Forced Response Sensitivity Analysis Using an Adjoint Harmonic Balance Solver

This paper describes the development and initial application of an adjoint harmonic balance solver. The harmonic balance method is a numerical method formulated in the frequency domain which is particularly suitable for the simulation of periodic unsteady flow phenomena in turbomachinery. Successful applications of this method include unsteady aerodynamics as well as aeroacoustics and aeroelasticity. Here we focus on forced response due to the interaction of neighboring blade rows. In the CFD-based design and optimization of turbomachinery components it is often helpful to be able to compute not only the objective values — e.g. performance data of a component — themselves, but also their sensitivities with respect to variations of the geometry. An efficient way to compute such sensitivities for a large number of geometric changes is the application of the adjoint method. While this is frequently used in the context of steady CFD, it becomes prohibitively expensive for unsteady simulations in the time domain. For unsteady methods in the frequency domain, the use of adjoint solvers is feasible, but still challenging. The present approach employs the reverse mode of algorithmic differentiation (AD) to construct a discrete adjoint of an existing harmonic balance solver in the framework of an industrially applied CFD code. The paper discusses implemen-tational issues as well as the performance of the adjoint solver, in particular regarding memory requirements. The presented method is applied to compute the sensitivities of aeroelastic objectives with respect to geometric changes in a turbine stage.

On the Simulation of Unsteady Turbulence and Transition Effects in a Multistage Low Pressure Turbine: Part III — Comparison of Harmonic Balance and Full Wheel Simulation

This is the third part of a series of three papers on the simulation of turbulence and transition effects in a multistage low pressure turbine. The third part of the series deals with the detailed comparison of the Harmonic Balance calculations with the full wheel simulations and measurements for the two-stage low-pressure turbine. The Harmonic Balance simulations were carried out in two confingurations, either using only the 0th harmonic in the turbulence and transition model or additional in all harmonics. The same Menter SST two-equation k–ω turbulence model along with Menter and Langtrys two-equation γ–Reθ transition model is used in the Harmonic Balance simulation as in the full wheel simulations. The measurements on the second stator ofthe low-pressure turbine have been carried out separately for downstream and upstream influences. Thus, a dedicated comparison of the downstream and upstream influences of the flow to the second stator is possible. In the Harmonic Balance calculations, the influences of the not directly adjacent blade, i.e. the first stator, were also included in the second stator In the first analysis, however, it was shown that the consistency with the full wheel configuration and the measurement in this case was not as good as expected. From the analysis ofthe full wheel simulation, we found that there is a considerable variation in the order ofmagnitude ofthe unsteady values in the second stator. In a further deeper consideration of the configuration, it is found that modes are reflected in upstream rows and influences the flow in the second stator. After the integration of these modes into the Harmonic Balance calculations, a much better agreement was reached with results ofthe full wheel simulation and the measurements. The second stator has a laminar region on the suction side starting at the leading edge and then transition takes place via a separation or in bypass mode, depending on the particular blade viewed in the circumferential direction. In the area oftransition, the clear difference between the calculations without and with consideration ofthe higher harmonics in the turbulence and transition models can be clearly seen. The consideration ofthe higher harmonics in the turbulence and transition models results an improvement in the consistency.

LES Investigation of Boussinesq Constitutive Relation Validity in a Corner Separation Flow

The present study aims at analysing the Boussinesq constitutive relation validity in a corner separation flow of a compressor cascade. The Boussinesq constitutive relation is commonly used in Reynolds-averaged Navier-Stokes (RANS) simulations for turbomachinery design. It assumes an alignment between the Reynolds stress tensor and the zero-trace mean strain-rate tensor. An indicator that measures the alignment between these tensors is used to test the validity of this assumption in a high fidelity large-eddy simulation. Eddy-viscosities are also computed using the LES database and compared. A large-eddy simulation (LES) of a LMFA-NACA65 compressor cascade, in which a corner separation is present, is considered as reference. With LES, both the Reynolds stress tensor and the mean strain-rate tensor are known, which allows the construction of the indicator and the eddy-viscosities. Two constitutive relations are evaluated. The first one is the Boussinesq constitutive relation, while the second one is the quadratic constitutive relation (QCR), expected to render more anisotropy, thus to present a better alignment between the tensors. The Boussinesq constitutive relation is rarely valid, but the QCR tends to improve the alignment. The improvement is mainly present at the inlet, upstream of the corner separation. At the outlet, the correction is milder. The eddy-viscosity built with the LES results are of the same order of magnitude as those built as the ratio of the turbulent kinetic energy k and the turbulence specific dissipation rate ω. They also show that the main impact of the QCR is to rotate the mean strain-rate tensor in order to realign it with the Reynolds stress tensor, without dilating it.

On the Use of the Automatic Differentiation for Developing a Linear Harmonic Solver

The linear and nonlinear harmonic methods are efficient frequency domain methods for analyzing time periodic unsteady flow fields. They have been widely used in both academia and industry. But the cost and complexity of developing a linear harmonic solver has been limiting its wider applications. On the other hand, the automatic differentiation (AD) has long been used in the CFD community with a focus on generating adjoint codes in a reverse mode. All those AD tools can do a much better job in generating linearized codes in a tangent mode, but so far very little, if any, attention is paid to using AD for developing linear harmonic solvers. The linear harmonic method, in comparison with the harmonic balance method, has its own advantages. For example, it can capture small disturbances very effectively, and avoids aliasing errors which can lead to solution instability since each wave component is solved for separately. This paper presents the effort of using an AD tool to generate major source codes for the development of a linear harmonic solver for analyzing time periodic unsteady flows. It includes the procedures and advice of using AD for such a purpose. A case study is also presented to validate the developed linear harmonic solver.

Correlations for the Prediction of Intermittency and Turbulent Spot Production Rate in Separated Flows

Experimental data describing laminar separation bubbles developing under strong adverse pressure gradients, typical of Ultra-High-Lift turbine blades, have been analyzed to define empirical correlations able to predict the main features of the separated flow transition. Tests have been performed for three different Reynolds numbers and three different free-stream turbulence intensity levels. For each condition, around 4000 Particle Image Velocimetry (PIV) snapshots have been acquired. A wavelet based intermittency detection technique, able to identify the large scale vortices shed as a consequence of the separation, has been applied to the large amount of data to efficiently compute the intermittency function for the different conditions. The transition onset and end positions, as well as the turbulent spot production rate are evaluated. Thanks to the recent advancements in the understanding on the role played by Reynolds number and free-stream turbulence intensity on the dynamics leading to transition in separated flows, guest functions are proposed in the paper to fit the data. The proposed functions are able to mimic the effects of Reynolds number and free-stream turbulence intensity level on the receptivity process of the boundary layer in the attached part, on the disturbance exponential growth rate observed in the linear stability region of the separated shear layer, as well as on the nonlinear later stage of completing transition. Once identified the structure of the correlation functions, a fitting process with own and literature data allowed us to calibrate the unknown constants. Results reported in the paper show the ability of the proposed correlations to adequately predict the transition process in the case of separated flows. The correlation for the spot production rate here proposed extends the correlations proposed in liter-ature for attached (by-pass like) transition process, and could be used in γ–Reϑ codes, where the spot production rate appears as a source term in the intermittency function transport equation.

CFD Study on the Influence of Geometric Parameters on the Aerodynamics Within an Ejector Injection System for Compressor Stabilization

Injection of high-momentum air into the tip-gap region of rotor stages is a measure of active aerodynamic stabilization of turbo compression systems. The Institute of Jet Propulsion at the University of the German Federal Armed Forces Munich advanced the concept of conventional tip air injection by deliberately deploying the ejector effect in order to increase the mass flow rate of the air injected. A novel Ejector Injection System (EIS) has been developed for the Larzac 04 jet engine and its intended ejector performance was proven in experimental pre-investigations. In addition to that, the corresponding CFD setup has been validated and an approach for highly efficient CFD simulations of the EIS ejector aerodynamics (node number reduction > 90%) was developed, verified, and validated. Thus, optimization of the ejector geometry in order to enhance the ejector aerodynamics and subsequently the stabilization performance of the EIS comes into focus now. Within this paper, a parametric CFD study is conducted to determine the influence of three main geometry parameters of the EIS ejector design on the ejector’s performance. The parameters, namely the injection nozzle spacing, the mixing duct length, and the ejector nozzle height, are introduced in the context of the overall EIS design and functionality. For efficiency purposes, a script-based procedure which deploys ANSYS ICEM CFD and ANSYS CFX has been developed in order to conduct the CFD parameter study covering 205 simulations fully automated. Each ejector geometry is thereby simulated with five different primary air mass flow rates supplied to the EIS covering a range from low-speed to transonic operation. It is revealed that all three geometry parameters investigated show partially significant impact on the ejector performance in terms of the entrainment ratio μ. In order to get a detailed insight into the inner EIS aerodynamics, also primary air Mach and Reynolds numbers, the state of mixing between primary and secondary air, and velocity profiles in the LPC’s tip region are subjects of investigation. Based on these findings and the general aerodynamic coherences discovered, recommendations for optimizing the current EIS ejector design are presented.

High Fidelity Spectral/hp Element Methods for Turbomachinery

The high order spectral/hp element methods implemented in the software framework Nektar++ are investigated for scale-resolving simulations of LPT profiles. There is a growing demand for high fidelity methods for turbomachinery to move towards numerical “experiments”. The study contributes at building best practices for the use of emerging high fidelity spectral element methods in turbomachinery predictions, with focus on the numerical details that are specific of these classes of methods. For this reason, the T106A cascade is used as a base reference application because of availability of data from previous investigations. The effects of polynomial order (p-refinement), spanwise domain extent and spanwise Fourier planes are considered, looking at flow statistics, convergence and sensitivity of the results. The performance of the high order spectral/hp element method is also assessed through validation against experimental data at moderately high Reynolds number. Thanks to the reduced computational cost, the proposed methods will have a strong impact in turbomachinery, paving the way to its use for design purposes and also allowing for a deeper understanding of the flow physics.

Axial Compressor Performance Prediction Using Improved Streamline Curvature Approach

This paper presents in detail the improved streamline curvature approach (SLC) to the performance evaluation and internal flow field calculation of subsonic and transonic axial compressors. Based on previous research, the diverse incidence, deviation and total pressure loss models, generally existing in the form of fitting curves and semi-empirical correlations, are discussed respectively. Typically, transonic flow in axial compressor results in the variation of several flow parameters and particularly the appearance of shock waves compared with subsonic flow. In this paper, the revision and improvement of loss models are applied to reach higher accuracy, especially considering the loss component due to actual incidence angle. Several modifications have been made as well considering the influence of three-dimensional flow. For the purpose of validating this approach, two test cases, including a single-stage transonic axial compressor NASA Stage37 and a 3-stage subsonic axial compressor P&W 3S1, are calculated. The overall characteristics and spanwise aerodynamic parameters for blade rows are demonstrated at both design and off-design conditions. Furthermore, the results agree well with both experimental data and computational fluid dynamic (CFD) results. This throughflow method is verified as an applicable and convenient tool for aerodynamic analysis and performance prediction of subsonic and transonic axial compressors.

Aero-Thermal Coupled Predictive Model for Preliminary Gas Turbine Blade Cooling Analysis

The turbine inlet temperature has been increasing over the years to improve gas turbine efficiency and specific power. Blade cooling technology is essential to keep component temperatures below their critical value, and this makes the aero-thermal coupled phenomenon more significant. Blade life assessment is closely related to blade metal temperature distribution and gradients, and blade cooling analysis is always considered starting from the preliminary design stage. However, traditional blade cooling analysis for preliminary design is always based on external boundary conditions determined by experience, which affects the prediction accuracy as the interaction effect between the main flow and coolant is not considered. In this paper, an aero-thermal coupled blade cooling model is further developed by combining the improved streamline curvature method with a one-dimensional thermo-fluid network. This model is capable of predicting blade surface temperature distribution and internal coolant flow conditions in the preliminary phase of blade cooling design with a limited amount of input information. Experimental data for the NASA C3X profile with film cooling was selected for validation. In addition, a sensitivity analysis was performed on different film cooling mass flow rates to demonstrate the model flexibility for different boundary conditions.