Strongly Heated Turbulent Flow in a Channel with Pin Fins

Abstract

Large-eddy simulations (LES) were performed to study the turbulent flow in a channel of height H with a staggered array of pin fins with diameter D = H/2 as a function of heating loads that are relevant to the cooling of turbine blades and vanes. The following three heating loads were investigated—wall-to-coolant temperatures of T_w/T_c = 1.01, 2.0, and 4.0—where the Reynolds number at the channel inlet was 10,000 and the back pressure at the channel outlet was 1 bar. For the LES, two different subgrid-scale models—the dynamic kinetic energy model (DKEM) and the wall-adapting local eddy-viscosity model (WALE)—were examined and compared. This study was validated by comparing with data from direct numerical simulation and experimental measurements. The results obtained show high heating loads to create wall jets next to all heated surfaces that significantly alter the structure of the turbulent flow. Results generated on effects of heat loads on the mean and fluctuating components of velocity and temperature, turbulent kinetic energy, the anisotropy of the Reynolds stresses, and velocity-temperature correlations can be used to improve existing RANS models.

1. Introduction

The thermal efficiency of gas turbines can be increased by increasing the temperature of the gas entering the turbine component. In advanced gas turbines, the gas temperature entering the turbine can far exceed the melting point of the turbine’s material. Thus, the turbine component must be cooled to maintain its mechanical strength for reliable operation. One part that is especially difficult to cool is the trailing-edge regions of the turbine’s blades and vanes. In those regions, embedded cooling passages with pin fins have been found to be effective [1,2,3,4].

Since the physical processes that take place in channels with pin fins are quite complicated, many investigators have performed experimental and computational studies to understand how design and operating parameters affect heat transfer and pressure drop (see e.g., [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22] and references cited there). Parameters studied include pin-fin shape (circular, square, diamond), pin-fin hydraulic diameter to channel height, arrangement of pin fins (staggered and inline), pin-to-pin streamwise and spanwise spacings, and the Reynolds number of the flow. Of the experimental studies, most were based on time-averaged measurements. Only Ames et al. [9,10,11] and Ostanek and Thole [16] examined time-resolved measurements; these studies showed the flow in channels with a staggered array of pin fins at Reynolds numbers from 3000 to 30,000 to be highly unsteady due to vortex shedding and unsteady separation.

Most computational studies were also based on steady analyses by using steady RANS (i.e., Reynolds-averaged continuity, Navier-Stokes, and energy equations) closed by one or more of the following models: k-ω, Shear-Stress Transport (SST), Explicit Algebraic Reynolds Stress (EARS), realizable k-ε, Renormalization Group (RNG) k-ε, and ν²-f [15,19]. Steady RANS was found to predict trends with reasonable accuracy. However, quantitatively, relative errors could be 10–30% or more.

Relatively few investigators performed time-accurate simulations of the unsteady flow and heat transfer in channels with pin fins. Delibra et al. [12,13,14] used unsteady RANS (URANS) with the elliptic-relaxation eddy-viscosity (ζ-f) turbulence model. By capturing the large-scale wake shedding structures, the relative error was reduced to less than 10%. Paniagua et al. [18] used large-eddy simulation (LES) and a hybrid LES-RANS method with LES away from walls and URANS next to walls. Their study showed that the hybrid LES-RANS was able to capture the dominant large-scale eddies and mean flow quantities with reasonable accuracy. Their study also assessed three LES subgrid-scale (SGS) models–WALE, QR, and VMS and found the WALE model to predict the mean velocity distribution the best and the QR model to predict the mean pressure coefficient the best.

The aforementioned studies only investigated low heat loads where the wall-to-coolant temperature was near unity. However, that ratio can be as high as two or more in cooling passages in turbine vanes and blades. Shih et al. [20] and Lee et al. [21,22,23] studied the effects of heating load on flow, heat transfer, and pressure loss by using steady RANS, URANS, LES, and hybrid LES. Shih et al. [20] showed scaling formulas that accounted for heating developed for passages with smooth walls to yield large errors when applied to passages with pin fins. They developed a method to scale experimental data obtained at low heat loads in laboratory conditions to high heat loads in gas turbines. Lee et al. [21,22,23] showed that high heating load significantly increased the length of the entrance region and reduced the Nusselt number (Nu) in the entrance and post-entrance regions. Lee et al. [22] used LES to examine the turbulence statistics created by high heating loads. Lee et al. [23] showed RANS to underpredict Nu because it did not account for vortex shedding and URANS to first underpredict and then overpredict Nu because the RANS models were unable to predict at which rows shedding occurred and then overpredicted its effects on heat transfer. Thus, better RANS models are needed for this class of flows with unsteady separation under high heat loads.

Though the study by Lee et al. [21,22,23] provided considerable understanding on how heat load affects flow and heat transfer, the details of the turbulent flow field, such as Reynolds stresses and velocity-temperature correlations, were not provided and analyzed. Since such information could assist the development of better RANS models to study this class of flows as well as enable improved designs of pin-fin arrays under high heat loads, the objective of this study is to use LES to study the statistics of the turbulent flow structure created by high heat loads.

The remainder of this paper is organized as follows. The problem studied is first described. Then, the problem formulation, the numerical method of solution, and the verification and validation are presented. This is followed by results obtained on the effects of heat load and the turbulent statistics.

2. Problem Description

A schematic of the problem studied is shown in Figure 1, where all dimensions are given in terms of D = 2.54 cm, the diameter of the pin fin. In this figure, the cooling channel is bound by two flat plates with length 130D and height H = 2D. This channel is made up of three sections – a test section of length 12.5D and two smooth sections connected to the test section. The test section has five rows of pin fins that are arranged in a staggered fashion. The center of the first pin fin is located at x = 1.25D, and the spacing between centers of pin fins is S_x = 2.5D in the streamwise (x) and S_z = 2.5D in the spanwise directions (z). All solid surfaces of the test section, including the surfaces of the pin fins, are isothermal. For the two smooth sections, where there are no pin fins, all solid surfaces are adiabatic. The smooth section of length 115D is attached to the inlet of the test section to ensure that flow entering the test-section has a “fully developed” velocity profile in the sense that it is no longer affected by the entrance region. The smooth section of length 2.5D is attached to the exit of the test section to ensure no reversed flow at the outflow boundary.

For this cooling channel, the cooling flow that enters at x = −115D is air and has a uniform temperature of T_c = 300 K and a mass flow rate of ${\dot{m}}_c$ = 0.014 kg/s. The pressure at the channel’s outlet located at x = 15D is maintained at P_b = 1 bar. The Reynolds number (Re_D) is 10,000, which is based on the pin-fin diameter D, the averaged momentum of $4{\dot{m}}_c/\left({\mathsf{\pi }\mathrm{D}}^2\right)$, and the dynamic viscosity evaluated at T_c. As noted, all solid surfaces of the test section are isothermal, and the following surface or wall temperatures were investigated: T_w = 303 K, 600 K, and 1200 K, representing negligible, moderate, and strong heat loads, respectively.

Table 1. Summary of simulations performed.

Case	Tw (K)	Tw/Tc	SGS Model
NH-DKEM MH-DKEM SH-DKEM	303 600 1200	1.01 (negligible heating, NH) 2.0 (moderate heating, MH) 4.0 (strong heating, SH)	DKEM
NH-WALE MH-WALE SH-WALE	303 600 1200	1.01 (NH) 2.0 (MH) 4.0 (SH)	WALE

T_c = 300 K for all cases.

summarizes all of the cases studied.

The configuration just described matches the experimental study of Ames and Dvorak [11] with detailed measurements of the mean turbulent flow field. Their data was used to validate this computational study. Here, it is also noted that this configuration differs from the configurations studied by Lee and Shih [23] by having a channel height of 2D instead of D so that the pin fins have height-to-diameter ratio of two instead of one.

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]

$$\frac{\partial U}{\partial t}+\nabla ·\left(F_I-F_V\right)=0$$

(1)

where $U$, $F_I=\left(F_{I,1},F_{I,2},F_{I,3}\right)$, and $F_V=\left(F_{V,1},F_{V,2},F_{V,3}\right)$ are given by

$$U=\left[\begin{array}{c}\overline{\rho }\\ \overline{\rho }\widetilde{u_1}\\ \begin{array}{c}\overline{\rho }\widetilde{u_2}\\ \overline{\rho }\widetilde{u_3}\\ \overline{\rho }\tilde{e}\end{array}\end{array}\right]$$

$$F_{I,1}=\left[\begin{array}{c}\overline{\rho }\widetilde{u_1}\\ \overline{\rho }\widetilde{u_1}\widetilde{u_1}+\overline{p}\\ \overline{\rho }\widetilde{u_1}\widetilde{u_2}\\ \overline{\rho }\widetilde{u_1}\widetilde{u_3}\\ (\overline{\rho }\tilde{e}+\overline{p})\widetilde{u_1}\end{array}\right] F_{I,2}=\left[\begin{array}{c}\overline{\rho }\widetilde{u_2}\\ \overline{\rho }\widetilde{u_1}\widetilde{u_2}\\ \overline{\rho }\widetilde{u_2}\widetilde{u_2}+\overline{p}\\ \overline{\rho }\widetilde{u_2}\widetilde{u_3}\\ (\overline{\rho }\tilde{e}+\overline{p})\widetilde{u_2}\end{array}\right] F_{I,3}=\left[\begin{array}{c}\overline{\rho }\widetilde{u_3}\\ \overline{\rho }\widetilde{u_1}\widetilde{u_3}\\ \overline{\rho }\widetilde{u_2}\widetilde{u_3}\\ \overline{\rho }\widetilde{u_2}\widetilde{u_3}+\overline{p}\\ (\overline{\rho }\tilde{e}+\overline{p})\widetilde{u_3}\end{array}\right]$$

$$F_{V,1}=\left[\begin{array}{c}0\\ {\sigma }_{11}\\ \begin{array}{c}{\sigma }_{12}\\ {\sigma }_{13}\\ {\sigma }_{1k}\widetilde{u_k}-\widetilde{q_1}\end{array}\end{array}\right] F_{V,2}=\left[\begin{array}{c}0\\ {\sigma }_{12}\\ \begin{array}{c}{\sigma }_{22}\\ {\sigma }_{23}\\ {\sigma }_{2k}\widetilde{u_k}-\widetilde{q_2}\end{array}\end{array}\right] F_{V,3}=\left[\begin{array}{c}0\\ {\sigma }_{13}\\ \begin{array}{c}{\sigma }_{23}\\ {\sigma }_{33}\\ {\sigma }_{jk}\widetilde{u_k}-\widetilde{q_3}\end{array}\end{array}\right]$$

$${\sigma }_{ij}=2\mu S_{ij}^{*}+{\tau }_{ij}, S_{ij}^{*}=S_{ij}-\frac{1}{3}\frac{\partial \widetilde{u_k}}{\partial x_k}{\delta }_{ij}, S_{ij}=\frac{1}{2}\left(\frac{\partial \widetilde{u_i}}{\partial x_j}+\frac{\partial \widetilde{u_j}}{\partial x_i}\right), K=\frac{1}{2}\widetilde{u_k^{’}{u}_k^{’}}$$

$$\widetilde{q_j}=-\left(k+C_p\frac{{\mu }_t}{{\mathrm{Pr}}_t}\right)\frac{\partial \tilde{T}}{\partial x_j}$$

In the above equations, $\overline{p}$ 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,

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

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

For RANS, ${\mu }_{t,RANS}$ is modelled by the shear-stress transport (SST) model [25]. For LES, two different subgrid-scale models were used for ${\mu }_{t,LES}$: 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 P_b. 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 10⁻⁵ for the continuity equation and the three momentum equations, less than 10⁻⁷ for the energy equation and less than 10⁻⁵ 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.

4.1. LES of Flow in Channel without Pin Fins

To ensure that the synthetic turbulence inflow BC imposed at x = −15D could provide the correct turbulent flow to enter into the test section with pin fins at x = 0, three grids were examined along with time-step sizes that would yield stable and accurate solutions. The three grids used, denoted as coarse, baseline, and fine, are shown in Figure 2 and summarized in

Table 2. Summary of grid used for channel without pin fins.

Grid		Coarse	Baseline	Fine
Total Number of Cells (million)		1.64	2.79	5.08
Cell Spacing (max./min.)	Δx+	$\approx$16/16	$\approx$16/8	$\approx$16/4
	Δy+	13.9/0.9	13.7/0.5	13.5/0.3
	Δz+	$\approx$16/16	$\approx$16/8	$\approx$16/4
y+ of Cell next to Wall		0.9	0.5	0.3

. Note that h-refinement was used when finer resolution was needed next to the wall to maintain a consistent aspect ratio as the grid is refined. The coarse grid in the LES region from x = −100D to x = 0 consisted of 1.64 million cells with maximum/minimum non-dimensional cell spacings of 16/16 in the streamwise direction, 13.9/0.9 in the wall-normal direction, and 16/16 in the spanwise direction. The baseline grid consisted of 2.79 million cells with max./min. non-dimensional cell spacings of 16/8, 13.7/0.5, and 16/8 in the streamwise, wall-normal, and spanwise directions, respectively. The fine grid consists of 5.08 million cells with max./min. cell spacings of 16/4, 13.5/0.3, and 16.4/0.3 in the streamwise, wall-normal direction, and spanwise directions, respectively. Here, it is noted that the region upstream of the LES region (x = −115D to −15D), where RANS with the SST model was used to provide the “fully-developed” mean flow at x = −15D for the LES region, had a grid with 2.5 M cells. This grid provided grid-independent solutions for the RANS region, where the first cell next to the channel walls had y+ less than unity, and there were five cells within y+ of five.

Figure 3 shows the mean flow profiles normalized by the friction velocity at Re_τ = 365 obtained by using the coarse, baseline, and fine grids along with results from DNS [30] and experimental measurements [31]. From this figure, the solution from the coarse grid can be seen to overpredict the DNS and experimental data in the log-law region with y+ between 10 and 100, indicating inadequate resolution in the near-wall region. Solutions from both the baseline and fine grids were in good agreement with the log-law equation and the DNS and experimental data.

Figure 4 shows the Reynolds stress and the streamwise and spanwise velocity fluctuations normalized by the friction velocity. The solution from the coarse grid can be seen to overpredict the DNS data, whereas solutions from the baseline and fine grids match well.

Figure 5 shows the power spectrum of turbulent kinetic energy at (x, y) = (56.25H, 0.5D) in the “fully-developed” region of the channel obtained by the baseline grid. The spectrum in the inertial subrange follows the Kolmogorov’s −5/3 energy decay slope, indicating that most of the energetic large-scale eddies were resolved by the grid. According to these comparisons, the baseline grid had sufficient near-wall resolution to resolve the turbulent boundary layer and required less computation costs than the fine grid. Thus, the grid resolution based on the baseline grid was used for all LES simulations in the region between x = −15D and x = 0.

4.2. LES of Flow in Channel Flow with Pin Fins

Figure 6 shows a close-up view of the grid used in the test section with pin fins. As shown in the figure, a wrap-around grid was used about each pin fin. Also, grid points were clustered to all solid surfaces. The grid spacings used were guided by the “baseline grid” described in the previous section to ensure that the turbulent structures about all solid surfaces were adequately resolved, including those about pin fins.

Table 3. Summary of grid and time-step size used for channel with pin fins.

Total Number of Cells: 11.5 Million		Tw/Tc = 1.01 (Negligible Heating)	Tw/Tc = 2.0 (Moderate Heating)	Tw/Tc = 4.0 (Strong Heating)
Cell Spacing (max.)	Δx+	11.4	10.2	8.4
	Δy+	16.1/0.5	14.3/0.4	10.3/0.3
	Δz+	7.6/0.5	6.8/0.4	5.1/0.3
Wall y+ (endwall/pin-wall)		0.9/0.5	0.8/0.4	0.5/0.3
Time-step Size, Δt+		0.1378	0.0841	0.0111

summarizes the grid spacings and time-step sizes used for all the LES cases performed. Note that the grid spacings and time-step sizes had to be reduced to get the required spatial and temporal accuracy when the heating load was increased.

Figure 7 shows the power spectra of turbulent kinetic energy at (x/D, y/D, z/D) = (5, 1, 0) and (10, 1, 0) behind Pin 2 and Pin 4 at different heating loads. In the figure, the energy spectra can be seen to follow the Kolmogorov’s -5/3 energy decay slope in the inertial subrange before their cut-off wave numbers.

To further assess the grid resolutions, the index of resolution quality for LES, $LESI{Q}_{\nu }$, introduced by Celik et al. [32], given by the equation below was computed:

$$LESI{Q}_{\nu }=\frac{1}{1+0.05{\left(\frac{\left(\mu +{\mu }_{SGS}\right)}{\mu }\right)}^{0.53}}$$

(2)

Figure 8 shows the $LESI{Q}_{\nu }$ distributions from the LES solutions obtained under three levels of heating. Since the minimum value of $LESI{Q}_{\nu }$ is 0.94, which is higher than 0.8, it shows that the grids used were able to resolve a significant portion of turbulent kinetic energy.

To validate the LES solutions, the results generated were compared with the experimental data of Ames & Dvorak [9,10,11]. Figure 9 shows a comparison between computed and measured turbulent boundary layer profiles normal to the pin-fin wall and normal to the endwall. From that figure, LES can be seen to predict the mean flow velocity well with maximum relative difference less than 2%. Note that the profile normal to the pin-wall at Row#1 does show noticeable overprediction in the outer region with about 8% relative difference, but matches well to the near-wall region.

4.3. DKEM vs. WALE Models

DKEM sub-grid scale model was used in the grid-sensitivity and validation study described in the previous section. In this section, DKEM is compared with the wall-adapting local eddy-viscosity sub-grid scale model (WALE). Figure 10 shows profiles of the mean velocity, streamwise velocity fluctuation, and velocity-temperature correlations obtained by using the DKEM and WALE SGS models. From this figure, it can be seen that both DKEM and WALE models predict similar results in all of these quantities. Also, the results obtained match the experimental data well, which gives confidence to the results obtained by LES with DKEM.

5. Results

As noted in the introduction, the objective of this study was to examine the details of the turbulent flow created by increasing heat load from T_w/T_c = 1.01 to 4.0. In this section, the results are presented in the following order. First, the structure of turbulent flow induced by heat loads is described. Next, results for the mean velocity and temperature profiles and their root-mean-square of fluctuation are described. This is followed by results obtained for the turbulent kinetic energy, Reynolds stresses and measures of their anisotropy, and velocity-temperature correlations. The results obtained are plotted in two midplanes (an x-z plane at y = H/2 and an x-y plane at z = 0) and along several coordinate lines emanating from the endwall and from pin fins with row numbers 1 to 5 designating x/D = 1.25, 3.75, 6.25, 9.75, and 11.25 locations, respectively.

5.1. Flow Structures

Figure 11 shows the turbulent structures in the cooling channel studied by using iso-surfaces of the velocity gradient tensor based on the λ₂-criterion [33] colored by the magnitude of the mean velocity. The turbulent structures shown in this figure are dominated by three flow mechanisms and their interactions, and they are as follows: (1) horseshoe vortices about each pin fin; (2) jet impingement on the leading edge of each pin fin; and (3) the recirculating flows in the wake of each pin fin due to flow separation. Of these, the recirculating flows in the wake were found to be the most affected by heat load. At all heat loads, recirculating flows about pin fins in the first row did not shed. At low heat loads, those recirculating flows shed for pin fins in the second and higher rows. However, at high heat loads, those recirculating flows did not shed until the third row of pin fins. This is because when the heat load was high, the temperature of the air in the recirculating flows about second-row pin fins was significantly higher than the temperature of the air that flowed around them. With higher temperature, density is lower. As a result, even though the magnitudes of the velocity of the recirculating flows and the flow around them were similar where they interacted, the momentum of the recirculating flows was significantly less and so could not shed. When recirculating flows did shed, the vortical structures created were large. At low heat loads, the large vortical structures broke up into clusters of smaller-scale structures by the third and fourth pin fins. However, at high heat loads, this breakup did not occur until the fifth-row pin fins. High heat loads delayed breakup because they reduced density, which increased velocity magnitude and acceleration, and acceleration reduced turbulence. Thus, high heat loads reduced mixing and heat transfer by delaying the shedding and breakup of the large-scale vortical structures. Additional details of the flow field can be found in Lee & Shih [23].

Figure 12 shows the mean streamwise velocity contours and profiles ($u^{*}$) as a function of heat load. From this figure, it can be seen that the magnitude of the streamwise velocity increased as the heat load increased. This is because increasing the heat load increased the cooling flow’s bulk temperature, which reduced its density and hence increased its velocity. Though the velocity magnitude of the flow increased with heat load, the velocity gradient next to the wall decreased as heat load increases. This is because heating causes expansion. This expansion also created jet-like flows about each pin fin, and the strength of the jet increased with heat load.

Figure 13 shows the mean temperature distributions ($T^{*}$) along the cooling channel as a function of heat load. From this figure, higher heating loads can be seen to lead to not just higher temperature in the flow but also higher temperature gradients in the thermal boundary layers. In addition, because of the expansion of the cooling air next to the walls, the thicknesses of the thermal boundary layer increased with increases in heat load.

Figure 14 shows the velocity fluctuations—streamwise, wall normal, and spanwise—at three heat loads. From Figure 14a, it can be seen that the streamwise fluctuating velocity ($u_{rms}^{*}$) about the pin fin decreased as the heat load increased for the first two rows of pin fins. This is because the flow there was highly accelerated by the expansion of the cooling air created by heating, and acceleration dampens turbulence. However, once shedding started about the pin fins in the third row, streamwise fluctuating velocity about the pin fin increased as the heat load increased. In the region midway between pin fins, the effect of heat loads was less because most of the acceleration created by the heating occurred near the pin fins. Figure 14b shows that heating of the endwall increased velocity fluctuation normal ($v_{rms}^{*}$) to the wall as the heat load increased. This was caused by the expansion of the air and jet-like flow created by the heating. However, like streamwise velocity fluctuation, normal velocity fluctuation about pin fins in the second row decreased as the heat load increased, but increased with the heat load once shedding started. Figure 14c shows that the spanwise velocity fluctuations ($w_{rms}^{*}$) had the same trend as the normal velocity fluctuations ($v_{rms}^{*}$).

Figure 15 shows the variance of the temperature fluctuation (${T^{″}}^{*}$). With T_w/T_c = 1.01 (NH), there was no temperature variance as expected. As T_w/T_c increased, the fluctuating temperature got higher, particularly in the near-wall region, where the mean temperature gradient was also large. For the moderate and strong heating cases, the peak temperature fluctuations next to the endwall increased steadily along the channel. After pin fins in the fourth row, the mean temperature and the normalized temperature fluctuation became periodic from row to row.

5.2. Turbulent Statistics

Figure 16 shows the “total” turbulent kinetic energy normalized by the reference velocity (K*) as a function of heat load. The “total” K is the sum of the turbulent kinetic energy resolved by LES and the “unresolved” turbulent kinetic energy modelled by the SGS. The turbulent kinetic energy was generated in regions of high mean velocity gradients – although there was a delay between the gradients and the production. Regions of high gradients were next to walls created by vortex shedding, horseshoe vortices, and the shear layer where different flow structures interacted. With the heat load at T_w/T_c = 1.01 (which is negligible heating), K was essentially periodic in the streamwise direction with the periodicity repeating from one row of pin fins to the next. With the heat load increased to T_w/T_c = 2.0, K increased along the channel, and the periodicity from one row of pin fins to the next no longer existed. With heat load further increased to T_w/T_c = 4.0, K in the first two rows decreased significantly. This is because once the heat load reaches a critical value, shedding about pin fins is suppressed. With T_w/T_c = 4.0, shedding did not start until pin fins in the third row. Once shedding did start, K with T_w/T_c = 4.0 was significantly higher than the K at lower heat loads. As noted, once there was heat load with shedding, K increased from row to row because the heat load increased the velocity along the channel. This steady increase in K along the channel explains why Lee and Shih [23] observed the row-averaged Nusselt number to be nearly constant though the Reynolds number was steadily decreasing along the channel.

Figure 17 shows the Reynolds stresses normalized by the turbulent kinetic energy ($-\widetilde{u_i^{″}{u}_j^{″}{}^{*}}$). From this figure, it can be seen that only $-{\widetilde{u^{″}{w}^{″}}}^{*}$ was significantly affected by the heat load. However, this is only true at the few coordinate lines where data are shown. Figure 18 shows the distribution of the anisotropy flatness parameter at two planes for the three heating loads. The anisotropy flatness parameter, A, proposed by Lumley and Newman [34] is as follows:

$$A=1-\frac{9}{8}\left(A_2-A_3\right)$$

(3)

$$A_2=a_{ij}{a}_{ji}$$

$$A_3=a_{ij}{a}_{jk}{a}_{ki}$$

$$\mathrm{where} a_{ij}=\frac{\overline{u_i^{’}{u}_j^{’}}}{k}-\frac{2}{3}{\delta }_{ij}$$

where A = 1 implies isotropic turbulence, and A = 0 implies maximum anisotropy. From Figure 18, it can be seen that anisotropy in the Reynolds stresses was highest in the separated regions about the pin fins due to shedding of the recirculating flows and the horseshoe vortices. With T_w/T_c = 1.01, the anisotropy was highest next to the pin fins. Increasing heat load from T_w/T_c = 1.01 to 2.0 increased anisotropy throughout the flow field. However, further increase to T_w/T_c = 4.0 decreased anisotropy in the first three rows. Downstream of the third row, anisotropy did increases with the increase in heat load from T_w/T_c = 2.0 to 4.0.

Figure 19 shows the velocity-temperature correlations, which represent turbulent heat flux components, normalized by the product of the reference velocity and temperature ($-{\widetilde{u^{″}{T}^{″}}}^{+}$_, $-{\widetilde{v^{″}{T}^{″}}}^{+}$, and $-{\widetilde{w^{″}{T}^{″}}}^{+}$). This figure shows the velocity-temperature correlations to be highest next to walls where shedding, horseshoe vortices, and shear-layer interactions took place. Of the correlations, $-{\widetilde{u^{″}{T}^{″}}}^{+}$ was the highest, and it is higher about pin-fin walls because of shedding and about endwalls because of horseshoe vortices, where streamwise fluctuating components played a dominant role. $-{\widetilde{v^{″}{T}^{″}}}^{+}$ was higher about the endwall because of the horse vortices, which had dominant fluctuations normal to the wall. $-{\widetilde{w^{″}{T}^{″}}}^{+}$ was higher about pin fins because of shedding, which had dominant fluctuations in the spanwise direction. Figure 20 shows the distribution of $-{\widetilde{w^{″}{T}^{″}}}^{*}$ in two midplanes, where the effects of shedding about pin fins can be clearly seen.

6. Conclusions

Results obtained from large-eddy simulations of the unsteady flow and heat transfer in a channel with a staggered array of pin fins show the heating load to significantly affect the nature of the flow and the statistics of the turbulence. Though the turbulent structures were dominated by horseshoe vortices at the bases of each pin fin, jet impingement on the leading edge of each pin fin, and vortex shedding about each pin fin, only vortex shedding was the most affected by heat load. Basically, once the heat load reached a critical value, it could suppress vortex shedding about pin fins and breakup of large vortical structures into clusters of smaller-scale structures, which significantly affect turbulence statistics. Velocity fluctuations in the streamwise direction were found to decrease as the heat load increased for the first two rows of pin fins because the flow there was highly accelerated by the expansion of the cooling air created by heating. However, once vortex shedding started, streamwise fluctuating velocity increased as the heat load increased. Increase heating of the endwall also increased velocity fluctuation normal to the wall. Velocity fluctuations in the normal and spanwise directions had similar trends. As expected, temperature fluctuations increased as heat load increased When there was negligible heat load, the turbulent kinetic energy was found to be essentially periodic in the streamwise direction. However, once heat load was increased, the turbulent kinetic energy increased along the channel, and periodicity broke down. When shedding about pin fins in the first few rows was suppressed by high heat loads, the turbulent kinetic energy was significantly reduced. On the Reynolds stresses, the anisotropy was highest next to pin fins if the heat load was negligible. Increasing heat load increased anisotropy throughout the flow field. However, further increase in heat load decreased anisotropy in the first three rows because shedding was suppressed. Of the velocity-temperature correlations, the streamwise-component correlation was the highest, and it was higher about pin fins because of shedding and about endwalls because of horseshoe vortices. Normal-component correlation was higher about endwalls because of horseshoe vortices, and spanwise-component correlations were higher about pin fins because of vortex shedding.