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International Journal of Turbomachinery Propulsion and Power Article Reynolds Sensitivity of the Wake Passing Effect on a LPT Cascade Using Spectral/hp Element Methods 1 2 1 3 4 Andrea Cassinelli , Andrés Mateo Gabín , Francesco Montomoli , Paolo Adami , Raul Vázquez Díaz 1, and Spencer J. Sherwin * Department of Aeronautics, Imperial College London, London SW7 2AZ, UK; andrea.cassinelli.93@gmail.com (A.C.); f.montomoli@imperial.ac.uk (F.M.) School of Aeronautical and Space Engineering, Universidad Politécnica de Madrid, 28040 Madrid, Spain; andres.mgabin@upm.es Rolls-Royce Deutschland, 15827 Dahlewitz, Germany; Paolo.Adami2@Rolls-Royce.com Rolls-Royce plc., Derby DE24 8ZF, UK; Raul.Vazquez@Rolls-Royce.com * Correspondence: s.sherwin@imperial.ac.uk † This paper is an extended version of our paper published in Proceedings of the European Turbomachinery Conference ETC14 2021, Paper No. 606, Gdansk, Poland, 12–16 April 2021. Abstract: 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 Citation: Cassinelli, A.; Mateo implemented in the spectral/hp element framework Nektar++. The bar passing effect is modelled Gabín, A.; Montomoli, F.; Adami, P.; by leveraging a spectral-element/Fourier Smoothed Proﬁle Method. The Reynolds sensitivity is Vázquez Díaz, R.; Sherwin, S.J. Reynolds Sensitivity of the Wake analysed, focusing in detail on the dynamics of the separation bubble on the suction surface as well Passing Effect on a LPT Cascade as the mean ﬂow properties, wake proﬁles and loss estimations. The main ﬁndings are compared Using Spectral/hp Element Methods. with experimental data, showing agreement in the prediction of wake traverses and losses across the Int. J. Turbomach. Propuls. Power 2022, entire range of ﬂow regimes, the latter within 5% of the experimental measurements. 7, 8. https://doi.org/ 10.3390/ijtpp7010008 Keywords: wake-passing effect; reynolds sensitivity; spectral/hp element method; high-order; Academic Editor: Tony Arts smoothed proﬁle method Received: 20 June 2021 Accepted: 7 February 2022 Published: 22 February 2022 1. Introduction Publisher’s Note: MDPI stays neutral In a gas turbine engine, the pressure expansion through high- and low-pressure with regard to jurisdictional claims in turbines (LPT) is achieved in a number of subsequent stages. The interaction of multiple published maps and institutional afﬁl- stages of rotors and stators is a crucial source of deterministic unsteadiness, which has iations. 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. Copyright: © 2022 by the authors. Subsequently, a number of Direct Numerical Simulation (DNS) studies were per- Licensee MDPI, Basel, Switzerland. formed on LPTs, providing further contributions to the understanding of the effect of This article is an open access article incoming wakes on the pressure- and suction-side boundary layers [3–6]. A detailed re- distributed under the terms and conditions of the Creative Commons view by Hodson and Howell [7] summarises the wake-induced boundary layer transition Attribution (CC BY-NC-ND) license mechanisms in LPTs. (https://creativecommons.org/ The impact of the wake passing frequency on loss mechanisms has been numerically licenses/by-nc-nd/4.0/). investigated by several authors. Michelassi et al. [8] built on top of previous work [9] Int. J. Turbomach. Propuls. Power 2022, 7, 8. https://doi.org/10.3390/ijtpp7010008 https://www.mdpi.com/journal/ijtpp Int. J. Turbomach. Propuls. Power 2022, 7, 8 2 of 15 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 = 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 mixed- out losses, further conﬁrming that wake merging (which produces a free-stream turbulence- like 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 [12–14] in the use of the Nektar++ open-source software frame- work [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 experi- mental 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. 2. Methods 2.1. Numerical Approach The incompressible Navier–Stokes equations ¶u = (ur)urp + nr u inW , (1) ¶t r u = 0 inW , (2) 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; there- fore, 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 Int. J. Turbomach. Propuls. Power 2022, 7, 8 3 of 15 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 el- ements, where the C continuous pressure ﬁeld is computed at one polynomial order lower than the C continuous velocity variables. Further details on the numerical methodology, mesh resolution and convergence properties were discussed by Cassinelli et al. [12,22]. 2.2. 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 = 86,000, 157,000, and 297,000, based on mixed-out exit velocity U and suction surface perimeter S . The research LPT 2 0 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, con- trolled by an electric motor. To analyse this phenomenon, two nondimensional parameters are of fundamental importance: the reduced frequency F and ﬂow coefﬁcient F, deﬁned red as: U C U ¥,x F = , F = , (3) red P U U b 2 b where U indicates the bar velocity, P is the vertical distance between bars and U is the ¥,x b b 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 to be a multiple of the distance between bars P . 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 = P ). The numerical values imposed in the simulations b y 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 = P /U , time can be expressed as function b b b of the number of periods m and the phase 0 j 1: t = mT + jT . After transient times b b of at least 12C/U , time-averaged statistics were subsequently sampled over m = 20 bar passing periods (corresponding to T = 24C/U ). At phase j = 0, the bar is situated at the same pitchwise location as the leading edge, y/C = 0. Int. J. Turbomach. Propuls. Power 2022, 7, 8 4 of 15 Table 1. LPT bar passing setup with cylinder parameters in the upper portion of the table. Re = 86,000 was simulated both with inﬂow wakes (IW) and inﬂow turbulence (IT), while other regimes analyse IW alone. The compute time is estimated on 1000 cores on the Archer supercomputer, including runtime post-processing and thus providing a conservative estimate. Re 86,000 (IW, IW+IT) 157,000 (IW) 297,000 (IW) F (S -based) 0.624132 0.627675 0.633188 red sim U /U 0.705339 0.706414 0.708116 sim F 1.17731 1.17414 1.16966 a [ ] 33.86 33.96 34.08 5 5 5 Dt 2.5 10 2.5 10 2 10 Compute time for T = 1C/U 8 h 40 min 8 h 40 min 10 h 45 min The x–y planes were discretised with a high-order expansion at P = 6; the span- wise domain was set to L = P /4 = 0.21164 C and discretised with N = 72 Fourier z y z planes, resulting in 52.67 M DoF per variable. As shown in Figure 1, this numerical setup + + + yields a wall resolution of Dx < 20, Dy < 0.6, Dz < 8 at Re = 297,000. The space- and 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 indepen- dence 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. Δx DNS limit Δz DNS limit Δy DNS limit −1 Δx Δy wall Δz −2 Re = 86,000 Re = 157,000 Re = 297,000 −3 0.0 0.2 0.4 0.6 0.8 1.0 s/S Figure 1. Wall resolution along the suction surface with increasing Re and incoming wakes. Enforcing the pitchwise domain to be a multiple of the spanwise domain allowed adopting a modiﬁed version of the inﬂow turbulence generation algorithm by David- son [23], introduced and validated in Nektar++ [13]. This implementation allows to generate Wall resolution Int. J. Turbomach. Propuls. Power 2022, 7, 8 5 of 15 a spanwise- and pitchwise-periodic synthetic velocity signal, requiring a choice of spanwise domain such that P is a multiple of L . y z Case Re = 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 = 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 discreti- sation 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%. 2.3. 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 aux- iliary study focused on varying the diameter of the cylinders at ﬁxed interface thickness. SPM A smaller bar diameter of d = 0.6d (with d /C = 0.023) produced wake proﬁles and b b 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. Figure 2. Iso-surfaces of Q-Criterion (Q = 200) contoured by velocity magnitude in case Re = 297,000. The computational domain is duplicated for graphical purposes. 2.4. Low-Speed Experimental Testing of LPTs The present generation of LPTs typically operates at subsonic conditions, characterised by exit Mach numbers of Ma 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 Int. J. Turbomach. Propuls. Power 2022, 7, 8 6 of 15 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 num- ber 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 condi- tions. However any compressibility effects, even if overall small, are not accounted in the present work. 3. Results 3.1. 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 time- averaged 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 j for Re = 86,000. Similar considerations could be discussed for the higher ﬂow regimes Re = 157,000 and 297,000 but they are here omitted for brevity. The wakes impinge on the suction surface separation bubble between j = 0.25 and j = 0.375. The extended region of weakly negative recirculation (the pale yellow region) visible at j = 0–0.125 between s/S = 0.7–0.95 is largely suppressed by the incoming disturbance. Concurrently, at j = 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 0.85, 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 j 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 j 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], Int. J. Turbomach. Propuls. Power 2022, 7, 8 7 of 15 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 j 0.25 and highlighted in Figure 4 left; this region is identiﬁed by the dashed black rectangle in the third panel of the C 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. Figure 3. Instantaneous time–space skin friction on the suction surface at eight consecutive phases, Re = 86,000. The dashed boxes identify the regions shown in detail in Figure 4. 2 Int. J. Turbomach. Propuls. Power 2022, 7, 8 8 of 15 Figure 4. Instantaneous suction surface statistics for Re = 86,000 in the regions identiﬁed by the black dashed boxes of Figure 3. The top subﬁgure shows spanwise vorticity in the blade-normal plane denoted with dash-dotted lines of Figure 3, superimposed with ﬂuctuating velocity vectors. The middle and bottom ﬁgures show respectively wall-parallel and wall-normal ﬂuctuating velocity n/C = 0.01 away from the wall. The solid and dashed lines in the bottom subﬁgures are iso-lines of w = 0.15. (Left): j = 0.25; (right): j = 0.625. 3.2. 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 periodsT 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. Figure 5. Instantaneous space–time contour of suction surface BL shape factor, superimposed with iso-lines of wall-shear stress at two levels: C = 0 (continuous line), C = 0.024 (dashed line). f f (Left): Re = 86,000; (middle): Re = 157,000; (right): Re = 297,000. 2 2 2 The presence of the separation bubble is highlighted by the solid iso-lines of C = 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 = 297,000 is signiﬁcantly smaller compared to Re = 86,000; however, it is 2 2 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. Int. J. Turbomach. Propuls. Power 2022, 7, 8 9 of 15 This suggests that the dynamics of the boundary layer between s/S 0.7 0.9 is largely affected by the presence of streak instabilities trailing the passing of the wake in every cycle. 3.3. Blade Wall Distributions The time-averaged and spanwise-averaged pressure distribution and skin-friction co- efﬁcient 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 high- light 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 eff cases without wake passing, from a = 33.86 to a = 31.73 in Re = 86,000, a = 33.96 1 2 1 eff eff to a = 32.17 in Re = 157,000 and a = 34.08 to a = 32.18 in Re = 297,000. 2 1 2 1 1 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 0.05, 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. Re = 86,000 IW Re = 157,000 IW Re = 297,000 IW 2 2 2 Re = 83,000 BF Re = 155,000 BF Re = 290,000 BF 2 2 2 0.80 0.85 0.90 0.95 1.00 0.80 0.85 0.90 0.95 1.00 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 s/S s/S 0 0 Figure 6. Blade wall distributions with increasing Re , compared with momentum forcing cases at the same ﬂow regime. (Left): pressure distribution; (right): skin-friction coefﬁcient. Y-axis tick labels are omitted due to data sensitivity. 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 distribution the BF cases present a short plateau region at s/S 0.85, indicative p 0 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 = 86,000 and Re = 157,000. In both cases, ﬂow separation occurs at s/S 0.74, and the extent of separation is almost halved compared to the cases with body forcing. The negative C 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 = 86,000 and Re = 157,000, the ﬂow is attached at the trailing 2 2 edge but not fully developed. Further increasing the Reynolds number (Re = 297,000) −C f Int. J. Turbomach. Propuls. Power 2022, 7, 8 10 of 15 removes separation in a time-averaged sense and yields a fully turbulent boundary layer at the trailing edge, as highlighted by the ﬂat C proﬁle for s/S > 0.9. f 0 3.4. 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 at x ˆ = x x = 0.513, and compared with the experi- TE ments in Figure 7. IW, Re = 86,000 IW, Re = 157,000 IW, Re = 297,000 IW + IT, Re = 86,000 Exp. S1, Re = 157,000 Exp. S1, Re = 297,000 Exp. S1, Re = 86,000 Exp. S2, Re = 157,000 Exp. S2, Re = 297,000 Exp. S2, Re = 86,000 || || Re = 86,000 Re = 157,000 Re = 297,000 2 2 2 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 Re = 86,000 Re = 157,000 Re = 297,000 2 2 2 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 Re = 86,000 Re = 157,000 Re = 297,000 2 2 2 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 y/P y/P y/P y y y Figure 7. Wake traverses at x ˆ = 0.513 with increasing Reynolds numbers. (Left): Re = 86,000 (in yellow); (middle): Re = 157,000 (in red); (right): Re = 297,000 (in burgundy). Three differ- 2 2 ent wake proﬁles are shown. (Top): velocity magnitude; (middle): total pressure loss coefﬁcient; (bottom): turbulence kinetic energy. The solid lines indicate IW simulations, and the dashed line indicates IW+IT, for Re = 86,000 only. The two experimental traverses S1 and S2 are represented with squares and circles. Y-axis tick labels are omitted due to data sensitivity. 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 2 2 ΔP / ρU k /U u /U 2d 0 2 2 2 2 Int. J. Turbomach. Propuls. Power 2022, 7, 8 11 of 15 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 dis- crepancy 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 = 157,000 and Re = 297,000, compared to 2 2 Re = 86,000. The experimental values, however, remain approximately constant across the three ﬂow regimes. At higher Re , 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 = 157,000 and Re = 297,000 have very similar proﬁles. 2 2 3.5. 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 = 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 = 86,000 and Re = 297,000 achieve the minimum error, and are within the range of 2 2 experimental uncertainty. IW IW (ST-correction) IW IW+IT IW+IT (ST-correction) IW + IT Exp. Exp. 50 100 150 200 250 300 50 100 150 200 250 300 3 3 Re [× 10 ] Re [× 10 ] 2 2 Figure 8. Mixed-out wake traverse measurements: (left): total pressure loss coefﬁcient; (right): exit angle, with associated streamtube correction shown with red markers. The orange area indicates the uncertainty associated with the measurement chain, of respectively 2.5% and0.2 . Y-axis tick labels are omitted due to data sensitivity. 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 M ◦ α [ ] 2 Int. J. Turbomach. Propuls. Power 2022, 7, 8 12 of 15 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 can be estimated as: R R p p p y y y 1 S1 S2 ( u dy + u dy) u n ˆ dy Exp Exp Exp 2 0 0 k = = , (4) R R p p y y u n ˆ dy u dy IW IW 0 0 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: mod u = k u , (5) IW IW and the exit angle is recalculated. The associated spanwise domain contraction can be estimated by imposing mass conservation: L = L /k . (6) z s 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. Table 2. Summary of percentage relative error between experimental and computational mixed-out quantity, as well as summary of the streamtube contraction factors. Parameter Re = 86,000 Re = 157,000 Re = 297,000 2 2 2 M M jjw w jj IW Exp [%] 2.919 5.388 2.516 Exp M M jja a jj 2,IW 2,Exp [%] 0.563 0.757 0.841 2,Exp M,mod jja a jj 2,IW 2,Exp [%] 0.129 0.171 0.187 2,Exp L L z z [%] 1.164 1.564 1.734 4. 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 roll- up 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 Int. J. Turbomach. Propuls. Power 2022, 7, 8 13 of 15 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. Author Contributions: Conceptualization, A.C., F.M., P.A., R.V.D., S.J.S.; data curation, A.C.; formal analysis, A.C.; funding acquisition, F.M., P.A., R.V.D., S.J.S.; investigation, A.C.; methodology, A.C., S.J.S.; project administration, F.M., S.J.S.; resources, A.C., F.M., P.A., R.V.D., S.J.S.; software, A.C., A.M.G., S.J.S.; supervision, F.M., P.A., R.V.D., S.J.S.; validation, A.C.; visualization, A.C.; writing— original draft, A.C.; writing—review and editing, A.C.; All authors have read and agreed to the published version of the manuscript. Funding: This research received funding from Rolls-Royce plc. and the Aerospace Technology Institute (ATI)/Innovate UK programme “FANFARE”. Data Availability Statement: Due to the nature of funding and the proprietary set of data used, supporting data cannot be made openly available. Acknowledgments: A.C. would like to acknowledge Vittorio Michelassi, Giacomo Castiglioni and Yuri Frey for the technical discussions. The Authors would also like to acknowledge HPC support from Imperial College Research Computing Service (DOI: 10.14469/hpc/2232) and Archer under the UK Turbulence Consortium (EP/R029326/1). Conﬂicts of Interest: The authors declare no conﬂict of interest. Abbreviations The following abbreviations are used in this manuscript: BF Body Forcing BL Boundary Layer CFD Computational Fluid Dynamics DNS Direct Numerical Simulation IW Inﬂow Wakes IT Inﬂow Turbulence LDA Laser Doppler Anemometry LES Large-Eddy Simulation LPT Low Pressure Turbine RANS Reynolds-Averaged Navier–Stokes SVV Spectral Vanishing Viscosity TI Turbulence Intensity URANS Unsteady RANS Nomenclature a Flow angle w Total pressure loss coefﬁcient F Flow coefﬁcient j Wake passing phase C, (C ) Blade (axial) chord length ax C Skin friction coefﬁcient C Static pressure coefﬁcient F Reduced frequency red H Boundary layer shape factor Int. J. Turbomach. Propuls. 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International Journal of Turbomachinery, Propulsion and Power – Multidisciplinary Digital Publishing Institute

**Published: ** Feb 22, 2022

**Keywords: **wake-passing effect; reynolds sensitivity; spectral/*hp* element method; high-order; smoothed profile method

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