Christian Frey

Time-Linearized and Time-Accurate 3D RANS Methods for Aeroelastic Analysis in Turbomachinery

Graham Ashcroft Institute of Propulsion Technology, German Aerospace Center (DLR), Linder HA¶he, 51147 Cologne, Germany A computational method for performing aeroelastic analysis using either a time-linearized or an unsteady time-accurate solver for the compressible Reynolds averaged Navier--Stokes (RANS) equations is described. The time-linearized solver employs the assumption of small time-harmonic perturbations and is implemented via finite differences of the nonlinear flux routines of the time-accurate solver. The resulting linear system is solved using a parallelized generalized minimal residual (GMRES) method with block-local preconditioning. The time accurate solver uses a dual time stepping algorithm for the solution of the unsteady RANS equations on a periodically moving computational grid. For either solver, and both flutter and forced response problems, a mapping algorithm has been developed to map structural eigenmodes, obtained from finite element structural analysis, from the surface mesh of the finite element structural solver to the surface mesh of the finite volume flow solver. Using the surface displacement data an elliptic mesh deformation algorithm, based on linear elasticity theory, is then used to compute the grid deformation vector field. The developed methods are validated first using standard configuration 10. Finally, for an ultra-high bypass ratio fan, the results of the time-linearized and the unsteady module are compared. The gain in prediction time using the linearized methods is highlighted.

The Discrete Adjoint of a Turbomachinery RANS Solver

Since adjoint flow solvers allow for the computation of sensitivities of global flow parameters under geometric variations in an amount of time which is nearly independent of the number of geometric parameters, automatic shape optimization can be accelerated considerably by the use of an adjoint solver. In this article, a systematic approach for the development of an exact discrete adjoint of a turbomachinery flow solver is described. By using finite differences to differentiate the numerical fluxes, the problems associated with automatic and hand differentiation are circumvented. Moreover, a general treatment of the adjoint numerical boundary conditions is presented. As a result, an exact adjoint boundary condition for the conservative mixing planes is obtained. In combination with nonreflecting boundary conditions the latter are crucial for accurate flow simulations in turbomachinery. The adjoint is validated on the basis of a transonic compressor stage.Copyright

A Harmonic Balance Technique for Multistage Turbomachinery Applications

This article describes a nonlinear frequency domain method for the simulation of unsteady blade row interaction problems across several blade rows in turbomachinery. The capability to efficiently simulate such interactions is crucial for the improvement of the prediction of blade vibrations, tonal noise, and the impact of unsteadiness on aerodynamic performance.The simulation technique presented here is based on the harmonic balance approach and has been integrated into an existing flow solver. A nontrivial issue in the application of harmonic balance methods to turbomachinery flows is the fact that various fundamental frequencies may occur simultaneously in one relative system, each one being due to the interaction of two blade rows. It is shown that, considering the disturbances corresponding to different fundamental frequencies as mutually uncoupled, one can develop an unsteady simulation method which from a practial view point turns out to be highly attractive. On the one hand, it is possible to take into account arbitrarily many nonlinear interaction terms. On the other, the computational efficiency can be increased considerably once it is known that the nonlinear coupling between certain subsets of the harmonics plays only a minor role.To validate the method and demonstrate its accuracy and efficiency a multistage compressor configuration is simulated using both the method described in this article and a conventional time-domain solver.Copyright

Gradient Enhanced Surrogate Models Based on Adjoint CFD Methods for the Design of a Counter Rotating Turbofan

This paper studies the use of adjoint CFD solvers in combination with surrogate modelling in order to reduce the computational cost of the optimization of complex 3D turbomachinery components. The method is applied to a previously optimized counter rotating turbofan, with a shape parameterized by 104 CAD parameters.Through random changes on the reference design, a small number of design variations are created to serve as training samples for the surrogate models. A steady RANS solver and its discrete adjoint are then used to calculate objective function values and their corresponding sensitivities. Kriging and neural networks are used to build surrogate models from the training data. To study the impact of the additional information provided by the adjoint solver, each model is trained with and without the sensitivity information. The accuracy of the different surrogate model predictions is assessed by comparison against CFD calculations.The results show a considerable improvement of the fitness function approximation when the sensitivity information is taken into account. Through a gradient based optimization on one of the surrogate models, a design with higher isentropic efficiency at the aerodynamic design point is created. This application demonstrates that the improved surrogate models can be used for design and optimization.Copyright

Advanced Numerical Methods for the Prediction of Tonal Noise in Turbomachinery—Part II: Time-Linearized Methods

This is the second part of a series of two papers on unsteady computational fluid dynamics (CFD) methods for the numerical simulation of aerodynamic noise generation and propagation. It focuses on the application of linearized RANS methods to turbomachinery noise problems. The convective and viscous fluxes of an existing URANS solver are linearized and the resulting unsteady linear equations are transferred into the frequency domain, thereby simplifying the solution problem from unsteady time-integration to a complex linear system. The linear system is solved using a parallel, preconditioned general minimized residual (GMRES) method with restarts. In order to prescribe disturbances due to rotor stator interaction, a so-called gust boundary condition is implemented. Using this inhomogeneous boundary condition, one can compute the generation of the acoustic modes and their near field propagation. The application of the time-linearized methods to a modern high-bypass ratio fan is investigated. The tonal fan noise predicted by the time-linearized solver is compared to numerical results presented in the first part and to measurements.

Compressor Leading Edge Sensitivities and Analysis with an Adjoint Flow Solver

Based on the results of a prior study about fan blade degradation, which state a noticeable influence of small geometric changes on the fan performance, an adjoint computational fluid dynamics method is applied to systematically analyze the sensitivities of fan blade performance to changes of the leading edge geometry.As early as during manufacture, blade geometries vary due to fabrication tolerances. Later, when in service, engine operation results in blade degradation which can be reduced but not perfectly fixed by maintenance, repair and overhaul processes. The geometric irregularities involve that it is difficult to predict the blade’s aerodynamic performance. Therefore, the aim of this study is to present a systematic approach for analyzing geometric sensitivities for a fan blade.To demonstrate the potential, two-dimensional optimizations of three airfoil sections at different heights of a transonic fan blade are presented. Although the optimization procedure is limited to the small area of the leading edge, the resulting airfoil sections can be combined to a three-dimensional fan blade with an increased isentropic efficiency compared to the initial blade.Afterwards, an adjoint flow solver is applied to quasi-three-dimensional configurations of an airfoil section in subsonic flow with geometric leading edge variations in orders representative for realistic geometry changes. Validations with non-linear simulation results demonstrate the high quality of the adjoint results for small geometric changes and indicate physical effects in the leading edge region that influence the prediction quality.© 2013 ASME

Nonreflecting Boundary Conditions for Aeroelastic Analysis in Time and Frequency Domain 3D RANS Solvers

This paper describes the implementation of a set of nonreflecting boundary conditions of increasing approximation quality for time-accurate and time-linearized 3D RANS solvers in the time and frequency domain. The implementations are based on the computation of eigenfunctions, either analytically or numerically, of the linearized Euler or Navier-Stokes equations for increasingly complex background flows. This results in a hierarchy of nonreflecting boundary conditions based on 1D characteristics, 2D circumferential mode decomposition, and 3D circumferential and radial mode decomposition, including viscous effects in the latter, for the frequency domain solver. By applying a Fourier transform in time at the boundaries the frequency domain implementations can be employed in the time domain solver as well. The limitations of each approximation are discussed and it is shown that increasing the precision of the boundary treatment the nonreflecting property of the boundary conditions is preserved for more complex flows without incurring an excessive increase in computing time.Results of a flutter analysis of a low pressure turbine blade obtained by time and frequency domain simulations are validated against each other and against reference results obtained with a 3D Euler frequency domain solver. The comparison of the results for different boundary conditions reveals the importance of using high quality boundary conditions.Copyright

Advanced Numerical Methods for the Prediction of Tonal Noise in Turbomachinery—Part I: Implicit Runge–Kutta Schemes

This is the first part of a series of two papers on unsteady CFD methods for the numerical simulation of aerodynamic noise generation and propagation. In this part, the stability, accuracy and efficiency of implicit Runge-Kutta schemes for the temporal integration of the compressible Navier-Stokes equations are investigated in the context of a CFD code for turbomachinery applications. Using two model academic problems, the properties of two Explicit first stage, Singly Diagonal Implicit Runge-Kutta (ESDIRK) schemes of second- and third-order accuracy are quantified and compared with more conventional second-order multi-step methods. Finally, to assess the ESDIRK schemes in the context of an industrially relevant configuration, the schemes are applied to predict the tonal noise generation and transmission in a modern high bypass ratio fan stage and comparisons with the corresponding experimental data are provided.

Simulations of Unsteady Blade Row Interactions Using Linear and Non-Linear Frequency Domain Methods

This paper compares various approaches to simulate unsteady blade row interactions in turbomachinery. Unsteady simulations of turbomachinery flows have gained importance over the last years since increasing computing power allows the user to consider 3D unsteady flows for industrially relevant configurations. Furthermore, for turbomachinery flows, the last two decades have seen considerable efforts in developing adequate CFD methods which exploit the rotational symmetries of blade rows and are therefore up to several orders of magnitude more efficient than the standard unsteady approach for full wheel configurations.This paper focusses on the harmonic balance method which has been developed recently by the authors. The system of equations as well as the iterative solver are formulated in the frequency domain.The aim of this paper is to compare the harmonic balance method with the time-linearized as well as the non-linear unsteady approach. For the latter the unsteady flow fields in a fan stage are compared to reference results obtained with a highly resolved unsteady simulation. Moreover the amplitudes of the acoustic modes which are due to the rotor stator interaction are compared to measurement data available for this fan stage.The harmonic balance results for different sets of harmonics in the blade rows are used to explain the minor discrepancies between the time-linearized and unsteady results published by the authors in previous publications. The results show that the differences are primarily due to the neglection of the two-way coupling in the time-linearized simulations.Copyright

a code-coupling approach to the implementation of discrete adjoint solvers based on automatic differentiation

We propose a method for selectively applying automatic differentiation (AD) by operator overloading to develop the discrete adjoint of a turbomachinery flow solver. A fully differentiated version of the solver is generated by operator overloading using the tapeless tangent mode of ADOL-C. The differentiated solver is coupled to an undifferentiated version of the same code using message passing. The automatic differentiation is used to calculate derivatives of the flux calculation routines. The flux derivatives depending on inner cell states are sparse, and this sparsity is exploited using analytical differentiation of the spatial discretization scheme. Subsequently the sparse matrix is communicated to the undifferentiated code for solution. Turbomachinery boundary conditions may have dense Jacobians and are therefore only evaluated during the solution process. The solution of the adjoint system of equations is achieved through a preconditioned GMRES, implemented inside the undifferentiated code. A modern three dimensional contra-rotating fan stage with engineering parameterization serves as application example in order to demonstrate the technique and to perform numerical validations. The validation of gradient results is performed by comparing against results from finite differences, and the tangent forward mode.