Film Cooling Performance Prediction for Air and Supercritical CO₂

Supercritical CO₂ (sCO₂) power cycles have gained a lot of interest in recent years. Some of the benefits include higher efficiencies, reduced fuel usage, carbon capture, and compact systems that will potentially reduce cost. While indirect cycles rely on heat transfer surfaces to heat the CO₂ working fluid, direct-fired cycles rely on a CO₂ diluted oxy-combustor to directly heat the working fluid. The direct sCO₂ cycle can produce a 99% purity CO₂ stream and still have more favorable efficiency and economic benefits than conventional power cycles that remove 90% of the CO₂ [1]. The CO₂ removal in a direct sCO₂ cycle is accomplished by simply condensing the water vapor, so no solvents or other CO₂ separation technology is required. Furthermore, parasitic power for CO₂ compression is minimal due to the high operating pressure for this cycle. Figure 1 shows a conceptual flow diagram for a direct sCO₂ cycle [1] that can utilize any fuel, including biomass fuels in a carbon-negative application.

Uncertainties in the turbine cooling approach have been linked to the significant variations in predicted cycle efficiency [2]. The turbine inlet temperature and pressure targets for these turbines are in the range of 1150–1200 °C and 30 MPa, respectively [1–4]. Although these turbine inlet temperatures are lower than today’s advanced gas turbines, the Reynolds numbers could be 10–15 times higher. Conventional gas turbines operating above 1200 °C require advances like film cooling and thermal barrier coatings to keep the material temperature below the allowable limits [5].

Therefore, direct sCO₂ turbines may require film cooling due to the higher external heat loads. Although turbine cooling has been studied extensively for gas turbines in the past five decades [6], little attention has been given to cooling engine components using supercritical CO₂. To the authors’ knowledge, this is one of the first studies to explore film cooling using supercritical carbon dioxide.

The purpose of this paper is to investigate film cooling performance differences between conventional gas turbine operating conditions and the operating conditions expected in a direct-fired sCO₂ turbine application. Although numerous film cooling hole geometries have been proposed previously [7–15], this study will only consider conventional cylindrical film cooling holes. This effort has focused on using computational fluid dynamics (CFD) models to compare the fundamental differences in film cooling performance due to the following variables: (1) density ratio, (2) blowing ratio, (3) Reynold’s number, and (4) method of supplying coolant to the film cooling holes.

In recent years, a few studies have considered the problem of cooling turbines for direct-cycle sCO₂ turbines. These prior studies have focused on internal cooling, and not film cooling. Searle and coauthors [16,17] conducted experimental and numerical investigations of three variants of internal cooling configurations—dimples only, ribs only, and ribs with dimples using sCO₂ as the coolant. Numerical analyses of these three internal cooling designs were performed over a range of Reynolds numbers from 80,000 to 400,000. Based on the experimental results and numerical predictions, the authors found that the Nusselt number augmentation increases as higher Reynolds numbers were approached, whereas prior work on internal cooling of air-breathing gas turbines predicted a decay in the heat transfer enhancement as the Reynolds number increases.

Khadse et al. [18] investigated the heat transfer effects on the first stage vane of a sCO₂ turbine. A single vane with 6 internal circular channels along with the freestream was modeled numerically as a conjugate problem. Film cooling holes were not included in this study. The free-stream temperature and pressure were modeled at 1350 K and 28 MPa, respectively, but the authors acknowledged the need to achieve higher turbine inlet temperature for future oxy-combustion sCO₂ cycles. For the coolant mass flowrates used in the study, the wall temperature was not sufficiently lowered, and the authors have discussed film cooling as a possible way to reduce the airfoil surface temperature.

Due to interactions between the coolant jet and the free-stream gas flow, the local heat transfer coefficients can also be changed by the presence of these cooling jets. Earlier film cooling studies [20,21] focused on measuring the film effectiveness since adiabatic wall temperature varied more than the heat transfer coefficient. This was a reasonable approach, particularly at lower blowing ratios. Liess [22] investigated the performance of a 35-deg inclined film cooling holes by measuring both the adiabatic effectiveness and the heat transfer coefficient on a flat plate with copper strips under various conditions. The upstream boundary layer displacement thickness (δ₁) was found to have a strong influence on effectiveness, but no measurable effect on heat transfer coefficient. For example, doubling the δ₁/d from 0.25 to 0.5 reduced the adiabatic effectiveness at x/d = 10 by a factor of 2 (i.e., from 0.2 to 0.1). Although the observed change in heat transfer coefficient due to coolant injection was found to vary by only 20–25% for a majority of the film cooling region. Hay et al. [23] explored a novel technique based on a heat-mass transfer analogy to measure local heat transfer coefficients over film-cooled surface with a similar hole inclination angle. At higher blowing ratios, a 35% increase in the laterally averaged heat transfer coefficient ratio was observed.

Due to the temperature differences between the coolant and the hot-gas path, density differences between the coolant and the hot gas can also impact the local film cooling effectiveness. Most of the film cooling studies that have investigated density ratio effects utilized CO₂ as a surrogate for air at engine conditions where the coolant-to-free-stream density ratio is expected to be closer to 1.8. The applicability of using a secondary fluid with higher density has been studied by Teekaram et al. [24]. Eckert [25] also reported findings on similarity analysis of film cooling experiments typically conducted at near ambient conditions and its scaling at high temperatures in gas turbines.

Sinha et al. [26] conducted experiments to measure film cooling effectiveness on a flat plate made of polystyrene foam (Styroform, k = 0.027 W/mK) using a single row of holes. The test section had cylindrical film cooling holes inclined at 35 deg to the surface with an L/D of 1.75. Similar to Ref. [21], the study focused on understanding the relative importance of density ratio, blowing ratio, velocity ratio, and momentum flux ratio on film effectiveness. The authors found that the centerline effectiveness scales with mass flux ratio and momentum flux ratio at lower and higher blowing ratios, respectively. A single parameter was not sufficient to scale either the centerline or the laterally averaged effectiveness when the density ratio was varied. Schmidt et al. [27] investigated a 60-deg compound angle on film cooling effectiveness of cylindrical and forward expanded holes using the same test setup. Ekkad et al. [28] conducted transient film cooling experiments using a liquid crystal technique to study the effect of compound angle on film effectiveness. Using a single transient experimental test section made of plexiglass (k ∼ 0.18 W/mK), film effectiveness, as well as heat transfer coefficient due to film cooling, was measured experimentally.

The effects of hole length and free-stream turbulence were studied in detail by Bons et al. [30] and Burd et al. [31]. Gritsch et al. [32] measured the adiabatic effectiveness of shaped holes on a test plate made of a relatively higher temperature plastic material (TECAPEK, k = 0.2 W/mK) using an IR camera system. Drost and Bolcs [33] investigated the film cooling performance of cylindrical holes on a nozzle guide vane under different free-stream conditions. Heat transfer experiments were conducted over a test surface made of plexiglass using a transient liquid crystal technique.

Baldauf et al. [34] conducted flat plate experiments to obtain a spatial resolution of adiabatic (film) effectiveness due to cylindrical film cooling holes. Experiments were conducted on a semi-crystalline thermoplastic material (Tecapek) at moderately high hot-gas temperatures (T_g = 550 K) and mass flowrate of 1.3 kg/s. Greiner et al. [35] explored the scaling parameters to compare the film cooling performance from ambient and near engine conditions using CFD. They found that thermophysical properties or property ratios do affect the scaling of adiabatic effectiveness. In addition to matching density and blowing ratio, Prandtl number and Reynolds numbers needed to be matched to scale film cooling effectiveness accurately.

In most experimental and numerical film cooling studies, the coolant is fed through a quiescent plenum. However, in many gas turbine applications, the coolant is fed from an internal serpentine channel in which the direction of coolant flow is perpendicular to the cooling hole axis. Although numerous studies have investigated the effects of the compound angle coolant jets where the coolant jet angle relative to the main hot-gas flow direction was varied, only a few known studies have investigated the effects of the angle between the coolant jet and the coolant channel flow direction.

Burd and Simon [36] found the coolant supply geometry (co-flow, counter-flow, and short hole with unrestricted plenum) to have a noticeable effect on both the centerline and laterally averaged effectiveness. Steady-state heat transfer experiments were conducted on a test plate made of silicon phenolic laminate plate (k = 0.25 W/mK). Interestingly, the adiabatic effectiveness was measured from extrapolated wall temperature obtained by a thermocouple traverse under film cooling conditions. As a result, thermocouple measurements were made at discrete locations (e.g., at 6 points in the streamwise direction and 23 points along the lateral direction about a single hole).

Gritsch et al. [37] conducted a comprehensive study investigating the effects of internal channel Mach number on adiabatic effectiveness distribution over a flat surface. Three types of film cooling hole shapes (cylindrical, fan-shaped, and laid-back fan-shaped hole) were studied for a range of blowing ratios (0.5–2) and coolant crossflow Mach numbers (0, 0.3, and 0.6). The density ratio was maintained close to 1.85 and assumed to be representative of typical engine conditions. Experiments were conducted on a thermoplastic material with a thermal conductivity of 0.2 W/mK. In all cases studied, film cooling effectiveness distribution was affected in the near hole region (x/d < 8) when the coolant channel Mach number was increased from 0 to 0.3 and 0.6. In the case of a cylindrical hole, the coolant lateral coverage was found to increase with the peak effectiveness shifting toward the coolant channel upstream side (z/d < 0), thereby resulting in a skewed distribution. The coolant crossflow increased the laterally averaged effectiveness at all blowing ratios except for the smallest blowing ratio of 0.5. This suggests that the jet detachment typically expected at higher momentum flux ratios was absent. In the case of the fan-shaped hole, at higher crossflow Mach numbers, the effectiveness distribution was skewed with the peak values occurring near (z/d ≥ 1) when compared to a plenum fed hole where peak values are usually present toward the hole center $(−1<z/d<1)$⁠.

A three-part study was carried out by Saumweber et al. [38–40] where the effects of free-stream turbulence and internal coolant passage were studied for a range of geometric variations. Over a range of blowing ratios studied for a 6-degree fan-shaped hole, higher coolant crossflow Mach numbers (0.3 & 0.6) were found to have a tendency to skew the coolant distribution over the surface and the local effectiveness in the negative z-direction (z/d < 0) except for at the highest Ma_c and lowest blowing ratio where the effectiveness distribution was skewed in the +z direction (z/d > 0). Consistent with Gritsch et al., the extent of skewness in the local adiabatic cooling effectiveness and the subsequent laterally averaged value, though not monotonic, was a function of the crossflow Mach number, blowing ratio, and the diffuser angle.

More recently, McClintic et al. [41] looked at a range of crossflow coolant channel (and coolant jet) to hot-gas velocity ratios for the 7–7–7-shaped film cooling hole. A significant finding was that most of the key film cooling performance metrics seemed to scale with the coolant channel-to-coolant jet velocity ratio (VRi). This suggests that the velocity at the inlet to the film cooling hole can play a significant role when scaling film cooling performance. Stratton et al. [42] conducted a CFD study of the internal flow patterns in the film cooling holes to show the presence of secondary flows within the cooling holes that generate an internal swirl in the coolant jet.

Several research groups have compared the performances of various turbulence models in predicting the adiabatic cooling effectiveness and heat transfer enhancement due to film cooling. Conjugate effects of film cooling have also been studied previously [43–47]. The studies pertinent to the current work will be explained in the following paragraphs.

Harrison et al. [48] studied realizable k−ɛ (RKE), Standard k−ω (SKW), and Reynold's stress models (RSM) where the RKE and RSM models were coupled with an enhanced wall treatment approach to model the near-wall region. Of these three models, the SKW model provided the best prediction of the laterally averaged effectiveness, and the RKE model produced the worst predictions. For the centerline film effectiveness, the opposite trend was observed. In other words, the RKE model produced the best predictions for centerline film effectiveness and the SKW model produced the worst predictions. Consistent with other studies, the RKE and SKW did not predict the lateral spread of the coolant. The SKW model was recommended for predicting laterally averaged effectiveness which is one of the more commonly studied and reported outcomes in the film cooling literature. Arguably, cooled gas turbine models often rely on laterally averaged or area-averaged effectiveness in estimating the performances of a particular cooling configuration.

Na et al. [49] looked at the effect of three different RANS-based eddy diffusivity models (realizable k−ɛ, shear-stress transport (SSTKW), and Spalart–Allmaras (SA)) to predict film cooling performance over a flat plate and a semi-cylindrical leading edge. A second-order upwind differencing scheme was used in the commercial solver Fluent. The eddy diffusivity models were primarily investigated since high fidelity models such as DES/LES/DNS are computationally too expensive for design purposes even though they offer better accuracy. Of the three turbulence models, the SSTKW and SA models were found to predict laterally averaged adiabatic effectiveness on a flat plate reasonably well. The RKE severely underpredicted the laterally averaged effectiveness.

Stratton et al. [50] studied the effects of crossflow in a setup similar to the one used in the current work. Film cooling effectiveness of eight compound angle holes $(β=45deg)$ was numerically predicted using two commonly used turbulence models (SSTKW and RKE) at three different blowing ratios (BR = 0.5, 1.0, and 1.5). The mainstream flow velocity and the boundary layer thickness upstream of the holes were 13.8 m/s and δ = 2.8d where d represents the film cooling hole diameter. Simulations were conducted using hot air at 303 K and nitrogen coolant at 202 K. The corresponding density ratio for this study was 1.5. At lower blowing ratio conditions; the authors concluded that the RKE model predicted the laterally averaged effectiveness better than the SSTKW. At higher blowing ratios, the opposite trend was observed.

Kampe et al. [51] performed experiments and CFD simulations to analyze the flow field and estimate the surface temperatures downstream of cylindrical and shaped holes. The SST k–ω turbulence model was employed for the numerical predictions. In terms of the cylindrical hole flow field, such as velocity distribution, magnitudes, jet expansion, and secondary flow, the authors found a very good match between the experiments and CFD simulations. From a cooling effectiveness perspective, the predictions were good even though the shape of the laterally averaged effectiveness was not the same.

Several studies [11,43,46,52–55] have relied on the k−ω-based turbulence models to predict the behavior of a jet in a crossflow problem. The k−ɛ-based equations have also been used with varying degrees of success in the film cooling literature [56–60]. In the current study, both turbulence models are briefly studied to understand their influence in predicting film cooling effectiveness. Given the lack of information on film cooling for sCO₂ turbines, the numerical predictions presented in the current work are expected to provide a preliminary understanding of what can be expected at the extreme operating conditions for a direct sCO₂ turbine application.

Although the thermal load on the airfoil depends on several factors, flat plate correlations provide a good estimate of heat transfer, as suggested by Cunha [61]. Flat plate film cooling geometries have been widely used in the open literature over the last several decades, and this paper will also study film cooling performance using a flat plate geometry. This paper will investigate three different design variations. The first design uses a crossflow coolant channel to supply coolant to a 5.1-mm diameter cylindrical film cooling hole (Fig. 2). A second design has the same hole diameter, but a quiescent coolant supply plenum was used instead of the crossflow channel. In these first two designs, the main objective was to validate the CFD results relative to existing data in the literature using air as the coolant. The temperatures and pressures used in these validation cases represented lab-scale conditions. The final design variant was a crossflow channel with a smaller film cooling hole and operating conditions that might be representative of engine conditions. Harrison et al. [48] found that a coolant channel height of 6d was sufficient to provide results similar to that of a channel of the height of 20d. In this study, a channel height of 10d was used. The numerical domain in the current study covers one-hole pitch in the lateral direction by using symmetric boundary conditions (Fig. 2). The remaining walls have been treated as adiabatic surfaces.

The numerical domains for the first two designs were meshed with roughly 1.14 million patch-conforming tetrahedral elements (Fig. 3). A refined mesh with an element size one-sixth of the cooling hole diameter was used for the flat surface upstream and downstream of the film cooling hole exit; the film cooling hole; and the surface flush with the hole inlet. The rest of the domain had a coarse mesh. Neighboring cells in the vicinity of these surfaces (i.e., a region within the range of 2.5d) were also influenced by the element size. To reduce the mesh skewness and improve the orthogonal quality of the mesh inside the hole, small fillets (r_f = d/10) were introduced in all film cooling configurations studied. Reynolds-Averaged Navier–Stokes equations were solved using a steady-state solver instead of resorting to DNS or large eddy simulations (LES). The shear-stress transport variant of the k–ω and the realizable k–ɛ (RKE) model provided the turbulence closure. An enhanced wall treatment method was employed for near-wall modeling. The prism layers of varying first layer thicknesses were added depending on the mainstream Reynolds number. The wall y+ was maintained close to 1 for all the cases. A green gauss node-based gradient approach was selected since a node-based gradient is known to be more accurate than the cell-based gradient, particularly on irregular unstructured meshes. A second-order upwind spatial discretization scheme was used for all the variables. A coupled solver with a pseudo transient formulation was selected instead of pressure-based segregated solver (and pressure–velocity coupling schemes) as it was found to accelerate convergence. Convergence criteria were set to 1E-6 for both momentum and energy equations and 1E-5 for the turbulent kinetic energy and specific dissipation rate equations. The area-averaged temperature on the surface downstream of the film cooling hole exit was also monitored in addition to the residuals. For most cases, the area-averaged temperature was observed to reach a steady-state value within 150–200 iterations, and the residuals were found to converge within 400 iterations. Additionally, 600 iterations were carried out to be on the conservative side.

The fluid properties as a function of temperature and pressure for both air and sCO₂ (Fig. 4) were estimated using the CoolProp [63] library and a python routine.

To enable an efficient way of calculating and comparing laterally averaged profiles from several test cases, the unstructured data set on a particular surface from the simulation was converted to a structured array format. The wall temperature and nodal coordinates on the surface downstream of the film cooling hole exit were exported as a text file. As mentioned earlier, the numerical domain meshed with roughly 1.14 million tetrahedral elements. Surface data exported from the simulation typically contained the nodal values on the base triangular face of these tetrahedrons. A temperature value is associated with every {x, y} corresponding to the location of the mesh node. Using the triangulation API within the Matplotlib library, the triangular mesh data were interpolated to a structural grid of size that was one-tenth of the hole diameter.

Figure 5 shows a scatter plot comparison between the original (unstructured data from the simulation, Fig. 5(a)) and the interpolated (structured data set, Fig. 5(b)) surface temperature. For reference, the trailing edge of the film cooling hole exit is located at x/d = z/d = 0 with the hole lying between −0.5 ≤ z/d < 0.5. Some high frequency, noise-like components were observed in the interpolated results, so a mild gaussian filter (low pass filter) was used to smooth the interpolated data. Satisfactory reproduction of the original prediction was key to this comparative study.

A statistical comparison between the two data sets is shown in Table 3. The difference in the minimum, maximum, and mean values were 0.16%, −0.01%, and −0.46%, respectively. Finally, Fig. 6 shows the laterally averaged film effectiveness data estimated directly from the CFD Post data set compared to the film effectiveness calculated from the interpolated data using this post-processing methodology. The adiabatic effectiveness was calculated using Eq. (2).

Three different element sizes were chosen initially to estimate the impact of mesh on film cooling performance prediction. A mesh size of d/6 implies that roughly six elements are present within a space equal to one film cooling hole diameter. Although the sas model was not included in Table 2, this model was used to inspect the accuracy of the predicted adiabatic effectiveness. One of these simulations employed a hybrid RANS–LES turbulence model, named Scale Adaptive Simulation (sas). This model has been described as an improved version of the unsteady RANS turbulence model since it can resolve the turbulence spectrum (smaller eddies) in flows that have inherent unsteadiness. A fine time-step size of 10⁻⁵ s was chosen. Simulations were run for 700 time-steps which is roughly more than twice the time taken for the flow to travel from the inlet to the outlet of the numerical domain. Numerical data were sampled to estimate a time-averaged solution for the 700 additional time-steps. For a free-stream velocity of 60 m/s, the Strouhal number based on film cooling hole diameter was roughly 8.5. The grid-independent study was extended with the finest mesh (size = d/12) using SST k–ω and sas SST k–ω turbulence models, respectively. The adiabatic effectiveness is presented in Fig. 7.

The marginal difference in laterally averaged effectiveness (Fig. 7) between the SST k–ω and a Scale Adaptive Simulation (sas) model indicates that the initial grid (d/6) chosen was sufficiently fine. To be on the conservative side, a grid size = d/8 or d/10 was chosen for all the cases.

After obtaining a grid-independent solution, numerical predictions were compared against the experimental data published in the gas turbine film cooling literature. The laterally averaged effectiveness $(η¯)$ predicted using the SST k–ω RANS model was close to the experimentally reported values (Fig. 8). The SST k–ω model predictions were also found to lie close to the effects predicted using the standard k–ω model by Harrison et al. [48]. There was a noticeable variation in $η¯$ reported by various studies, and the experimental error bars were not shown in Fig. 8.

As mentioned earlier, it is critical to understand the impact of the coolant injection mechanism on adiabatic effectiveness. Equally important is the ability of the turbulence model to correctly predict the trend, i.e., whether crossflow injection will increase or decrease the laterally averaged effectiveness $(η¯)$⁠. Since the focus of the current study is to investigate film cooling for sCO₂ turbines, only one crossflow velocity ratio $(VRc=Vch/Vg=0.2)$ was chosen. The blowing ratio and density ratio for these simulations can be identified by referring to Tables 1 and 2. Figures 9(a) and 9(b) show the effect of crossflow coolant injection and choice of turbulence model on laterally averaged effectiveness and local adiabatic effectiveness, respectively.

Gritsch et al. [37] found that when the coolant was supplied through a crossflow channel, the laterally averaged effectiveness increased as seen in Fig. 9(a) for the case of SST k−ω model. Under crossflow conditions, the peak adiabatic effectiveness was found to shift away from the centerline (z/d = 0), toward the coolant flow direction beneath the film cooling hole (+ve z-direction). Also, the SST k−ω clearly predicts the jet separation that has been typically reported at higher blowing ratios. The RKE model, on the other hand, was found to underpredict the jet detachment. These findings were observed by Stratton et al. [64] as well. The RKE also predicted a lower $η¯$ under crossflow conditions. As a result, the SST k−ω model was used for the remaining test cases. Having said that, the impact of the turbulence model on film cooling behavior fed through a crossflow channel needs to be studied in greater detail in conjunction with other relevant factors shown in Eq. (3).

Reynolds numbers used in experimental film cooling studies are typically in the range of 3000–20,000 as it is expected to represent the engine conditions. The characteristic length scale used in the definition of Reynolds number for this application is the film cooling hole diameter. Schroeder and Thole [15] studied the performance of a baseline-shaped hole for a Reynolds number range of 2800–20600. McClintic et al. [41] studied the effect of crossflow on axial-shaped holes at a mainstream approach Reynolds number of 6000. Anderson et al. [65] investigated the effects of free-stream Mach number, Reynolds number (5500–13,000), and boundary layer thickness on film cooling effectiveness of shaped holes. The range of Reynolds numbers in Baldauf et al.’s [66] study was in the range of 6800–14,000. These previous studies were largely conducted at a lab-scale or near ambient conditions. In this section, the effect of free-stream Reynolds number on laterally averaged adiabatic effectiveness, especially at near engine conditions, will be reviewed. Figure 10(a) shows that, in the near hole region (x/d < 10), the laterally averaged effectiveness $(η¯)$ is largely independent of the Reynolds number when Re_d ≥ 5872. For distances greater than 15 hole diameters downstream of the film cooling hole, the $η¯$ at Re_d = 5872 is slightly higher than $η¯$ at a higher Reynolds number by roughly 10%. It must be noted that typical uncertainties reported in the experimentally measured adiabatic effectiveness are often between 5 and 10%. Irrespective of the streamwise location, Fig. 10 shows $η¯$ is independent of the free-stream Reynolds number when $Red≥∼8800$⁠.

Of all the factors [67] influencing film cooling effectiveness, blowing ratio and density ratio have been widely studied since they are the most influential scaling parameters. Scaling these two quantities also scales the other two important ratios, namely, the velocity and momentum flux ratio. The velocity ratio has been found useful in understanding the coolant–mainstream interaction, especially the shear layers that influence the shape of the coolant jet once it exits the hole. The momentum flux ratio, on the other hand, has been found useful in identifying conditions that could potentially result in the coolant jet lift-off or separation from the surface [68].

In most previous studies of film cooling, the coolant air has been fed through a plenum to study the effects of blowing ratio and density ratio on $η¯$⁠. The peak effectiveness was found to occur at a blowing ratio $∼0.6−0.85$⁠. Increasing the density ratio from 1.0 to 2.0 increases $η¯$ since the higher density coolant has the tendency to stay close to the surface.

Several velocity planes through a round cooling hole (i.e., l/d = 2, 3, 4, and 5) have been shown in Fig. 11. The perspective used in Fig. 11 was chosen to look downstream in the direction of the hot-gas flow (i.e., flow into the page). Another velocity plane located five hole diameters downstream is shown in Fig. 11. All the velocity vectors are colored by temperature. Finally, an iso-temperature surface has also been shown to visualize the coolant jet coverage downstream. The temperature on this surface (1100 K) was approximately the average temperature between the coolant and the free-stream gas temperature for a density ratio of 2.0. All the cases shown in Fig. 11 have a density ratio of 2.0.

As the coolant enters the cooling hole from the crossflow supply, a non-uniform velocity distribution was generated. Secondary flows similar to those observed by Stratton et al. [42] were evident within the cooling hole. After the cooling jet interacts with the hot gas, the cooling jet divides into two lobes along the shear region in the cooling hole. In other words, the coolant jet divides into a high-velocity lobe and a lower-velocity lobe. It is possible that counter-rotating vortices could be mitigated using this inlet coolant configuration, but more detailed studies would be needed to make that conclusion.

At low blowing ratio conditions, the velocity in the cooling holes seemed to be more uniform and the velocity vectors inside the hole were closer to a bulk swirling flow. The secondary flows inside the hole were not as strong at the lower blowing ratio condition. At a blowing ratio of 0.5, both the CO₂ and the air-cooling jets stay very close to the downstream surface. At a blowing ratio of 1.0, the velocity bias in the hole becomes more significant, and the high-velocity lobe of the coolant jet may separate from the surface since some high-temperature velocity vectors can be seen traveling right-to-left along the surface.

To investigate the effects of density ratio, the same perspective and interrogation planes were used (Fig. 12). The momentum ratio was constant for all cases, so as the density ratio was increased, the velocity decreased to maintain the same momentum ratio and blowing ratio. As a result, the highest coolant jet velocities occurred for the lowest density ratio and the highest blowing ratio (i.e., DR = 1.5, BR = 1.0). From the velocity vectors at x/d = 5 in Fig. 12, jet lift-off, or separation, could be present due to the flow of hot gas along the surface on the right side of the figure. All cases showed some entrainment of hot gas along the right (high-velocity) lobe, but no entrainment of hot gas along the left (low-velocity) lobe. The swirl generated by the crossflow coolant supply seems to generate some outward flow as the cooling jet expands.

The primary objective of this study was to investigate whether film cooling can be pursued as a potential thermal solution for sCO₂ turbines. Figure 13 shows the laterally averaged film effectiveness, $η¯$⁠, for air and sCO₂ turbines under a mild crossflow velocities (i.e., VR_c = 0.2). At higher blowing ratio (BR = 1.0), increasing the density ratio from DR = 1.5–2.0 increases the laterally averaged effectiveness consistent with the expectations. The decay in $η¯$ near the hole exit (i.e., x/d < 5) could be caused by mild separation near the hole. This decay was more significant for the CO₂ case than it was for the air cases, but the gradients were also more severe for the CO₂ cases (Fig. 12). For air at the higher blowing ratio (BR = 1.0), a peak in effectiveness occurred between 5 < x/d < 10, due to the nature of crossflow coolant injection.

Film cooling with supercritical CO₂ showed similar features and trends when compared to gas turbines. The laterally averaged film effectiveness at BR = 0.5 is greater than at BR = 1.0_, and the effect of density ratio seems to be quite similar when compared to air. At higher blowing ratio (BR = 1.0), the effect of coolant jet separation in the near hole region, 0 < x/d < 10, is visible even at higher density ratio (DR = 2.0). The differences between sCO₂ and air at engine conditions was more pronounced at the higher blowing ratio (BR = 1.0) condition. The role of momentum flux ratio and crossflow injection for film cooling with sCO₂ needs more study.

Irrespective of the blowing ratio, the $η¯$ with sCO₂ was found to be lower than with air at any given density ratio. However, at the lowest blowing ratio (BR = 0.5), the difference in $η¯$ appears to be low (Fig. 13(b)), especially for (x/d > 15). The maximum difference was found to occur between 4 ≤ x/d ≤ 6. Interestingly, the laterally averaged effectiveness was still higher than 0.2 very far downstream $x/d∼20$ at both density ratios, indicating that film cooling can be viable at these conditions for the sCO₂ turbines.

At a higher blowing ratio (BR = 1.0), significant reduction in the $η¯$⁠, especially in the near hole region (2 ≤ x/d ≤ 7), would make it difficult to recommend a cylindrical hole for film cooling sCO₂ turbines. At a higher density ratio (DR = 2.0), the $η¯$ consistently stayed higher than 0.2 over x/d ≥ 8. Future studies can look at the effect of hole exit shaping (shaped holes) or alternate geometries at higher blowing ratios to improve effectiveness near the hole exit.

Figure 14 shows the local distribution of adiabatic effectiveness at different blowing ratios and density ratios. The coolant flow direction beneath the flat plate was toward the +ve z-direction. At BR = 0.5 and DR = 1.5, the location of the peak effectiveness for air and sCO₂ was shifted from the centerline (z/d = 0). This observation was consistent with the double-lobe shape of the coolant jets described in Figs. 11 and 12. These double-lobe shapes were observed in most of the CFD results, but these effects may not be present in the adiabatic effectiveness contours, particularly if the coolant jet separates from the surface. At higher density ratios, the streaks in film effectiveness were shifted toward the coolant downstream side of the coolant hole (i.e., positive z-value shift). This was consistent with the location of the lower velocity lobes of the cooling jet. As discussed, the coolant velocity distribution inside the film cooling hole varied with the inlet velocity, which resulted in a swirling action in addition to secondary flows.

Conventional film cooling studies often rely on the scaling laws based on non-dimensional parameters such as blowing ratio and density ratio. Often both parameters need to be used in conjunction to scale the velocity and momentum flux ratio. The film cooling effectiveness contours presented in this study highlight the influence of crossflow injection, as well as coolant properties on the existing scaling parameters. As mentioned earlier, film cooling using supercritical carbon dioxide has not been studied in detail. Future studies on film cooling for sCO₂ turbines should consider the effects of the coolant injection mechanism as well as the nature of the supercritical fluid for scaling film effectiveness in direct-cycle sCO₂ gas turbines.

The technical findings presented in this study are one of the earliest works investigating the feasibility of film cooling for sCO₂ turbines. The operating conditions of a direct-fired sCO₂ cycle and the thermophysical properties of the fluid at those conditions can alter the flow field characteristics of the coolant jet. A computational fluid dynamic analysis was used to study cylindrical film cooling holes in which the coolant was supplied in a crossflow configuration. Spatial distribution of cooling effectiveness and laterally averaged adiabatic effectiveness were presented and the following conclusions can be made:

• The Reynolds number (based on the cooling hole diameter) was found to have a marginal effect on laterally averaged film cooling effectiveness. These effects were particularly small for Reynolds numbers greater than 8000.

• In many ways, the film cooling behavior observed for sCO₂ was similar to the expectations from prior work using air. At low blowing ratios (BR = 0.5) and especially at higher density ratios (DR = 2.0), film cooling could be a viable approach for sCO₂ turbines. At higher blowing ratios (BR = 1 or higher), shaped cylindrical holes or other cooling geometries may need to be considered.

• The impact of the crossflow coolant supply may be more severe for sCO₂ applications at the constant momentum ratio that was investigated in this study. Numerical predictions revealed that reduced film coverage area, or jet separation, effects were more pronounced for sCO₂ under similar conditions.

  • More work to understand and mitigate the effects of crossflow coolant supplies for direct-cycle sCO₂ film cooling is needed.

  • The crossflow coolant supply generated secondary flows inside the cooling hole. This work has shown that the interactions between the secondary flows generated within the cooling hole and the subsequent interactions with the main hot-gas flow can significantly impact the film cooling effectiveness.

  • This study has also shown that a coolant jet can separate into two distinct lobes (i.e., a high-velocity and a low-velocity region). In this study, the dividing line between these lobes occurred along the shear layer generated within the film cooling hole between the high-velocity and low-velocity regions of the coolant jet.

This work was performed in support of the U.S. Department of Energy’s Fossil Energy Turbines Program and the guidance and support from Rich Dennis, Nate Weiland, and Pete Strakey are greatly appreciated. The thoughtful discussions and input from Dr. James Black were also appreciated and of great value in this effort.

This project was funded by the United States Department of Energy, National Energy Technology Laboratory, in part, through a site support contract. Neither the United States Government nor any agency thereof, nor any of their employees, nor the support contractor, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of the authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof.

• The United States Department of Energy’s Fossil Energy Turbines Program.

There are no conflicts of interest.

The data sets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

d =

film cooling hole diameter

I =

momentum flux ratio

V =

velocity

DES =

detached eddy simulation

DNS =

direct numerical simulations

sCO₂ =

supercritical carbon dioxide

η =

film effectiveness

ρ =

density

c =

coolant

ch =

channel

f =

film

g =

hot gas