**3. Governing Equations and Solution Procedure**

Since the cooling air that flowed through the channel with pin fins was subjected to high heat loads, the density and temperature of the air were expected to change considerably along the channel. Thus, though the Mach number of the flow throughout the channel was low, the compressible formulation with temperature-dependent properties was needed. In this study, steady RANS was used from x = −115D to −15D, and LES was used at x = −5D and downwards. The density-weighted Reynolds-averaged and spatially-filtered continuity, momentum, and energy equations for RANS and for LES can be written as [24]

$$\frac{\partial L}{\partial t} + \nabla \cdot (F\_I - F\_V) = 0 \tag{1}$$

where *U*, *FI* = (*FI*,1, *FI*,2, *FI*,3), and *FV* = (*FV*,1, *FV*,2, *FV*,3) are given by

*U* = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ *ρ ρu*,1 *ρu*,2 *ρu*,3 *ρ*,*e* ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ *FI*,1 = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ *ρu*,1 *<sup>ρ</sup>u*,1*u*,<sup>1</sup> <sup>+</sup> *<sup>p</sup> <sup>ρ</sup>u*,1*u*,<sup>2</sup> *<sup>ρ</sup>u*,1*u*,<sup>3</sup> (*ρ*,*<sup>e</sup>* <sup>+</sup> *<sup>p</sup>*)*u*,<sup>1</sup> ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ *FI*,2 = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ *ρu*,2 *<sup>ρ</sup>u*,1*u*,<sup>2</sup> *<sup>ρ</sup>u*,2*u*,<sup>2</sup> <sup>+</sup> *<sup>p</sup> <sup>ρ</sup>u*,2*u*,<sup>3</sup> (*ρ*,*<sup>e</sup>* <sup>+</sup> *<sup>p</sup>*)*u*,<sup>2</sup> ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ *FI*,3 = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ *ρu*,3 *<sup>ρ</sup>u*,1*u*,<sup>3</sup> *<sup>ρ</sup>u*,2*u*,<sup>3</sup> *<sup>ρ</sup>u*,2*u*,<sup>3</sup> <sup>+</sup> *<sup>p</sup>* (*ρ*,*<sup>e</sup>* <sup>+</sup> *<sup>p</sup>*)*u*,<sup>3</sup> ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ *FV*,1 = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ 0 *σ*<sup>11</sup> *σ*<sup>12</sup> *σ*<sup>13</sup> *<sup>σ</sup>*1*ku*,*<sup>k</sup>* <sup>−</sup> *<sup>q</sup>*,<sup>1</sup> ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ *FV*,2 = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ 0 *σ*<sup>12</sup> *σ*<sup>22</sup> *σ*<sup>23</sup> *<sup>σ</sup>*2*ku*,*<sup>k</sup>* <sup>−</sup> *<sup>q</sup>*,<sup>2</sup> ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ *FV*,3 = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ 0 *σ*<sup>13</sup> *σ*<sup>23</sup> *σ*<sup>33</sup> *<sup>σ</sup>jku*,*<sup>k</sup>* <sup>−</sup> *<sup>q</sup>*,<sup>3</sup> ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ *σij* = 2*μS*<sup>∗</sup> *ij* + *τij*, *S*<sup>∗</sup> *ij* <sup>=</sup> *Sij* <sup>−</sup> <sup>1</sup> 3 *∂u*,*k ∂xk <sup>δ</sup>ij*, *Sij* <sup>=</sup> <sup>1</sup> 2 - *∂u*,*i ∂xj* + *∂u*,*j ∂xi* . , *<sup>K</sup>* <sup>=</sup> <sup>1</sup> 2 *u* 0 *ku k <sup>q</sup>*,*<sup>j</sup>* <sup>=</sup> <sup>−</sup> - *k* + *Cp μt* Pr*t <sup>∂</sup>T*, *∂xj*

In the above equations, *p* is the Reynolds-averaged or spatially-filtered pressure and is connected to density and temperature through the thermally perfect equation of state. The turbulent stresses in RANS and LES both invoke the Boussinesq concept, namely,

$$\pi\_{ij,RANS} = 2\mu\_{t,RANS} S\_{ij}^\* - \frac{2}{3} \overline{\rho} k \delta\_{ij}$$

$$\pi\_{ij,LES} = 2\mu\_{t,LES} S\_{ij}^\* - \frac{2}{3} \overline{\rho} k \delta\_{ij}$$

For RANS, *μt*,*RANS* is modelled by the shear-stress transport (SST) model [25]. For LES, two different subgrid-scale models were used for *μt*,*LES*: the dynamic kinetic energy model (DKEM) [26] and the wall-adapting local eddy-viscosity model (WALE) [27]. Two subgrid models were examined because of the complexity of the unsteady shedding about pin fins, the horseshoe vortices about the bases of pin fin, and the interactions among them in the near-wall region under high heat-load conditions. In this study, the temperature-dependence of the constant pressure specific heat, dynamic viscosity, and thermal conductivity were accounted for. The turbulent thermal conductivity was modelled by connecting it to the turbulent viscosity through the turbulent Prandtl number, Pr*t*, which was set to 0.85 for both RANS and LES.

The boundary conditions imposed are as follows: At the inflow boundary (x = −115D), uniform mean temperature and mass flow rate were imposed for the RANS. RANS was used from x = −115D to −15D to obtain a solution that was no longer affected by the entrance region (referred to as "fully" developed flow if the flow was incompressible). LES was used for x > −15D. To get the turbulent fluctuations started and self-sustaining for LES, the synthetic turbulence generator (STG) method of Shur et al. [28], a Fourier based method, was applied at x = −15D. The input into STG was the RANS solution obtained at x= −15D. 15D was the distance between where LES started and where the test section started. That distance, 15D, was obtained by numerical experiments to ensure that the correct turbulent structures were produced before the cooling flow entered the test section, as will be explained in the section on verification and validation. The boundary condition

imposed at the outflow boundary (x = 15D) was constant static pressure at Pb. At periodic boundaries (x = −1.25D and +1.25D), periodic conditions were imposed.

Solutions to the governing equations were obtained by using version 19.2 of the ANSYS Fluent code [29]. For both RANS and LES, the finite-volume method with the SIMPLE algorithm was used. For RANS, the fluxes at cell faces were interpolated by using the second-order upwind scheme, and the Poisson equation for pressure was computed by using a second-order scheme. For LES, the time derivatives were approximated by a second-order accurate in time bounded implicit scheme. The fluxes at the cell faces were interpolated by using a bounded central difference scheme, and the pressure equation was computed by using a second-order central scheme.

For RANS, only steady-state solutions were of interest. Iterations were continued until all residuals plateaued. At convergence, the scaled residuals were always less than 10−<sup>5</sup> for the continuity equation and the three momentum equations, less than 10−<sup>7</sup> for the energy equation and less than 10−<sup>5</sup> for the turbulent transport equations. For LES, time-accurate solutions were of interest. The number of iterations needed to get a converged solution at each time step ranged from 15 to 20 once initial transients washed out. The time-step size used was obtained via a time-step-size sensitivity study described in the next section.

Although the flow was highly unsteady, the flow field was statistically stationary. Thus, all results presented on the flow field and the turbulence statistics were obtained by time-averaging the time-resolved solutions. The time-averaging started once the turbulent flow became statistically stationary and continued until the time-averaged values no longer changed. The maximum number of flow-through time needed to achieve this was nine. One cycle time was defined as L/V, where L = 30D was the length of the LES domain and V was the mean flow speed at the inlet of the LES domain. Since the length of the test section with pin fins was 12.5D, nine flow-through time for the entire LES domain was equivalent to 21.6 flow-through time for the test section. Also, since the flow speed increased along the duct because of heating, the maximum number of flow-through time for the test section based on actual flow speed was higher than 21.6.

#### **4. Verification and Validation**

To verify this study, a grid-and time-step-size sensitivity study was performed. On validation, it was conducted in two parts. Data from direct numerical simulation (DNS) was used to validate the turbulent flow predicted in the smooth part of the channel upstream of the test section, and available experimental data was used to validate the flow predicted in the test section with pin fins. The details of these studies are given below.
