**Abbreviations**

The following abbreviations are used in this manuscript:


#### **References**


**Chien-Shing Lee 1, Tom I. -P. Shih 1,\*, Kenneth Mark Bryden 2, Richard P. Dalton <sup>3</sup> and Richard A. Dennis <sup>3</sup>**


**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 Tw/Tc = 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.

**Keywords:** LES; internal cooling; pin fin; heat loads; turbulence

## **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 [1–4].

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., [1–22] 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. [9–11] 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

**Citation:** Lee, C.-S.; Shih, T.I.-P.; Bryden, K.M.; Dalton, R.P.; Dennis, R.A. Strongly Heated Turbulent Flow in a Channel with Pin Fins. *Energies* **2023**, *16*, 1215. https:// doi.org/10.3390/en16031215

Academic Editors: Artur Bartosik and Dariusz Asendrych

Received: 30 December 2022 Revised: 18 January 2023 Accepted: 19 January 2023 Published: 22 January 2023

**Copyright:** © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

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. [12–14] 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. [21–23] 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. [21–23] 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. [21–23] 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 Sx = 2.5D in the streamwise (x) and Sz = 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.

**Figure 1.** Schematic of cooling channel studied.

For this cooling channel, the cooling flow that enters at x = −115D is air and has a uniform temperature of Tc = 300 K and a mass flow rate of . *mc* = 0.014 kg/s. The pressure at the channel's outlet located at x = 15D is maintained at Pb = 1 bar. The Reynolds number (ReD) is 10,000, which is based on the pin-fin diameter D, the averaged momentum of 4 . *mc*/ πD<sup>2</sup> , and the dynamic viscosity evaluated at Tc. As noted, all solid surfaces of the test section are isothermal, and the following surface or wall temperatures were investigated: Tw = 303 K, 600 K, and 1200 K, representing negligible, moderate, and strong heat loads, respectively. Table 1 summarizes all of the cases studied.

**Table 1.** Summary of simulations performed.


Tc = 300 K for all cases.

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.
