Linear Aerodynamics Research Articles

Data-driven aeroelastic analyses of structures in turbulent wind conditions using enhanced Gaussian Processes with aerodynamic priors

Recent advancements in data-driven aeroelasticity have been driven by the wealth of data available in the wind engineering practice, especially in modeling aerodynamic forces. Despite progress, challenges persist in addressing free-stream turbulence and incorporating physics knowledge into data-driven aerodynamic force models. This paper presents a hybrid Gaussian Process (GPs) methodology for non-linear modeling of aerodynamic forces induced by gusts and motion on bluff bodies. Building on a recently developed GP model of the motion-induced forces, we formulate a hybrid GP aerodynamic force model that incorporates both gust- and motion-induced angles of attack as exogenous inputs, alongside a semi-analytical quasi-steady (QS) model as a physics-based prior knowledge. In this manner, the GP model incorporates the absent physics of the QS model, and the non-dimensional hybrid formulation enhances its appeal from an aerodynamic perspective. We devise a training procedure that leverages simultaneous input signals of gust angles, based on random free-stream turbulence, and motion angles, based on random broadband signals. We verify the methodology through analytical linear aerodynamics of a flat plate and non-linear aerodynamics of a bridge deck using Computational Fluid Dynamics (CFD). The standout feature of the presented methodology is its applicability for aeroelastic buffeting analyses, showcasing robustness when handling broadband excitation. Importantly, the non-linear hybrid model preserves its capability to capture higher-order harmonics in the motion-induced forces and remains applicable for flutter analysis, while incorporating both motion and gust angles as input. Applications of the methodology are anticipated in the aeroelastic analysis and monitoring of slender line-like structures.

Multifidelity Comparison of Supersonic Wave Drag Prediction Methods Using Axisymmetric Bodies

Low-fidelity analytic and computational wave drag prediction methods assume linear aerodynamics and small perturbations to the flow. Hence, these methods are typically accurate for only very slender geometries. The present work assesses the accuracy of these methods relative to high-fidelity Euler, compressible computational-fluid-dynamics solutions for a set of axisymmetric geometries with varying radius-to-length ratios (R/L). Grid-resolution studies are included for all computational results to ensure grid-resolved results. Results show that the low-fidelity analytic and computational methods match the Euler CFD predictions to around a single drag count ( ∼1.0×10−4) for geometries with R/L≤0.05 and Mach numbers from 1.1 to 2.0. The difference in predicted wave drag rapidly increases, to over 30 drag counts in some cases, for geometries approaching R/L≈0.1, indicating that the slender-body assumption of linear supersonic theory is violated for larger radius-to-length ratios. All three methods considered predict that the wave drag coefficient is nearly independent of Mach number for the geometries included in this study. Results of the study can be used to validate other numerical models and estimate the error in low-fidelity analytic and computational methods for predicting wave drag of axisymmetric geometries, depending on radius-to-length ratios.

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.

Kutta–Joukowski Theorem for Unsteady Linear Aerodynamics

This paper presents the extension of the Kutta–Joukowski theorem to unsteady linear aerodynamics. Starting from the formulation developed by Theodorsen for the solution of the velocity potential for circulatory flows around thin, rectilinear airfoils, the frequency response function between bound circulation and circulatory lift is derived. To provide a formulation suitable for time-domain applications, two finite-state approximations of the Kutta–Joukowski frequency response function, with different degrees of complexity and accuracy, are also proposed. The validity of the derived unsteady Kutta–Joukowski theorem is verified by correlation with numerical predictions of the circulatory lift given by a validated boundary-element-method solver for potential flows. Furthermore, the proposed finite-state approximations of the unsteady Kutta–Joukowski theorem are applied to problems concerning the circulatory lift response to damped oscillatory airfoil motion and gust perturbation. The higher-accuracy finite-state model provides predictions that correspond perfectly to those obtained by the convolution integrals of the step responses given by the theories of Wagner and Küssner. The results given by the simpler finite-state model derived from the linear approximation of the frequency response function are satisfactory for low-frequency problems, and are compared with those provided by a widely used approximate unsteady version of the Kutta–Joukowski theorem available in the literature.

Analytical Model for Ring-Wing Propulsors at Angle of Attack

A simple, accurate, analytical model is developed for predicting the forces and moments on subsonic ring-wing propulsors (aka fan ducts and shrouded propellers) at angle of attack. The method is applicable to drones, aircraft, ships, and more from hover to cruise. The new model is made up from elements of three previously published, closely related models, herein combined and modified for the current purpose: an empty shroud ring-wing force and moment linear aerodynamics model; a ring-wing propulsor axial force model employing linear aerodynamic, axial momentum, and propeller actuator disk theories; and a jet engine normal force model as semi-empirically adapted to ring-wing propulsors. The model’s efficacy is assessed using three independent datasets: two experimental and one Reynolds-averaged Navier–Stokes computational fluid dynamics. It is shown that the model’s algebraic solutions provide good engineering estimates for the lift, drag, normal, and axial forces as well as pitching moments at all operating conditions from hover to cruise and angles of attack up to 85°.

Torsional Aeroelasticity of a Flexible VAWT Blade using a Combined Aerodynamic Method by Considering Post-stall and Local Reynolds Regime

The present research investigates the torsional aeroelasticity of the blade of an H-type vertical axis wind turbine subject to stall and post-stall conditions in various Reynolds regimes, which is experienced by the blade in a full revolution. In order to simulate the aerodynamics, a new model based on a combination of the Double Multi Streamtubes (DMST) model and the nonlinear multi-criteria Cl-a equations, which is depended on the local Reynolds number of the flow, has been proposed. The results indicate that using of multi-criteria function dependent on the Reynolds number for the Cl-a curve has improved the prediction of the torsional behavior of the blade in azimuthal rotation of the blade compared to using single-criterion functions and linear aerodynamics. The blade’s aeroelastic torsion has been studied for various TSR values.

Investigation of direct force control for aerocapture at Neptune

In this work, a direct force control numerical predictor-corrector guidance architecture is developed to enable Neptune aerocapture using blunt body aeroshells. A linear aerodynamics model is formulated for a Mars Science Laboratory-derived aeroshell. The application of optimal control theory shows that the ΔV-minimizing angle of attack and side-slip angle control laws are bang-bang. A closed-loop numerical predictor-corrector direct force control guidance algorithm is developed and numerically simulated using the Program to Optimize Simulated Trajectories II. A series of Monte Carlo simulations are conducted to assess the guidance robustness to uncertainties in vehicle aerodynamics, atmospheric density, and entry state. For the reference set of uncertainties, the direct force control vehicle achieves 99.7% successful science orbit insertion within a 330 m/s total ΔV budget for periapsis raise, apoapsis, inclination, and ascending node corrections. Improved atmospheric knowledge and delivery state accuracy are shown to improve the success to 100% and reduce the ΔV to 230 m/s. Direct force control is demonstrated to be an enabling technology for blunt body aerocapture at Neptune while providing comparable performance to existing slender body vehicles studied in literature.

Changing Bridge Aerodynamics under Nonstationary Winds

The nonstationary winds during tropical cyclones and non-synoptic events have been extensively observed and analysed; however, the significance of nonstationarity in the consideration of wind load effects has not been widely investigated yet. In this study, the effects of nonstationarity on changing bridge aerodynamics have been discussed from both linear and nonlinear viewpoints. For linear aerodynamics change, the conventional 1-D indicial response function has been extended to a 2-D case since an additional time scale, resulting from the time-varying transient nonstationarity, is introduced in the wind–structure interactions. For nonlinear aerodynamics change, the conventional hybrid model has been generalized to consider the additional effective angle of attack due to the time-varying mean wind speed. The bridge buffeting response under the tropical-cyclone and downburst winds were comprehensively examined based on 1-D semi-empirical linear model, 2-D semi-empirical linear model, 1-D hybrid nonlinear model, 1-D generalized hybrid nonlinear model and 2-D generalized hybrid nonlinear model, where the ability of these models to capture the changing linear and/or nonlinear bridge aerodynamics under nonstationary winds is highlighted. The results demonstrated the important effects of the transient nature of nonstationary winds on the changing bridge aerodynamics and hence on the structural response.

Comparison of Comprehensive Analyses Predicting Whirl Flutter Stability of the Wing and Rotor Aeroelastic Test System

Whirl flutter stability is a critical limitation for tiltrotor aircraft. This paper investigates whirl flutter predictions for the Wing and Rotor Aeroelastic Test System (WRATS) using comprehensive analysis, focusing on the comparison of the whirl flutter stability predictions between Comprehensive Analytical Model of Rotorcraft Aerodynamics and Dynamics (CAMRAD) II and the Rotorcraft Comprehensive Analysis System (RCAS). The analytical models were created using a modular approach to systematically validate the modeling process of the WRATS tiltrotor. Comparison of nonrotating frequencies for blade, flexbeam, flexbeam and cuff, and blade with flexbeam and cuff shows excellent agreement between CAMRAD II and RCAS. The assembled model is then used to predict the whirl flutter stability boundary for various configurations with varying levels of fidelity. Results show near exact agreement between the two analyses for a rigid rotor and linear aerodynamics, and good to fair agreement when an elastic rotor is used. Predicted wing beam mode frequencies and damping values are also compared against the wind tunnel test data, and the frequencies are shown to be reasonably well-predicted. However, damping values, and thus stability boundaries, are not accurately predicted.

Multiobjective optimization of an aircraft wing design with respect to structural and aeroelastic characteristics using neural network metamodel

The multidisciplinary design optimization (MDO) is an important concept that focuses on bringing together disciplines involved with machine design. It is possible to use MDO in any stage of an aircraft design, that is in the conceptual, preliminary, or detailed design, as long as the numerical models are fitted to each of these stages. This work describes the development of a multidisciplinary design optimization applied to flexible aircraft wings, with respect to structural and aeroelastic characteristics. As tools for the aircraft designer, the numerical models must be fairly accurate and fast. Therefore, metamodels for the critical flutter speed prediction of aircraft wings were considered, thereby reducing significantly the computational cost of the optimization. For this purpose, artificial neural networks (NN) metamodeling is evaluated, based on their inherent properties in dealing with complex mappings. The NN metamodel is prepared using an aeroelastic code based on finite-element model coupled with linear potential aerodynamics. Results of the metamodel performance are presented, from where one can note that the NN is well suited for flutter prediction. Multiobjective optimization (MOO) using the genetic algorithm (GA) based on non-dominance approach is considered. The objectives were the maximization of critical flutter speed and minimization of structural mass. One case study is presented to evaluate the performance of the MOO, revealing that overall optimization process actually achieves the search for the Pareto frontier.

New Look at Nonlinear Aerodynamics in Analysis of Hypersonic Panel Flutter

A simply supported plate fluttering in hypersonic flow is investigated considering both the airflow and structural nonlinearities. Third-order piston theory is used for nonlinear aerodynamic loading, and von Karman plate theory is used for modeling the nonlinear strain-displacement relation. The Galerkin method is applied to project the partial differential governing equations (PDEs) into a set of ordinary differential equations (ODEs) in time, which is then solved by numerical integration method. In observation of limit cycle oscillations (LCO) and evolution of dynamic behaviors, nonlinear aerodynamic loading produces a smaller positive deflection peak and more complex bifurcation diagrams compared with linear aerodynamics. Moreover, a LCO obtained with the linear aerodynamics is mostly a nonsimple harmonic motion but when the aerodynamic nonlinearity is considered more complex motions are obtained, which is important in the evaluation of fatigue life. The parameters of Mach number, dynamic pressure, and in-plane thermal stresses all affect the aerodynamic nonlinearity. For a specific Mach number, there is a critical dynamic pressure beyond which the aerodynamic nonlinearity has to be considered. For a higher temperature, a lower critical dynamic pressure is required. Each nonlinear aerodynamic term in the full third-order piston theory is evaluated, based on which the nonlinear aerodynamic formulation has been simplified.

Transonic limit cycle oscillation behavior of an aeroelastic airfoil with free-play

The limit cycle oscillation (LCO) behaviors of an aeroelastic airfoil with free-play for different Mach numbers are studied. Euler equations are adopted to obtain the unsteady aerodynamic forces. Aerodynamic and structural describing functions are employed to deal with aerodynamic and structural nonlinearities, respectively. Then the flutter speed and flutter frequency are obtained by V-g method. The LCO solutions for the aeroelastic airfoil obtained by using dynamically linear aerodynamics agree well with those obtained directly by using nonlinear aerodynamics. Subsequently, the dynamically linear aerodynamics is assumed, and results show that the LCOs behave variously in different Mach number ranges. A subcritical bifurcation, consisting of both stable and unstable branches, is firstly observed in subsonic and high subsonic regime. Then in a narrow Mach number range, the unstable LCOs with small amplitudes turn to be stable ones dominated by the single degree of freedom flutter. Meanwhile, these LCOs can persist down to very low flutter speeds. When the Mach number is increased further, the stable branch turns back to be unstable. To address the reason of the stability variation for different Mach numbers at small amplitude LCOs, we find that the Mach number freeze phenomenon provides a physics-based explanation and the phase reversal of the aerodynamic forces will trigger the single degree of freedom flutter in the narrow Mach number range between the low and high Mach numbers of the chimney region. The high Mach number can be predicted by the freeze Mach number, and the low one can be estimated by the Mach number at which the aerodynamic center of the airfoil lies near its elastic axis. Influence of angle of attack and viscous effects on the LCO behavior is also discussed.

A Quasi-Steady Lifting Line Theory for Insect-Like Hovering Flight.

A novel lifting line formulation is presented for the quasi-steady aerodynamic evaluation of insect-like wings in hovering flight. The approach allows accurate estimation of aerodynamic forces from geometry and kinematic information alone and provides for the first time quantitative information on the relative contribution of induced and profile drag associated with lift production for insect-like wings in hover. The main adaptation to the existing lifting line theory is the use of an equivalent angle of attack, which enables capture of the steady non-linear aerodynamics at high angles of attack. A simple methodology to include non-ideal induced effects due to wake periodicity and effective actuator disc area within the lifting line theory is included in the model. Low Reynolds number effects as well as the edge velocity correction required to account for different wing planform shapes are incorporated through appropriate modification of the wing section lift curve slope. The model has been successfully validated against measurements from revolving wing experiments and high order computational fluid dynamics simulations. Model predicted mean lift to weight ratio results have an average error of 4% compared to values from computational fluid dynamics for eight different insect cases. Application of an unmodified linear lifting line approach leads on average to a 60% overestimation in the mean lift force required for weight support, with most of the discrepancy due to use of linear aerodynamics. It is shown that on average for the eight insects considered, the induced drag contributes 22% of the total drag based on the mean cycle values and 29% of the total drag based on the mid half-stroke values.

Hopf bifurcation analysis of an aeroelastic model using stochastic normal form

We investigate the effects of parameter uncertainties on the dynamical response of an aeroelastic model representing an oscillating airfoil in pitch and plunge with linear aerodynamics and cubic structural nonlinearities. An approach based on the stochastic normal form is proposed to determine the effects due to the variations in the flow speed and the structural stiffness terms on the stability of the aeroelastic system near the Hopf bifurcation point. This approach allows us to study analytically the bifurcation scenario and to predict the amplitude and frequency of the limit cycle oscillation (LCO). The results show that the amplitude of LCO corresponding to the supercritical Hopf bifurcation increases with the intensity of the noise perturbing the pitch and plunge cubic terms, but there is almost no effect on the LCO frequency. Uncertainties in the flow speed produce a shift in the bifurcation point, and unstable subcritical behavior may occur for values of parameters for which the corresponding deterministic model is stable. The stochastic normal form confirms and extends previously known numerical results regarding the effect of parameter variations, and offers an effective way to perform sensitivity analysis of the system's response.

Helicopter blade flapping with and without small angle assumption in the presence of dynamic stall

The flapping equation for a rotating rigid helicopter blade is typically derived by considering (1) small flap angle, (2) small induced angle of attack and (3) linear aerodynamics. However, the use of nonlinear aerodynamics such as dynamic stall can make the assumptions of small angles suspect as shown in this paper. A general equation describing helicopter blade flap dynamics for large flap angle and large induced inflow angle of attack is derived. A semi-empirical dynamic stall aerodynamics model (ONERA model) is used. Numerical simulations are performed by solving the nonlinear flapping ordinary differential equation for steady state conditions and the validity of the small angle approximations are examined. It is shown that the small flapping assumption, and to a lesser extent, the small induced angle of attack assumption, can lead to inaccurate predictions of the blade flap response in certain flight conditions for some rotors when nonlinear aerodynamics is considered.

Response Surface Technique for Static Aeroelastic Optimization on a High-Aspect-Ratio Wing

I T IS a key subject in aircraft design to have the maximum lift and the minimum drag. Computational fluid dynamics (CFD)-based optimization design has been widely used to optimize rigid configurations in aircraft engineering [1,2]. As is well known, for a highaspect-ratio wing, structural deformation can reduce its cruise aerodynamic performance. Because the aeroelastic characteristic has not been considered in the original optimum design, the real flight performances will deviate from the rigid-design results. Therefore, an optimum design that contains the aeroelastic effect needs to be developed. However, the static aeroelastic optimization is usually coupled with the aerodynamicmodel, structural model, optimization algorithms, and even such related issues as response surface method (RSM), fluid–structure interface, and moving grid. Therefore, it is muchmore difficult and time-consuming than the CFD aerodynamic optimization. High-fidelity CFD tools are now available to aircraft designers and are commonly used in the design of aerodynamic configuration. CFD codes are capable of accurately predicting flowfields about complex aircraft configurations. Compared with linear aerodynamics, CFD tools are characterized by large computational costs due to complex geometrical modeling and grid generation. They require a higher level of proficiency from the users in defining run parameters and interpreting results. Endeavors to exploit CFD solvers for aeroelastic analyses and aircraft structural design are relatively advanced, due to the main hindrance of large computational cost. They are typically characterized by iterations between CFD and a structure solver, which aggravate the problem of computational cost. Bennett and Edwards [3] reviewed the current state of computational aeroelasticity (CA), where CA is defined as the numerical analyses of coupled CFD and structural dynamics. They listed the main efforts needed in the development of CA: that is, the reduction of elapsed run times, improvement of the credibility of CFD tools, and simplification of methods for further applications. For the interface between fluid and structure, Smith et al. [4] provided a comprehensive review of spline methods, their mathematical formulation, and practical applications, which contain the infinite plate spline (IPS) for plate configuration and the beam spline for fuselage configuration. A large amount of literature exists on the subject of grid deformation, which mainly contains the algebraic transfinite interpolation (TFI) [5], the springnetwork analogy by Batina [6] and Farhat et al. [7], and the boundary element method by Chen and Hill [8]. Even fluid–structuremultidisciplinary optimizationmay cut down the design cycles and reduce the reliance on wind-tunnel and flight tests; the drawbacks are the repeated fluid–structure coupling analyses during the optimization process. At present, there are very few studies that consider the adaptation of CFD-based aeroelastic analyses in structural-design optimization. Applications of CFDbased structural and multidisciplinary design optimization were reviewed by Guruswamy and Obayashi [9]. Martins et al. [10–12] presented a CFD-based methodology for aerodynamic-structural optimization. Cross sensitivity is computed by the adjoint method, which was shown to be vastly more computationally efficient for a supersonic business jet optimization involving a larger number of design variables than ever. Reuther et al. [13] presented a structural optimization in which the aerodynamic and structural analyses were performed separately. CFD analyses were used to generate a response surface, and the response surface was incorporated in the structural optimization to improve the roll maneuverability, in which only the generation of a response surface required an amount of CFD-based analyses. In this Note, a quadratic polynomial response surface model is put forward based on the CFD-based static aeroelastic calculations. The structure is represented by a finite element model and holds fixed in the optimization process. The genetic algorithm (GA) optimization method is used for the improvement of static aeroelastic performance through the optimization of the spanwise sectional airfoil shapes. To reduce the amount of CFD-based static aeroelastic analyses for the generation of RSM, only two sectional airfoils are chosen for the optimizations. Either for singleor multi-objectives optimizations, the study shows that the optimization of aeroelastic performances can be greatly improved compared with the original static aeroelastic results.

Modeling Helicopter Rotor Blade Flapping Motion Considering Nonlinear Aerodynamics

The flapping equation for a rotating rigid helicopter blade is typically derived by considering 1) small flap angle, 2) small induced angle of attack and 3) linear aerodynamics. However, the use of nonlinear aerodynamics can make the assumptions of small angles suspect. A general equation describing helicopter blade flap dynamics for large flap angle and large induced inflow angle of attack is derived in this paper with nonlinear aerodynamics . Numerical simulations are performed by solving the nonlinear flapping ordinary differential equation for steady state conditions and the validity of the small angle approximations are examined. It is shown that the small flapping assumption, and to a lesser extent, the small induced angle of attack assumption can lead to inaccurate predictions of the blade flap response in certain flight conditions for some rotors when nonlinear aerodynamics is considered.

Combined Direct/Adjoint Reduced-Order Approximations for Design-Oriented Structural-Acoustics

The accuracy of modal-order reduction methods in structural dynamics and structural acoustics depends on the particular problem analyzed and particular modal bases selected. Modal-order reduction methods can become quite inaccurate when sensitivities of response functions with respect to design variables are required. Past research in linear aerodynamics and structural dynamics had shown how direct and adjoint solutions can be combined to yield reduced-order analysis and sensitivity approximations of the problems at hand with second-order accuracy. Efforts are reported to extend the methodology to structural acoustic problems of fluid-filled cavities enclosed by flexible walls. In this case, stress and pressure response to excitation has to be evaluated over a range of frequencies. Improved and more consistent accuracy is reported over a range of frequencies with the second-order method when accuracy of both approximate direct and approximate adjoint solutions is comparable. The second-order method cannot improve accuracy of first-order methods in general when one of the approximate first-order solutions is highly inaccurate.

Aeroelastic analysis through linear and non-linear methods: a summary of flutter prediction in the PUMA DARP

AbstractThis paper presents a comparison of linear and non-linear methods for the analysis of aeroelastic behaviour and flutter boundary prediction. The methods in question include NASTRAN and ZAERO, based on linear aerodynamics, and the non-linear coupled CFD-CSD methods RANSMB and PMB, developed at the Universities of Bristol and Glasgow respectively. The test cases used for this comparison are the MDO and AGARD 445.6 weakened wing. In general, it was found that the non-linear methods demonstrate excellent agreement with respect to pressure distributions, deflections, dynamic behaviour, and flutter boundary locations for both cases. This is in contrast to previous studies involving similar methods, where notable differences across the MDO span were found, and is taken to imply good performance of the CFD-CSD interpolation schemes employed here. While the linear methods produce similar flutter boundaries to the coupled codes for the aerodynamically simple AGARD 445.6 wing, results for the transonic ‘rooftop’ MDO wing design did not agree as well.

Identification of Computational-Fluid-Dynamics Based Unsteady Aerodynamic Models for Aeroelastic Analysis

Three approaches for reduced-order modeling of computational-fluid-dynamics-(CFD) based unsteady aerodynamics, employing system-identification methods, are presented, and used for generation of three models: A frequency-domain model, a time-domain autoregressive-moving-average model, and a discrete-time state-space model. All models are identified based on the same identification data, which consists of the time histories of the generalized aerodynamic forces developed in response to filtered white-Gaussian-noise modal excitation, computed in a CFD analysis. The models are used for rapid flutter analysis via traditional frequency-domain methods, linear stability analysis, and time simulation. The method is applied for flutter analysis of the AGARD 445.6 wing. The filtered white-Gaussian-noise input is found to be applicable within the framework of CFD, yielding informative identification data sets. The identification process is simple, and the resulting reduced-order models closely reproduce the CFD system response to various excitations. Reduced-order model-based flutter analysis is rapid and yields accurate results compared with wind-tunnel test, CFD, and linear aerodynamics results.