Time domain simulation of airfoil flutter using fluid structure coupling through FEM based CFD solver

The objective of this work is to study the flutter characteristics of an airfoil in a two-dimensional subsonic flow by externally coupling a Reynolds Average Navier Stokes (RANS) based CFD solver with the in-house structural code in time domain. The ANSYS FLOTRAN CFD, a finite element based computational fluid dynamics solver, is adopted here to generate the aerodynamic pressure distributions on an airfoil section in subsonic regime. The airfoil dynamics is accordingly simulated through external coupling of the ANSYS FLOTRAN CFD solver with a 2DOF airfoil structural model through a Newmarks’s implicit time integration scheme. A symmetric NACA 0012 airfoil of unit chord is chosen for the analysis, with suitable spring stiffness and inertia values so that flutter instability occurs in the subsonic regime. Unsteady motion in the fluid-structure system is numerically simulated with small initial conditions. In the present analysis, the airfoil is not allowed to move and the pitch angle of airfoil is assigned to the air flow at the inlet boundary of the domain. Flutter boundary is indicated by the critical free stream flow velocity (and dynamic pressure) beyond which oscillation amplitudes diverge in time

Time domain simulation of airfoil flutter in the subsonic regime using fluid structure coupling through panel method

This report presents a brief theory of aeroelastic flutter of airfoils and the relevant algorithm for the development of a computer code in FORTRAN for dynamic coupling of the airfoil structure, in the time domain, to a two-dimensional subsonic aerodynamic flow, so that the aeroelastic motion can be simulated in the time domain and the flutter boundary be determined for a typical rigid airfoil (of heave and pitch degrees of freedom) in the subsonic flow. The relevant computer code with fluid structure coupling has been developed at the Structural Technologies Division (STTD) for the purpose. The present work starts with a brief introduction to the fundamental concepts of airfoil flutter. The relevant equations of motion of the airfoil in subsonic airflow have been derived from the first principles. First, the classical method, based on the eigenvalue approach is used to solve the equations of motion and to determine the flutter boundary of the airfoil in the subsonic flow regime. A symmetric NACA 0012 airfoil profile of unit chord and width is chosen for analysis, with suitable spring stiffness and inertia values so that flutter instability can occur in the subsonic regime. Results from the panel code for steady flow conditions over the NACA 0012 confirm the validity of the code. For the purpose of time domain flutter simulation the panel code with the Prandtl-Glauert compressibility correction factor is suitably coupled to the airfoil through a Newmark's implicit time integration scheme. Unsteady motion in the fluid-structure system is numerically simulated through the code with small initial conditions. Flutter boundary is indicated by the critical free stream flow speed (and dynamic pressure) beyond which oscillation amplitudes show divergence in time. Flutter frequencies and flutter velocities obtained by the various methods are then compared, and good agreement is observed. Present analysis with a simple airfoil coupled to a 2D subsonic flow (simulated by the panel method) indicate that it is possible, in principle, to simulate flutter even in the transonic and supersonic regimes, using more sophisticated aerodynamic codes (Navier Stokes)

A successive robust flutter prediction technique for aeroelastic systems using µ method

In this work, a successive robust flutter prediction technique is developed by coupling nominal analysis, ground vibration test, wind tunnel test, uncertainty model updation and robust analysis based on the structured singular value method to predict the worst flutter boundary of a swept back wing in transonic flow regime. Here, uncertainties in both structural and unsteady aerodynamics parameters are considered in the generalized coordinates. These uncertainties are introduced in the nominal aeroelastic system in a linear fractional transformation framework. The magnitudes of structural uncertainties are estimated based on the difference in natural frequencies between ground vibration test and nominal analysis. The magnitudes of aerodynamic uncertainties are estimated using a model updation technique based on the structured singular value method considering the difference in damping values between wind tunnel test and nominal analysis. The capability of the present successive robust flutter prediction technique is investigated by estimating the robust flutter boundary of a swept back wing in transonic flow regime. From the results, it is observed that the uncertainty model updation provides a reasonable estimate of aerodynamic uncertainty magnitude. Further, the present flutter prediction approach gives a good estimate of transonic flutter boundary (transonic dip) by successively updating the aerodynamic uncertainty bounds using wind tunnel data for various set of test Mach numbers

Robust Flutter Prediction of an Airfoil Including Uncertainties

This work presents the robust stability analysis of 2DoF airfoil by including various uncertainties. These uncertainties arise due to several factors such as modeling and manufacturing errors as well as disturbances in the flight conditions. The approach adopted to study the uncertain aeroelastic system is based on the structured singular value (µ-method). In this approach, the aeroelastic system is formulated in a robust stability framework by parameterizing around dynamic pressure and introducing uncertainties in the system parameters to account for errors and disturbances. This results in the perturbed aeroelastic system which is then represented using Linear Fractional Transformation (LFT). Then, the nominal and robust stability analysis of the perturbed aeroelastic system is carried out using µ method. In this work, first the validation of µ method is done for 2DoF airfoil with quasi-steady aerodynamics having uncertainties in the structural and aerodynamic properties. Further, the robust flutter boundary of 2DoF airfoil with Theodorsen’s unsteady aerodynamics is studied using µ method in the presence of stiffness, damping, and aerodynamic uncertainties

Multicriteria optimization of variable thickness plates using adaptive weighted sum method

In this paper, a multicriteria design framework for variable thickness isotropic plates using the adaptive weighted sum method is developed. The design objectives are the minimization of weight and static displacement and the design variables are the elemental thicknesses of plates modelled using finite elements. Here, the multicriteria optimization framework is constructed by integrating the finite element method, ana- lytical sensitivity technique along with optimization algorithms. The first-order shear deformation theory is used in the static and dynamic analyses of plates. Both single and multiobjective optimization studies are conducted to study the optimal thickness distributions of variable thickness plates under static and dynamic constraints. To study multicriteria optimization of plates, the weighted sum method is first applied which gives sparsely distributed Pareto optimal solutions. Then, the adaptive weighted sum method is employed where a coarser representation of Pareto optimal solutions is generated using the weighted sum method and less populated regions are identified for further refinement. The suboptimization problems are solved in these regions to determine a new set of Pareto optimal solutions. The Pareto optimal curves obtained using the adaptive weighted sum method are also compared with the conventional weighted sum method under different constraints. The effect of boundary conditions on the Pareto optimal solutions and thickness distributions of plates is also investigated

Frequency Domain Based Robust Flutter Analysis of Swept Back Wing Using μ Method

The present work deals with the robust flutter analysis of a sweptback wing in frequency domain in the presence of various parametric uncertainties. The methodology adopted for the studies is based on the structured singular value ($\mu$) method. $\mu$ method requires valid description of various uncertainties associated with the aeroelastic system and then introducing these uncertainties to the nominal aeroelastic system in the form of a feedback loop. This feedback representation of uncertainties results in the Linear Fractional Transformation (LFT) model of the uncertain aeroelastic system which is then used for robust stability studies using $\mu$ method. This method is implemented in MATLAB and validation studies are carried out for 3DOF airfoil system in the presence of various structural and aerodynamic uncertainties. Further, the present method is extended to study the robust flutter of AGARD 445.6 sweptback wing in the presence of structural and aerodynamic uncertainties at various Mach numbers

Probabilistic Flutter Analysis of a Cantilever Wing

A probabilistic flutter analysis of geometrically coupled cantilever wing is carried out using first-order perturbation approach by considering bending and torsional rigidities as Gaussian random variables. The unsteadiness in the aerodynamic flow is modeled using Theodorsen’s thin airfoil theory. The probabilistic response of the wing is obtained in terms of mean, standard deviation, and coefficient of variation (COV) of real and imaginary parts of the eigenvalues at various free stream velocities. The perturbation results are also compared with Monte Carlo simulations. It is observed that the probabilistic response obtained from the perturbation approach is very accurate up to 7% COV in bending rigidity but in the case of torsional rigidity, it starts losing accuracy after 3%

Stochastic Modeling and Reliability Analysis of Wing Flutter

In this work, a physics-based first-order reliability method (FORM) algorithm is proposed for the flutter reliability analysis of an aircraft wing in the frequency domain. The limit state function, which is an implicit function of random variables, is defined in terms of the damping ratio of the aeroelastic system in a conditional sense on flow velocity. Two aeroelastic cases, namely, an airfoil section model and a cantilever wing model, are considered for carrying out the studies. These aeroelastic models have well separated mean bending and mean torsional modal frequencies. The geometric, structural, and aerodynamic parameters of airfoil and wing systems are modeled as independent Gaussian random variables. The effects of these on the statistics of frequency and damping ratio, and the cumulative distribution functions (CDFs) of flutter velocity are studied. In the case of the wing, the effects of modeling stiffness parameters as Gaussian random fields on the CDFs of flutter velocity are also studied. Here, spectral stochastic finite element method (SFEM) based on Karhunen–Loeve (K–L) expansion is used to discretize the random fields into random variables. From the study of an airfoil system, it is observed that parameters like torsional stiffness, elastic axis location, free stream density, and mass moment of inertia are more sensitive as compared with other parameters. However, in the case of the wing parameters such as torsional stiffness, free stream density, mass moment of inertia, and mass are observed to be more sensitive. The CDFs of flutter velocity obtained using the proposed algorithm are compared with Monte Carlo simulations (MCS) and found to be accurate. A comparative study of aeroelastic reliability for the wing is also carried out by treating stiffness parameters as random variables and random fields. It is observed that the CDFs of flutter velocity in the tail region are conservative when stiffness parameters are treated as random variables

Frequency domain approach for probabilistic flutter analysis using stochastic finite elements

In this work, a stochastic finite element method based on first order perturbation approach is developed for the probabilistic flutter analysis of aircraft wing in frequency domain. Here, both bending and torsional stiffness parameters of the wing are treated as Gaussian random fields and represented by a truncated Karhunen–Loeve expansion. The aerodynamic load on the wing is modeled using Theodorsen’s unsteady aerodynamics based strip theory. In this approach, Theodorsen’s function, which is a complex function of reduced frequency, is also treated as a random field. The applicability of the present method is demonstrated by studying the probabilistic flutter of cantilever wing with stiffness uncertainties. The present method is also validated by comparing results with Monte Carlo simulation (MCS). From the analysis, it is observed that torsional stiffness uncertainty has significant effect on the damping ratio and frequency of the flutter mode as compared to bending stiffness uncertainty. The probability density functions of damping ratio and frequency using perturbation technique and MCS are also discussed at various free stream velocities due to stiffness uncertainties. Furthermore, the flutter probability of the cantilever wing is studied by defining implicit limit state function in conditional sense on flow velocity for the flutter mode. Both perturbation and MCS are considered to study the flutter probability of the wing. From the cumulative distribution functions of flutter velocity, it is noticed that the presence of uncertainty in torsional rigidity lowers the predicted flutter velocity in comparison to uncertainty in bending rigidity