## 3 Computational Procedure

3.1 Overview

The Reynolds-Averaged Navier-Stokes equations in conjunction with a near wall Reynolds stress turbulence model are solved using the chimera RANS method of Chen et al. [23, 24]. The governing equations with the second-moment closure turbulence model were described in detail by Chen et al. [23, 24] and will not be repeated here. The flow is considered to be incompressible since the Mach number is quite low. However, the density in the centrifugal force terms is approximated by p=poTo/T to account for the density variations caused by the temperature differences. po and To are the density and temperature at the inlet of the cooling channel. In general, the density is also a function of the rotating speed because the centrifugal force creates a pressure gradient along the duct. In the experiments of Griffith et al. , the maximum pressure variation between the channel inlet and the exit is approximately 0.0113 atm for the highest rotation number of 0.28 (i.e., W = 550 rpm) considered in the present study. This gives a maximum density variation of only about 1.1% from the inlet to the exit of the duct at the highest rotation number. It is therefore reasonable to omit the density variation caused by the pressure gradients induced by the channel rotation. For completeness, the numerical method will be briefly summarized in the following subsection.

### 3.2 Chimera RANS Method

The present method solves the mean flow and turbulence quantities in arbitrary combinations of embedded, overlapped, or matched grids using a chimera domain decomposition approach. In this approach, the solution domain was first decomposed into a number of smaller blocks to facilitate efficient adaptation of different block geometries, flow solvers, and boundary conditions for calculations involving complex configurations and flow conditions. Within each computational block, the finite-analytic numerical method of Chen et al.  was employed to solve the unsteady RANS equations on a general curvilinear, body-fitted coordinate system. The coupling between the pressure and velocity was accomplished using the hybrid PISO/SIMPLER algorithm of Chen and Patel . The method satisfied continuity of mass by requiring the contravariant velocities to have a vanishing divergence at each time step. Pressure was solved by using the concept of pseudo-velocities and, when combined with the finite-analytic discretization gives the Poisson equation for pressure. To ensure the proper conservation of mass and momentum between the linking grid blocks, the grid-interface conservation techniques of Chen and Chen  were employed to eliminate the unphysical mass source resulting from the interpolation errors between the chimera grid blocks. In the present study, the numerical grids in the block overlap region are fully matched. Therefore, the grid interface conservation is automatically satisfied.

### 3.3 Boundary conditions

A uniform velocity profile was used at the inlet of the duct (Z = 0). The unheated length (Lj) was long enough for the velocity profile to be fully developed turbulent profile before the heating start-point (Z = Lj). At the exit of the duct, zero-gradient boundary conditions were specified for the mean velocity and all turbulent quantities, while linear extrapolation was used for the pressure field. The coolant fluid at the inlet of the duct is air at uniform temperature T = To (i.e., 6 = (T - To) / (Tw - To) = 0). The wall temperature of the unheated sections is kept constant at T = To (6 = 0) while the wall temperature of the heated section is kept constant at T = Tw (6= 1).

### 3.4 Computational grid details

Figure 3.2a and 3.2b show the computational grid for the smooth duct and around the ribs for the ribbed duct. The grid was generated using an interactive grid generation code GRIDGEN . It was then divided into five overlapped chimera grid blocks (three for the case of smooth duct) to facilitate the implementation of the near-wall turbulence model and the specification of the boundary conditions. To provide adequate resolutions of the viscous sublayer and buffer layer adjacent to a solid surface, the minimum grid spacing in the near-wall region is maintained at 10-3 of the hydraulic diameter which corresponds to a wall coordinate y+ of the order of 0.5. The number of grid points in the streamwise direction from inlet to outlet is 50 for the smooth case and 394 for the ribbed duct. Whether smooth or ribbed, the number of grid points in the cross-stream plane is 33 x 75. The number of grid points and their distributions in the present smooth duct were obtained based on extensive grid-refinement studies that were performed in Chen et al. [23, 24] and Al-Qahtani  for similar smooth channels of a square and rectangular cross sections. The interested reader is referred to references [23-25] for the details of the grid refinement studies performed on the smooth surface channels. The number of grid points and their distributions in the present ribbed duct were obtained based on extensive grid-refinement studies that were performed in Jang et al. [31 and 32]. They performed grid refinement studies on 90° and 60° ribbed two-pass square channels with nine ribs and a total grid points of 1,060,000 and 1,020,000 (Re =30,000) respectively. Also, Jang et al.  performed a grid independence study on a 45° ribbed one-pass square channel with 13 ribs (Re =25,000). The numerical grids used in these studies were shown to yield nearly grid-independent results. Also, their results were in close agreement with the experimental data. Note that the number of grid points used in the present rectangular configuration is 33x75x394 grid points with a total number of approximately 1,000,000 points which is comparable to the above mentioned geometries. In addition, the Reyno lds number used in the present study (Re = 10,000) is lower than the one used in the previous studies. Therefore, it is believed that the present grid will produce nearly grid-independent results with accurate resolution of the boundary layer profile and Nusselt number distribution. In all calculations, the root-mean-square (rms) and maximum absolute errors for both the mean flow and turbulence quantities were monitored for each computational block to ensure complete convergence of the numerical solutions and a convergence criterion of 10-5 was used for the maximum rms error.

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