## 3 Results and Discussions
### U-Shaped Channel with a Larger Temperature Difference  
In this section, we will demonstrate the results at a temperature difference of 300-273 K between the incoming fluid and the channel wall of the U-shaped channel. Figure 4 shows the pressure and the vorticity at the middle-height plane of the channel at uncoolingand cooling temperature boundary conditions and at different flow rates, respectively. Comparing the cooling flow and the uncooled flow at the lower flow rate, there is no obvious difference. However, at the higher flow rate, both the pressure and the vorticity distributions change as the cooling processes. The pressure for the uncooled fluid decreases more gently along the channel, whereas the pressure of the cooled fluid drops more suddenly, especially behind the bend. From this, we can suspect that the cooled and quicker fluid dissipates the energy most at the bend, which can be partly evidenced by the fact that the vorticity of the cooled fluid is generally larger in space than that of the uncooled fluid, although the maximum vorticity of the cooled fluid is smaller. If we compare the data at the same temperature conditions but at different flow rates, the uncooled results demonstrate no obvious change, whereas the cooled fluid does. For the cooled fluid, as the flow rate increases, the pressure drops more sharply, and the vorticity is universally larger, the latter of which suggests that the secondary flow for the quicker fluid is stronger. The following results of the transverse flow structures will verify this. To represent the flow details in the bend, Figure 5 shows the 3D streamlines, vorticity magnitude, velocity magnitude, and the velocity vector arrows on the middle-height plane. Again, let us first compare the results at the same flow rate but under different temperature conditions. In the cooled fluid, the maximum vorticity value seems to decrease, and the overlap of the streamlines occurs at an earlier position, which corresponds to the earlier twisted velocity vectors. As the fluid processes cool, the maximum velocity zones become wider. Secondly, let us compare the results at the same temperature conditions but at different flow rates. As the flow rate increases, the vorticity at the inner wall within the bend is less symmetric, the maximum velocity zone shrinks, and the turn of the flow is larger. Interestingly, at the higher flow rate, and especially in the cooled fluid, the flow deflects towards the outer wall behind the bend but then recovers slightly back to the inner wall, which is presumably because that the more viscous fluid has more motion in the transverse direction since it is more difficult to be sheared in the stream direction. This will be confirmed by the results of the transverse flow structures discussed in the following. Figure 2: (**a**) Pressure in the straight channels along the axial line, at the inlet velocities of 0.5 m/s and 1 m/s, respectively. In these simulations, the temperature at the channel wall is 273 K, while the temperature of the incoming fluid at the inlet is 300 K. The black lines are the results with no temperature difference, and in these cases, the temperature of the fluid is constant at 300 K. (**b**) Similar to (**a**), but the temperature at the channel wall is 283 K.  
Figure 4: The pressure and the vorticity magnitude at the middle-height plane of the U-shaped channel. The pressure is displayed in the left column, and the vorticity magnitude is displayed in the right column. The first row shows the data at the inlet velocity of 0.5 m/s with no temperature difference. The second row shows the results at the inlet velocity of 0.5 m/s but at the temperature conditions of 273 K on the wall and 300 K of the incoming fluid at the inlet. The third row shows the results at the inlet velocity of 1 m/s with no temperature difference. The fourth row shows the results at the inlet velocity of 1 m/s but at the temperature conditions of 273 K on the wall and 300 K of the incoming fluid at the inlet.