 5.1. Effect of Cylinder Position and Magnetic Field on Streamlines and Isotherms

^{3}–10

^{4}, correspondingly, for flow and thermal field in Figure 5 and Figure 6. The impact of the presence of a magnetic field for streamlines and isotherms is also demonstrated in Figure 5a and Figure 6, respectively, for cavity configuration (LBC). Figure 5a shows that one eddy circulation cell formed inside the cavity. The flow strength decreases and streamlines close to the heated cylinder configurations due to the enhancement of the Hartmann number, which is shown in Figure 5b–d. The effect of the Hartmann number (Ha = 0–200) on the distributions of the velocity and temperature contours for right bottom configuration (RBC), while Ra = 10

^{3}and Pr = 0.71 is also shown in Figure 5. A tiny recirculation cell appeared in the center of the square of the cavity and the recirculation cell was smaller, owing to the increase in the Hartmann number, which is shown in Figure 5. Figure 5 also illustrates the streamlines for the left top heated cylinder configurations (LTC), along with variations of the Hartmann number (Ha = 0–200), when Ra = 10

^{3}and Pr = 0.71. Figure 5 shows in the LTC configuration that one cell was created inside the center of the square cavity in the absence of magnetic field. In addition to this, one large vortex also formed in the left bottom side of the cavity. The cell became bigger and oval shaped in the cavity with the increase of the Hartmann number and also, a tiny vortex was found in the left top side of the cavity. The variation of the Hartmann number for the right top cylindrical heat source (RTC) configuration is shown in Figure 5 for the square cavity. It can be seen that the smaller cell was formed in the square cavity due to both the absence and presence of Ha, compared with the LTC. Figure 6 shows that the isotherms for the left bottom heated cylinder configuration (LBC) are likely linear, as well as nonlinear close to the upper wall and base wall, correspondingly, with the increase in the Hartmann number when Ra = 10

^{3}and Pr = 0.71 (see Figure 6a–d). Furthermore, Figure 6a–d shows the thermal increases, owing to the increase of the magnetic field parameter: the Hartmann number (Ha). The temperature distributions for the right bottom cylindrical heat source configuration (RBC) with magnetic field’s effect on the parameter Hartmann number (Ha = 0–200) is shown in Figure 6a–d for fixed Ra = 10

^{3}and Pr = 0.71. The isotherms were parallel to the upper wall of the cavity. On the other hand, a nonlinearity effect was found near to the base wall of the cavity. The isotherms for LTC shown in Figure 6 were almost as linear as those near to the top wall. However, bend isotherms could be seen near the base wall through the effect of the Hartmann number. By increasing the Hartmann number, it could be seen that the isotherms in the RTC transform slightly in the cavity, as shown in Figure 6. When the Rayleigh number increased, that is, for higher Ra = 10

^{5}, streamlines and isotherms were analyzed, as shown in Figure 7 and Figure 8 for various configurations of heated cylinders (LBC, RBC, LTC and RTC) within the square cavity for Ha = 0–200 and Pr = 0.71. As shown in Figure 7, by analyzing all configurations of the heated cylinders (LBC, RBC, LTC and RTC), it can be understood that one primary larger eddy circulation cell was created inside the cavity when Ha = 0. However, due to the increase in the Hartmann number (0 ≤ Ha ≤ 200), the velocity flow strength dwindled. Therefore, likely larger secondary recirculation cells with tiny vortices were created inside the square cavity. At higher Ra = 10

^{5}–10

^{3}and when Pr = 0.71, it is shown in Figure 8 that isotherms for every arrangement of heated cylinders (LBC, RBC, LTC and RTC) looked parallel and non-parallel, respectively, near to the upper and beneath wall of the cavity for the impact of magnetic field Ha = 0–200. However, due to the increased Hartmann number and strength of flow of convection, more compacted and non-parallel isotherm lines were seen in the cavity. In addition to this, fewer bond isotherm lines were also observed near the side walls of the cavity.

#### 5.2. Velocity and Temperature Profiles

#### 5.3. Heat Transfer

## 6. Conclusions

- ■
- The distributions of flow field and isotherm patterns, velocity and temperature profiles, rate of heat transport for various cylinder configurations within the cavity fully depended on the Prandlt number (Pr), the Rayleigh number (Ra) and the Hartmann number (Ha) and the heated bottom wall of the cavity.
- ■
- The number of vortices increased within the streamlines for various configurations of the cavity due to enhance of the Hartmann number.
- ■
- The bonding of isotherm lines reduced close to the side walls of the cavity.
- ■
- The bend isotherm lines were observed adjacent to the base wall of the cavity.
- ■
- The velocity decreased for each heated cylinder configurations (LBC, RBC, LTC and RTC), as well as for the increasing value of the Hartmann number below the central portion of the cavity, but the velocity increased with the decrease in the Hartmann number.
- ■
- For the LBC and LTC configurations, the local Nusselt number decreased with the increase of the Hartmann number, but for the RBC and RTC configurations, the local Nusselt number increased with the increase in the Hartmann number.
- ■
- The mean Nusselt number for the LBC, RBC, LTC and RTC configurations increased due to the absence of the Hartmann number, but the mean Nusselt number decreased due to the increase in the Hartmann number.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

B_{0} | Magnetic field |

Cp | Specific heat at constant pressure (J/kg·K) |

g | Gravitational acceleration (m/s^{2}) |

h | Convective heat transfer coefficient (W/m^{2}·K) |

Ha | Hartmann number |

k | Thermal conductivity of fluid (W/m·K) |

K | Thermal conductivity ratio fluid |

N | Non-dimensional distance |

Nu_{av} | Mean Nusselt number |

Nu_{local} | Local Nusselt number |

P | Non-dimensional pressure |

p | Pressure |

Pr | Prandtl number |

Ra | Rayleigh number |

T | Non-dimensional temperature |

U | Dimensionless horizontal velocity |

u | Velocity in x-direction (m/s) |

V | Dimensionless vertical velocity |

v | Velocity in y-direction (m/s) |

x, y | Cartesian coordinates |

X, Y | Dimensionless Cartesian coordinates |

Greek symbols | |

α | Thermal diffusivity (m^{2}/s) |

β | Coefficient of thermal expansion (K^{−1}) |

θ | Temperature of fluid |

∆θ | Discrepancy of temperature |

μ | Dynamic viscosity of the fluid (Pa·s) |

ν | Kinematic viscosity of the fluid (m^{2}/s) |

r | Fluid density (kg/m^{3}) |

σ | Fluid electrical conductivity (Ω^{−1}m^{−1}) |

Abbreviations | |

LBC | Left bottom heated cylinder |

LTC | Left top heated cylinder |

RTC | Right top heated cylinder |

RBC | Right bottom heated cylinder |

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**Figure 2.**Comparison of streamlines and isotherms of by Jani et al. [10] and the present work with Pr = 0.71, Ha = 50 and Ra = 10

^{5}.

**Figure 5.**Streamlines for different orientations of heated cylinders for Ha = 0–200, Ra = 10

^{3}and Pr = 0.71.

**Figure 6.**Isotherms for different orientations of heated cylinders for Ha = 0–200, Ra = 10

^{3}and Pr = 0.71.

**Figure 7.**Streamlines for different orientations of heated cylinders for Ha = 0–200, Ra = 10

^{5}and Pr = 0.71.

**Figure 8.**Isotherms for different orientations of heated cylinders for Ha = 0–200, Ra = 10

^{5}and Pr = 0.71.

**Figure 9.**Variations of velocity vs. distance for different orientations of heated cylinders for Ha = 0–200, Ra = 10

^{5}and Pr = 0.71.

**Figure 10.**Variations of temperature with distance for different orientations of heated cylinders for Ha = 0–200, Ra = 10

^{5}and Pr = 0.71.

**Figure 11.**Variations of local Nusselt number with distance for different orientations of heated cylinders for Ha = 0–200, Ra = 10

^{5}and Pr = 0.71.

**Figure 12.**Variations of average Nusselt number with the Hartmann number for different orientations of heated cylinders for Ha = 0–200, Ra = 10

^{5}and Pr = 0.71.

Nodes | 16,030 | 19,099 | 21,560 | 23,780 | 32,945 | 37,682 |

Elements | 2408 | 2878 | 3258 | 3568 | 4978 | 5696 |

Nu_{av} | 0.130212 | 0.130203 | 0.137988 | 0.141502 | 0.141502 | 0.1415 |

Time (s) | 15.913 | 19.308 | 22.568 | 26.879 | 36.5135 | 38.495 |

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**MDPI and ACS Style**

Hossain, M.S.; Fayz-Al-Asad, M.; Mallik, M.S.I.; Yavuz, M.; Alim, M.A.; Khairul Basher, K.M.
Numerical Study of the Effect of a Heated Cylinder on Natural Convection in a Square Cavity in the Presence of a Magnetic Field. *Math. Comput. Appl.* **2022**, *27*, 58.
https://doi.org/10.3390/mca27040058

**AMA Style**

Hossain MS, Fayz-Al-Asad M, Mallik MSI, Yavuz M, Alim MA, Khairul Basher KM.
Numerical Study of the Effect of a Heated Cylinder on Natural Convection in a Square Cavity in the Presence of a Magnetic Field. *Mathematical and Computational Applications*. 2022; 27(4):58.
https://doi.org/10.3390/mca27040058

**Chicago/Turabian Style**

Hossain, Muhammad Sajjad, Muhammad Fayz-Al-Asad, Muhammad Saiful Islam Mallik, Mehmet Yavuz, Md. Abdul Alim, and Kazi Md. Khairul Basher.
2022. "Numerical Study of the Effect of a Heated Cylinder on Natural Convection in a Square Cavity in the Presence of a Magnetic Field" *Mathematical and Computational Applications* 27, no. 4: 58.
https://doi.org/10.3390/mca27040058