Reynolds Sensitivity of the Wake Passing Effect on a LPT Cascade Using Spectral/ hp Element Methods

: Reynolds-Averaged Navier–Stokes (RANS) methods continue to be the backbone of CFD-based design; however, the recent development of high-order unstructured solvers and meshing algorithms, combined with the lowering cost of HPC infrastructures, has the potential to allow for the introduction of high-ﬁdelity simulations in the design loop, taking the role of a virtual wind tunnel. Extensive validation and veriﬁcation is required over a broad design space. This is challenging for a number of reasons, including the range of operating conditions, the complexity of industrial geometries and their relative motion. A representative industrial low pressure turbine (LPT) cascade subject to wake passing interactions is analysed, adopting the incompressible Navier–Stokes solver implemented in the spectral/ hp element framework Nektar++ . The bar passing effect is modelled by leveraging a spectral-element/Fourier Smoothed Proﬁle Method. The Reynolds sensitivity is analysed, focusing in detail on the dynamics of the separation bubble on the suction surface as well as the mean ﬂow properties, wake proﬁles and loss estimations. The main ﬁndings are compared with experimental data, showing agreement in the prediction of wake traverses and losses across the entire range of ﬂow regimes, the latter within 5% of the experimental measurements.


Introduction
In a gas turbine engine, the pressure expansion through high-and low-pressure turbines (LPT) is achieved in a number of subsequent stages. The interaction of multiple stages of rotors and stators is a crucial source of deterministic unsteadiness, which has effects on the loss production mechanisms, and it is thus of great importance to designers. One of the early computational investigations of the wake-passing effect was carried out by Wu and Durbin [1,2], who presented evidence that incoming wakes are responsible for the formation of longitudinal structures on the pressure side.
Subsequently, a number of Direct Numerical Simulation (DNS) studies were performed on LPTs, providing further contributions to the understanding of the effect of incoming wakes on the pressure-and suction-side boundary layers [3][4][5][6]. A detailed review by Hodson and Howell [7] summarises the wake-induced boundary layer transition mechanisms in LPTs.
The impact of the wake passing frequency on loss mechanisms has been numerically investigated by several authors. Michelassi et al. [8] built on top of previous work [9] to analyse the effects of various types of inflow disturbances on losses in a series of compressible DNS simulations. In particular, they observed the effect of three reduced frequencies and two Reynolds numbers (Re 2 = 60,000 and 100,000) on various types of loss indicators. The behaviour of high reduced frequencies was found to be most effective in suppressing the suction surface separation bubble, by manifesting a very similar behaviour to high levels of inflow turbulence, due to the constant-area mixing prior to the leading edge.
The comparisons of mixed-out losses with profile losses as defined by Denton [10] allowed distinguishing the losses generated in the boundary layer from those generated in the flow core by the variable-area wake mixing. The largest difference was found at low levels of reduced frequency (1 bar per pitch): in this case, the wakes remain distinct within the blade passage and experience significant wake distortion losses.
Further LES simulations [11] demonstrated the combined importance of the reduced frequency and flow coefficient on losses. These important design parameters affect the frequency with which incoming wakes impact on the cascade as well as the wake inclination. The normal distance between incoming wakes was found to correlate very well with mixedout losses, further confirming that wake merging (which produces a free-stream turbulencelike behaviour) is generally beneficial for the reduction of unsteady losses. This approach can importantly be adopted to inform design optimisation, as the various combinations of reduced frequency and flow coefficient are effectively a consequence of design parameters, such as the revolution speed and flow-through velocity.
As part of an ongoing effort to develop a validated incompressible DNS capability for the turbomachinery Industry using spectral/hp element methods, the present work builds on consolidated expertise [12][13][14] in the use of the Nektar++ open-source software framework [15] for industrial applications. The wake-passing effect is analysed in the context of a representative research LPT profile, with a particular focus on the Reynolds sensitivity of the transition mechanism on the suction surface and the loss prediction. The optimal and most realistic degree of reaction corresponds to the reduced frequency range where unsteady losses are highest [11], which is targeted in this paper. The availability of experimental data in the wake traverses at the flow regimes analysed allows quantification of the accuracy of the numerical approach.
The paper is organised as follows: first, a brief introduction on the numerical setup and wake modelling strategy is presented. Subsequently, the flow dynamics on the suction surface are characterised in the various phases of the wake passing cycle. The following sections focuses on the time-averaged effects of the wake passing, analysing the blade wall distributions, wake profiles and mixed-out measurements and establishing a comparison with highly accurate experimental data. The paper concludes by providing remarks on the validity of the results presented.

Numerical Approach
The incompressible Navier-Stokes equations were discretised using the spectral/hp element framework Nektar++ [15]. The solver adopts a stiffly stable time discretisation [16], which decouples the velocity and pressure fields, also known as the Velocity-Correction Scheme [17]. This formulation treats the convective terms explicitly, while pressure and the viscous contributions are treated implicitly; therefore, addressing stability constraints that would otherwise be associated with the viscous time stepping. Analysis of test cases at high Reynolds numbers with under-resolved meshes requires the use of stabilization techniques: if the dissipative scales at high wavenumbers are not fully resolved, a buildup of energy might cause the solution to diverge. Spectral Vanishing Viscosity (SVV) was introduced by Tadmor et al. [18] as a technique to add artificial diffusion to potentially unstable under-resolved scales. Since the oscillations causing divergence of the solution emerge at sub-elemental level, SVV is a stabilization technique that acts at the subgrid-scale level, without introducing any explicit model.
The DG-Kernel SVV formulation introduced by Moura et al. [19,20] was adopted to ensure numerical stability in the spectral/hp planes, while the traditional exponential kernel was employed in the Fourier expansion in the spanwise direction. In adopting such a stabilisation approach, better resolution ability is achieved by increasing the polynomial order (both in terms of diffusion and dispersion properties) as opposed to reducing the base mesh size. As shown in the earlier work of Cassinelli et al. [12] at Re = 88,000, the polynomial order adopted is sufficiently high for SVV not to affect the flow physics of interest. The spectral/hp dealiasing approach by [21] was adopted for all computations. The spectral/hp element implementation of the present work relies on Taylor-Hood type elements, where the C 0 continuous pressure field is computed at one polynomial order lower than the C 0 continuous velocity variables. Further details on the numerical methodology, mesh resolution and convergence properties were discussed by Cassinelli et al. [12,22].

LPT Setup with Wake Passing
The availability of experimental data motivated the choice of the flow regimes in analysis. Three Reynolds numbers were considered: Re 2 = 86,000, 157,000, and 297,000, based on mixed-out exit velocity U 2 and suction surface perimeter S 0 . The research LPT cascade in analysis is scaled by the true chord and inflow velocity, so that C = 1 and U ∞ = 1.
The wake-passing effect is experimentally reproduced by including a mechanism to introduce upstream-generated wakes shed from uniformly spaced cylindrical bars, controlled by an electric motor. To analyse this phenomenon, two nondimensional parameters are of fundamental importance: the reduced frequency F red and flow coefficient Φ, defined as: where U b indicates the bar velocity, P b is the vertical distance between bars and U ∞,x is the axial inlet velocity. These parameters control the bar speed (which affects the inclination of the wake when it impinges on the blade) and their relative distance. Together, they affect the frequency at which the wakes impact on the blade. Adjusting the cylinder distance and speed allows for analysis of realistic configurations where the rotor and stator count are different. However, the numerical experiments are typically carried out with pitchwise periodic boundary conditions. Limitations owing to the computational cost that would be required to simulate multiple blades require the cascade pitch P y to be a multiple of the distance between bars P b . Therefore, from a computational perspective, the control parameter in the simulations is the velocity of the cylinders. In this study, the exact value of the reduced frequency was simulated, while the flow parameter was enforced as a consequence of single-pitch periodicity (i.e., imposing P b = P y ). The numerical values imposed in the simulations are reported in Table 1. The relative error between the nominal flow coefficient and effective flow coefficient is 6.748%. The lower flow coefficient results in slightly higher wake inclination compared to the experiments.
Introducing the wake passing period T b = P b /U b , time can be expressed as function of the number of periods m and the phase 0 ≤ ϕ ≤ 1: t = mT b + ϕT b . After transient times of at least 12C/U ∞ , time-averaged statistics were subsequently sampled over m = 20 bar passing periods (corresponding to T = 24C/U ∞ ). At phase ϕ = 0, the bar is situated at the same pitchwise location as the leading edge, y/C = 0. The x-y planes were discretised with a high-order expansion at P = 6; the spanwise domain was set to L z = P y /4 = 0.21164 C and discretised with N z = 72 Fourier planes, resulting in 52.67 M DoF per variable. As shown in Figure 1, this numerical setup yields a wall resolution of ∆x + < 20, ∆y + < 0.6, ∆z + < 8 at Re 2 = 297,000. The spaceand time-convergence properties of a similar numerical setup on a range of statistical distributions and properties were previously assessed extensively in a clean inflow case at Re = 88,000 [12]. In particular, x − y plane convergence was analysed by means of P-refinement, reporting negligible discrepancies for expansion orders P > 5. The independence of the results with respect to the span was also discussed, and further evaluated by analysis of the correlation function [22], which showed that the blade wake is fully contained within the chosen spanwise domain. It should be noted that the aforementioned analyses were carried out in a previous study in the context of a disturbance-free inflow environment, at a flow regime corresponding to the lower end of the Reynolds envelope considered in the present study. Furthermore, since the same mesh is employed for all Reynolds numbers, the effective resolution is lower at high Reynolds numbers. Enforcing the pitchwise domain to be a multiple of the spanwise domain allowed adopting a modified version of the inflow turbulence generation algorithm by Davidson [23], introduced and validated in Nektar++ [13]. This implementation allows to generate a spanwise-and pitchwise-periodic synthetic velocity signal, requiring a choice of spanwise domain such that P y is a multiple of L z .
Case Re 2 = 86,000 is analysed with and without inflow turbulence (on top of the discrete disturbances shed by passing bars). The inflow turbulence case (IW+IT) required a refined computational mesh to resolve the turbulent structures introduced at the inlet. This mesh differs from the IW mesh only in the region comprised between the inflow and the vertically refined line where the bar passing occurs. A nominal inflow turbulence intensity TI = 3.5% was prescribed in combination with a length scale L t = 0.05C to guarantee domain independence.
The algorithm selected 30 modes to discretise the modified von Kármán spectrum, compared to 1024 specified previously analysed [13]. As a consequence of a coarser discretisation of the wavenumber space, the level of turbulence intensity effectively introduced at the inlet of the computational domain is lower than prescribed: TI = 2.578%.

Modelling the Bar Passing Effect
To model the bar passing effect, the Smoothed Profile Method (SPM), first introduced by Nakayama and Yamamoto [24], was selected. The method was further developed to adopt a time discretisation based on a high-order semi-implicit splitting scheme [25,26], and incorporated in the Nektar++ framework. Extensive preliminary validation was carried out in the PhD thesis by Cassinelli [22] to ensure the generation of a realistic cylinder wake.
As part of the SPM formulation, an interface thickness parameter must be selected to represent the rigid particles. The interface thickness was selected to ensure accurate representation of the SPM boundaries (thus driven by resolution requirements). An auxiliary study focused on varying the diameter of the cylinders at fixed interface thickness.
023) produced wake profiles and spectral characteristics matching those of a corresponding DNS simulation over the entire range of Reynolds numbers analysed.
The qualitative result of the wake-passing effect at high Reynolds number is shown in Figure 2. The initial stage of wake bending is visible in the Figure, while the suction surface transition leads to fully turbulent boundary layer at the trailing edge.

Low-Speed Experimental Testing of LPTs
The present generation of LPTs typically operates at subsonic conditions, characterised by exit Mach numbers of Ma 2 ∼0.6. The cost of experimental programs can be significantly reduced by performing tests in low-speed wind tunnels, at essentially incompressible conditions. However, the pressure distribution is in general strongly dependent on the Mach number [27]: the peak suction Mach number ultimately controls the adverse pressure gradient flow region, where most losses are generated. Therefore, experimental testing cannot be carried out on the same blade shape and cascade setup.
A number of scaling techniques as well as more advanced redesign strategies were developed to derive a modified profile shape and flow conditions [28,29], allowing to compensate for the effects of compressibility.
The linear cascade in analysis was redesigned to match the design high Mach number distribution when tested at incompressible conditions, modifying profile shape and increasing the inlet flow angle. This has important implications for the applicability of the capability presented in the paper. One of the main research purposes is the validation of a virtual wind tunnel capability, and the use of an incompressible flow solver is not a road blocker: leveraging scaling techniques, realistic LPT geometries with a peak suction Mach number up to 0.6-0.7 can be accurately simulated in their low-speed testing conditions. However any compressibility effects, even if overall small, are not accounted in the present work.

Evidence of the Transition Mechanism
The time averaging operation masks the wealth of flow phenomena occurring in the boundary layer (BL) of LPTs subject to incoming disturbances, and it particularly fails to highlight the presence of high-amplitude events that precede the onset of turbulence. A discussion on some instantaneous flow statistics is presented before analysing timeaveraged statistics, to highlight some of the flow phenomena occurring on the suction surface of the cascade. Figure 3 provides qualitative insight into the suction surface dynamics at consecutive phases ϕ for Re 2 = 86,000. Similar considerations could be discussed for the higher flow regimes Re 2 = 157,000 and 297,000 but they are here omitted for brevity. The wakes impinge on the suction surface separation bubble between ϕ = 0.25 and ϕ = 0.375. The extended region of weakly negative recirculation (the pale yellow region) visible at ϕ = 0-0.125 between s/S 0 = 0.7-0.95 is largely suppressed by the incoming disturbance. Concurrently, at ϕ = 0.25 the shear layer rolls up into two separate regions of recirculating flow, as previously highlighted by Michelassi et al. [11]. The upstream recirculation region is found at s/S 0 ≈ 0.85, while the downstream one merges into the trailing edge. As the wake impacts on the separation bubble, the region of high shear in the upstream portion of the suction surface (the dark blue region) moves towards the trailing edge, following the impinging disturbance. At ϕ ≈ 0.375, the attached flow region develops a spanwise pattern of alternating high-and low-speed flow, called Klebanoff streaks, which play an important role in the bypass transition mechanism [30,31].
Streaks are found at all the flow regimes analysed, and their spanwise length scale is inversely proportional to the Reynolds number. The generation mechanism of elongated streaks in the suction surface in presence of inflow turbulence was recently discussed in detail by Zhao and Sandberg [32]. In the context of wake passing, the generation of streaks occurs periodically, and its effect on the transition mechanism in the aft portion of the suction surface is evident at 0.375 ≤ ϕ ≤ 0.75.
The nature of the disturbances that penetrate the boundary layer shear is determined by the shear sheltering mechanism [33,34]: the boundary layer shear acts as a filter for the high-frequency vortical disturbances. The low-frequency perturbations that penetrate into the boundary layer promote strong shear associated with the streaks, which may develop into turbulent spots through localized secondary streak instability. These steps characterise the early stages of the bypass transition mechanism, described as a secondary instability of lifted shear layers when they reach the top of the boundary layer, and they are subject to high-frequency free-stream disturbances [30,35].
Liu et al. [36] discussed how the physical mechanisms governing streak instability can be revealed through Floquet analysis of secondary instability. As discussed by Zaki [31], in the presence of Klebanoff streaks the two most unstable modes of the boundary layer are the inner mode (varicose instability) and the outer mode (sinuous instability).
In the early stages of the wake passing cycle, the penetration of streaks within the boundary layer has not reached the trailing edge (TE) region yet. However, the formation of instabilities is detected at ϕ ≈ 0.25 and highlighted in Figure 4 left; this region is identified by the dashed black rectangle in the third panel of the C f carpet plots shown on the left. The vorticity contour suggests the presence of part-span K-H instability, which rapidly lifts up from the wall and develops across the boundary layer edge.
The alternating parallel fluctuating velocity patterns are consistent with the findings of Zhao and Sandberg [32], where they are argued to be evidence of varicose streak instability. However, in this case the instability is most likely due to part-span K-H type instability which occurs naturally in the separation bubble. At the later phases of the wake passing cycle, the presence of developed turbulent spots is detected. However, the evidence here presented is not sufficient to uniquely identify the physical streak instability mechanism (i.e., sinuous or varicose) preceding the inception of the spots.

Space-Time Boundary Layer Behaviour
The time evolution of the spanwise-averaged effect of the wakes on the suction surface boundary layer is revealed in the space-time plots of the shape factor of Figure 5. The BL evolution is captured over 4 wake passing periods T b with a resolution of 80 flow snapshots per period. Instantaneous statistics are here preferred over phase-averaged realisations to capture the highly unsteady nature of the wake interactions. The presence of the separation bubble is highlighted by the solid iso-lines of C f = 0, which identify the spanwise-averaged extent of separation. When the wake interacts with the separation bubble, wake-induced transition promotes reattachment of the flow over the whole extent of the suction surface, as indicated by the dashed arrow on Figure 5. Following the wake-driven reattachment and the subsequent calmed region, the separation bubble recovers as the separation line moves upstream. The separation bubble captured in Figure 5 at Re 2 = 297,000 is significantly smaller compared to Re 2 = 86,000; however, it is also subject to larger relative variability in terms of both streamwise and temporal extent: the second instance of separation visualised in Figure 5 is roughly twice as persistent compared to the following instance.
This suggests that the dynamics of the boundary layer between s/S 0 ≈ 0.7 − 0.9 is largely affected by the presence of streak instabilities trailing the passing of the wake in every cycle.

Blade Wall Distributions
The time-averaged and spanwise-averaged pressure distribution and skin-friction coefficient are shown in Figure 6. Additional data obtained introducing a momentum forcing near the leading edge (labelled "BF") as described by Cassinelli et al. [13] is included to highlight the impact of the periodic disturbances on time-mean flow performance indicators. Due to the presence of passing bars, the effective inflow angle is decreased compared to the cases without wake passing, from α 1 = 33.86 • to α eff 1 = 31.73 • in Re 2 = 86,000, α 1 = 33.96 • to α eff 1 = 32.17 • in Re 2 = 157,000 and α 1 = 34.08 • to α eff 1 = 32.18 • in Re 2 = 297,000. The difference translates into discrepancies in the front portion of the suction surface between the cases with and without passing wakes, evident in the suction surface skin friction peak at s/S 0 ≈ 0.05, where the IW profiles are slightly lower than the cases without incoming wakes. In this same region of the suction surface, lower Reynolds number corresponds to higher wall-shear stress. The major differences due to the introduction of periodic disturbances are in the adverse pressure gradient part of the suction surface. Especially at low Reynolds numbers, in the C p distribution the BF cases present a short plateau region at s/S 0 ≈ 0.85, indicative of a weak separation bubble. This region is delayed and shortened with the introduction of wake passing. The skin-friction coefficient distribution further clarifies the differences. Due to the periodically modified transition mechanism on the suction surface, the wake passing causes an upstream shift of the time-averaged separation bubble at Re 2 = 86,000 and Re 2 = 157,000.
In both cases, flow separation occurs at s/S 0 ≈ 0.74, and the extent of separation is almost halved compared to the cases with body forcing. The negative C f peak denoting the presence of roll-ups and anticipating reattachment is moved upstream, flattened and only visible at low Reynolds number. This is a consequence of the wake motion along the suction surface and it does not correspond to stationary flow features.
In both cases, Re 2 = 86,000 and Re 2 = 157,000, the flow is attached at the trailing edge but not fully developed. Further increasing the Reynolds number (Re 2 = 297,000) removes separation in a time-averaged sense and yields a fully turbulent boundary layer at the trailing edge, as highlighted by the flat C f profile for s/S 0 > 0.9.

Wake Traverses and Experimental Comparison
Extensive experimental data available from LDA measurements are presented in this section. The experimental setup was described in detail by Bolinches-Gisbert et al. [37]. The wake traverses are extracted atx = x − x TE = 0.513, and compared with the experiments in Figure 7. The velocity solution accurately captures all the relevant wake flow features, i.e., flank width and negative peak: the latter is within 1% of the experimental value. The presence of background turbulence (only simulated at Re = 86,000) does not modify the wake profiles, which overlap to the curves obtained without incoming turbulence, with a marginal discrepancy observed in the mean passage on the pressure side: discrete disturbances are the dominant mechanism in modifying the dynamics of the suction surface separation bubble.
The loss profiles are also reported in the Figure, showcasing good agreement with the experiments especially at higher Reynolds numbers, where the peak is captured within 6% of the experiments. The entity of losses in the mean passage remains unvaried in the three flow regimes explored. The computed results capture a slight decrease of the loss profile peak by ∼6% and 10% in cases Re 2 = 157,000 and Re 2 = 297,000, compared to Re 2 = 86,000.
The experimental values, however, remain approximately constant across the three flow regimes. At higher Re 2 , the wake width decreases, owing to the smaller momentum thickness at the trailing edge typical of high-Reynolds configurations, combined with a fully suppressed separation bubble due to bypass transition.
The turbulence length scale in the TE wake was measured from a spanwise line of probe points, yielding a smaller value in the high-Reynolds case, where the two-point correlation decays faster. This in turn promotes lower levels of fluctuations in the wake. At low Re the separation bubble is also periodically open, giving rise to large-scale vortex shedding which results in higher levels of fluctuating turbulence kinetic energy. The higher Reynolds cases Re 2 = 157,000 and Re 2 = 297,000 have very similar profiles.

Mixed-Out Measurements
The mixed-out loss coefficient is shown in Figure 8. The numerical results match the experimental data, with an uncertainty of 2.5% in the total pressure loss coefficient and 0.4 deg of exit angle. At low Reynolds numbers, where the separation periodically extends to the trailing edge, it is traditionally very challenging to accurately capture losses. Case IW + IT at Re 2 = 86,000 does not improve the agreement with experiments, which were conducted in absence of background fluctuations. The relative error with respect to the experimental measurements is reported in Table 2, which highlights that the two regimes Re 2 = 86,000 and Re 2 = 297,000 achieve the minimum error, and are within the range of experimental uncertainty.  The mixed-out exit angle is shown in Figure 8. As highlighted quantitatively in Table 2, the flow turning is consistently overestimated across the flow regimes analysed, suggesting the presence of a bias. The streamwise component of the velocity wakes (not shown) is in deficit compared to the experimental data, especially in the mean passage. Among the various sources of uncertainty between numerical simulations and experiments, the contraction of the experimental streamtube is the most dominant effect. Physically, this is due to the growth of a boundary layer on the experimental sidewalls, accelerating the flow in the axial direction in the midspan.
A first-order correction is introduced to validate this conjecture, showing that the streamtube effect is mostly responsible for the lack of agreement in the prediction of the exit angle. The ratio between the experimental and computational mass-flow rate k s can be estimated as: considering the average of the mass flow rate calculated in the experimental passages S1 and S2. The streamwise velocity extracted from the numerical simulations is corrected: and the exit angle is recalculated. The associated spanwise domain contraction can be estimated by imposing mass conservation: The exit angle corrected with the experimental streamtube is consistently shifted to a region within the error bounds of the experimental measurements, therefore suggesting that the streamtube contraction is the physical mechanism responsible for the discrepancy of numerical and experimental measurements. Quantitatively, the mixed-out exit angle error is reduced by ≈80%. The streamtube contraction estimated to yield the correct mass flow rate is between 1-2% and it is larger at higher Reynolds number.

Conclusions
In this paper, we discussed the introduction of the wake-passing effect on an LPT cascade focusing, in particular, on the Reynolds sensitivity of the flow on the suction surface and introducing a comparison of wake measurements with experiments. The three Reynolds numbers analysed correspond to subcritical, critical and supercritical states of the suction surface boundary layer in a clean inflow setup.
The analysis of the LPT flow features first focused on instantaneous statistics. The aft portion of the suction surface features a range of different transition mechanisms, driven by the periodic impingement of the wake passing. At low Reynolds numbers, two rollup regions were periodically identified where the wake suppresses the suction surface separation bubble; however, at higher Reynolds numbers, this mechanism was not retained. This highlights the high sensitivity of the suction surface separation to external disturbances in the subcritical flow regime.
The comparison of wake traverses from the numerical simulations demonstrates close agreement with experimental measurements with less than a 1% relative error in the profile peak. The total pressure loss coefficient was also predicted accurately, within 5% of the experimental measurements across the flow regimes in analysis. The low-and high-Reynolds cases were within experimental measurement uncertainty. A first-order estimation suggested that introducing a model of the streamtube contraction of 1-2% of the spanwise domain would yield the same levels of flow turning as in the experiments.
There is a growing need for high-fidelity simulations to train deep neural networks to develop new insights into flow physics. The DNS capability presented in the paper can be used to develop new turbulence closures [38] and promote a better physical understanding of the mechanisms of loss generation in turbomachinery.

Conflicts of Interest:
The authors declare no conflict of interest.

Abbreviations
The following abbreviations are used in this manuscript: