**2. Numerical Methods**

In the simulation, an in-house code was used. This code was based on the finite volume incompressible Navier–Stokes solver [31] at Stanford University, USA. All spatial derivatives were discretized by the central difference with second-order accuracy. The semiimplicit fractional step method based on the Crank–Nicolson and Runge–Kutta methods with third-order accuracy was used as the time integration method. The code was revised to deal with solids in the flow field with the IBM [32] and supplemented to solve the convective heat transfer between solids and fluids [33]. Later, it was revised twice, once to perform an LES of turbulent heat transfer [18,19], and again to handle conjugate heat transfer (CHT) [34].

The computational domain analyzed in this study is presented in Figure 1b. Ribs with a square cross-section and a height (*e*) that was 0.1 of the channel height (*H*) are arranged at intervals of 10 *e* on the upper and lower sides of the channel. By comparing the computational domain, including three periods in the *x* direction and one including single period, the same time average result was obtained, and finally, the computational area was set to include single period as shown in Figure 1c. The spanwise domain was set to be 2.5 π *e*, for which zero fall-off was observed with two-point correlation in a smooth channel simulation [18,19]. The thickness of the solid wall was set to be three times *e*, which corresponds to 30% of *H*.

Periodic boundary conditions were imposed in the main flow (*x*) and spanwise (*z*) directions. In the wall normal direction (*y*), no slip conditions or isothermal conditions were imposed on the upper and lower surfaces in the computational domain. The grid system comprised 128, 256, and 48 meshes in the *x*, *y*, and *z* directions (Figure 1c). A nonuniform grid was used in the *x* and *y* directions, while a uniform grid was used in the *z* direction. This was similar to the resolution obtained by [35] for a grid independent solution by performing a resolution test on the LES of a ribbed duct.

The grid-filtered incompressible Navier–Stokes equation and energy equation were adopted as the dimensionless governing equations, and they can be expressed as follows [34]:

$$\frac{\partial \overline{u}\_{l}}{\partial \chi\_{l}} - ms = 0,\tag{2}$$

$$\frac{\partial \overline{u}\_{i}}{\partial t} + \frac{\partial \overline{u}\_{i} \overline{u}\_{j}}{\partial \mathbf{x}\_{j}} = -\frac{d\overline{p}}{d\mathbf{x}\_{i}} + \frac{1}{\text{Re}} \frac{\partial^{2} \overline{u}\_{i}}{\partial \mathbf{x}\_{j} \partial \mathbf{x}\_{j}} + \frac{\partial \tau\_{ij}}{\partial \mathbf{x}\_{j}} + f\_{i\nu} \tag{3}$$

$$\frac{\partial\overline{\theta}}{\partial t} + \omega \frac{\partial}{\partial \mathbf{x}\_{\dot{j}}} (\overline{\mathbf{u}}\_{\dot{j}} \overline{\theta}) = \frac{\mathbb{C}^\* K^\*}{\text{Re } \text{Pr}} \frac{\partial^2 \overline{\theta}}{\partial \mathbf{x}\_{\dot{j}} \partial \mathbf{x}\_{\dot{j}}} + \frac{\partial q\_{\dot{j}}}{\partial \mathbf{x}\_{\dot{j}}} + \xi. \tag{4}$$

Under the assumption of fully developed flow, the mean streamwise pressure and temperature gradient were decoupled as follows to impose periodic boundary conditions in the streamwise direction [36]:

$$P(\mathbf{x},t) = -\beta \mathbf{x} + p(\mathbf{x},t),\tag{5}$$

$$T(\mathbf{x}, t) = \gamma \mathbf{x} + \theta(\mathbf{x}, t), \tag{6}$$

where β and γ are the mean streamwise pressure and temperature gradients, respectively. These two parameters were determined to satisfy the conservation of global momentum and energy, respectively [36].

In the energy equation, the thermal properties of the solid were distinguished from those of the fluid by defining the heat capacity ratio C\* and the thermal conductivity ratio K\*, and by introducing the concept of effective thermal conductivity (*k*e). The effective thermal conductivity was determined to satisfy the continuity of temperature and heat flux at the interface [34]. The parameter ω was a convection correction factor, and it was 0 in the cell containing the solid–fluid boundary and 1 in the remaining cells when conduction between the solid and the fluid was considered. In Equation (4), *ξ* was introduced to maintain second-order accuracy in the cell containing the interface [34]. The code was verified through the CHT problem involving a ribbed duct and a circular cylinder [34].

Turbulent flow was analyzed using an LES. In Equations (3) and (4), τij and *q*<sup>j</sup> are the sub-grid scale turbulent stress and turbulent heat flux, respectively, and τij was determined as a dynamic sub-grid model by using scale similarity and setting a test filter around the grid [37,38]. The dynamic sub-grid model provided better results than the constant model for the ribbed channel problem [35]. Similar to τij, *q*<sup>j</sup> was determined dynamically; this approach yields better results in problems where the flow and heat transfer are dissimilar [39]. The simulation were performed for 10,000 time steps to reach a steady state. After that, additional 10,000 time steps (*t U*b/*D*<sup>h</sup> = 5) were carried out to obtain the statistics.

Numerical analysis was performed with thermal conductivity ratios of 1, 10, 100, and 566.26 to examine the thermal resistance effect of the solid wall. The value 566.26 is the thermal conductivity ratio between the gas turbine blade material and air [22]. The remaining flow and geometry conditions are summarized in Table 1. In this study, an LES

was performed to obtain the instantaneous flow field and turbulence statistics along with the time average flow and temperature field. Furthermore, the turbulent heat transfer on the fluid side and changes in temperature fluctuations on the solid side according to the thermal conductivity ratio were observed.


**Table 1.** Parameters related to conjugate heat transfer.
