## Abstract

Design of film-cooled engine components requires the ability to predict behavior at engine conditions through low-temperature testing. The adiabatic effectiveness, η, is one indicator film cooling performance. An experiment to measure η in a low-temperature experiment requires appropriate selection of the coolant flowrate. The mass flux ratio, M, is usually used in lieu of the velocity ratio to account for the fact that the coolant density is larger than that of the hot freestream at engine conditions. Numerous studies have evaluated the ability of M to scale η with mixed results. The momentum flux ratio, I, is an alternative also found to have mixed success, leading some to recommend matching the density ratio to allow simultaneous matching of M and I. Nevertheless, inconsistent results in the literature regarding the efficacy of these coolant flowrate parameters to scale the density ratio suggest other properties also play a role. Experiments were performed to measure η on a flat plate with a 7-7-7-shaped hole. Various coolant gases were used to give a large range of property variations. We show that a relatively new coolant flowrate parameter that accounts for density and specific heat, the advective capacity ratio, far exceeds the ability of either M or I to provide matched η between the various coolant gases that exhibit extreme property differences. With the specific heat of coolant in an engine significantly lower than that of the freestream, advective capacity ratio (ACR) is appropriate for scaling η with non-separating coolant flow.

## Introduction

Modern gas turbine airfoils are required to survive in an environment where freestream gas temperatures exceed the melting point of metallic turbine components. Film cooling is used to reduce the heat flux into the material and lower the components’ surface temperatures. Bogard and Thole [1] define many of the basic film cooling equations used in the literature. The heat flux with film cooling present (qf can be quantified
(1)
where Taw is the adiabatic wall temperature, Tw is the actual wall temperature, and hf is the heat transfer coefficient with film cooling present. The adiabatic wall temperature is nondimensionalized to yield the adiabatic effectiveness (η)
$η=T∞−TawT∞−Tc,exit$
(2)
The density of the coolant is typically much larger than the density of the freestream gas and is characterized in terms of the density ratio, DR:
$DR=ρcρ∞$
(3)
To characterize the coolant flowrate leaving the film cooling holes, multiple parameters have been in use including the mass flux or blowing ratio (M), the momentum flux ratio (I), and the velocity ratio (VR).
$M=ρcUcρ∞U∞$
(4)
$I=ρcUc2ρ∞U∞2$
(5)
$VR=UcU∞$
(6)

If DR is matched, then matching any one of those parameters would mean that all three are matched; however, matching the density ratio adds some complexity to many experiments. Extensive research has been conducted to determine which of these parameters best scales the effects of density ratio with the thought being that if an appropriate parameter is identified, then it would be unnecessary to match the density ratio in experimental work. However, no single parameter has been able to perfectly scale η results. Instead, most researchers will use a combination of the three in an attempt to scale their η results, and in some circles, it has become commonplace to go through the effort of matching DR so that all three parameters are matched.

A study carried out by Sinha et al. [2] attempted to quantify the effects of DR on η using a simple flat-plate geometry and an axially oriented cylindrical cooling hole. They quantified the coolant flowrate using the three previously defined parameters, M, I, and VR, and then varied the DR between 1.2 and 2.0. The DR changes were achieved using cold coolant, chilled by liquid nitrogen, with the coldest temperatures reaching 150 K in the DR = 2.0 case. If the same value of a flowrate parameter produced the same η profiles at different DR values, then a perfect scaling parameter has been found. As shown in Fig. 1, neither M, I, nor VR was found to be a perfect scaling parameter. The authors claim that I best predicts η with a maximum deviation of 0.15. This was compared with the matched M experiments which had a maximum deviation of 0.25, and the matched VR cases with a maximum deviation of 0.20. However, it should be noted that the researchers only matched I values up to 0.5, but higher I values occurred when M was matched at 1.0 with a DR = 1.2, resulting in I = 0.83. Previous research has shown that cylindrical holes are very prone to jet separation, so it is possible that the large change in η for the matched M = 1.0 case could have been caused by jet separation. This separation was not seen in the matched I cases because the matched values were less than 0.5, resulting in inconclusive results as to which parameter truly scales η best.

Fig. 1
Fig. 1
Close modal

The inability of the defined scaling parameters to properly scale η has also been seen more recently with a shaped hole geometry. After creating a new, non-proprietary-shaped hole design, the 7-7-7 hole, Schroeder and Thole tested different DR values by matching both M and I [3]. They found that, “effectiveness for the shaped holes did not scale well with blowing ratio or momentum flux ratio over the range evaluated” [3]. Their inconclusive results are shown in Fig. 2. Similar to the Sinha et al.’s study, the DR was manipulated using liquid nitrogen–cooled coolant air and room temperature freestream. All of these results show that although the density ratio does influence the scaling performance, it is not the only factor to consider when attempting to scale η.

Fig. 2
Fig. 2
Close modal

Owing to the obvious role of density in the momentum equation, a great number of other studies have also been conducted to characterize the ability of M, I, and VR to scale density differences between experimental conditions. Ethridge et al. [4], for example, found that neither M nor I sufficiently accounted for coolant density variations on the suction side of a turbine blade.

Stratton and Shih [5] recently considered the effect of the density ratio on the adiabatic effectiveness and also the turbulent flow structure, the likes of which are governed by the momentum equation in an incompressible flow. They found that matching VR allowed the best matching of the turbulence characteristics, but their thermal results were presented with M characterized instead.

Others have used pressure-sensitive paint (PSP) and the mass transfer analogy as a proxy for adiabatic effectiveness in their studies of density ratio effects. Using an anaerobic coolant gas, the measurement of the partial pressure of oxygen (through proper data reduction) is indicative of the concentration of the coolant gas. In this way, the molecular mixing and diffusion is a stand-in for thermal mixing and diffusion. In one such study, Li et al. [6] characterized adiabatic effectiveness on a leading edge showerhead region in terms of blowing ratio but note that increasing DR tends to increase the adiabatic effectiveness. Similar results have been found with other geometries in Vinton et al. [7], but while those authors characterize their results in terms of blowing ratio, they note that momentum flux ratio tends to predict jet lift-off. We also note that at low I values on their shaped hole for which the jet is well attached, the blowing ratio appears to scale their results decently.

The inconsistent and inconclusive nature of previous scaling research drove researchers to seek out alternative parameters that may better scale η results. This was the motivation behind a study performed by Rutledge and Polanka [8]. The authors looked at the effects of matching the Reynolds number ratio (ReR), as well as a new parameter defined by the authors: the heat capacity ratio, or as it is now known, the advective capacity ratio (ACR), shown in Eq. (9). This new parameter includes the specific heat ratio (CpR) between the coolant and freestream gases and takes into consideration physics present in the energy equation which had been previously largely ignored
$ReR=μ∞ρcUcμcρ∞U∞$
(7)
$CpR=cp,ccp,∞$
(8)
$ACR=cp,cρcUccp,∞ρ∞U∞$
(9)

Advective capacity ratio was developed due to the suspected impact that cp has on η distributions. To test this new parameter, the authors simulated a turbine blade leading edge in computational fluid dynamics (CFD) and then used various coolant flowrates with air and CO2. The density ratio for a hypothetical low-temperature film cooling experiment using room temperature air for both the coolant and freestream was set to 1.0, and the DR, when using CO2, was 1.5. They matched all five parameters (M, I, ACR, VR, and ReR) at a value of 1.0, then analyzed the results at an x/D location of 3. The authors found that I best matched the location of the coolant plume to the baseline case. This makes sense when one considers that the coolant was injected at a 90-deg compound angle. Even though the hole was angled at only 20 deg to the surface, the coolant was still ejected perpendicular to the freestream flow. Thus, the well-established physics of jets informs us that the penetration of the jet into the freestream would indeed be governed by the ratio of momentum fluxes.

While the utility of ACR with this geometry was not clear, Rutledge and Polanka [8] did show that the specific heat of the coolant had a noticeable effect on η by creating an artificial gas in the CFD simulation that was like air in every way but with a different cp. Therefore, Wiese et al. [9] extended that work with an experimental study using a leading edge model like the one simulated in Ref. [8]. The authors collected adiabatic effectiveness data using both thermal (infrared thermography) and mass transfer (pressure-sensitive paint) techniques. To simultaneously alter multiple gas properties, they ran two foreign gases (argon and CO2) as well as air for the coolant flow. The five coolant flowrate parameters (M, I, ACR, VR, and ReR) were then matched at values ranging from 0.25 to 2.0 for each coolant gas in order to evaluate which parameter, if any, scaled η values. A sample of their results with matched M and I cases are shown in Fig. 3. I was found to best predict the location of peak values of η, but not the magnitude of η. This corroborated the earlier CFD work of Ref. [8], but a new important observation was made in Ref. [9]: even with matched coolant plume location, the argon jet yielded η values far lower than CO2 or air.

Fig. 3
Fig. 3
Close modal

However, when Wiese et al. repeated the experiment using PSP, the disparity was not apparent, showing that there was something inherent with argon that made it perform differently than the other gases. In fact, argon's specific heat is only half that of air and 40% less than that of CO2, indicating a strong effect of cp. The effect of this is evident in the thermal experiments but not in the PSP experiments which are insensitive to the amount of heat absorbed by the coolant. This leads one to expect that η scaling experiments performed using PSP measurements will not capture the thermal transport phenomena, but will effectively isolate the density ratio effects when other properties are not controlled (cf. Ref. [7]).

In another paper, Wiese et al. presented the matched ACR cases using air, argon, CO2, and N2 as coolant [10]. As shown in Fig. 4, matching ACR resulted in approximately the same η profiles for air, CO2, and N2, but a very different profile for argon. The authors attributed this to jet separation effects since matching ACR required much larger I values for argon than the other gases (I = 2.53 instead of I ≈ 1). Hence, the coolant plume is shifted to more negative y/d values and the η magnitude decreases due to the greater penetration of the coolant plume into the freestream.

Fig. 4
Fig. 4
Close modal

Luque et al. [11] also demonstrated the importance of cp in their study of overall cooling effectiveness on a fully cooled turbine blade. They performed experiments on a conducting film-cooled component with two different coolant gases (air and a mixture of argon and SF6). The foreign gas mixture had a specific heat 44% less than that of the air coolant, and the authors developed a mathematical model that provided a first-order approximation for the influence of ACR mismatch on overall effectiveness, ϕ, when I was matched between the two cases.

With physics and experimental results clearly pointing to the importance of specific heat, but without a single scaling parameter evident for the compound angle injection described above, the investigation of cooling geometries without compound angle injection was desired. Gritsch et al. [12] studied cylindrical, fan-shaped, and laid-back fan-shaped cooling holes to investigate the effects that each geometry had on the adiabatic effectiveness. They found that the laid-back fan-shaped holes allowed the jet to smoothly transition and cool the surface at higher flowrates. This resulted in an increase in η values with increased flowrate since the laid-back fan-shaped holes were less prone to coolant separation.

The laid-back fan-shaped hole design proposed by Schroeder and Thole [3], the 7-7-7 hole, was selected for the present work due to its open source nature and thorough characterization in the literature. The 7-7-7 hole exit expands 7 deg in the positive and negative lateral directions and includes a 7-deg laid-back angle. The complete geometry and parameters can be seen in Table 1 and Fig. 5.

Fig. 5
Fig. 5
Close modal
Table 1

7-7-7 Hole parameters [3]

 Injection angle, α 30 deg Lm/D 2.5 Llat/D, Lfwd/D 3.5 L/D 6 Laid-back angle, βfwd 7 deg Lateral angle, βlat 7 deg P/D 6 Coverage ratio, t/P 0.35 Area ratio, AR 2.5 Rounding of four edges inside diffuser, R/D 0.5
 Injection angle, α 30 deg Lm/D 2.5 Llat/D, Lfwd/D 3.5 L/D 6 Laid-back angle, βfwd 7 deg Lateral angle, βlat 7 deg P/D 6 Coverage ratio, t/P 0.35 Area ratio, AR 2.5 Rounding of four edges inside diffuser, R/D 0.5

The objective of the present work is to establish an appropriate scaling methodology for adiabatic effectiveness resulting from a fan-shaped cooling hole whose jet penetration is not strong and thus not dominated by momentum effects. The traditional coolant flowrate parameters (M, I, and VR) are used in addition to the two newer parameters introduced in Ref. [8] (ReR and ACR) to determine their efficacy at providing matched η distributions with four coolant gases (Ar, CO2, He, and N2). Both CO2 and Ar have densities higher than that of the freestream air, but Ar has a much lower cp to ensure that cp effects can be separately evaluated. Furthermore, helium is a particularly unorthodox choice for film cooling experiments since its density is far lower than that of the freestream air, opposite the case in an engine. However, if a scaling technique can be shown to handle that extreme density ratio condition, it lends credibility to the efficacy of the scaling technique.

## Theory of the Advective Capacity Ratio

The legitimacy of the advective capacity ratio as a nondimensional parameter of importance lies in the energy equation. The energy equation for steady low-speed flow but with nonuniform properties may be written [13]
$ρcpvi∂T∂xi=∂∂xi(k∂T∂xi)+μ(∂vj∂xi∂vi∂xj+∂vi∂xj∂vi∂xj)+βTvi∂P∂xi$
(10)
The importance of ρcp is quite evident since the only place that ρ appears in the energy equation, it is multiplied by cp, but we shall add further rigor to the development of ACR. It is reasonable to limit our scope to flows of ideal gases for which β = 1/T. Introduce the nondimensional variables
$Ji*=ρcpviρ∞cp,∞U∞,vi*=viU∞,T*=T∞−TT∞−Tc,k*=kk∞,μ*=μμ∞,P*=P−P∞ρ∞U∞2,Φ*=μ(∂vj∂xi∂vi∂xj+∂vi∂xj∂vi∂xj)μ∞U∞2/L2,xi*=xiL$
so that the relevant energy equation may be written in nondimensional form
$Ji*∂T*∂xi*=1Re∞Pr∞∂∂xi*(k*∂T*∂xi*)−Ec∞Re∞Φ*−Ec∞vi*∂P*∂xi*$
(11)
where
$Ec∞=U∞2cp,∞(T∞−Tc)andRe∞=ρ∞U∞Lμ∞$

The nondimensionalized energy equation reveals that the freestream Reynolds number, Prandtl number, and Eckert number must theoretically be matched in an experiment designed to imitate engine conditions. It is noteworthy, however, that the Eckert number may be cast in terms of the Mach number, which reveals that the final two terms on the right-hand side of Eq. (11) may be negligible for many low Mach number flows. The nondimensional boundary conditions must be matched for the nondimensional parameters to be matched within the interior of the domain of interest. In the case of T*, this is straightforward since T* = 0 in the freestream and T* = 1 at the coolant inlet. The nondimensional parameters Ji* and k* are also straightforward in the freestream (i.e., unity) but the theory suggests that they must also be matched at the coolant inlet. That is, kc/k, and ρccp,cUc/ρcp,∞U (defined as ACR in Eq. (9)) must be matched according to Eq. (11). Nevertheless, the justification for matching ACR is grounded in physics.

The momentum equation places additional requirements on conditions that must be matched (including the momentum flux ratio). Of course, the relative importance of all of the nondimensional parameters that theory suggests must be matched is unclear until tested experimentally, nor is it reasonable to expect that one may simultaneously matched all nondimensional parameters in a low-temperature experiment. The large component of coolant momentum perpendicular to the freestream flow direction with the 90 deg compound angle injection used by Weise et al. [9] meant that the momentum flux ratio, I, had great importance and tended to dominate much of the coolant flow physics in lieu of ACR. It thus stands to reason that coolant jets inclined closer to the direction of the freestream flow such as laid-back fan-shaped holes aligned in the flow direction would be more scalable with ACR. This is the hypothesis studied presently.

## Experimental Methodology

A new test rig was designed to run the experiments for this study. The rig was designed to add an extended afterbody to the 8.89-cm-diameter leading edge model of Wiese [5]. This provided a flat surface to locate the 7-7-7 cooling holes. The holes were placed at a distance of x/D = 69 downstream of the leading edge in an effort to match the boundary layer conditions, and thus the coolant plume behavior, in the original 7-7-7 study [3] as pictured in Fig. 6. The test section was constructed of low thermal conductivity foam (k = 0.03 W/m K) to reduce the conduction through the material. An infrared (IR) camera was mounted to view the flat-plate section outlined in Fig. 6 and pictured in Fig. 7, resulting in a full view of the cooling holes and 18 hole diameters downstream. The coolant temperature was held between 295 K and 310 K and was collected with a thermocouple mounted at the entrance to the cooling hole, inside the plenum.

Fig. 6
Fig. 6
Close modal
Fig. 7
Fig. 7
Close modal

The rig was placed in an open loop, temperature-regulated tunnel, pictured in Fig. 8. A control circuit allowed the tunnel Reynolds number, ReD, to be independent of any changing ambient conditions. By monitoring the freestream temperature collected with a thermocouple inserted into the flow away from the tunnel wall, and the velocity obtained with a pitot-static tube, the Reynolds number was monitored and adjusted. The freestream flow temperature was set to 327 K for all cases. The freestream Reynolds number, ReD, was subsequently held to within 1% of 5000.

Fig. 8
Fig. 8
Close modal

All test points were run with two different freestream turbulence intensities. The “clean” configuration with no turbulence grid in place resulted in a very low freestream turbulence intensity 0.67%. The experiments were then repeated after installing a turbulence grid 1.2 m upstream (x/D = −206) of the cooling hole exit. The addition of the turbulence grid increased the freestream turbulence intensity to a value of 4.40%.

The data collection process began by setting the freestream temperature and velocity to match ReD = 5000. After the tunnel wall temperature achieved a steady-state value, the coolant gas was selected and the coolant flow was dialed into the desired value. After a two-minute settling time to allow for a steady-state measurement, an IR image was captured along with the thermocouple, pressure transducer, and flowrate measurements.

To evaluate each flowrate parameter's performance, four different gases with varying properties were used as coolant: argon (Ar), carbon dioxide (CO2), helium (He), and nitrogen (N2). The freestream gas for all data points was air. The coolant and freestream properties are given in Table 2.

Table 2

Properties of gases at engine and testing conditions

GasTemp. (K)Density, ρ (kg/m3)Specific heat, cp (kJ/(kg K))Dynamic viscosity, µ 10−6 (Pa·s)Thermal conductivity, k 10−3 (W/m K)
Engine freestream (air, 30 bar)20005.221.3468.9137
Engine coolant (air, 30 bar)100010.451.1442.466.7
Freestream air3271.081.0119.728.4
Ar3001.660.5222.317.2
CO23001.770.8514.916.6
He3000.165.1919.9152
N23001.121.0417.825.9
GasTemp. (K)Density, ρ (kg/m3)Specific heat, cp (kJ/(kg K))Dynamic viscosity, µ 10−6 (Pa·s)Thermal conductivity, k 10−3 (W/m K)
Engine freestream (air, 30 bar)20005.221.3468.9137
Engine coolant (air, 30 bar)100010.451.1442.466.7
Freestream air3271.081.0119.728.4
Ar3001.660.5222.317.2
CO23001.770.8514.916.6
He3000.165.1919.9152
N23001.121.0417.825.9

Typical freestream and coolant properties at engine conditions are also given for comparison. The coolant flow was varied to match values from the five different coolant flowrate parameters, given in Table 3. These set points represent a wide spread of coolant flowrates, allowing for detailed analysis and fine resolution of the final trend curves. The uncertainty in the flowrates was within 3%.

Table 3

Experimental conditions

Test parameterValue(s) during testing
ReD5000
Tu0.67% and 4.40%
DR, CpR for Ar1.41–1.54, 0.51–0.54
DR, CpR for CO21.58–1.72, 0.83–0.84
DR, CpR for He0.15–0.16, 5.00–5.57
DR, CpR for N21.04–1.09, 1.02–1.04
Matched M values0.25, 0.50, 1.00, 1.50, 2.00
Matched I values0.25, 0.50, 1.00, 1.50, 2.00
Matched ACR values0.25, 0.50, 1.00, 1.50, 2.00
Matched ReR values0.25, 0.50, 1.00, 1.50, 2.00
Matched VR values0.25, 0.50, 1.00, 1.50, 2.00
Test parameterValue(s) during testing
ReD5000
Tu0.67% and 4.40%
DR, CpR for Ar1.41–1.54, 0.51–0.54
DR, CpR for CO21.58–1.72, 0.83–0.84
DR, CpR for He0.15–0.16, 5.00–5.57
DR, CpR for N21.04–1.09, 1.02–1.04
Matched M values0.25, 0.50, 1.00, 1.50, 2.00
Matched I values0.25, 0.50, 1.00, 1.50, 2.00
Matched ACR values0.25, 0.50, 1.00, 1.50, 2.00
Matched ReR values0.25, 0.50, 1.00, 1.50, 2.00
Matched VR values0.25, 0.50, 1.00, 1.50, 2.00

After running the experiment, the raw IR counts and thermocouple readings were then reduced into η values. The IR camera was calibrated using thermocouples on the surface of the experimental model downstream of a second coolant hole that was utilized only for the calibration and shut off during data collection. The use of a separate independently fed cooling hole upstream of the calibration thermocouples allowed for a clean surface downstream of the hole of interest. The IR camera calibration curve is shown in Fig. 9. The resulting surface temperature was nondimensionalized as an effectiveness which was then corrected for conduction, using an established one-dimensional conduction correction, previously used by a number of researchers including Williams et al. [14]. The uncertainty in the resulting η distribution was a function of Ts and the difference, TTc, and was calculated using the technique of Kline and McClintock [15]. The calculated uncertainty in η was within ±0.04 for all testing and was corroborated through the observed repeatability of the measurements.

Fig. 9
Fig. 9
Close modal

The first parameter analyzed was the blowing ratio M. As shown in Fig. 10, at a low value of 0.25 and low Tu, M did not scale well between the four gases. Using He obviously produced higher η values, and noticeable differences are apparent between the other coolants. Plotting the centerline effectiveness in Fig. 11 shows the disparity between all four gases in this non-separating condition. Also listed in the figure's legend are the values of the other scaling parameters. It is clearly seen that the η values increase with increasing ACR values from Ar at a value of 0.13 to He at 1.29. Since ACR is M multiplied by CpR, the increase in η from Ar to He is attributed to an increase in CpR. It is also important to note that very little difference was seen between the two different turbulence intensities, as the dashed lines (high Tu case) almost exactly match the solid lines (low Tu case).

Fig. 10
Fig. 10
Close modal
Fig. 11
Fig. 11
Close modal

The dependence on CpR was less evident at higher flowrates as shown in Fig. 12, where M = 1.50. The flowrate required for He to match M = 1.50 was above the facility max of 50 SLPM, so no He data could be collected at this point. As with M = 0.25, Ar produced the lowest η values, but the CO2 flow rose above the N2 values despite having a lower ACR. It was theorized that the break in CpR dependence could be a result of jet separation effects. A slight difference was shown between the two turbulence cases especially with CO2 and N2. However, the maximum difference between the two cases was still within the established ±0.04 uncertainty. With the indications toward jet separation effects, the next parameter analyzed was the momentum flux ratio.

Fig. 12
Fig. 12
Close modal

Figure 13 shows that like M, I did not effectively collapse the data. Much like the work of Sinha et al. [2], the CO2 and N2 data were matched within their experimental uncertainty. These two gases, however, have an ACR within 0.01 of each other for the matched I = 0.25 cases. The gases with the highest and lowest ACR values corresponded to significantly different η values. Therefore, the trend of increasing η with increasing CpR is maintained. Unlike the matched M cases, at higher values such as I = 1.50, the η values were still ordered with increasing ACR values as shown in Fig. 14. I still does not scale well as shown by the large gap between Ar and the other gases, but this can be explained by its ACR being significantly lower than the others. Note that the necessarily high flowrate of He precluded its use at I = 1.50.

Fig. 13
Fig. 13
Close modal
Fig. 14
Fig. 14
Close modal

Even though I was found to not be a sufficient scaling parameter, it was found to be an important parameter at indicating jet separation. By plotting the area-averaged η values $(η¯¯)$ verses I for all the collected data points in Fig. 15, each of the $η¯¯$ distributions peaked at approximately the same I value, I = 1.1. Because jet separation is a momentum-driven effect, the jet leaving the hole would be expected to begin to separate at the same critical value for each gas. However, jet separation is independent of the specific heat of the fluid, which explains the differing magnitudes of $η¯¯$ for each gas. Additionally, the high turbulence cases resulted in higher $η¯¯$ values, especially after the peak. This turbulence effect was expected as it has been observed in previous studies (cf. Ref. [16]). While He was not able to be pushed to its peak $η¯¯$ value due to flowrate constraints, the initial shape of the curve and the other evidence indicate that helium would likely peak at a similar value of I.

Fig. 15
Fig. 15
Close modal

The results from both the M and I data sets suggest a strong dependence on CpR, especially at lower flowrates, so the ACR was analyzed next. As shown in Fig. 16, at a low value of 0.25, ACR was able to collapse the centerline η profiles of all four gases at both turbulence intensities to within 0.05 for x/d > 5.

Fig. 16
Fig. 16
Close modal

Despite having DR values ranging from 0.15 to 1.67, the inclusion of the CpR allows ACR to account for each gas's ability to cool the surface, ultimately resulting in matched η profiles. This was seen to an even greater extent at ACR = 0.50, as shown in Figs. 17 and 18. The dramatic superiority of the scaling capability of ACR relative to M is evident when comparing the spatial contours of η in Figs. 10 and 18.

Fig. 17
Fig. 17
Close modal
Fig. 18
Fig. 18
Close modal

Figure 18 also offers an opportunity to compare those contours of η to similar results obtained by Schroeder and Thole [3]. Since Schroeder and Thole used 295 K freestream and cold coolant to achieve elevated DR, their CpR ≈ 1, such that ACR ≈ M in their work. A comparison of the N2 case in Fig. 18 to the Schroeder and Thole’s case with M = 0.51, DR = 1.2 reveals that they match each other within a contour level or better. The difference is well within the combined experimental uncertainties and demonstrates that the hole is behaving appropriately for a 7-7-7 hole. Although not shown, similar matching to the data of Schroeder and Thole was achieved at higher coolant flowrates.

As the ACR values were increased beyond 0.50, more variation in η between the various gases was present. Although not shown, at ACR = 0.75, the argon yields lower values of η than the other gases. N2 and CO2 yield lower values of η than He once ACR is increased to 1.0. At ACR = 1.50, as shown in Fig. 19, the familiar trend of lower η values for Ar and higher values for He is evident.

Fig. 19
Fig. 19
Close modal

Previously in the M and I results, if there was a separation between the η profiles, the lines on the plot would be ordered with increasing ACR. With the matched ACR lines, the order was now with decreasing I values. All gases except helium had an I value greater than the established peak point of I = 1.1, suggesting that the reason for the variation was that three of the gases had separated. With an I = 5.44, Ar had clearly separated leading to the lower η values, while helium's relatively high η values were due to this gas still being fully attached with I = 0.54.

The utility of ACR is best seen when $η¯¯$ versus ACR is plotted for all collected data points. Figure 20 illustrates ACR's ability to scale η no matter the density ratio, specific heat ratio, or turbulence intensity at low values of ACR. All gases follow the same curve at the corresponding low I values and then, one-by-one, the $η¯¯$ values for individual gases deviate from the general trend and decrease as the ACR is increased. Interestingly, each of these deviations occurred at similar I values. Each of the dashed arrows in Fig. 20 is placed at the ACR value where the trend line for an individual gas deviates from the general trend. The corresponding I values at each arrow location ranged from 0.5 to 0.7. The 50 SLPM flow limit prevented helium testing above ACR = 1.50, so the corresponding arrow was placed at the approximate expected deviation point based on I = 0.6. This indicates that at ACR values less than those indicated by the arrows, the coolant jet is in a fully attached state since the peak $η¯¯$ values occurred at approximately I = 1.1. Given this compelling evidence, it was concluded that ACR is the best scaling parameter for the adiabatic effectiveness for I < 0.5, where the coolant jet is assumed to be fully attached. It should be noted that the specific threshold of I < 0.5 is expected to be geometry and orientation dependent; however, it is anticipated that similar results would be replicated with other fully attached jets with zero-compound angle injection.

Fig. 20
Fig. 20
Close modal

In addition to M, I, and ACR, matched VR and ReR cases were also completed with similar results to M and I. As shown in Figs. 21 and 22, matching VR does not account for DR changes and thus yields a large spread of η values between the four coolant gases at both low and high flowrates. The utility of using the various gases is again seen in the VR cases. If the argon and helium lines were removed from Figs. 21 and 22, one may incorrectly conclude that VR is an acceptable scaling parameter for η.

Fig. 21
Fig. 21
Close modal
Fig. 22
Fig. 22
Close modal

The Reynolds number ratio was also found to be an inappropriate scaling parameter highlighted by the use of helium and argon. Figure 23 demonstrates that at ReR = 0.25 there is a large spread of η values between the four gases.

Fig. 23
Fig. 23
Close modal

## Applicability to Previously Published Data

The utility of ACR in scaling η is also evident by determining the ACR of the coolant flow in studies that predated this investigation. By calculating ACR through knowledge of the gases and temperatures used in the study conducted by Sinha et al. [2], ACR comparisons of that data set are possible. Reevaluating the Fig. 1 data for the I = 0.2 case, the two ACR values were calculated to be 0.49 for DR = 1.2 case and 0.57 for DR = 1.6 case. A higher ACR value likely explains the observed higher η values in Fig. 24. In the I = 0.3 case, the difference between the η distributions is greater than the first, and this is mirrored by a larger difference between ACR values of 0.60 and 0.78, respectively. As seen in the current study, ACR is less effective when the jet has been separated from the surface, as seen in I = 0.5 case. Although this I value is the same as the critical I value of 0.5 found in the present study, the study by Sinha et al. used cylindrical holes which are more prone to coolant separation than the 7-7-7 hole. The calculated ACR values were 0.77, 0.90, and 1.01 for the three DR values at I = 0.5. Beyond where the jet appears to reattach to the surface, the η profiles become ordered according to the increasing ACR.

Fig. 24
Fig. 24
Close modal

When Sinha et al. matched low M values, the η profiles matched extremely well between the two presented DR values, as shown in Fig. 25. Since the authors achieved those two DR values by changing the coolant air temperature between two low temperatures, the coolant specific heat was nearly the same in both cases, reflected in Table 2. So by happenstance, matching M also had the effect of nearly matching ACR—the ACR turns out to be 0.49 and 0.51 for the DR = 1.2 and 2.0 cases, respectively. Once the jet becomes fully separated, as in the M = 1.0 case, ACR does not explain the variation in η values between density ratios, consistent with the findings of this study. While Sinha et al. concluded that M was effective for scaling DR with non-separating coolant flows, it is likely they would have reached a less conclusive result had they used a foreign gas that would have resulted a non-unity cp ratio.

Fig. 25
Fig. 25
Close modal

If ACR can explain η variations from cylindrical holes, then it should definitely explain differences from the original 7-7-7 hole study conducted by Schroeder and Thole [3]. For the matched M cases shown previously in Fig. 2, only the first point resulted in I values less than 0.5. As shown in Fig. 26, for the M = 0.5 point, there was a close match between the measured $η¯¯$ values. This is explained because the ACR values were also matched at this point; the calculated ACR values were 0.50 for both DR = 1.2 and for DR = 1.5 conditions. A similar trend was seen when calculating ACR for the matched I cases in Fig. 27. The first two points on the matched I plot were approximately 0.01 apart, and the ACR values also differed slightly between 0.49 and 0.55. Even though the second-matched I point lies beyond I = 0.5 separation point, ACR predicts the slight difference yielding values of 1.10 and 1.23. These results demonstrate that ACR is a powerful tool for scaling η on flat-plate geometries without compound angle injection.

Fig. 26
Fig. 26
Close modal
Fig. 27
Fig. 27
Close modal

## Conclusion

Through testing five coolant flowrate parameters (M, I, ACR, VR, and ReR), this study provided comprehensive results for scaling adiabatic effectiveness with an axially oriented, shaped cooling hole on a flat plate. First, it was confirmed that I best predicts jet separation. For this particular geometry, peak area-averaged η values $(η¯¯)$ occur at a value of I = 1.1, regardless of coolant gas properties or turbulence intensities. This observation about I was consistent with many prior film cooling studies. The new and significant result from this study is that ACR will almost exactly scale adiabatic effectiveness results, as long as the coolant jet remains fully attached (cases with I < 0.5). At greater values of I, the coolant is not attached to the surface, and since jet separation is governed by I, ACR becomes less effective for scaling η.

A result of this definitive about the efficacy of a scaling parameter for η has not been seen in prior studies. Clearly, there is a need to consider the specific heat of the coolant and freestream gases. ACR is the only quantity needed to accurately quantify the cooling capacity of any given gas, even those with drastically different properties such as with helium and argon, provided that the cooling flow is not separated and ejected predominantly in the direction of the freestream as with axially aligned fan-shaped holes. We thus emphasize that the result is geometry dependent as geometries with 90-deg compound angle injection, for instance, will have coolant flow behavior dominated far more by momentum effects simply dictating the placement of the coolant jet rather than the jet's cooling capacity [9].

Decades of research have been conducted searching for a way to make either M, I, or VR scale η across multiple density ratios with little success leading some to believe that density ratio must be matched in experimental work. While DR is important, this misses the fact that the only place density appears in the energy equation, it is multiplied by cp. The cp ratio is of equal importance, and cp turns out to be significantly lower for the coolant in an engine than it is for the hot freestream (see Table 2). This research demonstrates that as long as ACR is matched, there is no need to match the density ratio for certain geometries and low values of I. Indeed, even helium coolant with its very low density gave the same η distributions as with far heavier gases provided I was low and ACR was matched.

## Acknowledgment

The authors thank the Air Force Research Laboratory for their support to 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.

## Nomenclature

• h=

convective heat transfer coefficient (W/(m2 K))

•
• l=

arbitrary length scale (m)

•
• q=

heat flux (W/m2), Eq. (1)

•
• x=

first ordinate (m)

•
• y=

second ordinate (m)

•
• D=

cooling hole diameter (m)

•
• I=

momentum flux ratio, Eq. (5)

•
• M=

mass flux (blowing) ratio, Eq. (4)

•
• T=

temperature (K)

•
• U=

freestream velocity (m/s)

•
• $m˙$=

mass flowrate, SLPM

•
• cp=

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

•
• DR=

density ratio, Eq. (3)

•
• Re=

Reynolds number, ρUl/μ

•
• Tu=

freestream turbulence intensity

•
• VR=

velocity ratio, Eq. (6)

•
• CpR=

specific heat ratio, Eq. (8)

•
• ACR=

•
• ReR=

Reynolds number ratio, Eq. (7)

•
• $δ*$=

boundary layer thickness (m)

•
• ΔT=

temperature difference (K)

•
• Δη=

•
• ɛ=

measurement uncertainty

•
• η=

•
• $η¯$=

•
• $η¯¯$=

•
• μ=

dynamic viscosity (Pa·s)

•
• Ρ=

density (kg/m3)

•
• ϕ=

overall effectiveness

### Subscripts

• app=

apparent

•
• aw=

•
• c=

coolant

•
• D=

based on hole diameter

•
• f=

with film cooling present

•
• s=

surface

•
• w=

wall

•
• ∞=

freestream

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