A Reliable Update of the Ainley and Mathieson Profile and Secondary Correlations

Yumin Liu; Patrick Hendrick; Zhengping Zou; Frank Buysschaert

doi:10.3390/ijtpp7020014

A Reliable Update of the Ainley and Mathieson Profile and Secondary Correlations

Liu, Yumin;Hendrick, Patrick;Zou, Zhengping;Buysschaert, Frank 2022-04-21 00:00:00 International Journal of Turbomachinery Propulsion and Power Article A Reliable Update of the Ainley and Mathieson Proﬁle and Secondary Correlations 1, 1 2 3 Yumin Liu * , Patrick Hendrick , Zhengping Zou and Frank Buysschaert Department of Aero-Thermo-Mechanics, Université Libre de Bruxelles, 1050 Brussels, Belgium; patrick.hendrick@ulb.be School of Energy & Propulsion Engineering, Beihang University, Beijing 100191, China; zouzhengping@buaa.edu.cn Department of Applied Mechanics & Energy Conversion, Katholieke Universiteit Leuven, 8200 Brugge, Belgium; frank.buysschaert@kuleuven.be * Correspondence: yu.m.liu@ulb.be Abstract: Empirical correlations are still fundamental in the modern design paradigm of axial turbines. Among these, the prominent Ainley and Mathieson correlation (Ainley D. and Mathieson G., 1951, “A Method of Performance Estimation for Axial-Flow Turbines,” ARC Reports and Memoranda No. 2974) and its derivatives, plays a crucial role. In this paper, the underlying assumptions of the aforementioned models are discussed by means of a descriptive review, whilst an attempt is made to enhance their reliability and, potentially, accuracy in performance estimations. Closer investigation reveals an intriguing misuse of the lift coefﬁcient in the secondary loss. In light of this, an enhanced model that, notably, builds upon the Zweifel criterion and the vortex penetration depth concept is developed and discussed. The obtained accuracy is subsequently assessed through CFD computations, employing a database comprising 109 cascades. The results indicate a 50% probability of achieving the 15% error interval, which is twice as good as the most recent Aungier model (Aungier R., 2006, “Turbine Aerodynamics: Axial-Flow and Radial-Inﬂow Turbine Design and Analysis”, ASME Press, New York). Furthermore, the reliability of the proposed model is demonstrated by a reconstruction of the Smith chart, on the one hand, and a performance analysis, Citation: Liu, Y.; Hendrick, P.; Zou, on the other. The reconstruction exhibits contours that conform to the original. The results of the Z.; Buysschaert, F. A Reliable Update performance study are compared with the CFD solutions of eight cascades working in off design of the Ainley and Mathieson Proﬁle conditions and conﬁrm the need of the additionally included turbine design parameters, such as the and Secondary Correlations. Int. J. axial velocity and the meanline radius ratios. Turbomach. Propuls. Power 2022, 7, 14. https://doi.org/10.3390/ Keywords: loss correlation; axial-ﬂow turbine; turbine performance ijtpp7020014 Received: 2 December 2021 Accepted: 3 April 2022 Published: 21 April 2022 1. Introduction Publisher’s Note: MDPI stays neutral The axial turbine became incontestably the accustomed device for medium to high with regard to jurisdictional claims in mechanical power generation in modern powerplants and therefore, not surprisingly, it published maps and institutional afﬁl- still remains the subject of extensive research. Throughout the past decades, experience iations. and knowledge have grown continuously, adding reﬁnements to the art of turbine design. In this study, the 1D aerodynamic design, more speciﬁcally, the loss correlations employed in the meanline analysis used in the early design phase, are reviewed. These models are crucial, as they set the preliminary design parameters that can be maintained up to the ﬁnal Copyright: © 2022 by the authors. design stage, that is, if the model incorporates sufﬁcient levels of reliability [1]. Among Licensee MDPI, Basel, Switzerland. these, it is evident to note the renown Ainley and Mathieson (AM) correlation, published This article is an open access article in 1957, which serves this purpose [2] well. For the record, Ainley and Mathieson’s model distributed under the terms and conditions of the Creative Commons was considered to be a signiﬁcant innovation, as it enabled the quantiﬁcation of local Attribution (CC BY-NC-ND) license losses. It was, e.g., successfully employed in the design process of the Rolls Royce Olympus (https://creativecommons.org/ engine [3]. Being the cradle leading to great successes and broad acceptance, it underwent licenses/by-nc-nd/4.0/). multiple updates [4–6], which expanded into a veritable genealogy. Int. J. Turbomach. Propuls. Power 2022, 7, 14. https://doi.org/10.3390/ijtpp7020014 https://www.mdpi.com/journal/ijtpp Int. J. Turbomach. Propuls. Power 2022, 7, 14 2 of 27 Over time, the design of axial turbines underwent several improvements and the geometry became much more diverse, causing the accuracy and reliability of the current historic correlation based (open source) 1D models to become less effective. An important issue is their mathematical foundation, which relies on a fuzzy blend of classical ﬂow theory and heuristic manipulations matching the early cascade measurements [7]. Hence, it is not surprising that modern turbines may become excluded from their range of applica- tion. In light of this, studies have evaluated the correlation reliability through parametric analysis [8–10]. Although inconsistencies have been identiﬁed and reported, no remedies have been proposed. Accuracy has been evaluated on few test cases, which is insufﬁcient to afﬁrm any statistical signiﬁcance. These two aspects (reliability and accuracy) have to be investigated with statistical evidence to clarify the true potential of the available 1D correlations with the latest turbines. Moreover, it is deemed mandatory to upgrade the correlations with a more robust physics based backbone, compatible with modern turbines. The objective of this paper is, therefore, twofold. The ﬁrst is to provide a comprehen- sive review of the AM family correlations. The second is to enhance the AM correlation based on recent breakthroughs in turbine aerodynamics and loss mechanisms, and boost- ing compatibility with current design workﬂows. In particular, the Zweifel criterion [11] and the depth penetration [12,13] are integrated into the latest Aungier (Ag) proﬁle and secondary loss models [6]. The latter are validated on a data set composed of 109 cascades (static and rotating) to achieve statistical signiﬁcance in terms of accuracy and reliability. High ﬁdelity simulations and their numerical solutions serve as the benchmark during the evaluations. The correlation accuracy is determined in terms of probability according to the Gaussian distribution of the relative errors, whilst the correlation reliability is proven by its ability to reproduce the Smith chart [14] and maintain consistent trends in off design conditions under the variation of the newly introduced axial velocity ratio. 2. Review The importance of empirical loss correlations still remains high in the design process of modern axial turbines. Despite their restricted ﬁdelity, they can effectively offer guidance in decision making during the preliminary design stages [1]. In general, the development of empirical models is built upon on a turbine database, which is produced by means of experimental tests where a multitude of measurements are performed and linked to several, mostly dimensionless, design parameters. When parameters are found to correlate well, dedicated design laws can consequently be established. Quite recently, there was a spirited debate on the capability of cascade tests to approxi- mate real turbine ﬂow. The proponents claimed that cascades could offer a satisfactory basis for estimating losses in real machines and should be preferred thanks to their simplicity, ﬂexibility and low cost [15]. The opponents, however, believe that cascades are inherently limited since they cannot reproduce key aspects of the real operation environment, involv- ing, e.g., rotational, curvature and inlet ﬂow distortion effects [16]. Despite this contention, the current state of the art research has clearly joined the supporting side, considering the solid foothold of cascade tests in public literature. The authors of this work adhere to the standpoint of Craig and Cox (CC) [17], which holds a certain relativism. In fact, the extreme situation in which application of the correlations fails in both accuracy and reliability is rather unlikely. However, careless use must be avoided, as this could lead to an entirely erroneous result [18,19]. Hence, the user has to be fully aware of the features and limitations involved, so that the correlations can be applied effectively in real turbine design. This aspect is the crux of the following discussions. Correlations are typically determined by the investigator ’s own judgment, experience and interpretation of ﬂow physics, test quality and database size [20]. The challenge resides in the way the wide information of a complex 3D ﬂow is ﬁltered and condensed into a 1D empirical model. During its setup, certain parameters could be deliberately biased, with the objective to reﬂect speciﬁc turbine characteristics. The reason behind this intervention is quite straightforward. For example, aero-engine and land based steam turbines do not Int. J. Turbomach. Propuls. Power 2022, 7, 14 3 of 27 share the same design requirements, as is the case with high and low pressure turbines. They differ by design and aerodynamics and so will the correlations. The comparative studies of several public domain correlations [9,21] emphasized this issue clearly. As a consequence, there are no universal turbine correlations at this point. Hence, they only serve a speciﬁc range of applications adequately. Since it is common to use empirical models for a wide range of turbine design applica- tions, it is not surprising that the parameters estimated from the cascade correlation may disagree with real machine measurements by a considerable order of magnitude. In these circumstances, one resorts typically to a heuristic calibration. Otherwise, the correlation is of little value [22]. However, there are some risks involved with such an approach. First, simple multipliers/polynomials are often deemed sufﬁcient, even though the resulting correlation would much better suit turbines akin to those used in the database from which the model was established. Second, its use inherently leads to designs whose aerodynamics are similar to that of the turbines in its original database. The fundamental issue is the statistical bias engendered by the limited size of the database. This could be prevented by the construction of a large public database collecting all cascade measurements performed up to the present day [7]. Unfortunately, early investigators did not share or even lost their databases. New investigators often had to generate a new database from the ground up, with neither access to previous data, nor methodologies. In particular, insights that could have revealed a stronger interaction between design parameters, were curtailed. As a consequence, 1D correlations revert to quite simple mathematics or charts and are biased, which, thus, leads to results with rather low ﬁdelity. In contrast, the models established in large industries do not suffer from the aforementioned issues, where one can afford the calibration of public domain or in-house correlations using the vast amount of data accumulated over time for dedicated conﬁgurations. From this data, new versions of earlier models can be developed, which enables a safer and cheaper enhancement of the design process. In [23], it was advanced that any loss correlation exhibits the same probability to accurately predict turbine performance once out of its range of application. Considering the current discussion, it is rather irrational to expect outstanding accuracy from models that attempt to cover complex systems with elemental information. With full awareness of the higher ﬁdelity 3D methods intervening at later design stages, correlation accuracy should be less prioritized, but the focus must initially go towards including all relevant design parameters in the model. Therefore, the ability to cope coherently with the true physics, i.e., reliability, must become the primary concern. 2.1. Ainley and Mathieson Family Correlations The empirical AM correlation [2] has been so successful that it owns a lineage (the AM correlation and its derivatives will not be differentiated in this framework, as these are fundamentally equivalent). Chronologically, it was established through cascade tests using conventional proﬁles stemming from the 1940s and claimed a nominal accuracy of 2% in stage efﬁciency estimation. The latter corresponds to about 15% total pressure loss error. It was published at a time when the expertise and the understanding of axial compressors were more profound than those of turbines [24]. As a matter of fact, before AM was published, one had to resort to expensive experimental trial and error processes when the available methods revealed efﬁciencies largely inferior to those of the actual turbines. The AM correlation was, therefore, considered to be groundbreaking, as it provides a systematic way to quantify turbine aerodynamic loss and its components. The Dunham and Came (DC) correlation [4] was the ﬁrst and a canonical update to the model of AM, which used 16 1960s technology Rolls Royce turbines. It extended the ap- plication range to small turbines, while retaining the same accuracy level. Dunham claimed that their model was still the most reliable design tool during the 1970s, in a context where there was a progressive adoption of potential ﬂow calculation in industry [3]. However, the investigators did not have access to the database of AM and, thus, the improvement Int. J. Turbomach. Propuls. Power 2022, 7, 14 4 of 27 resulted from model calibration based on the overall performance of a restricted number of turbines. It is noteworthy to mention that CC [17], along with Traupel [25], published new correlations with different approaches with regard to turbine aerodynamics in that same period. The Kacker and Okapuu (KO) correlation [5] came as a second update based on 33 1980s technology Pratt & Whitney turbines and showed an accuracy up to 1.5% in stage efﬁciency estimation in design conditions. Furthermore, KO, having high conﬁdence in the validity of the Smith chart [14] which contains 70 1960s technology Rolls Royce turbines [26], calibrated their correlation to ﬁttingly reproduce the same chart. Ag [6] sub- sequently revised KO to handle recompression produced in extreme off design conditions and real working ﬂuids. In addition, it can cope with hub to shroud S2 calculations (the hub to shroud S2 and blade to blade S1 calculations proposed by Wu [27] refer to stream surfaces on which 2D ﬂow equations are solved by mass-averaging the third coordinate). This has recently been conﬁrmed, notably in ORC turbines [28,29]. Despite many the critics of its limitations, the choice of the AM correlation over other public correlations is driven by three particular aspects. Firstly, it owns a comparatively broader mathematical basis and is less reliant on empirical charts. Secondly, it does not require speciﬁc assumptions on the blade surface velocity distribution nor the need to perform cumbersome mathematical computations. As a consequence, faster solution and better accessibility are ensured, in contrast to the semi-empirical models of Baljé and Binsley [30], Denton [31] and Coull and Hodson [32]. Lastly, transparent and valuable details of its development can be retraced in the successive works [2,24,33,34]. There are ﬁve sources of total pressure loss encountered in aerodynamics, notably, skin friction, pressure drag, shock, leakage and mixing losses. However, this classiﬁcation was rearranged to associate these losses to their location instead of their phenomenological origin. The AM family correlations assumed the turbine loss system as follows: p p tr,out,is tr,out Y = , p p tr,out out (1) = Y + Y + Y . p s cl • Proﬁle loss Y is generated by the growth of the boundary layer and ﬂow mixing over the blade surface. This encompasses skin friction and pressure drag. In case the exit ﬂow reaches supersonic velocities and triggers shocks near the trailing edge, then an additional supersonic loss Y is to be accounted for. These losses exclusively arise ex from blade to blade S1 ﬂow and thus assume spanwise uniformity. • Secondary loss Y originates from the interaction between endwall ﬂow and the pressure difference across the blade passage. The main actors are the 3D induced vortices and endwall boundary layers. AM, drawing on the model of Carter [35], regarded both ﬂows and the annulus loss as one entity. This local loss is responsible for spanwise non-uniformity at blade tip and hub. • Tip leakage loss Y arises from the presence of a gap between the blade tip and the cl shroud, preventing the rotating blade from rubbing against the casing. Although this is indispensable to complete the loss system, this loss is not considered within the scope of this work, where priority is given to the more fundamental proﬁle and secondary losses. Although it was not speciﬁed, the correlations implicitly refer to losses of fully mixed ﬂow. However, to be precise, there is neither theoretical nor experimental evidence that can justify the breakdown of Equation (1), as this inherently assumes little interaction between the components [36]. However, its effectiveness might be due to the comparatively small chord of axial turbine blades and the short ﬂow transit time in the blade passage inhibiting strong interaction between the secondary ﬂows (all ﬂows that differ from the primary inviscid ﬂow) [7]. Int. J. Turbomach. Propuls. Power 2022, 7, 14 5 of 27 2.2. Proﬁle Losses The AM 2D proﬁle loss [37] is in q q 5t out max in in Y = Y Y Y , p,A M p,q =0 p,q =b p,q =0 in in out in (2) b b c out out = f (s/c, t /c, q , b ). max out in AM blended the proﬁle losses of a slightly compressible 50% reaction nozzle Y p,q =0 in and impulse nozzle Y , as depicted in their Figure 4 on page 24 in [2], to cover all p,q =b out in intermediate designs. This correlation inherently comprises the trailing edge loss taken at standard t /s = 0.02. Equation (2) was obtained from low speed wind tunnel tests te and focuses on parabolically cambered conventional proﬁles, i.e., the British T.6 type proﬁles. Their experimental results showed the notable sensitivity of the losses to the proﬁle thickness in impulse proﬁles and, thus, a correction for any deviation from standard t /c = 0.2 [34] was introduced in the model. In addition, there was a variation according max to (q /b ) for proﬁles in-between the impulse and nozzle blades, hence justifying the out in arrangement/blending in Equation (2). The velocity distribution of the nozzle and impulse blades was presented in AM’s Figure 5 on page 26 in [34]. This distribution shows that a considerable portion of the suction side of the blade extending from midchord to trailing edge is affected by ﬂow diffusion, which is nowadays regarded as unacceptable in design standards. Use of AM would, therefore, implicitly assume a similar velocity distribution over modern turbine proﬁles, which is likely to result in a signiﬁcant loss mismatch if not handled with care [38]. This speciﬁc problem with associated implications has to be addressed when enhancing Equation (2). In order to improve the reliability of the model of AM for recent turbine conﬁgurations, several correction factors—in this work designated as the auxiliary factorsK—have been proposed by different authors [4,5]. For example, in his latest update, Ag [6] adapted the proﬁle loss correlation of KO (and, thus, AM). This equation possesses several auxiliary factors and is expressed as: Y = K K K K K Y Y + Y , p,Ag mod M p Re i p, A M t =0.02s sh te (3) = f (e /s, q , q , M , M , Re, e/c, k, i)Y . c out r,out in r,in p,A M • Mach correction factor K 1 reﬂects the ﬂow acceleration in the blade passage reducing viscous loss and acts in the subsonic range M 1. According to Ag [6], r,out K , introduced for the ﬁrst time by KO, is ﬂawed in extreme off-design conditions, especially when recompression occurs at the blade hub. It was known that KO yielded only satisfactory results at the design point [15]. Wei [9] tested KO and exposed a curious slope disruption and a nonphysical reduction in the predicted proﬁle losses in cases of higher positive incidences i resulting in recompression, because of spurious K values. It is also noteworthy that the impact of the Mach number was investigated by AM [24] and is consistent with the purpose of K , but it was, surprisingly, not adopted in the earlier Equation (2). • Expansion correction factor K 1 deals with local weak shocks on the blade suction side in the subsonic exit Mach number band M 2 [0.6–1] and was rendered r,out more physics driven. This parameter was ﬁrst introduced by DC [4] for supersonic ﬂows and was derived from the overall turbine performance established by means of experiments. • Reynolds correction factor K applies to Reynolds numbers outside the transition Re 5 5 range [10 –5 10 ] and includes surface roughness, as seen in the CC correlation [17]. A standard ﬁnish of e/c = 10 is assumed throughout the analysis. • Technology correction factor K copes with the technology mismatch of Y when mod p,A M examining the losses in post-1980s turbine proﬁles [37]. K takes the value of 0.825 mod for meridional entry proﬁles and 2/3 for the other. Int. J. Turbomach. Propuls. Power 2022, 7, 14 6 of 27 • Incidence correction factor K 1 conditions the proﬁle loss in off design conditions and was proven to be successful in pre-1980s turbines [39] but overly conservative in modern turbines [15]. In summary, the purpose of the auxiliary factors is to calibrate Equation (2) with its underlying velocity distribution, without the need for an in depth understanding or elaboration of the loss mechanisms. This is evidently caused by a lack of knowledge transfer between AM, DC, KO and Ag [7]. The shock factor Y 0 accounts for local sh shocks situated on hub proﬁles with a thicker leading edge under comparatively high inlet velocities and is exclusively considered as an independent loss contributing to the proﬁle loss, as introduced by KO. However unlike that of KO, it is unaffected by the auxiliary factors. If M 1, then K and K are held at M = 1 and the supersonic loss Y r,out p r,out ex has to be supplemented to Y in Equation (3). Ag proposed M 1 r,out Y = . (4) ex,Ag r,out It must be recognized that this is a rather arbitrary formulation. However, this is re- tained throughout the analysis because there are no more reliable alternatives at the moment. As mentioned earlier, Equation (2) encompasses the trailing edge loss for standard t /s = 0.02. For any variation of t /s, AM proposed a trailing edge correction factor te te affected to Equation (1), such that Y = K (Y + Y + Y ). (5) t te p s cl This expression interrelates the trailing edge to the proﬁle, and secondary and tip clearance losses and was criticized by KO for its weak physical soundness regarding the latter two options. In fact, neither of the endwall nor tip leakage vortices remain closely attached to the blade surface upon reaching the blade exit, rendering interaction between the vortices and the trailing edge wake ﬂow unlikely. Instead, KO employed a more intuitive approach and proposed the trailing edge loss Y as an independent te component [5], such that Y = Y + Y + Y + Y . (6) t p s te cl However, Y still remains part of Y in a broader sense, so that Equation (1) is not te p contracted. To enable this expression, one has to virtually bring Equation (2) to t /s = 0 te beforehand. However, Ag carried on this approach and formulated the trailing edge loss as the consequence of an abrupt area enlargement: te Y = . (7) te,Ag o t te Correction to zero of the trailing edge of Equation (2) is performed by the subtraction of Y . t =0.02s te 2.3. Secondary Losses The AM 3D gross secondary loss is presented by the following equation [2]: Y = l Z, s,A M A M (8) = f (b , b , q , A / A , r /r ). in out in out in hub ti p This compact expression was the result of continuous efforts to interpret turbine endwall secondary ﬂows. l and Z are intrinsic parameters of the AM correlations and A M originate from the work of Carter [35], who drew on the classical secondary ﬂow theory. The intent of AM was to deﬁne a common and unique basis between axial compressors and turbines to quantify secondary losses, as one had (historically) the objective to employ mostly similar blades for both devices. The theory associates endwall vortices with passage Int. J. Turbomach. Propuls. Power 2022, 7, 14 7 of 27 vortices produced in curved channels [40]. The latter originate from the distortion of the inlet endwall boundary layer by the near-wall velocity deﬁcit and the blade passage pressure gradient, causing the inception of a helical cross-ﬂow motion spreading throughout the blade passage, as shown in Figure 1a. (a) (b) Figure 1. Secondary ﬂow mechanisms considered by Carter [35]. (a) Classical secondary ﬂow model. (b) Cascade lifting line model. As Carter strived for a practical model to quantify the losses incurred by these passage vortices, he made an analogy with the lifting line theory to obtain a ﬁrst order approxima- tion. The idea that Carter used leans on the trailing edge inviscid vortex sheet produced by ﬂow circulating around a blade. Being unstable, the closed vortex ﬁlaments of the vortex sheet roll up into trailing vortices at the blade ends, as shown in Figure 1b. Based on qualitative analyses of some experimental evidence taken downstream of the blade, the modelled trailing and passage vortices were made equivalent, to establish a secondary loss correlation [35]. Hence, the induced drag under uniform spanwise circulation is: 2 2 C C 1 2H L,m L,m C = 1 = l , (9) D,s A M 4 s/c H s/c with H the distance from blade midspan to the core of the endwall vortices. This loss coherently reﬂects lost work by the product of blade loading and the width of the trailing vortices; an analogy to force time displacement. AM subsequently replaced H by the empirical Ainley parameter l , which is shown in their Figure 16 on page 31 in [34], and A M which reﬂects the vortex characteristics implicitly. The involved lift coefﬁcient C refers L,m to the mean velocity and relates to the mean angle b , where C = 2 (tan b tan b ) cos b , (10) L,m out m in and tan b + tan b out in b = tan . (11) This was transformed into the Ainley loading Z with: 2 3 Y = C (cos b / cos b )(c/s) = l Z, (12) s,A M D,s out m A M where: 2 2 cos b c cos b L,m out 2 out Z = = 4(tan b tan b ) . (13) out in s/c cos b s cos b m m This equation assumes an axial velocity ratio AVR and meanline radius ratio k, with: AVR = W /W = k = r /r = 1 and is fully independent of s/c. Note that a change z,out out z,in in in blade loading due to a pitch to chord variation would not be identiﬁable [41]. As a matter Int. J. Turbomach. Propuls. Power 2022, 7, 14 8 of 27 of fact, this expression is much closer to the lift coefﬁcient of a single airfoil. The chosen lifting line theory is also used under steady and incompressible conditions and is applied to low camber/loading blades to guarantee the linearity of the problem. As a result, a large disagreement in secondary loss may arise in impulse turbine blades [8] and Equation (13) may wrongfully override other parameters under high loadings [10]. AM were aware of this issue, given the shortage of high-loading blades in their database [34]. It is also interesting to note that a mathematical “mistake” occurred with respect to Equation (12). Normally, the outcome should have been [40]: Y = C (1/ cos b )(c/s), (14) s,A M D,s m and consequently, Equation (13): Z = 4(tan b tan b ) cos b (15) in out m which yields results that are several orders of magnitude higher. This indicates that Equation (12) was most likely adapted heuristically to match the measurements. Therefore, as they possess little physical meaning, l and Z cannot be taken apart in the analysis [7]. A M This is not surprising, since the secondary loss of AM in Equation (8) was calibrated with: Y = Y Y (16) s t p This corresponds to Equation (1) in absence of clearance losses but has another impli- cation. In fact, the secondary loss was calibrated to the subtraction of the measured true total loss by the modelled proﬁle loss of Equation (2). Hence, Equation (8) embodies the errors associated to Equation (2). As is the case with l and Z, secondary and proﬁle A M losses should also not be used separately, as they are bound together by Equation (16). Dunham [19] criticized the approach of Carter but still relied on it to produce cos b out Y = 0.0334K Z, s,A MDC AR cos q in (17) = f ( H/c, b , b , q ). out in in The aspect ratio factor K 1 relates the intensity of the endwall vortices with blade AR height and was reported to have questionable reliability [42]. DC simpliﬁed the multipliers in Equation (13) using a cascade conﬁguration ! ! ( A cos b / A cos q ) 1 cos b cos b n,out out n,in in out out l = f = f = 0.0334K (18) A M AR (1 + d /d ) 2 cos q cos q ti p in in hub The constant 0.0334 is expected to cover most inlet boundary layer thicknesses and was obtained by calibration to overall turbine performance data [22]. Even though this crude approach is not able to estimate the downstream endwall loss of cascade ﬂows [18], it was, however, proven to be more accurate than most reﬁned correlations tested and compared under diverse inlet boundary layer proﬁles [43]. Eventually, Ag improved Equation (17) to ! 1 s,A MDC Y = K K , s,Ag p Re (19) 1 + 7.5Y s,A MDC = f ( M , M , Re, e/c)Y , r,out r,in s,A MDC which shares the K and K of Equation (3) [6]. Nevertheless, K was inherited from DC p Re Re and was later removed by KO. They certainly did not agree on the impact of viscosity on the endwall ﬂow and, interestingly, the debate is still ongoing. It is true that the endwall vortices are mostly driven by inviscid mechanisms. However, their growth in the blade passage highly depends on the endwall boundary layer ﬂow preceding the blade leading edge Int. J. Turbomach. Propuls. Power 2022, 7, 14 9 of 27 and continuous interaction with the blade surface boundary layer. Thereby, the approach of DC and Ag is more plausible. The square root manipulation prevents loss overshoots in extreme off design conditions. This feature has proven to be particularly crucial in optimization. In fact, the search algorithm could inadvertently exploit inconsistencies of the correlation to reach its objective [44]. Again, the auxiliary factors act as moderate multipliers to the basic Equation (17). It is interesting to note that the aptness of the axial turbine loss correlations was investigated in the context of centrifugal turbines, and, more speciﬁcally, the interaction of the auxiliary factors [23]. Drastic changes in the aerodynamic parameters did not trigger a comparable variation in the auxiliary factors, which could have guaranteed reliable solutions. Hence, these auxiliary factors can only sustain minor and individual adjustment and, thus, axial turbine correlation should only be used with axial turbines. 3. Proﬁle Loss Enhancement The Ag correlation [6], being the latest complete loss system, constitutes the most suitable vessel for successive enhancements. Thus, the proﬁle loss of Equation (3) is revamped into Y = 0.914K K K K Y + Y + Y , M p Re p,Enh mod p,A M sh inc (20) = f (e /s, q , q , M , M , Re , e/c, k, i, d /s, We , AVR)Y , c out r,out s LE LE p,A M in r,in in which Equation (2) is rewritten as in b b 5t out in in max Y = Y Y Y , p,A M p,b =0 p,b =b p,b =0 in in out in (21) b b c out out = (s/c, t /c, b , b , y ). max in out Z Carrying on with the auxiliary factor arrangement of Equation (3), several minor changes were performed. 1. Metal angle q in Equation (21) was replaced by a design ﬂow angle b . Practically, a in in better continuity in design workﬂow is gained by focusing on the ﬂow angles ﬁrst and deriving the metal angles afterward. Modern turbines tend to integrate little negative incidence to enable low loss in off designs. Proceeding with q would, rather, require in extra iterations in the preliminary design process. 2. Y and Y in Equation (21) were scaled by a common s/c. Before p,b =0 p,b =b in in out developing the modiﬁcations, the context in which they were built is reviewed and compared to current practice. During the 1940s, blade proﬁling was an empirical science. It consisted of a selection from a family of standardized proﬁles and was arranged into cascade conﬁguration with little allowable change. This approach was inherited from axial compressors and guaranteed cost-effectiveness as it facilitated manufacture. It was impossible to grasp the trend of cascade performance without resorting to cumbersome trial and error wind tunnel tests until the determination of a competent proﬁle. This substantially contrasts with modern proﬁling techniques, which offer close monitoring and automated optimization of the blade surface velocity distribution. Y and Y were certainly obtained from competent but p,b =0 p,b =b out in in obsolete proﬁles, in which blade surface velocity distributions or efﬁciencies have become unacceptable according to modern state of the art standards [34]. As they form the cornerstone of Equation (2), it would be unreasonable to initiate large modiﬁcation at the risk of disrupting the auxiliary factor arrangement in Equations (3) and (20). Instead, a scaling technique with a reference s/c, which is analogous to that applied to compressor performance maps, is proposed. In a design process, s/c is set following a trade off between efﬁciency and cost. If efﬁciency consideration prevails, the optimum s/c that minimizes proﬁle loss, in this case of Equation (2), is chosen, so that any s/c departure results in proﬁle loss increase or simply efﬁciency deterioration. However, Int. J. Turbomach. Propuls. Power 2022, 7, 14 10 of 27 the optimum s/c of Y and Y are ﬁxed and speciﬁc to competent p,b =0 p,b =b in in out proﬁles. The issue would then be its insensitivity to other types of proﬁle. With this in mind, a reasonable approach that maintains the Y and Y trend and p,b =0 p,b =b in in out scales the optimum pitch to chord ratio s/c of Equation (2) to a new optimum A M s/c is proposed. The purpose is to extend Equation (21) to other proﬁles in which optimum s/c abides by different design rules. Among these, the prominent Zweifel criterion [11] is chosen to inherit its ﬂexibility and its incompressible is generalized to: 1 b y s/c = , (22) 2 c j tan b tan b j cos b in out out AVR introducing the axial velocity ratio AVR as an additional design parameter. An y beyond the range of [0.6–0.8] recommended by Zweifel is chosen, to achieve the higher optimum pitch to chord ratio s/c in modern turbines. The vital scaling factor is: s/c f = , s/c s/c A M (23) = f (s/c, b , b , AVR, y , g), in out Z which is multiplied to the s/c of Equation (21). Note that Y and Y A M p,b =0 p,b =b in in out are ﬁtted as polynomials about their respective optimum s/c (the ﬁtting formulae of Tournier and El-Genk [45], Korpela [46], Aungier [6] and Concepts/ETI [47] were tested at extreme b = 30 and b = 80 ; the Tournier and El-Genk performed best), out out so that the scaling process does not alter their slope. As demonstrated, the original and scaled Y and Y are depicted at exit ﬂow angles b = 50 and p,b =0 p,b =b out in in out 70 in Figure 2, with the y = 1 typical of modern turbines. 0.1 0.3 0.08 = 70 o out = 70 0.2 out 0.06 0.04 0.1 = 50 0.02 out = 50 out 0 0 0.4 0.6 0.8 1 1.2 0.4 0.6 0.8 1 1.2 (a) (b) Figure 2. Optimal pitch to- hord ratio matching of AM proﬁle losses [2] with y = 1. (a) Nozzle b = 0 . (b) Impulse b = b . in in out Both Figure 2a,b are consistent on the minimum shift. At b = 70 , a larger s/c can be out achieved with better boundary layer ﬂow control during blade proﬁling. However, at b = 50 , a lower s/c is resorted tp, in order to satisfy y = 1. This is an undesirable out solution, as this would add structural weight. A cost-effective constraint is imposed to refrain f 1.1. s/c 3. The Reynolds correction factor K was adjusted to suction surface length and becomes Re a function of the suction surface Reynolds number Re . Suction surface momentum thickness could be responsible of up to 90% of the proﬁle loss [32], the suction surface ﬂow should logically be prioritized. Int. J. Turbomach. Propuls. Power 2022, 7, 14 11 of 27 4. Incidence loss Y replaces the incidence factor K of Equation (3) and acts as a fully inc i independent, contributing to the proﬁle loss. According to Moustapha et al. [15], modern turbines are designed with relatively thicker leading edges and smoother front curvatures, to retard boundary layer separation over a wide range of incidence angles and to keep loss at low level. This feature can not be reﬂected in the conservative K . In this regard, Benner et al. [48] formulated a polynomial incidence loss Y compatible inc with KO and used in Equation (20). 5. The trailing edge loss Y of Equation (7) is insensitive to change of turbine operating te,Ag conditions and, thus, was deemed inappropriate for modiﬁcations. As an alternative, the Y of KO, which encompasses many key aerodynamic parameters, is considered: te b b in in DF = DF DF DF . (24) te,KO te,b =0 te,b =b te,b =0 out in in in b b out out The replacement of q by b follows the same reasoning behind Equation (21). Draw- in in ing on the proﬁle loss of AM in Equation (2), KO blended the exit boundary layer mixing and base pressure losses of nozzle and impulse blades [5]. Instead of a subtrac- tion, as in Equation (3), virtual correction of Y to t /s = 0 produces a multiplication p te by 0.914 in Equation (20). The latter corresponds to the value of K proposed by AM te in Equation (5) at standard t /s = 0 [2]. Now, the conversion of Equation (24) to Y te te involves M [49]. This is a property speciﬁc to the total pressure loss coefﬁcient and r,out cannot be counted as part of the correlation. However, recent literature has proven its reliance on M but has not delivered any correlation compatible to the AM r,out family [50]. Conveniently, Equation (24) was acquired on low speed tests in the same way as Equation (2). This implies that auxiliary factors deﬁned for the proﬁle loss can be reasonably extended to the trailing edge loss. Considering the relevance of compressibility and Reynolds effects, Equation (24) is revamped into Y = K K K Y , te,Enh M p Re te,KO (25) = f (b , b , t /o, M , M , Re , e/c). out te r,out s in r,in 4. Revised Secondary Loss In light of the issues disclosed earlier in Equation (9), its terms are overhauled. The purpose is to determine a better product of loading and displacement supported by a robust physical basis and depart from the approach used in Equation (16). The latter implies that each contribution has to fully be independent, as in Equation (1). Interestingly, there was a formulation anterior to Equation (8) found in early works [24,33]: in Y = 0.04 1 C , (26) s,A M L,out out using the lift coefﬁcient based on the outlet velocity s cos b out C = 2 (tan b tan b ) . (27) L,out in out c cos b Unfortunately, Equation (26) was abandoned in favor of the model of Carter [35] because it owns a ﬁrmer theoretical background and provides common ground for future compressor and turbine models, according to AM [34]. Contrary to their expectation, axial compressors and turbines have not taken the same course of evolution. As odd as it may be, there is a soundness in Equation (26) in the present context. In particular, C is consistent L,out with turbine analysis, which is not the case with Z of Equation (8), which relies on C in L,m Equation (10). Int. J. Turbomach. Propuls. Power 2022, 7, 14 12 of 27 The secondary loss improvement begins with C rearranged and generalized into L,out the lift solidity coefﬁcient C 1 k cos b L,out out = 1 + tan b tan b . (28) out in s/c AVR AVR cos b Assuming AVR = k = 1, Equation (27) is recovered. Modern turbines possess a meridional ﬂowpath that forces the radial shift of streamlines such that k 6= 1. The latter triggers the Coriolis force and may lower work output [51]. Moreover, expansion through each row inevitably accelerates the ﬂow and, thus, invariance of AVR even in early design stages, which thus constitutes a poor assumption. This aspect would serve in the subsequent evaluation of the correlation reliability. In addition, b is also generalized to: tan b + tan b in out AVR b = tan . (29) 1 + AVR Elaborated models of the endwall vortices have been produced since then, to substitute the classical secondary ﬂow theory. Among these, the passage vortex penetration depth of Sharma and Butler [12], later enhanced by Benner et al. [13], is proposed as the replacement to l : A M 0.79 2 0.1F Z d te = + 32.7 , 0.55 H H CR( H/c) (30) = f ( H/c, d / H, b , b , g, s/c, AVR). out in Once the incident endwall ﬂow impinges on the blade, a ﬁrst separation occurs at the leading edge, creating a horseshoe vortex, as illustrated in Figure 3a. Under passage pressure gradient and crossﬂow friction, the pressure leg is progressively strengthened and pushed towards the adjacent blade suction side. Meanwhile, the suction leg remains near the suction side wall. Eventually, both legs of opposite vorticity coincide at the minimum pressure point. The suction leg starts to orbit around the dominant pressure leg, also identiﬁed as the passage vortex. The latter draws a demarcation line S4 on the blade suction side. Sharma and Butler assumed symmetric and linear S4, beginning at the leading edge and forming a triangle, as shown in Figure 3b. Z / H corresponds to the width of the te passage vortex taken at the trailing edge. It also differs from l by the weight attributed A M to the parameters. For instance, AM prioritized the area ratio A / A over blade turning out in b + b , as there was only weak evidence on the inﬂuence of blade loading [34]. On the in out opposite, the genetic algorithm used by Benner et al. identiﬁed a stronger inﬂuence of the blade tangential force F , rather than the convergence ratio CR, analogous to A / A . q out in (b) (a) Figure 3. Endwall vortices model. (a) New secondary loss model [12]. (b) Passage vortex penetration depth [13]. Int. J. Turbomach. Propuls. Power 2022, 7, 14 13 of 27 Coull and Hodson [10] pointed out the absence of directives in the calculation of the upstream displacement thickness d / H. This is a minor issue, as classical formulae or rea- sonable estimates could be resorted to. However, it is still unable to consider downstream endwall loss increase [18]. In a turbine environment, this issue becomes less relevant, as vortex ﬂow is disrupted by succeeding blade rows distanced by a short row gap. The threshold of Equation (30) is Z / H = 0.5, above which the merging of the endwall vortices te takes place and produces distinct ﬂow dynamics. An additional consideration is made on the aspect ratio H/c. Enlarging blade height H and shortening blade chord c both increase H/c but affect secondary loss in different ways [34]. This problem was raised in the early cascade tests of Kraft [52], which exposed the insensitivity of the secondary loss under the change of c. The traditional interpretation reﬂected by the auxiliary factor K in Equations (17) and (19) is that secondary loss is AR inversely proportional to H/c. This is valid when H is varied but no special regard to c has been made. However, as H/c approaches 0, K tends to inﬁnity and vice versa. This AR contradicts the real physics of the secondary ﬂow. Teia even showed that the structure of the endwall vortices is unaltered by the change of H beyond a certain critical aspect ratio in his Figures 10–12, page 25–26 [42]. He deduced that interaction between the endwall vortices is absent and, thus, secondary loss should remain constant. Below a certain critical aspect ratio varying H, the vortices merge together and amplify loss. His observation offers new insights into the highlighted inconsistency but lacks key veriﬁcation. Notably, correlation between vortices’ interaction and loss was not established and this can hardly argue secondary loss insensitivity for all H/c before merging. In addition, no means were provided to estimate his critical aspect ratio. Drawing partially on his observation, invariance to H/c above H/c = 2 in Equation (30) is imposed. H/c = 2 corresponds to the traditional threshold below which the slope of K becomes sharper. Concurrently, the AR database size bias of Equation (30) reduces its certainty beyond the same threshold [13]. At this stage, it is cautiously advanced that loss ampliﬁcation occurs even before the merging of the vortices within H/c 2. The problem with c remains unsolved. Replacing Equation (8), the basic secondary loss is 1 Z C te L,out Y = , s,b 2 H s/c (31) = f ( H/c, d / H, b , b , g, s/c, AVR, k), in out accounting for the pair of endwall vortices and their core located at Z /2H as in Figure 3b. te This respects the structure of lost work with loading and displacement parameters. The square of C /(s/c), as in Equation (26), is inherited from the classical lifting-line theory, L,out which associates the endwall and induced vortices. It yields values which are lower than Z by several order of magnitude, hence predominance of C /(s/c) over other parameters L,out is naturally prevented The independence to s/c in Equation (19) has been criticized, as this does not ﬁt reality [41,53]. Practice refers to the optimum s/c with proﬁle loss consideration only, there are still no available studies, to this day, that have addressed an overall optimum for both proﬁle and secondary losses [19]. The tangential force F of Equation (30) utilizes s/c as loading indicator and is unable to serve this purpose. As a reasonable attempt to incorporate the intensiﬁcation of secondary loss caused by deviation from the proﬁle optimum s/c, an auxiliary factor is proposed s/c K = 1 + 10 1 , sc s/c (32) = f (s/c, b , b , AVR, y , g). in out Z This polynomial expression was acquired by ﬁtting the cascade data of Perdichizzi and Dossena [41], as shown in Figure 4. Int. J. Turbomach. Propuls. Power 2022, 7, 14 14 of 27 2.5 1.5 0.5 0.6 0.8 1 1.2 1.4 Figure 4. Optimum pitch to chord ratio auxiliary factor for secondary loss. Tests were conducted for s/c = 0.58, 0.73, 0.87 and i 2 [60–+35] . The highlighted points result from averaging over the incidence range and normalizing by s/c with y = 0.6 such that a clear parabola could be drawn. The data of Hodson and Dominy [53] were also exposed and normalized with y = 0.9, for comparison. According to the trend, it is seen that their LPT proﬁles were not optimized at nominal s/c and, thus, their results are omitted. The magnitude of Equation (32) is restrained to 1.5 beyond the range [0.8–1.2] as a precaution, given the scarcity of data. Lastly, the enhanced secondary loss, keeping the same arrangement as Equation (19), is updated as 2 2 s,b Y = K K K , sc p s,Enh Re (33) 1 + 7.5Y s,b = f (s/c, y , g, M , M , Re , e/c, H/c, d / H, b , b , AVR, k). Z r,in r,out s in out Further discussions regarding recent works are carried on. Teia [42] stipulated that proﬁle loss should increase with H/c. He argued that increasing H incurs additional loss, as the boundary layer covers more surface area. This implies a 3D effect which violates the 2D spanwise uniformity of the proﬁle loss assumed in Equation (1). A similar attempt was done by Benner et al. [54], with their alternative loss breakdown te Y = Y 1 + Y . (34) t p s The boundary layer ﬂow was treated separately in accordance to the area division on the suction side in Figure 3b. They stated that their secondary loss correlation includes the proﬁle loss of the secondary regions, which is unlikely since their genetic algorithm search removed the blade skin friction during the development of their correlation. In that sense, the net proﬁle loss is conﬁned within the primary region delimited by S4. Although intuitive, these results are not compatible with older correlations, especially when the Y of Equation (34) is still that of KO. To ascertain the credibility of the update, it has to achieve statistical signiﬁcance and thus requires a large database. A major challenge in this framework is to gather a sufﬁcient number of public turbine test cases to possibly enable unbiased analysis. A previous study [23] collected 33 subsonic cascades (15 static and 18 rotating) partitioned from single stage turbines. The same approach is adopted and the previous cascade database is enlarged to 109 specimens (54 static and 55 rotating) to enhance representativeness. Their key design parameter range is summarized in Table 1. Int. J. Turbomach. Propuls. Power 2022, 7, 14 15 of 27 Table 1. Design parameter range of the cascade database. Parameters Min Max s/c 0.498 0.974 H/c 0.456 6.493 M 0.071 0.649 r,in M 0.149 1.280 r,out b + b 45.6 138.9 in out 4 6 Re 4.538 10 6.025 10 4 7 Re 5.487 10 1.1185 10 Although it was advanced that reliability should be prioritized over the accuracy in the early review, both aspects would be investigated. For this purpose, high ﬁdelity CFD simulations were conducted on the cascades and the solutions would serve as a benchmark in the analysis. Although the clearance leakage loss Y is not addressed in this paper, it is funda- cl mental to select a compatible leakage loss model to complete the system of Equation (1). For this purpose, the recent correlations of Yaras and Sjolander [55] or Farokhi [56] are recommended. 5. Numerical Method Since numerical steady solutions serve in determining Y , their credibility has to be guaranteed. At issue is their overriding dependence upon numerical discretization and turbulence modelling [57]. Without proper veriﬁcation and validation, any produced solutions would be untrustworthy [58]. In this regard, a previous study [23] adopted the systematic V and V procedure [59] to identify the numerical scheme (mesh and turbulence model) best suited for the simulation of turbine aerodynamics. It was performed on the Aachen turbine rotor test case [60] with NUMECA FINE/Turbo [61] and presented an on design numerical error of 3.79% the experimental value. The latter is considerably lower than the accuracy standard of 15% of the correlations [34,38], therefore justifying the use of CFD in the assessment. The computations were run with the commercial package NUMECA FINE/Turbo [61] whose code architecture is capable of maximum second order accuracy. In order to cap- ture anisotropy, and impact of curvature and body forces on turbulence, the proprietary separation sensitive corrected explicit algebraic Reynolds stress model (SSC-EARSM) was chosen. Compared to the conventional EARSM model [62], this was calibrated by scale adaptive simulation to increase turbulent mixing in the ﬂow separation region and enhance the near wall behavior of anisotropy. Single blade passage meshes were generated with the semi-automatic mesher AutoGrid5. O4H topology, which comprises batches of curvilinear structured hexahedral blocks, was adopted. A large portion of cells were clustered at the wall boundaries and plane intersections to enable, ﬁrst, inner cell spacing characterized by + 6 y 0.8 within the viscous sublayer. The employed mesh resolution was about 2.7 10 nodes. To ensure algorithm robustness, a uniform proﬁle of absolute total temperature and pressure was imposed at the inlet patch, placed at one chord upstream of the leading edge. This also guarantees the control of the endwall boundary layer over the distance separating the inlet patch to the blade leading edge and enables the use of classical formula in the d estimation in Equation (30). Static pressure was subsequently prescribed at the outlet patch placed at one chord downstream of the trailing edge. This distance should enable mixed out solutions at the outlet. Periodic boundary conditions were imposed on the circumferential patches of the control volume. All simulations were performed with real gas in on design conditions, unless speciﬁed otherwise. 6. Correlation Accuracy Although it has been advanced that correlation accuracy should not be prioritized, the extreme case with 100% error is not tolerable either. In that sense, the limit at which Int. J. Turbomach. Propuls. Power 2022, 7, 14 16 of 27 inaccuracy fails the correlation has to be evaluated. For this purpose, the previous gauge with the probability DP of reaching a tolerance interval of 15% error is reused [23], tol this is depicted in Figure 5. This conforms to the aforementioned accuracy standard of 15% [38] and accommodates numerical error uncertainties. DP tol E 10[ %] 3 2 0 2 3 4 1 1 4 Figure 5. Tolerance interval for correlation accuracy evaluation. The assessment relies on straightforward descriptive statistics. The latter require a bounded quantity following a near normal distribution and including Equation (1). Thus, the relative error normalized by the benchmark CFD solutions is Y Y t,num t E = . (35) t,num with the large database available, a z-distribution is assumed [63]. The update is assessed together with its original form in Figure 6. By comparing the predicted loss of each cascade, the depicted trend in Figure 6a indicates a higher estimation by the enhanced correlation. Distinct disparities occur at high loss levels, whereas most points are clustered below the bisection at low levels, within a 0.05 band. Hence, Ag and its update moderately agree for low loss only. As per the convention for scatter plots, the centered bar represents the mean and the other smaller and larger bars delimit the 99% conﬁdence interval and standard deviation, respectively. Figure 6b uncovers a signiﬁcant difference between Ag and its update, with approximately 10% distance separating their conﬁdence interval. Both populations have comparable spreads but that of Ag is largely decentered, unveiling a systematic underestimation from the Ag correlation. Their difference produces a coherent shift of its mean to the negative side. However, contrary to any expectation, its spread has not shrunk, implying that the update has inherently departed from its original model in terms of turbine aerodynamics interpretation. A priori, if accuracy has to be improved, then the correlation should have its relative error centered on 0 and standard deviation within 15%, to deliver at least DP = 66%. The numerical values are gathered in Table 2 and include the DP of each tol tol distribution. 0.5 100 0.4 0.3 0.2 0.1 -50 0 0.1 0.2 0.3 0.4 0.5 -100 Ag Enhanced (a) (b) Figure 6. Comparison of Ag and enhanced correlations. (a) Total loss. (b) Total loss relative error. Int. J. Turbomach. Propuls. Power 2022, 7, 14 17 of 27 Table 2. Statistical solutions of the axial cascade database. Method Descriptive Statistics Parameters E [%] s [%] DP [%] tol Ag 30.255 21.308 22.018 Enhanced 8.548 23.409 45.136 D 21.707 22.634 33.107 With the Ag correlation, it is possible to achieve the required accuracy with approx- imately 25% chance. This result is opposite to the previous analysis, which reported an optimistic DP = 45.13% while retaining a comparable s = 23.41 over 33 cascades [23]. tol It is very likely that the mean is susceptible to sample size bias, while the spread remains invariant. The enhanced correlation outperforms by doubling DP with a better position tol of its mean error. This result has to be handled with care, as this would change with another comparable cascade database with a shift of the mean error. On the other hand, the spread manifests as a barrier marking the limit of low ﬁdelity models. For this reason, accuracy should be relegated to higher order methods. 7. Correlation Reliability Adopting the approach of Horlock [64] and Coull and Hodson [32], trend consistency is evaluated with the ﬂow coefﬁcient f and stage loading y. The latter condition the turbine stage velocity triangle and efﬁciency. Their early selection is guided by the famous Smith chart [14], after ﬁxing the stage number. As a reminder, Figure 7 was established by tests on 70 cold single stage turbines which reactions varied from 20% to 60%. These were designed with AVR = 1 and zero incidence. Coull and Hodson supplemented the blade proﬁles associated to different areas of the chart. The top left area is characterized by the highest turning and lowest efﬁciency airfoils to satisfy the loading requirement. In this regard, accrued interblade passage convergence is required to guide reduced momentum ﬂows. Nevertheless, ﬂow turning can be alleviated for the same loading and efﬁciency by increasing f and thus transiting to the top right area. Eventually, the bottom area is characterized by low turning and higher efﬁciency airfoils and the same consideration concerning the ﬂow passage conﬁguration. Figure 7. Smith chart calibrated to zero tip loss and 50% reaction [14,32]. To demonstrate reliability, the enhanced correlation is expected to reproduce the same topology by varying f and y, as attempted by CC [17] and KO [5]. For this purpose, the analysis begins with the design of a single LPT stage using the setting of Coull and Hodson to acquire a common basis for comparison and explore the LPT for which the AM correlations are known to fail [10]. Repeating stage, 50% reaction, a constant hub to tip 0.94 0.89 0.92 Int. J. Turbomach. Propuls. Power 2022, 7, 14 18 of 27 ratio of 0.75 and parabolic camber are assumed. The ﬁrst stage power requirement and inlet boundary conditions are taken from the GE E3 LPT [65]. The design parameters s/c, b , velocity ratio VR= W /W and H/c of the LPT rotor are plotted in Figure 8 over out out in the same y and f range as the Smith chart, for reference. Consecutively, the LPT isentropic efﬁciency is derived. 3 3 2.5 2.5 2 2 1.5 1.5 1 1 0.6 0.8 1 1.2 0.6 0.8 1 1.2 (a) (b) 3 3 2.5 2.5 2 2 1.5 1.5 1 1 0.6 0.8 1 1.2 0.6 0.8 1 1.2 (c) (d) Figure 8. Velocity triangle and geometric parameter ranges of the LPT rotor. (a) Pitch to chord ratio s/c. (b) Exit ﬂow angle b . (c) Velocity ratio VR. (d) Aspect ratio H/c. out First, considering the proﬁle loss efﬁciency in Figure 9, both contour plots unsurpris- ingly share the same topology. Equations (2) and (21) differ by minor calibrations involving the Zweifel criterion [11] and extending the auxiliary factors to Equation (24). Furthermore, their bottom right area is identical, since Zweifel criterion matching is disabled for low turning proﬁles at f > 1.1. The abrupt break of the contour 0.02 arises from K , which s/c mod alters its value when departing from nozzle proﬁles at low y [37]. In Figure 9a, proﬁle loss is minimized for most of the bottom area, reﬂecting low turning proﬁles, as shown in Figure 8b. Greater y and fewer losses are feasible by increasing f but at expense of lower s/c, as in Figure 8a. Loss is aggravated towards the top left area, which conforms to Figure 7. Figure 9b stands out with its comparatively lighter gradient, induced by the adequate application of f . With the exception of the bottom right area and according to s/c the results of Coull and Hodson [32], a loss level comparable to that of the semi-analytical correlations of Denton [31] and Coull and Hodson [10], in Figure 10a,b, page 7, respectively, is achieved. Then, secondary losses differ in all possible aspects in Figure 10. The Ag secondary loss [6] remains low for most of the domain and increases towards the top left area in Figure 10a. This pattern is not shared by other AM family correlations [10]. Further insights are offered by analyzing the components of Equation (8). Here, l displays a topology A M similar to that of Figure 7 in Figure 11a. Its minimum is reached at the bottom left area 0.94 0.96 0.92 0.99 1.01 41.69 45.81 1.48 49.92 54.04 54.04 1.7 1.7 58.16 58.16 4.04 1.91 62.28 62.28 1.91 2.13 3.78 3.78 66.4 66.4 66.4 2.13 2.35 3.52 3.52 0.89 0.94 70.52 70.52 70.52 2.35 0.92 0.96 0.94 3.26 3.26 3.26 0.96 2.56 2.35 0.99 0.99 1.01 3 3 3 1.01 0.99 74.64 74.64 2.56 0.96 1.04 1.04 2.56 2.74 2.74 1.06 1.06 2.78 1.08 2.49 2.49 1.11 78.76 2.78 2.23 1.97 3.22 1.71 3.22 3.43 0.02 0.05 0.03 0.04 Int. J. Turbomach. Propuls. Power 2022, 7, 14 19 of 27 and, curiously, remains invariant towards the top right direction. Meanwhile, ampliﬁcation takes place in all other directions, switching from curved to straight contours. On the other hand, Z progressively grows accordingly, along y, to reach the steepest straight contour in the top left area in Figure 11c. With a difference of O(10 ), it naturally plays a predominant role in the product of Equation (8) and dictates the resulting topology. Thus, Figure 10a disregards most features of Figure 11a and highlights losses with contours that are curved by the auxiliary factors and the square root of Equation (19). Conversely, the lowest loss is produced at greater y and gradually intensiﬁes towards the bottom right area in Figure 10b. 3 3 2.5 2.5 2 2 1.5 1.5 1 1 0.6 0.8 1 1.2 0.6 0.8 1 1.2 (a) (b) Figure 9. Proﬁle loss efﬁciency with y = 1.1, AVR = 1, i = 0, H/b = 6.5, 4.6 for stator and rotor. (a) Aungier [6]. (b) Enhanced. 3 3 2.5 2.5 2 2 1.5 1.5 1 1 0.6 0.8 1 1.2 0.6 0.8 1 1.2 (a) (b) Figure 10. Secondary loss efﬁciency with y = 1.1, AVR = 1, H/b = 6.5, 4.6 for stator and rotor. (a) Aungier [6]. (b) Enhanced. The central question to be posed now is which correlation conforms to reality? Back to Figure 11a, l is supposed to reﬂect the endwall vortices. In this regard, the increase A M in the top left area is consistent, since it is backed by a sharper exit angle in Figure 8b and lesser aspect ratio in Figure 8d. However, the other increase in the bottom right area is unlikely. It is impossible to magnify the endwall vortices by decreasing turning/loading, increasing aspect ratio and also lowering VR in Figure 8c. Moreover, a higher f should not favor the growth of secondary ﬂows in the ﬂow passage, as these are downwashed through considerable momentum convection [7]. With these points in mind, it is seen that the monotonous penetration depth of Figure 11b better matches the topology of Figure 8b–d. The contour slope disruption in the top left area is triggered by the aspect ratio variation for H/c < 2, ensuring discussion over the results of Teia [42]. Once again, there is a clear divergence between the blade loadings in Figure 11c,d. Contrasting with Figure 11c, the maximum is located in the bottom right area. This feature comes from the s/c at the denom- inator in Equation (28) and is substantiated by Figure 8a, in which the minimum occupies 0.03 0.04 0.05 0.02 0.05 0.05 0.04 0.06 0.08 0.05 0.06 0.07 0.06 0.05 0.02 0.06 0.05 0.02 0.02 0.04 0.04 0.04 0.03 0.03 0.03 0.02 0.03 0.03 0.04 0.03 0.02 0.07 0.08 0.08 0.05 0.06 0.05 0.02 0.02 0.06 0.04 0.09 0.07 0.03 0.04 0.02 Int. J. Turbomach. Propuls. Power 2022, 7, 14 20 of 27 the same area. Z originally accounted for s/c as uncovered in Equation (13) but dispensed with its contribution, as conﬁrmed herein. Furthermore, secondary loss increase under high f and low y was also produced by the trusted CC [17] and Traupel [25] correlations [10]. Despite the absence of similarity with Figure 7, consistency in the enhanced correlation is comparatively augmented. In particular, the difference between the components of Equation (19) is mitigated to O(10). 3 3 2.5 2.5 2 2 1.5 1.5 1 1 0.6 0.8 1 1.2 0.6 0.8 1 1.2 (a) (b) 3 3 2.5 2.5 2 2 1.5 1.5 1 1 0.6 0.8 1 1.2 0.6 0.8 1 1.2 (c) (d) Figure 11. Rotor secondary loss components with y = 1.1, AVR = 1, H/b = 4.6. (a) Ainley parameter l . (b) Vortex penetration depth Z /2H. (c) Ainley loading Z. (d) Generalized lift te A M solidity coefﬁcient C /s/c. Eventually, the isentropic efﬁciency key to correlation reliability is depicted in Figure 12. The trend of Ag in Figure 12a is clearly driven by the proﬁle loss in Figure 9a. In that respect, the bowed contours that are supposed to be inscribed in the domain of Figure 7 are stretched towards higher f. As a consequence, losses induced by large f are under- estimated by the correlation. In addition, the update reveals to be more conservative in the quantiﬁcation of the losses, as the ﬁrst identiﬁed contour is 0.91 in Figure 12b. The latter exhibits an almost complete bow but is not comparable to those of the Smith chart, as it covers too much area. Hence, it can only enable a global trend consistency. Although overstretching to high f is reduced, its expansion occasions over and under estimations in the bottom and top areas, respectively. Variation of contour level or sensitivity to design parameters is lessened. These weaknesses are occasioned by the secondary loss in which H/c = 2 is held constant for most of the domain, as shown in Figure 11b. This is somewhat expected, as Equation (30) of Benner et al. [13] was not intended for the current treatment regarding H/c. The model’s conservativeness could be alleviated through a larger H/c, if justiﬁed. For now, the threshold H/c = 2 is untouched, as the reasonable conservativeness in preliminary design provides an adequate margin for subsequent modiﬁcations. 1.43 1.55 1.37 1.49 0.1 1.37 1.19 1.31 1.25 0.11 0.0058 0.09 0.09 3.22 3.22 0.0056 0.0054 0.0052 0.1 0.005 0.0052 0.0048 4.84 0.005 4.84 0.11 0.0046 0.0048 0.12 0.0046 0.0042 6.47 6.47 6.47 0.0044 0.12 0.0044 0.0044 8.09 8.09 1.49 8.09 1.43 1.55 0.0044 1.37 1.31 1.43 1.31 1.25 1.25 9.71 9.71 0.0046 1.37 1.19 1.19 0.005 1.25 1.13 0.0048 1.13 0.0052 11.34 11.34 1.07 0.13 1.01 12.96 14.59 0.0042 16.21 1.31 1.49 0.91 Int. J. Turbomach. Propuls. Power 2022, 7, 14 21 of 27 3 3 2.5 2.5 2 2 1.5 1.5 1 1 0.6 0.8 1 1.2 0.6 0.8 1 1.2 (a) (b) Figure 12. Isentropic efﬁciency with y = 1.1, AVR = 1, H/b = 6.5, 4.6 for stator and rotor. (a) Aungier [6]. (b) Enhanced. The enhanced correlation has introduced additional design parameters or degrees of freedom, in hopes of reaching a better control on design. These are summarized in Appendix A Table A1 and their number amounts to three to ﬁve for the proﬁle and secondary losses, respectively. The following analysis retains four static and rotating cascades [26,60,65,66] with distinct k and with their operating conditions altered with a back pressure within10% of its nominal value. This aims to evaluate the change of Y over the prescribed operation range, drawing on the approach of Wei [9]. Again, CFD solutions form the benchmark. AVR determined in the early velocity triangle and knowingly set to unity in the AM correlations is concurrently highlighted to demonstrate its relevance as a design parameter. The results of the static and rotating cascades normalized by on design CFD values are depicted in Figure 13. The cascades are not identiﬁed to their references in the plots to foster randomness, as reliability must not be biased by the turbine origin. 1.8 1.2 1.6 1.4 1.2 0.8 0.8 0.6 0.6 0.9 0.95 1 1.05 1.1 1.15 0.9 0.95 1 1.05 1.1 1.15 (a) (b) Figure 13. Total loss variation of four cascades. (a) Stators. (b) Rotors. The distance separating each pair of curves relates to accuracy. It is worth mentioning that two out of four cascades achieve the early tolerance interval and this conform with the estimated DP = 45%. AVR varies for different operating conditions and extends as far tol as 15% its nominal value in Figure 13b. This parameter has demonstrated its nontriviality and hence justiﬁes the generalization of Equations (22) and (28). By diminishing the back pressure, AVR increases and vice-versa. For the static cascades in Figure 13a, the trend of each pair of curves similarly results in an acceptable global consistency. If strict consistency is imposed, then the curves should have synchronous slopes that conserve their separation at each variation. Whereas, for the rotating cascades in Figure 13b, serious trend disruptions are displayed introducing undesirable uncertainties. One cascade undergoes slope switchover to negative values at AVR = 1.05. Moreover, there are two cascades for 0.95 0.94 0.93 0.91 0.9 0.91 0.96 0.91 0.96 0.96 0.94 0.95 0.88 0.89 0.96 0.92 0.95 0.93 0.94 0.94 0.95 0.91 0.93 0.92 0.91 0.9 0.89 0.91 Int. J. Turbomach. Propuls. Power 2022, 7, 14 22 of 27 which a considerable slope mismatch is seen. The ﬁrst starts from AVR = 1.05 along the back pressure increase. The second even covers the entire range around AVR = 1. Contrasting with Figure 13a, the reason for these divergences becomes rather straightforward. CFD can capture the impacts of rotation on secondary ﬂows and, thus, loss mechanism, whereas correlations built upon cascade tests and calibrated to turbine data cannot. As a matter of fact, parameters or factors relating to rotation are absent in the correlations. In this view, reliability is globally guaranteed in stators and compromised in rotors because of rotation. Clearly, this observation constitutes the most sound argument against the use of cascades as benchmark of the correlations, as presented in the early review. 8. Conclusions In this paper, historical insight was provided into the numerous features of the AM correlations, whose most authoritative contributions were given by AM, DC, KO, and Ag. These models were established empirically using measurements on turbine cascades and deliver a standardized solution for the conﬁgurations considered in the experiments. As a consequence, the range of applications of the models are biased towards the types of turbine cascades used to develop the correlations. An important restriction of the AM correlations is related to the proﬁle loss, which is established through interpolation between nozzle and impulse blade empirical charts. This implies the prior setting of the velocity distribution over the blade surface, which follows from measurements on a restricted set of turbine cascade conﬁgurations. Another restriction is related to the secondary loss, which draws on the classical lifting-line theory of Carter and associates wingtip trailing vortices with blade passage vortices. Further investigation revealed a mathematical mistake in the secondary loss formulation and an undesirable link with the proﬁle loss. In light of these issues, an update to the proﬁle and secondary losses was proposed in this work, building upon the correlations of Ag. For the proﬁle loss, the addressed points are: • The substitution of metal angle q by the ﬂow angle b . in in • The scaling of the AM optimum pitch to chord s/c by the Zweifel optimum pitch A M to chord s/c . This, notably, preserves of the slope of the AM proﬁle losses Y Z p,b =0 in and Y and adapts the AM correlation to other design rules. p,b =b in out • The use of a Reynolds number Re based on the suction surface length. • The substitution of the auxiliary factor K by a more suitable incidence loss Y , i inc proposed by Benner et al. • The reuse of the KO trailing edge loss Y affected by the profile loss auxiliary factors. te,KO As for the secondary loss, the addressed points are: • The replacement of the classical secondary ﬂow model of AM and Carter by the endwall vortices model of Sharma and Butler. • The use of the product of the elemental lift solidity coefﬁcient C /s/c and vortex L,out penetration depth Z instead of the heuristic Ainley loading Z and parameter l . te A M • The invariance of the auxiliary factor K beyond the threshold aspect ratio H/c = 2. AR • The deﬁnition of a new auxiliary factor K , to determine the variation of the secondary sc loss with the pitch to chord ratio s/c. With a database of 109 cascades available, the prediction of the new correlation is benchmarked against the cascade numerical solution. Using descriptive statistics and expressing accuracy as the probability of achieving a relative error within 15%, the im- provement yields DP = 45% against the DP = 22% of Ag. Although there is a clear tol tol improvement, these results are largely dependent on the choice of a ﬁnite database for which the representativeness of the population is unknown. Nevertheless, this quantiﬁ- cation should provide an estimate of the results and raise conﬁdence in the use of the enhanced correlation. Regarding reliability, the new correlation ensures global consistency by reproducing the Smith chart [14] and off design stator loss variation. In addition, the analysis points out a correlation conservativeness in the design of the typical LPT, which is Int. J. Turbomach. Propuls. Power 2022, 7, 14 23 of 27 due to insensitivity to the secondary loss correlations beyond H/c = 2. Severe trend dis- ruptions are identiﬁed during the application of the proposed enhanced correlation on four rotating cascades, the mismatches with CFD compromise its reliability in rotating ﬂows. Author Contributions: Conceptualization, Y.L.; methodology, Y.L.; software, Y.L.; validation, P.H. and Z.Z. and F.B.; formal analysis, Y.L.; investigation, Y.L.; resources, P.H. and Z.Z.; data curation, Y.L.; writing—original draft preparation, Y.L.; writing—review and editing, Y.L.; visualization, Y.L.; supervision, P.H. and Z.Z. and F.B.; project administration, P.H. and Z.Z.; funding acquisition, P.H. and Z.Z. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: The cascade dataset and its CFD results are available on demand. Conﬂicts of Interest: The authors declare no conﬂict of interest. Nomenclature Acronyms Ag Aungier AM Ainley and Mathieson BSM Benner and Sjolander and Moustapha CFD Computational ﬂuid dynamics CFL Courant–Friedrich–Levy DC Dunham and Came E3 Energy efﬁcient engine KO Kacker and Okapuu LPT Low pressure turbine MKT Moustapha and Kacker and Tremblay VandV Veriﬁcation and validation Z Zweifel Symbols b Relative ﬂow angle [ ] DF Kinetic loss coefﬁcient [-] DP Tolerance interval probability [-] tol d Endwall displacement thickness [m] g Stagger [ ] k Meanline radius ratio [-] l Ainley parameter [-] A M f Flow coefﬁcient [-] y Blade loading [-] y Zweifel loading [-] s Standard deviation [-] q Metal angle [ ] A Cross section area [m ] b Axial chord [m] c True chord [m] C Lift coefﬁcient [-] C Secondary drag coefﬁcient [-] D,s d Annulus diameter [m] d Leading edge diameter [m] LE E Relative error [-] e Surface roughness [m] e Back curvature [m ] c Int. J. Turbomach. Propuls. Power 2022, 7, 14 24 of 27 F Tangential loading [-] f Scaling factor [-] s/c H Blade height [m] H Distance to endwall vortices [m] i Incidence [ ] K Auxiliary factor [-] k Heat ratio coefﬁcient [-] K Aspect ratio correction [-] AR K Incidence factor [-] K Technology factor [-] mod K Expansion correction factor [-] K Proﬁle correction factor [-] K Mach correction factor [-] K Reynolds correction factor [-] Re K Pitch to chord ratio correction [-] sc K Trailing edge correction factor [-] te M Mach number [-] o Throat width [m] p Pressure [Pa] r Meanline radius [m] Re Reynolds number [-] Re Suction length Reynolds number [-] s Pitch [m] t Maximum thickness [m] max t Trailing edge thickness [m] te W Relative velocity [m/s] We Leading edge wedge [ ] LE Y Proﬁle loss [-] Y Secondary loss coefﬁcient [-] Y Total loss coefﬁcient [-] Y Tip clearance loss coefﬁcient [-] cl Y Supersonic loss coefﬁcient [-] ex Y Incidence loss coefﬁcient [-] inc Y Shock factor [-] sh Y Trailing edge loss coefﬁcient [-] te Z Ainley loading [-] Z Vortex penetration depth [m] te AVR Axial velocity ratio [-] CR Convergence ratio [-] VR Velocity ratio [-] Superscripts Average Subscripts Enh Enhanced hub Hub in Inlet is Isentropic m Mean n Normal num Numerical out Outlet r Relative t Total ti p Tip z Axial Int. J. Turbomach. Propuls. Power 2022, 7, 14 25 of 27 Appendix A Table A1. Dimensionless parameters of the AM correlations. Parameters Y Y Y Y Y Y Y Y Y p,AM s,AM p,Ag s,Ag te,Ag ex,Ag p s te s/c 2 2 2 2 t /c 2 2 2 max q 2 2 2 2 2 in q 2 2 out b 2 2 2 2 in b 2 2 2 2 2 2 2 out t /s 2 te o/s 2 e /s 2 2 M 2 2 2 2 2 r,in M 2 2 2 2 2 2 r,out Re 2 2 Re 2 2 2 e/c 2 2 2 2 2 k 2 2 i 2 2 2 d /s 2 LE We 2 LE A / A 2 out in r /r 2 hub ti p H/c 2 2 AVR 2 2 y 2 2 t /o 2 2 te d / H 2 g 2 2 k 2 Reference Equation (2) Equation (8) Equation (3) Equation (19) Equation (7) Equation (4) Equation (20) Equation (33) Equation (24) Total 5 5 12 8 3 1 17 13 7 References 1. Zou, Z.P.; Wang, S.T.; Liu, H.X.; Zhang, W.H. Axial Turbine Aerodynamics for Aero-Engines: Flow Analysis and Aerodynamics Design, 1st ed.; Springer Nature: Singapore, 2017. 2. Ainley, D.; Mathieson, G. A Method for Performance Estimation for Axial-Flow Turbines; Technical Report R&M No. 2974; Aeronautical Research Council: London, UK, 1951. 3. 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References (33)

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Abstract

International Journal of Turbomachinery Propulsion and Power Article A Reliable Update of the Ainley and Mathieson Proﬁle and Secondary Correlations 1, 1 2 3 Yumin Liu * , Patrick Hendrick , Zhengping Zou and Frank Buysschaert Department of Aero-Thermo-Mechanics, Université Libre de Bruxelles, 1050 Brussels, Belgium; patrick.hendrick@ulb.be School of Energy & Propulsion Engineering, Beihang University, Beijing 100191, China; zouzhengping@buaa.edu.cn Department of Applied Mechanics & Energy Conversion, Katholieke Universiteit Leuven, 8200 Brugge, Belgium; frank.buysschaert@kuleuven.be * Correspondence: yu.m.liu@ulb.be Abstract: Empirical correlations are still fundamental in the modern design paradigm of axial turbines. Among these, the prominent Ainley and Mathieson correlation (Ainley D. and Mathieson G., 1951, “A Method of Performance Estimation for Axial-Flow Turbines,” ARC Reports and Memoranda No. 2974) and its derivatives, plays a crucial role. In this paper, the underlying assumptions of the aforementioned models are discussed by means of a descriptive review, whilst an attempt is made to enhance their reliability and, potentially, accuracy in performance estimations. Closer investigation reveals an intriguing misuse of the lift coefﬁcient in the secondary loss. In light of this, an enhanced model that, notably, builds upon the Zweifel criterion and the vortex penetration depth concept is developed and discussed. The obtained accuracy is subsequently assessed through CFD computations, employing a database comprising 109 cascades. The results indicate a 50% probability of achieving the 15% error interval, which is twice as good as the most recent Aungier model (Aungier R., 2006, “Turbine Aerodynamics: Axial-Flow and Radial-Inﬂow Turbine Design and Analysis”, ASME Press, New York). Furthermore, the reliability of the proposed model is demonstrated by a reconstruction of the Smith chart, on the one hand, and a performance analysis, Citation: Liu, Y.; Hendrick, P.; Zou, on the other. The reconstruction exhibits contours that conform to the original. The results of the Z.; Buysschaert, F. A Reliable Update performance study are compared with the CFD solutions of eight cascades working in off design of the Ainley and Mathieson Proﬁle conditions and conﬁrm the need of the additionally included turbine design parameters, such as the and Secondary Correlations. Int. J. axial velocity and the meanline radius ratios. Turbomach. Propuls. Power 2022, 7, 14. https://doi.org/10.3390/ Keywords: loss correlation; axial-ﬂow turbine; turbine performance ijtpp7020014 Received: 2 December 2021 Accepted: 3 April 2022 Published: 21 April 2022 1. Introduction Publisher’s Note: MDPI stays neutral The axial turbine became incontestably the accustomed device for medium to high with regard to jurisdictional claims in mechanical power generation in modern powerplants and therefore, not surprisingly, it published maps and institutional afﬁl- still remains the subject of extensive research. Throughout the past decades, experience iations. and knowledge have grown continuously, adding reﬁnements to the art of turbine design. In this study, the 1D aerodynamic design, more speciﬁcally, the loss correlations employed in the meanline analysis used in the early design phase, are reviewed. These models are crucial, as they set the preliminary design parameters that can be maintained up to the ﬁnal Copyright: © 2022 by the authors. design stage, that is, if the model incorporates sufﬁcient levels of reliability [1]. Among Licensee MDPI, Basel, Switzerland. these, it is evident to note the renown Ainley and Mathieson (AM) correlation, published This article is an open access article in 1957, which serves this purpose [2] well. For the record, Ainley and Mathieson’s model distributed under the terms and conditions of the Creative Commons was considered to be a signiﬁcant innovation, as it enabled the quantiﬁcation of local Attribution (CC BY-NC-ND) license losses. It was, e.g., successfully employed in the design process of the Rolls Royce Olympus (https://creativecommons.org/ engine [3]. Being the cradle leading to great successes and broad acceptance, it underwent licenses/by-nc-nd/4.0/). multiple updates [4–6], which expanded into a veritable genealogy. Int. J. Turbomach. Propuls. Power 2022, 7, 14. https://doi.org/10.3390/ijtpp7020014 https://www.mdpi.com/journal/ijtpp Int. J. Turbomach. Propuls. Power 2022, 7, 14 2 of 27 Over time, the design of axial turbines underwent several improvements and the geometry became much more diverse, causing the accuracy and reliability of the current historic correlation based (open source) 1D models to become less effective. An important issue is their mathematical foundation, which relies on a fuzzy blend of classical ﬂow theory and heuristic manipulations matching the early cascade measurements [7]. Hence, it is not surprising that modern turbines may become excluded from their range of applica- tion. In light of this, studies have evaluated the correlation reliability through parametric analysis [8–10]. Although inconsistencies have been identiﬁed and reported, no remedies have been proposed. Accuracy has been evaluated on few test cases, which is insufﬁcient to afﬁrm any statistical signiﬁcance. These two aspects (reliability and accuracy) have to be investigated with statistical evidence to clarify the true potential of the available 1D correlations with the latest turbines. Moreover, it is deemed mandatory to upgrade the correlations with a more robust physics based backbone, compatible with modern turbines. The objective of this paper is, therefore, twofold. The ﬁrst is to provide a comprehen- sive review of the AM family correlations. The second is to enhance the AM correlation based on recent breakthroughs in turbine aerodynamics and loss mechanisms, and boost- ing compatibility with current design workﬂows. In particular, the Zweifel criterion [11] and the depth penetration [12,13] are integrated into the latest Aungier (Ag) proﬁle and secondary loss models [6]. The latter are validated on a data set composed of 109 cascades (static and rotating) to achieve statistical signiﬁcance in terms of accuracy and reliability. High ﬁdelity simulations and their numerical solutions serve as the benchmark during the evaluations. The correlation accuracy is determined in terms of probability according to the Gaussian distribution of the relative errors, whilst the correlation reliability is proven by its ability to reproduce the Smith chart [14] and maintain consistent trends in off design conditions under the variation of the newly introduced axial velocity ratio. 2. Review The importance of empirical loss correlations still remains high in the design process of modern axial turbines. Despite their restricted ﬁdelity, they can effectively offer guidance in decision making during the preliminary design stages [1]. In general, the development of empirical models is built upon on a turbine database, which is produced by means of experimental tests where a multitude of measurements are performed and linked to several, mostly dimensionless, design parameters. When parameters are found to correlate well, dedicated design laws can consequently be established. Quite recently, there was a spirited debate on the capability of cascade tests to approxi- mate real turbine ﬂow. The proponents claimed that cascades could offer a satisfactory basis for estimating losses in real machines and should be preferred thanks to their simplicity, ﬂexibility and low cost [15]. The opponents, however, believe that cascades are inherently limited since they cannot reproduce key aspects of the real operation environment, involv- ing, e.g., rotational, curvature and inlet ﬂow distortion effects [16]. Despite this contention, the current state of the art research has clearly joined the supporting side, considering the solid foothold of cascade tests in public literature. The authors of this work adhere to the standpoint of Craig and Cox (CC) [17], which holds a certain relativism. In fact, the extreme situation in which application of the correlations fails in both accuracy and reliability is rather unlikely. However, careless use must be avoided, as this could lead to an entirely erroneous result [18,19]. Hence, the user has to be fully aware of the features and limitations involved, so that the correlations can be applied effectively in real turbine design. This aspect is the crux of the following discussions. Correlations are typically determined by the investigator ’s own judgment, experience and interpretation of ﬂow physics, test quality and database size [20]. The challenge resides in the way the wide information of a complex 3D ﬂow is ﬁltered and condensed into a 1D empirical model. During its setup, certain parameters could be deliberately biased, with the objective to reﬂect speciﬁc turbine characteristics. The reason behind this intervention is quite straightforward. For example, aero-engine and land based steam turbines do not Int. J. Turbomach. Propuls. Power 2022, 7, 14 3 of 27 share the same design requirements, as is the case with high and low pressure turbines. They differ by design and aerodynamics and so will the correlations. The comparative studies of several public domain correlations [9,21] emphasized this issue clearly. As a consequence, there are no universal turbine correlations at this point. Hence, they only serve a speciﬁc range of applications adequately. Since it is common to use empirical models for a wide range of turbine design applica- tions, it is not surprising that the parameters estimated from the cascade correlation may disagree with real machine measurements by a considerable order of magnitude. In these circumstances, one resorts typically to a heuristic calibration. Otherwise, the correlation is of little value [22]. However, there are some risks involved with such an approach. First, simple multipliers/polynomials are often deemed sufﬁcient, even though the resulting correlation would much better suit turbines akin to those used in the database from which the model was established. Second, its use inherently leads to designs whose aerodynamics are similar to that of the turbines in its original database. The fundamental issue is the statistical bias engendered by the limited size of the database. This could be prevented by the construction of a large public database collecting all cascade measurements performed up to the present day [7]. Unfortunately, early investigators did not share or even lost their databases. New investigators often had to generate a new database from the ground up, with neither access to previous data, nor methodologies. In particular, insights that could have revealed a stronger interaction between design parameters, were curtailed. As a consequence, 1D correlations revert to quite simple mathematics or charts and are biased, which, thus, leads to results with rather low ﬁdelity. In contrast, the models established in large industries do not suffer from the aforementioned issues, where one can afford the calibration of public domain or in-house correlations using the vast amount of data accumulated over time for dedicated conﬁgurations. From this data, new versions of earlier models can be developed, which enables a safer and cheaper enhancement of the design process. In [23], it was advanced that any loss correlation exhibits the same probability to accurately predict turbine performance once out of its range of application. Considering the current discussion, it is rather irrational to expect outstanding accuracy from models that attempt to cover complex systems with elemental information. With full awareness of the higher ﬁdelity 3D methods intervening at later design stages, correlation accuracy should be less prioritized, but the focus must initially go towards including all relevant design parameters in the model. Therefore, the ability to cope coherently with the true physics, i.e., reliability, must become the primary concern. 2.1. Ainley and Mathieson Family Correlations The empirical AM correlation [2] has been so successful that it owns a lineage (the AM correlation and its derivatives will not be differentiated in this framework, as these are fundamentally equivalent). Chronologically, it was established through cascade tests using conventional proﬁles stemming from the 1940s and claimed a nominal accuracy of 2% in stage efﬁciency estimation. The latter corresponds to about 15% total pressure loss error. It was published at a time when the expertise and the understanding of axial compressors were more profound than those of turbines [24]. As a matter of fact, before AM was published, one had to resort to expensive experimental trial and error processes when the available methods revealed efﬁciencies largely inferior to those of the actual turbines. The AM correlation was, therefore, considered to be groundbreaking, as it provides a systematic way to quantify turbine aerodynamic loss and its components. The Dunham and Came (DC) correlation [4] was the ﬁrst and a canonical update to the model of AM, which used 16 1960s technology Rolls Royce turbines. It extended the ap- plication range to small turbines, while retaining the same accuracy level. Dunham claimed that their model was still the most reliable design tool during the 1970s, in a context where there was a progressive adoption of potential ﬂow calculation in industry [3]. However, the investigators did not have access to the database of AM and, thus, the improvement Int. J. Turbomach. Propuls. Power 2022, 7, 14 4 of 27 resulted from model calibration based on the overall performance of a restricted number of turbines. It is noteworthy to mention that CC [17], along with Traupel [25], published new correlations with different approaches with regard to turbine aerodynamics in that same period. The Kacker and Okapuu (KO) correlation [5] came as a second update based on 33 1980s technology Pratt & Whitney turbines and showed an accuracy up to 1.5% in stage efﬁciency estimation in design conditions. Furthermore, KO, having high conﬁdence in the validity of the Smith chart [14] which contains 70 1960s technology Rolls Royce turbines [26], calibrated their correlation to ﬁttingly reproduce the same chart. Ag [6] sub- sequently revised KO to handle recompression produced in extreme off design conditions and real working ﬂuids. In addition, it can cope with hub to shroud S2 calculations (the hub to shroud S2 and blade to blade S1 calculations proposed by Wu [27] refer to stream surfaces on which 2D ﬂow equations are solved by mass-averaging the third coordinate). This has recently been conﬁrmed, notably in ORC turbines [28,29]. Despite many the critics of its limitations, the choice of the AM correlation over other public correlations is driven by three particular aspects. Firstly, it owns a comparatively broader mathematical basis and is less reliant on empirical charts. Secondly, it does not require speciﬁc assumptions on the blade surface velocity distribution nor the need to perform cumbersome mathematical computations. As a consequence, faster solution and better accessibility are ensured, in contrast to the semi-empirical models of Baljé and Binsley [30], Denton [31] and Coull and Hodson [32]. Lastly, transparent and valuable details of its development can be retraced in the successive works [2,24,33,34]. There are ﬁve sources of total pressure loss encountered in aerodynamics, notably, skin friction, pressure drag, shock, leakage and mixing losses. However, this classiﬁcation was rearranged to associate these losses to their location instead of their phenomenological origin. The AM family correlations assumed the turbine loss system as follows: p p tr,out,is tr,out Y = , p p tr,out out (1) = Y + Y + Y . p s cl • Proﬁle loss Y is generated by the growth of the boundary layer and ﬂow mixing over the blade surface. This encompasses skin friction and pressure drag. In case the exit ﬂow reaches supersonic velocities and triggers shocks near the trailing edge, then an additional supersonic loss Y is to be accounted for. These losses exclusively arise ex from blade to blade S1 ﬂow and thus assume spanwise uniformity. • Secondary loss Y originates from the interaction between endwall ﬂow and the pressure difference across the blade passage. The main actors are the 3D induced vortices and endwall boundary layers. AM, drawing on the model of Carter [35], regarded both ﬂows and the annulus loss as one entity. This local loss is responsible for spanwise non-uniformity at blade tip and hub. • Tip leakage loss Y arises from the presence of a gap between the blade tip and the cl shroud, preventing the rotating blade from rubbing against the casing. Although this is indispensable to complete the loss system, this loss is not considered within the scope of this work, where priority is given to the more fundamental proﬁle and secondary losses. Although it was not speciﬁed, the correlations implicitly refer to losses of fully mixed ﬂow. However, to be precise, there is neither theoretical nor experimental evidence that can justify the breakdown of Equation (1), as this inherently assumes little interaction between the components [36]. However, its effectiveness might be due to the comparatively small chord of axial turbine blades and the short ﬂow transit time in the blade passage inhibiting strong interaction between the secondary ﬂows (all ﬂows that differ from the primary inviscid ﬂow) [7]. Int. J. Turbomach. Propuls. Power 2022, 7, 14 5 of 27 2.2. Proﬁle Losses The AM 2D proﬁle loss [37] is in q q 5t out max in in Y = Y Y Y , p,A M p,q =0 p,q =b p,q =0 in in out in (2) b b c out out = f (s/c, t /c, q , b ). max out in AM blended the proﬁle losses of a slightly compressible 50% reaction nozzle Y p,q =0 in and impulse nozzle Y , as depicted in their Figure 4 on page 24 in [2], to cover all p,q =b out in intermediate designs. This correlation inherently comprises the trailing edge loss taken at standard t /s = 0.02. Equation (2) was obtained from low speed wind tunnel tests te and focuses on parabolically cambered conventional proﬁles, i.e., the British T.6 type proﬁles. Their experimental results showed the notable sensitivity of the losses to the proﬁle thickness in impulse proﬁles and, thus, a correction for any deviation from standard t /c = 0.2 [34] was introduced in the model. In addition, there was a variation according max to (q /b ) for proﬁles in-between the impulse and nozzle blades, hence justifying the out in arrangement/blending in Equation (2). The velocity distribution of the nozzle and impulse blades was presented in AM’s Figure 5 on page 26 in [34]. This distribution shows that a considerable portion of the suction side of the blade extending from midchord to trailing edge is affected by ﬂow diffusion, which is nowadays regarded as unacceptable in design standards. Use of AM would, therefore, implicitly assume a similar velocity distribution over modern turbine proﬁles, which is likely to result in a signiﬁcant loss mismatch if not handled with care [38]. This speciﬁc problem with associated implications has to be addressed when enhancing Equation (2). In order to improve the reliability of the model of AM for recent turbine conﬁgurations, several correction factors—in this work designated as the auxiliary factorsK—have been proposed by different authors [4,5]. For example, in his latest update, Ag [6] adapted the proﬁle loss correlation of KO (and, thus, AM). This equation possesses several auxiliary factors and is expressed as: Y = K K K K K Y Y + Y , p,Ag mod M p Re i p, A M t =0.02s sh te (3) = f (e /s, q , q , M , M , Re, e/c, k, i)Y . c out r,out in r,in p,A M • Mach correction factor K 1 reﬂects the ﬂow acceleration in the blade passage reducing viscous loss and acts in the subsonic range M 1. According to Ag [6], r,out K , introduced for the ﬁrst time by KO, is ﬂawed in extreme off-design conditions, especially when recompression occurs at the blade hub. It was known that KO yielded only satisfactory results at the design point [15]. Wei [9] tested KO and exposed a curious slope disruption and a nonphysical reduction in the predicted proﬁle losses in cases of higher positive incidences i resulting in recompression, because of spurious K values. It is also noteworthy that the impact of the Mach number was investigated by AM [24] and is consistent with the purpose of K , but it was, surprisingly, not adopted in the earlier Equation (2). • Expansion correction factor K 1 deals with local weak shocks on the blade suction side in the subsonic exit Mach number band M 2 [0.6–1] and was rendered r,out more physics driven. This parameter was ﬁrst introduced by DC [4] for supersonic ﬂows and was derived from the overall turbine performance established by means of experiments. • Reynolds correction factor K applies to Reynolds numbers outside the transition Re 5 5 range [10 –5 10 ] and includes surface roughness, as seen in the CC correlation [17]. A standard ﬁnish of e/c = 10 is assumed throughout the analysis. • Technology correction factor K copes with the technology mismatch of Y when mod p,A M examining the losses in post-1980s turbine proﬁles [37]. K takes the value of 0.825 mod for meridional entry proﬁles and 2/3 for the other. Int. J. Turbomach. Propuls. Power 2022, 7, 14 6 of 27 • Incidence correction factor K 1 conditions the proﬁle loss in off design conditions and was proven to be successful in pre-1980s turbines [39] but overly conservative in modern turbines [15]. In summary, the purpose of the auxiliary factors is to calibrate Equation (2) with its underlying velocity distribution, without the need for an in depth understanding or elaboration of the loss mechanisms. This is evidently caused by a lack of knowledge transfer between AM, DC, KO and Ag [7]. The shock factor Y 0 accounts for local sh shocks situated on hub proﬁles with a thicker leading edge under comparatively high inlet velocities and is exclusively considered as an independent loss contributing to the proﬁle loss, as introduced by KO. However unlike that of KO, it is unaffected by the auxiliary factors. If M 1, then K and K are held at M = 1 and the supersonic loss Y r,out p r,out ex has to be supplemented to Y in Equation (3). Ag proposed M 1 r,out Y = . (4) ex,Ag r,out It must be recognized that this is a rather arbitrary formulation. However, this is re- tained throughout the analysis because there are no more reliable alternatives at the moment. As mentioned earlier, Equation (2) encompasses the trailing edge loss for standard t /s = 0.02. For any variation of t /s, AM proposed a trailing edge correction factor te te affected to Equation (1), such that Y = K (Y + Y + Y ). (5) t te p s cl This expression interrelates the trailing edge to the proﬁle, and secondary and tip clearance losses and was criticized by KO for its weak physical soundness regarding the latter two options. In fact, neither of the endwall nor tip leakage vortices remain closely attached to the blade surface upon reaching the blade exit, rendering interaction between the vortices and the trailing edge wake ﬂow unlikely. Instead, KO employed a more intuitive approach and proposed the trailing edge loss Y as an independent te component [5], such that Y = Y + Y + Y + Y . (6) t p s te cl However, Y still remains part of Y in a broader sense, so that Equation (1) is not te p contracted. To enable this expression, one has to virtually bring Equation (2) to t /s = 0 te beforehand. However, Ag carried on this approach and formulated the trailing edge loss as the consequence of an abrupt area enlargement: te Y = . (7) te,Ag o t te Correction to zero of the trailing edge of Equation (2) is performed by the subtraction of Y . t =0.02s te 2.3. Secondary Losses The AM 3D gross secondary loss is presented by the following equation [2]: Y = l Z, s,A M A M (8) = f (b , b , q , A / A , r /r ). in out in out in hub ti p This compact expression was the result of continuous efforts to interpret turbine endwall secondary ﬂows. l and Z are intrinsic parameters of the AM correlations and A M originate from the work of Carter [35], who drew on the classical secondary ﬂow theory. The intent of AM was to deﬁne a common and unique basis between axial compressors and turbines to quantify secondary losses, as one had (historically) the objective to employ mostly similar blades for both devices. The theory associates endwall vortices with passage Int. J. Turbomach. Propuls. Power 2022, 7, 14 7 of 27 vortices produced in curved channels [40]. The latter originate from the distortion of the inlet endwall boundary layer by the near-wall velocity deﬁcit and the blade passage pressure gradient, causing the inception of a helical cross-ﬂow motion spreading throughout the blade passage, as shown in Figure 1a. (a) (b) Figure 1. Secondary ﬂow mechanisms considered by Carter [35]. (a) Classical secondary ﬂow model. (b) Cascade lifting line model. As Carter strived for a practical model to quantify the losses incurred by these passage vortices, he made an analogy with the lifting line theory to obtain a ﬁrst order approxima- tion. The idea that Carter used leans on the trailing edge inviscid vortex sheet produced by ﬂow circulating around a blade. Being unstable, the closed vortex ﬁlaments of the vortex sheet roll up into trailing vortices at the blade ends, as shown in Figure 1b. Based on qualitative analyses of some experimental evidence taken downstream of the blade, the modelled trailing and passage vortices were made equivalent, to establish a secondary loss correlation [35]. Hence, the induced drag under uniform spanwise circulation is: 2 2 C C 1 2H L,m L,m C = 1 = l , (9) D,s A M 4 s/c H s/c with H the distance from blade midspan to the core of the endwall vortices. This loss coherently reﬂects lost work by the product of blade loading and the width of the trailing vortices; an analogy to force time displacement. AM subsequently replaced H by the empirical Ainley parameter l , which is shown in their Figure 16 on page 31 in [34], and A M which reﬂects the vortex characteristics implicitly. The involved lift coefﬁcient C refers L,m to the mean velocity and relates to the mean angle b , where C = 2 (tan b tan b ) cos b , (10) L,m out m in and tan b + tan b out in b = tan . (11) This was transformed into the Ainley loading Z with: 2 3 Y = C (cos b / cos b )(c/s) = l Z, (12) s,A M D,s out m A M where: 2 2 cos b c cos b L,m out 2 out Z = = 4(tan b tan b ) . (13) out in s/c cos b s cos b m m This equation assumes an axial velocity ratio AVR and meanline radius ratio k, with: AVR = W /W = k = r /r = 1 and is fully independent of s/c. Note that a change z,out out z,in in in blade loading due to a pitch to chord variation would not be identiﬁable [41]. As a matter Int. J. Turbomach. Propuls. Power 2022, 7, 14 8 of 27 of fact, this expression is much closer to the lift coefﬁcient of a single airfoil. The chosen lifting line theory is also used under steady and incompressible conditions and is applied to low camber/loading blades to guarantee the linearity of the problem. As a result, a large disagreement in secondary loss may arise in impulse turbine blades [8] and Equation (13) may wrongfully override other parameters under high loadings [10]. AM were aware of this issue, given the shortage of high-loading blades in their database [34]. It is also interesting to note that a mathematical “mistake” occurred with respect to Equation (12). Normally, the outcome should have been [40]: Y = C (1/ cos b )(c/s), (14) s,A M D,s m and consequently, Equation (13): Z = 4(tan b tan b ) cos b (15) in out m which yields results that are several orders of magnitude higher. This indicates that Equation (12) was most likely adapted heuristically to match the measurements. Therefore, as they possess little physical meaning, l and Z cannot be taken apart in the analysis [7]. A M This is not surprising, since the secondary loss of AM in Equation (8) was calibrated with: Y = Y Y (16) s t p This corresponds to Equation (1) in absence of clearance losses but has another impli- cation. In fact, the secondary loss was calibrated to the subtraction of the measured true total loss by the modelled proﬁle loss of Equation (2). Hence, Equation (8) embodies the errors associated to Equation (2). As is the case with l and Z, secondary and proﬁle A M losses should also not be used separately, as they are bound together by Equation (16). Dunham [19] criticized the approach of Carter but still relied on it to produce cos b out Y = 0.0334K Z, s,A MDC AR cos q in (17) = f ( H/c, b , b , q ). out in in The aspect ratio factor K 1 relates the intensity of the endwall vortices with blade AR height and was reported to have questionable reliability [42]. DC simpliﬁed the multipliers in Equation (13) using a cascade conﬁguration ! ! ( A cos b / A cos q ) 1 cos b cos b n,out out n,in in out out l = f = f = 0.0334K (18) A M AR (1 + d /d ) 2 cos q cos q ti p in in hub The constant 0.0334 is expected to cover most inlet boundary layer thicknesses and was obtained by calibration to overall turbine performance data [22]. Even though this crude approach is not able to estimate the downstream endwall loss of cascade ﬂows [18], it was, however, proven to be more accurate than most reﬁned correlations tested and compared under diverse inlet boundary layer proﬁles [43]. Eventually, Ag improved Equation (17) to ! 1 s,A MDC Y = K K , s,Ag p Re (19) 1 + 7.5Y s,A MDC = f ( M , M , Re, e/c)Y , r,out r,in s,A MDC which shares the K and K of Equation (3) [6]. Nevertheless, K was inherited from DC p Re Re and was later removed by KO. They certainly did not agree on the impact of viscosity on the endwall ﬂow and, interestingly, the debate is still ongoing. It is true that the endwall vortices are mostly driven by inviscid mechanisms. However, their growth in the blade passage highly depends on the endwall boundary layer ﬂow preceding the blade leading edge Int. J. Turbomach. Propuls. Power 2022, 7, 14 9 of 27 and continuous interaction with the blade surface boundary layer. Thereby, the approach of DC and Ag is more plausible. The square root manipulation prevents loss overshoots in extreme off design conditions. This feature has proven to be particularly crucial in optimization. In fact, the search algorithm could inadvertently exploit inconsistencies of the correlation to reach its objective [44]. Again, the auxiliary factors act as moderate multipliers to the basic Equation (17). It is interesting to note that the aptness of the axial turbine loss correlations was investigated in the context of centrifugal turbines, and, more speciﬁcally, the interaction of the auxiliary factors [23]. Drastic changes in the aerodynamic parameters did not trigger a comparable variation in the auxiliary factors, which could have guaranteed reliable solutions. Hence, these auxiliary factors can only sustain minor and individual adjustment and, thus, axial turbine correlation should only be used with axial turbines. 3. Proﬁle Loss Enhancement The Ag correlation [6], being the latest complete loss system, constitutes the most suitable vessel for successive enhancements. Thus, the proﬁle loss of Equation (3) is revamped into Y = 0.914K K K K Y + Y + Y , M p Re p,Enh mod p,A M sh inc (20) = f (e /s, q , q , M , M , Re , e/c, k, i, d /s, We , AVR)Y , c out r,out s LE LE p,A M in r,in in which Equation (2) is rewritten as in b b 5t out in in max Y = Y Y Y , p,A M p,b =0 p,b =b p,b =0 in in out in (21) b b c out out = (s/c, t /c, b , b , y ). max in out Z Carrying on with the auxiliary factor arrangement of Equation (3), several minor changes were performed. 1. Metal angle q in Equation (21) was replaced by a design ﬂow angle b . Practically, a in in better continuity in design workﬂow is gained by focusing on the ﬂow angles ﬁrst and deriving the metal angles afterward. Modern turbines tend to integrate little negative incidence to enable low loss in off designs. Proceeding with q would, rather, require in extra iterations in the preliminary design process. 2. Y and Y in Equation (21) were scaled by a common s/c. Before p,b =0 p,b =b in in out developing the modiﬁcations, the context in which they were built is reviewed and compared to current practice. During the 1940s, blade proﬁling was an empirical science. It consisted of a selection from a family of standardized proﬁles and was arranged into cascade conﬁguration with little allowable change. This approach was inherited from axial compressors and guaranteed cost-effectiveness as it facilitated manufacture. It was impossible to grasp the trend of cascade performance without resorting to cumbersome trial and error wind tunnel tests until the determination of a competent proﬁle. This substantially contrasts with modern proﬁling techniques, which offer close monitoring and automated optimization of the blade surface velocity distribution. Y and Y were certainly obtained from competent but p,b =0 p,b =b out in in obsolete proﬁles, in which blade surface velocity distributions or efﬁciencies have become unacceptable according to modern state of the art standards [34]. As they form the cornerstone of Equation (2), it would be unreasonable to initiate large modiﬁcation at the risk of disrupting the auxiliary factor arrangement in Equations (3) and (20). Instead, a scaling technique with a reference s/c, which is analogous to that applied to compressor performance maps, is proposed. In a design process, s/c is set following a trade off between efﬁciency and cost. If efﬁciency consideration prevails, the optimum s/c that minimizes proﬁle loss, in this case of Equation (2), is chosen, so that any s/c departure results in proﬁle loss increase or simply efﬁciency deterioration. However, Int. J. Turbomach. Propuls. Power 2022, 7, 14 10 of 27 the optimum s/c of Y and Y are ﬁxed and speciﬁc to competent p,b =0 p,b =b in in out proﬁles. The issue would then be its insensitivity to other types of proﬁle. With this in mind, a reasonable approach that maintains the Y and Y trend and p,b =0 p,b =b in in out scales the optimum pitch to chord ratio s/c of Equation (2) to a new optimum A M s/c is proposed. The purpose is to extend Equation (21) to other proﬁles in which optimum s/c abides by different design rules. Among these, the prominent Zweifel criterion [11] is chosen to inherit its ﬂexibility and its incompressible is generalized to: 1 b y s/c = , (22) 2 c j tan b tan b j cos b in out out AVR introducing the axial velocity ratio AVR as an additional design parameter. An y beyond the range of [0.6–0.8] recommended by Zweifel is chosen, to achieve the higher optimum pitch to chord ratio s/c in modern turbines. The vital scaling factor is: s/c f = , s/c s/c A M (23) = f (s/c, b , b , AVR, y , g), in out Z which is multiplied to the s/c of Equation (21). Note that Y and Y A M p,b =0 p,b =b in in out are ﬁtted as polynomials about their respective optimum s/c (the ﬁtting formulae of Tournier and El-Genk [45], Korpela [46], Aungier [6] and Concepts/ETI [47] were tested at extreme b = 30 and b = 80 ; the Tournier and El-Genk performed best), out out so that the scaling process does not alter their slope. As demonstrated, the original and scaled Y and Y are depicted at exit ﬂow angles b = 50 and p,b =0 p,b =b out in in out 70 in Figure 2, with the y = 1 typical of modern turbines. 0.1 0.3 0.08 = 70 o out = 70 0.2 out 0.06 0.04 0.1 = 50 0.02 out = 50 out 0 0 0.4 0.6 0.8 1 1.2 0.4 0.6 0.8 1 1.2 (a) (b) Figure 2. Optimal pitch to- hord ratio matching of AM proﬁle losses [2] with y = 1. (a) Nozzle b = 0 . (b) Impulse b = b . in in out Both Figure 2a,b are consistent on the minimum shift. At b = 70 , a larger s/c can be out achieved with better boundary layer ﬂow control during blade proﬁling. However, at b = 50 , a lower s/c is resorted tp, in order to satisfy y = 1. This is an undesirable out solution, as this would add structural weight. A cost-effective constraint is imposed to refrain f 1.1. s/c 3. The Reynolds correction factor K was adjusted to suction surface length and becomes Re a function of the suction surface Reynolds number Re . Suction surface momentum thickness could be responsible of up to 90% of the proﬁle loss [32], the suction surface ﬂow should logically be prioritized. Int. J. Turbomach. Propuls. Power 2022, 7, 14 11 of 27 4. Incidence loss Y replaces the incidence factor K of Equation (3) and acts as a fully inc i independent, contributing to the proﬁle loss. According to Moustapha et al. [15], modern turbines are designed with relatively thicker leading edges and smoother front curvatures, to retard boundary layer separation over a wide range of incidence angles and to keep loss at low level. This feature can not be reﬂected in the conservative K . In this regard, Benner et al. [48] formulated a polynomial incidence loss Y compatible inc with KO and used in Equation (20). 5. The trailing edge loss Y of Equation (7) is insensitive to change of turbine operating te,Ag conditions and, thus, was deemed inappropriate for modiﬁcations. As an alternative, the Y of KO, which encompasses many key aerodynamic parameters, is considered: te b b in in DF = DF DF DF . (24) te,KO te,b =0 te,b =b te,b =0 out in in in b b out out The replacement of q by b follows the same reasoning behind Equation (21). Draw- in in ing on the proﬁle loss of AM in Equation (2), KO blended the exit boundary layer mixing and base pressure losses of nozzle and impulse blades [5]. Instead of a subtrac- tion, as in Equation (3), virtual correction of Y to t /s = 0 produces a multiplication p te by 0.914 in Equation (20). The latter corresponds to the value of K proposed by AM te in Equation (5) at standard t /s = 0 [2]. Now, the conversion of Equation (24) to Y te te involves M [49]. This is a property speciﬁc to the total pressure loss coefﬁcient and r,out cannot be counted as part of the correlation. However, recent literature has proven its reliance on M but has not delivered any correlation compatible to the AM r,out family [50]. Conveniently, Equation (24) was acquired on low speed tests in the same way as Equation (2). This implies that auxiliary factors deﬁned for the proﬁle loss can be reasonably extended to the trailing edge loss. Considering the relevance of compressibility and Reynolds effects, Equation (24) is revamped into Y = K K K Y , te,Enh M p Re te,KO (25) = f (b , b , t /o, M , M , Re , e/c). out te r,out s in r,in 4. Revised Secondary Loss In light of the issues disclosed earlier in Equation (9), its terms are overhauled. The purpose is to determine a better product of loading and displacement supported by a robust physical basis and depart from the approach used in Equation (16). The latter implies that each contribution has to fully be independent, as in Equation (1). Interestingly, there was a formulation anterior to Equation (8) found in early works [24,33]: in Y = 0.04 1 C , (26) s,A M L,out out using the lift coefﬁcient based on the outlet velocity s cos b out C = 2 (tan b tan b ) . (27) L,out in out c cos b Unfortunately, Equation (26) was abandoned in favor of the model of Carter [35] because it owns a ﬁrmer theoretical background and provides common ground for future compressor and turbine models, according to AM [34]. Contrary to their expectation, axial compressors and turbines have not taken the same course of evolution. As odd as it may be, there is a soundness in Equation (26) in the present context. In particular, C is consistent L,out with turbine analysis, which is not the case with Z of Equation (8), which relies on C in L,m Equation (10). Int. J. Turbomach. Propuls. Power 2022, 7, 14 12 of 27 The secondary loss improvement begins with C rearranged and generalized into L,out the lift solidity coefﬁcient C 1 k cos b L,out out = 1 + tan b tan b . (28) out in s/c AVR AVR cos b Assuming AVR = k = 1, Equation (27) is recovered. Modern turbines possess a meridional ﬂowpath that forces the radial shift of streamlines such that k 6= 1. The latter triggers the Coriolis force and may lower work output [51]. Moreover, expansion through each row inevitably accelerates the ﬂow and, thus, invariance of AVR even in early design stages, which thus constitutes a poor assumption. This aspect would serve in the subsequent evaluation of the correlation reliability. In addition, b is also generalized to: tan b + tan b in out AVR b = tan . (29) 1 + AVR Elaborated models of the endwall vortices have been produced since then, to substitute the classical secondary ﬂow theory. Among these, the passage vortex penetration depth of Sharma and Butler [12], later enhanced by Benner et al. [13], is proposed as the replacement to l : A M 0.79 2 0.1F Z d te = + 32.7 , 0.55 H H CR( H/c) (30) = f ( H/c, d / H, b , b , g, s/c, AVR). out in Once the incident endwall ﬂow impinges on the blade, a ﬁrst separation occurs at the leading edge, creating a horseshoe vortex, as illustrated in Figure 3a. Under passage pressure gradient and crossﬂow friction, the pressure leg is progressively strengthened and pushed towards the adjacent blade suction side. Meanwhile, the suction leg remains near the suction side wall. Eventually, both legs of opposite vorticity coincide at the minimum pressure point. The suction leg starts to orbit around the dominant pressure leg, also identiﬁed as the passage vortex. The latter draws a demarcation line S4 on the blade suction side. Sharma and Butler assumed symmetric and linear S4, beginning at the leading edge and forming a triangle, as shown in Figure 3b. Z / H corresponds to the width of the te passage vortex taken at the trailing edge. It also differs from l by the weight attributed A M to the parameters. For instance, AM prioritized the area ratio A / A over blade turning out in b + b , as there was only weak evidence on the inﬂuence of blade loading [34]. On the in out opposite, the genetic algorithm used by Benner et al. identiﬁed a stronger inﬂuence of the blade tangential force F , rather than the convergence ratio CR, analogous to A / A . q out in (b) (a) Figure 3. Endwall vortices model. (a) New secondary loss model [12]. (b) Passage vortex penetration depth [13]. Int. J. Turbomach. Propuls. Power 2022, 7, 14 13 of 27 Coull and Hodson [10] pointed out the absence of directives in the calculation of the upstream displacement thickness d / H. This is a minor issue, as classical formulae or rea- sonable estimates could be resorted to. However, it is still unable to consider downstream endwall loss increase [18]. In a turbine environment, this issue becomes less relevant, as vortex ﬂow is disrupted by succeeding blade rows distanced by a short row gap. The threshold of Equation (30) is Z / H = 0.5, above which the merging of the endwall vortices te takes place and produces distinct ﬂow dynamics. An additional consideration is made on the aspect ratio H/c. Enlarging blade height H and shortening blade chord c both increase H/c but affect secondary loss in different ways [34]. This problem was raised in the early cascade tests of Kraft [52], which exposed the insensitivity of the secondary loss under the change of c. The traditional interpretation reﬂected by the auxiliary factor K in Equations (17) and (19) is that secondary loss is AR inversely proportional to H/c. This is valid when H is varied but no special regard to c has been made. However, as H/c approaches 0, K tends to inﬁnity and vice versa. This AR contradicts the real physics of the secondary ﬂow. Teia even showed that the structure of the endwall vortices is unaltered by the change of H beyond a certain critical aspect ratio in his Figures 10–12, page 25–26 [42]. He deduced that interaction between the endwall vortices is absent and, thus, secondary loss should remain constant. Below a certain critical aspect ratio varying H, the vortices merge together and amplify loss. His observation offers new insights into the highlighted inconsistency but lacks key veriﬁcation. Notably, correlation between vortices’ interaction and loss was not established and this can hardly argue secondary loss insensitivity for all H/c before merging. In addition, no means were provided to estimate his critical aspect ratio. Drawing partially on his observation, invariance to H/c above H/c = 2 in Equation (30) is imposed. H/c = 2 corresponds to the traditional threshold below which the slope of K becomes sharper. Concurrently, the AR database size bias of Equation (30) reduces its certainty beyond the same threshold [13]. At this stage, it is cautiously advanced that loss ampliﬁcation occurs even before the merging of the vortices within H/c 2. The problem with c remains unsolved. Replacing Equation (8), the basic secondary loss is 1 Z C te L,out Y = , s,b 2 H s/c (31) = f ( H/c, d / H, b , b , g, s/c, AVR, k), in out accounting for the pair of endwall vortices and their core located at Z /2H as in Figure 3b. te This respects the structure of lost work with loading and displacement parameters. The square of C /(s/c), as in Equation (26), is inherited from the classical lifting-line theory, L,out which associates the endwall and induced vortices. It yields values which are lower than Z by several order of magnitude, hence predominance of C /(s/c) over other parameters L,out is naturally prevented The independence to s/c in Equation (19) has been criticized, as this does not ﬁt reality [41,53]. Practice refers to the optimum s/c with proﬁle loss consideration only, there are still no available studies, to this day, that have addressed an overall optimum for both proﬁle and secondary losses [19]. The tangential force F of Equation (30) utilizes s/c as loading indicator and is unable to serve this purpose. As a reasonable attempt to incorporate the intensiﬁcation of secondary loss caused by deviation from the proﬁle optimum s/c, an auxiliary factor is proposed s/c K = 1 + 10 1 , sc s/c (32) = f (s/c, b , b , AVR, y , g). in out Z This polynomial expression was acquired by ﬁtting the cascade data of Perdichizzi and Dossena [41], as shown in Figure 4. Int. J. Turbomach. Propuls. Power 2022, 7, 14 14 of 27 2.5 1.5 0.5 0.6 0.8 1 1.2 1.4 Figure 4. Optimum pitch to chord ratio auxiliary factor for secondary loss. Tests were conducted for s/c = 0.58, 0.73, 0.87 and i 2 [60–+35] . The highlighted points result from averaging over the incidence range and normalizing by s/c with y = 0.6 such that a clear parabola could be drawn. The data of Hodson and Dominy [53] were also exposed and normalized with y = 0.9, for comparison. According to the trend, it is seen that their LPT proﬁles were not optimized at nominal s/c and, thus, their results are omitted. The magnitude of Equation (32) is restrained to 1.5 beyond the range [0.8–1.2] as a precaution, given the scarcity of data. Lastly, the enhanced secondary loss, keeping the same arrangement as Equation (19), is updated as 2 2 s,b Y = K K K , sc p s,Enh Re (33) 1 + 7.5Y s,b = f (s/c, y , g, M , M , Re , e/c, H/c, d / H, b , b , AVR, k). Z r,in r,out s in out Further discussions regarding recent works are carried on. Teia [42] stipulated that proﬁle loss should increase with H/c. He argued that increasing H incurs additional loss, as the boundary layer covers more surface area. This implies a 3D effect which violates the 2D spanwise uniformity of the proﬁle loss assumed in Equation (1). A similar attempt was done by Benner et al. [54], with their alternative loss breakdown te Y = Y 1 + Y . (34) t p s The boundary layer ﬂow was treated separately in accordance to the area division on the suction side in Figure 3b. They stated that their secondary loss correlation includes the proﬁle loss of the secondary regions, which is unlikely since their genetic algorithm search removed the blade skin friction during the development of their correlation. In that sense, the net proﬁle loss is conﬁned within the primary region delimited by S4. Although intuitive, these results are not compatible with older correlations, especially when the Y of Equation (34) is still that of KO. To ascertain the credibility of the update, it has to achieve statistical signiﬁcance and thus requires a large database. A major challenge in this framework is to gather a sufﬁcient number of public turbine test cases to possibly enable unbiased analysis. A previous study [23] collected 33 subsonic cascades (15 static and 18 rotating) partitioned from single stage turbines. The same approach is adopted and the previous cascade database is enlarged to 109 specimens (54 static and 55 rotating) to enhance representativeness. Their key design parameter range is summarized in Table 1. Int. J. Turbomach. Propuls. Power 2022, 7, 14 15 of 27 Table 1. Design parameter range of the cascade database. Parameters Min Max s/c 0.498 0.974 H/c 0.456 6.493 M 0.071 0.649 r,in M 0.149 1.280 r,out b + b 45.6 138.9 in out 4 6 Re 4.538 10 6.025 10 4 7 Re 5.487 10 1.1185 10 Although it was advanced that reliability should be prioritized over the accuracy in the early review, both aspects would be investigated. For this purpose, high ﬁdelity CFD simulations were conducted on the cascades and the solutions would serve as a benchmark in the analysis. Although the clearance leakage loss Y is not addressed in this paper, it is funda- cl mental to select a compatible leakage loss model to complete the system of Equation (1). For this purpose, the recent correlations of Yaras and Sjolander [55] or Farokhi [56] are recommended. 5. Numerical Method Since numerical steady solutions serve in determining Y , their credibility has to be guaranteed. At issue is their overriding dependence upon numerical discretization and turbulence modelling [57]. Without proper veriﬁcation and validation, any produced solutions would be untrustworthy [58]. In this regard, a previous study [23] adopted the systematic V and V procedure [59] to identify the numerical scheme (mesh and turbulence model) best suited for the simulation of turbine aerodynamics. It was performed on the Aachen turbine rotor test case [60] with NUMECA FINE/Turbo [61] and presented an on design numerical error of 3.79% the experimental value. The latter is considerably lower than the accuracy standard of 15% of the correlations [34,38], therefore justifying the use of CFD in the assessment. The computations were run with the commercial package NUMECA FINE/Turbo [61] whose code architecture is capable of maximum second order accuracy. In order to cap- ture anisotropy, and impact of curvature and body forces on turbulence, the proprietary separation sensitive corrected explicit algebraic Reynolds stress model (SSC-EARSM) was chosen. Compared to the conventional EARSM model [62], this was calibrated by scale adaptive simulation to increase turbulent mixing in the ﬂow separation region and enhance the near wall behavior of anisotropy. Single blade passage meshes were generated with the semi-automatic mesher AutoGrid5. O4H topology, which comprises batches of curvilinear structured hexahedral blocks, was adopted. A large portion of cells were clustered at the wall boundaries and plane intersections to enable, ﬁrst, inner cell spacing characterized by + 6 y 0.8 within the viscous sublayer. The employed mesh resolution was about 2.7 10 nodes. To ensure algorithm robustness, a uniform proﬁle of absolute total temperature and pressure was imposed at the inlet patch, placed at one chord upstream of the leading edge. This also guarantees the control of the endwall boundary layer over the distance separating the inlet patch to the blade leading edge and enables the use of classical formula in the d estimation in Equation (30). Static pressure was subsequently prescribed at the outlet patch placed at one chord downstream of the trailing edge. This distance should enable mixed out solutions at the outlet. Periodic boundary conditions were imposed on the circumferential patches of the control volume. All simulations were performed with real gas in on design conditions, unless speciﬁed otherwise. 6. Correlation Accuracy Although it has been advanced that correlation accuracy should not be prioritized, the extreme case with 100% error is not tolerable either. In that sense, the limit at which Int. J. Turbomach. Propuls. Power 2022, 7, 14 16 of 27 inaccuracy fails the correlation has to be evaluated. For this purpose, the previous gauge with the probability DP of reaching a tolerance interval of 15% error is reused [23], tol this is depicted in Figure 5. This conforms to the aforementioned accuracy standard of 15% [38] and accommodates numerical error uncertainties. DP tol E 10[ %] 3 2 0 2 3 4 1 1 4 Figure 5. Tolerance interval for correlation accuracy evaluation. The assessment relies on straightforward descriptive statistics. The latter require a bounded quantity following a near normal distribution and including Equation (1). Thus, the relative error normalized by the benchmark CFD solutions is Y Y t,num t E = . (35) t,num with the large database available, a z-distribution is assumed [63]. The update is assessed together with its original form in Figure 6. By comparing the predicted loss of each cascade, the depicted trend in Figure 6a indicates a higher estimation by the enhanced correlation. Distinct disparities occur at high loss levels, whereas most points are clustered below the bisection at low levels, within a 0.05 band. Hence, Ag and its update moderately agree for low loss only. As per the convention for scatter plots, the centered bar represents the mean and the other smaller and larger bars delimit the 99% conﬁdence interval and standard deviation, respectively. Figure 6b uncovers a signiﬁcant difference between Ag and its update, with approximately 10% distance separating their conﬁdence interval. Both populations have comparable spreads but that of Ag is largely decentered, unveiling a systematic underestimation from the Ag correlation. Their difference produces a coherent shift of its mean to the negative side. However, contrary to any expectation, its spread has not shrunk, implying that the update has inherently departed from its original model in terms of turbine aerodynamics interpretation. A priori, if accuracy has to be improved, then the correlation should have its relative error centered on 0 and standard deviation within 15%, to deliver at least DP = 66%. The numerical values are gathered in Table 2 and include the DP of each tol tol distribution. 0.5 100 0.4 0.3 0.2 0.1 -50 0 0.1 0.2 0.3 0.4 0.5 -100 Ag Enhanced (a) (b) Figure 6. Comparison of Ag and enhanced correlations. (a) Total loss. (b) Total loss relative error. Int. J. Turbomach. Propuls. Power 2022, 7, 14 17 of 27 Table 2. Statistical solutions of the axial cascade database. Method Descriptive Statistics Parameters E [%] s [%] DP [%] tol Ag 30.255 21.308 22.018 Enhanced 8.548 23.409 45.136 D 21.707 22.634 33.107 With the Ag correlation, it is possible to achieve the required accuracy with approx- imately 25% chance. This result is opposite to the previous analysis, which reported an optimistic DP = 45.13% while retaining a comparable s = 23.41 over 33 cascades [23]. tol It is very likely that the mean is susceptible to sample size bias, while the spread remains invariant. The enhanced correlation outperforms by doubling DP with a better position tol of its mean error. This result has to be handled with care, as this would change with another comparable cascade database with a shift of the mean error. On the other hand, the spread manifests as a barrier marking the limit of low ﬁdelity models. For this reason, accuracy should be relegated to higher order methods. 7. Correlation Reliability Adopting the approach of Horlock [64] and Coull and Hodson [32], trend consistency is evaluated with the ﬂow coefﬁcient f and stage loading y. The latter condition the turbine stage velocity triangle and efﬁciency. Their early selection is guided by the famous Smith chart [14], after ﬁxing the stage number. As a reminder, Figure 7 was established by tests on 70 cold single stage turbines which reactions varied from 20% to 60%. These were designed with AVR = 1 and zero incidence. Coull and Hodson supplemented the blade proﬁles associated to different areas of the chart. The top left area is characterized by the highest turning and lowest efﬁciency airfoils to satisfy the loading requirement. In this regard, accrued interblade passage convergence is required to guide reduced momentum ﬂows. Nevertheless, ﬂow turning can be alleviated for the same loading and efﬁciency by increasing f and thus transiting to the top right area. Eventually, the bottom area is characterized by low turning and higher efﬁciency airfoils and the same consideration concerning the ﬂow passage conﬁguration. Figure 7. Smith chart calibrated to zero tip loss and 50% reaction [14,32]. To demonstrate reliability, the enhanced correlation is expected to reproduce the same topology by varying f and y, as attempted by CC [17] and KO [5]. For this purpose, the analysis begins with the design of a single LPT stage using the setting of Coull and Hodson to acquire a common basis for comparison and explore the LPT for which the AM correlations are known to fail [10]. Repeating stage, 50% reaction, a constant hub to tip 0.94 0.89 0.92 Int. J. Turbomach. Propuls. Power 2022, 7, 14 18 of 27 ratio of 0.75 and parabolic camber are assumed. The ﬁrst stage power requirement and inlet boundary conditions are taken from the GE E3 LPT [65]. The design parameters s/c, b , velocity ratio VR= W /W and H/c of the LPT rotor are plotted in Figure 8 over out out in the same y and f range as the Smith chart, for reference. Consecutively, the LPT isentropic efﬁciency is derived. 3 3 2.5 2.5 2 2 1.5 1.5 1 1 0.6 0.8 1 1.2 0.6 0.8 1 1.2 (a) (b) 3 3 2.5 2.5 2 2 1.5 1.5 1 1 0.6 0.8 1 1.2 0.6 0.8 1 1.2 (c) (d) Figure 8. Velocity triangle and geometric parameter ranges of the LPT rotor. (a) Pitch to chord ratio s/c. (b) Exit ﬂow angle b . (c) Velocity ratio VR. (d) Aspect ratio H/c. out First, considering the proﬁle loss efﬁciency in Figure 9, both contour plots unsurpris- ingly share the same topology. Equations (2) and (21) differ by minor calibrations involving the Zweifel criterion [11] and extending the auxiliary factors to Equation (24). Furthermore, their bottom right area is identical, since Zweifel criterion matching is disabled for low turning proﬁles at f > 1.1. The abrupt break of the contour 0.02 arises from K , which s/c mod alters its value when departing from nozzle proﬁles at low y [37]. In Figure 9a, proﬁle loss is minimized for most of the bottom area, reﬂecting low turning proﬁles, as shown in Figure 8b. Greater y and fewer losses are feasible by increasing f but at expense of lower s/c, as in Figure 8a. Loss is aggravated towards the top left area, which conforms to Figure 7. Figure 9b stands out with its comparatively lighter gradient, induced by the adequate application of f . With the exception of the bottom right area and according to s/c the results of Coull and Hodson [32], a loss level comparable to that of the semi-analytical correlations of Denton [31] and Coull and Hodson [10], in Figure 10a,b, page 7, respectively, is achieved. Then, secondary losses differ in all possible aspects in Figure 10. The Ag secondary loss [6] remains low for most of the domain and increases towards the top left area in Figure 10a. This pattern is not shared by other AM family correlations [10]. Further insights are offered by analyzing the components of Equation (8). Here, l displays a topology A M similar to that of Figure 7 in Figure 11a. Its minimum is reached at the bottom left area 0.94 0.96 0.92 0.99 1.01 41.69 45.81 1.48 49.92 54.04 54.04 1.7 1.7 58.16 58.16 4.04 1.91 62.28 62.28 1.91 2.13 3.78 3.78 66.4 66.4 66.4 2.13 2.35 3.52 3.52 0.89 0.94 70.52 70.52 70.52 2.35 0.92 0.96 0.94 3.26 3.26 3.26 0.96 2.56 2.35 0.99 0.99 1.01 3 3 3 1.01 0.99 74.64 74.64 2.56 0.96 1.04 1.04 2.56 2.74 2.74 1.06 1.06 2.78 1.08 2.49 2.49 1.11 78.76 2.78 2.23 1.97 3.22 1.71 3.22 3.43 0.02 0.05 0.03 0.04 Int. J. Turbomach. Propuls. Power 2022, 7, 14 19 of 27 and, curiously, remains invariant towards the top right direction. Meanwhile, ampliﬁcation takes place in all other directions, switching from curved to straight contours. On the other hand, Z progressively grows accordingly, along y, to reach the steepest straight contour in the top left area in Figure 11c. With a difference of O(10 ), it naturally plays a predominant role in the product of Equation (8) and dictates the resulting topology. Thus, Figure 10a disregards most features of Figure 11a and highlights losses with contours that are curved by the auxiliary factors and the square root of Equation (19). Conversely, the lowest loss is produced at greater y and gradually intensiﬁes towards the bottom right area in Figure 10b. 3 3 2.5 2.5 2 2 1.5 1.5 1 1 0.6 0.8 1 1.2 0.6 0.8 1 1.2 (a) (b) Figure 9. Proﬁle loss efﬁciency with y = 1.1, AVR = 1, i = 0, H/b = 6.5, 4.6 for stator and rotor. (a) Aungier [6]. (b) Enhanced. 3 3 2.5 2.5 2 2 1.5 1.5 1 1 0.6 0.8 1 1.2 0.6 0.8 1 1.2 (a) (b) Figure 10. Secondary loss efﬁciency with y = 1.1, AVR = 1, H/b = 6.5, 4.6 for stator and rotor. (a) Aungier [6]. (b) Enhanced. The central question to be posed now is which correlation conforms to reality? Back to Figure 11a, l is supposed to reﬂect the endwall vortices. In this regard, the increase A M in the top left area is consistent, since it is backed by a sharper exit angle in Figure 8b and lesser aspect ratio in Figure 8d. However, the other increase in the bottom right area is unlikely. It is impossible to magnify the endwall vortices by decreasing turning/loading, increasing aspect ratio and also lowering VR in Figure 8c. Moreover, a higher f should not favor the growth of secondary ﬂows in the ﬂow passage, as these are downwashed through considerable momentum convection [7]. With these points in mind, it is seen that the monotonous penetration depth of Figure 11b better matches the topology of Figure 8b–d. The contour slope disruption in the top left area is triggered by the aspect ratio variation for H/c < 2, ensuring discussion over the results of Teia [42]. Once again, there is a clear divergence between the blade loadings in Figure 11c,d. Contrasting with Figure 11c, the maximum is located in the bottom right area. This feature comes from the s/c at the denom- inator in Equation (28) and is substantiated by Figure 8a, in which the minimum occupies 0.03 0.04 0.05 0.02 0.05 0.05 0.04 0.06 0.08 0.05 0.06 0.07 0.06 0.05 0.02 0.06 0.05 0.02 0.02 0.04 0.04 0.04 0.03 0.03 0.03 0.02 0.03 0.03 0.04 0.03 0.02 0.07 0.08 0.08 0.05 0.06 0.05 0.02 0.02 0.06 0.04 0.09 0.07 0.03 0.04 0.02 Int. J. Turbomach. Propuls. Power 2022, 7, 14 20 of 27 the same area. Z originally accounted for s/c as uncovered in Equation (13) but dispensed with its contribution, as conﬁrmed herein. Furthermore, secondary loss increase under high f and low y was also produced by the trusted CC [17] and Traupel [25] correlations [10]. Despite the absence of similarity with Figure 7, consistency in the enhanced correlation is comparatively augmented. In particular, the difference between the components of Equation (19) is mitigated to O(10). 3 3 2.5 2.5 2 2 1.5 1.5 1 1 0.6 0.8 1 1.2 0.6 0.8 1 1.2 (a) (b) 3 3 2.5 2.5 2 2 1.5 1.5 1 1 0.6 0.8 1 1.2 0.6 0.8 1 1.2 (c) (d) Figure 11. Rotor secondary loss components with y = 1.1, AVR = 1, H/b = 4.6. (a) Ainley parameter l . (b) Vortex penetration depth Z /2H. (c) Ainley loading Z. (d) Generalized lift te A M solidity coefﬁcient C /s/c. Eventually, the isentropic efﬁciency key to correlation reliability is depicted in Figure 12. The trend of Ag in Figure 12a is clearly driven by the proﬁle loss in Figure 9a. In that respect, the bowed contours that are supposed to be inscribed in the domain of Figure 7 are stretched towards higher f. As a consequence, losses induced by large f are under- estimated by the correlation. In addition, the update reveals to be more conservative in the quantiﬁcation of the losses, as the ﬁrst identiﬁed contour is 0.91 in Figure 12b. The latter exhibits an almost complete bow but is not comparable to those of the Smith chart, as it covers too much area. Hence, it can only enable a global trend consistency. Although overstretching to high f is reduced, its expansion occasions over and under estimations in the bottom and top areas, respectively. Variation of contour level or sensitivity to design parameters is lessened. These weaknesses are occasioned by the secondary loss in which H/c = 2 is held constant for most of the domain, as shown in Figure 11b. This is somewhat expected, as Equation (30) of Benner et al. [13] was not intended for the current treatment regarding H/c. The model’s conservativeness could be alleviated through a larger H/c, if justiﬁed. For now, the threshold H/c = 2 is untouched, as the reasonable conservativeness in preliminary design provides an adequate margin for subsequent modiﬁcations. 1.43 1.55 1.37 1.49 0.1 1.37 1.19 1.31 1.25 0.11 0.0058 0.09 0.09 3.22 3.22 0.0056 0.0054 0.0052 0.1 0.005 0.0052 0.0048 4.84 0.005 4.84 0.11 0.0046 0.0048 0.12 0.0046 0.0042 6.47 6.47 6.47 0.0044 0.12 0.0044 0.0044 8.09 8.09 1.49 8.09 1.43 1.55 0.0044 1.37 1.31 1.43 1.31 1.25 1.25 9.71 9.71 0.0046 1.37 1.19 1.19 0.005 1.25 1.13 0.0048 1.13 0.0052 11.34 11.34 1.07 0.13 1.01 12.96 14.59 0.0042 16.21 1.31 1.49 0.91 Int. J. Turbomach. Propuls. Power 2022, 7, 14 21 of 27 3 3 2.5 2.5 2 2 1.5 1.5 1 1 0.6 0.8 1 1.2 0.6 0.8 1 1.2 (a) (b) Figure 12. Isentropic efﬁciency with y = 1.1, AVR = 1, H/b = 6.5, 4.6 for stator and rotor. (a) Aungier [6]. (b) Enhanced. The enhanced correlation has introduced additional design parameters or degrees of freedom, in hopes of reaching a better control on design. These are summarized in Appendix A Table A1 and their number amounts to three to ﬁve for the proﬁle and secondary losses, respectively. The following analysis retains four static and rotating cascades [26,60,65,66] with distinct k and with their operating conditions altered with a back pressure within10% of its nominal value. This aims to evaluate the change of Y over the prescribed operation range, drawing on the approach of Wei [9]. Again, CFD solutions form the benchmark. AVR determined in the early velocity triangle and knowingly set to unity in the AM correlations is concurrently highlighted to demonstrate its relevance as a design parameter. The results of the static and rotating cascades normalized by on design CFD values are depicted in Figure 13. The cascades are not identiﬁed to their references in the plots to foster randomness, as reliability must not be biased by the turbine origin. 1.8 1.2 1.6 1.4 1.2 0.8 0.8 0.6 0.6 0.9 0.95 1 1.05 1.1 1.15 0.9 0.95 1 1.05 1.1 1.15 (a) (b) Figure 13. Total loss variation of four cascades. (a) Stators. (b) Rotors. The distance separating each pair of curves relates to accuracy. It is worth mentioning that two out of four cascades achieve the early tolerance interval and this conform with the estimated DP = 45%. AVR varies for different operating conditions and extends as far tol as 15% its nominal value in Figure 13b. This parameter has demonstrated its nontriviality and hence justiﬁes the generalization of Equations (22) and (28). By diminishing the back pressure, AVR increases and vice-versa. For the static cascades in Figure 13a, the trend of each pair of curves similarly results in an acceptable global consistency. If strict consistency is imposed, then the curves should have synchronous slopes that conserve their separation at each variation. Whereas, for the rotating cascades in Figure 13b, serious trend disruptions are displayed introducing undesirable uncertainties. One cascade undergoes slope switchover to negative values at AVR = 1.05. Moreover, there are two cascades for 0.95 0.94 0.93 0.91 0.9 0.91 0.96 0.91 0.96 0.96 0.94 0.95 0.88 0.89 0.96 0.92 0.95 0.93 0.94 0.94 0.95 0.91 0.93 0.92 0.91 0.9 0.89 0.91 Int. J. Turbomach. Propuls. Power 2022, 7, 14 22 of 27 which a considerable slope mismatch is seen. The ﬁrst starts from AVR = 1.05 along the back pressure increase. The second even covers the entire range around AVR = 1. Contrasting with Figure 13a, the reason for these divergences becomes rather straightforward. CFD can capture the impacts of rotation on secondary ﬂows and, thus, loss mechanism, whereas correlations built upon cascade tests and calibrated to turbine data cannot. As a matter of fact, parameters or factors relating to rotation are absent in the correlations. In this view, reliability is globally guaranteed in stators and compromised in rotors because of rotation. Clearly, this observation constitutes the most sound argument against the use of cascades as benchmark of the correlations, as presented in the early review. 8. Conclusions In this paper, historical insight was provided into the numerous features of the AM correlations, whose most authoritative contributions were given by AM, DC, KO, and Ag. These models were established empirically using measurements on turbine cascades and deliver a standardized solution for the conﬁgurations considered in the experiments. As a consequence, the range of applications of the models are biased towards the types of turbine cascades used to develop the correlations. An important restriction of the AM correlations is related to the proﬁle loss, which is established through interpolation between nozzle and impulse blade empirical charts. This implies the prior setting of the velocity distribution over the blade surface, which follows from measurements on a restricted set of turbine cascade conﬁgurations. Another restriction is related to the secondary loss, which draws on the classical lifting-line theory of Carter and associates wingtip trailing vortices with blade passage vortices. Further investigation revealed a mathematical mistake in the secondary loss formulation and an undesirable link with the proﬁle loss. In light of these issues, an update to the proﬁle and secondary losses was proposed in this work, building upon the correlations of Ag. For the proﬁle loss, the addressed points are: • The substitution of metal angle q by the ﬂow angle b . in in • The scaling of the AM optimum pitch to chord s/c by the Zweifel optimum pitch A M to chord s/c . This, notably, preserves of the slope of the AM proﬁle losses Y Z p,b =0 in and Y and adapts the AM correlation to other design rules. p,b =b in out • The use of a Reynolds number Re based on the suction surface length. • The substitution of the auxiliary factor K by a more suitable incidence loss Y , i inc proposed by Benner et al. • The reuse of the KO trailing edge loss Y affected by the profile loss auxiliary factors. te,KO As for the secondary loss, the addressed points are: • The replacement of the classical secondary ﬂow model of AM and Carter by the endwall vortices model of Sharma and Butler. • The use of the product of the elemental lift solidity coefﬁcient C /s/c and vortex L,out penetration depth Z instead of the heuristic Ainley loading Z and parameter l . te A M • The invariance of the auxiliary factor K beyond the threshold aspect ratio H/c = 2. AR • The deﬁnition of a new auxiliary factor K , to determine the variation of the secondary sc loss with the pitch to chord ratio s/c. With a database of 109 cascades available, the prediction of the new correlation is benchmarked against the cascade numerical solution. Using descriptive statistics and expressing accuracy as the probability of achieving a relative error within 15%, the im- provement yields DP = 45% against the DP = 22% of Ag. Although there is a clear tol tol improvement, these results are largely dependent on the choice of a ﬁnite database for which the representativeness of the population is unknown. Nevertheless, this quantiﬁ- cation should provide an estimate of the results and raise conﬁdence in the use of the enhanced correlation. Regarding reliability, the new correlation ensures global consistency by reproducing the Smith chart [14] and off design stator loss variation. In addition, the analysis points out a correlation conservativeness in the design of the typical LPT, which is Int. J. Turbomach. Propuls. Power 2022, 7, 14 23 of 27 due to insensitivity to the secondary loss correlations beyond H/c = 2. Severe trend dis- ruptions are identiﬁed during the application of the proposed enhanced correlation on four rotating cascades, the mismatches with CFD compromise its reliability in rotating ﬂows. Author Contributions: Conceptualization, Y.L.; methodology, Y.L.; software, Y.L.; validation, P.H. and Z.Z. and F.B.; formal analysis, Y.L.; investigation, Y.L.; resources, P.H. and Z.Z.; data curation, Y.L.; writing—original draft preparation, Y.L.; writing—review and editing, Y.L.; visualization, Y.L.; supervision, P.H. and Z.Z. and F.B.; project administration, P.H. and Z.Z.; funding acquisition, P.H. and Z.Z. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: The cascade dataset and its CFD results are available on demand. Conﬂicts of Interest: The authors declare no conﬂict of interest. Nomenclature Acronyms Ag Aungier AM Ainley and Mathieson BSM Benner and Sjolander and Moustapha CFD Computational ﬂuid dynamics CFL Courant–Friedrich–Levy DC Dunham and Came E3 Energy efﬁcient engine KO Kacker and Okapuu LPT Low pressure turbine MKT Moustapha and Kacker and Tremblay VandV Veriﬁcation and validation Z Zweifel Symbols b Relative ﬂow angle [ ] DF Kinetic loss coefﬁcient [-] DP Tolerance interval probability [-] tol d Endwall displacement thickness [m] g Stagger [ ] k Meanline radius ratio [-] l Ainley parameter [-] A M f Flow coefﬁcient [-] y Blade loading [-] y Zweifel loading [-] s Standard deviation [-] q Metal angle [ ] A Cross section area [m ] b Axial chord [m] c True chord [m] C Lift coefﬁcient [-] C Secondary drag coefﬁcient [-] D,s d Annulus diameter [m] d Leading edge diameter [m] LE E Relative error [-] e Surface roughness [m] e Back curvature [m ] c Int. J. Turbomach. Propuls. Power 2022, 7, 14 24 of 27 F Tangential loading [-] f Scaling factor [-] s/c H Blade height [m] H Distance to endwall vortices [m] i Incidence [ ] K Auxiliary factor [-] k Heat ratio coefﬁcient [-] K Aspect ratio correction [-] AR K Incidence factor [-] K Technology factor [-] mod K Expansion correction factor [-] K Proﬁle correction factor [-] K Mach correction factor [-] K Reynolds correction factor [-] Re K Pitch to chord ratio correction [-] sc K Trailing edge correction factor [-] te M Mach number [-] o Throat width [m] p Pressure [Pa] r Meanline radius [m] Re Reynolds number [-] Re Suction length Reynolds number [-] s Pitch [m] t Maximum thickness [m] max t Trailing edge thickness [m] te W Relative velocity [m/s] We Leading edge wedge [ ] LE Y Proﬁle loss [-] Y Secondary loss coefﬁcient [-] Y Total loss coefﬁcient [-] Y Tip clearance loss coefﬁcient [-] cl Y Supersonic loss coefﬁcient [-] ex Y Incidence loss coefﬁcient [-] inc Y Shock factor [-] sh Y Trailing edge loss coefﬁcient [-] te Z Ainley loading [-] Z Vortex penetration depth [m] te AVR Axial velocity ratio [-] CR Convergence ratio [-] VR Velocity ratio [-] Superscripts Average Subscripts Enh Enhanced hub Hub in Inlet is Isentropic m Mean n Normal num Numerical out Outlet r Relative t Total ti p Tip z Axial Int. J. Turbomach. Propuls. Power 2022, 7, 14 25 of 27 Appendix A Table A1. Dimensionless parameters of the AM correlations. Parameters Y Y Y Y Y Y Y Y Y p,AM s,AM p,Ag s,Ag te,Ag ex,Ag p s te s/c 2 2 2 2 t /c 2 2 2 max q 2 2 2 2 2 in q 2 2 out b 2 2 2 2 in b 2 2 2 2 2 2 2 out t /s 2 te o/s 2 e /s 2 2 M 2 2 2 2 2 r,in M 2 2 2 2 2 2 r,out Re 2 2 Re 2 2 2 e/c 2 2 2 2 2 k 2 2 i 2 2 2 d /s 2 LE We 2 LE A / A 2 out in r /r 2 hub ti p H/c 2 2 AVR 2 2 y 2 2 t /o 2 2 te d / H 2 g 2 2 k 2 Reference Equation (2) Equation (8) Equation (3) Equation (19) Equation (7) Equation (4) Equation (20) Equation (33) Equation (24) Total 5 5 12 8 3 1 17 13 7 References 1. Zou, Z.P.; Wang, S.T.; Liu, H.X.; Zhang, W.H. Axial Turbine Aerodynamics for Aero-Engines: Flow Analysis and Aerodynamics Design, 1st ed.; Springer Nature: Singapore, 2017. 2. Ainley, D.; Mathieson, G. A Method for Performance Estimation for Axial-Flow Turbines; Technical Report R&M No. 2974; Aeronautical Research Council: London, UK, 1951. 3. 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International Journal of Turbomachinery, Propulsion and Power – Multidisciplinary Digital Publishing Institute

Published: Apr 21, 2022

Keywords: loss correlation; axial-flow turbine; turbine performance

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A Reliable Update of the Ainley and Mathieson Profile and Secondary Correlations

A Reliable Update of the Ainley and Mathieson Profile and Secondary Correlations

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A Reliable Update of the Ainley and Mathieson Profile and Secondary Correlations

A Reliable Update of the Ainley and Mathieson Profile and Secondary Correlations

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