A Computational Technique to Evaluate the Relative Influence of Internal and External Cooling on Overall Effectiveness

Gas turbine components require active cooling in order to remain operational. There are three ways a blade is cooled. Plenum cooling, such as impingement, cools the inside of the blade. The coolant then flows through film cooling holes, absorbing heat via convection. Finally, external cooling, such as a showerhead of film cooling holes, injects a thin film over the external surface.

The importance of ACR is evident when one considers the energy equation and the fact that when the only place density appears, it is multiplied by the specific heat. Note that many experiments and simulations utilize low-temperature air as both coolant and freestream such that the c_p ratio is unity. For that special case, ACR = M.

Albert et al. [1] found adiabatic effectiveness on a leading edge using a low thermal conductivity model and overall effectiveness distributions for an identical geometry, but with a higher thermal conductivity such that the Biot number was approximately matched to engine conditions. They found that overall effectiveness values were higher than the adiabatic effectiveness between cooling holes due to convective cooling within the holes and internal cooling.

Mensch and Thole [4] looked at the effect of impingement on the end wall of a cascade. They tested film and impingement cooling separately and together. They did not separate the effects of convection within the holes and film cooling over the top of the conducting surface, but the combined effects of both forms of cooling were evaluated. Film cooling showed high effectiveness directly around the coolant holes. The authors credited convection within the holes as a primary contributor to the increased effectiveness in those areas and impingement tended to provide more uniform effectiveness, which increased with increasing coolant flowrate.

Ravelli et al. [5] investigated the effects of impingement on a leading edge with showerhead film cooling. They found that overall effectiveness was only slightly improved by impingement, possibly because convective cooling within the cooling holes had a greater effect than previously recognized.

Bryant et al. [6] conducted experiments on a film-cooled leading edge showerhead region to determine ϕ distributions. They were able to confirm the predictions of Ravelli et al. [5] that impingement cooling had only a negligible effect on ϕ. Further, by selectively blocking coolant holes, they were able to confirm that conduction directly to the coolant within the coolant holes dominated over the impingement effects. In fact, there are locations in the leading edge region where conduction to the coolant holes is the dominant form of cooling beyond that of external film cooling.

Terrell et al. [7] further studied the influence of convective heat transfer through film cooling holes in the leading edge. Using a series of thermocouples to measure the heat rise through coolant holes, they found cooling within the holes accounted for 50–80% of internal cooling in a leading edge without impingement.

Chavez et al. [8] also studied hole cooling on a leading edge. They found that even with poor adiabatic effectiveness (due to poor film cooling), hole and impingement cooling combined gave adequate overall effectiveness. Since Bryant et al. [6] showed that impingement is not a primary contributor to overall cooling in a leading edge region, the most effective form of cooling, in this case, was likely hole cooling.

As described above, there has been some effort to determine the relative contributions of various forms of cooling; however, the coupled nature of internal and external cooling makes it difficult to experimentally determine the individual influence of the cooling techniques on ϕ. Though some progress has been made, it has been hampered by the physical realities of the experimental work. For instance, it is impossible to completely isolate internal cooling from film cooling since the internal coolant must be ejected as the film coolant. Blocking cooling holes is only partially effective since it precludes convection within the cooling hole, which Refs. [5,6] both showed is a dominant form of cooling. While it is impossible to have the coolant flow through the coolant holes without ejecting onto the surface to segregate the hole cooling and film cooling, this can be accomplished computationally by setting a nonphysical boundary condition at the hole exit that causes the coolant exiting the holes to vanish. Additionally, selected surfaces on the model can be made adiabatic in a computational model while maintaining conduction elsewhere in order to segregate the effects of various forms of cooling. This novel use of computational fluid dynamics (CFD) to model nonphysical effects can be used to complement the experimental work and inform designers on the most beneficial ways of improving cooling.

The model used in these simulations is a large-scale flat plate with five staggered shaped film cooling holes. The holes are 7-7-7 shaped holes which were characterized by Schroeder and Thole [9]. The holes are at a 30 deg angle to the freestream and have 7 deg layback and 7 deg fan at each side. The holes are in a staggered array with a row of three holes in the middle and two half holes on either side, as shown in Fig. 1. Periodic boundary conditions were used on the boundaries going through the half holes. The bottom portion of the hole is a cylindrical hole at a 30 deg angle to the freestream with a diameter of 0.643 cm. The solid conducting material has a thermal conductivity of 3.5 W/m/K. This thermal conductivity allows for the Biot number to be approximately matched to a typical engine condition. A schematic of the hole geometry is shown in Fig. 2.

An adiabatic flat plate (not shown in Fig. 1) was modeled extending 50 hole diameters upstream of the conducting material in order to allow a hydrodynamic boundary layer to develop upstream of the region of interest. The inlet boundary condition was set to a uniform velocity inlet at 11.36 m/s at 320 K, which matched the Reynolds number based on the hole diameter used by Fischer et al. [3] of Re_d = 5000. One large plenum was underneath the film cooling hole inlets. The bottom of the plenum was a uniform mass flow inlet which allowed the coolant advective capacity ratio to be set. The advective capacity ratios used for this study were ACR = 0.5, 1.0, and 1.5 which correspond to advective capacity ratios used by Fischer et al. [3]. The coolant temperature of air was set to 300 K which means the coolant’s specific heat is virtually the same as that of the freestream. Therefore, in the present study, the ACR is equal to the mass flux ratio, M, but we characterize the coolant flowrate using ACR based upon the updated knowledge we have demonstrated its superiority over M for this geometry [3].

The model was meshed using Pointwise. It had 2.2 million structured cells with a boundary layer refinement such that the first cell within the fluid had z⁺ < 1. Because only structured cells were used, the cell count is lower than an unstructured grid with the same resolution. The drawback of this method is that cells in the farfield can have high aspect ratios. A smoothing algorithm in the solver was used to reduce the maximum aspect ratio. After the model was meshed, it was solved using ansys fluent. First, the model was set up with the appropriate boundary conditions then solved using the realizable k–ɛ turbulence model with enhanced wall treatment. After the Reynolds-averaged Navier–Stokes equations solution had converged, it was used as the starting point for the large eddy simulation (LES) turbulence solution. The model was first run until it converged, then data sampling was enabled and the solution was further run for 100 iterations with a 0.005 s time-step. The total solution averaging time was 0.5 s, about 10 flow throughs. The Courant–Friedrichs–Lewy (CFL) number was set to unity using Fluent’s built-in capability. The outputs of these LES simulations are the final results. The incompatibility of the LES algorithm with symmetry boundary conditions is the reason for the symmetric domain with periodic boundary conditions shown in Fig. 1.

After the first grid was generated with the appropriate z⁺ < 1, grid convergence was tested by creating grids with higher resolution using Fluent’s built-in grid adaptation ability. The grid adaption was solution dependent based on gradients of the velocity and temperature. The primary areas of grid refinement were around the film cooling hole exits and inlets. The medium grid had 2.7 million cells and the fine grid had 3.2 million cells. Both refined grids had nearly identical temperatures on the top surface, as shown in Fig. 3; therefore, the coarse grid was sufficient.

The grid was further verified by measuring the velocity fluctuations at a point just upstream of the first cooling hole exit. The energy frequency spectrum of the fluctuations displays a −5/3 slope which indicates that turbulence is being resolved in the inertial range and the dissipative scales in the flow are being modeled.

To investigate the effects of the various contributions to the cooling, three nonphysical modifications were made to the computational model. The purpose of the modified boundary conditions was to distinguish the effects of each type of cooling, providing information that would be impossible to obtain physically. Figure 4 shows each boundary condition modification.

The first modification was made in order to isolate the effect of internal surface cooling, so the bottom of the conducting surface was made adiabatic. The flow path was not affected—the flow still impinged on the bottom surface, flowed through the cooling holes, and formed a film over the top of the surface. Heat transfer between the fluid and the solid occurred everywhere except on the internal surface exposed to the plenum.

The second modification was made to isolate cooling within the holes. The walls of each coolant hole were made adiabatic, much like the internal surface of the previously described modification. By having adiabatic holes, the coolant was prevented from gaining heat or cooling the solid as it passed through the coolant holes.

The third modification was made to isolate the effect of film cooling. In this case, the planes at the exit of each film cooling hole, facing the hole exits, were set as pressure outlets. The other sides of the planes, facing the freestream, were set as no-slip, adiabatic walls. In this case, every surface transferred heat like the baseline case and only the external flow was affected. Furthermore, the sink at the hole exit only affected the coolant flow by simply removing it from the computational model. The wall condition in the opposing direction looked like a flat plate to the freestream hydrodynamic boundary layer. This nonphysical modification allows for all of the internal cooling mechanisms in the baseline case but removes external film cooling. In this way, the computational model can be used to determine ϕ₀, the theoretical overall effectiveness without film cooling as conceived in Refs. [10,11] and used more recently in Ref. [12]. Prior to the present research, ϕ₀ has only been estimated by blocking coolant holes on experimental models, which alters the internal coolant flow path in addition to precluding the flow through the coolant holes themselves.

To show cooling effectiveness, the temperature of the external wall was nondimensionalized using Eq. (2) and is shown in the following figures. In all the figures, the freestream flow is from left to right and the coolant is coming from behind to out of the page.

Adiabatic effectiveness of the model is shown in Fig. 5 for all three advective capacity ratios. Adiabatic effectiveness was found by making all solid–fluid boundaries adiabatic; therefore, it only shows the effect of film cooling on the external surface. The top plot shows film effectiveness at ACR = 0.5. The centerline film becomes more effective with each row of cooling holes. This is due to the film from the upstream holes keeping the coolant attached longer and spreading wider. The coolant exiting the first hole is funneled inward to the centerline by the next row of staggered coolant holes. Similarly, the coolant exiting the second row is pushed to each hole’s centerline by the next row of staggered holes. The coolant has a wider spread and stays attached longer after the third, fourth, and fifth rows due to interactions with holes directly upstream. The same trends are evident at higher coolant flowrates. ACR = 1.0 shows higher effectiveness than ACR = 0.5, but effectiveness decreases slightly, as the coolant is increased even more to ACR = 1.5. The decrease in effectiveness is due to coolant liftoff. As the coolant flow is increased, the jet momentum increases and the film begins to liftoff the surface, thereby decreasing coolant effectiveness.

It is also evident in Fig. 5 that there is asymmetry in the flow issuing from the coolant hole. This is a consequence of the well-documented asymmetries that can occur in diffusing channels, the dependability of which even allows for the function of fluidic oscillators [13]. Further, very similar asymmetry in the flow issuing from 7-7-7 holes is also evident in the experimental data of Schroeder and Thole [9] and Fischer et al. [3] (as shown in Fig. 6). The fact that the CFD results capture the presence of this asymmetry is a testament to the fidelity of the LES model.

In order to validate the accuracy of the computational results, the adiabatic effectiveness contours at ACR = 1.0 were compared with the previous experimental work done by Fischer et al. [3]. The geometry used in the experiment was identical to the computational model in terms of Reynolds number Re_d = 5000 and ACR = 1.0; however, the experimental geometry employed only a single 7-7-7 shaped hole. Figure 6 shows a direct comparison of the experimental and computational η results. Comparatively, the computational results show only slightly higher effectiveness on the centerline, likely due to the array’s aforementioned tendency to funnel the coolant to the centerline. Note that the experimental data are not valid upstream of the cooling hole due to the large amount of conduction in that reason and the invalidity of the conduction correction there.

There are two takeaways from the adiabatic effectiveness contours in Fig. 5. First, the η profiles from this geometry show that jet liftoff causes decreasing adiabatic effectiveness with increased advective capacity ratio due to the greater momentum of the coolant jet at the high coolant flowrates. Second, there is high η due solely to film cooling downstream of the coolant holes. However, η is insufficient to determine how well a part is cooled. The conduction in the solid material must also be taken into consideration. The overall effectiveness of the full model with conjugate heat is presented next.

Figure 7 shows the baseline ϕ contours for all three advective capacity ratios. This is a fully conjugate model, where all surfaces transfer heat, and the solid material was fully meshed. As can be discerned through Fig. 1, the flow enters from the bottom of the plenum, cools the bottom of the conducting material, goes through the film holes, and forms a film over the top of the surface.

At ACR = 0.5, there is increasing ϕ with downstream distance from the first row to the last row. Downstream of the last row, ϕ decreases again. Upstream of the first row, ϕ gradually increases, more so directly upstream of the first hole exit. There is also a local increase in ϕ directly downstream of each hole exit. As ACR increases, so does ϕ. ACR = 1.0 shows the same trends as ACR = 0.5, but with higher ϕ values throughout. ACR = 1.5 has nearly the same values, but slightly different trends. At this coolant flowrate, increased ϕ downstream of the coolant hole exits is less pronounced, but an increase in ϕ directly upstream of exits is evident. Coolant jet liftoff, evident through the η profiles in Fig. 5, likely causes a decrease in ϕ downstream of the coolant hole exits at the high coolant flowrates.

Note that when a baseline adiabatic and overall effectiveness are established for this model, the first boundary condition modification was made. The overall effectiveness plots for the first boundary condition modification, removing plenum cooling, are shown in Fig. 8. This case shows the effect of making the internal surface adiabatic, thereby removing the effects of cooling within the plenum. All three coolant flows show lower ϕ than the baseline case in Fig. 7. Since plenum cooling is removed, these plots show only the effect of hole and film cooling. Centerline ϕ is again greater in-line with hole exits compared with between the holes. This is due to both film cooling, which acts downstream of the hole exit where the film lays over the surface, and upstream of the hole exits, where conduction from the cold hole walls just below the surface cools the top surface. That is, cooling effectiveness increased with the coolant flowrate, though not linearly. The increase in ϕ from ACR = 0.5–1.0 is greater than the increase from ACR = 1.0–1.5. Coolant jet liftoff, identified in the η plots, contributes to this. Overall, there is lower ϕ upstream of the coolant hole exits compared with the baseline case. The effect of the coolant in the first rows of holes is entirely responsible for all cooling upstream of the first hole when plenum cooling is not present. The cooling effect in this region increases with the blowing ratio. Similarly, the effect downstream of the last hole is also only an effect of film cooling. The decrease in effectiveness between ACR = 1.0 and 1.5 supports the theory that film liftoff is responsible.

The Δϕ contours for plenum cooling removed are shown in Fig. 9. A negative Δϕ shows where the modified case is now hotter than the baseline case or where overall effectiveness is degraded by removing the indicated cooling mechanism. Therefore, regions of negative Δϕ show where the cooling mechanism that is removed is effective.

In all three plots, plenum cooling is most effective upstream and downstream of the group of coolant hole exits. The holes interfere with the conduction path that plenum cooling must take to affect the top surface. Therefore, the primary areas of plenum cooling effectiveness are outside the group of coolant holes. Effectiveness in these regions increases with the coolant flowrate.

Plenum cooling is most effective at the extreme ends of solid in Fig. 9 because cooling within the holes dominates the cooling in the middle. Plenum cooling only affects the internal surface and the effect is conducted through the material to the top surface. Away from the cooling holes, there is little to interfere with the conduction, but in the center of the solid, the holes change the conduction path. Heat transfer from the walls of the holes occurs along the length of the entire hole through the entire thickness of the solid material. The length of the conducting wall and relative closeness to the top surface compared with the bottom surface where plenum cooling occurs means that the effect from plenum cooling is not significant. A geometry with a less dense clustering of film cooling holes would likely see greater effectiveness attributed to plenum cooling.

The takeaway from Fig. 9 is that plenum cooling is not the primary factor in overall effectiveness since the Δϕ contours are generally not far from zero. This result corroborates an experimental result described in Ref. [6] that indicated cooling on the internal surface had a negligible contribution to ϕ in a region with closely spaced holes through the solid material. For this reason, impingement cooling was found to have little positive benefit in such a region. Instead, cooling directly from the internal surface has its greatest effect on the leading edge region, where it is the only effective cooling form.

The next case is with hole cooling removed; that is, where the walls of the film cooling holes are set to be adiabatic. Figure 10 shows the ϕ distributions that occur without hole cooling. In other words, these plots show the effects of only the film and plenum cooling. As evident earlier in Fig. 9, plenum cooling is the most significant upstream of the first coolant hole and now, without hole cooling, it is the only cooling form affecting the region upstream of the holes. In this area, effectiveness increases monotonically with the coolant flowrate. There is a small spot of lower effectiveness directly upstream of the first coolant hole exit. This region is warmer than the baseline because it is missing the cooling from the hole wall directly beneath. Because the material is so thin here, the heat from the freestream is able to have a greater effect on it than the thicker regions away from the centerline. Downstream of the first hole exit, the cooling effects are due to a combination of plenum and film cooling. The contours clearly show the outlines of the film cooling plumes (see η plots in Fig. 5) as they tend to dominate over plenum cooling effects. This downstream region looks very similar between ACR = 1.0 and 1.5 because η is very similar in these two cases.

The corresponding Δϕ contours demonstrating the effect of removing hole cooling are shown in Fig. 11. Hole cooling also increases with the coolant flowrate, as it is not susceptible to coolant flow separation like external film cooling is.

When comparing Figs. 11 and 9, it is clear that removing hole cooling has a greater effect than removing plenum cooling. Removing hole cooling significantly decreased ϕ in regions near coolant holes. The effect of cooling within the holes is the highest directly upstream of each coolant hole exit where the material is very thin. As the coolant flows through the cooling holes, it cools the surrounding wall via convection. The top surface is then cooled via conduction from the cooling hole walls. The less material between the top surface and the cooling holes, the cooler the top surface. This occurs directly upstream of the cooling holes, where most effectiveness is seen. Past the first coolant hole exit, Δϕ slightly increases, then at the exit to the next row of coolant holes, decreases again as the solid material becomes thin. This trend continues for each row of coolant holes. Lateral conduction from one hole to another also cools the model between hole exits.

At ACR = 1.5, the effect of hole cooling is increased around the hole exits relative to the lower ACR values. The greater ACR results in a lower bulk average temperature of the coolant in the holes, thereby increasing convective heat transfer in the holes. We therefore see more negative Δϕ values, even though the adiabatic effectiveness is reduced at the high coolant flowrates.

The final case of removing film cooling is perhaps the most unique aspect of this study. In this case, the coolant travels through the internal passages to include the coolant holes, as normal, but disappears through an outlet at the exit of each cooling hole leaving the top surface non-film-cooled and affected by the freestream and conduction only. Figure 12 shows the velocity field along the centerline (where it is zero in the solid material). The velocity field within the external surface boundary layer is clearly affected since the film cooling does not exit the cooling hole. The velocity within the cooling holes is affected very little, although the elliptic nature of the momentum equation and the interaction of the coolant with the freestream at the exit of the hole does have a very small effect on the flow within the holes.

Figure 13 shows the overall effectiveness contours for the film cooling removed case and Fig. 14 shows the Δϕ plots for the effect of removing film cooling. Removing film cooling significantly impacts the overall effectiveness particularly where η is nonzero, but even in lateral regions adjacent to the first row of holes due to conduction that would otherwise be occurring to the film-cooled region.

Film cooling is very effective directly downstream of cooling holes, and more so laterally after the first two rows of cooling holes as the jets begin to merge and form a film over the entire surface. After the fourth row of holes, the film stays better attached to the surface because the film from the second row has detached and is flowing over the top, keeping the jets from the fourth row attached. The same is true for the third and fifth rows: the film stays better attached due to the flow from the first and third rows flowing over the top.

ϕ is reduced greatly downstream of each film cooling hole when film cooling is removed, with an increasing effect further downstream. This finding agrees with the adiabatic film cooling effectiveness in Fig. 5. After each hole exit, the film grows, increasing the thermal and viscous boundary layers. As the film grows, the less the layer closest to the top surface is warmed by the freestream and the more it cools the surface via convection.

In Fig. 14, it is also clear that effectiveness increases with the downstream distance as the coolant film builds. Finally, the effectiveness increases from ACR = 0.5 to 1.0, but decreases from ACR = 1.0 to 1.5 due to jet liftoff. But, one trend which was not predicted by the adiabatic effectiveness plots is also evident. Between the holes and laterally, film cooling is effective because of conduction within the material.

Additional insight into the effects of each form of cooling may be realized through the examination of cross-sections of the temperature within the solid. Figure 15 shows contours of the solid’s nondimensional temperature along the centerline for the baseline case and each of the cooling mechanisms removed individually for ACR = 1.0. Removal of plenum cooling is confirmed with the temperature contours intersecting the interior wall perpendicular to the wall, indicating a zero temperature gradient normal to that surface since that surface is adiabatic. Likewise, in the hole cooling removed case, the temperature contours intersect the hole surfaces perpendicularly.

Removal of plenum cooling has a pronounced effect of increasing the temperature in the upstream region of the plate; but closer to the first coolant hole, the temperature decreases significantly due to the large effect of cooling within the coolant hole. On a similar note, the removal of the hole cooling causes the thin region of the wall immediately upstream of the first hole opening to be much warmer due to the convection from the freestream and lack of proximity to the interior surface cooling. Removal of film cooling causes the exterior surface downstream of the holes to be much warmer than the baseline case and it is evident that this also substantially increases the heat load to the internal surface and cooling holes.

It is also helpful to compare the boundary condition effects through centerline ϕ and spanwise ϕ plots, as shown in Figs. 16 and 17. The pronounced effect of losing film cooling downstream of the holes and the very significant effect of losing hole cooling upstream of the holes are clear in Fig. 16. The presence of the film cooling issuing from the coolant hole is predicted to have an effect just upstream of the leading edge of each hole as evident with the sharp rises in the “hole cooling removed” line at each of those locations in Fig. 16. This could be due to streamwise conduction within the fluid and/or an effect of a horseshoe vortex or similar structure that can sometimes form at the leading edge of a coolant hole. Notable in Fig. 17 is the substantial depression in ϕ at y/d = 0 with film cooling removed immediately upstream of each hole (x/d = 12, 24, and 35). ϕ remains higher off the centerline at those x/d positions since the off-centerline positions have more direct conduction paths to regions receiving other forms of cooling.

The technique of using modified boundary conditions to identify the pertinent regions and strengths of each form of cooling has been demonstrated in this particular model. This technique could be extended to any cooling design, including geometries with different Biot numbers resulting from different thermal conductivities. In addition to the earlier results with k = 3.5 W/m/K, we now examine a lower thermal conductivity, k = 1 W/m/K, to match DuPont Corian, a material commonly used in film cooling experiments with an intention to approximately match the Biot number of certain types of turbine components. The purpose of using another model is to show the effect of changing thermal conductivity on the effectiveness from each of the three forms of cooling. The simulation of film cooling flow with a DuPont Corian model also offers a practical opportunity to compare the conjugate CFD results to some experimental data. A Corian model with geometrically similar full-coverage film cooling was tested in the manner of the adiabatic effectiveness experiments of Ref. [3]. The experimentally measured centerline ϕ is compared with the computational determined values in Fig. 18 for ACR = 0.5, k = 1 W/m/K. In the figure, ϕ is shown inside the exit of the cooling hole with the experimental data, but is not shown for the computational model since the computational data were extracted right at the surface. The computational results matched the experimental results extremely closely upstream of the hole, indicating that the CFD simulated the convection within the hole and conduction through the wall remarkably well. Downstream of the hole, where both film cooling and conduction have a large effect on ϕ, there were differences in the magnitude of ϕ, but the computational results matched the experimental results within 0.05 along the centerline, which is within the experimental uncertainty. Also notable is the way in which the level of ϕ increases from one row of holes to the next, as the internal cooling becomes more significant just upstream of the second row of holes.

Returning to the computational model alone, the baseline overall effectiveness for the lower thermal conductivity (k = 1 W/m/K) model is shown in Fig. 19. There is less lateral heat conduction between holes than the higher thermal conductivity case (Fig. 7). The other trends, however, are similar. The Δϕ between the two baseline cases, as shown in Fig. 20, shows where changing k decreased the overall effectiveness on the top surface. The locations with negative Δϕ are areas where we know that film and hole cooling dominate, both of which rely on conduction. This caused an overall detriment to ϕ because the lower k decreased hole and plenum cooling without a matching increase in film cooling effectiveness. By breaking out each form of cooling, the effect of changing the thermal conductivity becomes even more clear.

Figure 21 shows the effect of plenum cooling for the lower thermal conductivity case. Compared with the higher k case in Fig. 9, these plots show the same trend but with slightly lower Δϕ magnitudes. Plenum cooling is still only effective on the leading and trailing edges.

Δϕ contours with hole cooling removed are shown in Fig. 22. Again, these plots look similar to the high thermal conductivity case in Fig. 11, except directly in front of the hole exits. In this region, effectiveness is much higher. Therefore with decreasing thermal conductivity, hole cooling becomes more effective directly upstream of the hole exits.

Figure 23 shows the effect of removing film cooling on the lower thermal conductivity model. These Δϕ contours show that the external film cooling has a greater contribution to the overall greater impediment to conduction cooling the external surface. The film cooling effect is also more localized directly downstream of each cooling hole exit.

The area average Δϕ values for both thermal conductivities are shown in Fig. 24. This figure makes it easy to compare the effect of removing each type of cooling. In this figure, the more negative the Δϕ value, the greater the effect that the form of cooling had on overall effectiveness. Removing plenum cooling had the smallest effect on overall effectiveness whereas removing film cooling had the most effect at all three advective capacity ratios. However, at ACR = 1.5, film cooling clearly had less effect than at ACR = 1.0 due to coolant liftoff. Hole cooling had the next most significant contribution to the overall cooling, but unlike film cooling, its contribution becomes monotonically greater as ACR increases.

The relative contributions to ϕ from each form of cooling are clearly evident in Fig. 24, and an examination of them reveals a coupling between the cooling mechanisms. Consider ACR = 1.0 at k = 3.5 W/m/K for example, the area average ϕ is 0.621, but the sum of the Δϕ for each cooling form removed is only −0.287 even though the resulting ϕ with all cooling removed must be zero. This reveals that if one cooling mechanism is disabled, the heat load increases on the others. For instance, if film cooling is disabled, the increased external heat load will cause higher temperatures on internal surfaces, thus increasing the internal heat loads. Likewise, if hole cooling is disabled, the coolant will gain less heat as it passes through the cooling hole, thereby exiting at a lower temperature and improving the external film cooling, partially compensating for the loss of hole cooling. This partial compensation for the removal of a particular cooling mechanism by the increased cooling from the remaining cooling mechanism explains when the sums of the individual Δϕ values do not add to the overall ϕ. Indeed, this phenomenon would also occur in a real engine should a certain cooling feature fail as could occur with a clogged film cooling hole. That is, the loss of film cooling from a clogged film cooling hole will be compensated in part by increased conductive cooling.

The coupling becomes further apparent when we examine the effect of decreasing the thermal conductivity in Fig. 24. Film cooling became more effective (more negative Δϕ) with lower k, while plenum and hole cooling became less effective. This is because plenum and hole cooling depend upon conduction, which is hampered with the lower k material. But since conduction was reduced, the coolant ejected from the coolant hole is at a lower temperature. This caused film cooling to have a greater effect with the lower thermal conductivity material.

To utilize this information, a designer should consider the trends shown in Fig. 24. Film cooling is the most important form of cooling (for this geometry), hole cooling is next, and plenum cooling least important. Rather than optimizing designs for increased plenum cooling (the least relative contribution to ϕ) with designs like impingement, which in some cases can lead to decreased film cooling effectiveness (cf. Ref. [6]), designers should optimize all three forms of cooling without causing a detriment to the more influential forms of cooling.

A new technique was demonstrated to segregate the effects of plenum cooling, hole cooling, and film cooling on overall effectiveness. This technique can be used to identify the most significant contributions to overall effectiveness to help designers focus on the forms that are likely to provide the greatest incremental benefit when they are improved. The technique uses conjugate heat transfer CFD simulations with nonphysical boundary conditions to in effect, turn off individual cooling mechanisms. The resulting ϕ values are compared with the baseline case using Δϕ. The technique was demonstrated on a flat plate with five staggered shaped cooling holes arranged in an array with periodic boundary conditions.

In the present use of the technique, three nonphysical boundary conditions were applied to investigate the relative contributions of plenum cooling, convective cooling within the holes, and external film cooling. Plenum cooling and hole cooling were removed by setting the internal surface and cooling hole surfaces as adiabatic, respectively. An additional effect of making a solid boundary adiabatic is that the boundary no longer warms the coolant. To determine the effect of film cooling, film cooling was removed by creating a one-sided outlet at the exit to the film cooling holes to in effect, dispose of the internal coolant before it was ejected onto the surface. This type of operation has been attempted experimentally in the past, but it always required blocking coolant holes which interfere with the internal coolant flow and also negate the cooling benefit of the flow within the blocked coolant holes.

Plenum cooling had the smallest effect on Δϕ in all cases. Film cooling was the most effective form of cooling, but had a decreasing effect as the coolant jets started to liftoff. Adiabatic cooling effectiveness simulations confirmed jet liftoff contributed to decreasing cooling effectiveness at advective capacity ratios higher than ACR = 1.0. The film was only effective downstream of the exit of each cooling hole. In all cases, the film built up with each hole exit causing increased cooling with increasing downstream distance. Close to the hole exits, the material is thinnest and is, therefore, cooled most effectively by hole cooling which is dependent upon conduction. Hole cooling was most effective just upstream the hole exit and increased monotonically with increasing advective capacity ratio. The same model was tested at a different thermal conductivity to show the technique’s utility and the coupling between the different cooling mechanisms.

Using this technique, it is possible to ascertain the relative contributions of each form of cooling, leading to better optimized cooling strategies.

The authors thank the Air Force Research Laboratory for their support of this work and most of all, express their thanks to Luke McNamara and Jacob Fischer for performing the experimental work used to validate the computational techniques employed in this work. Many thanks also to Mitchell Scott and Marta Kernan for their assistance with the laboratory work.

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.

d=

hole diameter, m

h=

heat transfer coefficient, W/m/K

x=

surface distance downstream, m

y=

spanwise distance from hole centerline, m

z=

distance normal to surface into fluid, m

M=

mass flux ratio

T=

temperature, K

c_p=

specific heat, J/kg/K

Bi=

Biot number

ACR=

advective capacity ratio, Eq. (4)

η=

adiabatic effectiveness, (T_∞−T_aw)/(T_∞−T_ce)

ρ=

density, kg/m³

ϕ=

overall effectiveness, (T_∞−T_s)/(T_∞−T_c)

χ=

coolant warming factor, (T_∞−T_c,e)/(T_∞−T_c)

Δ=

change operator

c=

coolant

e=

coolant hole exit

i=

internal

s=

surface

∞=

freestream conditions

aw=

adiabatic wall