4.4 Results and Discussion
4.4.1 Effect of Rayleigh number
Streamlines (on the left column) and isotherms (on the right column) for horizontal position of solid strip and Ha = 0 (no magnetic field) are presented in Figure 4.3 (Ra=103 to 106) to understand the effects of Rayleigh number on flow field and temperature distribution. At Ra = 103 and Ha = 0 (in the absence of the magnetic field) two vortices are formed, one of them is above and another is below of the solid strip in the enclosure. At Ra = 104 and 105 the result is almost same but here we observed that the flow strength gradually increases. For higher Rayleigh number i.e. at Ra=106, it is observed that the upper vortex is bigger than lower one and both of the vortices are shifted to the right cold wall. We also observed that for higher Rayleigh numbers, the shape of vortices are changed and flow strength increased significantly. Stream function has symmetrical value about the horizontal central line as the heated solid strip is symmetrical for Ra=103 to 104. Conduction dominant heat transfer is observed from the isotherms in Figure 5.6 at Ra = 103 and Ra = 104. With increase in Rayleigh numbers, isotherms are concentrates near the right vertical cold wall and isotherm lines are more bending which means increasing heat transfer through convection. Formation of thermal boundary layers can be found and increases from the isotherms for Ra = 105 and Ra = 106 at Ha = 0.
Streamlines (on the left column) and isotherms (on the right column) for horizontal position of solid strip and Ha = 100 (with magnetic field) are presented in Figure 4.9 (Ra=103 to 106). Two vortices and mainstream flow are found for Ra = 103 shown in Figure 4.9. Value of the stream function and flow strength increases with increasing of Rayleigh numbers shown in Figure 4.9. For Ra=103 to 105, it is seen that on the upper side of solid strip the isotherms lines are gradually scattered towards semicircular top wall but below the strip the isotherms lines are bending almost in the same natured. For Ra=106, we observed that the isotherm lines are reduced on the upper side and increased on the lower side of the solid strip very significantly. With increasing of Rayleigh numbers, isotherms are condensed to the right vertical wall.
Streamlines (on the left column) and isotherms (on the right column) for inclined position of solid strip and Ha = 0 (no magnetic field) are presented in Figure 4.4. From streamlines,
we can observed that at Ra = 103 to 106 and Ha=0 (in the absence of the magnetic field) two vortices are generated. With increase in Rayleigh number and buoyancy force, flow strength increases on the upper side and decreases on the lower side, that’s why upper vortices are becoming larger and lower vortices are smaller, but for value of Ra = 106 the upper and lower vortices are same in size. In the Figure 4.4, it is observed that the isotherms are uniformly scattered on the both sides of the solid strip and condensed near to the right vertical wall.
At Ra = 103 to 106 and in the presence of the magnetic field (Ha = 100) two vortices with mainstream flow formed shown in Figure 4.10. At Ra = 106, it is observed that both vortices are changed in size significantly because of flow strength increases for higher Rayleigh number. Isotherm lines are gradually reduced on upper side of the enclosure with increase of Rayleigh number.
Streamlines (on the left column) and isotherms (on the right column) for vertical position and Ha = 0 (no magnetic field) are presented in Figure 4.5. Only one vortex is formed here on the right side of solid strip. Size of the vortex becomes larger and stream lines more condensed near to the right vertical wall with increasing of Rayleigh numbers. Flow strength increases also with increasing rotation. Isotherms density increases near to the right vertical wall and more bends lower-left side of the strip with increasing of Rayleigh numbers.
Streamlines (on the left column) and isotherms (on the right column) for vertical position and Ha = 100 (with magnetic field) are presented in Figure 4.11. Pattern of the streamlines are similar as vertical position and Ha = 0 but flow strength decreases with increasing rotation of solid strip. Isotherms are also similar but flow strength decreases with increasing rotation. Isotherms are also similar type of changes compared with vertical position and Ha = 0.
Velocity Profiles
Variations of the velocity component along the horizontal line Y=0.07 of the cavity with the Rayleigh number and for Ha = 0 are shown in Figure 4.6. It can be seen from the Figure 4.6 (Horizontal Position) that the absolute value of maximum and minimum value of velocity increases with increasing the Rayleigh number i.e. with increasing the buoyant force. The curves are symmetrical parabolic shaped as the heated solid strip is symmetrical. For lower Rayleigh number value of velocity has smaller changed but for higher Rayleigh number value of velocity has larger changed. In Figure 4.6 for inclined
and vertical position, value of velocity has significant changes for lower Rayleigh number and minimum point of velocity increases and maximum point of velocity decreases compared with horizontal position.
Variations of the velocity component along the horizontal line Y=0.07 of the enclosure with the Rayleigh number and for Ha = 100 are shown in Figure 5.12. It can be observed from the Figure (for horizontal position) that velocity increases with increasing the Rayleigh number. In Figure 4.6 for inclined and vertical position, value of velocity changes in identical scenario for lower Rayleigh number compared with Figure 4.12 (for horizontal position) and the absolute value of maximum and minimum value of velocity increases with increasing the rotations.
Temperature profiles
Figure 4.7 presents the temperature profile along the horizontal line Y=0.07 of the enclosure with the Rayleigh number and for Ha = 0. As seen from the Figure 4.7 (for horizontal position), temperature value is decreased from the increasing of Rayleigh numbers. For lower Rayleigh number temperature value has less significant changed but for higher Rayleigh number temperature value has more significant. The plume like temperature distribution is observed and temperature value is increased from the increasing of higher Rayleigh numbers (Ra=105, Ra=106). Also lower value of temperature is decreased with increasing rotation. A similar temperature distribution is observed in Figure 4.7 (for vertical position).
Figure 4.13 (for different positions of solid strip) presents the temperature profiles along the horizontal line Y=0.07 of the enclosure with the Rayleigh number and for Ha = 100.
As seen from the Figure 4.13 (for horizontal position), temperature value is decreased with increasing of Rayleigh numbers similar as Ha = 0 in Figure 4.7(for horizontal position) but temperature value is increased than Ha=0. For lower numbers temperature value is insignificant change are shown in Figure 4.13 (for inclined and vertical position).
Local Nusselt number along the horizontal line
The local Nusselt number along the horizontal line Y=0.07 of the enclosure with the Rayleigh number and for Ha = 0 are shown in Figure 4.8. Minimum and maximum shape curves are obtained here. Absolute value of local Nusselt number (that is heat transfer rate) is increased with the increasing of Rayleigh numbers is shown in Figure 4.8 (for horizontal position). For inclined and vertical position only minimum shape curves are found and
local Nusselt number is decreased from the increasing of Rayleigh numbers are shown in Figure 4.8 (for inclined and vertical position). For lower Rayleigh numbers shape of the curves are almost linear but for higher Rayleigh numbers shape of the curves are nonlinear for all positions.
The local Nusselt number along the horizontal line Y=0.07 of the enclosure with the Rayleigh number and for Ha = 100 are shown in Figure 4.14(For horizontal, inclined and vertical position). A similar pattern is observed but absolute value of local Nusselt numbers are decreased compared with Figure 4.8.