## Abstract

Ceramic matrix composites (CMCs) show promise as higher temperature capable alternatives to traditional metallic components in gas turbine engine hot gas paths. However, CMC components will still require both internal and external cooling, such as film cooling. The overall cooling effectiveness is determined not only by the design of coolant flow, but also by the conduction through the materiel itself. CMCs have anisotropic thermal conductivity, giving rise to heat flow that differs somewhat relative to what we have come to expect from experience with traditional metallic components. Conjugate heat transfer computational fluid dynamics (CFD) simulations were performed in order to isolate the effect anisotropic thermal conductivity has on a cooling architecture consisting of both internal and external cooling. Results show the specific locations and unique effects of anisotropic thermal conduction on overall effectiveness. Thermal conductivity anisotropy is shown to have a significant effect on the resulting overall effectiveness. As CMCs begin to make their way into gas turbine engines, care must be taken to ensure that anisotropy is characterized properly and considered in the thermal analysis.

## Introduction

Higher gas turbine performance requirements and reduced core sizes are driving increases in the turbine inlet temperature, surpassing the melting point of advanced materials. Ceramic matrix composites (CMCs) are a leading technology to enable higher turbine inlet temperatures, but cooling CMC parts via film cooling is a multifaceted problem with many challenges. One unique aspect of CMCs is they have a bulk anisotropic thermal conductivity; that is, the thermal conductivity differs between the through-thickness (ky) and in-plane (kx) directions. The magnitude of the anisotropy varies widely, depending on the material, matrix, and processing method (compared with Refs. 15) as presented in Table 1.

Table 1

Comparison of in-plane (kx) to through-plane (ky) thermal conductivity

Materialkx/ky
Carbon/epoxy [1]10.4
Graphite epoxy weave fabric composite [2]11.5
Hexcel F593 carbon epoxy prepreg laminate [3]4.2
Graphite/epoxy matrix lamina [4]9.8
SiC/SiC 2D fabric CVI-MI [5]2.8
SiC/SiC 2D fabric full PIP [5]10.0
Materialkx/ky
Carbon/epoxy [1]10.4
Graphite epoxy weave fabric composite [2]11.5
Hexcel F593 carbon epoxy prepreg laminate [3]4.2
Graphite/epoxy matrix lamina [4]9.8
SiC/SiC 2D fabric CVI-MI [5]2.8
SiC/SiC 2D fabric full PIP [5]10.0

Traditional film cooling analysis involves the assumption that the material thermal conductivity is isotropic. This is a very good assumption when the material used is metallic alloy, but poor for a CMC. The purpose of this article is to describe the effect of anisotropic thermal conductivity overall effectiveness in gas turbine film cooling.

One method of evaluating a film cooling design is to measure the adiabatic effectiveness η or the temperature of the gas at the surface downstream of a film cooling hole. It is defined as follows:
$η=T∞−TawT∞−Tce$
(1)
where T is the freestream temperature, Taw is the adiabatic wall temperature, and Tce is the temperature of the coolant at the exit of the film hole. The adiabatic effectiveness only considers the role the film has on cooling the outside surface and is therefore commonly used to evaluate different cooling hole configurations. Conversely, overall effectiveness ϕ is measured on a conducting part and characterizes the overall reduction of the surface temperature from all forms of cooling, including internal and film cooling. It is defined as follows:
$ϕ=T∞−TsT∞−Tc$
(2)
where Ts is the temperature of the outside surface and Tc is the temperature of the coolant in the plenum. Albert et al. [6] showed through a simplified one-dimensional analysis how ϕ is dependent on several factors, to include η:
$ϕ=χη(Bi+hhi)+1hhi+Bi+1$
(3)
In Eq. (3), χ is the coolant warming factor, or how much the coolant warms as it travels through the internal cooling design, Bi is the Biot number, and h/hi is the ratio of the external to internal heat transfer coefficients. Most notable, in addition to η, the Biot number must be matched to scale ϕ laboratory tests to engine conditions. Stewart and Dyson [7] demonstrate that this implies that the ratio of the freestream air thermal conductivity to the material thermal conductivity must be matched. These matched Biot number experiments have been performed by a number of researchers including Terrell et al. [8] and Chavez et al. [9].
For a lab-scale experiment to match engine conditions, the coolant flowrate must also be matched in an appropriate nondimensional sense. Coolant gases are denser than the freestream flow, so the density ratio must be considered. As such, it has become rather common to match the mass flux ratio or the momentum flux ratio rather than the velocity ratio. However, the specific heat of the coolant can be significantly lower than the specific heat of the freestream. Recently, Fischer et al. [10] show that the energy equation dictates that the advective capacity ratio (ACR) must be matched:
$ACR=cp,cρcUccp,∞ρ∞U∞$
(4)
They went on to show that for a laidback fan shaped coolant hole on a flat plate, the ACR scaled adiabatic effectiveness for a variety of coolant gases with widely varying properties. Note that for experiments where the both coolant and freestream gases are air at low temperatures, the cp ratio is essentially unity and the advective capacity ratio is thus equal to the blowing ratio.
Heat flux in isotropic solids obeys Fourier’s law, given in Eq. (5), where thermal conductivity is independent of direction and the heat flux is normal to the surface.
$q=−k∇T$
(5)
In anisotropic solids, the component of heat flux depends on the linear combination of the temperature gradients in each direction. The thermal conductivity of 3D solid can be expressed as a coefficient of a second-order tensor $k¯¯$
$k¯¯=[k11k12k13k21k22k23k31k32k33]$
(6)
Written in index notation, the heat flux is expressed as follows:
$qi=−kij∂T∂xj$
(7)
Therefore, for an anisotropic solid, the heat flux is not necessarily aligned with the temperature gradient [11].
There are limitations to what each coefficient can be. Onsager [12,13] shows that when fluxes (q) and gradients (∂T/∂x) are related to each other linearly, the coefficients obey the reciprocity relation. Casimir [14] further applied this logic to heat conduction in anisotropic solids. Therefore, the thermal conductivity coefficients obey reciprocity, that is, the thermal conductivity tensor is symmetric:
$kij=kjii,j=1,2,3$
(8)
Furthermore, according to irreversible thermodynamics, the diagonal components (k11, k22, k33) are positive; however, off-diagonal components may be negative. Most research on CMCs for film cooling applications has focused on the material strength and durability (compared with Refs. 15 and 16).

Little research has been done on the effects of anisotropic conduction on overall cooling effectiveness. Tu et al. [17] performed a numerical study looking at the effect of anisotropic thermal conductivity on the heat distribution on a CMC turbine vane with cooling holes. The authors found that by changing the anisotropic thermal conductivity, the location and magnitude of the max temperature varied. In one case, they found that anisotropic thermal conductivity had a nearly 100 K effect of relative to the isotropic case. They concluded that it is important to take anisotropic thermal conductivity into account when conducting thermal analysis on a composite hot component.

Another study by the same group [18] looked at the effect of anisotropic thermal conductivity on a film-cooled flat plate. In the study, they varied the thermal conductivity in one direction and the angle of the “fiber” to determine the effect on cooling effectiveness. By using this method, they were able to keep the magnitude of the k tensor constant with each change in the hole angle. Most significantly, they found that by increasing the component of heat transfer in the direction of the holes, hole cooling was increased. The area around the hole had the greatest effect from changing the angle of the principle thermal conductivity component. Further downstream, orienting the fibers perpendicular to the flow resulted in the greatest changes in cooling effectiveness. Changing the component in the spanwise direction did not have a significant effect.

Further research is needed to better understand the role of anisotropic thermal conduction on overall effectiveness. The previous work cited earlier only looks at increased thermal conductivity in one direction as one would have with an unidirectional fiber; however, many turbine-relevant CMC parts are made by stacking layers of fabric, made by weaving fibers in two directions. Such CMCs have increased thermal conductivity in 2D along the plane of fabric weave. Another area of interest is the effect of shaped holes on CMCs. Shaped holes have a compound angle that intersect the layers at a different angle depending on the depth. From the previous research on cylindrical holes, conduction around the hole is affected by the orientation of the k tensor, so changing the k tensor will affect a shaped hole differently at different depths. Finally, the computational technique of artificially removing cooling on various surfaces shown in Ref. [19] is particularly useful for examining anisotropic thermal conductivity because changing the k tensor has a different effect on cooling from each surface. Through using these techniques, we can better quantify the expected anisotropic thermal conductivity effects on turbine-relevant hardware.

## Model Description

The geometry used in these simulations is a flat plate with shaped film cooling holes. The model was previously used in the study by Bryant and Rutledge [19]. The film cooling holes are the 7-7-7 shaped holes first characterized by Schroeder and Thole [20]. They are at a 30 deg angle to the freestream with a 7 deg layback and 7 deg fan to each side. The holes are in a staggered array with a spanwise pitch of 6.32 hole diameters and a row spacing of 5.91 hole diameters, where the metering diameter of the hole is 0.643 cm. A diagram of the model is shown in Fig. 1, and a schematic of the hole geometry is shown in Fig. 2. Although not shown in this study, an adiabatic flat plate was modeled that extended 50 hole diameters upstream of the first cooling hole to ensure a fully formed hydrodynamic boundary layer in the region of interest.

Fig. 1
Fig. 1
Close modal
Fig. 2
Fig. 2
Close modal

The mesh, shown in Fig. 3, has 2.2 million structured cells with z+ ≤ 1 along the outside surface and in the coolant holes. Because the entire mesh is structured, it has a lower cell count than many unstructured meshes. One drawback to this is that some cells in the far field had high aspect ratios. A smoothing algorithm was used to reduce the maximum aspect ratio. Grid convergence was tested by creating higher resolution grids using fluent’s built in grid adaptation ability. The adaption algorithm was solution dependent based on the velocity and temperature gradients. The areas refined were primarily around the film cooling hole exits and inlets in both the fluid and solid zones. The medium grid had 2.7 million cells and the fine grid had 3.2 million cells. Both grids yielded nearly identical to the original coarse grid, as shown in Fig. 4, and so the coarse grid was sufficient and used for all simulations.

Fig. 3
Fig. 3
Close modal
Fig. 4
Fig. 4
Close modal

The flow was solved using ansys fluent’s realizable k-ɛ solver with enhanced wall treatment. The model has full conjugate heat transfer with both conduction and convection, where both the fluid and solid zones are solved simultaneously. The freestream air was set to 11.36 m/s at 300 K at the velocity inlet to match the Reynolds number based on hole diameter in the experimental work of Fischer et al. [10], Red = 5000. Convergence was monitored by examining the surface temperature on the top wall. Once the temperature at several points on the surface, both in and out of the film cooling flow, was invariant for 100 iterations, convergence was achieved. After convergence was reached, the flow field was saved and used to initialize each run. Because only the k tensor was changed with each case, the flow field was nearly unchanged so each case was able to converge very quickly.

The mesh was validated by comparing the computational results to experimentally obtained overall effectiveness on an array of 7-7-7 shaped holes in Corian. The experimental array had only three rows of holes due to limited coolant airflow; however, the computational model has five rows of coolant holes. To compare to experiment, the isotropic conductivity in the mesh was set to a constant 1.007 W/m K (the conductivity of Corian) and run to convergence. A comparison of the overall effectiveness at ACR = 0.5 is shown in Fig. 5. There is very good agreement around the first and second rows of holes. Because the third row of holes is the final row in the experiment, the overall effectiveness is lower than the computational model. Overall, the excellent agreement between the experiment and computational results indicates the mesh and computational solver used are sufficient. Large eddy simulations would likely yield more accurate results; however, there is already good agreement between Reynolds-averaged Navier–Stokes (RANS) and experiment. Furthermore, large eddy simulations are very computationally expensive, and this study does not examine any variations in airflow; therefore, it was determined that the RANS realizable k-ɛ solver is sufficient and was used for the all the cases discussed in the article.

Fig. 5
Fig. 5
Close modal

The thermal conductivity of the model was set using fluent’s materiel property settings. Because the goal of this research is to examine anisotropy, like that found in CMCs, the thermal conductivity was modeled as if layers of materials were stacked together. Within the layers, the in-plane thermal conductivity is uniform in both directions. The through-thickness conductivity, or the component of thermal conductivity normal to the conductive layers, is much lower. To discuss the thermal conductivities in higher fidelity, coordinate systems must be described. For the flat-plate model, the global coordinate system is (X, Y, Z). Let a local coordinate system (ζ, η, ν) be oriented with the weave layup. When the coordinate systems are aligned, X = ζ, Y = η, Z = ν (Fig. 6).

Fig. 6
Fig. 6
Close modal
The through-thickness and in-plane thermal conductivities are constant; therefore, the thermal conductivity tensor in the local coordinate system will remain constant. The through-thickness thermal conductivity was chosen to match the experiment using Corian, $kη$ = 1 W/m K, and the in-plane k was chosen to be four times $kη$, as in Ref. [3], $kζ$ = $kν$ = 4 W/m k. The thermal conductivity tensor in the local coordinate system will remain constant, no matter what orientation the layers are in compared to the global coordinate system:
$kζην=[400010004]W/mK$
(9)
To set with thermal conductivity tensor in fluent, kζην had to be rotated back to the global coordinate system. Tensor rotation is accomplished using the standard technique [21] where the tensor in the original unrotated or “unprimed” coordinate system can be specified in a rotated “primed” coordinate system:
$ki′j′=cmi′cn′jkmn$
(10)
In Eq. (10), $cij′$ is the direction cosine matrix. The thermal conductivity tensor in global coordinates kXYZ was then employed in fluent’s anisotropic material setting.

Table 2 lists the first five cases tested. The ACR is 1.0 for every case tested. In each case, the thermal conductivity tensor is rotated about at least one axis. In case of multiple rotations, each rotation is with respect to the global coordinate system, not the newly rotated local axes. In addition, the rotations are order specific: Z, X, then Y.

Table 2

Cases tested with each rotation angle listed in degrees

CaseRotation in degrees
ZXY
1000
23000
39000
40900
509028
CaseRotation in degrees
ZXY
1000
23000
39000
40900
509028

## Results

The first plate modeled is the standard 0/90 deg layup commonly seen in flat plates. In this orientation, the layers are parallel to the outside surface, along the X, Z plane. The thermal conductivity tensor is therefore simple kXYZ = kζην in Eq. (9). This case is the baseline to which the remaining cases will be compared. The resulting contours of overall effectiveness are shown in Fig. 7.

Fig. 7
Fig. 7
Close modal

It is impossible to directly compare overall effectiveness for otherwise equivalent isotropic and anisotropic cases because the change in thermal conductivity eradicates any semblance of equivalency. The anisotropic cases have a different thermal conductivity in at least one direction from any isotropic case. Increasing the conductivity in one direction, while reducing it in another, does not even out the “bulk” thermal conductivity. This is because heat flux occurs due to a directional temperature gradient; therefore, the directionality of the thermal conductivity tensor cannot be “averaged-out.”

To better understand where heat transfer is occurring in each case, area-specific cooling was isolated using the technique shown in Ref. [19]. This is a computational technique using nonphysical boundary conditions to remove heat transfer on individual surfaces or from film cooling itself. The result is the overall effectiveness without that specific form of cooling. In this case, three types of cooling are removed: plenum cooling, hole cooling, and film cooling. To remove plenum cooling, the internal surface of the flat plate facing the plenum is computationally set to adiabatic. To remove hole cooling, only the interior surfaces of the film cooling holes are made adiabatic. Finally, removing film cooling involves placing pressure outlets at the exits of the film cooling holes, thereby removing the coolant so that it cannot flow over the external surface of the model, but internal flow of the coolant remains unimpeded. Through this technique, one may determine the relative importance of each form of cooling to the overall effectiveness. The ϕ contours for each form of cooling removed and the baseline are shown in Fig. 8.

Fig. 8
Fig. 8
Close modal

The top contour is identical to the baseline shown in Fig. 7. The next contour shows the effect on ϕ from removing plenum cooling. Compared to the top figure, removing plenum cooling shows decreased effectiveness on the leading edge and trailing edge in particular. The general shape of the contours is not affected since the contour shapes are driven by the hole cooling and film cooling. The third contour shows effectiveness with hole cooling removed. The main difference between this contour and the baseline shows much lower effectiveness directly upstream of the first hole where the thin material with cool coolant flowing through the hole right under the surface would normally provide a substantial cooling effect. The final contour shows film cooling removed. This contour shows far lower effectiveness downstream of each cooling hole.

To better evaluate each case, delta phi (Δϕ) plots were also generated. The delta phi method, first applied in Ref. [22], quantifies the nondimensional change in surface temperature due to some modification in the cooling scheme.
$Δϕ=ϕmodified−ϕbaseline$
(11)
or
$Δϕ=Ts,baseline−Ts,modifiedT∞−Tc$
(12)

By using Δϕ, one may discern the magnitude and location of changes in overall effectiveness due to changes in the cooling scheme. Of present interest is the effect of removing the various contributions to cooling and later, we shall use the technique to examine the effects of rotating the thermal conductivity tensor.

The Δϕ contours for the baseline thermal conductivity tensor are shown in Fig. 9. Each contour is the difference between each form of cooling removed and no cooling removed. The more negative Δϕ is, the more important that form of cooling is in that particular location. The top contour shows plenum cooling is most effective on the leading and trailing edges. The middle contour shows hole cooling is most effective in front of each film cooling exit, with decreasing effectiveness with downstream distance. The last contour shows that film cooling is very effective downstream of each film cooling hole, with increasing effectiveness with downstream distance.

Fig. 9
Fig. 9
Close modal
In case 2, the thermal conductivity tensor was rotated as if the layers were rotated 30 deg around the Z axis, such that the layers are aligned with the metering section of the coolant holes. Figure 10 shows how the layers would be aligned within the film-cooled plate. The thermal conductivity tensor does not change in the local coordinate system because it is aligned with the layers; however, in the global coordinate system, it now has nonzero off-diagonal components. According to Özışık [11], such tensors are also common in monoclinic crystal systems, for which two sides of the crystal are at right angles and the third is inclined, much as we have done here by inclining the CMC planes. The thermal conductivity in the global coordinate system is given by
$kXYZ=[3.251.29901.2991.750004]W/mK$
(13)
Fig. 10
Fig. 10
Close modal

Figure 10 also shows the resulting ϕ contours for case 2 as well as a Δϕ plot comparing ϕ with case 1. The alignment of the CMC layers with the hole somewhat impedes hole cooling in the region upstream of the cooling holes; thus, the effectiveness is lower in the upstream region indicated by the negative Δϕ upstream of the first hole. As the Δϕ contours shown in Fig. 11 demonstrate, removing the plenum cooling indeed generally causes a greater decrease in ϕ compared to case 1 (see Fig. 9). However, removing hole cooling resulted in a smaller decrease in ϕ, indicating that hole cooling is a smaller contributor to the overall cooling when the CMC planes are aligned with the film cooling holes. Nevertheless, CMC planes do intersect both the holes and the surface laterally to the cooling holes, providing a small region of more substantial Δϕ just adjacent to the cooling holes. Referring back to the Δϕ contour in Fig. 10, we can also see that rotating the CMC layup 30 deg results in improved cooling in between cooling hole exits within the array. This improvement in ϕ is attributable to the fact that the CMC planes provide an improved conductive path to the cool plenum as evidenced by the more negative Δϕ with plenum cooling removed for the 30 deg rotation compared to case 1. Since the 30 deg rotation allows for improved internal cooling relative to the baseline case, one would expect the coolant exiting the film cooling holes to be slightly warmer with the 30 deg rotation. Indeed, comparison of the Δϕ contours shows that Δϕ with film cooling removed is less negative for the 30 deg rotation than it is for case 1. To be clear, this means that a loss of film cooling with the 30 deg rotation is not as deleterious as it would be in the baseline case, i.e., the film cooling is a less important contributor to the overall effectiveness due to the improved internal cooling and resulting warmer film coolant ejection.

Fig. 11
Fig. 11
Close modal
Case 3 completes the rotation around the Z axis to the full 90 deg. The layers are now stacked vertically so the layers are aligned spanwise when viewed from the top. In the global coordinate system, the layers are oriented in the Y, Z plane so the thermal conductivity in the Y direction is now four times that in the X and Z directions. The layup orientation, ϕ and Δϕ, contours are shown in Fig. 12.
$kXYZ=[100040004]W/mK$
(14)
Case 3 is quite remarkable when compared to case 1 because it is entirely cooler than the baseline case. The upstream leading edge is nearly identical, until area in front of the first cooling hole. This area is now much cooler because the layers allow for increased hole cooling by creating improved conduction paths straight down from the hot surface to the cooler air travelling through the film cooling holes. This is repeated down the length of the plate, through the array of film cooling holes. The resulting average ϕ is 0.626 compared to the baseline average ϕ = 0.581. In essence, the overall effectiveness has increased nearly 0.05 on average, without any changes to the flowrate, type of holes, etc. The only change is the way the layers of conductive material are stacked.
Fig. 12
Fig. 12
Close modal

The Δϕ contours for the 90 deg Z axis rotation are shown in Fig. 13. Plenum cooling is much more effective in this case, particularly between holes and upstream of the first hole. Hole cooling is also much more effective in this case, compared to case 1. Upstream of the first and second rows of holes Δϕ is 0.1 lower (more negative), and in fact, the average Δϕ is 0.05 more negative than the baseline average with hole cooling removed. Hole cooling averages out to be nearly as effective as film cooling. Compared to the baseline, film cooling is slightly less effective again due to the higher film coolant temperature resulting from the more substantial conductive cooling.

Fig. 13
Fig. 13
Close modal
In case 4, the layers were rotated 90 deg around the X axis instead of the Z axis. Therefore, the layers were now aligned horizontally, stretching down the length of the plate when viewed from the top. In the global coordinate system, the layers are aligned with the Y, X plane. The k tensor has zero off-diagonal coefficients again because it was rotated the full 90 deg, shown in Eq. (15) and Fig. 14. Case 4 is also cooler than case 1, but differs in most other ways.
$kXYZ=[400040001]W/mK$
(15)
First, there is elevated ϕ upstream and downstream of each coolant hole, extending much farther downstream than in case 3. Because the layers are aligned with the stream-wise direction (and the direction of the holes in the material), heat flows directly to the plenum and coolant holes within the material. The result is increased effectiveness on the leading edge, trailing edge, and between holes in the array.
Fig. 14
Fig. 14
Close modal

Figure 15 shows the Δϕ contours for case 4. Plenum cooling is significantly increased on the leading edge. There is also more cooling from the plenum between holes in the array although less than case 3 with 90 deg rotation about the Z axis. Hole cooling is also much more significant in this case than the baseline. However, unlike case 3, the increased effect of hole cooling is localized to directly upstream of the hole exits. Because the layers are oriented such that the greatest conductivity is in the X-Y plane most conductive in the stream-wise direction, it follows that the increased effectiveness does not spread laterally in the Y direction as it did in all previous cases with fibers oriented in the spanwise direction. The film cooling effectiveness is slightly decreased again.

Fig. 15
Fig. 15
Close modal
The final example, case 5, consists of two rotations. The first rotation in this case was 90 deg about the X axis (same as case 4), but then the layers were rotated 28 deg around the Y axis, so the layers aligned with the angle of the offset film hole array. The resulting k tensor, Eq. (16), has a different structure than the previous rotations. kXZ and kZX are the only nonzero off-diagonal coefficients. The resulting ϕ distribution is shown in Fig. 16.
$kXYZ=[3.31830−1.2571040−1.257101.6817]W/mK$
(16)
Overall, case 5 is cooler than case 1 and is similar to case 4. The far upstream leading edge has similar contour levels to case 4 with the 90 deg rotation about the X axis. This makes sense because this case has the same 90 deg rotation around the X axis. Unlike case 4, the additional rotation about the Y axis means this case has more lateral conduction across the plate, so the increase in effectiveness is not limited to the spanwise direction from the holes. The most notable feature of case 5 is the ϕ distribution is no longer symmetric. The airflow, and therefore temperature gradient, is still nearly identical to the previous cases, but the k tensor has shifted the heat flow to the side.
Fig. 16
Fig. 16
Close modal

The Δϕ contours for this case are shown in Fig. 17. The magnitudes of these contours are similar and show the same trends as Fig. 15; however, the asymmetry is very obvious in Fig. 17. Perhaps most notably is the effect that hole cooling has in that it not only provides a great deal of cooling upstream of the holes, but the upstream hole cooling effect also has a significant component in the +z direction. Asymmetry is also pronounced in the upstream region for the plenum cooling removed Δϕ contour. These contours reveal that in regions without a clear diagonal conductive path to cooling holes, the cooling is more dependent on plenum cooling. Since the k tensor is the cause of the asymmetry, it makes sense that types of cooling most affected by conduction would most contribute to the asymmetry. The film cooling removed contour also shows some asymmetry from heat conducting within the surface, but the asymmetry due to removing film cooling is less pronounced than for the hole cooling and plenum cooling.

Fig. 17
Fig. 17
Close modal

The area-averaged Δϕ value for each form of cooling, in each case, is depicted in Fig. 18. In each case, plenum cooling was more effective than in case 1. In particular, as the rotation around the Z axis increased from 0 deg to 90 deg, plenum cooling increased. Average hole cooling effectiveness slightly decreased when the angle of the layers matched the angle of the holes, thereby impeding the conduction path from the holes and top surface. However, when the CMC planes were rotated to 90 deg about the Z axis, hole cooling effectiveness became very high, nearly as high as the film cooling effectiveness. The two cases with rotation about the X axis, cases 4 and 5, have nearly identical Δϕ values for each form of cooling because rotating the k tensor around the Y axis did not change the conduction paths except to make them at an angle. In all cases, the film cooling effectiveness decreased due to increased heat transfer on the plenum and hole walls.

Fig. 18
Fig. 18
Close modal

Next, a rotation sweep of each axis was conducted. The solved airflow was used to initialize each case, then the tensor was rotated in increments of 10 deg around the X and Z axes, separately. The computational regimen required 35 unique CFD simulations, so it was imperative to consider computational efficiency. Since the velocity field was only minimally impacted by changing the solid’s thermal conductivity tensor, a frozen velocity solution was used to solve the energy equation for each case. The converged energy solution and the frozen velocity field were then used to initialize the fully coupled simulation, which was then run to convergence. The resulting ϕ distributions were area averaged and plotted in Fig. 19.

Fig. 19
Fig. 19
Close modal

The maximum effectiveness for rotations about the X axis occurred at ±90 deg. At these equivalent angles, the layers were aligned as shown in Fig. 14. For rotations around the Z axis, the maximum effectiveness occurred at −70 deg. At this angle, the layers would be nearly perpendicular to the upstream side of the film holes. This angle allows the heat from the surface to flow at a sharp angle to the film holes, so more effective conduction occurs from the surface to the film holes, compared to other angles. Thus, hole cooling becomes even more effective. Plenum cooling is also improved by the high rotation angle about the Z axis. The question left by this study was, what would happen if the two most effective angles were combined? Would these rotations combined produce the most effective cooling layup for this film cooling geometry?

While Fig. 19 shows the optimum X and Z rotation angles for independent rotations, it does not address compound rotations about both axes simultaneously. A sweep of two-axis rotations was also accomplished in 10 deg increments. This regimen required 307 total simulations (eliminating duplicate equivalent angles). Note that rotation about the third axis is unnecessary as it would only create redundant layup angles. The resulting ϕ distributions were again area averaged and plotted in Fig. 20 as a topographical map. A significant finding is that the standard 0/90 layup (no rotation) has nearly the worst overall effectiveness. Nearly any rotation will improve the baseline effectiveness. The rotations with the highest effectiveness occur with any large negative rotation around the Z axis. Recall that case 3 also had very high effectiveness and was rotated 90 deg around the Z axis.

Fig. 20
Fig. 20
Close modal

The lowest and highest phi values in this data set show several interesting points. The lowest area-averaged ϕ occurs at 10 deg around Z axis and 0 deg around X axis. The highest area-averaged ϕ is at −70 deg around Z axis and 0 deg around X axis. The Δϕ plot for the best case is shown in Fig. 21.

Fig. 21
Fig. 21
Close modal

The worst case is only slightly worse than baseline. The ϕ contour shows some small changes, such as the angles in the leading edge due to X axis rotation. But according to the Δϕ plot, all the changes except the decrease in effectiveness on the leading edge are small, less than 0.05. The only change that shows up on the Δϕ plot is the decrease in effectiveness at the spanwise edges of the leading edge.

The best case shows several improvements. The Δϕ contour shows that most of the improvements are on the leading edge, especially in the hole cooling region upstream of the first and second rows of cooling holes. There is also a region of slightly worse cooling effectiveness downstream of the last coolant hole exit. Figure 19 also shows that a −70 deg rotation about the Z axis is the best layup. Because the layers are nearly perpendicular to the holes, the cooling hole array is able to conduct nearly all the heat from the upper surface to the film cooling holes, thereby increasing hole cooling and increasing the overall cooling effectiveness. The average ϕ for the best case is 0.048 higher than the baseline 0/90 orientation case.

## Conclusion

The effect of anisotropic thermal conductivity on a film-cooled flat plate and leading edge have been examined. First, several cases examined the effect of rotating the thermal conductivity tensor so that heat was conducted in different directions. In a physical sense, rotating the k tensor would be accomplished by stacking the composite layers in different orientations. It was found that simple one-axis rotations could have a profound effect on the overall effectiveness. Next, a compound two-axis rotation was performed to find the optimal two-axis orientation for the specific geometry. The optimized orientation has a 0.048 improvement in overall effectiveness. Moreover, the baseline case, representing the standard techniques for making CMC components, had nearly the worst orientation for cooling performance.

This research provides an improved understanding of how to design CMC parts with the aim of maximizing overall cooling effectiveness. To increase cooling performance in layered CMCs, designs should incorporate as many tows and/or plys in the through-thickness direction as possible. Conduction paths directly through the material and intersecting coolant holes lead to more effective cooling designs. Naturally, any recommendations based on cooling performance must also be evaluated in terms of manufacturability and structural performance, which were out of the scope of the present research.

## Acknowledgment

The authors thank the Air Force Research Laboratory for their support of this research. The authors also thank Luke McNamara, Jacob Fischer, Mitchell Scott, and Marta Kernan for conducting the experimental work used to validate computational simulations.

## 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 =

hole diameter, m

•
• h =

external heat transfer coefficient, W/(m2 K)

•
• k =

thermal conductivity, W/(m K)

•
• q =

heat flux, W/m2

•
• x =

surface distance downstream, m

•
• y =

distance normal to surface into fluid, m

•
• z =

spanwise distance from hole centerline, m

•
• T =

temperature, K

•
• U =

velocity, m/s

•
• cp =

specific heat, J/(kg K)

•
• ACR =

advective capacity ratio, cp,cρcUc/cp,∞ρU

•
• Bi =

Biot number, hL/ks

•
• Re =

Reynolds number, ρUD/μ

•
• Δϕ =

change in overall effectiveness, (Ts,baselineTs,modified)/(TTc)

•
• ζ =

local distance downstream oriented with materiel layers, m

•
• η =

adiabatic effectiveness, (T—Taw)/(T—Tce)

•
• η =

local distance perpendicular to materiel layers, m

•
• ν =

spanwise distance oriented with materiel layers, m

•
• ρ =

density, kg/m3

•
• ϕ =

overall effectiveness, (T—Ts)/(T—Tcp)

•
• χ =

coolant warming factor, (T—Tce)/(T—Tcp)

### Subscripts

• ∞ =

freestream conditions

•
• aw =

adiabatic wall

•
• c =

coolant conditions

•
• ce =

coolant at the hole exit

•
• i =

internal

•
• s =

surface

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