## Abstract

For many years, there has been interest in evaluating the effect of density differences between the coolant and the freestream in terms of the cooling effectiveness. Numerous experiments have been conducted with different cooling gases or different temperature gases to evaluate the effect of the density ratio. With little agreement on the best way to scale the density ratio effect, it has become commonplace for some researchers to insist upon matching the density ratio for the experimental work. Unfortunately, the density is not the only property that differs between the various coolant gases used in experiments, and it is certainly not the only property difference between the coolant and the freestream in actual engines. In the present work, we isolate some of these effects through film-cooling experiments with carefully selected and conditioned coolant gases at near-identical densities but exhibiting other property differences. Most significantly, coolant specific heat varied, but subtle viscosity and thermal conductivity effects were present. Through measurements of the adiabatic effectiveness from a film-cooling hole on a leading-edge model, we are able to show that the specific heat effect is just as important as the density effect, providing more evidence that effects in prior research attributed to density differences are actually a combination of density and other property differences.

## Introduction

As the turbine inlet temperature increases with each new generation of gas turbine engines, the thermal load on engine components has increased in kind. To reduce the thermal load on these components, relatively cool air from the final stage of the compressor bypasses the combustor and is reintroduced into the cycle as hot-section component coolant. This coolant, however, is not without cost, as the reduction of combustor mass flow and turbine temperatures that result from coolant use decreases the performance of the machine. Furthermore, as temperatures continue to rise, the cycle benefit that results from an increased turbine inlet temperature may be offset by the detriment that manifests due to the increase in the requisite compressor bleed needed for coolant.

As a result, developing coolant schemes that provide sufficient cooling flow to maintain component integrity while requiring the minimum cooling flow is paramount. Component cooling is accomplished using flow passages inside of turbine components, as well as numerous holes on the exterior.

The purpose of the film-cooling holes is twofold. First is to eject the heat that was absorbed while the coolant air traveled through the internal channels. Second, the coolant that exits the coolant hole provides an insulating layer of relatively cool air on the surface of the engine part, which reduces the convective heat transfer into the part. This heat transfer is defined in Eq. (1). External coolant films reduce the heat transfer from the freestream gas by reducing the driving temperature for convective heat transfer, Taw, below the gas recovery temperature, T. Nondimensionalization of the adiabatic wall temperature using Eq. (2) yields the adiabatic effectiveness:
$q=h(Taw−Ts)$
(1)
$η=T∞−TawT∞−Tc,e$
(2)

In Eq. (2), Tc,e is the temperature that the coolant exits the film-cooling hole. Additionally, the examination of Eq. (2) reveals that the selection of T and Tc,e are arbitrary; there is no material difference—at least in the form of η shown Eq. (2)—between evaluating η near room temperature, with a difference between T and Tc,e of 10 K and at realistic turbine conditions where the difference between T and Tc,e is 1000 K. Gas turbine heat transfer researchers leverage the seemingly arbitrary temperature selection to scale the results of experimentation to engine conditions. Often, researchers elect to perform experimentation near room temperature rather than that of a realistic engine due to the cost-prohibitive nature and difficulty of performing experiments at high temperatures.

Despite the arbitrary nature of the selected temperatures in adiabatic effectiveness experimentation, the effect of the temperature selection has a significant influence on the resultant effectiveness distribution. When adiabatic effectiveness experiments are performed near room temperature, the physical transport properties of the coolant and freestream cannot simultaneously match that of the engine condition. This result is the basis for investigations of film-cooling scaling.

Typically, coolant and freestream properties are tracked via the ratio between them. In the current work, these property ratios are denoted using the superscript (*) on the property in question. Equation (3) demonstrates this notation using the property ratio that has been most often considered historically, the density ratio, or $ρ*$; however, similar constructs can be built with other properties, such as dynamic viscosity, μ; and transport parameters, such as the Prandtl number, Pr:
$ρ*=ρcρ∞$
(3)
$ρ*$ significantly affects the fluid dynamics of a coolant flow, and as a result, it is a variable that often manifests in coolant flowrate parameters. The coolant-to-freestream velocity ratio, VR—defined in Eq. (4), is the building block of the parameters used to describe coolant flow. From the products of $ρ*$ and VR, the commonly used flowrate parameters: the mass flux (or blowing) ratio—M and the momentum flux ratio—I are defined as shown in Eqs. (4) and (5), respectively:
$VR=ucu∞$
(4)
$M=ρ*(VR)$
(5)
$I=ρ*(VR)2$
(6)

Pietrzyk et al. [1] provides an overview of the above flowrate parameters and their effect on film-cooling effectiveness. VR was found to properly scale velocity field very near the coolant hole exit. While such a result is not necessarily surprising, it does offer a basis for using VR to describe some relevant fluid dynamics of film-cooling flows. However, once attention is turned to regions outside of the immediate vicinity of the cooling hole, VR is not able to properly predict coolant flow behavior. Additionally, Thole et al. [2] showed that VR cannot accurately scale the thermal effects of a coolant flow. Sinha et al. [3] evaluated the effects of M and I on cooling effectiveness downstream of axially injected coolant flows and found that when the coolant remained attached to the wall, M provided proper scaling of the adiabatic effectiveness, but in cases where the coolant was separated from the wall, I was found to be the proper scaling parameter. Wiese et al. [4] demonstrated that for compound coolant injection—i.e., on a leading edge, I best tracks the location of the coolant jet over a range of density ratios.

An additional conclusion made by Sinha et al. [3] was that M yielded good scaling performance due to the ability of M to approximate the thermal capacity of the coolant flow relative to the freestream flow. However, Eq. (5) contains neither the coolant nor the freestream specific heat, and the good agreement in Ref. [3] was likely due to using only air as the coolant, so wide variations in specific heats did not occur. Realizing that cp was unaccounted for, Rutledge and Polanka [5] introduced the advective capacity ratio, or ACR, defined in Eq. (7):
$ACR=cp*M$
(7)

In addition to evaluating the effects of $ρ*$ and $cp*$, Ref. [5] also evaluated the effects of coolant properties beyond density and specific heats on adiabatic effectiveness. The effects of the coolant properties were evaluated computationally by setting the individual properties to the value that would result if the coolant was carbon dioxide. Variations of $cp*$ and $k*$ resulted in variations in adiabatic effectiveness magnitude, while manipulation of the coolant dynamic viscosity resulted in small differences in the spatial distribution of key adiabatic effectiveness features, particularly the location of peak effectiveness.

Reference [4] provided an initial demonstration of the findings of Ref. [5] on a single hole model with 90 deg compound injection. First, the distance that the coolant effectiveness region—the area on the surface that the thermal effect of the coolant is realized—was displaced from the coolant hole, was similar across various values of $ρ*$ when I was matched. However, it is important to note that this result is only applicable in momentum dominated regions, i.e., in the leading-edge showerhead. Additionally, simply matching I between different cooling gases did not ensure matched effectiveness magnitudes. In fact, at matched I conditions, the resultant magnitude of ACR appeared to scale the magnitude of η, and the coolant-to-freestream viscosity ratio, $μ*$, accounted for in the coolant-to-freestream Reynolds number ratio, ReR—defined in Eq. (8), appeared to shift the thermal influence of the coolant laterally and influence the spreading of the coolant effect:
$ReR=Mμ*$
(8)

Fischer et al. [6] demonstrated the efficacy of ACR scaling in flow regions where the fluid dynamics of the coolant flow is not momentum driven, such as the regions downstream of axially aligned coolant holes. Specifically, Fischer et al. [6] investigated the effect of matching ACR on a flat plate with the 7-7-7 hole developed by Schroeder and Thole [7]; finding that at I ≤ 0.5, matched ACR test cases maintained similar area-averaged and centerline adiabatic effectiveness. However, once separation occurred, ACR was no longer able to directly account for the differences in cooling effectiveness.

## Experimental Methodology

This work utilized a model similar to that used by Ekkad et al. [8]. The model simulated a turbine blade leading edge as a semi-cylinder with a flat afterbody. The model was constructed out of high-density, low thermal conductivity (k = 0.03 W/(m K)) foam, reducing the uncertainty due to heat transfer to the model. The external surface of the model was sealed with a thin layer of epoxy to provide a stable surface for the black paint used to produce a uniform surface emissivity.

Figure 1 details the relevant geometric definitions for the model utilized in this study. The model in this study utilized a single cooling hole so as to accurately characterize the coolant flowrate, and by extension, coolant flowrate parameters. The model leading edge to coolant hole diameter ratio, D/d, was 18.7, and the coolant channel length-to-diameter ratio, L/d, was 11.69, as described in Ref. [8]. The coolant was ejected from the coolant plenum at an angle to the stagnation line, β, of 21.5 deg, and the angle between the cooling hole axis and the surface y-axis, γ, was 20 deg, resulting in a 90 deg compound cooling hole.

Fig. 1
Fig. 1
Close modal

The model was evaluated in an open loop wind tunnel with a straightening and settling chamber upstream of the test section. The test section, detailed in Fig. 2, was of rectangular cross section, and the model (A) was placed near the exit of the tunnel. In addition to the hardware shown in Fig. 2, a single J-type thermocouple was placed approximately one coolant hole diameter into the coolant hole from the plenum side to obtain the coolant temperature. The freestream temperature was maintained within 0.4 K of the target T through the use of a tube-and-fin cold-water heat exchanger for the T = 295 K cases and a 70 kW duct heater for the T = 315 K cases. Coolant temperatures were controlled to within 0.6 K of the target Tc using liquid–gas stacked plate heat exchangers, where the liquid was heated for the Tc = 315 K cases and chilled for the Tc = 295 K cases using a circulating bath.

Fig. 2
Fig. 2
Close modal

The wind tunnel was fed by a single 50 hp blower, and tunnel speed was controlled by diverting a portion of the blower output mass flow to an unused leg of the wind tunnel system. Tunnel speed was measured using a pitot-static system and was set so that the leading-edge diameter based Reynolds number, ReD, was 60,000 in all cases evaluated. Additionally, the wind tunnel was set up without a turbulence grid installed, and the resulting freestream turbulence intensity, Tu, was 0.67% with a length scale, Λ/d, of 24.2 [9].

Coolant flow was supplied to the model via high-pressure gas cylinders. Coolant flowrates were controlled using a programmable mass flow controller, calibrated for various gases and gas mixtures, with a maximum flowrate of 50 SLPM.

Surface temperature measurements were acquired using an infrared camera, which was calibrated in situ using a purpose-built calibration model. The camera was calibrated each day that testing was conducted, as well as for each freestream temperature setting. A separate calibration curve was required for each freestream temperature due to reflections of the tunnel wall surfaces shifting the number of counts observed at a particular surface temperature. While the calibration was sensitive to the wind tunnel wall temperature, it was confirmed to be insensitive to any emission or absorption within the thin coolant gas plumes. The calibration curves and their respective uncertainties are shown in Fig. 3.

Fig. 3
Fig. 3
Close modal
With an adiabatic model, the surface temperature obtained through experiment would necessarily be Taw, and the adiabatic effectiveness obtained through the application of Eq. (2). However, since no physical material is adiabatic, conduction into the model was accounted for via a one-dimensional conduction correction. This correction corrects apparent adiabatic effectiveness, ηapp—defined in Eq. (9), for apparent effectiveness that manifests due to conduction from the coolant plenum and from coolant flowing through the coolant channel, η0, which is defined similarly to ηapp. The application of this correction is shown in Eq. (10):
$ηapp=T∞−TsT∞−Tc$
(9)
$η=ηapp−η01−η0$
(10)

Argon, carbon dioxide, and nitrogen were the coolant gases examined in this study. Argon and carbon dioxide were selected due to their elevated densities relative to the nitrogen coolant, as well as the differences that they exhibit in other thermodynamic properties.

Table 1 shows the gas coolant-to-freestream ratios for each of the coolants investigated in this study at gas temperatures of 295 K and 315 K. In addition, the coolant-to-freestream property ratios at a representative engine condition—Tc = 1000 K and T = 2000 K—are included in Table 1. Individual cases will be referred to by the value of Tc. Density ratios were calculated using the ideal gas law, and assuming equal coolant and freestream pressures. $μ*$, $k*$, and $cp*$ were determined from linear interpolation of published thermodynamic property data [1012], which assumed an ambient pressure of 101.3 kPa. These properties, however, were invariant over the range of pressures experimentally observed in this study.

Table 1

Coolant-to-freestream property ratios and reference values at Tc = 295 K and 315 K and T = 315 K and 295 K

GasTc (K)$T∞(K)$$ρ*$$μ*$$k*$$cp*$$C*$$α*$$ν*$$Pr*$
Ar2953151.471.170.640.520.760.840.790.94
3152951.291.290.720.520.671.071.000.93
CO22953151.620.770.600.831.350.440.471.07
3152951.420.860.700.861.220.570.601.06
N22953151.030.920.941.031.070.880.891.01
3152950.911.011.051.030.941.121.121.00
Freestream reference valuesT(K)$ρ∞(kg/m3)$$μ∞(Pa⋅s)$$k∞(W/(mK))$$cp,∞(J/(kgK))$$C∞(J/(m3K))$$α∞(m2/s)$$ν∞(m2/s)$$Pr∞(−)$
Air3151.0919.3 × 10−627.3 × 10−31.00 × 1031.09 × 10325.0 × 10−617.7 × 10−60.71
2151.1618.3 × 10−625.8 × 10−31.00 × 1031.17 × 10322.1 × 10−615.7 × 10−60.71
GasTc (K)$T∞(K)$$ρ*$$μ*$$k*$$cp*$$C*$$α*$$ν*$$Pr*$
Ar2953151.471.170.640.520.760.840.790.94
3152951.291.290.720.520.671.071.000.93
CO22953151.620.770.600.831.350.440.471.07
3152951.420.860.700.861.220.570.601.06
N22953151.030.920.941.031.070.880.891.01
3152950.911.011.051.030.941.121.121.00
Freestream reference valuesT(K)$ρ∞(kg/m3)$$μ∞(Pa⋅s)$$k∞(W/(mK))$$cp,∞(J/(kgK))$$C∞(J/(m3K))$$α∞(m2/s)$$ν∞(m2/s)$$Pr∞(−)$
Air3151.0919.3 × 10−627.3 × 10−31.00 × 1031.09 × 10325.0 × 10−617.7 × 10−60.71
2151.1618.3 × 10−625.8 × 10−31.00 × 1031.17 × 10322.1 × 10−615.7 × 10−60.71

As high-density coolants, argon and carbon dioxide exhibited density ratios of 1.47 and 1.62 at Tc = 295 K, respectively, and 1.29 and 1.42 at Tc = 315 K. Compared with the nitrogen coolant at the same Tc condition, the argon and carbon dioxide coolants were 37% and 51% more dense, respectively. Notably, the $ρ*$ for the argon case at Tc = 295 K, when compared with the carbon dioxide case at Tc = 315 K differed by only 3.5%, which provided a case wherein $ρ*$ was approximately matched, while the other salient coolant-to-freestream properties remained significantly different.

Both argon and carbon dioxide thermal conductivities and specific heats were lower than that of nitrogen, with $k*$ 33% and 37% lower than nitrogen for argon and carbon dioxide at Tc = 295 K, respectively, and 32% and 34% lower at Tc = 315 K. $cp*$, on the other hand, was 50% lower for argon than nitrogen at both Tc conditions, while $cp*$ for carbon dioxide was 19% and 17% lower than nitrogen at Tc = 295 K and Tc = 315 K, respectively. This difference in $cp*$ indicates that the argon coolant can absorb approximately 1/3 less energy from—or transfer energy to, in the case of Tc = 315 K—the freestream before changing temperature by 1 K than carbon dioxide, all else being equal. Combining $cp*$ with $ρ*$ into the volumetric heat capacity ratio, $C*$, via Eq. (11), the difference in $cp*$ is magnified, where $C*$ for carbon dioxide is 78% and 82% greater than of argon at Tc = 295 K and 315 K, respectively:
$C*=ρccp,cρ∞cp,∞=ρ*cp*$
(11)

$k*$, when divided by $C*$, yields the thermal diffusivity ratio, $α*$, which describes the relative amount of heat transferred through the coolant vice the amount of heat absorbed. At Tc = 295 K, the argon coolant $α*$ was 6% less than that of the nitrogen coolant, while the carbon dioxide $α*$ was 51% lower. At Tc = 315 K, the coolant’s $α*$ values were 5% and 50% lower for argon and carbon dioxide, respectively.

$ν*$, or the coolant-to-freestream kinematic viscosity ratio, describes the strength of the shear layer that forms at the coolant-freestream fluid interface. At the test conditions, carbon dioxide’s $ν*$ was the lowest of the coolants, 46–47% lower than the nitrogen coolant. This was due to the large $ρ*$ and low $μ*$. With argon, the high $μ*$ partially offsets the high $ρ*$, resulting in $ν*$ values 11% lower than nitrogen coolant at both Tc conditions.

The final parameter of interest in this study was the Prandtl number ratio, $Pr*$. The Prandtl number describes the relative thicknesses of the momentum and thermal boundary layers of a fluid, and Prandtl numbers did not change significantly between Tc = 295 K and 315 K conditions. However, the coolant with the lowest Prandtl number was argon, at 6% less than nitrogen; while carbon dioxide had the highest Prandtl number—6–7% greater than nitrogen.

Figure 4 shows the adiabatic effectiveness distribution contours at I = 1.0 in order to orient the reader to the coolant flow situation. In this figure and subsequent figures, the ellipse shows the outline of the coolant hole exit and the coolant exits the hole in the −y direction before turning in the direction of the freestream, which is the +x direction.

Fig. 4
Fig. 4
Close modal

Often, blowing ratio is selectively matched between coolants of different densities. On a leading-edge model, however, matching M does not result in matching the location of the coolant effectiveness region. Unlike Fig. 4, Fig. 5 shows the y/d locations of maximum η, or $η^$ for argon, carbon dioxide, and nitrogen at M = 1.0 conditions. At these flow conditions, the argon and carbon dioxide peak effectiveness locations were fairly well aligned, but the nitrogen jet penetrated farther in the −y/d direction due to greater I. To better scale the effect of $ρ*$ on $η^$ location, the coolant gases were evaluated at matched I conditions of 0.5, 1.0, and 2.0, and the resultant $η^$ locations are shown in Fig. 6, with matched I groupings labeled. In general, the y/d location of $η^$ varied by less than 0.2 across gas and Tc condition at matched I.

Fig. 5
Fig. 5
Close modal
Fig. 6
Fig. 6
Close modal

Table 2 shows the resultant VR, M, ACR, and ReR values for each gas at the matched I conditions—which are bolded—and at each Tc condition at ReD = 60,000. Again, air coolant at relevant engine conditions is included for comparison purposes. Over the course of this study, the measured I value was within 0.024 of the target I. Therefore, the values of the resultant flowrate parameters that would result at exact target I values are presented for discussion purposes.

Table 2

Coolant flowrate parameter values at matched I conditions, Tc = 295 K and 315 K

GasTc (K)$VR$$M$$I$$ReR$$ACR$
Ar2950.580.860.500.740.45
3150.620.800.500.620.42
CO22950.560.900.501.170.75
3150.590.840.500.980.72
N22950.700.720.500.780.74
3150.740.670.500.660.69
Ar2950.821.211.001.030.63
3150.881.141.000.880.59
CO22950.791.271.001.651.05
3150.841.191.001.381.02
N22950.991.011.001.101.04
3151.050.951.000.940.95
Ar2951.171.712.001.460.89
3151.251.612.001.250.84
CO22951.111.802.002.341.49
3151.191.692.001.971.45
N22951.391.442.001.571.48
3151.481.352.001.341.39
GasTc (K)$VR$$M$$I$$ReR$$ACR$
Ar2950.580.860.500.740.45
3150.620.800.500.620.42
CO22950.560.900.501.170.75
3150.590.840.500.980.72
N22950.700.720.500.780.74
3150.740.670.500.660.69
Ar2950.821.211.001.030.63
3150.881.141.000.880.59
CO22950.791.271.001.651.05
3150.841.191.001.381.02
N22950.991.011.001.101.04
3151.050.951.000.940.95
Ar2951.171.712.001.460.89
3151.251.612.001.250.84
CO22951.111.802.002.341.49
3151.191.692.001.971.45
N22951.391.442.001.571.48
3151.481.352.001.341.39

Note: The bold values show resultant VR, M, ACR, and ReR values for each gas at the matched I conditions.

Since argon and carbon dioxide are elevated density coolants when compared with nitrogen, VR is lower and M is elevated at each I condition. Additionally, when comparing argon at Tc = 295 K and carbon dioxide at Tc = 315 K, the M and VR values at matched I were similar, differing by no more than 0.02. This was the primary result of the closely matched $ρ*$ conditions. However, the argon coolant ACR and ReR at these conditions were approximately 36% and 25% lower, respectively, than the carbon dioxide coolant.

In addition to the approximately matched $ρ*$ case, carbon dioxide coolant exhibited similar ACR to nitrogen at matched I conditions—varying no more than 0.1. However, due to the high density and low viscosity, ReR for carbon dioxide coolant was significantly greater than the nitrogen coolant. Argon, on the other hand, matched ReR fairly well to the nitrogen coolant—deviating no more than 0.1 when at matched Tc conditions, but with a suppressed ACR.

## Uncertainty and Repeatability

Uncertainty was determined using the method of Kline and McClintock [13]. Freestream Reynolds number uncertainty was determined to be approximately 2%. Additionally, ReD for each data point was set within 2% of the target 60,000 in this study. As a result, the freestream Reynolds number can be reasonably considered to be within 4% of 60,000.

The uncertainty in the gas properties was reported in Refs. [1012], and Table 3 shows the maximum uncertainties in coolant-to-freestream property ratios observed in this work. Since the static pressure at the exit of the coolant hole was assumed to be equal to that of the freestream, which in turn, was assumed to be near ambient pressure, the uncertainty in $ρ*$ to be a function of only the thermocouple uncertainty, which was 0.3 K.

Table 3

Coolant-to-freestream property ratio maximum uncertainties

$ϵρ*$$ϵμ*$$ϵk*$$ϵcp*$$ϵC*$$ϵα*$$ϵν*$$ϵPr*$
1.5%2.8%2.2%1.0%1.0%2.5%4.7%3.8%
$ϵρ*$$ϵμ*$$ϵk*$$ϵcp*$$ϵC*$$ϵα*$$ϵν*$$ϵPr*$
1.5%2.8%2.2%1.0%1.0%2.5%4.7%3.8%

Maximum flowrate parameter uncertainties are shown in Table 4. ReR had the largest uncertainty of the flowrate parameters. This was due to the relatively high uncertainty in kinematic viscosity ratio.

Table 4

Flowrate parameter maximum uncertainties

εVR$ϵM$$ϵI$$ϵReR$$ϵACR$
0.8%0.8%1.7%3.0%1.3%
εVR$ϵM$$ϵI$$ϵReR$$ϵACR$
0.8%0.8%1.7%3.0%1.3%

Uncertainty in η measurements was influenced primarily by the absolute difference between the coolant and freestream temperatures, and increases as the difference approaches zero. The minimum absolute temperature difference recorded for this study was 19.6 K, which was combined with the thermocouple uncertainties of 0.3 K and the surface temperature uncertainty of 0.4 K to determine $ϵηapp$ and $ϵη0$. By inspection of Eq. (10), as η0 approaches unity, $ϵη$ approaches ∞. However, η0 only is large near the cooling hole exit in the present work. In the primary region of interest, however, η0 lies near 0.04 and decreases below 0.035 in locations where elevated effectiveness was observed, i.e., beneath the coolant plume.

Repeatability for this facility and experiment type was discussed in Ref. [4], and peak adiabatic effectiveness was found to be repeatable within 0.02 when using air as a coolant.

## Results and Discussion

Figure 7 shows the spanwise adiabatic effectiveness profile at x/d = 3 for I = 0.5, 1.0, and 2.0. Note that the data are relatively dense, with approximately 25 data points per hole diameter in the figure, thus lines were used to represent the data instead of discrete data points. Despite the elevated $ρ*$, the argon coolant was less effective at each flow condition than the carbon dioxide and nitrogen coolants. Since the argon coolant ACR in both Tc cases was approximately 40% lower than the nitrogen and carbon dioxide coolants, the argon coolant changed temperature more rapidly, reducing effectiveness at each I condition. The nitrogen coolant, however, performed similarly to the carbon dioxide coolant jet, despite its decreased $ρ*$ value due to the fairly well matched ACR—differing from the carbon dioxide coolant jet by no more than 3% across the various I conditions.

Fig. 7
Fig. 7
Close modal

Similar to Fig. 6, it is apparent in the spanwise extractions of adiabatic effectiveness in Fig. 7 that I does indeed match the peak effectiveness locations fairly well between coolants, effectively scaling the main hydrodynamic effect of the coolant $ρ*$. However, closer examination of Figs. 7(b) and 7(c)I = 1.0 and 2.0, respectively—reveals a slight shift in the −y/d direction for the carbon dioxide coolants relative to argon and nitrogen.

Furthermore, when comparing a single gas at a particular I, the higher ACR gas generally produces a more effective coolant flow. This is apparent at all I values with argon coolant and at I ≥ 1.0 (Figs. 7(b) and 7(c)) for carbon dioxide and nitrogen. At I = 0.5, this improvement is less apparent as the improvement is primarily not in peak effectiveness, but rather at the edges of the coolant effectiveness profile.

Focusing attention on the results for argon at Tc = 295 K (dashed argon line) and carbon dioxide at Tc = 315 K (solid carbon dioxide line), which yielded $ρ*$ values of 1.47 and 1.42, respectively, it is apparent that the $ρ*$ does not effectively scale the thermal performance of the cooling jet. This is despite the relatively well matched VR and M at these matched I conditions. The ACR for the argon jet, however, was approximately 38% lower than the carbon dioxide jet for all I conditions, directly a result of the diminished $C*$.

Figure 8 shows the streamwise $η^$ profile over 2 ≤ x/d ≤ 6 at each I condition examined in this study. As was evident in Fig. 7, the carbon dioxide and nitrogen jets performed similarly at each I condition when ACR was approximately matched. At I = 0.5, the two nitrogen cases and the ACR = 0.75 carbon dioxide case maintained similar peak effectiveness across the entire profile, while the ACR = 0.72 case diverged slightly as x/d increased. This result may have been due to additional jet dynamics that I and ACR together do not fully describe. At I ≥ 1.0, however, the elevated ACR carbon dioxide and nitrogen jets improved approximately equally when compared with their lower ACR counterparts. Likewise, throughout the region of interest, the elevated ACR argon jets improved $η^$ performance relative to the lower ACR condition. Furthermore, the decay rate was consistent between the high and low ACR cases at matched I conditions. Increasing I, however, decreased $η^$ for each gas. This decrease occurred despite the increases in ACR. This detrimental effect was the result of increased coolant jet separation as I increased—indicating that direct scaling of film cooling based on ACR is not reasonable for compound injected cooling holes, but rather multipart scaling is required.

Fig. 8
Fig. 8
Close modal

The comparison of the streamwise $η^$ profiles at the near-matched $ρ*$ condition yields similar conclusions to the spanwise profiles discussed prior. While the argon $ρ*$ is slightly greater than that of the carbon dioxide, the thermal performance is significantly reduced at each I condition: approximately a 0.08–0.1 reduction in $η^$ performance across the region of interest. Meanwhile the much lower $ρ*$ nitrogen coolant performs similarly to the carbon dioxide when ACR is closely matched.

Peak adiabatic effectiveness, however, does not adequately describe how well a cooling jet performs. Laterally averaged adiabatic effectiveness profiles are shown in Fig. 9. Effectiveness data were averaged at each x/d location from −4 ≤ y/d ≤ 2 for each I value. For each gas, increasing ACR resulted in increased laterally averaged adiabatic effectiveness throughout the profile. Curiously, despite the close match in $η^$ between nitrogen and carbon dioxide at each I condition when ACR was closely matched, the $η¯$ was greater for carbon dioxide at each x/d location at similar ACR conditions. In fact, the $η¯$ profile for carbon dioxide at the lowest ACR condition met or exceeded nitrogen at its highest ACR condition at matched I. Since $η^$ was reasonably well matched when ACR between the gases was equivalent, the only explanation of this phenomena was that the carbon dioxide coolant jets were more effective at y/d values away from $η^$, i.e., maintained a wider jet. Additionally, despite the near-matched $ρ*$, the argon jet underperformed compared with carbon dioxide by approximately 0.05 throughout the profile at each I condition despite close matches in M and VR as well.

Fig. 9
Fig. 9
Close modal

Area-averaged adiabatic effectiveness is shown in Fig. 10. First, for a given cooling gas and coolant temperature, as I increases, $η¯¯$ decreases despite the increased ACR. As I increases, however, the coolant jet separation and liftoff increases, since the coolant jet penetrates farther into the freestream and does not return to the model surface as effectively. At constant I and comparing within a cooling gas, as ACR increased—due to increased $C*$, $η¯¯$ increased. However, when comparing argon at $ρ*$ = 1.47 to carbon dioxide at $ρ*$ = 1.42, it is apparent that $ρ*$ is not the driving factor in scaling $η¯¯$. ACR is not a perfect scaling parameter for $η¯¯$, as at each I condition, the elevated ACR nitrogen jet $η¯¯$ was, at best, equal to the low ACR carbon dioxide jet, even though the nitrogen jet ACR and $η^$ were equal to or greater than the low ACR carbon dioxide jet. However, this result was somewhat expected considering the generally elevated $η¯$ for the low ACR carbon dioxide jet compared with the high ACR nitrogen jet at matched I conditions.

Fig. 10
Fig. 10
Close modal

### Reynolds Number Ratio Effects.

Since nitrogen and carbon dioxide exhibited approximately matched $η^$ when ACR was approximately matched, but the carbon dioxide jets were more effective when laterally and area-averaged, the width of the carbon dioxide jet must have been larger. This is suggested in Fig. 7 for each I condition. However, the actual magnitude of η in these profiles can mask the jet width. In order to better visualize the shape of the coolant effectiveness profiles, the peak-normalized adiabatic effectiveness, $η/η^$, is presented in Fig. 11 at x/d = 3. At I = 0.5 (Fig. 11(a)), the carbon dioxide jets maintained greater $η/η^$ at y/d locations on either side of the peak, indicating a wider jet. Additionally, normalized adiabatic effectiveness increased slightly for carbon dioxide at the elevated Reynolds number condition. Argon and nitrogen performed similarly at y/d locations below the peak when their Reynolds numbers were similar. However, at y/d locations above the peak, the argon jet maintained greater normalized effectiveness.

Fig. 11
Fig. 11
Close modal

At I = 1.0 and 2.0, both carbon dioxide jets settled farther in the −y/d direction than the argon and nitrogen jets. Since the shear layers that form between the coolant and freestream flows are ultimately responsible for turning the coolant jet downstream, the less viscous carbon dioxide jets—which form weaker shear layers—established themselves at more negative y/d positions. This corroborates some of the computational results of Rutledge and Polanka [5], as well as the experimental results of Voet et al. [14]; though the authors of that work did not attribute differences cooling jet penetration depth at matched I to $μ*$ or $ν*$. This subtle effect of viscosity on coolant effectiveness region placement was likely not observed at I = 0.5 since the bulk of the coolant effectiveness was directly behind the coolant hole and did not have the required relative momentum to extend farther in the −y/d direction.

Figure 12 shows the normalized cooling jet effectiveness width, w/d, at I = 0.5, 1.0, and 2.0. The width of the effectiveness region was determined by finding distance between the y/d locations where the local η was 50% that of the $η^$ at a particular x/d. At each I condition, w/d increased with ReR. For the gases in question, elevated Reynolds numbers occurred for each gas when $ρ*$ was elevated and $μ*$ was depressed, resulting in lower $ν*$. Lower $ν*$ facilitated jet spreading because the lower $ν*$ likely resulted in weaker shear layers, which would not induce as much mixing at the edges of the jet. As a result, the energy transfer into the coolant jet due to mixing with the freestream likely was reduced, which would result in a more coherent jet at the edge of the coolant plume. Furthermore, w/d was generally inversely related to I when comparing gases at equal Tc. As the jet momentum increased, the jets became more resistant to spreading their thermal effect. This result was most likely due to both the high I jets separating from the surface of the model and ejecting into the freestream to a greater degree, as well as the enhanced mixing due to stronger shear layers.

Fig. 12
Fig. 12
Close modal

At I = 0.5, the jets generally maintained a similar w/d on the range 2 ≤ x/d ≤ 6, or exhibited a slight decrease. However, the argon case at a ReR of 0.62 actually increased in width over the x/d range. This result, however, was not likely due to an increase in jet spreading, but rather the decay in the peak effectiveness reduced the normalizing factor in $η/η^$, rather than an increase of effectiveness at regions above and below the coolant peak.

At I = 1.0 and 2.0, the jet width decreased with streamwise distance at each ReR condition—due to the separation of the coolant jet from the surface. Since these high I jets penetrate farther into the freestream flow, additional turbulence would return more coolant flow to the surface. At matched I conditions, the carbon dioxide jet has a higher Reynolds number than the other gases, and thus more turbulence is generated within the shear layer than the other coolants.

This is particularly evident in Fig. 13, which shows the average normalized coolant jet effectiveness width, $w¯/d$ for each coolant jet examined in this study. At matched I, the jet width scales fairly linearly with the jet Reynolds number. As I increased, however, the influence of jet Reynolds number on jet width diminished, so with increased I, greater differences in ReR were required to increase the jet width. When VR was approximately matched, as was the case for argon coolant at Tc = 295 K and carbon dioxide at Tc = 315 K. The difference in the coolant jet Reynolds number was inversely dependent on $μ*$ since $ρ*$ was nearly matched. The carbon dioxide jet $μ*$ was 26% lower than the argon jet at these coolant conditions, resulting in an approximately 32% increase in ReR.

Fig. 13
Fig. 13
Close modal

Figure 13 also includes lines of best fit for the data at each I condition. As ReR approaches zero, the lines of best fit show a general trend toward a $w¯/d$ of unity and the lines of best fit intercept the $w¯/d$ axis within 0.15 hole diameters of unity.

## Conclusion

Of particular interest in this study was the comparison between argon and carbon dioxide adiabatic effectiveness when $ρ*$ was approximately matched. Based on the bulk of the literature, these conditions should yield similar η results since $ρ*$ was similar. However, regardless of the test condition, the argon coolant significantly underperformed. Furthermore, the poor cooling performance of the argon coolant occurred despite the fact that M, I, and VR—the primary film-cooling scaling parameters that have been investigated in the literature were matched within 2.5% between the argon and carbon dioxide coolants.

Additionally, despite the relatively large difference in $ρ*$ between carbon dioxide and nitrogen, the adiabatic effectiveness magnitudes, particularly the peak magnitudes matched quite well, as did ACR, at matched I conditions. It should be noted, however, that merely matching ACR in those cases did not result in the close η matching observed by Fischer et al. [6]. The researchers in Ref. [6], however, provided the caveat that if a coolant jet was not separated—specifying a separation IACR provided good effectiveness scaling, essentially removing the hydrodynamics of the cooling jet from the cooling process. Due to the model geometry used in this study, the hydrodynamics and thermodynamics of the cooling flow could not be separated. For this reason, when I and ACR were closely matched—e.g., carbon dioxide and nitrogen, both the hydrodynamics and thermodynamics of coolant peak performance were similar. However, some subtle effects that appear to be influenced by the coolant jet Reynolds number were observed: namely the coolant effectiveness region width. Additionally, at I ≥ 1.0, the placement of the carbon dioxide coolant jet experimentally corroborated the previous computational work in Ref. [5], where the lower viscosity gas penetrated farther in the −y/d direction than the more viscous gases. Furthermore, this effect of coolant viscosity on jet placement corroborates the experimental work in Ref. [14], where carbon dioxide cooling jets penetrated farther off of the surface into the freestream.

## Acknowledgment

The authors thank the Air Force Research Laboratory for their support of this research. The views expressed in this article are those of the authors and do not reflect the official policy or position of the United States Air Force, the Department of Defense, or the US Government.

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request. The authors attest that all data for this study are included in the paper.

## Nomenclature

d =

coolant hole diameter, m

h =

heat transfer coefficient, W/(m2 K)

k =

thermal conductivity, W/(m K)

q =

convective heat flux, W/m2

u =

velocity, m/s

w =

coolant effectiveness region half-peak span, m

x =

streamwise distance from hole center, m

y =

vertical distance from hole center, m

C =

volumetric heat capacity, J/(m3 K)

D =

I =

momentum flux ratio, $ρcuc2/ρ∞u∞2$

L =

coolant hole length, m

M =

blowing ratio, ρcuc/ρu

T =

temperature, K

cP =

specific heat at constant pressure, J/(kg K)

$w¯$ =

mean effectiveness region half-peak span, m

ACR =

Pr =

Prandtl number, μcp/k

ReD =

ReR =

Reynolds number ratio, ρcucμ/ρuμc

Tu =

freestream turbulence intensity

VR =

velocity ratio, uc/u

α =

thermal diffusivity, m2/s, k/ρcP

β =

angle between stagnation line and hole axis, deg

γ =

angle between y-axis and hole axis, deg

ε =

uncertainty of subscripted measured quantity

η =

$η^$ =

peak adiabatic effectiveness at constant x/d

$η¯$ =

$η¯¯$ =

Λ =

integral length scale, m

μ =

dynamic viscosity, Pa · s

ν =

kinematic viscosity, m2/s

ρ =

density, kg/m3

### Subscripts

app =

apparent, uncorrected for conduction

aw =

c =

coolant

e =

at cooling hole exit

s =

surface

∞ =

freestream

0 =

underlying conduction baseline

### Superscript

* =

coolant-to-freestream ratio of gas property

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