Abstract

Unsteadiness, in the form of both broadband background disturbances and discrete coherent wakes, can have a strong effect on the performance of turbomachinery blades. The influence of the incoming flow has received much interest as it inevitably affects the blade boundary layers and develops as it passes through the machine. In the present work, we investigate the effect of unsteady flow on high-pressure turbines (HPTs), using high-fidelity datasets produced by wall resolved large-eddy simulation of an HPT stage. The effects of incident wakes from an upstream stator, compounded by the presence of freestream turbulence, on the downstream rotor are investigated. Based on analyzing cases with different turbulence intensities and length scales prescribed at the inlet, we show that changing the freestream turbulence characteristics has a direct effect on the unsteady behavior of the stator wakes. As a result, the performance of the rotor is also significantly affected. By detailing the influence of the wake–turbulence interaction, we aim to distinguish driving forces on rotor performance, be it changes in the incident wakes or direct influence from the freestream turbulence. Furthermore, the aerothermal behaviors of the rotor blades have been extensively investigated, showing that the blade boundary layers on the suction and pressure sides respond differently to external disturbances. The insights gained can provide designers with guidelines in understanding the unsteady flow effects of a given flow state, and how the unsteadiness present, either broadband or deterministic, will affect the performance of downstream blades.

1 Introduction

Unsteady effects, in turbine stator–rotor interactions, have been extensively studied due to their important role in advanced aero-design. Previous studies have focused mainly on two aspects regarding the adjacent blade-rows interaction. First, on the influence of downstream blockage on the upstream blade-row, second on the effects of incoming wakes on the downstream blades [1]. The incoming wake interaction on downstream blades has been investigated via different methods. Wu and Durbin [2] applied direct numerical simulation (DNS) to study the T106A low-pressure turbine (LPT) profile with upstream moving cylindrical bars, shedding light on the wake-induced transition. The complex transition mechanism was further investigated by Coull and Hodson [3], who came to the conclusion that boundary layer transition is triggered in different ways that alter throughout the wake-cycle. Particle image velocimetry was also used in experiments to investigate the wake distortion process in LPT passages, where the effects of negative jetting by the wake on the suction-side transition were highlighted [4,5]. In several studies [1,69], the distortion of incoming wakes by the pressure gradient in the passage was identified as one key mechanism responsible for loss amplification.

Previous studies on unsteady effects are mostly limited to LPT stator–rotor interactions, as the flow conditions of high-pressure turbines (HPTs) can be extremely challenging. HPT usually operate at much higher Reynolds and Mach numbers compared to LPT, which makes both experiments and scale-resolving numerical simulations such as DNS and large-eddy simulations (LES) exceedingly costly. In addition, HPT typically experiences high levels of incoming unsteadiness and turbulence, as it is directly affected by the highly unsteady exit flow of upstream combustion chambers, requiring much larger computational domains. The high Reynolds and Mach numbers and the strong freestream turbulence (FST), as well as other factors, can significantly influence the stator–rotor interaction in HPT and challenge computational methods. However, continuing improvements in high performance computing are opening up new possibilities to study turbomachinery flows using high-fidelity methods, and many new insights into the flow dynamics associated with such flows are discovered via simulations [10].

Considering the unsteady effects on stage performance can be affected by both broadband freestream turbulence and discrete deterministic wakes, and common computational fluid dynamics tools based on Reynolds-averaged Navier–Stokes (RANS) and unsteady RANS are insufficient as they fail to correctly represent the complex interactions. Despite the significant computational efforts required, high-fidelity simulations including DNS and LES are becoming more and more feasible in gas turbine studies [11]. Such simulations allow detailed analysis of a range of physical phenomena impacting the performance of a design to be studied. In a recent study [12], leveraging cutting-edge computational techniques and resources, and high-fidelity simulations of a HPT stage operating at engine-relevant conditions were performed, focusing on loss mechanisms using an entropy-based analysis.

In the present study, a detailed analysis is performed on the effects of unsteady rotor–stator interactions in HPT operating at engine-relevant conditions. The study utilizes the data produced using high-fidelity simulations of a HPT stage subject to different inlet turbulence conditions. The outline of the present study is summarized as follows. The configurations and the parameters of the HPT stages are introduced in the case configuration section. Thereafter, the highly resolved flow fields are analyzed to study the effects of varying inlet turbulence on the rotor–stator interactions. These are investigated using time-averaged, instantaneous, phase-lock averaged and statistical quantities to highlight variations in the aerothermal behavior of the HPT.

2 Cases Configuration

The configuration of the present linear HPT stage, shown in Fig. 1(a), consists of a VKI LS-89 stator vane [13] and two rotor blades from the open literature [14]. A brief outline of the numerical setup and simulation setup is presented here but for full details regarding the HPT stage cases the reader is pointed to the original study [12]. The computational domain is setup with periodic boundary conditions in the pitch-wise and spanwise directions, with a fixed inflow super-imposed with freestream turbulence, and with a non-reflective exit boundary condition. To achieve relative pitch-wise motion between rotor and stator, a sliding interface is used [15], while an overset mesh interpolation method [16] is employed to reduce mesh complexity and ensure accurate resolution of the blade boundary layers. The grid spacings on the blade surface in the tangential, wall-normal, and spanwise directions can be normalized with the local viscous length scale and denoted as Δs+, Δn+, and Δz+, respectively. The corresponding near-wall grid resolution is indicative of highly resolved LES, given the values 〈Δn+〉 ≈ 1.5, 〈Δs+〉 ≈ 25, and 〈Δz+〉 ≈ 10.

Fig. 1
(a) Configuration of the HPT stage cases. Every 15 points of the mesh shown for clarity, (b) convective wall heat transfer coefficient around the stator boundary layer in the Re = 1 × 106 cases (B0, B1, and B2), validated against the VKI experiment [13], (c) Nusselt number distribution around the rotor boundary layer in the Re = 1 × 106 cases (B0, B1, and B2), validated against Ref. [14]. Pressure-side: surface distance s < 0; suction-side s > 0 [12].
Fig. 1
(a) Configuration of the HPT stage cases. Every 15 points of the mesh shown for clarity, (b) convective wall heat transfer coefficient around the stator boundary layer in the Re = 1 × 106 cases (B0, B1, and B2), validated against the VKI experiment [13], (c) Nusselt number distribution around the rotor boundary layer in the Re = 1 × 106 cases (B0, B1, and B2), validated against Ref. [14]. Pressure-side: surface distance s < 0; suction-side s > 0 [12].
Close modal

The simulations are performed using the in-house code HiPSTAR developed for compressible flow simulations and optimized for turbomachinery flows. The code allows for both DNS or LES using the wall-adaptive local eddy viscosity model [17] to close the filtered compressible Navier–Stokes equations. For LES, no additional modeling is employed for near-wall turbulence, which requires that the LES meshes resolve near-wall flow structures. The code applies a fourth-order finite difference scheme for spatial discretization and a ultra-low storage frequency optimized fourth-order explicit Runge-Kutta method [18] for time integration, ensuring high-order numerical accuracy for the solved flow fields. HiPSTAR has previously been used to perform in-depth studies on both turbines and compressors, such as low pressure turbines and high-pressure turbines [1921], including rotor–stator setups [1] and axial compressors [22,23]. Nevertheless, the present simulations of HPT stages are extremely challenging due to the complex configuration, including the overset setup and the sliding interface method, and the large number of total grid points (up to 8.2 billion) required by the engine-relevant Reynolds numbers. GPU-acceleration, along with other efficiency-optimization methods, have been exploited to make the current large-scale simulations possible, more details of the code can be found in Ref. [19].

A series of cases have been performed, using the stage configuration shown in Fig. 1(a), at Reynolds number Re = 1.1 × 106, covering a wide range of inlet turbulent states including turbulence intensities of Tu = 8–20% and integral length scales of LS = 8–20%Cax. Here, Cax represents the stator blade axial chord. Moreover,
Tu=u2+v2+w23/Ui
where Ui is the inlet time-averaged velocity, and u′, v′, and w′ are fluctuating velocities in the stream-wise, pitch-wise, and spanwise directions, respectively. It should be noted that the spanwise extent varies based on the different inlet boundary conditions, which is due to the requirement to resolve the evolution of large-scale turbulence introduced at the inlet. The parameters of the cases are presented in Table 1. The synthetic digital filtering method [24] is applied in the turbulence cases B1 and B2, while one case with no inlet turbulence, case B0, is also included for comparison. The fluid is modeled as a perfect gas with properties of cold air and the ratio of specific heats is equal to 1.4.
Table 1

Case setup parameters

CasesReTuLS/CaxLz/CaxPoints
B01.1 × 1060N.A.0.11.0 × 109
B11.1 × 1068% 8%0.44.1 × 109
B21.1 × 10620%20%0.88.2 × 109
CasesReTuLS/CaxLz/CaxPoints
B01.1 × 1060N.A.0.11.0 × 109
B11.1 × 1068% 8%0.44.1 × 109
B21.1 × 10620%20%0.88.2 × 109
The HPT stage cases listed in Table 1 have been carefully validated in the original paper [12]. As briefly shown in Figs. 1(b) and 1(c), the distributions of the convective wall heat transfer coefficient, given as
H=μcpPr(TtiTw)Tn
and the Nusselt number as
Nu=CaxTtiTwTn
from the three cases are plotted around the stator and rotor blades, respectively. The in-house solver HiPSTAR used for the present simulations is a non-dimensional code, and thus the quantities discussed in the present paper, unless otherwise stated, are inherently non-dimensional. The reference length scale is selected as Cax, and the reference velocity is selected as Ui. However, the convective wall heat transfer coefficient H is in dimensional form in order to be directly compared to the experimental data [13]. Comparing to the previous results in open literature [13,14], we can see that the high-fidelity simulations show good agreement for both the stator and rotor blades. For detailed characterization and validation of the cases, the reader is referred to the original study [12].

3 Analysis of Unsteady Effects

The analysis is broken down into different sections. First, the effects of changes in freestream turbulence on the stator vane wake are studied. In particular we focus on the changes in the wake structure, turbulent content, and strength as is important to understand their influence on the rotor flow. Thereafter, the rotor boundary layer properties are detailed based on instantaneous flow fields, phase-lock averaged data, and statistical variations of shear and thermal properties.

3.1 Characterization of Stator Wake.

Time-averaged profiles of the stator wake for the different inflow turbulence levels are presented in Fig. 2. Figure 2(a) shows the kinetic wake loss profiles given as
Ω(y)=PtiPt(y)PtiPo
where Pti, Po, and Pt(y) denote the inlet total pressure, outlet static pressure, and local total pressure at the stator exit plane, respectively. Figure 2(b) shows the turbulent kinetic energy (TKE). It is clear that the changes in inflow turbulence have minimal effect on the kinetic wake loss, with differences limited to the suction-side boundary layer (near y = 0 in Fig. 2(a)), due to changes to the stator suction-side transition. Deviations in other parts are negligible. Comparing the TKE plots for cases B1 and B2, Fig. 2(b), the wake is more energetic with stronger TKE in the peak region of case B2 in addition to the obviously stronger FST in the passage region. The wake of case B2 is also shown to be marginally more diffusive with a wider wake width.
Fig. 2
Time-averaged stator wake statistics from cases B1 and B2 at plane-cut of x = x2 as indicated in Fig. 1 (10% downstream of the stator trailing edge): (a) kinetic wake loss Ω and (b) TKE profiles. Negative y values from the wake center are the pressure-side boundary layer of the vane, while positive y values from the wake center are the suction-side of the vane.
Fig. 2
Time-averaged stator wake statistics from cases B1 and B2 at plane-cut of x = x2 as indicated in Fig. 1 (10% downstream of the stator trailing edge): (a) kinetic wake loss Ω and (b) TKE profiles. Negative y values from the wake center are the pressure-side boundary layer of the vane, while positive y values from the wake center are the suction-side of the vane.
Close modal

In Fig. 3, histograms of total temperature (Tt) and total pressure (Pt) for the cases B1 and B2 are presented. The y-axis in the plots is the pitch-wise location at the plane-cut of x = x2 as indicated in Fig. 1 (10% downstream of the stator trailing edge), while the contour shows the histograms of Pt or Tt at each pitch-wise location, based on 2400 instantaneous snapshots collected across three flow-through times. Here, the flow-through time is defined as Cax/Ui. The histograms are also sampled across the span, and 200 bins are used in the analysis. The sub-figures (a) and (c) show the Pt and Tt, respectively, for case B1, and the histograms of Tt are shown in log scale. We can see that the distribution of Pt is more concentrated on the pressure-side, especially in the area close to the wake region (∼y = −0.5). This is presumably because the stator boundary layer stays laminar on the pressure-side while it exhibits laminar-turbulent transition on the suction-side, which is indicated in Fig. 1(b) and also has been extensively discussed in Ref. [20]. This leads to the vortical structures caused by vortex shedding near the blade trailing edge being more coherent on the wake pressure-side, as opposed to the less coherent and more diffusive part of the wake resulting from the suction-side turbulent boundary layer. The wake, being inherently more turbulent than the passage region, shows a very broad distribution, and the Pt and Tt values inside the wake region are for the most part lower, obviously due to turbulent mixing loss, apart from a region of higher values of Tt, discussed below.

Fig. 3
Histograms of stator wakes at the plane-cut of x = x2 as indicated in Fig. 1 (10% downstream of the stator trailing edge): (a) Pt of case B1, (b) Pt of case B2, (c) Tt of case B1, histograms in log scale, and (d) Tt of case B2, histograms in log scale. The dashed lines indicate the Pt and Tt values at the inlet of the HPT stage.
Fig. 3
Histograms of stator wakes at the plane-cut of x = x2 as indicated in Fig. 1 (10% downstream of the stator trailing edge): (a) Pt of case B1, (b) Pt of case B2, (c) Tt of case B1, histograms in log scale, and (d) Tt of case B2, histograms in log scale. The dashed lines indicate the Pt and Tt values at the inlet of the HPT stage.
Close modal

In Figs. 3(b) and 3(d) for case B2 with stronger inlet turbulence, the distribution of Pt and Tt is far less concentrated compared to the B1 case. This is because the FST has higher intensity and larger length scale, thus has stronger diffusive effects in case B2. It is noted that the FST tends to diffuse at similar rates independent of the pressure gradient across the passage, which is shown by the nearly constant TKE distribution outside the wake region shown in Fig. 2(b). The TKE remains stronger as shown by the broader more diffused distribution, which again causes the wake to show a broad histogram distribution, marginally more diffused than in case B1, as expected from the wider wake width shown in Fig. 2(b).

The stator wake distribution also highlights another process, the scattered events that result in the separated Tt distribution known as energy separation. As indicated by the dashed circle in Fig. 3(c), the phenomenon is captured by revealing the significantly higher local Tt than the inlet total temperature Ttin. This phenomenon mainly occurs in the area close to the edge of the blade wake, while the Tt near the wake centerline is much lower. The energy separation here is mainly due to unsteady effects of the strong coherent structures in the wake, i.e., the vortex movement separates the instantaneous total temperature into hot and cold spots located around the coherent vortices in the near wake [25]. Once time-averaged, however, the total temperature distribution conceals the presence of hot spots, so that the energy separation cannot be captured by RANS calculations.

3.2 Wake Interaction With Rotor.

In order to quantitatively investigate the interactions of the stator wakes with the rotor boundary layers, we divide the sliding period Tp into 20 consecutive phases, namely, phases P0–P19. Performing a phase-lock averaged analysis allows us to study the mean dynamics of the rotor–stator interactions. It is noted that the phase-lock averaged data presented here are collected over 12 sliding periods. Moreover, the two rotor blades are identical except for a fixed phase difference, so that the phase-lock data can be further averaged across the span as well as over the two blades. As a result, the phase-lock averaged data in the present analysis are sufficiently converged, though more data might be needed if higher order statistics were to be investigated.

Phase-lock averaged data of wakes passing through the rotor passage are shown by the TKE contour plots in Fig. 4. The wake first approaches the rotor leading edge in Fig. 4(a), followed by the interaction between the wake and the leading edge seen in plot (b). Thereafter, the wake passes through the passage in plots (c)–(e), continuously deforming under the mean shear, with the wake distortion highest on the suction-side of the passage. On the pressure side, the wake remains more intact and passes more slowly. Finally in plot (f) the wake exits the passage. It is noted that due to the deformation and jetting, the wake has a stronger effect on the blade suction-side, shown in Figs. 4(c) and 4(d) by the higher concentration of TKE.

Fig. 4
Evolution of the stator wake through the rotor passage presented by the contour of turbulent kinetic energy in case B1, showing phases: (a) P0, (b) P3, (c) P6, (d) P10, (e) P13, and (f) P17, respectively.
Fig. 4
Evolution of the stator wake through the rotor passage presented by the contour of turbulent kinetic energy in case B1, showing phases: (a) P0, (b) P3, (c) P6, (d) P10, (e) P13, and (f) P17, respectively.
Close modal
Presented in Fig. 5 is the temporal–spatial distribution of the spanwise-averaged wall shear stress variation Δτw. It clearly shows a periodic pattern on the rotor blade, and the periodicity is
Tp=Ly/US
with Ly denoting the pitch-wise size of the computational domain and US the sliding speed of the rotor blades. The wall shear stress is calculated as
τw=μutn
where ut denotes the velocity in the blade wall tangential direction and n represents the local wall-normal direction. Finally, Δτw denotes the deviation between the time-averaged τw and the spanwise-averaged instantaneous wall shear stress. During one period, we observe a high-shear region convecting downstream along the suction surface as indicated by the dashed arrow in Fig. 5(b). This development of the high-shear region on the blade suction-side, therefore, can be directly associated with the stator wakes traveling through the rotor passages. Compared to the suction-side boundary layer, even though the Δτw on the pressure-side shows a similar periodic pattern, the amplitude of the variation is much reduced. This again suggests that the effects of the stator wake are not as significant on the pressure-side. Furthermore, we can see from Figs. 5(b) and 5(d) that the amplitudes of the high-shear streaks caused by the stator wake are weaker in case B1 compared to B2. This is because the stator wake in case B2 is more diffusive due to the stronger freestream turbulence, as also suggested by Figs. 2(b) and 3.
Fig. 5
Temporal–spatial distribution of the spanwise-averaged instantaneous wall shear stress variation Δτw around the rotor blade: (a) pressure-side in case B1, (b) suction-side in case B1, (c) pressure-side in case B2, and (d) suction-side in case B2. The x-axis denotes the surface length s normalized by Cax increasing along the blade, starting from s = 0 at the leading edge. The y-axis represents the time normalized by the sliding period t/Tp.
Fig. 5
Temporal–spatial distribution of the spanwise-averaged instantaneous wall shear stress variation Δτw around the rotor blade: (a) pressure-side in case B1, (b) suction-side in case B1, (c) pressure-side in case B2, and (d) suction-side in case B2. The x-axis denotes the surface length s normalized by Cax increasing along the blade, starting from s = 0 at the leading edge. The y-axis represents the time normalized by the sliding period t/Tp.
Close modal

In addition to the stator wake, one other important phenomenon that can be observed is streak-like features traveling upstream as indicated by the dotted arrows in Fig. 5(b). These streaks, with amplitudes much smaller than those caused by downstream traveling wake structures, are presumably due to pressure waves traveling in the passage, similar to those observed in compressors [26]. The waves passing through the passage can be further visualized in Fig. 6. Here, case B0 with no inlet turbulence is used, in order to highlight the pressure waves without contamination from the background freestream turbulence. On the one hand, the pressure waves caused by the vortex shedding at the stator trailing edge can be seen to interact with the rotor blades. On the other hand, the pressure distribution inside the passage varies as the rotor blades move, and the positions of the shocks on the rotor are altered accordingly. These waves, as implied in Fig. 5, seem to induce extra fluctuations in the blade boundary layer.

Fig. 6
Instantaneous contours of the density gradient magnitude on a spanwise plane-cut in case B0, with subfigures (a) and (b) showing different instantaneous snapshots to highlight the variation of the shock positions on the rotor blades
Fig. 6
Instantaneous contours of the density gradient magnitude on a spanwise plane-cut in case B0, with subfigures (a) and (b) showing different instantaneous snapshots to highlight the variation of the shock positions on the rotor blades
Close modal

To further investigate the interactions between the stator wake and the rotor blades, instantaneous contours in case B1 are presented in Fig. 7 for several time instants, showing the evolution of the flow field around the rotor blade. The contours on the blade, on the side plane-cut, and on the front plane-cut show wall shear stress, spanwise vorticity, and total pressure, respectively. Along with the pitch-wise motion of the stator wake relative to the rotor blades, as visible on the front plane-cut, we can observe the deformation and convection of the wake inside the passage on the side plane-cut. The rotor blades are thus affected by the wake, and the evolution of the boundary layer can be observed by contours of the instantaneous wall shear stress on the suction side. We can see that the wakes clearly disturb the boundary layer and cause high-shear regions, as shown in Figs. 7(a) and 7(b) near the suction peak. After the passing of the wake, the suction peak is becalmed as seen in Figs. 7(c)7(e). Passed the suction peak location, laminar-turbulent transition is observed across all phases near the shock position, downstream of which the boundary layer is subject to an adverse pressure gradient for a period. The effects of the passing wake on the transition can be observed, showing that the onset location of transition changes slightly and the amplitude of wall shear is also altered. Nevertheless, the deviations across different phases are not significant, which is also shown by the phase limits in Fig. 8.

Fig. 7
Evolution of the instantaneous flow fields in case B1, showing phases: (a) P0, (b) P3, (c) P6, (d) P10, (e) P13, and (f) P17, respectively. The contours on the blade, on the side plane-cut, and on the front plane-cut show wall shear stress, spanwise vorticity, and total pressure, respectively. The black dashed line and the solid arrow indicate the approximate locations of the shock and suction peak, respectively.
Fig. 7
Evolution of the instantaneous flow fields in case B1, showing phases: (a) P0, (b) P3, (c) P6, (d) P10, (e) P13, and (f) P17, respectively. The contours on the blade, on the side plane-cut, and on the front plane-cut show wall shear stress, spanwise vorticity, and total pressure, respectively. The black dashed line and the solid arrow indicate the approximate locations of the shock and suction peak, respectively.
Close modal
Fig. 8
Time-averaged flow statistics around the rotor blade boundary layer in cases B1 and B2, with the corresponding shaded areas showing phase-lock averaged extremes: (a) Isentropic Mach number, (b) convective heat flux coefficient distribution, and (c) wall shear stress. The negative values of surface length s on the left denote pressure-side boundary layer, while the right-hand side represents the blade suction-side.
Fig. 8
Time-averaged flow statistics around the rotor blade boundary layer in cases B1 and B2, with the corresponding shaded areas showing phase-lock averaged extremes: (a) Isentropic Mach number, (b) convective heat flux coefficient distribution, and (c) wall shear stress. The negative values of surface length s on the left denote pressure-side boundary layer, while the right-hand side represents the blade suction-side.
Close modal

We remark here that the major difference between the current study and bypass transition in flat-plate boundary layers (see, e.g., Ref. [27]) is that the turbine blade boundary layer is significantly affected by the mean flow pressure gradient and surface curvature. Along with the passing wake, these factors cause complex transitional phenomena in turbomachinery flows [2832]. In the present study, fluctuations in the boundary layer triggered by freestream turbulence and wakes obviously have an impact, sometimes inducing scattered turbulent spots as can be seen in Fig. 7(d). However, the boundary layer stays laminar in the favorable pressure gradient region, and becalmed regions can be observed. These observations only persist until the boundary layer is subjected to the adverse pressure gradient downstream of the shock which leads to the onset of transition.

To quantify the interaction between the wake and the rotor blade, the phase-lock averaged variations around the rotor blade in cases B1 and B2 are presented in Fig. 8, with solid lines indicating time-averaged quantities while the shaded areas represent the limits of all phases based on phase-lock averaged data. The isentropic Mach number Mais, which is related to the pressure distribution around the blade Pw as follows
Mais=(2γ1((Pw/Pti)((γ1)/γ)1))1/2
is shown in Fig. 8(a), with Pti representing the inlet total pressure and γ = 1.4 denoting the ratio of specific heats. Overall, the isentropic Mach number loading for the two cases is almost identical for both the time-averaged and phase-lock limit averages. However, there is a small deviation in the phase limits near the pressure-side leading edge. Here, case B1 shows a larger variation in loading associated with the wake passing. This is a consequence of the stator wake in case B1 being less diffused than the stator wake in case B2, see Fig. 2. Additionally, there is a larger variation in the distribution in the FST length scales in the flow between the wakes for the two cases, as shown in Fig. 3. The difference in the incident passage flow has an effect on the wake calming and transition between wake passing. Given the large pressure gradient around the leading edge of the blade which has strong calming properties, the greater difference between wake and FST forcing of the boundary layer in the B1 case results in the larger variation in loading.

The convective heat transfer coefficient H for the B1 and B2 cases is shown in Fig. 8(b). The surface heat flux shows substantial variations among the different phases, highlighting the effects of the stator–rotor interactions. On the pressure-side, the time-averaged values are higher for the B2 case, while the suction-side shows closer agreement. The difference in the phase limits is more substantial along the pressure-side, with the B2 case seeing an increase in the phase peak heat flux coinciding with the higher time-averaged values. Furthermore, in case B2, the increase in the phase limits is greater and sustained further along the pressure surface. On the suction-side, the time-averaged values of H show close agreement between the two cases with different FST, even though the phase limits present noticeable differences. The difference of the phase limits on the suction-side here is presumably mainly due to the fact that the wake of case B2 is more turbulent. The pronounced differences observed on the pressure side highlight the substantial impact of varying inflow turbulence on the rotor heat flux.

The wall shear stress is shown for both cases B1 and B2 in Fig. 8(c). It is noted that the time-averaged values of τw suggest that the blade boundary layer remains laminar along the pressure-side while laminar-turbulent transition takes place on the suction-side. Moreover, we observe no noticeable phase variations on the pressure-side from s = −0.5 to s = −1.0. This suggests that even though the convective heat flux is significantly affected by the different levels of FST as discussed for Fig. 8(b), the velocity boundary layer on the pressure-side remains unaffected by the FST and the wake behaviors across different phases. On the suction side, the phase limits of τw for the two cases show quite close agreement, especially for the laminar region (s = 0–0.8). However, the differences between the phase limits between the two cases start to amplify from the throat region (suction peak) around s ≈ 0.8, where the adverse pressure gradient dominates and onset of laminar-turbulent transition occurs.

3.3 Rotor Boundary Layer Statistics.

In Fig. 9, histograms of the convective wall heat transfer coefficient H around the rotor blades for cases B1 and B2 are presented. The x-axis in the plots is the surface length s increasing along the blade, starting from s = 0 at the leading edge, while the contours show the level of the histograms of H at each s location with the y-axis showing the corresponding H value. The histograms are obtained based on 2400 instantaneous snapshots collected across three flow-through time units. The histograms are also sampled across the span, and 200 bins are used for analysis. It is noted that the x and y axes and the corresponding scales are identical to the plots in Fig. 8(b), enabling a direct comparison to the phase variations while the histograms here include statistics regarding instantaneous fluctuations.

Fig. 9
Histograms of the convective heat flux coefficient H around the rotor blade boundary layer: (a) suction-side in case B1, (b) suction-side in case B2, (c) pressure-side in case B1, and (d) pressure-side in case B2. The x-axis denotes the surface length s increasing along the blade, starting from s = 0 at the leading edge. The contour shows the histograms of H at the corresponding s.
Fig. 9
Histograms of the convective heat flux coefficient H around the rotor blade boundary layer: (a) suction-side in case B1, (b) suction-side in case B2, (c) pressure-side in case B1, and (d) pressure-side in case B2. The x-axis denotes the surface length s increasing along the blade, starting from s = 0 at the leading edge. The contour shows the histograms of H at the corresponding s.
Close modal

The suction-side boundary layers from cases B1 and B2 are presented in Figs. 9(a) and 9(b), which can be roughly divided into two different stages. The boundary layer flow is first affected by the favorable pressure gradient and strong acceleration in the range of s = 0–0.8 as is also clear from Fig. 8(a). In this region, the histograms show progressively more concentrated distributions of H which can be related to development of the laminar boundary layer. Thereafter, the onset of transition occurs and causes a sudden increase of H, resulting in the diffusive distributions in the turbulent boundary layer in the range of s > 0.8. Comparing the cases with different FST, we can see that the H distributions on the suction-side show close agreement, except that case B2 shows more diffusive distributions in the laminar region.

In Figs. 9(c) and 9(d), however, the flow is under favorable pressure gradient across the whole pressure-side boundary layer. Therefore, the H distributions on the pressure-side show a continuous concentration due to the flow acceleration, which reduces the amplitudes of fluctuations. Nevertheless, the stronger FST in case B2 obviously has an impact on the heat flux in the laminar pressure-side boundary layer, showing much more diffusive distributions and also elevated H levels as indicated by Fig. 8(b).

Furthermore, histograms of the wall shear stress τw around the rotor blades are presented for cases B1 and B2 in Fig. 10. We can see that the τw from these two cases show similar distributions around the rotor blade, including the laminar boundary layer of pressure-side and early suction-side and also the turbulent states on the late suction-side. In particular, the fluctuations of wall shear stress are quite small in the pressure-side boundary layer, and the distributions are relatively concentrated, which agree with the negligible phase variations shown in Fig. 8(c). Taking the H and τw distributions into consideration at the same time, it seems that the velocity boundary layer on the pressure-side remains nearly laminar, even for very significant levels of freestream forcing, while the heat flux is significantly affected by the FST.

Fig. 10
Histograms of the wall shear stress τw around the rotor blade boundary layer in the Re = 1M cases: (a) suction-side in case B1, (b) suction-side in case B2, (c) pressure-side in case B1, and (d) pressure-side in case B2. The x-axis denotes the surface length s increasing along the blade, starting from s = 0 at the leading edge. The contour shows the histograms of H at the corresponding s.
Fig. 10
Histograms of the wall shear stress τw around the rotor blade boundary layer in the Re = 1M cases: (a) suction-side in case B1, (b) suction-side in case B2, (c) pressure-side in case B1, and (d) pressure-side in case B2. The x-axis denotes the surface length s increasing along the blade, starting from s = 0 at the leading edge. The contour shows the histograms of H at the corresponding s.
Close modal

3.4 Boundary Layer Fluctuations.

The fluctuations in wall-normal heat flux and shear in the near-wall boundary layer are presented in Figs. 11 and 12, showing the histograms for both the suction and pressure surfaces at given locations. The histograms are obtained based on 3200 instantaneous snapshots collected across four flow-through time units. They are further sampled across the span, and 200 bins are used with limits set to the maximum and minimum values in the dataset. Here, the y-axis represents wall-normal distance to the blade surface, while the contour shows histograms at each wall-normal distance with the x-axis presenting the corresponding H and τw levels. Figure 11 shows the distribution of fluctuations in the suction surface boundary layer at s/Cax = 0.67 near transition. Sub-plots (a) and (b) show H for the B1 and B2 cases, for which it can be seen that the mean profiles shown by the dashed lines are in close agreement between the two cases, though the distribution of fluctuations in the histogram for the B2 case is larger, as suggested by Fig. 9. This location on the suction surface is just upstream of transition onset, and it is clear that the wake and higher FST levels of the B2 case are able to force the boundary layer more than in the B1 case, resulting in the broader distribution in H in the near-wall boundary layer and at the wall. The inflection in H is brought about by transition which affects the near wall flow.

Fig. 11
Wall-normal heat flux and shear stress on suction surface for cases B1 and B2 at s = 0.67. Plots (a and b) show heat flux and (c and d) shear stress. Black dashed line shows time-averaged data.
Fig. 11
Wall-normal heat flux and shear stress on suction surface for cases B1 and B2 at s = 0.67. Plots (a and b) show heat flux and (c and d) shear stress. Black dashed line shows time-averaged data.
Close modal
Fig. 12
Wall-normal heat flux and shear stress on pressure surface for cases B1 and B2 at s = −0.68. Plots (a and b) show heat flux and (c and d) shear stress. Black dashed line shows time-averaged data.
Fig. 12
Wall-normal heat flux and shear stress on pressure surface for cases B1 and B2 at s = −0.68. Plots (a and b) show heat flux and (c and d) shear stress. Black dashed line shows time-averaged data.
Close modal

The shear stress for the two cases is shown in Figs. 11(c) and 11(d), which display many of the same characteristics as the heat flux. Again the time-averaged profiles agree closely between the two cases with the B2 case showing a broader histogram distribution. The same driving forces affecting the heat flux, being the wake passing and higher FST levels, produce the same broader distribution in the shear stress of the B2 case.

The shear stress and heat flux distributions on the pressure-side surface of the rotor blades are shown in Fig. 12, taken at a surface location of s/Cax = −0.68. The heat flux distribution is presented in sub-plots (a) and (b) for the B1 and B2 cases, respectively. The time-averaged profiles show similar trends, although, as shown in Fig. 8, the time-averaged heat flux is higher in the B2 case. Furthermore, the B2 case shows a much broader distribution near the wall, suggesting the wake and FST interaction generate larger disturbances in the boundary layer. For the shear stress, on the other hand, the time-averaged profiles agree well in both trends and magnitude. The histograms of the distribution of shear stress fluctuations in the boundary layer also show similar trends between the two cases, although in the B2 case the distribution is slightly broader in the near-wall region. This coincides with the trends seen in the heat flux further supporting the idea that the higher FST intensity drives fluctuations in the boundary layer that affect the time-averaged heat flux but have limited effect on the time-averaged shear stress.

Comparing the pressure and suction surface distributions, it is suggested that the local pressure gradient has an effect on the heat flux response. In Figs. 11 and 12, the respective x-axis scales for heat flux coefficient and shear stress levels are identical. Therefore, we can easily see that the heat flux distributions in the pressure-side boundary layer are broader compared to the suction-side, while the shear stress distribution on the pressure-side is similar to that of the suction-side boundary layer. This implies that the heat flux fluctuations in the pressure-side boundary layer are more directly affected by the freestream turbulence. On the other hand, the FST effects on the velocity fluctuations inside the boundary layer show no significant difference between the suction and pressure sides. Nevertheless, the pressure-side boundary layer remains laminar, while at the s/Cax = 0.67 location on the suction-side it is close to the onset of transition so that the velocity fluctuations start to amplify.

4 Conclusions

A set of high-fidelity LES of a high-pressure turbine stage at a Reynolds number of 1.1 × 106 are investigated, focusing on the unsteady flow effects of freestream turbulence and incoming wakes on the aerothermal performance of the rotor blades. First, the wake profiles of the stator vane are investigated that constitute the inflow to the rotor blades. It is shown that increases in the freestream turbulence intensity and length scale result in an increase in stator wake TKE and a marginally wider wake profile. The statistical results also show this and results in a broader distribution of turbulent fluctuations in the passage.

Furthermore, the stator–rotor interaction effects under different inflows have been discussed in detail based on instantaneous flow fields, phase-lock averaged data, and statistical variations of the wall shear stress and heat flux coefficient. Summarizing the effects on mean performance of HPT blades, the larger and more intense inflow turbulent fluctuations cause higher levels of heat flux on the pressure-side as shown in Fig. 8, while the loading and wall shear stress are not significantly affected. For unsteady statistics, however, the more diffusive stator wake and stronger FST in the case with higher inlet turbulence intensity have a direct impact, mainly resulting in stronger variations inside the rotor boundary layer. This has been shown in different ways, including phase-lock averaged results and histograms of blade boundary layer statistics. Overall, the study suggests that in addition to changes in mean aerothermal performance, the increase in variability associated with inflow changes may affect other performance metrics such as service life of HPT and is likely a valuable consideration during design.

Acknowledgment

Yaomin Zhao was supported by the National Natural Science Foundation of China (Grant No. 92152102). This research used data generated on the Oak Ridge Leadership Computing Facility, which is a DOE Office of Science User Facility supported under Contract DE-AC05-00OR22725. This work was also supported by a grant from the Swiss National Supercomputing Centre (CSCS) under project ID s977.

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.

Nomenclature

s =

surface distance

x =

axial coordinate, normalized by axial chord

y =

pitch-wise coordinate, normalized by axial chord

H =

convective wall heat transfer

R =

gas constant

Cax =

axial chord length

Lz =

spanwise domain size

LS =

turbulence integral length scale

Pw =

blade surface pressure

Tw =

blade surface wall temperature

Po =

outlet pressure

Ui =

inlet time-averaged velocity

Pti =

inlet total pressure

Tti =

inlet total temperature

Ma =

Mach number

Mais =

isentropic Mach number

Pr =

Prandtl number

Re =

Reynolds number

Tu =

turbulence intensity

μ =

viscosity

References

1.
Pichler
,
R.
,
Michelassi
,
V.
,
Sandberg
,
R.
, and
Ong
,
J.
,
2018
, “
Highly Resolved Large Eddy Simulation Study of Gap Size Effect on Low-Pressure Turbine Stage
,”
ASME J. Turbomach.
,
140
(
2
), p.
021003
.
2.
Wu
,
X.
, and
Durbin
,
P.
,
2001
, “
Evidence of Longitudinal Vortices Evolved From Distorted Wakes in a Turbine Passage
,”
J. Fluid Mech.
,
446
, pp.
199
228
.
3.
Coull
,
J. D.
, and
Hodson
,
H. P.
,
2011
, “
Unsteady Boundary-Layer Transition in Low-Pressure Turbines
,”
J. Fluid Mech.
,
681
, pp.
370
410
.
4.
Stieger
,
R. D.
, and
Hodson
,
H. P.
,
2003
, “
The Transition Mechanism of Highly-Loaded LP Turbine Blades
,”
ASME Turbo Expo
, Paper No. GT2003-38304.
5.
Stieger
,
R. D.
, and
Hodson
,
H. P.
,
2005
, “
The Unsteady Development of a Turbulent Wake Through a Downstream Low-Pressure Turbine Blade Passage
,”
ASME J. Turbomach.
,
127
(
2
), pp.
388
394
.
6.
Michelassi
,
V.
,
Wissink
,
J. G.
, and
Rodi
,
W.
,
2003
, “
Direct Numerical Simulation, Large Eddy Simulation and Unsteady Reynolds-Averaged Navier—Stokes Simulations of Periodic Unsteady Flow in a Low-Pressure Turbine Cascade: A Comparison
,”
Proc. Inst. Mech. Eng. Part A
,
217
(
4
), pp.
403
411
.
7.
Praisner
,
T. J.
,
Clark
,
J. P.
,
Nash
,
T. C.
,
Rice
,
M. J.
, and
Grover
,
E. A.
,
2006
, “
Performance Impacts Due to Wake Mixing in Axial-Flow Turbomachinery
,”
ASME Turbo Expo
, Paper No. GT2006-90666.
8.
Michelassi
,
V.
,
Chen
,
L. W.
,
Pichler
,
R.
, and
Sandberg
,
R. D.
,
2015
, “
Compressible Direct Numerical Simulation of Low-Pressure Turbines Part II: Effect of Inflow Disturbances
,”
ASME J. Turbomach.
,
137
(
7
), pp.
71005
71012
.
9.
Michelassi
,
V.
,
Chen
,
L.
,
Pichler
,
R.
,
Sandberg
,
R. D.
, and
Bhaskaran
,
R.
,
2016
, “
High-Fidelity Simulations of Low-Pressure Turbines: Effect of Flow Coefficient and Reduced Frequency on Losses
,”
ASME J. Turbomach.
,
138
(
11
), p.
111006
.
10.
Sandberg
,
R. D.
, and
Michelassi
,
V.
,
2022
, “
Fluid Dynamics of Axial Turbomachinery: Blade- and Stage-Level Simulations and Models
,”
Annu. Rev. Fluid Mech.
,
54
, pp.
255
285
.
11.
Sandberg
,
R. D.
, and
Michelassi
,
V.
,
2019
, “
The Current State of High-Fidelity Simulations for Main Gas Path Turbomachinery Components and Their Industrial Impact
,”
Flow Turbul. Combust.
,
102
(
4
), pp.
797
848
.
12.
Zhao
,
Y.
, and
Sandberg
,
R. D.
,
2021
, “
High-Fidelity Simulations of a High-Pressure Turbine Stage: Effects of Reynolds Number and Inlet Turbulence
,”
ASME Turbo Expo
, Paper No. GT2021-58995.
13.
Arts
,
T.
,
Lambertderouvroit
,
M.
, and
Rutherford
,
A. W.
,
1990
, “Aero-Thermal Investigation of a Highly Loaded Transonic Linear Turbine Guide Vane Cascade,” Tech. Rep.,
von Karman Institute for Fluids Dynamics
,
Brussels
.
14.
Kopriva
,
J. E.
,
2017
, “
The Role of Free-Stream Turbulence on High Pressure Turbine Aero-thermal Stage Interaction
,” PhD thesis,
Northeastern University
,
Boston, MA
.
15.
Johnstone
,
R.
,
Chen
,
L.
, and
Sandberg
,
R. D.
,
2015
, “
A Sliding Characteristic Interface Condition for Direct Numerical Simulations
,”
Comput. Fluids
,
107
, pp.
165
177
.
16.
Deuse
,
M.
, and
Sandberg
,
R. D.
,
2020
, “
Implementation of a Stable High-Order Overset Grid Method for High-Fidelity Simulations
,”
Comput. Fluids
,
211
, p.
104449
.
17.
Nicoud
,
F.
, and
Ducros
,
F.
,
1999
, “
Subgrid-Scale Stress Modelling Based on the Square of the Velocity Gradient Tensor
,”
Flow Turbul. Combust.
,
62
(
3
), pp.
183
200
.
18.
Kennedy
,
C. A.
,
Carpenter
,
M. H.
, and
Lewis
,
R. M.
,
2000
, “
Low-Storage, Explicit Runge–Kutta Schemes for the Compressible Navier–Stokes Equations
,”
Appl. Numer. Math.
,
35
(
3
), pp.
177
219
.
19.
Sandberg
,
R. D.
,
Michelassi
,
V.
,
Pichler
,
R.
,
Chen
,
L.
, and
Johnstone
,
R.
,
2015
, “
Compressible Direct Numerical Simulation of Low-Pressure Turbines Part I: Methodology
,”
ASME J. Turbomach.
,
137
(
5
), p.
051011
.
20.
Zhao
,
Y.
, and
Sandberg
,
R. D.
,
2020
, “
Bypass Transition in Boundary Layers Subject to Strong Pressure Gradient and Curvature Effects
,”
J. Fluid Mech.
,
888
, p.
A4
.
21.
Zhao
,
Y.
, and
Sandberg
,
R. D.
,
2021
, “
High Fidelity Simulations of a High Pressure Turbine Vane Subject to Large Disturbances: Effect of Exit Mach Number on Losses
,”
ASME J. Turbomach.
,
143
(
9
), p.
091002
.
22.
Leggett
,
J.
,
Priebe
,
S.
,
Shabbir
,
A.
,
Michelassi
,
V.
,
Sandberg
,
R.
, and
Richardson
,
E. S.
,
2018
, “
Loss Prediction in an Axial Compressor Cascade at Off-Design Incidences With Free Stream Disturbances Using Large Eddy Simulation
,”
ASME J. Turbomach.
,
140
(
7
), p.
071005
.
23.
Leggett
,
J.
,
Richardson
,
E. S.
,
Priebe
,
S.
,
Shabbir
,
A.
,
Michelassi
,
V.
, and
Sandberg
,
R. D.
,
2020
, “
Loss Analysis of Unsteady Turbomachinery Flows Based on the Mechanical Work Potential
,”
ASME J. Turbomach.
,
142
(
11
), p.
111009
.
24.
Klein
,
M.
,
Sadiki
,
A.
, and
Janicka
,
J.
,
2003
, “
A Digital Filter Based Generation of Inflow Data for Spatially Developing Direct Numerical or Large Eddy Simulations
,”
J. Comput. Phys.
,
186
(
2
), pp.
652
665
.
25.
Kurosaka
,
M.
,
Gertz
,
J. B.
,
Graham
,
J. E.
,
Goodman
,
J. R.
,
Sundaram
,
P.
,
Riner
,
W. C.
,
Kuroda
,
H.
, and
Hankey
,
W. L.
,
1987
, “
Energy Separation in a Vortex Street
,”
J. Fluid Mech.
,
178
, pp.
1
29
.
26.
Leggett
,
J.
, and
Sandberg
,
R. D.
,
2020
, “
Exploring the Presence of Pressure Waves in Axial Compressor Cascades
,”
ASME Turbo Expo
, Paper No. GT2020-16022.
27.
Jacobs
,
R. G.
, and
Durbin
,
P. A.
,
2001
, “
Simulations of Bypass Transition
,”
J. Fluid Mech.
,
428
, pp.
185
212
.
28.
Halstead
,
D. E.
,
Wisler
,
D. C.
,
Okiishi
,
T. H.
,
Walker
,
G. J.
,
Hodson
,
H. P.
, and
Shin
,
H. W.
,
1997
, “
Boundary Layer Development in Axial Compressors and Turbines: Part 1 of 4-Composite Picture
,”
ASME J. Turbomach.
,
119
(
1
), pp.
114
127
.
29.
Halstead
,
D. E.
,
Wisler
,
D. C.
,
Okiishi
,
T. H.
,
Walker
,
G. J.
,
Hodson
,
H. P.
, and
Shin
,
H. W.
,
1997
, “
Boundary Layer Development in Axial Compressors and Turbines: Part 4 of 4-Computations and Analyses
,”
ASME J. Turbomach.
,
119
(
1
), pp.
128
139
.
30.
Durbin
,
P.
, and
Wu
,
X.
,
2007
, “
Transition Beneath Vortical Disturbances
,”
Annu. Rev. Fluid Mech.
,
39
, pp.
107
128
.
31.
Wu
,
X.
,
Moin
,
P.
, and
Hickey
,
J. P.
,
2014
, “
Boundary Layer Bypass Transition
,”
Phys. Fluids
,
26
, p.
91104
.
32.
Dupuy
,
D.
,
Gicquel
,
L.
,
Odier
,
N.
,
Duchaine
,
F.
, and
Arts
,
T.
,
2020
, “
Analysis of the Effect of Intermittency in a High-Pressure Turbine Blade
,”
Phys. Fluids
,
32
, p.
095101
.