Fully Coupled Large Eddy Simulation of Conjugate Heat Transfer in a Ribbed Channel with a 0.1 Blockage Ratio

Large eddy simulations are performed to analyze the conjugate heat transfer of turbulent flow in a ribbed channel with a heat-conducting solid wall. An immersed boundary method (IBM) is used to determine the effect of heat transfer in the solid region on that in the fluid region in a unitary computational domain. To satisfy the continuity of the heat flux at the solid–fluid interface, effective conductivity is introduced. By applying the IBM, it is possible to fully couple the convection on the fluid side and the conduction inside the solid and use a dynamic subgrid scale model in a Cartesian grid. The blockage ratio (e/H) is set at 0.1, which is typical for gas turbine blades. Through conjugate heat transfer analysis, it is confirmed that the heat transfer peak in front of the rib occurs because of the impinging of the reattached flow and not the influence of the thermal boundary condition. When the rib turbulator acts as a fin, its efficiency and effectiveness are predicted to be 98.9% and 8.32, respectively. When considering conjugate heat transfer, the total heat transfer rate is reduced by 3% compared with that of the isothermal wall. The typical Biot number at the internal cooling passage of a gas turbine is <0.1, and the use of the rib height as the characteristic length better represents the heat transfer of the rib.


Introduction
One way to ensure that a gas turbine engine is highly efficient is to increase the turbine inlet temperature. However, this temperature is limited by the maximum allowable temperature of the material of construction and intensive cooling is necessary to overcome this limitation. For this purpose, turbine blades are typically designed with internal cooling channels, with ribs placed on the blades' surfaces to enhance the heat transfer and cooling performance, as illustrated in Figure 1a [1]. If ribs are periodically installed on the wall, the flow is separated and reattached to enhance the heat transfer. As a result, it is possible for one to obtain a heat transfer coefficient that is two to three times higher on the wall compared with that of a smooth channel [2].
The average heat transfer is increased by the ribs, but it becomes non-uniform locally. On the wall just behind the rib, heat transfer is minimized, and a region with a high-heattransfer coefficient appears near the reattachment point [3]. In many experimental data in the literature, a secondary heat transfer maximum appears in front of the rib. Behind the ribs, the cooling fluid is not supplied smoothly to the wall because of the recirculation flow generated by the flow separation, so heat transfer is low. Various attempts have been made to solve this problem [4][5][6][7].
There has been some controversy surrounding the subject of high-heat transfer. As explained by previous researchers, the primary peak is reached when the main flow is supplied to the wall surface as the separated flow is reattached to the channel wall to increase heat transfer. The exact heat transfer peak occurs slightly upstream of the reattachment point, but it has been considered to exist in the same region because it is a There has been some controversy surrounding the subject of high-heat transfer. As explained by previous researchers, the primary peak is reached when the main flow is supplied to the wall surface as the separated flow is reattached to the channel wall to increase heat transfer. The exact heat transfer peak occurs slightly upstream of the reattachment point, but it has been considered to exist in the same region because it is a gentle peak. The secondary peak in front of the rib occurs as a result of the corner vortex that is present in the same location [3].
Ahn et al. [8,9] explained the above two peaks by observing the instantaneous flow field obtained by large eddy simulation (LES). By observing the instantaneous temperature field in the xy plane, they found that the primary heat transfer peak occurs upstream of the reattachment point because the main flow is entrained by eddies at the shear layer. In addition, by observing the instantaneous temperature field in the xz plane near the wall, they found that the reattached cold fluid collides with the rib that follows and promotes heat transfer in front of the rib.
In some experiments, e.g., those of [10], the heat transfer peak in front of the rib did not appear. According to [11], this is because of the difference in thermal boundary conditions. They postulated that when the ribs are heated, the cooling fluid impinging on the ribs is heated and supplied to the wall, meaning that the heat transfer coefficient cannot increase. They verified this hypothesis by conducting a series of Reynolds-averaged Navier-Stokes (RANS) simulations. However, many experimental data, including those of [10], cannot be explained by this hypothesis.
In the ribbed channel problem, LES predicts the local heat transfer distribution more accurately than RANS simulations [8]. In addition, the change in turbulence that occurs due to rotation, which is difficult to predict with RANS simulations, is well predicted by LES [9,12,13]. Therefore, the effect of the thermal boundary condition on the local heat transfer distribution must be clarified with LES rather than RANS simulations.
Fedrizi et al. [14] investigated the problem of thermal boundary conditions by performing conjugate heat transfer experiments on triangular channels with angled ribs. Ahn et al. [8,9] explained the above two peaks by observing the instantaneous flow field obtained by large eddy simulation (LES). By observing the instantaneous temperature field in the xy plane, they found that the primary heat transfer peak occurs upstream of the reattachment point because the main flow is entrained by eddies at the shear layer. In addition, by observing the instantaneous temperature field in the xz plane near the wall, they found that the reattached cold fluid collides with the rib that follows and promotes heat transfer in front of the rib.
In some experiments, e.g., those of [10], the heat transfer peak in front of the rib did not appear. According to [11], this is because of the difference in thermal boundary conditions. They postulated that when the ribs are heated, the cooling fluid impinging on the ribs is heated and supplied to the wall, meaning that the heat transfer coefficient cannot increase. They verified this hypothesis by conducting a series of Reynolds-averaged Navier-Stokes (RANS) simulations. However, many experimental data, including those of [10], cannot be explained by this hypothesis.
In the ribbed channel problem, LES predicts the local heat transfer distribution more accurately than RANS simulations [8]. In addition, the change in turbulence that occurs due to rotation, which is difficult to predict with RANS simulations, is well predicted by LES [9,12,13]. Therefore, the effect of the thermal boundary condition on the local heat transfer distribution must be clarified with LES rather than RANS simulations.
Fedrizi et al. [14] investigated the problem of thermal boundary conditions by performing conjugate heat transfer experiments on triangular channels with angled ribs. They demonstrated that the heat transfer of the rib itself significantly affects the total heat transfer rate, and the thermal conductivity of the rib has a great influence on the thermal performance. Cukurel et al. [15,16] provided the temperature distribution inside the rib obtained from experimental data. Scholl et al. [17,18] performed LES considering conjugate heat transfer under conditions tested by [15,16]. LES predicted the experimental results well on the channel wall; however, the local heat transfer distribution at the front and rear sides of the rib indicated some variation [17,18]. Scholl et al. [17,18] pointed out that, as a result of this variation, optical problems occur at the edges. Cukurel et al. [15,16] addressed the adequacy of the bulk temperature weighted by streamwise velocity at the front and rear faces of the rib.
Cukurel et al. [15,16] adopted a blockage ratio (e/H) of 0.3, which appears near the trailing edge of the turbine blade [19,20]. In the mid-chord region, its typical value is approximately 0.1, and most previous studies have used this value. In conjugate heat transfer, because the length scales of the solid and fluid should be different, the Biot number (Bi) must be considered [19]. The thickness of the blade was suggested as the characteristic length of Bi [16,19]. When ribs are installed, however, the rib height (e) must be considered as a characteristic length. Cukurel et al. [15,16] performed experiments on rigs with the same rib height and floor plate thickness, but the thickness of the gas turbine blades was approximately two to seven times the rib height [21,22]. Therefore, in this study, LES that included conjugate heat transfer was performed on a gas turbine rib generator with a typical blockage ratio of 0.1. The computational domain was set based on typical gas turbine blades, i.e., the thickness of the solid wall was three times the rib height [21,22]. We examined whether the rib height or blade thickness was appropriate as the characteristic length of Bi. In addition, we introduced the immersed boundary method (IBM) to perform a fully coupled conjugate heat transfer analysis. The IBM with a Cartesian grid also enables the use of a dynamic subgrid model, which produces better results regarding the ribbed channel problem [23].
In this paper, a series of LESs were conducted under conjugate and isothermal conditions, and the following issues were reviewed by comparing our data with those in the literature and with LES data. First, the effect of thermal boundary conditions on the secondary heat transfer peak in front of the rib is examined. To investigate issues related to heat transfer around the rib edges, experiments of [24] with different optical access are compared. Subsequently, turbulence statistics and instantaneous temperature fields are analyzed to understand the mechanism of conjugate heat transfer. To the best of our knowledge, transient thermal behavior in solids in a ribbed channel through fully coupled simulation is presented for the first time. Finally, we present the thermal performance of the rib as a fin in the ribbed channel and discuss the appropriate characteristic length defining the Biot number.

Numerical Methods
The grid-filtered incompressible Navier-Stokes equation and energy equation were adopted as dimensionless governing equations and can be expressed as follows [25]: The solid region was generated using the IBM. In Equation (2), the momentum forcing (f i ) was imposed to ensure a velocity of zero in the solid region. In the continuity equation (Equation (1)), mass source/sink (ms) was imposed to satisfy the conservation of mass in cells containing a solid-fluid interface. The amounts of momentum forcing and mass source/sink were determined using a procedure proposed by [26].
In Equation (2), τ ij is the subgrid stress, which is obtained from the strain rate tensor. Germano et al. [27] proposed a method of dynamically determining the proportionality coefficient between the strain rate and τ ij , and [28] improved upon this method. Tafti [23] Energies 2021, 14, 2096 4 of 17 confirmed that using the dynamic model better predicts heat transfer in the ribbed channel. Since the dynamic model predicts τ ij through scale similarity by setting a test filter around the grid, it is difficult for one to obtain good results in body-fitted coordinates including sharp corners. This simulation secured scale similarity by adopting a Cartesian grid while introducing IBM.
In the energy equation (Equation (3)), the subgrid heat flux q j must be obtained. In this simulation, q j was dynamically determined from the scale similarity between the heat flux obtained from the grid filter and the test filter. When isothermal conditions on a solid wall were creating, as suggested by [29], a heat source/sink was imposed on the cell containing the solid. When considering conjugate heat transfer, the continuity of temperature and the heat flux must be satisfied at the solid-fluid interface.
In the present simulation, IBM, which handles conjugate heat transfer using the method proposed by [25], was employed. First, momentum (f i ) and mass (ms) were imposed inside the solid so that the velocity became zero, and Equation (3) became the heat diffusion equation. The difference in physical properties between the fluid and the solid was reflected by introducing the heat capacity ratio (C*) and the thermal conductivity ratio (K*) while maintaining the basis of the Prandtl number in Equation (3). A convection correction factor (τ) was introduced to maintain the second-order accuracy at the interface, and effective thermal conductivity was introduced to satisfy continuity of the heat flux at the interface. An explanation of how to determine each factor according to the interface configuration in the cell can be found in [25,30].
A series of simulations was performed using an in-house code. The code originated from the finite volume incompressible Navier-Stokes solver [31] at Stanford University, in the USA. Kim et al. [29] added IBM to this code, and the authors modified it to handle conjugate heat transfer [25]. All spatial discretization was formulated by the central difference scheme with second-order accuracy. The Crank-Nicolson method and the semiimplicit fractional step method based on the third-order Runge-Kutta method were used as time integration methods. The heat conduction inside the solid was also fully coupled with the flow simulation using the IBM.
The computational domain of this study is presented in Figure 1b. The ribs of a square cross section with a height (e) of 0.1 times the channel height (H) are arranged at intervals of 10 e at the top and bottom of the channel. The time-averaged results obtained from the computational domain containing three cycles ( Figure 1b) and the domain containing one cycle coincided [8]. Therefore, as illustrated in Figure 1c, the computational domain was set to include one period. The domain in the spanwise direction was set to 2.5π e, which indicated zero fall-off in the two-point correlation in the smooth channel simulation [8]. The thickness of the solid wall was made to be three times the rib height (e), which is a typical ratio found in a turbine blade [21,22].
As for the boundary conditions, periodic conditions were imposed in the main flow direction (x) and the spanwise direction (z). In the vertical direction (y), non-slip conditions and isothermal conditions were imposed on the top and bottom surfaces of the computational domain. The grid system was composed of 128 × 256 × 48 grid points, and in the fluid domain, the resolution was the same as that in a previous study [8] using the same code.
Ahn et al. [8] and Tafti [15] obtained a grid independent solution with 96 × 96 and 128 × 128 grids in the xy plane of the ribbed channel as in this study. Based on these solutions, a 128 × 128 grid was placed on the xy plane. To accurately predict heat transfer, the grid was clustered near the wall. In the solid region having 60% of the volume of the fluid region, 128 × 128 grids of the same number as the fluid region were constructed. In the z direction, 48 uniform grids were arranged that gave results that were consistent with direct numerical simulation (DNS) in turbulent channel flow of the same Reynolds number. Table 1 summarizes the solid thermal properties and solid region thickness adopted in ribbed channel studies considering conjugate heat transfer. In this study, the thermal conductivity and specific heat were set to be close to that of the gas turbine material. The thermal conductivity ratio (K*) was set to 566.26 with reference to [14]. Liou et al. [24] used brass with high thermal conductivity and made it resemble isothermal conditions. Cukurel et al. [15,16] and [17,18] used the properties of AISI304 steel and a K* of 618.32, which is within 10% similarity to the current study. The heat capacity ratio (C*) values of the present study and the three previous studies compared were all within a similar range. To determine the heat transfer at the rib edge, which was addressed in the introduction, it is necessary for one to validate the prediction of conjugate heat transfer at the sharp edge using the IBM. Song et al. [25] validated the code used in this study with a finite element solution for conjugate heat transfer of laminar ribbed channels used by [32]. The results obtained using the IBM predicted the heat transfer well near the edge of the rib regardless of the thermal conductivity ratio [25].

Time-Averaged Flow Fields and Heat Transfer Coefficient
Time-averaged streamlines ( Figure 2a) are nearly identical to the particle image velocimetry (PIV) data obtained by [33]. A recirculation vortex occurs behind the rib, and there is a corner vortex near the downstream corner. There is an additional corner vortex in the upstream corner of the rib (8.5 < x/e < 9). Compared with the PIV data, both of the corner vortices obtained by LES are slightly larger, seemingly because of the differences in the blockage ratio and the aspect ratio of the channel (see Table 2). The size of the recirculation bubble and the location of the reattachment point are almost identical to those in the PIV data. When the time-averaged temperature field ( Figure 2b) is compared with those in the streamlines, the temperature in the recirculation zone is higher than those in other places and results in warmer colors.     [15,16] 40,000 IR Camera 0.3 10 Conjugate 1 Scholl et al. [17,18] 40,000 LES 0.3 10 Conjugate 1 Casarsa et al. [33] 40,000 PIV 0.3 10 Conjugate 1 In Figure 2c, six data obtained through experiments or LES were compared to analyze the heat transfer distribution of the channel wall with ribs. The heat transfer coefficient (h) in the channel flow is defined as follows based on the bulk temperature (T b ): The heat transfer coefficient was compared with the Nusselt number ratio, where Nu 0 is the Nusselt number of the smooth channel wall obtained using the following Dittus-Boelter correlation: The flow and geometry conditions and the method of obtaining data are different, as summarized in Table 2, but the distribution of local heat transfer coefficients is generally consistent within a similar range. Typically, the heat transfer coefficient has a maximum value of approximately x/e = 3.5 and a minimum value of approximately x/e = 0.5. In five data, excluding Rau et al. [10], the heat transfer coefficient had secondary peaks in front of the rib (8.5 < x/e < 9). In four of the six data, the minimum value of the heat transfer coefficient ratio was less than 1.
In Figure 3a, the temperature distribution of the conjugate case is compared with that of the isothermal case. Most of the temperature changes occur in the fluid region, so the two temperature distributions are generally similar. The temperature change in the solid region is concentrated around the ribs; the temperature distribution around the rib is presented in Figure 3b. Heat transfer is active at both corners of the rib, so in Figure 3b, the isotherm line inside the rib becomes convex upward, and the heat flux vector points toward both corners. At the bottom of the rib, conduction occurs toward the rib with low thermal resistance, and the heat flux vector is directed toward the rib. In the vicinity of y/e = 0, the isotherm becomes convex downward as heat is conducted to the bottom surface. Looking at the curvature of the isotherm, the upstream side has a large curvature because heat transfer is active at the upstream side of the corner. The experimental data of [15] compared in Figure 3c indicate a pattern similar to the LES data illustrated in Figure 3b.
In the RANS data of [11], a heat transfer peak occurs before the insulated rib, the peak disappears for heated rib. In all five of the experiments and LES data presented in Figure 4, the ribs were heated, but high heat transfer occurred in front of the ribs; this cannot be explained by the thermal boundary conditions. When the heat transfer peak does not appear in front of the rib, a corner vortex occurs, so the phenomenon cannot be adequately explained even with a corner vortex. As explained by [8], the secondary peak should be described as cold fluid impinging, so that there is no contradiction with the foregoing data.
When a conjugate heat transfer simulation (red solid line in Figure 4) is performed, local heat transfer change is slightly reduced compared with that in the isothermal case (black dotted line in Figure 5). In particular, when conjugate heat transfer is performed around x/e = 3.5, where the heat transfer coefficient is at its maximum, the heat transfer coefficient decreases by 10% compared with the isothermal case. In the 7 < x/e < 8 region, conjugate heat transfer has a slightly larger heat transfer coefficient than that in the isothermal case.
The Nusselt number ratio obtained by LES in the present study is in good agreement with the experimental data of [24] (green square) with the same blockage ratio (0.1). A Energies 2021, 14, 2096 7 of 17 study by [15,16], in which an iso-flux (pink triangle) and conjugate heat transfer (blue circle) experiment was performed by installing a rib with a blockage ratio of 0.3, was also compared. The results of [15,16] indicate that the Nusselt number ratio was 20-30% smaller than that in the present LES and [24] in the 3 < x/e < 6 region near the reattachment point. In the upstream region of 3 < x/e < 4, convective wall has a higher heat transfer coefficient than conjugate one, but the trend reverses as it flows downstream, similarly to what is indicated by the present LES data. coefficient ratio was less than 1.
In Figure 3a, the temperature distribution of the conjugate case is compared with that of the isothermal case. Most of the temperature changes occur in the fluid region, so the two temperature distributions are generally similar. The temperature change in the solid region is concentrated around the ribs; the temperature distribution around the rib is presented in Figure 3b. Heat transfer is active at both corners of the rib, so in Figure 3b, the isotherm line inside the rib becomes convex upward, and the heat flux vector points toward both corners. At the bottom of the rib, conduction occurs toward the rib with low thermal resistance, and the heat flux vector is directed toward the rib. In the vicinity of y/e = 0, the isotherm becomes convex downward as heat is conducted to the bottom surface. Looking at the curvature of the isotherm, the upstream side has a large curvature because heat transfer is active at the upstream side of the corner. The experimental data of [15] compared in Figure 3c indicate a pattern similar to the LES data illustrated in Figure 3b. In the RANS data of [11], a heat transfer peak occurs before the insulated rib, the peak disappears for heated rib. In all five of the experiments and LES data presented in Figure  4, the ribs were heated, but high heat transfer occurred in front of the ribs; this cannot be explained by the thermal boundary conditions. When the heat transfer peak does not appear in front of the rib, a corner vortex occurs, so the phenomenon cannot be adequately explained even with a corner vortex. As explained by [8], the secondary peak should be described as cold fluid impinging, so that there is no contradiction with the foregoing data.   Table 1 for information related to the compared experimental data.).
When a conjugate heat transfer simulation (red solid line in Figure 4) is performed, local heat transfer change is slightly reduced compared with that in the isothermal case (black dotted line in Figure 5). In particular, when conjugate heat transfer is performed around x/e = 3.5, where the heat transfer coefficient is at its maximum, the heat transfer coefficient decreases by 10% compared with the isothermal case. In the 7 < x/e < 8 region, conjugate heat transfer has a slightly larger heat transfer coefficient than that in the isothermal case.  Table 1 for information related to the compared experimental data.). (black dotted line in Figure 5). In particular, when conjugate heat transfer is performed around x/e = 3.5, where the heat transfer coefficient is at its maximum, the heat transfer coefficient decreases by 10% compared with the isothermal case. In the 7 < x/e < 8 region, conjugate heat transfer has a slightly larger heat transfer coefficient than that in the isothermal case.  Table 1 for information related to the compared data.).
The Nusselt number ratio obtained by LES in the present study is in good agreement with the experimental data of [24] (green square) with the same blockage ratio (0.1). A study by [15,16], in which an iso-flux (pink triangle) and conjugate heat transfer (blue circle) experiment was performed by installing a rib with a blockage ratio of 0.3, was also compared. The results of [15,16] indicate that the Nusselt number ratio was 20-30% smaller than that in the present LES and [24] in the 3 < x/e < 6 region near the reattachment point. In the upstream region of 3 < x/e < 4, convective wall has a higher heat transfer  Table 1 for information related to the compared data.). Figure 5, the local heat transfer distributions of conjugate heat transfer (red solid line) and isothermal heat transfer (black dotted line) are similar even on the rib's surface. On the upstream surface (0 < s/e < 1), the heat transfer is at its minimum at the corner (s/e = 0) and increases along the surface before reaching its maximum at the edge (s/e = 1). The experiment of [24] (green square) and LES of [18] (orange dotted line) indicate the same trend. The experiment conducted by [15,16] indicates a maximum value of approximately s/e = 0.7. Scholl et al. [17,18] explained that this difference occurs when the infra-red (IR) camera is placed on the top of the test section, and optical access at the edge was not smooth. The data of [24], who obtained hologram images from the side, indicated a peak at s/e = 1, thus supporting the explanation of [17,18].

As illustrated in
At the upstream surface (0 < s/e < 1), the heat transfer coefficient (red solid line) of conjugate heat transfer is lower than that of isothermal heat transfer (black dotted line) near s/e = 0.5. In the experimental data of [15,16], the heat transfer coefficient of the conjugate case (blue circle) is lower than that of the iso-flux case (pink triangle) on the upstream surface, indicating the greatest difference at around s/e = 0.5.
On the upper surface of the rib (1 < s/e < 2), the minimum is at the upstream edge, where flow separation occurs, and the maximum is near the downstream edge, where reattachment occurs. The LES (orange dotted line) of [18] indicates a similar trend, but this trend is not evident in the experimental data (symbols). In contrast with the upstream surface, the heat transfer coefficient in the conjugate case (red solid line) is higher than that of the isothermal wall (black dotted line). The reversal of the heat transfer coefficient of conjugate and convective cases was also observed in the experimental data of [15,16]. In other words, in the 1 < s/e < 2 region, the blue symbols are distributed above the pink symbols.
In the region on the back side of the rib (2 < s/e < 3), the heat transfer coefficient decreases from the edge (s/e = 2) to the corner (s/e = 3). This trend can be observed consistently in the six data compared in Figure 5. The present data and LES of [18] agree well. In the region 2 < s/e < 3, the conjugate and isothermal data of the present LES were almost identical. However, in the experiments of [15,16], the conjugate data (blue circle) are significantly lower than the iso-flux data (pink triangle). The difference in the present LES seems to have occurred because of the influence of the blockage ratio and the thickness of the solid wall. transfer is active, which is indicated in yellow in Figure 6a. In the case of conjugate heat transfer (lower part of Figure 6a), the overall distribution is similar to that of isothermal heat transfer, but the turbulent heat flux near the edge of the rib is slightly weaker than that of isothermal wall. In the vicinity of the channel wall, the level is similar to that of the isothermal wall, and on the rib's surface, the location differs somewhat; it was slightly weaker at the front surface and slightly stronger at the downstream position of the top surface (−0.5 < x/e < 0). Overall, the turbulent heat flux in Figure 6a supports the heat transfer coefficient distribution illustrated in Figures 4 and 5.  Figure 6b illustrates the turbulent stress distribution. The flow fields of the isothermal and conjugate transfers are identical, so the turbulent stress is the same. The turbulent stress has a very different distribution from that of the turbulent heat flux illustrated in Figure 6a. The turbulent stress appears in the form of a band with a strong negative value along y/e = 1. The positive part occurs in a narrow area near the upstream edge and corner. Figure 6c illustrates the temperature fluctuation distribution. Most of the temperature fluctuations occur in the fluid domain and have values of 2% and higher (red color in Figure 6c). In the solid region, the fluctuation stays within 0.5% and occurs mostly in the rib. Inside the rib, temperature fluctuation is maximized at the upstream edge.

Turbulence Statistics and Instantaneous Thermal Fields
In the instantaneous temperature field in the xy plane (Figure 7a), it is difficult to clearly see the difference between the conducting wall and the isothermal wall. The temperature difference in the solid region of the conducting wall appears faintly only in the rib. In both cases, most of the temperature changes occur in the fluid region, and the distribution patterns are similar.
In the instantaneous temperature field in the xz plane near the wall (Figure 7b), the difference according to the thermal boundary conditions of the wall is more evident. It is common for high-temperature regions to appear upstream (0 < x/e < 1.5), and cold streaks are mainly distributed in 2 < x/e < 7. It is also typical for the cold streak to change to a spanwise elongated shape in front of the rib. However, on the conjugate wall, the upstream high-temperature region (0 < x/e < 1.5) is weakened, and the cold streak is weakened in the 2 < x/e < 5 region. In the 5 < x/e < 8 region, it can be seen that the cold streak is wider on the conjugate wall.  Figure 6b illustrates the turbulent stress distribution. The flow fields of the isothermal and conjugate transfers are identical, so the turbulent stress is the same. The turbulent stress has a very different distribution from that of the turbulent heat flux illustrated in Figure 6a. The turbulent stress appears in the form of a band with a strong negative value along y/e = 1. The positive part occurs in a narrow area near the upstream edge and corner. Figure 6c illustrates the temperature fluctuation distribution. Most of the temperature fluctuations occur in the fluid domain and have values of 2% and higher (red color in Figure 6c). In the solid region, the fluctuation stays within 0.5% and occurs mostly in the rib. Inside the rib, temperature fluctuation is maximized at the upstream edge.
In the instantaneous temperature field in the xy plane (Figure 7a), it is difficult to clearly see the difference between the conducting wall and the isothermal wall. The temperature difference in the solid region of the conducting wall appears faintly only in the rib. In both cases, most of the temperature changes occur in the fluid region, and the distribution patterns are similar.
In the instantaneous temperature field in the xz plane near the wall (Figure 7b), the difference according to the thermal boundary conditions of the wall is more evident. It is common for high-temperature regions to appear upstream (0 < x/e < 1.5), and cold streaks are mainly distributed in 2 < x/e < 7. It is also typical for the cold streak to change to a spanwise elongated shape in front of the rib. However, on the conjugate wall, the upstream high-temperature region (0 < x/e < 1.5) is weakened, and the cold streak is weakened in the 2 < x/e < 5 region. In the 5 < x/e < 8 region, it can be seen that the cold streak is wider on the conjugate wall. Compared with the heat transfer distribution presented in Figure 4, at 0 < x/e < 1.5, the isothermal and conjugate Nusselt numbers are almost identical, so the difference illustrated in Figure 7b cannot be recognized. The Nusselt number is higher for the isothermal wall at 2 < x/e < 5 and for the conjugate wall at 5 < x/e < 8, which can be explained using Figure 7b. Figure 8 illustrates how the instantaneous temperature field changes over time in the xy plane. Here t * is the dimensionless time defined as t e/Ub, and when it reaches 10, the bulk flow passes once through the computational domain. A mechanism that promotes heat transfer by entraining the cold core fluid at the shear layer and flowing it along the wall after reattachment is also effective for conducting walls. At t * = 2, 4, and 6, it can be observed that cold fluid flows in the shear layer. Compared with the heat transfer distribution presented in Figure 4, at 0 < x/e < 1.5, the isothermal and conjugate Nusselt numbers are almost identical, so the difference illustrated in Figure 7b cannot be recognized. The Nusselt number is higher for the isothermal wall at 2 < x/e < 5 and for the conjugate wall at 5 < x/e < 8, which can be explained using Figure 7b. Figure 8 illustrates how the instantaneous temperature field changes over time in the xy plane. Here t* is the dimensionless time defined as t e/U b , and when it reaches 10, the bulk flow passes once through the computational domain. A mechanism that promotes heat transfer by entraining the cold core fluid at the shear layer and flowing it along the wall after reattachment is also effective for conducting walls. At t* = 2, 4, and 6, it can be observed that cold fluid flows in the shear layer. Figure 9 illustrates how the instantaneous temperature field around the rib changes over time. Compared with that in the fluid region, the temperature in the solid region does not change significantly. The position of the isotherm on the windward surface in the solid region barely changes over time. On the leeward surface, the isotherm moves up and down; when t* = 0, the isotherm is located near the top, but at t* = 4, it descends to the middle of the surface.
The isotherm within the solid near the corner moves back and forth over time before the upstream corner. At t* = 2, the isotherm in the solid near the upstream corner comes out in front of the corner, but at t* = 6, it retreats toward the corner. In the vicinity of the downstream corner, the isotherm is almost fixed at the corner and does not move over time. Figure 10 illustrates change of the Nusselt number ratio over time on the rib's surface. Red high-heat-transfer streaks can be seen on the windward surface (AB); this is caused by cold fluid impinging on the rib. Heat transfer is maximized at position B, the upstream edge. On the top surface (BC), there is a sky-blue area on the upstream side where heat transfer is not active as the flow is separated. At t* = 0 and t* = 8, a thin high-heat-transfer  Figure 9 illustrates how the instantaneous temperature field around the rib changes over time. Compared with that in the fluid region, the temperature in the solid region does not change significantly. The position of the isotherm on the windward surface in the solid region barely changes over time. On the leeward surface, the isotherm moves up and down; when t * = 0, the isotherm is located near the top, but at t * = 4, it descends to the middle of the surface.    Figure 9 illustrates how the instantaneous temperature field around the rib changes over time. Compared with that in the fluid region, the temperature in the solid region does not change significantly. The position of the isotherm on the windward surface in the solid region barely changes over time. On the leeward surface, the isotherm moves up and down; when t * = 0, the isotherm is located near the top, but at t * = 4, it descends to the middle of the surface.   Red high-heat-transfer streaks can be seen on the windward surface (AB); this is caused by cold fluid impinging on the rib. Heat transfer is maximized at position B, the upstream edge. On the top surface (BC), there is a sky-blue area on the upstream side where heat transfer is not active as the flow is separated. At t * = 0 and t * = 8, a thin high-heat-transfer streak across BC is observed. In the leeward surface, heat transfer decreases from the edge (C) to the corner (D).  Figure 11 presents the change in thermal performance due to wall conduction compared with data from experiments by [15,16]. In their experiments, assuming a high-blockage ratio of 0.3, when considering wall conduction, the heat transfer rate decreased by 15% compared with convection alone. In the present LES with a blockage ratio of 0.1, when conjugate heat transfer is considered, the average Nusselt number and the total heat transfer rate were predicted to decrease by 3% compared with that those during pure convection (Figure 11a). When the high-blockage ratio is adopted, the heat transfer promotion effects are similar to that of the present study, but the thermal performance decreases due to an increase in pressure drop. The thermal performance degradation due to wall conduction is much more severe at a high-blockage ratio.  Figure 11 presents the change in thermal performance due to wall conduction compared with data from experiments by [15,16]. In their experiments, assuming a highblockage ratio of 0.3, when considering wall conduction, the heat transfer rate decreased by 15% compared with convection alone. In the present LES with a blockage ratio of 0.1, when conjugate heat transfer is considered, the average Nusselt number and the total heat transfer rate were predicted to decrease by 3% compared with that those during pure convection (Figure 11a). When the high-blockage ratio is adopted, the heat transfer promotion effects are similar to that of the present study, but the thermal performance decreases due to an increase in pressure drop. The thermal performance degradation due to wall conduction is much more severe at a high-blockage ratio. Since ribs can be considered to be extended surfaces or fins, the performance of fins was analyzed. Fin performance is evaluated by examining fin efficiency and fin effectiveness [33]. Fin efficiency is the actual fin heat transfer rate (qf) divided by the maximum possible heat transfer rate (qmax). Here, since the rib plays the role of a fin. Fin efficiency can be obtained by using Equation (6)   Since ribs can be considered to be extended surfaces or fins, the performance of fins was analyzed. Fin performance is evaluated by examining fin efficiency and fin effectiveness [33]. Fin efficiency is the actual fin heat transfer rate (q f ) divided by the Energies 2021, 14, 2096 13 of 17 maximum possible heat transfer rate (q max ). Here, since the rib plays the role of a fin. Fin efficiency can be obtained by using Equation (6) [34]:

Thermal Performance and the Biot Number
Fin efficiency is 98.9% close to pure convection in the present LES but 75.7% in the experimental data of [15,16].
Another measure of fin performance is fin effectiveness [34]. It is defined as the ratio of the fin heat transfer rate to the heat transfer rate that would exist without the fin and can be obtained using the following equation: In Equation (7), A c,b is the fin cross-sectional area at the base. In the present study, A c,b per unit width is the bottom area e of the rib. Fin effectiveness is calculated to be~8 in pure convection for both the data from the experiments by [15,16] and the present LES. At a blockage ratio of 0.1 (present LES), fin efficiency is 98.9%, so fin effectiveness is almost maintained even in conjugate heat transfer. However, at a blockage ratio of 0.3 (as in the experiments conducted by [15,16]), fin effectiveness decreases to 6.19 by conduction, but it is still a much larger value than 2, which is considered to be effective.
In conjugate heat transfer, [16] suggested that the Nusselt number becomes a function of the Reynolds number (Re), conductivity ratio (K*), and Bi using dimensional analysis. Coletti et al. [19] also reported that Bi should be considered because the length scales of the fluid and solid regions are different. In both papers, Bi is defined as in Equation (8) using the thickness of the solid wall (d) as the characteristic length: In the heat transfer of the gas turbine blade, the typical Bi is reported to be~0.3 [35]. Coletti et al. [19] stated that experiments were conducted with Bi values ranging from 0.1 to 1.7, and [15,16] described that Bi was set to 1. When Bi = 1, the temperature drops in the solid and fluid regions should be similar. In the solid temperature distribution of [15] (Figure 3c), the temperature difference between the heating surface and the channel wall is~2 K, which is much smaller than the average temperature difference between the solid and the fluid, which is~25 K.
In pure convection, the temperature inside the solid is uniform, and conduction does not occur, so it can be seen as a condition where Bi = 0. Substituting the average Nu of [16] into Equation (8) to results in a Bi of 0.094. This value well describes the ratio of the temperature drop between the solid and the fluid in the experimental data. In the present LES, Bi is calculated as 0.05, which supports the result that most of the temperature changes occur on the fluid side.
By integrating the local heat transfer coefficients presented in Figures 4 and 5, the conjugate and pure convection data are compared in Figure 12 by location. For conduction, the heat transfer coefficient generally decreases compared with that during pure convection, except for the top surface of the rib. In the present LES, the heat transfer rate for each location indicates a difference of less than 10% in conjugate and isothermal heat transfer, but the data from [15,16] indicate a large difference in the front and rear faces of the rib.
conjugate and pure convection data are compared in Figure 12 by location. For conduction, the heat transfer coefficient generally decreases compared with that during pure convection, except for the top surface of the rib. In the present LES, the heat transfer rate for each location indicates a difference of less than 10% in conjugate and isothermal heat transfer, but the data from [15,16] indicate a large difference in the front and rear faces of the rib. When Bi is defined, the blade thickness (d) is mainly used as the characteristic length (Equation (8)). However, as illustrated in Figure 3b, most of the temperature change that occurs inside the solid takes place in the rib. To examine this, the local Bi values of all the rib faces are compared in Figure 13. In Figure 13a, the data of [24], which were tested with a brass specimen with high thermal conductivity, indicate that Bi approaches 0 and is close to that in isothermal wall. The present LES has a range similar to that of the data of [16] or [18] in the front surface (0 < s/e < 1) and the back surface (2 < s/e < 3) of the rib. On the top surface (1 < s/e < 2), the local Bi of the present LES is much smaller than that in the high-blockage data. When Bi is defined, the blade thickness (d) is mainly used as the characteristic length (Equation (8)). However, as illustrated in Figure 3b, most of the temperature change that occurs inside the solid takes place in the rib. To examine this, the local Bi values of all the rib faces are compared in Figure 13. In Figure 13a, the data of [24], which were tested with a brass specimen with high thermal conductivity, indicate that Bi approaches 0 and is close to that in isothermal wall. The present LES has a range similar to that of the data of [16] or [18] in the front surface (0 < s/e < 1) and the back surface (2 < s/e < 3) of the rib. On the top surface (1 < s/e < 2), the local Bi of the present LES is much smaller than that in the high-blockage data.  Table 1 for information related to the compared data.).
As discussed in Figure 11, the fin efficiency obtained from the present LES data differs considerably from that obtained from the data of [16], but it is difficult to explain this difference using Bi (Figure 13a). In Figure 13b, Bi values based on rib height were compared. The Bi value of the present LES is within 0.05, which can explain why the results are so similar to those under isothermal conditions. As the Bi value obtained from the data of [16] becomes larger than 0.1, the conduction effect appears to affect fin efficiency. Therefore, use of the rib height as the characteristic length of Bi in the internal cooling passage of a gas turbine blade results in clearer conjugate heat transfer characteristics.

Conclusions
In this study, IBM-based LES was performed for conjugate heat transfer inside a ribbed channel. The use of the IBM meant that conduction and convection could be fully  Table 1 for information related to the compared data.).
As discussed in Figure 11, the fin efficiency obtained from the present LES data differs considerably from that obtained from the data of [16], but it is difficult to explain this difference using Bi (Figure 13a). In Figure 13b, Bi values based on rib height were compared. The Bi value of the present LES is within 0.05, which can explain why the results are so similar to those under isothermal conditions. As the Bi value obtained from the data of [16] becomes larger than 0.1, the conduction effect appears to affect fin efficiency. Therefore, use of the rib height as the characteristic length of Bi in the internal cooling passage of a gas turbine blade results in clearer conjugate heat transfer characteristics.

Conclusions
In this study, IBM-based LES was performed for conjugate heat transfer inside a ribbed channel. The use of the IBM meant that conduction and convection could be fully coupled, and a dynamic subgrid scale model could be applied. The blockage ratio was fixed at 0.1, which is typically used in the mid-chord region of the gas turbine blade. The wall thickness was set at three times the rib height with reference to the actual turbine blade. The main results obtained through the simulation can be summarized as follows: (1) The heat transfer peak that occurs in front of the rib is not caused by unheated ribs but rather by impinging cold fluid. When the thermal properties of the gas turbine blade are applied, secondary heat transfer peaks occur in front of the ribs, even in cases of conjugate heat transfer. (2) In conjugate heat transfer, the average heat transfer rate and thermal performance were reduced by 3% compared with those during pure convection. On the channel wall, there appeared to be slight decreases in the variation of the local heat transfer in the windward face of the rib and on the top surface. Even for the conducting rib, high heat transfer was predicted at the upstream edge of the rib. (3) In the conjugate heat transfer, the overall distribution of the turbulent heat flux was similar to that in the isothermal heat transfer and was consistent with the local heat transfer distribution on the front and rear surfaces of the rib perpendicular to the main flow. The temperature fluctuation inside the solid was much smaller than that in the fluid region, and most of the fluctuation occurred in the rib. (4) When the thermal performance was evaluated using the rib as an extended surface, fin effectiveness and efficiency were 8.32 and 98.9%, respectively, under typical gas turbine operating conditions. Both indices are recommended values in fin design, meaning that the rib performs well as a fin. Data Availability Statement: Data is contained within this article.

Conflicts of Interest:
The authors declare no conflict of interest.