Comparative Analysis of Methodologies for Uncertainty Propagation and Quantification

In engineering design, uncertainty is inevitable and can cause a significant deviation in the performance of a system. Uncertainty in input parameters can be categorized into two groups: aleatory and epistemic uncertainty. The work presented here is focused on aleatory uncertainty, which can cause natural, unpredictable and uncontrollable variations in performance of the system under study. Such uncertainty can be quantified using statistical methods, but the main obstacle is often the computational cost, because the representative model is typically highly non-linear and complex. Therefore, it is necessary to have a robust tool that can perform the uncertainty propagation with as few evaluations as possible. In the last few years, different methodologies for uncertainty propagation and quantification have been proposed. The focus of this study is to evaluate four different methods to demonstrate strengths and weaknesses of each approach. The first method considered is Monte Carlo simulation, a sampling method that can give high accuracy but needs a relatively large computational effort. The second method is Polynomial Chaos, an approximated method where the probabilistic parameters of the response function are modelled with orthogonal polynomials. The third method considered is Mid-range Approximation Method. This approach is based on the assembly of multiple meta-models into one model to perform optimization under uncertainty. The fourth method is the application of the first two methods not directly to the model but to a response surface representing the model of the simulation, to decrease computational cost. All these methods have been applied to a set of analytical test functions and engineering test cases. Relevant aspects of the engineering design and analysis such as high number of stochastic variables and optimised design problem with and without stochastic design parameters were assessed. Polynomial Chaos emerges as the most promising methodology, and was then applied to a turbomachinery test case based on a thermal analysis of a high-pressure turbine disk.

Ported Shroud Flow Processes and Their Effect on Turbocharger Compressor Operation

The ported shroud (PS) self-recirculating casing treatment is widely used to delay the onset of the surge by enhancing the aerodynamic stability of the turbocharger compressor. The increase in the stable operation region of the turbocharger compressor is achieved by recirculating the low momentum fluid that blocks the blade passage to the compressor inlet through a ported shroud cavity. While the ported shroud design delays surge, it comes with a small penalty in efficiency. This work presents an investigation of the flow processes associated with a ported shroud compressor and quantifies the effect of these flow mechanisms on the compressor operation. The full compressor stage is numerically modelled using a Reynolds Averaged Navier-Stokes (RANS) approach employing the shear stress transport (SST) turbulence model for steady state simulations at the design and near surge conditions. The wheel rotation is modelled using a multiple reference frame (MRF) approach. The results show that the flow exits the PS cavity at the near surge condition in the form of three jet-like structures of varying velocity amplitudes. Net entropy generation in the compressor model is used to assess the influence of the ported shroud design on the compressor losses, and the results indicate a small Inlet-PS mixing region is the primary source of entropy generation in the near surge conditions. The analysis also explores the trends of entropy generation at the design and the near surge condition across the different speed lines. The results show that the primary source of entropy generation is the impeller region for the design condition and the inlet-PS cavity region for the near surge condition.

Design Optimization of a 3D Parameterized Vane Cascade With Non-Axisymmetric Endwall Based on a Modified EGO Algorithm and Data Mining Techniques

The design of turbomachinery cascades is a typical high dimensional and computationally expensive problem, a metamodel-based global optimization and data mining method is proposed to solve it. A modified Efficient Global Optimization (EGO) algorithm named Multi-Point Search based Efficient Global Optimization (MSEGO) is proposed, which is characterized by adding multiple samples at per iteration. By testing on typical mathematical functions, MSEGO outperforms EGO in accuracy and convergence rate. MSEGO is used for the optimization of a turbine vane with non-axisymmetric endwall contouring (NEC), the total pressure coefficient of the optimal vane is increased by 0.499%. Under the same settings, another two optimization processes are conducted by using the EGO and an Adaptive Range Differential Evolution algorithm (ARDE), respectively. The optimal solution of MSEGO is far better than EGO. While achieving similar optimal solutions, the cost of MSEGO is only 3% of ARDE. Further, data mining techniques are used to extract information of design space and analyze the influence of variables on design performance. Through the analysis of variance (ANOVA), the variables of section profile are found to have most significant effects on cascade loss performance. However, the NEC seems not so important through the ANOVA analysis. This is due to the fact the performance difference between different NEC designs is very small in our prescribed space. However, the designs with NEC are always much better than the reference design as shown by parallel axis, i.e., the NEC would significantly influence the cascade performance. Further, it indicates that the ensemble learning by combing results of ANOVA and parallel axis is very useful to gain full knowledge from the design space.

Numerical Investigation of an Asymmetric Double Suction Centrifugal Compressor With Different Backswept Angle Matching for a Wide Operating Range

The work presented here investigates the characteristics of the different impeller backswept angle matchings for a wide stable operating range in an asymmetric double suction centrifugal compressor. The numerical simulation was employed to investigate the influence of different backswept angle matchings on the stable operating range. The aim is to propose a proper change of the backswept angle matching between two impeller sides to improve the impeller power capability and mass flow distribution, furthermore, to delay the operating mode transition and widen the stable operating range of the compressor. Firstly, the method to determine the optimum backswept angle matching obtained by the theory calculation. Then, three matching models were proposed and analyzed in detail. In three matching models, the backswept angle differences between the front and rear impeller side are 0°, 10° and 20°, respectively. The analysis mainly focused on the influence of the different backswept angle matchings on the compressor flow field characteristics and the mass flow distribution characteristics. The results show that the change of the impeller backswept angle matching can improve the mass flow distribution characteristics for two impeller sides and further reduce the stall mass flow rate of the double suction compressor. The model that the backswept angle difference is 10° can delay the operating mode transition and reduce the stall mass flow of the double suction compressor. The model that the backswept angle difference is 20° can also reduce the stall mass flow and finally enable the front impeller into the stall condition. Therefore, the proper change of the backswept angle matching can achieve the purpose of reducing the stall mass flow and widening the operating range for the double suction centrifugal compressor.

Stall Inception in a High Speed Centrifugal Compressor During Speed Transients

The inception and evolution of rotating stall in a high-speed centrifugal compressor are characterized during speed transients. Experiments were performed in the Single Stage Centrifugal Compressor (SSCC) facility at Purdue University and include speed transients from sub-idle to full speed at different throttle settings while collecting transient performance data. Results show a substantial difference in the compressor transient performance for accelerations versus decelerations. This difference is associated with the heat transfer between the flow and the hardware. The heat transfer from the hardware to the flow during the decelerations locates the compressor operating condition closer to the surge line and results in a significant reduction in surge margin during decelerations. Additionally, data were acquired from fast-response pressure transducers along the impeller shroud, in the vaneless space, and along the diffuser passages. Two different patterns of flow instabilities, including mild surge and short-length-scale rotating stall, are observed during the decelerations. The instability starts with a small pressure perturbation at the impeller leading edge and quickly develops into a single-lobe rotating stall burst. The stall cell propagates in the direction opposite of impeller rotation at approximately one third of the rotor speed. The rotating stall bursts are observed in both the impeller and diffuser, with the largest magnitudes near the diffuser throat. Furthermore, the flow instability develops into a continuous high frequency stall and remains in the fully developed stall condition.

Investigation on Effect of Curvilinear Element Blades on Centrifugal Impeller Performance

This study experimentally and numerically investigates the effect of application of curvilinear element blades to fully-shrouded centrifugal compressor impeller on the performance of centrifugal compressor stage. Design suction flow coefficient of compressor stage investigated in this study is 0.125. The design guidelines for the curvilinear element blades which had been previously developed was applied to line element blades of a reference conventional impeller and a new centrifugal compressor impeller with curvilinear element blades was designed. Numerical calculations and performance tests of two centrifugal compressor stages with the conventional impeller and the new one were conducted to investigate the effectiveness of application of the curvilinear element blades and compare the inner flowfield in details. Despite 0.5% deterioration of the impeller efficiency, it was confirmed from the performance test results that the compressor stage with the new impeller achieved 1.7% higher stage efficiency at the design point than that with the conventional one. Moreover, it was confirmed that the compressor stage with the new impeller achieved almost the same off-design performance as that of the conventional stage. From results of the numerical calculations and the experiments, it is considered that this efficiency improvement of the new stage was achieved by suppression of the secondary flows in the impeller due to application of negative tangential lean. The suppression of the secondary flows in the impeller achieved uniformalized flow distribution at the impeller outlet and increased the static pressure recovery coefficient in the vaneless diffuser. As a result, it is thought that the total pressure loss was reduced downstream of the vaneless diffuser outlet in the new stage.

Comparing Human Driven and Automatic Blade Row Aerodynamic Designs

In this paper a fast automatic design environment is developed, making use of a well established and validated turbomachinery design software system for geometry generation and flow analysis. The design is updated via a gradient based algorithm, where gradients are obtained via the adjoint method. The computational advantages of Graphics Processing Units are used to accelerate the mesh generation and flow analysis stages. The capabilities of the system are illustrated by automatically generating two Low Pressure Turbine vanes, and comparing them to the ones arrived at by a human designer, respecting the same explicit design criteria. The quality of the automatically designed airfoils is assessed against the human generated ones, and insight on the influence of implicit criteria is extracted. It is concluded that acceptable quality geometries can be designed automatically in a short time. For instance, the automatic procedure takes of the order of two days for an equivalent human driven case, where the designer took of the order of two weeks.

A Centrifugal Compressor Impeller: A Multidisciplinary Optimization to Improve its Mass, Strength, and Gas-Dynamic Characteristics

The high-loaded centrifugal compressor blisk-type impeller, one of the main low-sized gas-turbine engine components, strongly affects engine efficiency. However, its design is a time-consuming and complex task for several reasons, including its high loading, the large number of structural and technological constraints, and the variety of requirements needed for application to a gas-turbine engine centrifugal compressor impeller (e.g., increased efficiency and strength, minimized weight requirements, etc.). The imposition of several constraints for structure modification of the centrifugal wheels can improve one characteristic but can worsen others. The standard solution for this problem is to use an iterative approach, whereby the design process is reduced to a consistent set of impeller element design problem statements and decisions; these are separate for different analysis disciplines. The main drawbacks to this approach are that it is labor intensive and can cause deterioration of the design quality because this procedure does not consider the design object as a unit. The present work considers a centrifugal compressor wheel design approach based on the use of an integrated multidisciplinary parameterized 3D model. This model includes a number of specialized sub-models that describe the necessary design areas as well as physical process features and phenomena occurring in the designed object. The model also realizes the integration and interaction of sub-models used in an integrated computing space. The proposed approach allows the optimization of the structure based on several criteria, such as the mass of the wheel, stage efficiency, strength, economic indicators, etc. The result of multi-criteria optimization is not a single product design, but a set of optimal Pareto points, which describes a number of centrifugal wheel models. The optimal configuration is selected from this set, based on what is considered the most important criterion. Optimization criteria may vary depending on the problem formulation, but the design technology, parameterization scheme, and choice of multidisciplinary integrated mathematical model are retained. Therefore, in the case of a product requirement correction, a new optimal design will require less time. In aggregate, with the nonlinear constrained optimization application, this approach reduces the total time of the design cycle, decreases development costs, and improves quality.

Design and Optimization of Multi-Stage Centrifugal Compressors With Uncertainty Quantification of Off Design Performance

The use of multi-stage centrifugal compressors carries out a leading role in oil and gas process applications. Green operation and market competitiveness require the use of low-cost reliable compression units with high efficiencies and wide operating range. A methodology is presented for the design optimization of multi-stage centrifugal compressors with prediction of the compressor map and estimation of the uncertainty limits. A one-dimensional (1D) design tool has been developed that automatically generates a multi-stage radial compressor satisfying the target machine requirements based on a few input parameters. The compressor performance map is then assessed using the method proposed by Casey-Robinson [1], and the approach developed by Al-Busaidi-Pilidis [2]. The off-design performance method relies on empirical correlations calibrated on the performance maps of many single-stage centrifugal compressors. An uncertainty quantification study on the predicted performance maps was conducted using Monte Carlo method (MCM) and generalized Polynomial Chaos Expansion (gPCE). Finally, the design procedure has been coupled to an in-house optimizer based on evolutionary algorithms. The complete design procedure has been applied to a multi-stage industrial compressor test case. A multi-objective optimization of a multi-stage industrial compressor has been performed targeting maximum compressor efficiency and flow range. The results of the optimization show the existence of optimum compressor architectures and how the Pareto fronts evolve depending on the number of stages and shafts.

LPC Blade and Non-Axisymmetric Hub Profiling Optimization Using Multi-Fidelity Non-Intrusive POD Surrogates

The present contribution proposes a Reduced Order Model based multi-fidelity optimization methodology for the design of highly loaded blades in low pressure compressors. Environmental, as well as, economical limitations applied to engine manufacturers make the design of modern turbofans an extremely complex task. A smart compromise has to be found to guarantee both a high efficiency and a high average stage loading imposed for mass reduction constraints, while satisfying stability requirements. The design of compressor blades, usually involves at the same time a dedicated parametrization set-up in highdimensional space and high-fidelity simulations capturing, at least, efficiency and stability as most impacting phenomena. Despite recent advances in the high-performance computing area, introducing high-fidelity simulations into automated optimization, or even surrogate assisted optimization, loops still stands as a endeavor for engineers. In this framework, the proposed methodology is based on multi-fidelity surrogate models capable of representing the physics at hand in reduced spaces inferred from both precise, albeit costly, high-fidelity simulations and abundant, yet less accurate lower-fidelity data. Finally, we investigate the coupling of the proposed hierarchised multi-fidelity non-intrusive Proper Orthogonal Decomposition based surrogates with an evolutionary algorithm to reduce the number of high-fidelity simulation calls towards the targeted optimum.