1 Improved Modelling Capabilities of The Airflow Within Turbine Case Cooling Systems Using Smart Porous Media

Impingement cooling is commonly employed in gas turbines to control the turbine tip clearance. During the design phase, computational fluid dynamics (CFD) is an effective way of evaluating such systems but for most turbine case cooling (TCC) systems resolving the small scale and large number of cooling holes is impractical at the preliminary design phase. This paper presents an alternative approach for predicting aerodynamic performance of TCC systems using a “smart” porous media (PM) to replace regions of cooling holes. Numerically CFD defined correlations have been developed, which account for geometry and local flow field, to define the PM loss coefficient. These are coded as a user-defined function allowing the loss to vary, within the calculation, as a function of the predicted flow and hence produce a spatial variation of mass flow matching that of the cooling holes. The methodology has been tested on various geometrical configurations representative of current TCC systems and compared to full cooling hole models. The method was shown to achieve good overall agreement while significantly reducing both the mesh count and the computational time to a practical level.


INTRODUCTION
In modern aero-engines, control of the over-tip leakage flow in turbines is crucial in reducing secondary flow losses and hence improving specific fuel consumption.
Various tip or shroud geometries have been utilized in an attempt to minimize these losses but in all cases the tip clearance varies notably through the flight cycle due to the differential expansion of the rotor and the casing under the action of centrifugal and thermal loads. Therefore, it is common practice to employ a Turbine Case Cooling (TCC) system to control the thermal expansion of the casing and hence actively manage the tip clearance. See, for example, Lattime and Steinetz [1,2] and Melcher and Kypuros [3].
A typical TCC system, as shown in Fig. 1, consists of an off-take, an inlet duct, a butterfly valve(s) and air transfer boxes connected to a set of circumferential manifolds. These feed several arrays of impingement cooling holes which deliver the cooling air to the surface of the turbine casing. However, design of TCC systems is, in practice, challenging. TCC systems are complex and comprise of many interacting sub-systems all with different aero thermal and mechanical constraints.
The aim of the TCC systems is to optimize tip clearance across different engine operating conditions [4] and around the casing circumference. To achieve this a key requirement is to provide a near-uniform delivery of cooling air to the turbine casing.
The cooling flow is commonly extracted from the bypass duct; directed towards the turbine casing through an inlet duct; the mass flow rate of the cooling flow is controlled by the valve and the cooling flow is then delivered to the turbine casing through the manifold (Fig. 1). According to Miller [5], one of the main factors affecting the uniformity of flow distribution of the impingement cooling holes in the manifold is the loss ratio. This can be expressed as either (i) the ratio of the inlet dynamic pressure of the manifold to the losses across the impingement cooling holes or (ii) the ratio of the total impingement cooling hole area to the manifold cross-sectional area. Ideally at the design stage the loss ratio should not be greater than 1 as this results in poor uniformity of the flow distribution resulted as reported by Miller [5] and illustrated in Fig. 2.
Theoretically, the required flow distribution could be achieved by adjusting total impingement cooling hole areas, manifold areas and loss coefficients. Another parameter that affects the flow distribution of impingement cooling holes is the length of the manifold. Miller [5] (Fig. 2) reported that with a low loss ratio, less than 0.5, the uniformity of the flow distribution is fairly good regardless of the length of the manifold.
However, with a medium to high loss ratio the uniformity begins to vary notably.
Developing TCC systems through the use of an experimental campaign is time consuming, expensive and yields limited data due to the difficulties of installing suitable instrumentation. Use of existing correlations and in-house experience is limited by different engine architectures and requirements leading to varying design styles being employed. Furthermore, during engine development it is common to undertake several iterations as, for example, cooling flow budgets change. To save development costs, or due to space constraints, certain features, such as the manifold cross-sectional area, are often fixed at an early stage. As other parameters change, this can often compromise the final design. It is therefore attractive to employ Computational Fluid Dynamics (CFD) to support the design process, particularly in the early phases. However, the complex nature of the geometry and existence of multiple small features such as the impingement cooling holes make generating and running a high fidelity CFD model too computationally intensive to be of practical use. The major difficulty for CFD simulations to investigate this type of problem is the large disparity in scales both in terms of the geometry and the flow topology. The turbine case is of order one to two meters diameter whilst the several thousand impingement cooling holes are typically of order one millimeter in diameter. The challenge is how to capture the general flow within the whole TCC system without the need to model the individual impingement cooling holes.
To overcome the problem, this paper describes the development of a numerical methodology which allows the impingement cooling holes to be replaced by porous media zones which mathematically simulate the same pressure drop (see Fig. 3). The main advantage of this approach is that it simplifies the computational domain and reduces computational cost to such a level that CFD can be used as a practical design tool. This type of porous media approach has been used in several other fields including will play a major role in determining the quality of the flow delivered to the impingement cooling holes. The current work aims to address this and develop a robust methodology for replacing the small-scale impingement cooling holes with porous media thus facilitating the modelling of the whole upstream TTC system. The paper presents a theoretical derivation of this porous media approach followed by an assessment of the effect of various global geometrical parameters. Throughout, where feasible, the results are compared to, and validated against, cases with the corresponding fully featured geometry which includes the impingement cooling holes.

Theoretical Derivation
In the current work, the commercial CFD code ANSYS Fluent has been used for all predictions. All solutions were converged based on second order accuracy utilizing a RANS approximation, a pressure-based solver with k-ω SST turbulence closure which is commonly used for predicting impingement cooling hole problems in the literature (e.g. Andreini et al. [4]). Convergence was judged by monitoring the residuals within the solutions and the flow parameters at the inlet of the domain. All cases with impingement cooling hole features utilized a compressible solver as the flow in the holes exceeds a Mach number of 0.3. However, in the porous media cases the flow never reaches velocities associated with compressibility and as such those cases employed an incompressible solver; the inlet Mach number is typically around 0.1. The typical mesh density used had a node spacing in the order of 0.5-1.0 mm with significantly increased resolution applied inside and near the impingement cooling hole regions. The boundary conditions used were based on typical cruise conditions for a large civil turbofan with uniform pressures applied at the inlet and outlet.
For simplicity in developing the porous media methodology all simulations were assumed to be isothermal and heat transfer was not considered. It will be shown later that this is a valid assumption and the effects of heat transfer are small, and arguably negligible, when compared to the effects of geometry. However, the methodology can be easily adapted to account for the effects heat transfer. Eq (1) shows that the porous media source term accounts for changes in density and velocity resulting from heat transfer. However, this would require prior knowledge of the heat loads on the turbine casing which is generally not available at the preliminary design stage for which this method is intended. Additionally, this would probably also require a conjugate heat transfer solution to capture the two-way coupled heating-cooling effects between the air and the metal casing. Although possible this would add significant numerical expense which will somewhat negate the aim of this methodology providing a rapid, numerically fast design tool.
Porous media is commonly modelled as a volume in many CFD codes by adding a source term in the momentum equations which replicates the pressure loss across the media [20]. The source term normally consists of a viscous loss term and an inertial term which need to be defined by the user. According to the ANSYS Fluent User's Guide [20], for perforated plate problems (such as the impingement cooling holes in a TCC system) only the inertial term typically needs to be considered. This is because that the viscous term is commonly used for solving problems with laminar flows and it is negligible for flow with high velocities. The benefit of omitting the viscous term is that it reduces the mathematical complexity of the current model presented. The momentum source term, , can be now written as [20]: where is the source term for the th ( , , or ) momentum equation, is the inertial resistance factor, is the fluid density in the porous media, is the magnitude of the velocity, and v is the velocity components in the , , and directions respectively.
Fundamentally, the mass flow through the impingement cooling holes is set by the pressure drop across them. Therefore, for a porous media to be representative of the impingement cooling holes the pressure drop must be the same across both the porous media and the impingement cooling holes, i.e.:

∆ ∆ 2
By expanding the equation and applying mass flow conservation the inertial resistance factor, , can be written as: 3 Where is the loss coefficient and is defined as:

∆ ⁄
is a function of the total pressure drop ∆ and the dynamic head across the impingement cooling holes. is the area ratio between the porous media used and all the impingement cooling holes, and is the thickness of the impingement cooling holes. Importantly, in this representation the resistance in the porous media becomes a function of the flow field as well as the geometry of the TCC. It should be possible to define the model in terms of the parameters discussed in Section 1 and the model should also be able to react to variations in system mass flow (for example when cooling demand changes across the engine operating cycle).
The resistance factor has to be defined in two directions for 2D problems or three directions for 3D problems. For perforated plate problems the flow through the as it would be through actual impingement cooling holes. In practice, this can be achieved by setting the value of in the other two directions to be ~100 times the value in flow direction.

Influence Factors of the Porous Media Pressure Drop
As shown in equation 3 is a function of the loss coefficient and geometrical parameters (the area ratio and thickness) of the TCC manifold. As is unknown prior to the simulation it needs to be estimated through a sensitivity study of the related parameters and it is only feasible to do this numerically. These parameters include: (i) the area ratio of the manifold cross-section to the impingement cooling holes as this sets the magnitude of the cross-flow ratio, To develop a robust method for estimating these parameters were examined using a simplified model comprising of a single row of impingement cooling holes. An example model is shown in Fig. 4. Note that for a stand-off distance set to typical engine size it was found that the actual impingement has negligible effect of the IH mass flow and as such a pressure outlet was applied to the exit of the impingement cooling holes, saving computation expense. The typical thickness (or length) to diameter ratio, / , of the impingement cooling holes was 0.58 and the pitch (spacing) to diameter ratio, / , was 8.33. The cross-sectional area of manifold was variable to alter the area ratio between the manifold and the impingement cooling holes (with the latter being fixed in the simplified model).
It was found that the area ratio of the manifold cross-section to the impingement cooling holes, , and the flow topology generated by the upstream components, , have the most significant influence on . Miller [5] also reported the effect of length and area ratio as shown in Fig 1) and (2) the first one or two holes also have a higher cross-hole velocity. As decreases ( Fig. 6 (3), (4) and (5)), it negatively influences the blockage inside impingement cooling holes close to the inlet leading to reduced effective area and thus lowers discharge coefficients. Thus, it causes the mass flow deficit in Fig. 5.
Towards the end wall of the manifold the system was losing mass flow through impingement cooling holes and thus experienced reduced manifold velocity. With the latter resulting in localized increased values of discharge coefficient and hence increased mass flow through the impingement cooling holes towards the end of the manifold (Fig.  5). Meanwhile the results also suggested that it followed the trend of flow distribution reported by Miller (Fig. 2) [5].
In terms of developing a suitable expression for a porous media loss coefficient, , it can be seen in Fig. 5  2. An initial guess is needed for the average value of the loss coefficient to set the bulk mass flow. An initial estimate of was chosen to be 2 for all cases with porous media. This can then be iterated numerically until a target TCC mass flow (obtained here from the corresponding predictions with impingement cooling holes) is achieved.

PROOF OF CONCEPT
In this section verification of the porous media methodology is presented by applying it to several different configurations broadly representative, non-dimensionally, of typical engine geometries. By keeping the model as simple as possible results from these proof-of-concept predictions can be directly compared to cases which still include the impingement cooling holes. Hence the validity of the porous media representation can be demonstrated over a range of geometrical and aerodynamic parameters. These includes varying the manifold cross-section to impingement hole area ratio , the manifold length, and the hole thickness-todiameter ratio ⁄ . The change was made to one parameter at a time with the rest of parameters kept the same. These were all varied in line with typical values seen in current TCC systems. Additionally, better assess the effect of inlet flow angle a generic junction (or feed pipe) was added to generate a more realistic distribution of flow angles instead of imposing a single artificial inlet angle. For simplicity, from herein, IH will be used to refer to cases with impingement cooling hole features whilst PM will be used to refer to cases where these are replaced by porous media.

Computational Domains and Boundary Conditions
The computational domains are illustrated in Fig. 7 and 8. Each incorporate a single row of 20 impingement cooling holes (or the porous media representation) with a thickness-to-diameter ratio, / , of 0.58 and a pitch-to-diameter ratio ⁄ of 6.67 and 8.33 in the x and y directions with periodic side-walls in the x direction. Four different manifold cross-sections to IH area ratios were modelled, of 1 to 4, and these were again achieved by altering the manifold cross-sectional area. A T-shaped inlet was modelled with using only half of the geometry and a symmetry boundary condition (Fig.  7). This essentially represents the same length of the manifold on each side of the Tshaped inlet. This was also modelled without the symmetry condition enabling different lengths of manifold to be incorporated either side of the inlet (Fig. 8). A short length was included upstream of the junction with a length twice that of the hydraulic diameter.
This was done to ensure the fixed inlet boundary condition did not constrain the flow in the junction. Several inlet lengths were examined, and it was found that two hydraulic diameters were sufficient. Finally, another model was also generated which included circumferential curvature close to that expected in the TCC system of a large civil turbofan.

Results and Discussions
Sensitivity to Area Ratio, To investigate the flow features inside the manifold, Fig. 10 shows contours of normalized velocity magnitude for different area ratios. Note that the plot shows only the first 7 out 20 IH and that the CFD only simulated half of the T-shaped inlet (using a symmetry boundary condition). Each subfigure shows the comparison between the IH and PM cases with the same area ratio, . Note that to alter the area ratio , by altering the manifold area, it is somewhat difficult to keep all the other parameters consistent. For example, the area ratio between the junction and the manifold was kept the same as was the IH area and spacing. The position of the first IH was maintained at the same non-dimensional location with respect to inlet which unfortunately changes the initial IH spacing. However, regarding a direct comparison with the performance of the PM model this is not a major issue. In general, the PM method correctly predicts the flow in the manifold with AR from 1 to 4. The PM method does not predict the mass flow distribution well close to the inlet (Fig. 9). Here the flow is close to a total feed which has (as yet) not been incorporated into the PM model. However, it should be possible to construct a PM treatment to account for this or perhaps keep a small number of IH in this region is the final model.

Sensitivity to ⁄ Ratio
In practical TCC systems it is often necessary to have regions of holes with different diameters to control local cooling flows. As mentioned previously the thickness of the manifold casing is often defined early in the design process which then fixes the length of the impingement cooling holes. If a change in flow rate is then desired, it is achieved by varying the hole diameter. In some other cases, the manifold configurations are designed different with different casing thickness for different engine configurations while cooling hole diameters may be fixed by tooling specification. Hence these constraints lead to variations in the thickness-to-diameter ratio, ⁄ , and it is important that a robust PM methodology should have the capability to model this. To investigate the effect of varying ⁄ ratio several IH models were generated varying both the hole diameter and the manifold thickness. With the initial thickness of the holes fixed the diameter was varied to achieve ⁄ ratios of 0.23, 0.35 and 0.70. Similarly, with the hole diameter fixed the thickness was varied to achieve / ratios of 0.42, 0.83 and 1.25. See Table 1. Note that the pitch, , remained constant. Two different area ratios , were modelled: of 2 which is above the critical value (1.41) and of 1 which is below that value.
The effect of ⁄ ratio on the mass flow distribution of the impingement cooling holes with a T-shaped inlet is shown in Fig. 11. These cases were also predicted using a symmetry boundary condition at the center of the T-shaped inlet junction. The mass flow distribution is comparatively flat except when the ⁄ ratio is lower than 0.4.
However, even with ⁄ ratio higher than 0.4 the maximum deficit of mass flow rate is around 17% difference from the average value. As shown in Fig. 12 (normalized velocity magnitude contours over the first 7 out of 20 IH) with lower ⁄ ratios (0.23 and 0.35), the discharge coefficients of the cooling holes (thus the mass flow rate) are much more sensitive to the local flow feed. In the region close to the inlet, the total feed and low ⁄ ratio (larger diameter) together will result in higher mass flow rates. With fed by the cross flow in high velocity just after the junction, the discharge coefficient is low due to the blockage in the cooling holes and reduction in the effective area of the vena contracta. As the manifold flow velocity decreases from losing flow through cooling holes, both the discharge coefficient and mass flow rates increase. The cases with of 1 have similar results as shown in Fig. 12 but the ⁄ ratio effect is worse. This phenomenon observed after the junction is the same as seen in the IH cases without the junction.
Based on previously experience (from the industrial partner) it is unlikely that a ⁄ ratio will be lower than 0.35 in the current engine architectures. Therefore, for the porous media methodology it is important to verify its validity at this lower limit of ⁄ = 0.35. Hence, Fig. 13 shows a comparison between the IH and PM case with a ⁄ ratio of 0.35 at two different area ratios . Although the porous media methodology does not predict the exact flow behavior close to the junction region there is still reasonable agreement in the trend of the mass flow distribution as for the IH cases for of 2 (Fig. 13). However, the porous media methodology meets its limit with of 1 together with a low ⁄ ratio of 0.35.

Sensitivity to T-Shaped Inlet with Different Length of Manifold on Each Side
Previous sections have discussed the sensitivity of the porous media methodology to several parameters. All those parameters have been investigated on computational domains with the same length of manifold on both sides of the T junction. In some TCC systems the T-shaped inlet may feed manifolds of differing lengths and having different arrangement of cooling holes and thus different mass flow (Fig. 8). Additionally, TCC systems will have curvature which is also thus far been omitted. This section also aims to address how the porous media approach copes with this by examining the models shown in Fig. 8; both with impingement cooling holes and porous media. Predictions were made for cases with different area ratios, of 1 and 2 and for various levels of curvature. In addition to the no curvature cases two levels of curvature were examined which were broadly representative of those seen in the HP and LP turbines in a large civil turbofan. The level curvature is defined, nondimensionally [23], as the ratio of the hydraulic diameter, , of the manifold passage to its radius of curvature, , about the engine center line:

9
Where is the hydraulic diameter of manifold passage and is the radius of the manifold about the engine center-line. For a typical HP turbine the higher level of curvature gives rise to ~ 0.02 whilst for the LP turbine the increased radius reduces to ~0.01.
There are 77 impingement cooling holes in the computational domain (Fig. 8).
The porous zone was divided into two parts from the middle of the junction and the PM method was applied to each part separately to match the mass flow split after the junction.