Aerodynamic Influence Coefficients Research Articles

The broad study of blade cascade under controlled torsional flutter: Dynamics of the flow and stability analysis

The experimental and numerical investigation of the flow instabilities acting on rigid blades and vice versa was conducted for both compressor and turbine configuration. The blade cascade consisted of five rectangular NACA 0010 blades, with three middle blades capable of performing harmonic motion with one degree of freedom (pitching) using force excitation. The base case (all blades fixed) and excited regime were examined. The influence of various angles of attack, harmonic frequency values, amplitude values, inter-blade phase angles and Reynolds numbers (Re) were tested. The mean flow properties as well as the fluid - structure interaction (FSI) were studied using Particle Image Velocimetry (PIV), Reynolds-averaged Navier-Stokes (RANS) CFD methods and using force measurement. Additionally, two different approaches, namely traveling wave mode (TWM) and aerodynamic influence coefficient (AIC), were adopted to estimate the aeroelastic stability of the blade cascade, and the results were compared. The results show significant aeroelastic coupling between the blades in both compressor and turbine configuration. However, the aerodynamic coupling effect for torsional flutter is more prominent in turbine configuration.

Experimental Investigations on Aeroelastic Stability of Combined Plunge-Pitch Mode Shapes in a Linear Compressor Cascade

Abstract The effect of mode shape and reduced frequency on the flutter stability of a linear low-speed compressor cascade was investigated experimentally, employing the aerodynamic influence coefficient (AIC) approach. This paper describes in detail the methodology, experimental setup, and measurement techniques. The paper presents experimentally determined influence coefficients, and discusses the findings with regard to the aeroelastic design parameter “plunge-to-twist incidence ratio” (PTIR), which combines reduced frequency and torsion axis setback in an attempt a design variable reduction. The influence of the vibrating blade on itself was always stabilizing, while other blades, mainly the first pressure side neighbor, vibrating at forward-traveling, low nodal diameter inter-blade phase angles (IBPAs), can destabilize the reference blade. As PTIR is increased, the phase of the complex modal force coefficients (AICs) of all blades is decreased, increasing the stabilizing effect of the vibrating blade on itself and overall cascade stability. Within the considered parameter range, the impact of reduced frequency is eliminated to a large extent if the incidence ratio is kept constant, requiring the torsion axis setback to be adjusted accordingly. The findings show that the plunge-to-twist incidence ratio is a meaningful aeroelastic design parameter that can explain the global effect of mode shape on flutter stability and should be considered early in the aeromechanical blade design process to ensure flutter-free blading.

Identification of the Aeromechanical Parameters of a Compressor Rotor from Blade Tip Timing Data

Abstract This article presents a maximum likelihood approach to identifying the mistuning, aerodynamic forcing functions, and aerodynamic influence coefficients of compressor rotor blades from noisy experimental measurements of blade deflection. The maximum likelihood estimation approach leads naturally to a nonlinear least squares optimization problem that can be solved using numerical optimization techniques such as quasi-Newton methods. An eigenmode decomposition approach is presented that permits the optimization objective function and gradient to be efficiently calculated so that the optimization is computationally feasible, even for experiments involving very large data sets. Simulation results show that the method has the potential to accurately identify the mistuning and aerodynamic nodal diameter forcing functions, but obtaining accurate results for influence coefficients may not be possible if there is significant disk flexibility. We demonstrate the method using both simulated and experimental blade vibration data from rotor 2 of the Purdue University three-stage axial compressor research facility collected using a light probe tip timing system as the rotor is slowly accelerated or decelerated through a Campbell diagram crossing.

A two-level strategy for aeroelastic optimization of a 3D wing with constant and variable stiffness skins

In this work, optimization strategies employing a standard heuristic free-derivative algorithm combined with a first-order gradient approach are used to maximize the flutter or divergence airspeed, using the fibre material orientations as design variables. The aeroelastic stability analysis calculations are performed by a classical PK method. The finite element method is used to obtain the structural modes and frequencies and the unsteady aerodynamic influence coefficient matrices are obtained with a doublet-lattice model. Numerical studies are performed with a complex 3D model that includes curved surfaces and internal elements such as spars and ribs. The main goal of the article is to use the heuristic part with a small number of parameters, to reduce the design space and provide an efficient initial guess for the gradient search, so that the optimization complexity is substantially reduced. Results for constant and variable stiffness composite wings are presented for laminates with different numbers of plies. It is shown by several examples that the pres.

Uncertainty quantification in free vibration and aeroelastic response of variable angle tow laminated composite plate

The influence of uncertainty in material properties on free vibration and aeroelastic response of variable angle tow (VAT) laminated plates is presented in this work. Free vibration analysis is conducted by a finite element model based on third-order shear deformation theory. The FE model is further coupled with MSC. Nastran to carry out aeroelastic analysis of VAT laminates in the subsonic regime. MSC. Nastran helps to extract the aerodynamic influence coefficient matrix at discrete values of reduced frequencies using Doublet Lattice Method. The accuracy and adaptability of the highly efficient radial basis neural network (RBFN) based surrogate model developed for uncertainty analysis are checked by comparing the result with that of the Monte Carlo simulation. Stochastic analysis of natural frequencies and flutter characteristics of VAT laminates are conducted through various parametric studies. Variance-based sensitivity analysis is used to calculate the contribution of individual material properties to the free vibration and aeroelastic response of VAT laminates. Further, the influence of the highly sensitive properties on the structural failure probability is estimated. This research can be leveraged to predict the probability of failure of the structure at different levels of material uncertainty present in it.

The influence and application of nonlinear aerodynamics on static derivatives in transonic regime

This paper details two static aeroelastic analysis methods applied to a passenger aircraft model with high aspect ratio wing. The influence of nonlinear aerodynamic force on static aeroelastic derivatives in the transonic regime is analysed. The traditional aerodynamic influence coefficient (AIC) matrix method can produce fast and reliable aerodynamic force and is widely used in aeroelastic analysis. However, the AIC matrix computed by linear aerodynamics will lead to some errors in transonic regime because of the nonlinear effect of aerodynamics. By generating the correction matrices, the AIC matrix is modified, and the accuracy of transonic static aeroelastic correction of aerodynamic data can be improved. The static derivatives are compared to the results of the computational fluid dynamics (CFD) / computational structural (CSD) interaction method.

A Modification to the Enhanced Correction Factor Technique for the Subsonic Wing–Body Interference Model

In this work, the enhanced correction factor technique (ECFT) is modified for a subsonic wing–body interference model, which can consider the forces on both the lifting boxes and the body elements of an idealized airplane, termed the advanced ECFT method. A passenger aircraft model is chosen as the simulation model, and the longitudinal static aeroelasticity at the transonic situation for two-degree freedom, including the α (angle of attack) degree and ϕ (angle of horizontal tail) degree, is simulated in this paper. The corresponding CFD results are used to correct the aerodynamic influence coefficients (AIC) matrix, which is then simulated by MSC.NASTRAN. The pressure distribution results of different aircraft components received by the advanced ECFT method indicate that it is suitable for the subsonic wing–body interference model. Compared with the uncorrected linear method and the diagonal corrected method, it is generally more consistent with the CFD/CSD coupling method, not only for the lifting boxes, but also for the body elements. In addition, the aerodynamic derivative results also show good agreement with the flight test data, which solidly verifies the advance ECFT method.

Geometrically Nonlinear Effects on the Aeroelastic Response of a Transport Aircraft Configuration

This paper investigates geometrically nonlinear aeroelastic effects on an industry-level aircraft configuration. Results are obtained via a new approach that uses geometrical information to compute nonlinear effects and seamlessly integrates with conventional, linear aeroelastic load packages. The starting point is a generic finite-element model and frequency-domain aerodynamic influence coefficient matrices, and the resulting system includes control inputs, longitudinal trim, or gust disturbances in time domain, which can be employed separately or in combination to obtain a nonlinear dynamic model for loads and stability analysis. The aeroelastic response of a long range aircraft is studied, highlighting the difference between linear and nonlinear approaches in the calculations of the aircraft flying equilibrium, dynamic perturbations, and variations to the flutter boundary. Results justify the need for geometrically nonlinear analysis in the production environment of airplanes with ultrahigh-aspect-ratio wings.

Subsonic source and doublet panel methods

This work presents a source and doublet surface panel method for solving unsteady subsonic flows around wing and bodies. The approach is based on Morino’s theory but features three modifications. The impermeability boundary condition is defined as zero mass flux normal to the surface instead of zero normal velocity, a new methodology is introduced to calculate the aerodynamic influence coefficient matrices and the pressure distribution around the surface is evaluated using a nonlinear unsteady Bernoulli equation. The method is demonstrated on three experimental test cases from the literature; it is shown that it can predict more accurately the steady and oscillating pressure distribution around the surface than Morino’s original linear approach. Crucially, the modified approach can represent asymmetries in the pressure distribution due to both asymmetric motion and shape. While higher fidelity alternatives exist for solving such flows, the present method has very low computational cost and would be suitable for optimization procedures during preliminary design, as long as the flow remains subcritical and attached at all times.

Aeroelastic Analysis of a Supercritical Airfoil with Free-Play in Transonic Flow

An investigation has been made into the nonlinear aeroelastic behavior of a supercritical airfoil (NLR 7301) considering free-play in transonic flow. Computational Fluid Dynamics (CFD) based on NavierStokes equations is implemented to calculate unsteady aerodynamic forces. A loosely coupled scheme with steady CFD and a graphic method are developed to obtain the static aeroelastic position. Frequency domain flutter solution with transonic aerodynamic influence coefficient is used to capture the linear flutter characteristic at different angle of attack (AoA). Time marching approach based on CFD is used to calculate the nonlinear aeroelastic response. The bifurcation diagram of pitching motion shows that as the airspeed increases, the limit cycle oscillation (LCO) appears then quenches, forming the first LCO branch; LCO occurs again and sustains until the divergence of the response, forming the second LCO branch. This phenomenon is essentially induced by the combined action of the static aeroelastic position caused by the camber effect of the airfoil and the S shape flutter boundary with respect to AoA.

A reduced order model for coupled mode cascade flutter analysis

A Reduced Order Model (ROM) based analysis method for turbomachinery cascade coupled mode flutter is presented in this paper. The unsteady aerodynamic model is established by a system identification technique combined with a set of Aerodynamic Influence Coefficients (AIC). Subsequently, the aerodynamic model is encoded into the state space and then coupled with the structural dynamic equations, resulting in a ROM of the cascade aeroelasticity. The cascade flutter can be determined by solving the eigenvalues of the ROM. Bending-torsional coupled mode flutter analysis for the Standard Configuration Eleven (SC11) cascade is used to validate the proposed method.

Mistuning and Damping of a Radial Turbine Wheel. Part 1: Fundamental Analyses and Design of Intentional Mistuning Pattern

Abstract The radial turbine impeller of an exhaust turbocharger is analyzed in view of both free vibration and forced response. Due to random blade mistuning resulting from unavoidable inaccuracies in manufacture or material inhomogeneities, localized modes of vibration may arise, which involve the risk of severely magnified blade displacements and inadmissibly high-stress levels compared to the tuned counterpart. Contrary, the use of intentional mistuning (IM) has proved to be an efficient measure to mitigate the forced response. Independently, the presence of aerodynamic damping is significant with respect to limit the forced response since structural damping ratios of integrally bladed rotors typically take extremely low values. Hence, detailed knowledge of respective damping ratios would be desirable while developing a robust rotor design. For this, far-reaching experimental investigations are carried out to determine the damping of a comparative wheel within a wide pressure range by simulating operation conditions in a pressure tank. Reduced-order models are built up for designing suitable intentional mistuning patterns by using the subset of nominal system modes approach introduced by Yang and Griffin (2001, “A Reduced-Order Model of Mistuning Using a Subset of Nominal System Modes,” J. Eng. Gas Turbines Power, 123(4), pp. 893–900), which conveniently allows for accounting both differing mistuning patterns and the impact of aeroelastic interaction by means of aerodynamic influence coefficients. Further, finite element analyses are carried out in order to identify appropriate measures of how to implement intentional mistuning patterns, which are featuring only two different blade designs. In detail, the impact of specific geometric modifications on blade natural frequencies is investigated. The first part of this three-part paper is focused on designing the IM pattern. The second and third part following, later on, will address the topics (i) experimental validation after implementation of the IM pattern at rest and under rotation, and (ii) the development of an approach for fast estimating damping ratios in the design phase.

Effects of Nonuniform Blade Spacing on Compressor Rotor Flutter Stability

A generalized flat plate cascade model is used in this study to predict the unsteady aerodynamics of a vibrating blade row with nonuniform blade spacing in subsonic compressible flow. The blade row is assumed to vibrate in an isolated family of blade-dominated modes. The effect of nonuniform blade spacing on compressor rotor flutter stability is demonstrated by case studies based on the geometric and flow conditions of a high-speed three-stage axial research compressor. The results show that nonuniform blade spacing can greatly alter the blades’ aerodynamic damping. At certain vibrational nodal diameters, some blades are destabilized so much that their aerodamping becomes negative. However, negative aerodamping of some blades do not necessarily lead to the instability of the whole blade row. A general multibladed system aeroelastic model is derived to study the effects of the nonuniform blade spacing on rotor stability through an eigenvalue approach. The aerodynamic influence coefficients matrix can be calculated using the generalized flat plate cascade model for a blade row with any user-specified blade spacing patterns. The case studies investigated in this paper show that alternating blade spacing and shifting only one blade position can slightly increase the stability of the least-stable eigenmode, whereas sinusoidal blade spacing has a slightly destabilizing effect. On the other hand, the eigenvectors of the least-stable mode for the nonuniformly spaced blade rows can be significantly different from the uniform blade spacing case.

A non-intrusive geometrically nonlinear augmentation to generic linear aeroelastic models

A new approach to build geometrically-nonlinear dynamic aeroelastic models is proposed that only uses information typically available in linear aeroelastic analyses, namely a generic (linear) finite-element model and frequency-domain aerodynamic influence coefficient matrices (AICs). Good computational efficiency is achieved through a two-step process: Firstly, a geometric reduction of the structure is carried out through static or dynamic condensation on nodes along the main load paths of the vehicle. Secondly, manipulation of the resulting linear normal modes (LNMs), the condensed stiffness and mass matrices, and the nodal coordinates provides the modal coefficients of the intrinsic beam equations along these load paths. This preserves the LNMs of the original problem and augments them with the geometrically nonlinear terms of beam theory. The structural description is in material coordinates and modal AICs are thus naturally included as follower forces. Numerical examples include cantilever wings built using detailed models, for which effects such as nonlinear aeroelastic equilibrium, nonlinear dynamics and structural-driven limit-cycle oscillations are shown. Results demonstrate the ability of the methodology to seamlessly and efficiently incorporate critical nonlinear effects to (linear) arbitrarily large aeroelastic models of high aspect ratio wings.

Analysis of Friction-Saturated Flutter Vibrations With a Fully Coupled Frequency Domain Method

Abstract Flutter stability is a dominant design constraint of modern turbines. Thus, flutter-tolerant designs are currently explored, where the resulting vibrations remain within acceptable bounds. In particular, friction damping has the potential to yield limit cycle oscillations (LCOs) in the presence of a flutter instability. To predict such LCOs, it is the current practice to model the aerodynamic forces in terms of aerodynamic influence coefficients for a linearized structural model with fixed oscillation frequency. This approach neglects that both the nonlinear contact interactions and the aerodynamic stiffness cause a change in the deflection shape and the frequency of the LCO. This, in turn, may have a substantial effect on the aerodynamic damping. The goal of this paper is to assess the importance of these neglected interactions. To this end, a state-of-the-art aero-elastic model of a low pressure turbine blade row is considered, undergoing nonlinear frictional contact interactions in the tip shroud interfaces. The LCOs are computed with a fully coupled harmonic balance method, which iteratively computes the Fourier coefficients of structural deformation and conservative flow variables, as well as the a priori unknown frequency. The coupled algorithm was found to provide excellent computational robustness and efficiency. Moreover, a refinement of the conventional energy method is developed and assessed, which accounts for both the nonlinear contact boundary conditions and the linearized aerodynamic influence. It is found that the conventional energy method may not predict a limit cycle oscillation at all while the novel approach presented here can.

Aeroelastic Flutter Prediction Using Multifidelity Modeling of the Generalized Aerodynamic Influence Coefficients

This work proposes a multifidelity modeling approach for predicting aeroelastic flutter of airfoils and wings. Using aerodynamic models based on the doublet-lattice method and time-accurate Euler equations, cokriging-based surrogates of the generalized aerodynamic influence coefficients are generated as functions of Mach number and reduced frequency. The surrogate-based matrix terms are then used in the method to determine flow conditions at flutter onset. To demonstrate the multifidelity process, a widely used pitching and plunging airfoil case is considered. Verification of the approach is done by comparing with results from a mode-based time-domain aeroelastic solver, as well as data from the literature. The approach draws inspiration from Timme et al., but focuses more on widely used industry tools (namely, the method, panel-based aerodynamics, and time-domain computational fluid dynamics). The benefit of using multiple aerodynamic fidelities, rather than high-fidelity kriging models, is also quantified by examining a flutter speed error metric as the number of high-fidelity samples is varied.

Multifidelity flutter prediction using local corrections to the generalized AIC

This work adopts the manifold mapping multifidelity modeling technique for frequency-domain flutter prediction. While this type of multifidelity model is usually employed in a trust-region optimization framework, the current approach uses it, in combination with the p-k method, to find a critical point in the high-fidelity Mach-reduced frequency space (namely, the matched flutter point for a given density and sound speed). This is done by first computing a grid of low-fidelity (doublet lattice-based) aerodynamic influence coefficients in modal coordinates. After starting at the low-fidelity match point, the process iteratively approaches the high-fidelity (Euler-based) match point using a single Mach and reduced frequency evaluation per iteration. The locally accurate multifidelity model operates by applying a correction matrix to the low-fidelity response using the changes in high- and low-fidelity response vectors between iterations; the correction matrix is computed using the Moore-Penrose pseudoinverse operator, making it reminiscent of various doublet lattice correction methods. When applied to a common test case, the process is shown to converge in 4 to 8 high-fidelity evaluations, depending on the fluid density and sound speed. For comparison, standard time- and frequency domain methods are estimated to be 3 to 10 times more costly in terms of time steps or frequency evaluations.

Multifidelity flutter prediction using regression cokriging with adaptive sampling

This work presents a flutter prediction approach that uses regression cokriging metamodels of generalized aerodynamic influence coefficients with adaptive sampling based on propagated model uncertainty along the flutter boundary. The use of regression cokriging models is compared to cokriging and regression cokriging with reinterpolation, as well as their single-fidelity counterparts. Comparisons to direct quantity-of-interest-based metamodeling are also shown. Several infill criteria based on the propagated flutter speed uncertainty are demonstrated on common flutter test cases. The value of adaptive sampling, multiple fidelity levels, and metamodeling of intermediate quantities is investigated by quantifying average cost and error metrics for the cases. Scalability with the number of structural modes is also investigated to gauge how the approach might fare for more conventional aircraft. Overall, the main benefits seen in this work stem from modeling intermediate quantities, with direct modeling costing six to eight times as much for multifidelity approaches, and three to five times as much for the single-fidelity comparators. In addition, using multiple fidelities was more accurate and required fewer infill points for convergence, leading to a cost savings of roughly 25% to 70%, depending on the case.

Nonlinear aeroelastic behavior of an airfoil with free-play in transonic flow

An investigation has been made into the nonlinear aeroelastic behavior of an airfoil system with free-play nonlinear stiffness in transonic flow. Computational Fluid Dynamics (CFD) and Reduced Order Model (ROM) based on Euler and Navier-Stokes equations are implemented to calculate unsteady aerodynamic forces. Results show that the nonlinear aeroelastic system experiences various bifurcations with increasing Mach number. Regular subcritical bifurcations are observed in low Mach number region. Subsequently, complex Limit Cycle Oscillations (LCOs) and even non-periodic motions appear at specific airspeed regions. When the Mach number is increased above the freeze Mach number, regular subcritical bifurcations occur again. Comparisons with inviscid solutions are used to identify and elaborate the effect of viscosity with the help of aeroelastic analysis techniques, including root locus, Single Degree of Freedom (SDOF) flutter and aerodynamic influence coefficient (AIC). For low Mach numbers in the transonic regime, the viscosity has little effect on the linear flutter characteristic because of limited influence on AIC, but a remarkable impact on the nonlinear dynamic behavior due to the sensitivity of the nonlinear structure. As the Mach number increases, the viscosity becomes significantly important due to the existence of shock-boundary layer interaction. It affects the unstable mechanism of linear flutter, impacts the aerodynamic center and hence the snap-through phenomenon, influences the AIC and consequently the nonlinear aeroelastic response. When the Mach number is increased further, the shock wave dominates the air flow and the viscosity is of minor importance.

Calculation of the Hinge Moments of a Folding Wing Aircraft during the Flight-Folding Process

A folding wing morphing aircraft should complete the folding and unfolding process of its wings while in flight. Calculating the hinge moments during the morphing process is a critical aspect of a folding wing design. Most previous studies on this problem have adopted steady-state or quasi-steady-state methods, which do not simulate the free-flying morphing process. In this study, we construct an aeroelastic flight simulation platform based on the secondary development of ADAMS software to simulate the flight-folding process of a folding wing aircraft. A flexible multibody dynamic model of the folding wing structure is established in ADAMS using modal neutral files, and the doublet lattice method is developed to generate aerodynamic influence coefficient matrices that are suitable for the flight-folding process. The user subroutine is utilized, aerodynamic loading is realized in ADAMS, and an aeroelastic flight simulation platform of a folding wing aircraft is built. On the basis of this platform, the flight-folding process of the aircraft is simulated, the hinge moments of the folding wings are calculated, and the influences of the folding rate and the aircraft’s center of gravity (c.g.) position on the results are investigated. Results show that the steady-state method is applicable to the slow folding process. For the fast folding process, the steady-state simulation errors of the hinge moments are substantially large, and a transient method is required to simulate the flight-folding process. In addition, the c.g. position considerably affects the hinge moments during the folding process. Given that the c.g. position moves aft, the maximum hinge moments of the inner and outer wings constantly increase.