International Journal of Aerospace Engineering

Volume 2017 (2017), Article ID 2571253, 17 pages

https://doi.org/10.1155/2017/2571253

## Bifurcation Analysis with Aerodynamic-Structure Uncertainties by the Nonintrusive PCE Method

School of Aeronautic Science and Engineering, Beihang University, Beijing 100191, China

Correspondence should be addressed to Yuting Dai

Received 18 October 2016; Revised 14 December 2016; Accepted 19 December 2016; Published 16 February 2017

Academic Editor: Enrico Cestino

Copyright © 2017 Linpeng Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

An aeroelastic model for airfoil with a third-order stiffness in both pitch and plunge degree of freedom (DOF) and the modified Leishman–Beddoes (LB) model were built and validated. The nonintrusive polynomial chaos expansion (PCE) based on tensor product is applied to quantify the uncertainty of aerodynamic and structure parameters on the aerodynamic force and aeroelastic behavior. The uncertain limit cycle oscillation (LCO) and bifurcation are simulated in the time domain with the stochastic PCE method. Bifurcation diagrams with uncertainties were quantified. The Monte Carlo simulation (MCS) is also applied for comparison. From the current work, it can be concluded that the nonintrusive polynomial chaos expansion can give an acceptable accuracy and have a much higher calculation efficiency than MCS. For aerodynamic model, uncertainties of aerodynamic parameters affect the aerodynamic force significantly at the stage from separation to stall at upstroke and at the stage from stall to reattach at return. For aeroelastic model, both uncertainties of aerodynamic parameters and structure parameters impact bifurcation position. Structure uncertainty of parameters is more sensitive for bifurcation. When the nonlinear stall flutter and bifurcation are concerned, more attention should be paid to the separation process of aerodynamics and parameters about pitch DOF in structure.

#### 1. Introduction

Hopf bifurcations can occur in aeroelastic models that are nonlinear just in the structural stiffness operator (e.g., panel LCO), but, nonlinear aerodynamics is, in most cases, the critical component of computational aeroelasticity that must be enhanced to enable dependable predictions of aeroelastic response and variability. Dynamic stall which is due to nonlinear phenomenon for aerodynamics is one of the most important reasons that leads the change in bifurcation. Calculating nonlinear dynamics correctly is the key to the investigation of LCO and bifurcation. For nonlinear dynamics, CFD method is a direct way to calculate which have a good accuracy. However, when the problem is complex and many conditions need to be calculated, using CFD method may cost much time [1]. There are other methods to calculate nonlinear aerodynamics such as Leishman–Beddoes (LB) [2]. This method can improve the calculation efficiency greatly. It is a semiempirical model and many of the model parameters are selected subjectively; there may be some variations of these parameters. These parameter uncertainties in the aerodynamic model may lead to conservative or optimistic prediction of stall flutter [3, 4].

Since uncertainty always exists in both aerodynamics part and structure part when a theoretical method is applied to the aeroelastic stability analysis, aeroelastic uncertainty quantification (AUQ) is somewhat unavoidable in the theoretical analysis [5]. Dai and Yang Reviewed the methods applied to aeroelastic uncertainty quantification [6], which is focused on investigating the effect of parameter uncertainty on the aeroelastic stability property [7, 8]. The methods can be divided into robust one [9] and probabilistic one [10]. When the uncertainty is a probabilistic variable, the stochastic aeroelastic analysis should be conducted to obtain both the stability boundary and its distribution [8, 11, 12]. The nonintrusive polynomial chaos expansions are to determine the evolution of uncertainty in a dynamic system, when a probabilistic uncertainty is embedded in the aeroelastic system [13, 14]. Beran et al. introduced this method to the nonlinear aeroelastic analysis, which has been proven to be a success [15]. Some related work has been carried out, including uncertainty quantification in aerodynamics and stochastic basis construction, Badcock et al. [16–18]. In AUQ, the parameter uncertainty exists both in the structural dynamics model and the aerodynamics [19, 20]. Most of these works are focused on the structural uncertainty in structure [21, 22]. Little work is concerned with aerodynamic uncertainty and structure uncertainty interactions in aeroelasticity [23, 24].

This current work is focused on the aerodynamic uncertainty quantification of aeroelastic stability containing limit cycle oscillation analysis and flutter analysis, subject to stochastic LB aerodynamic parameters at low Mach number. The modified unsteady aerodynamic model at low Mach number will be established, and uncertainty of parameters for aerodynamic model will be taken into consideration. Then the stochastic polynomial chaos expansion method for stall flutter will be investigated.

#### 2. Uncertain LCO Analysis by the Nonintrusive PCE Method

##### 2.1. Deterministic Aerodynamic Model

The Leishman–Beddoes (LB) dynamic stall model is a popular model that has been used to investigate dynamic stall of aerodynamics. The standard LB model is applicable above the Mach number of 0.3. But, it is not suitable in the incompressible flow. However, a huge effort has been undertaken for the aeroelastic characterization of next-generation aircraft; particularly, the low Mach number is important for high altitude very-long endurance solar power HALE UAVs [25, 26]. What is more, in most of stall flutter application cases, such as wind tunnel state, the Mach number is lower than 0.3. Hence, in this paper, a modified dynamic stall model for low Mach numbers model was built based on the work of Leishman and Beddoes and Sheng et al. [2, 27].

Under the conditions of low Mach number, an additional time lag is required for the disturbed flow, during which the disturbed flow can develop into vortex that is strong enough to cause the dynamic stall. Taking this effect into consideration, the additional lagged value , which is the substitute value of , is introduced.where is an intermediate variable, is the time lag constant, and is the distance traveled by airfoil in semichords.

###### 2.1.1. Modification of Normal Force and Pitching Moment Coefficients at Upstroke

Vortex forms and detaches near the airfoil leading edge, and the flow of the separated shear layer still attaches to the upper surface at low Mach number. Hence, it results in additional overshoot in normal force at low Mach numbers. The overshoot for normal force coefficient is given as follows:where is a coefficient related to airfoils; is separation location in terms of chord; is delayed separation location of in the original LB model; represents the shape function of normal force due to vortex, which is given bywhere is nondimensional time; is time constant of vortex traveling over chord; is vortex passage time constant.

The additional pitching moment coefficient is also the effect of vortex convection over the airfoil. It is as follows:where is the coefficient dependent on airfoil. is nondimensional time during vortex passage.

###### 2.1.2. Modification of Normal Force and Pitching Moment Coefficients during Return

During return, there is also an overshoot for normal force coefficient. A value of was given. It is assumed that it is the start of the convection process of overshoot. A value of was given to define the end of the convection process of overshoot. Their relationship is as follows:where is delay constant for reattachment process and is reduced pitch rate.

The overshoot for normal force coefficient at return process is given as follows:where represents the shape function of normal force due to vortex at return process, which is given bywhere is a nondimensional time variable at return process.

##### 2.2. Deterministic LCO Analysis in the Time Domain

The aeroelastic system considered here is a wing section with pitch and plunge motions, which is shown in Figure 1. Considering the unsteady aerodynamic force, the motion equation of the wing section is written aswhere is the airfoil pitch angle. is the vertical displacement. is the mass per unit length. and are spring stiffness in bending and torsion. and are damping in bending and torsion. is static mass moment. is the polar moment of inertia about 1/4 chord. and are external aerodynamic force and moment, respectively, which will be calculated by the above modified LB model at low Mach number. There expressions can be given as follows:where and are lift coefficient and moment coefficient, respectively, is the chord length, is the span length, is the distance between 1/4 chord and elastic axis, is air density, and is the free stream velocity.