Non-linear turbulence closures were developed that improve the prediction accuracy of wake mixing in low-pressure turbine (LPT) flows. First, Reynolds-averaged Navier-Stokes (RANS) calculations using five linear turbulence closures were performed for the T106A LPT profile at exit Mach number 0.4 and isentropic exit Reynolds numbers 60,000 and 100,000. None of these RANS models were able to accurately reproduce wake loss profiles, a crucial parameter in LPT design, from direct numerical simulation (DNS) reference data. However, the recently proposed kv2w transition model was found to produce the best agreement with DNS data in terms of blade loading and boundary layer behavior and thus was selected as baseline model for turbulence closure development. Analysis of the DNS data revealed that the linear stress-strain coupling constitutes one of the main model form errors. Hence, a geneexpression programming (GEP) based machine-learning technique was applied to the high-fidelity DNS data to train non-linear explicit algebraic Reynolds stress models (EARSM). In particular, the GEP algorithm was tasked to minimize the weighted difference between the DNS and RANS anisotropy tensors, using different training regions. The trained models were first assessed in an a priori sense (without running any CFD) and showed much improved alignment of the trained models in the region of training. Additional RANS calculations were then performed using the trained models. Importantly, to assess their robustness, the trained models were tested both on the cases they were trained for and on testing, i.e. previously not seen, cases with different flow features. The developed models improved prediction of the Reynolds stress, TKE production, wake-loss profiles and wake maturity, across all cases, in particular those trained on just the wake region.

In low-pressure turbines (LPTs), around 60–70% of losses are generated away from end-walls, while the remaining 30–40% is controlled by the interaction of the blade profile with the end-wall boundary layer. Experimental and numerical studies have shown how the strength and penetration of the secondary flow depends on the characteristics of the incoming end-wall boundary layer. Experimental techniques did shed light on the mechanism that controls the growth of the secondary vortices, and scale-resolving computational fluid dynamics (CFD) allowed to dive deep into the details of the vorticity generation. Along these lines, this paper discusses the end-wall flow characteristics of the T106 LPT profile at Re = 120 K and M = 0.59 by benchmarking with experiments and investigating the impact of the incoming boundary layer state. The simulations are carried out with proven Reynolds-averaged Navier–Stokes (RANS) and large-eddy simulation (LES) solvers to determine if Reynolds-averaged models can capture the relevant flow details with enough accuracy to drive the design of this flow region. Part I of the paper focuses on the critical grid needs to ensure accurate LES and on the analysis of the overall time-averaged flow field and comparison between RANS, LES, and measurements when available. In particular, the growth of secondary flow features, the trace and strength of the secondary vortex system, and its impact on the blade load variation along the span and end-wall flow visualizations are analyzed. The ability of LES and RANS to accurately predict the secondary flows is discussed together with the implications this has on design.

Machine learning was applied to large-eddy simulation (LES) data to develop nonlinear turbulence stress and heat flux closures with increased prediction accuracy for trailing-edge cooling slot cases. The LES data were generated for a thick and a thin trailing-edge slot and shown to agree well with experimental data, thus providing suitable training data for model development. A gene expression programming (GEP) based algorithm was used to symbolically regress novel nonlinear explicit algebraic stress models and heat-flux closures based on either the gradient diffusion or the generalized gradient diffusion approaches. Steady Reynolds-averaged Navier–Stokes (RANS) calculations were then conducted with the new explicit algebraic stress models. The best overall agreement with LES data was found when selecting the near wall region, where high levels of anisotropy exist, as training region, and using the mean squared error of the anisotropy tensor as cost function. For the thin lip geometry, the adiabatic wall effectiveness was predicted in good agreement with the LES and experimental data when combining the GEP-trained model with the standard eddy-diffusivity model. Crucially, the same model combination also produced significant improvement in the predictive accuracy of adiabatic wall effectiveness for different blowing ratios (BRs), despite not having seen those in the training process. For the thick lip case, the match with reference values deteriorated due to the presence of large-scale, relative to slot height, vortex shedding. A GEP-trained scalar flux model, in conjunction with a trained RANS model, was found to significantly improve the prediction of the adiabatic wall effectiveness.

In low-pressure turbines (LPT) at design point, around 60–70% of losses are generated in the blade boundary layers far from end walls, while the remaining 30–40% is controlled by the interaction of the blade profile with the end-wall boundary layer. Increasing attention is devoted to these flow regions in industrial design processes. This paper discusses the end-wall flow characteristics of the T106 profile with parallel end walls at realistic LPT conditions, as described in the experimental setup of Duden, A., and Fottner, L., 1997, “Influence of Taper, Reynolds Number and Mach Number on the Secondary Flow Field of a Highly Loaded Turbine Cascade,” Proc. Inst. Mech. Eng., Part A, 211(4), pp.309–320. Calculations are carried out by both Reynolds-averaged Navier–Stokes (RANS), due to its continuing role as the design verification workhorse, and highly resolved large eddy simulation (LES). Part II of this paper focuses on the loss generation associated with the secondary end-wall vortices. Entropy generation and the consequent stagnation pressure losses are analyzed following the aerodynamic investigation carried out in the companion paper (GT2018-76233). The ability of classical turbulence models generally used in RANS to discern the loss contributions of the different vortical structures is discussed in detail and the attainable degree of accuracy is scrutinized with the help of LES and the available test data. The purpose is to identify the flow features that require further modeling efforts in order to improve RANS/unsteady RANS (URANS) approaches and make them able to support the design of the next generation of LPTs.

The present paper describes the application of proper orthogonal decomposition (POD) to large eddy simulation (LES) of the T106A low-pressure-turbine profile with unsteady incoming wakes at two different flow conditions. Conventional data analysis applied to time averaged or phase-locked averaged flow fields is not always able to identify and quantify the different sources of losses in the unsteady flow field as they are able to isolate only the deterministic contribution. A newly developed procedure allows such identification of the unsteady loss contribution due to the migration of the incoming wakes, as well as to construct reduced order models that are able to highlight unsteady losses due to larger and/or smaller flow structures carried by the wakes in the different parts of the blade boundary layers. This enables a designer to identify the dominant modes (i.e., phenomena) responsible for loss, the associated generation mechanism, their dynamics, and spatial location. The procedure applied to the two cases shows that losses in the fore part of the blade suction side are basically unaffected by the flow unsteadiness, irrespective of the reduced frequency and the flow coefficient. On the other hand, in the rear part of the suction side, the unsteadiness contributes to losses prevalently due to the finer scale (higher order POD modes) embedded into the bulk of the incoming wake. The main difference between the two cases has been identified by the losses produced in the core flow region, where both the largest scale structures and the finer ones produces turbulence during migration. The decomposition into POD modes allows the quantification of this latter extra losses generated in the core flow region, providing further inputs to the designers for future optimization strategies.

This work shows the application of Gene &pression Pro­gramming to augment RANS turbulence closure modelling for flows through complex geometries, designed for additive manu­facturing. Specifically, for the design of optimised internal cool­ing channels in turbine blades. One of the challenges in internal coolant design is the heat transfer accuracy of the RANS formu­lation in comparison to higher fidelity methods, which are still not used in design on account of their computational cost. How­ever, high fidelity data can be extremely valuable for improving cu"ent lower fidelity models and this work shows the application of data driven approaches to develop turbulence closures for an internally ribbed duct. Different approaches are compared, and the results of the improved model are illustrated. The work shows the potential of using data driven models for accurate heat transfer predictions even in non-conventional configurations.

This paper presents development of accurate turbulence closures for wake mixing prediction by integrating a machine-learning approach with Reynolds Averaged Navier-Stokes (RANS)-based computational fluid dynamics (CFD). The data-driven modelling framework is based on the gene expression programming (GEP) approach previously shown to generate non-linear RANS models with good accuracy. To further improve the performance and robustness of the data-driven closures, here we exploit that GEP produces tangible models to integrate RANS in the closure development process. Specifically, rather than using as cost function a comparison of the GEP-based closure terms with a frozen high fidelity dataset, each GEP model is instead automatically implemented into a RANS solver and the subsequent calculation results compared with reference data. By first using a canonical turbine wake with inlet conditions prescribed based on high-fidelity data, we demonstrate that the CFD-driven machine-learning approach produces non-linear turbulence closures that are physically correct, i.e. predict the right downstream wake development and maintain an accurate peak wake loss throughout the domain. We then extend our analysis to full turbine blade cases and show that the model development is sensitive to the training region due to the presence of deterministic unsteadiness in the near wake region. Models developed including this region have artificially large diffusion coefficients to overcompensate for the vortex shedding steady RANS cannot capture. In contrast, excluding the near wake region in the model development produces the correct physical model behavior, but predictive accuracy in the near-wake remains unsatisfactory. We show that this can be remedied by using the physically consistent models in unsteady RANS, implying that the non-linear closure producing the best predictive accuracy depends on whether it will be deployed in RANS or unsteady RANS calculations. Overall, the models developed with the CFD assisted machine learning approach were found to be robust and capture the correct physical behavior across different operating conditions.

We report on a series of highly resolved large-eddy simulations of the LS89 high-pressure turbine (HPT) vane, varying the exit Mach number between Ma = 0:7 and 1:1. In order to accurately resolve the blade boundary layers and enforce pitchwise periodicity, we for the first time use an overset mesh method, which consists of an O-type grid around the blade overlapping with a background H-type grid. The simulations were conducted either with a synthetic inlet turbulence condition or including upstream bars. A quantitative comparison shows that the computationally more efficient synthetic method is able to reproduce the turbulence characterictics of the upstream bars. We further perform a detailed analysis of the flow fields, showing that the varying exit Mach number significantly changes the turbine efficiency by affecting the suction-side transition, blade boundary layer profiles, and wake mixing. In particular, the Ma = 1:1 case includes a strong shock that interacts with the trailing edge, causing an increased complexity of the flow field. We use our recently developed entropy loss analysis (Zhao and Sandberg, GT2019-90126) to decompose the overall loss into different source terms and identify the regions that dominate the loss generation. Comparing the different Ma cases, we conclude that the main mechanism for the extra loss generation in the Ma = 1:1 case is the shock-related strong pressure gradient interacting with the turbulent boundary layer and the wake, resulting in significant

The trailing edge slot is a canonical representation of the pressure-side bleed flow encountered in high pressure turbines. Predicting the flow and temperature downstream of the slot exit remains challenging for RANS and URANS, with both significantly over predicting the adiabatic wall effectiveness. This over prediction is attributable to the incorrect mixing prediction in cases where the vortex shedding is present. In case of RANS the modelling error is rooted in not properly accounting for the shedding scales while in URANS the closures account for the shedding scales twice, once by resolving the shedding and twice with the model for all the scales. Here, we present an approach which models only the stochastic scales that contribute to turbulence while resolving the scales that do not, i.e. scales considered as contributing to deterministic unsteadiness. The model for the stochastic scales is obtained through a data-driven machine learning algorithm, which produces a bespoke turbulence closure model from a high-fidelity dataset. We use the best closure (blowing ratio of 1.26) for the anisotropy obtained in the a priori study of Lav, Philip & Sandberg [A New Data-Driven Turbulence Model Framework for Unsteady Flows Applied to Wall-Jet and Wall-Wake Flows, 2019] and conduct compressible URANS calculations. In the first stage, the energy equation is solved utilising the standard gradient diffusion hypothesis for the heat-flux closure. In the second stage, we develop a bespoke heat-flux closure using the machine-learning approach for the stochastic heat flux components only. Subsequently, calculations are performed using the machine-learnt closures for the heat-flux and the anisotropy together. Finally, the generalisability of the developed closures is evaluated by testing them on additional blowing ratios of 0.86 and 1.07. The machine learnt closures developed specifically for URANS calculations show significantly improved predictions for the adiabatic wall-effectiveness across the different cases.

Turbulence modelling development has received a boost in recent years through assimilation of machine learning methods and increasing availability of high-fidelity datasets. This paper presents an approach that develops turbulence models for flows exhibiting organised unsteadiness. The novel framework consists of three parts. First, using triple decomposition, the high-fidelity data is split into organised motion and stochastic turbulence. A data-driven approach is then used to develop a closure only for the stochastic part of turbulence. Finally, unsteady calculations are conducted, which resolve the organised structures and model the unresolved turbulence using the developed bespoke turbulence closure. A case study of a wake with vortex shedding behind a normal flat plate, at a Reynolds number of 2,000, based on plate height and freestream velocity, is considered to demonstrate the method. The approach shows significant improvement in mean velocity and Reynold stress profiles compared with standard turbulence models.

The unsteady flow prediction for turbomachinery applications relies heavily on unsteady RANS (URANS). For flows that exhibit vortex shedding, such as the wall-jet/wake flows considered in this study, URANS is unable to predict the correct momentum mixing with sufficient accuracy. We suggest a novel framework to improve that prediction, whereby the deterministic scales associated with vortex shedding are resolved while the stochastic scales of pure turbulence are modelled. The framework first separates the stochastic from the deterministic length scales and then develops a bespoke turbulence closure for the stochastic scales using a data-driven machine-learning algorithm. The novelty of the method lies in the use of machine-learning to develop closures tailored to URANS calculations. For the walljet/wake flow, three different mass flow ratios (0.86, 1.07 and 1.26) have been considered and a high-fidelity dataset of the idealised geometry is utilised for the sake of model development. This study serves as an a priori analysis, where the closures obtained from the machine-learning algorithm are evaluated before their implementation in URANS. The analysis looks at the impact of using all length scales versus the stochastic scales for closure development, and the impact of the extent of the spatial domain for developing the closure. It is found that a two-layer approach, using bespoke trained models for the near wall and the jet/wake regions, produce the best results. Finally, the generalisability of the developed closures is also evaluated by applying a given closure developed using a particular mass flow ratio to the other cases.

Entropy loss is widely used to quantify the efficiency of components in turbomachines, and empirical relations have been developed to estimate the contribution of different mechanisms. However, further analysis is still needed to not only get a deeper insight of the physics, but also to more accurately quantify the loss generation caused by different terms. In the present study, the entropy transport equations based on averaged flow quantities are first derived, and the entropy generation process is fully decomposed into several terms representing different physical mechanisms, such as mean viscous dissipation, turbulence production, mean and turbulent heat flux, etc. This decomposition framework is then applied to available high-resolution LES and RANS results of a VKI LS-89 HPT vane, and a detailed quantification of different entropy generation terms is obtained. The results show that the entropy generation caused by mean flow features like mean viscous dissipation and mean heat flux are in close agreement between LES and RANS, indicating that RANS provides an overall good prediction for the mean flow. Furthermore, we find that turbulence production plays an important role in entropy generation as it represents the energy extracted from the mean flow to turbulent fluctuations. However, the difference between RANS and LES results for the turbulence production term is not negligible, particularly in the wake region. This implies that the failure of RANS to predict the correct total loss might be largely caused by errors in capturing the correct turbulence production in the near wake region.