In turbomachinery, the steady adjoint method has been successfully used for the computation of derivatives of various objective functions with respect to design variables in gradient-based optimization. However, the continuous advances in computing power and the accuracy limitations of the steady-state assumption lead toward the transition to unsteady computational fluid dynamics (CFD) computations in the industrial design process. Previous work on unsteady adjoint for turbomachinery applications almost exclusively rely upon frequency-domain methods, for both the flow and adjoint equations. In contrast, in this paper, the development the discrete adjoint to the unsteady Reynolds-averaged Navier–Stokes (URANS) solver for three-dimensional (3D) multirow applications, in the time-domain, is presented. The adjoint equations are derived along with the adjoint to the five-stage Runge–Kutta scheme. Communication between adjacent rows is achieved by the adjoint sliding interface method. An optimization workflow that uses unsteady flow and adjoint solvers is presented and tested in two cases, with objective functions accounting for the transient flow in a turbine vane and the periodic flow in a compressor three-row setup.

This work focuses on the application of multi-fidelity methods for the robust design optimization of engine components. The robust design optimization approach yields geometric designs that have high efficiencies and are less sensitive to uncertainties from manufacturing and wear. However, the uncertainty quantification techniques required to evaluate the robustness are computationally expensive, which limits their use in robust optimization. Multi-fidelity methods offer a promising solution to reduce the computational cost while maintaining accuracy in both uncertainty quantification and optimization. A Kriging and a multi-fidelity recursive Cokriging framework are developed, implemented, and applied to a test function. In addition, a multi-fidelity super efficient global optimization algorithm is developed. The optimizer is surrogate model-based and can handle constraints. The developed methods are then applied to a compressor test case of a high pressure compressor blade row with 9 uncertainty and 24 design parameters of the geometry. The 2.5% quantile of the stage efficiency is used as a robustness measure and it is therefore optimized. Design bounds and performance constraints are applied. In addition, various uncertainty quantification techniques are analyzed. A multi-fidelity uncertainty quantification approach is developed that combines simplified coarse-grid low-fidelity results with high-fidelity results to reduce the computational cost while maintaining high accuracy. Uncertainty quantification techniques of three fidelity levels are then developed and used for the multi-fidelity approach in the design space. The robust design optimization of the compressor is performed and the optimal designs obtained from the multi-fidelity approach show superior performance compared to existing robust design optima in the literature.

The increasing demands on jet engines require progressive thermodynamic process parameters, which typically lead to higher aerothermal loadings and accordingly to designs with high complexity. State-of-the-art high-pressure turbine (HPT) nozzle guide vane (NGV) design involves vane profiles with three-dimensional features including a high amount of film cooling and profiled endwalls (PEWs). Typically, the specific mass flow, also called capacity, which governs the engine's operation, is set by the HPT NGV. Hence, geometric variations due to manufacturing scatter of the HPT NGV's passage can affect relevant aerodynamic quantities and the entire engine behavior. Within the traditional deterministic design approach, the influences of those geometric variations are covered by conservative assumptions and engineering experience. This paper addresses the consideration of variability due to the manufacturing of HPT NGVs through probabilistic CFD investigations. To establish a statistical database, 80 HPT NGVs are digitized with a high precision optical 3D scanning system to record the outer geometry. The vane profiles are parametrized by a section-based approach. For this purpose, traditional profile theory is combined with a novel method that enables the description of NGV profile variability taking the particular leading edge (LE) shape into account. Furthermore, the geometric variability of PEWs is incorporated by means of principle component analysis (PCA). On this basis, a probabilistic system assessment including a sensitivity analysis in terms of capacity and total pressure loss coefficient is realized. Sampling-based methods are applied to conduct a variety of 3D CFD simulations for a typical population of profile and endwall geometries. This probabilistic investigation using realistic input parameter distributions and correlations contributes to a robust NGV design in terms of relevant aerodynamic quantities.

This paper will compare two approaches of sensitivity analysis, namely (i) the adjoint method which is used to obtain an initial estimate of the geometric sensitivity of the gas-washed surfaces to aerodynamic quantities of interest and (ii) a Monte Carlo type simulation with an efficient sampling strategy. For both approaches, the geometry is parameterized using a modified NACA parameterization. First, the sensitivity of those parameters is calculated using the linear (first-order) adjoint model. Since the effort of the adjoint computational fluid dynamics (CFD) solution is comparable to that of the initial flow CFD solution and the sensitivity calculation is simply a postprocessing step, this approach yields fast results. However, it relies on a linear model which may not be adequate to describe the relationship between relevant aerodynamic quantities and actual geometric shape variations for the derived amplitudes of shape variations. Second, in order to better capture nonlinear and interaction effects, a Monte Carlo type simulation with an efficient sampling strategy is used to carry out the sensitivity analysis. The sensitivities are expressed by means of the coefficient of importance (CoI), which is calculated based on modified polynomial regression and therefore able to describe relationships of higher order. The methods are applied to a typical high-pressure compressor (HPC) stage. The impact of a variable rotor geometry is calculated by three-dimensional (3D) CFD simulations using a steady Reynolds-averaged Navier–Stokes model. The geometric variability of the rotor is based on the analysis of a set of 400 blades which have been measured using high-precision 3D optical measurement techniques.

Modern high-pressure turbine (HPT) blade design stands out due to its high complexity comprising three-dimensional blade features, multipassage cooling system (MPCS), and film cooling to allow for progressive thermodynamic process parameters. During the last decade, probabilistic design approaches have become increasingly important in turbomachinery to incorporate uncertainties such as geometric variations caused by manufacturing scatter and deterioration. Within this scope, the first part of this two-part article introduces parametric models for cooled turbine blades that enable probabilistic finite element (FE) analysis taking geometric variability into account to aim at sensitivity and robustness evaluation. The statistical database is represented by a population of more than 400 blades whose external geometry is captured by optical measurement techniques and 34 blades that are digitized by computed tomography (CT) to record the internal geometry and the associated variability, respectively. Based on these data, parametric models for airfoil, profiled endwall (PEW), wedge surface (WSF), and MPCS are presented. The parametric airfoil model that is based on the traditional profile theory is briefly described. In this regard, a methodology is presented that enables to adapt this airfoil model to a given population of blades by means of Monte Carlo-based optimization. The endwall variability of hub and shroud are parametrized by radial offsets that are applied to the respective median endwall geometry. WSFs are analytically represented by planes. Variations of the MPCS are quantified based on the radial distribution of cooling passage centroids. Thus, an individual MPCS can be replicated by applying adapted displacement functions to the core passage centroids. For each feature that is considered within this study, the accuracy of the parametric model is discussed with respect to the variability that is present in the investigated blade population and the measurement uncertainty. Within the scope of the second part of this article, the parametric models are used for a comprehensive statistical analysis to reveal the parameter correlation structure and probability density functions (PDFs). This is required for the subsequent probabilistic finite element analysis involving real geometry effects.