*3.2. Time-Averaged Thermal Fields and Heat Transfer*

The time-averaged temperature fields for different thermal conductivity ratios (K\*) are compared in Figure 3. For a high conductivity ratio (K\* ≥ 100), the temperature distributions are similar to those for the isothermal wall condition (Figure 3a,b). The fluid region accounts for most of the thermal resistance, and therefore, the temperature distribution inside the solid wall is uniform and the temperature in the fluid region near the solid–fluid interface varies.

**Figure 3.** Time-averaged thermal fields: (**a**) K\* = 566.26, (**b**) K\* = 100.00, (**c**) K\* = 10.00, and (**d**) K\* = 1.00.

On the other hand, in the case of low conductivity ratios (K\* ≤ 10), the contour lines are narrowly packed in the solid region and the temperature distribution in the fluid region is relatively uniform since the thermal resistance inside the solid wall is greater than that of the fluid (Figure 3c,d).

Figure 4 shows the heat flux inside the solid wall for each thermal conductivity ratio; the contours are isotherms. In general, the thermal resistance of a fluid varies greatly spatially because of the recirculation of the flow and the collision of the cold core fluid [18,19]. On the other hand, the thermal resistance inside the solid wall is constant. When the thermal conductivity ratio is large (K\* ≥ 100), the thermal resistance in the fluid dominates. Therefore, the heat flux vector inside the solid wall is concentrated on the rib and directed toward both edges of the rib (Figure 4a,b).

**Figure 4.** Heat flux vectors inside the solid wall: (**a**) K\* = 566.26, (**b**) K\* = 100.00, (**c**) K\* = 10.00, and (**d**) K\* = 1.00.

When the thermal conductivity ratio is small (K\* ≤ 10), the thermal resistance inside the solid wall is dominant. Consequently, the heat flux vector flows toward the fluid in the wall normal direction (*y*), without being concentrated on the rib. In Figure 4c,d, the heat flux vector inside the rib is toward the upstream edge. However, the heat flux is smaller than that for a large thermal conductivity ratio. Furthermore, when the thermal conductivity ratio is 10 (Figure 4c), the fluid temperature is higher than that of the rib on the downstream side of the rib, and therefore, the heat flux cannot pass through the rib.

The cases of K\* = 100 and 1 were compared with RANS data [13] by adopting the same thermal conductivity ratio. At K\* = 100 (Figure 4b), the temperature inside the rib was almost uniform for both LES and RANS simulation. In the present LES, the isotherm inside the rib including the channel wall was close to the horizontal line, but in the RANS simulation in which only the conduction of the rib was considered, the isotherm inside the rib appeared in the diagonal direction. At K\* = 1, both LES and RANS simulation increased the number of isotherms inside the rib. The number of isotherms of the LES including the channel wall was considerably smaller than that of the RANS simulation. In the RANS simulation, isotherms occurred in a diagonal direction, while in the LES, they occurred in the form of a parabola that was convex upwards. If K\* = 100 or less, the conjugate effect can be accurately identified only when the channel wall is included in the computational domain.

Figure 5 shows the dependence of local temperature distributions at the solid–fluid interface on the thermal conductivity ratio. In the case of high conductivity ratios (K\* ≥ 100), the temperature at the interface between the ribs (Figure 5a) is considerably close to the outer wall temperature and uniform in space since the solid wall tends to maintain a uniform temperature distribution. Similar behavior is observed on the rib surface (Figure 5b). However, the upstream edge of the rib has the lowest temperature because of the high heat transfer rate. This is clearly observed for K\* = 100.

**Figure 5.** Local temperature distribution at the solid–fluid interface: (**a**) the temperature along the interface between the ribs and (**b**) the temperature on the rib.

In the case of low conductivity ratios (K\* ≤ 10), the temperature was considerably lower than that in the case of high conductivity ratios, and a relatively large temperature variation was observed. The reason is that the spatially nonuniform thermal resistance of the flow field mainly influenced the temperature at the interface. In particular, the temperature near the upstream corner of the rib increased (Figure 5a). For K\* ≤ 10, the fluid impinging on the upstream face of the rib was not very cold compared with the solid wall. However, the corner vortex prevented heat transfer from the solid to the fluid, resulting in the temperature near the upstream corner of the rib increasing.

Figure 6 shows the effect of the thermal resistance of the solid wall on the local heat transfer. Nu0 is the Nusselt number without ribs, given by Equation (7). The Nusselt number presented in Figure 6 was defined on the basis of *D*h. In Figure 6a, the local heat transfer coefficient evidently increases (8 ≤ *x/e* ≤ 9) near the upstream corner for all thermal conductivity ratios as the cold core fluid collides with the rib. However, as the thermal conductivity ratio increases, the local heat transfer coefficient decreases noticeably and becomes spatially uniform. Quantitatively, the local heat transfer coefficient depends strongly on the thermal conductivity ratio, but its qualitative distribution does not significantly depend on the local thermal conductivity ratio. Except in the vicinity of the rib, heat conduction in the flow direction is not significant since the heat flux vector inside the solid is directed in the +*y* direction (Figure 4).

**Figure 6.** Effect of thermal resistance on local convective heat transfer along the solid–fluid interface: (**a**) heat transfer along the interface between the ribs and (**b**) heat transfer on the rib.

On the other hand, the local heat transfer distribution over the rib was strongly dependent on the thermal conductivity ratio (Figure 6b). Quantitatively, even at K\* = 566.26, the heat transfer rate at both edges of the rib was considerably lower than that in the isothermal case, and the local difference was quite small. Nevertheless, for large thermal conductivity ratios (K\* ≥ 100), the distribution of local heat transfer was qualitatively similar to that of the isothermal wall. However, when the thermal conductivity ratio became small (K\* ≤ 10), the heat transfer distribution changed qualitatively. For large thermal conductivity ratios, the maximum value of heat transfer occurred at the upstream edge of the rib, while for small thermal conductivity ratios, the maximum value of heat transfer occurred near the upstream edge of the rib (*s/e* ≈ 0.2). Furthermore, in the vicinity of the downstream edge (*s/e* ≈ 2.1), as the temperature of the solid decreased below that of the fluid, a region with negative heat transfer was formed.
