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 kv2¯ω 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 gene-expression 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 (LPT) 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 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 = 120K 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, its impact on the blade load variation along the span and end-wall flow visualizations are analysed. The ability of LES and RANS to accurately predict the secondary flows is discussed together with the implications this has on design.

The design of low-pressure turbines (LPTs) must account for the losses generated by the unsteady interaction with the upstream blade row. The estimation of such unsteady wake-induced losses requires the accurate prediction of the incoming wake dynamics and decay. Existing linear turbulence closures (stress–strain relationships), however, do not offer an accurate prediction of the wake mixing. Therefore, machine-learnt, nonlinear turbulence closures (models) have been developed for LPT flows with unsteady inflow conditions using a zonal-based model development approach, with an aim to enhance the wake mixing prediction for unsteady Reynolds-averaged Navier–Stokes calculations. High-fidelity time-averaged and phase-lock averaged data at a realistic isentropic Reynolds number and two reduced frequencies, i.e., with discrete incoming wakes and with wake “fogging,” have been used as reference data for a machine learning algorithm based on gene expression programing to develop models. Models developed via phase-lock averaged data were able to capture the effect of certain prominent physical phenomena in LPTs such as wake–wake interactions, whereas models based on the time-averaged data could not. Correlations with the flow physics lead to a set of models that can effectively enhance the wake mixing prediction across the entire LPT domain for both cases. Based on a newly developed error metric, the developed models have reduced the a priori error over the Boussinesq approximation on average by 45%. This study thus aids blade designers in selecting the appropriate nonlinear closures capable of mimicking the physical mechanisms responsible for loss generation.

This paper describes a selection of Baker Hughes, a GE company (BHGE) activities to support Gas Turbine (GT) design and operation from simple to more elaborate applications of Machine Learning (ML).

A POD based procedure has been developed to identify and account for the different contributions to the entropy production rate caused by the unsteady aerodynamics of a low-pressure (LP) turbine blade. LES data of the extensively studied T106A cascade have been used to clearly highlight the capability of POD to identify deterministic incoming wake related modes, stochastic fine-scale structures embedded within the bulk of the wake carried during migration, and coherent structures originating in the boundary layer as a consequence of the wake-boundary layer interaction process. The POD modes computed by a kinematic kernel generate a full and complete basis, where both the velocity and enthalpy fields have been projected through an extended POD procedure to determine the relative coefficients. This allows to separately compute orthogonal sets of contributions to turbulent kinetic energy production, enthalpy-velocity correlation and turbulent dissipation of resolved structures, thus clearly identifying the dominating modes (i.e. phenomena) responsible for the overall entropy production rate. Moreover, low-order truncation of these different contributions have been grouped into three different parts: those arising from the deterministic incoming wake, those due to the turbulence carried by the wakes and its interaction with the boundary layer, and those related to boundary layer events. The spatial integration of these low-order truncations restricted to the time-mean boundary layer, wake mixing and the potential flow regions of the blade passage allows gathering further information on the unsteady loss generation mechanisms, and where they mainly act. Particularly, results show that the procedure is able to decompose losses into the dominant contributions, thus providing a new tool for a rapid and clear identification of the different sources of losses in complex unsteady flow fields.

This paper discusses the impact of inlet flow distortions on centrifugal compressors based upon a large experimental data base in which the performance of several impellers in a range of corrected flows and corrected speeds have been measured after been coupled with different inlet plenums technologies. The analysis extends to centrifugal compressor inlets including a side stream, typical of Liquefied Natural Gas (LNG) applications. The detailed measurements allow a thorough characterization of the flow field and associated performance. The results suggest that distortions can alter the head by as much as 3% and efficiency of around 1%. A theoretical analysis allowed to identify the design features that are responsible for this deviation. In particular, an extension of the so-called “reduced-frequency”, a coefficient routinely used in axial compressors and turbine aerodynamics to weigh the unsteadiness generated by upstream to downstream blade rows, allowed to determine a plenum-to-impeller reduced frequency that correlates very well with the measured performance. The theory behind the new coefficient is discussed together with the measurement details, and validate the correlation that can be used in the design phase to determine the best compromise between the inlet plenum complexity and impact on the first stage.

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.

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.

Blade-to-blade interactions in a low-pressure turbine were investigated using highly resolved compressible large eddy simulations. For a realistic setup, a stator and rotor configuration with profiles typical of low-pressure turbines was used. Simulations were conducted with an in-house solver varying the gap size between stator and rotor from 21.5% to 43% rotor chord. To investigate the effect of the gap size on the prevailing loss mechanisms, a loss breakdown was conducted. It was found that in the large gap size case, the turbulence kinetic energy levels of the stator wake close to the rotor leading edge were only one third of those in the small gap case, due to the longer distance of constant area mixing. The small time averaged suction side separation on the blade, found in the large gap case, disappeared in the small gap calculations, confirming how stronger wakes can keep the boundary layer attached. The higher intensity wake impinging on the blade, however, did not affect the time averaged losses calculated using the control volume approach of Denton. On the other hand, losses computed by taking cross sections upstream and downstream of the blade revealed a greater distortion loss generated by the stator wakes in the small gap case. Despite the suction side separation suppression, the small gap case gave higher losses overall due to the incoming wake turbulent kinetic energy amplification along the blade passage.

The paper analyzes losses and the loss generation mechanisms in a low-pressure turbine cascade by Proper-Orthogonal-Decomposition (POD) applied to measurements. Total pressure probes and time resolved particle image velocimetry (TR-PIV) are used to determine the flow field and performance of the blade with steady and unsteady inflow conditions varying the flow incidence. The total pressure loss co-efficient is computed by traversing two Kiel probes upstream and downstream of the cascade simultaneously. This procedure allows a very accurate estimation of the total pressure loss coefficient also in the potential flow region affected by incoming wake migration. The TR-PIV investigation concentrates on the aft portion of the suction side boundary layer downstream of peak suction. In this adverse pressure gradient region the interaction between the wake and the boundary layer is the strongest, and it leads to the largest deviation from a steady loss mechanism. POD applied to this portion of the domain provides a statistical representation of the flow oscillations by splitting the effects induced by the different dynamics. The paper also describes how POD can dissect the loss generation mechanisms by separating the contributions to the Reynolds stress tensor from the different modes. The steady condition loss generation, driven by boundary layer streaks and separation is augmented in presence of incoming wakes by the wake-boundary layer interaction and by the wake dilation mechanism. Wake migration losses have been found to be almost insensitive to incidence variation between nominal and negative (up to −9deg), while at positive incidence the losses have a steep increase due to the alteration of the wake path induced by the different loading distribution.

Numerical simulations based on the large eddy simulation approach were conducted with the aim to explore vaneless diffuser rotating stall instability in a centrifugal compressor. The effect of the impeller blade passage was included as an inlet boundary condition with sufficiently low flow angle relative to the tangent to provoke the instability and cause circulation in the diffuser core flow. Flow quantities, velocity and pressure, were extracted to accumulate statistics for calculating mean velocity and mean Reynolds stresses in the wall-to-wall direction. The paper focuses on the assessment of the complex response of the system to the velocity perturbations imposed, the resulting pressure gradient and flow curvature effects.

In the present paper direct numerical simulation data of a low pressure turbine is investigated in light of turbulence modelling. Many compressible turbulence models use Favre average transport equations of the conservative variables and turbulent kinetic energy along with other modelling equation. First a general discussion on the turbulence modelling error propagation prescribed by transport equations is presented, leading to the terms that are considered to be of interest for turbulence model improvement. In order to give turbulence modellers means of validating their models the terms appearing in the Favre averaged momentum equations are presented along pitchwise profiles at three axial position. These three positions have been chosen such that they represent regions with different flow characteristics.General trends indicate that terms related with thermodynamic fluctuations and Favre fluctuations are small and can be neglected for most of the flow field. The largest errors arise close to the trailing edge region where vortex shedding occurs. Finally, linear models and the scope for their improvement are discussed. Using locally optimized turbulence viscosities, the improvement potential of state of the art models is shown. On the other hand this study also highlights the danger of pure local optimization.

Large eddy simulations validated with the aid of direct numerical simulation (DNS) are used to study the concerted action of reduced frequency and flow coefficient on the performance of the T106A low-pressure turbine profile. The simulations are carried out by using a discretization in space and time that allows minimizing the accuracy loss with respect to DNS. The reference Reynolds number is 100,000, while reduced frequency and flow coefficient cover a range wide enough to provide valid qualitative information to designers. The various configurations reveal differences in the loss generation mechanism that blends steady and unsteady boundary layer losses with unsteady wake ingestion losses. Large values of the flow coefficient can alter the pressure side unsteadiness and the consequent loss generation. Low values of the flow coefficient are associated with wake fogging and reduced unsteadiness around the blade. The reduced frequency further modulates these effects. The simulations also reveal a clear trend of losses with the wake path, discussed by conducting a loss-breakdown analysis that distinguishes boundary layer from wake distortion losses.

Direct numerical simulations (DNS) on different grids for two Reynolds numbers of the T106A turbine cascade have been conducted to understand the influence of resolution on the solution. The work was motivated by different resolutions used in previously published studies that could not only be explained by different numerical methods. Most of these published studies used mean blade pressure distribution and wake loss profiles for validation and grid convergence purposes. DNS for canonical cases in general use more stringent criteria based on turbulence statistics considering that those require higher resolution. Therefore, in this study the influence of grid resolution on disparate scales of DNS is discussed. It is discussed that mean blade pressure distribution and mean wake loss already show good agreement with experiments for large eddy simulations (LES) provided a sufficient part of the turbulence kinetic energy is resolved. However, to avoid modelling uncertainties of turbulence quantities, DNS is required as LES only resolves part of the turbulence kinetic energy. It is found that in order to achieve grid convergence for turbulence dissipation a six times larger resolution and more than 10 times the computational cost is required than if grid convergence was required for turbulence kinetic energy.

Incompressible large eddy simulation and direct numerical simulation of a low-pressure turbine at Re=5.18×104 and 1.48×105 with discrete incoming wakes are analyzed to identify the turbulent kinetic energy generation mechanism outside of the blade boundary layer. The results highlight the growth of turbulent kinetic energy at the bow apex of the wake and correlate it to the stress-strain tensors relative orientation. The production rate is analytically split according to the principal axes, and then terms are computed by using the simulation results. The analysis of the turbulent kinetic energy is followed both along the discrete incoming wakes and in the stationary frame of reference. Both direct numerical and large eddy simulation concur in identifying the same production mechanism that is driven by both a growth of strain rate in the wake, first, followed by the growth of turbulent shear stress after. The peak of turbulent kinetic energy diffuses and can eventually reach the suction side boundary layer for the largest Reynolds number investigated here with higher incidence angle. As a consequence, the local turbulence intensity outside the boundary layer can grow significantly above the free-stream level with a potential impact on the suction side boundary layer transition mechanism.

Two stators of a multistage centrifugal compressor with progressively smaller outer radii have been designed, built, and tested. The aim was to achieve a significant reduction in the outer diameter of the compressor stage without compromising performance. The reduction in size was achieved by reducing the diffusion ratio (outer radius/inner radius) of the vaneless diffuser in two steps. In the first step, the outer diameter of the entire stage was reduced by 8% compared with the baseline design. In the second stage, the outer diameter was reduced by 14%. The outer radius of the smallest design was limited by the impeller exit diameter, which was kept constant, as was the axial length of the stage. The large radius baseline design has been tested on a rotating rig in a 1.5 stage setup. This setup aimed at simulating the multistage behavior by applying a pseudostage upstream of the main stage. The pseudostage consisted of a set of nonrotating preswirl vanes in order to mimic an upstream impeller and was followed by a scaled version of the return channel of the main stage. The experimental database was then used to calibrate a 1D analysis code and 3D–computational fluid dynamics methods for the ensuing design and optimization part. By applying an extensive design-of-experiments, the endwalls as well as the vanes of the stator part were optimized for maximum efficiency and operation range. In order to preserve the multistage performance, the optimization was constrained by keeping the circumferentially averaged spanwise flow profiles at the exit of the smaller radius stages within close limits to the original design. The reduced radius designs were then tested in the same 1.5 stage setup as the baseline design. The results indicate that the reduction in size was feasible without compromising the efficiency and operation range of the stage.

The flow around a low-pressure turbine rotor blade with periodically incoming wakes at a realistic Reynolds number is computed by means of LES. The computed results are discussed in terms of phase-averaged and mean quantities. A comparison is made with an existing DNS for the same geometry and operating conditions. Particular attention is devoted to flow structures associated with the incoming wakes, and their effect on the boundary layers. The analysis of the flow field reveals patterns similar to those encountered in DNS and in LES of flow in the same geometry at a lower Reynolds number. Noticeable differences occur in the suction side boundary layer, which exhibits a complete transition to turbulence for the present case. Nomenclature b two-dimensional anisotropy tensor 3 2 ij i j ij b u u k δ = − C chord Cf skin friction coefficient = ( ) 2 1 2 wall U / ρ τ Cp time-averaged static pressure coefficient ( ) 2 1 1 2 / p p U = − d diameter D velocity defect | | | | D = − v v f reduced frequency 2 (1/ ) / f T C U = ⋅ h span-wise extent H shape factor =δ/θ k turbulent kinetic energy l pitch n direction normal to the wall p static pressure Pk production rate of turbulent kinetic energy

A typical centrifugal impeller characterized by a low flow coefficient and cylindrical blades is redesigned by means of an intelligent automatic search program. The procedure consists of a feasible sequential quadratic programming algorithm (Fletcher, R. Practical Methods of optimization, 2000 (Wiley)) coupled to a lazy learning (LL) interpolator 1 to speed-up the process. The program is able to handle geometric constraints to reduce the computational effort devoted to the analysis of non-physical configurations. The objective function evaluator is an in-house developed structured computational fluid dynamics (CFD) code. The LL approx-imator is called each time the stored database can provide a sufficiently accurate performance estimate for a given geometry, thus reducing the effective CFD computations.The impeller is represented by 25 geometric parameters describing the vane in the meridional and s-0 planes, the blade thickness, and the leading edge shape. The optimization is carried out on the impeller design point maximizing the polytropic efficiency with nearly constant flow coefficient and polytropic head. The optimization is accomplished by maintaining unaltered those geometrical parameters which have to be kept fixed in order to make the impeller fit the original stage. The optimization, carried out on a cluster of 16 PCs, is self-learning and leads to a geometry presenting an increased design point efficiency. The program is completely general and can be applied to any component which can be described by a finite number of geometrical parameters and computed by any numerical instrument to provide performance indices.The work presented in this paper was done under the METHOD EC funded project for the implementation of new technologies for optimization of centrifugal compressors.