The Impact of Manufacturing Variations on the Aerothermal Performance of High-Pressure Turbine Blade Shrouds Open Access

Shrouded high-pressure turbine blades (HPTBs), such as shown in Fig. 1(a), reduce tip leakage flow using a platform with fins and fences. Manufacturing variability leads to misalignments between adjacent shrouds and variable inter-platform gaps. Shrouds are exposed to some of the highest loads in the engine, yet unlike hub platforms, the aerothermal effect of shroud platform manufacturing variations is nearly absent from the literature. Uncertainty in the sentencing of shroud manufacturing variations risks both scrapping good parts and using parts prone to wear, leading to early maintenance, with reported costs of up to $2M per shop visit [1]. To help reduce this uncertainty, this article presents a correlation for the shroud step heat transfer, which can be used to triage parts within manufacturing line timescales. Figures 1(b) and 1(c) highlight the key finding that the aftchord shroud step flow is quasi-2D (Q2D) and resembles the corresponding canonical step flow with heat transfer enhancement at the reattachment point. It will be shown that prediction errors below 20% are obtained for the majority of cases when the correlation is tested on 3D HPTB shroud endwall steps.

Prior studies on manufacturing variations can be divided into in-depth studies of a specific manufacturing variation applied to stylized geometry such as a cascade [2–4] and Monte Carlo simulations of the overall manufacturing variability occurring on the part of interest, represented by either a parametrized model [5–7] or principle component analysis modes [8,9]. In this first category, Cardwell et al. [4] studied platform misalignments in a nozzle guide vane cascade with rim seal cavity flow and no inter-platform purge flow. They showed that the cooling effectiveness decreases in the aftchord region for a local forward-facing (FWD) step and increases for a backward-facing (BWD) step. This has also been found by Hada and Thole [10] using Reynolds-averaged Navier–Stokes (RANS) simulations. Lange et al. [11] studied the influence of hub platform steps including both rim seal cavity flow and inter-platform purge flow using IR measurements and RANS simulations. They observed a decrease of area-averaged Nu on the aftchord suction side (SS) endwall of 4% for a local backward-facing step and an increase of 2% for a forward-facing step. The largest change in local Nu was observed at the throat, where the inter-platform flow gets ejected on the SS platform. The same observations where made by Zhang et al. [12] using RANS simulations.

The aligned shroud endwall flow study by Lehmann et al. [13] on an HPTB with an idealized three fin shroud showed that, as opposed to hub endwalls, main passage fluid gets ingested in the inter-platform gap over the entire chord due to the decrease in static pressure caused by the fins. Rushton [14] and Wallis et al. [15] showed that also the secondary flows differ from the hub endwall. The horseshoe vortex forms in the shroud inlet cavity and emerges onto the endwall at midpitch, away from the potential field of the leading edge (LE). The pressure side (PS) leg of the horseshoe vortex immediately gets swept toward the suction surface due to the cross-passage pressure gradient. Downstream of the swept vortex, Lehmann et al. [13] showed that the shroud endwall flow is initially aligned with the wedge face and from midchord onward traverses the platform from pressure to suction side, where the near-wall PS fluid gets ingested in the inter-platform gap. As a result, a new boundary layer forms on the aftchord SS endwall with a corresponding local maximum in Nu at the SS platform edge, as shown by IR measurements. Lehmann et al. [16] also studied the influence of gap width variations on shroud heat transfer, but the platform was not considered.

Even though the discussed prior art provides a good overview of the influence of hub platform steps on aerothermal performance, the different leakage and secondary flows on shroud endwalls suggests that there is an opportunity for studying shroud platform steps with a view to improved sentencing of these parts.

In this article, the influence of engine-representative shroud platform manufacturing variations on aerothermal performance are investigated using steady RANS simulations of a three passage engine part HPTB model. A step sentencing correlation for the reattachment Nu is then developed using a Q2D parametric model of the shroud endwall. The performance of the correlation is demonstrated against the full 3D flow field. Finally, the limitations of the RANS simulations used to build the correlation are addressed using experimental data.

First, the measurement of step heights and gap widths on two samples of 3D scanned blades is explained. Next, the numerical and experimental methods are discussed in turn.

Platform step heights and inter-platform gap widths are measured on structured light scans of 100 castings and 26 finished parts of an HPTB of a modern aircraft engine (the latter has been used in the study by Lee et al. [17]), scaled into the hot reference frame. The platform step height and inter-platform gap width are measured at the LE, $25%Cx$⁠, $50%Cx$⁠, and $75%Cx$ for all possible blade passages that can be created from the sample. Figures 2(a) and 2(b) show that the step height, $H$⁠, is measured on the intersection of an axial cutting plane and the shroud platforms at each axial location, whereas the gap width, w, is measured on the intersection of a plane cutting both wedge faces. For clarity, only the axial plane at $25%Cx$ is shown in Fig. 2(a). Since the wedge faces of the platform are machined on the parts after the casting stage, the inter-platform gap width cannot be measured in the casting sample.

Figure 2(c) shows the statistical distribution of the measured step heights, normalized by blade span, $S$⁠. A positive value indicates that the SS platform is more radially outward than the PS platform, as is the case shown in Fig. 2(a). Therefore, a positive value represents a forward-facing step (shown in Fig. 6(d)) at the LE where the flow crosses the step from SS to PS, and a backward-facing step (shown in Fig. 6(b)) for the locations further downstream, where the flow crosses the platform from PS to SS. The statistical gap width distribution is shown in Fig. 2(d). The standard deviation is larger for gap widths, illustrating the larger manufacturing variability. Note that even though negative gap widths occur in the hypothetical pairs, the shrouds would simply touch in reality.

A test matrix of three endwall misalignments and three inter-platform gap widths is created to study the corresponding effect on aerothermal performance. The manufacturing variations are chosen based on the statistical distribution shown in Figs. 2(c) and 2(d), such that their probability of occurrence can be determined. First, the histograms are compared to the equivalent normal probability density function, calculated using the respective mean, $μ$ and standard deviation, $σ$⁠. As such for the finished parts step height histograms, a multimodal distribution was needed, which suggests that the step heights are created by two or three distinct mechanisms. For the casting scan step height and the gap width distributions, however, a good match is obtained with a Gaussian.

The step height used in the following study is a $μ+4σ$ variation, indicated in Fig. 2(c), referred to as the $4σ$ backward-facing step. This represents a large step and somewhat rare occurrence, with a probability to occur once in 240 engines. The same deviation mirrored around an aligned endwall is studied as well, referred to as the $4σ$ forward-facing step. A test case with an aligned endwall provides a reference. The studied gap widths consist of the nominal and a $μ±σ$ gap, indicated in Fig. 2(d), of which the latter have a probability to occur in every engine.

Figures 2(e) and 2(f) illustrate how the manufacturing variations are applied to the StereoLithography (STL) geometry of the design-intent blade in matlab. The perturbations are applied using control points, shown in red, located where the manufacturing variations have been measured. An additional control point is added at the trailing edge (TE) to smooth the profile of the desired perturbation.

Steady RANS simulations of a three passage high-pressure turbine (HPT) rotor model, a Q2D shroud endwall step model, and the internal geometry of the step heat transfer rig discussed in Sec. 2.3 are conducted to calculate Nu. The simulations are run in star-ccm+ using the $k−ω$ shear stress transport (SST) turbulence model, where Nu is calculated using the two-run method. Meshing is performed using the automated unstructured polyhedral mesher in star-ccm+. Since the heat transfer coefficient (HTC) is a local property, grid refinement is applied in the vicinity of the step. A grid independence study is performed on the Q2D shroud platform step model for the mesh controls listed in Table 1. The grid-independent settings, shown in bold in Table 1, are obtained at the penultimate refinement, indicated by N-1, from which further refinement, indicated by N, leads to a relative change in reattachment Nu smaller than 2%. Note that the near step mesh refinement is mainly controlled by the local prism layer. The domain creation and boundary conditions for each model are now discussed.

The domain is created using the design-intent geometry of the blades transformed into the hot reference frame, including all shroud geometry but discarding cooling features except the shroud sealing flow. The tip gap is set to a typical mid-power value. Figure 3 shows the domain creation in star-ccm+, where four blades are subtracted from an empty periodic flow domain matching three blade passages. Manufacturing variations are applied to the red zone on the shroud platforms of the central passage. Using the grid-independent settings, a 51 million cell mesh is created.

The three passage HPT rotor model is run with temperature ratios of 0.92 and 0.96 using boundary conditions extracted from a full stage simulation at mid-power.

The parametric study is performed on a Q2D step model based on the streamwise cut in Fig. 1(b). The step height is equal to $0.75%S$⁠, which is approximately equal to the mean of the measured step heights at $50%Cx$⁠. The gap width equals half the step height and is comparable to the mean measured gap width of around $0.375%S$⁠. The domain extends for $20H$ both upstream and downstream of the step. Upstream, this distance is used to let the boundary layer develop and downstream to capture the reattachment point well within the domain. The Q2D step domain has a depth of $2.5H$⁠, such that flow angle variation with respect to the step can be simulated. The height of the domain is extended to $250H$ such that a step height to passage-width ratio representative of commonly occurring shroud platform steps is created.

The walls forming the step at the bottom of the domain are isothermal, with the temperature ratio of each run given in Table 2. A slip wall is applied on the top of the domain, and the sides are periodic to allow for flow angle variations. The bottom of the gap can be a wall or a mass flow outlet, depending on whether gap leakage flow is simulated. A wall-normal profile for both total pressure and turbulence properties is specified at the inlet in order to define the inlet boundary layer. Inlet boundary conditions are extracted from a streamwise slice through the aftchord shroud platform step in the three passage HPT model. A summary of the freestream flow conditions in the Q2D domain is provided in Table 2, where Tu represents the turbulence intensity in the streamwise direction and $TuL$ represents the turbulence length scale.

The internal geometry of the 2D step heat transfer rig described in Sec. 2.3 is modelled from the turbulence grid to the last pressure tapping in the test section. The experimental pressure and temperature data are used to set the boundary conditions, and the freestream turbulence at the step edge is matched to the experimental value. The step Reynolds number is 12,000, and the wall temperature ratios are 0.89 and 0.93, respectively.

Experimental Nu data are collected using a 2D step heat transfer test rig, shown in Fig. 4. The rig is designed to conduct a transient heat transfer experiment, where the HTC is calculated by measuring the thermal response of a test plate to an increase in freestream temperature of 85 K by heat addition through two 15 kW heater meshes. The flow is guided to the test section using an 8 to 1 contraction. The boundary layer is bled on all four walls of the test section in order to reduce flow acceleration by boundary layer blockage. The onset of boundary layer transition on the test plate is fixed using a Braslow trip [18]. The freestream turbulence intensity, Tu, induced by a turbulence grid has been measured using a hotwire and is equal to 4.8% in the vicinity of the step. The flow field in the test section is fixed by choking the flow downstream of the test section.

The temperature on the test plate is measured using a FLIR SC7000LW IR camera, which has a noise equivalent temperature of 25 mK and a measurement frequency of 200 Hz. The corresponding heat flux is calculated using the impulse-response method of Oldfield [19], which assumes 1D heat transfer in a semi-infinite substrate. Therefore, the step plates are designed such that errors introduced by violating these assumptions do not reach the reattachment point. This has been achieved by manufacturing the test plates out of polycarbonate, with measured thermal effusivity of $543Jm−2Ks−12$⁠. This allows to measure the thermal response of the test plate during 20 s, while limiting the lateral conduction errors to an area within $1.5H$ from the step edge.

The HTC is calculated using floating regression [20] of the measured temperature data, normalized by the time accurate $Taw$ increase, and the corresponding heat flux data at each pixel. The experimental uncertainty is estimated to be 5% based on the Monte Carlo analysis of uncertainties performed by Playford [21].

The 2D HTC contours are spanwise-averaged between the boundaries indicated in Fig. 5, where $Rex$ is measured from the plate LE, to avoid camera occlusion or corner flow effects.

In this section, the results of the simulation matrix are discussed for each applied platform step, starting from the aligned endwall. The influence of changing the inter-platform gap width is discussed with respect to every considered platform step.

Since the developed Q2D correlation will be tested against the results of this 3D simulation matrix, ideally the same methods should be used. Although an experimental parametric study of the Q2D shroud step model has been conducted, recreating the 3D endwall platform step flow in an experiment is impractical. Since the focus lies on wall-bounded shroud step flow, the closest alternative numerical method to RANS is a wall-resolved large eddy simulation (LES), of which the computational cost is comparable to a direct numerical simulation (DNS) [22]. A recent DNS study on a cascade style single passage HPT stage at engine-representative Re required an 8.2 billion cell mesh [23], illustrating the prohibitive computational cost for use in this type of parameter study. Hence, the 3D simulation matrix and Q2D shroud model are run using steady RANS and the associated uncertainties are mitigated using experimental results, which are presented in Sec. 6. For the current discussion, it is sufficient to know that the experiments showed that a correction factor of 1.37 is required on the predicted reattachment Nu, where the flow resembles canonical forward-facing step flow. No correction factor is required for backward-facing step flow.

The endwall flow in the vicinity of the wedge face depends on the pressure difference over the inter-platform gap, which is shown in Fig. 6(a), where $Cp=(pin−p)/12ρinVrel,in2$ with the subscript, in, referring to the rotor inlet conditions. The pressure drop downstream of the first fin initially causes flow to be driven to the over-shroud region. As the $Cp$ difference decreases between 20 to $40%Cx$⁠, the ingested main passage flow migrates parallel to the wedge face, as shown in the gap flow schematic in Fig. 6(a). At $40%Cx$⁠, the pressure difference reverses and the entrained gap flow gets driven back toward the main passage. From $50%Cx$ onward, the pressure drop downstream of the second fin leads to the ingestion of main passage flow.

The surface streamlines shown in Figs. 7(a)–7(c) confirm the conclusions drawn from the gap $Cp$ distribution. In the LE region, the nearly wedge face aligned flow gets ingested into the inter-platform gap from both platforms. At midchord, the ejected inter-platform gap flow reattaches on the SS endwall, indicated by reattachment line A. The reattachment occurs further downstream than $40%Cx$ where the pressure difference reverses due to the high axial momentum of the inter-platform gap flow. Further downstream, the near-endwall flow turns toward the wedge face due to the cross-passage pressure gradient such that the PS flow crosses the inter-platform gap. The near-endwall flow gets ingested in the inter-platform gap such that a new boundary layer forms on the SS endwall, as shown in Fig. 6(c). Similar to the study by Lehmann et al. [13], the streamlines show no evidence of the passage vortex crossing the wedge face.

The shroud endwall $NuC$ contours displayed in Figs. 8(a)–8(c) show that the flow ingestion in the LE region leads to high $NuC$ on both platform edges. Around midchord, the vortex formed by the ejected gap flow creates a streak of low $NuC$ on the SS endwall as it gets swept toward the suction surface. Further downstream, the restarting boundary layer leads to peak $NuC$ on the SS platform edge, followed by a decrease in the streamwise direction, which is in agreement with the experimental results by Lehmann et al. [13].

The endwall surface streamlines in Fig. 7(e) show that an additional reattachment line, marked B, occurs downstream of reattachment line A on the SS endwall. This suggests that the flow separates at the PS step edge and reattaches on the SS platform downstream forming a recirculation zone in between. This resembles canonical backward-facing step flow as is sketched in Fig. 6(b). This hypothesis is supported by the reduced pressure margin in the aftchord region compared to an aligned endwall shown in Fig. 6(a).

The $NuC$ contours displayed in Fig. 8(e) confirm the resemblance to canonical backward-facing step flow in the aftchord region, where a local maximum in $NuC$ occurs along reattachment line B. At $75%Cx$ a reattachment $NuC$ value of 3800 is obtained, which is an increase of 31% compared to the value obtained for an aligned endwall. The flow ingestion in the inter-platform gap leads to increased heat transfer in the recirculation zone compared to the canonical step flow without a gap since it accelerates the flow toward the step face. Due to this acceleration, the recirculating flow impinges on the PS wedge face, leading to a local $NuC$ of 3800, which is more than six times the value observed for an aligned endwall.

Figures 7(d)–7(f) show that the downstream reattachment length scales inversely to inter-platform gap width. This is likely to be caused by the ingestion rate, which controls the extent to which the reattaching streamline is pulled toward the wedge face. Since reattachment line B lies immediately downstream of the wedge face for the $+1σ$ inter-platform gap, it is likely that for a further reduced step height to gap width ratio, the reattachment point moves to the SS wedge face, such that a new boundary layer forms on the SS endwall.

Figures 8(d)–8(f) show that the peak $NuC$ scales with the inter-platform gap width and thus the leakage flowrate, giving a peak $NuC$ value of 2750 at $75%Cx$ for the $−1σ$ gap and a value of 4900 for the $+1σ$ gap. These correspond to a reduction of 28% and an increase of 29%, respectively, compared to the equivalent case with nominal gap.

Increasing the gap width also reduces the $NuC$ value on the PS wedge face resulting in a value of 2450 for the $+1σ$ gap, a reduction of 36% compared to the equivalent case with nominal gap. This is likely to be caused by the recirculating flow impinging on the PS wedge face with reduced intensity since most of this flow has been ingested in the gap before it reaches the wedge face.

Figure 7(h) shows that the reattachment line B corresponding to canonical step flow reoccurs for a local forward-facing step. This is consistent with the static pressure increase at the wedge face compared to an aligned endwall, shown in Fig. 6(a), caused by the impingement of the upstream flow on the SS wedge face as shown in the canonical forward-facing step flow in Fig. 6(d).

Unlike for local backward-facing steps, reattachment line B bends toward the wedge face at the TE. The streamlines in the recirculation bubble are nearly parallel to the wedge face before getting ingested in the inter-platform gap. This suggests that part of the high axial momentum flow ejected from the inter-platform gap gets entrained in the recirculation bubble. Another reason for the more wedge face aligned flow in the recirculation bubble is the reduced ingestion of recirculating fluid into the inter-platform gap compared to a local backward-facing step since mainly the upstream PS platform flow gets ingested.

The $NuC$ contours displayed in Fig. 8(h) show that $NuC$ enhancement along reattachment line B is smaller than for the equivalent backward-facing step, with a $NuC$ value of 2200, using the correction factor of 1.37, at $75%Cx$⁠. The largest $NuC$ enhancement now occurs at reattachment line A, resulting in a maximum $NuC$ value of 3100. This heat transfer increase is likely to be caused by the ejected gap flow now encountering a forward-facing step, which forces the flow to bend upward in order to emerge on the SS endwall. This upward movement leads to increased shear in the free shear layer forming at the SS platform edge and thus more energetic eddies impinging on the endwall at the reattachment point, leading to increased heat transfer.

The impingement of the PS endwall flow onto the SS wedge face leads to $NuC$ values of up to 4000, more than double the expected value for an aligned endwall. Such increased heat transfer rates increase metal temperatures and increase the risk of oxidation and erosion on the wedge face.

The shroud endwall streamlines shown in Figs. 7(g)–7(i) indicate that the reattachment length is less affected by a change in gap width compared to a local backward-facing step. This reinforces the hypothesis that the recirculating flow contained by reattachment line B is less influenced by the leakage flowrate.

Figures 8(g)–8(i) show that for both considered gap width variations, the $NuC$ levels at reattachment line B increase. For the $−1σ$ gap width, the reattachment $NuC$ value at $75%Cx$ is equal to 2660 using the correction factor of 1.37, which is an increase of 21% compared to the equivalent case with nominal gap. For the $+1σ$ gap width, the peak $NuC$ increases to a value of 2880 (using the correction factor of 1.37), which corresponds to a 31% increase.

The increase in $NuC$ for a $−1σ$ gap can be explained by the entrainment of the ejected gap flow in the recirculation bubble, which due to its high axial momentum increases the wall shear stress in the step-parallel direction. Figure 9 shows $Cf,//$ contours on the shroud endwall for both a nominal and a $−1σ$ inter-platform gap, where $Cf,//=τ///12ρinVrel,in2$⁠. It can be observed that $Cf,//$ increases by 35% at $75%Cx$ for the $−1σ$ gap width compared to a nominal gap. Following the Reynolds analogy, this increases $NuC$ in the aftchord region in addition to the increase in heat transfer caused by the reattaching flow.

The maximum $NuC$ value at reattachment line A decreases by 24% for a $−1σ$ gap width, which is likely to be a result of the reduced ejected flowrate. For the increased inter-platform gap width, the peak $NuC$ value changes by less than 10%. Also the peak $NuC$ on the SS wedge face changes by less than 10%, for both considered gap width variations.

The results of the 3D simulation matrix show that for an aligned endwall, the highest $NuC$ values occur on the aftchord SS endwall edge where the boundary layer is restarting. For a $4σ$ backward-facing step, the highest $NuC$ values are observed at the reattachment point on the aftchord SS endwall caused by the local canonical step flow. The reattachment $NuC$ equally is a strong function of the inter-platform gap width. For the $4σ$ forward-facing step, the ejected gap flow reattaching on the SS endwall at midchord leads to the highest $NuC$ enhancement.

The highest heat transfer enhancement has been observed on the aftchord region downstream of a local backward-facing step, which is therefore the most critical region from a part sentencing perspective, as this is where the part is most likely to get damaged. Since the flow is Q2D in the aftchord region, the flow parameters driving the change in heat transfer can be identified by conducting a parameter sweep on the Q2D shroud endwall step model for which both the domain and numerical methods are discussed in Sec. 2.2. These simulations are fast and easy to run, such that a simulation database is quickly acquired, but also allow to test the developed correlation on the HPTB shroud platform steps of the 3D simulation matrix using the same methods. However, since the correlation is ultimately used on manufactured parts, the final correlation is tuned using an experimental data base encompassing the required parameters, described in Sec. 6.

The parametrization of the shroud endwall step flow in the aftchord region uses a streamwise plane as shown in Fig. 1(c). The main parameter defining the flow field is the step height, which is included in the model as the characteristic length in the Reynolds number, $ReH$⁠. Vogel and Eaton [24] indicated that also the step height to boundary layer thickness ratio influences the heat transfer at the reattachment point. Section 3 highlighted that the influence of the step on aerothermal performance changes depending on the local flow angle with respect to the step, $α$⁠. Also the influence of the gap leakage flow, $m˙L$⁠, on both the reattachment length and the reattachment Nu has been demonstrated. Finally, the studies by Awasthi et al. [25], and Zahn and Rist [26] suggest that increasing the step edge radius reduces the reattachment length for forward-facing steps.

The boundary layer thickness is represented by momentum thickness since it is an integrated property and therefore more robust to define for the nonuniform freestream occurring in the HPTB passage.

A database of Nusselt number profiles, with $H$ as characteristic length, is created by conducting a parameter sweep on the Q2D shroud step domain for the variables listed in Table 3, using the numerical methods discussed in Sec. 2. Figure 10 shows the $NuH$ profile along the step for the reference case described in Table 3. All $NuH$ profiles have been normalized by the reference reattachment $NuH$ value, $NuH,ref$⁠, indicated by the blue dot in Fig. 10. Note that an equivalent reference case is used for the forward-facing step profiles.

The effect of each parameter is first considered in isolation. Since the shroud endwall step flow in the aftchord region is characterized by both a flow angle deviating from step-normal flow and inter-platform gap leakage, potential interactions between flow angle and leakage flow are studied afterward.

The normalized $NuH$ profiles and the corresponding reattachment $NuH/NuH,ref$ values are shown in Fig. 11 for all parameters for a backward-facing step. It can be observed that the reattachment $NuH$ scales almost linearly with $ReH$⁠, with an RMSN error below 2%. In order to provide further validation of the used numerical methods and the obtained dependency on $ReH$⁠, the Q2D model data points are displayed together with the results obtained using the 2D step heat transfer rig and the data from the study by Vogel and Eaton [24] in Fig. 12.

Figure 11 shows that both the peak $NuH$ and the reattachment length decreases with the decreasing flow angle. However, the reduction in reattachment length in the considered flow angle range does not match the constant reattachment length observed by Kaltenbach [27] based on LES results. This is likely to be a consequence of the turbulence model being benchmarked using the step-normal flow, such that it is unable to capture this effect.

Even though the normalized reattachment $NuH$ value is a strong function of flow angle, the decrease in $NuH$ for a certain $α$ is less than the decrease for the equivalent perpendicular flow with matched step-normal $ReH$⁠. This implies that the reduction of the reattachment $NuH$ with decreasing step-normal velocity component is counteracted by the increasing skin friction in the step-parallel direction, which following the Reynolds analogy increases the $NuH$ of the step-parallel flow component.

Different heat transfer rates can be observed upstream of the step in the Nu profiles for varying step height to momentum thickness ratio in Fig. 11. This is caused by the variations in upstream boundary layer thickness used to change $H/θ$⁠. When the reattachment $NuH$ values are expressed as a function of $H/θ$⁠, it can be observed that the peak $NuH$ varies less than 10% in the considered $H/θ$ range, indicating a weaker dependency compared to the previous two parameters. $H/θ$ ratios up to 35 have been observed in the aftchord region of an HPT shroud endwall for the large $4σ$ step, indicating that the considered range is representative for more commonly occurring shroud platform steps.

The dependency of $NuH$ on leakage flow rate is investigated using leakage flow rates representative of the aftchord region of an HPTB shroud endwall with a $4σ$ step, which ranges from 10% to 30% of $m˙H$⁠. The $NuH$ profiles in Fig. 11 show a decrease in reattachment length with the increased leakage flow ratio. The reattachment $NuH$ scales approximately linear with the leakage flow ratio, where a leakage flow ratio equal to 13% leads to a 45% increase in peak $NuH$⁠. The strong dependency of both the reattachment length and peak $NuH$ on leakage flow ratio is in agreement with the observation made for a 3D shroud endwall in Sec. 3 and the result of the RANS study on 2D backward-facing steps with leakage flow ingestion by Abu-Nada et al. [28].

Figure 13 shows the $NuH$ profiles and the corresponding peak values for all considered parameters for the Q2D forward-facing step model. The dependencies on $ReH$⁠, $α$⁠, and $H/θ$ are nearly identical to the dependencies observed for backward-facing steps, whereas the increase in $NuH$ for double the maximum leakage ratio considered for backward-facing steps is approximately 25%. The dependency on leakage flow ratio is thus weaker for forward-facing steps. This reduced sensitivity has also been observed in the aftchord region of an HPTB shroud endwall inSec. 3.

The final row of Fig. 13 shows the $NuH$ profiles for forward-facing steps with different step radii. It can be observed that the reattachment length reduces as the step radius increases, which is in agreement with the experimental results by Awasthi et al. [25] and the DNS results by Zahn and Rist [26]. Since assuming a constant $NuH$ for different step edge radii leads to an RMSN error below 1.5%, no dependency on $r/H$ is added to the correlation.

Figure 14(a) shows the downstream $NuH$ profiles for five different flow angles, with a leakage flow ratio of 8.9%. It can be observed that due to the reduced step-normal velocity for small $α$⁠, the flow gets pulled back more strongly toward the wedge face by the inter-platform leakage, which results in an additional shortening of the reattachment length compared to the profiles shown in Fig. 11. The reattachment $NuH$ is nearly constant for flow angles smaller than $45$ deg, which implies that the effect of leakage on the reattachment $NuH$ increases for small flow angles.

Figure 14(b) shows the peak $NuH$ as a function of both leakage flow rate and angle with respect to the step. It can be observed that for leakage flow ratios exceeding 8.8%, the reattachment $NuH$ increases with the decreasing flow angle. The peak $NuH$ asymptotes when the angle reduces toward the step-parallel flow. This illustrates that the flow mechanism changes from reattachment on the downstream platform to reattachment in the gap such that a new boundary layer starts on the downstream surface. The flow angle at which this change in flow reattachment occurs depends on the leakage flowrate, such that the dependency of peak $NuH$ on flow angle can be described using a blending function based on the leakage flowrate. Figure 14(b) shows that this blending function limits the RMSN error to 1.8% for $m˙L/m˙H$ equal to 13.3%.

Figure 15(a) shows that the interaction between leakage flowrate and flow angle is weaker for a forward-facing step. This can be attributed to the fact that mainly upstream platform flow gets ingested in the gap instead of the recirculating fluid, as has been observed in Sec. 3. However, for flow angles below $60$ deg, a disproportional increase in $NuH$ can still be noted, such that the RMSN error would increase to 3.5% for a leakage mass flow ratio of 26.6%. In order to account for this disproportional enhancement, a flow angle dependent factor is added to the leakage term, which increases the $NuH$ at small flow angles. Figure 15(b) shows that this additional factor reduces the RMSN error to 1.6%.

In order to assess the step correlation performance on the geometry it is meant to sentence, it is tested on all the aftchord shroud endwall steps discussed in Sec. 3. In order to minimize the additional resources required to use the step sentencing correlation on the production line, the input variables are calculated from a design-intent computational fluid dynamics (CFD) simulation and a step height measurement. Therefore, the input variables for these test cases are extracted on the shroud endwall from the aligned platform case with nominal gap. The detailed calculation of the input parameters is explained in the  Appendix.

The step correlation is tested on four streamwise slices on the aftchord shroud endwall, labeled A–D in Fig. 16. The extraction of the two-run $NuH$ to which the prediction is compared is shown for streamwise slice C. Note that the reattachment length has decreased from $6.5H$ to $2.5H$ due to the reduced flow angle relative to the step edge and the flow ingestion in the gap.

Figure 17 shows the $NuH$ prediction accuracy for the four streamwise slices for all step cases in the 3D simulation matrix. The influence of gap leakage flow for backward-facing steps is illustrated by the $NuH$ increase for all slices on a test case with the increasing gap width. Such a division in clusters cannot be observed for forward-facing steps.

All predictions for backward-facing steps are within 20%, where the highest relative error is observed for plane A in the case of a $+1σ$ gap. In this case, the flow is close to the changing point between reattaching backward-facing step flow and a restarting boundary layer downstream, where the correlation is sensitive to slight changes in the input parameters. Also the forward-facing step predictions are within 20%, except for the point representing plane A for a $−1σ$ gap, which lies closest to midchord where the ejected gap flow gets entrained in the recirculation bubble. It has been shown in Sec. 3 that due to the large axial momentum of the entrained flow and the reduced ingestion into the inter-platform gap, the step-parallel component of wall shear stress increases over the expected value for an equivalent Q2D forward-facing step. Nevertheless, the prediction error remains below 30%, which emphasizes the robustness of the correlation to deviations from the Q2D flow.

In this section, the shroud sentencing correlation developed using a numerical study on the Q2D step model is tuned using experimental results. This is done to mitigate the limitations of using steady RANS CFD for the physics associated with canonical backward- and forward-facing step flow, which are inherently unsteady flows due to the vortex shedding at the step edge [29,30]. Moreover, vortices comparable in size to the step height occur in the free shear layer formed between the freestream and the recirculation zone at the step edge, resulting in anisotropic turbulence, which is incompatible with the Boussinesq approximation [29]. However, the 2D backward-facing step has been used to tune turbulence models to improve their performance on turbulent separation [29,31,32]. Menter [33] showed that this has led to improved predictions of both reattachment length and shear stress downstream of the step using the $k−ω$ model, which is the near-wall model of the $k−ω$ SST model used in this study.

First, correction factors for the RANS prediction of the reattachment Nu are obtained by comparing the simulations results on the 2D step heat transfer rig model with both backward- and forward-facing steps to the experimental data. Afterwards, the dependencies in the step heat transfer correlation obtained using the Q2D shroud step model are tuned using experimental data.

The comparison of the $NuH$ profile along the middle of a backward-facing step between CFD and experiment is shown in Fig. 18(a), both as a function of $Rex$ and $X/H$⁠. The zero pressure gradient (ZPG) $Nux$ correlations taken from the review paper by Lienhard [34] are plotted to provide a reference, where the turbulent correlation is moved to the location of the boundary layer trip in the experiment. The $NuH$ profiles upstream of the step edge lie within 5% of each other and the ZPG turbulent correlation, which is within the experimental accuracy. The slope of the $NuH$ profiles match in the turbulent region upstream of the step, showing that the boundary layer growth is matched.

In order to compare the difference in $NuH$ at the reattachment point, the results are displayed as a function of $X/H$ and are both aligned and normalized using the $NuH$ value $20H$ upstream of the step. The $NuH$ at the reattachment point is underpredicted by only 3%, whereas the reattachment length is overpredicted by $2H$⁠. This shows that benchmarking turbulence models on a backward-facing step has led to predictions within 5% despite the occurring flow physics not being modeled correctly.

Figure 18(b) shows the comparison between the experimental and numerical $NuH$ profile for a forward-facing step. Again, the $NuH$ decay upstream of the step matches and both profiles lie within 10% of the turbulent ZPG correlation. The $NuH$ decay is steeper in this case since the upstream near-wall flow encounters an adverse pressure gradient created by the static pressure maximum at the step face. Immediately upstream of the step, the numerical $NuH$ prediction drops because the CFD does not capture the unsteady growing and shrinking of the upstream recirculation zone [35] but models it as a steady recirculation zone.

When the normalized $NuH$ are displayed as a function of $X/H$⁠, it can be observed that the experimentally measured reattachment point lies approximately $3H$ downstream of the step edge, which is in agreement with the literature [30,36]. The CFD, on the other hand, predicts a reattachment length of $7.2H$⁠. The corresponding reattachment $NuH$ value is underpredicted by 37% in the CFD. Therefore, a correction factor of 1.37 is used on forward-facing step cases in the 3D simulation matrix in Sec. 3.

The dependencies of the shroud step correlation developed using the Q2D simulations are tuned by comparing it to experimental reattachment $NuH$ values. In order to remove the effect of the non-engine representative $H/S$ in the rig, the correlation output for the reference case is scaled to match the corresponding experimental value. In order to further facilitate comparing the experimental data, the $NuH$ profiles are aligned $5H$ upstream of the step, where the boundary layer, and thus $NuH$⁠, is expected to be identical for all cases. Only the $NuH$ profiles for different $ReH$ are not scaled since the upstream boundary layer is affected by the change in flow conditions.

The measured $NuH$ profiles and the tuning of the correlation for all variables are shown in Fig. 19 for a backward-facing step. The experimental uncertainty of 5% is indicated by the vertical error bars, and the uncertainty of the measured leakage flowrate is indicated by a horizontal error bar where applicable. The Nu profiles for different $α$ in Fig. 19(a) show that the reattachment length is constant, which is in agreement with the study by Kaltenbach [27] and confirm that RANS simulations erroneously predict a shortening of the reattachment length in the considered range of flow angle. In the case of leakage flowrate, Fig. 19(a) shows that a leakage flow ratio of 9% reduces the reattachment length by $0.75H$⁠, which confirms the trend observed in both the 3D shroud endwall and the Q2D database simulations. The corresponding heat transfer increase is equal to 15.6%, which is underpredicted by the correlation and the dependency is tuned as shown by the black line in Fig. 19(b). For all other variables, including the insensitivity to step edge radius, the experimental data lie close to the trends in the correlation and no tuning is required.

The $NuH$ profiles and the tuning of the shroud step correlation are shown for all parameters in Fig. 20 for a forward-facing step. Figure 20(a) shows that for varying flow angle, the reattachment length only shortens when $α$ reduces from $60$ deg to $45$ deg. This behavior is not captured by RANS simulations, which predict a continuously reducing reattachment length. For the varying leakage flow ratio, the reattachment length reduces with $0.83H$ for a leakage flow ratio of 23.2%, which is similar to the decrease observed for backward-facing steps for a leakage flow ratio of 9.0%. This confirms the reduced dependency of the reattachment length on leakage flow rate for forward-facing steps observed in Sec. 3. Finally, the reducing reattachment length with increasing step edge radius can be observed and is in agreement with the Q2D CFD results and the available literature [25,26].

Figure 20(b) shows that the unmatched experimental $ReH$ data point exceeds the value predicted by the correlation. Therefore, the dependency on $ReH$ is adjusted as shown by the black line. The accuracy of this correction decreases for $ReH$ values outside of the two measured data points. However, since the $ReH$ for HPTB shroud platform steps representative of manufacturing variability is unlikely to exceed 15,000, this mainly risks overpredicting the reattachment $NuH$ value when $ReH<7600$⁠.

In the case of varying $α$⁠, the experimental reattachment $NuH$ closely matches the correlation from $90$ deg to $60$ deg followed by a deviation as the reattachment $NuH$ remains constant between $60$ deg and $45$ deg. Since this change in trend is unexpected, it is likely that the actual $NuH$ for a flow angle of $45$ deg lies toward the lower end of the error bar and the deviation from the correlation is small.

Finally, Fig. 20(b) shows that the Q2D CFD underpredicts the dependency of the reattachment $NuH$ on leakage flow rate and does not predict the increased reattachment $NuH$ with increasing step edge radius. Therefore, the correlation is tuned as shown by the black lines. Note that since platform edges tend to round off due to wear, the dependency on step edge radius suggests that the heat transfer enhancement caused by the step increases with time in service.

This study showed that a correlation based on a parametrized Q2D model of a shroud endwall step is able to predict the heat transfer enhancement caused by aftchord shroud platform steps with an accuracy of 20% for the majority of cases. This follows from the fact that the shroud platform flow is Q2D in the aftchord region, both with and without platform steps, as has been shown by a steady RANS study of engine-representative shroud platform manufacturing variations in a three passage HPT rotor model. The shroud step heat transfer correlation has been tuned using experimental data such that it is ready to triage manufactured HPTB on the production line.

The authors gratefully acknowledge InnovateUK via the Aerospace Technologies Institute for the numerical and experimental resources. The authors gratefully acknowledge Rolls-Royce for permission to publish. The first author was funded by a joint Rolls-Royce and EPSRC Ph.D. studentship.

There are no conflicts of interest.

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

The input variable extraction for the four planes on an HPTB endwall discussed in the article is illustrated in Fig. 21. The data extraction plane is parallel to the wedge face but is moved $1.25%H$ upstream in the direction perpendicular to the step. This is done in order to avoid the local acceleration caused by the leakage flow influencing the measured boundary layer and flow angle profile. The four wall-normal lines in Fig. 21 show where the four streamwise test planes intersect the data extraction plane. On each of these slices, the wall-normal profile of flow angle relative to the step, velocity, density, and dynamic viscosity is extracted.

Unfortunately, the leakage flow rate cannot be extracted from the aligned reference case since the driving pressure difference is dependent on both the local step and the inter-platform gap width. Therefore, the leakage flowrate is extracted using a local mass flow monitor positioned in the gap for every streamwise slice in the respective shroud platform step simulation. Since the leakage flow needs to be calculated assuming a unit width, the leakage flow is normalized by the length of the monitor in the direction parallel to the step.

$p$ =

pressure (Pa)

$r$ =

radius (m)

$w$ =

gap width (m)

$C$ =

chord (m)

$H$ =

step height (m)

$S$ =

span (m)

$T$ =

temperature (K)

$V$ =

velocity (⁠$ms−1$⁠)

$X$ =

streamwise coordinate (m)

$m˙H$ =

$=ρ∞V∞H$⁠, step blocked mass flow (⁠$kg/s$⁠)

$m˙L$ =

inter-platform gap leakage mass flow (⁠$kg/s$⁠)

$Cx$ =

axial chord (m)

$y+$ =

dimensionless wall distance

Tu =

turbulence intensity (%)

$α$ =

flow angle (deg)

$ϵ$ =

error

$θ$ =

momentum thickness (m)

$μ$ =

mean

$ρ$ =

density (⁠$kg/m3$⁠)

$σ$ =

standard deviation

$τ$ =

shear stress (⁠$Nm/2$⁠)

$Cp$ =

pressure coefficient

$Cf$ =

friction coefficient

 Nu =

Nusselt number

Re =

Reynolds number

aw =

adiabatic wall

 corr =

correlation value

in =

inlet value

is =

isentropic value

ref =

reference value

rel =

relative value

w =

wall

$∞$ =

freestream value

// =

step-parallel value

BWD =

backward-facing

CFD =

computational fluid dynamics

DNS =

direct numerical simulation

FWD =

forward-facing

HPTB =

high-pressure turbine blade

HTC =

heat transfer coefficient

IR =

infrared

LE =

leading edge

LES =

large eddy Simulation

PDF =

probability density function

PS =

pressure side

Q2D =

quasi-2D

RANS =

Reynolds-averaged Navier–Stokes

RMSN =

root mean square normalized

SS =

suction side

SST =

shear stress transport

STL =

StereoLithography

TE =

trailing edge

ZPG =

zero pressure gradient