*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 Tw/Tc = 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 Tw/Tc = 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 Tw/Tc = 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 Tw/Tc = 4.0, shedding did not start until pin fins in the third row. Once shedding did start, *K* with Tw/Tc = 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 16.** Total turbulent kinetic energy distribution for various wall heating levels.

Figure 17 shows the Reynolds stresses normalized by the turbulent kinetic energy (−*u* - *<sup>i</sup> u j* <sup>∗</sup>). From this figure, it can be seen that only <sup>−</sup>*u <sup>w</sup>* <sup>∗</sup> 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:

$$A = 1 - \frac{9}{8}(A\_2 - A\_3)\tag{3}$$

$$A\_2 = a\_{ij}a\_{ji}$$

$$A\_3 = a\_{ij}a\_{jk}a\_{ki}$$

$$\text{where } a\_{ij} = \frac{\overline{u\_i' u\_j'}}{k} - \frac{2}{3}\delta\_{ij}$$

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 Tw/Tc = 1.01, the anisotropy was highest next to the pin fins. Increasing heat load from Tw/Tc = 1.01 to 2.0 increased anisotropy throughout the flow field. However, further increase to Tw/Tc = 4.0 decreased anisotropy in the first three rows. Downstream of the third row, anisotropy did increases with the increase in heat load from Tw/Tc = 2.0 to 4.0.

**Figure 17.** Reynolds stress profiles of (**a**) *u v* <sup>∗</sup> , (**b**) *u w* <sup>∗</sup> , (**c**) *v w* <sup>∗</sup> components for various wall heating.

Figure 19 shows the velocity-temperature correlations, which represent turbulent heat flux components, normalized by the product of the reference velocity and temperature (−*u <sup>T</sup>* <sup>+</sup> , −*v <sup>T</sup>* <sup>+</sup> , and −*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, −*u <sup>T</sup>* <sup>+</sup> 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. −*v <sup>T</sup>* <sup>+</sup> was higher about the endwall because of the horse vortices, which had dominant fluctuations normal to the wall. −*w* - *T* + was higher about pin fins because of shedding, which had dominant fluctuations in the spanwise direction. Figure 20 shows the distribution of −*w* - *T* ∗ in two midplanes, where the effects of shedding about pin fins can be clearly seen.

**Figure 19.** Resolved turbulent heat flux distributions for various heat loads.

**Figure 20.** Resolved <sup>−</sup>*w T* ∗ in two midplanes (Tw/Tc = 4.0).

#### **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.

**Author Contributions:** C.-S.L.: Methodology, Validation, Analysis, Writing—original draft. T.I.-P.S.: Conceptualization, Analysis, Writing—original draft. K.M.B.: Writing—review—editing. R.P.D.: Supervision, Writing—review—editing. R.A.D.: Conceptualization, Writing—review & editing. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was supported by the US Department of Energy's Ames Laboratory and the National Energy Technology Laboratory. The Contract No. is DE-AC02-07CH11358, and the Agreement No. is 26110-AMES-CMI. The support is gratefully acknowledged.

**Data Availability Statement:** The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

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