Multidisciplinary multifidelity optimisation of a flexible wing aerofoil with reference to a small UAV

Abstract

The preliminary Multidisciplinary Design and Optimisation of a flexible wing aerofoil apropos a small Unmanned Air Vehicle is performed using a multifidelity model-based strategy. Both the passively adaptive structure and the shape of the flexible wing aerofoil are optimised for best aerodynamic performance under aero-structural constraints, within a coupled aeroelastic formulation. A typical flight mission for surveillance purposes is considered and includes the potential occurrence of wind gusts. A metamodel for the high-fidelity objective function and each of the constraints is built, based on a tuned low-fidelity one, in order to improve the efficiency of the optimisation process. Both metamodels are based on solutions of the aeroelastic equations for a flexible aerofoil but employ different levels of complexity and computational cost for modelling aerodynamics and structural dynamics within a modal approach. The high-fidelity model employs nonlinear Computational Fluid Dynamics coupled with a full set of structural modes, whereas the low-fidelity one employs linear thin aerofoil theory coupled with a reduced set of structural modes. The low-fidelity responses are then corrected according to few high-fidelity responses, as prescribed by an appropriate Design of Experiment, by means of a suitable tuning technique. A standard Genetic Algorithm is hence utilised to find the global optimum, showing that a flexible aerofoil is characterised by higher aerodynamic efficiency than its rigid counterpart. Wing weight reduction is also accomplished when a Multiobjective Genetic Algorithm is adopted.

Appendices

Appendix A: Aerodynamic models

1.1 Low-fidelity model: Peters’ thin aerofoil theory

For a uniform free-stream flow $\overset {\rightharpoonup }{V}_\infty =V_\infty \widehat {x}$, the aerodynamic load acting on the aerofoil is given, in terms of linear thin aerofoil theory, by (Peters 2007)

$$ \Delta p = \rho_\infty \left({V_\infty \gamma_a +\int\limits_{-b}^x {\dot{\gamma}_a d\xi}}\right), $$

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where $\rho _{\infty }$ is the constant density of the flow; γ _a =γ _a (x, t) is a vortex distribution, which is a solution of the Prandtl-Glauert equation (Glauert 1947) and gives the total bound circulation $\Gamma =\int \limits _{-b}^b {\gamma _a dx}$ around the aerofoil. The wake is assumed flat and shed from the aerofoil’s trailing edge, in accordance with the Kutta condition (Katz and Plotkin 2001), it does not bear any aerodynamic load (Peters 2007)

$$ \dot{\Gamma} =-V_\infty \gamma_{\mathit{w}} -\int\limits_b^x {\dot{\gamma}_{\mathit{w}} d\xi}, $$

(24)

where γ _w =γ _w (x, t) is a vortex distribution along the wake, which gives the total trailed circulation and influences the bound one around the aerofoil.

By introducing the Glauert change of variable x = b cos( φ) along the aerofoil chord, with 0 ≤φ≤π, the pressure distribution is rewritten as (Peters 2007)

$$ \Delta p=2\rho_\infty \left[ {p_0 \left( {\frac{1-\cos (\varphi)}{\sin (\varphi)}} \right)+\sum\limits_{i=1}^n {p_i \sin ({i\varphi})}}\right], $$

(25)

where all coefficients p _i =p _i (t) depend on the aerofoil’s motion and all normal goniometrical functions cos( i φ) are equivalent to Chebychev polynomials. With the flow velocity and the aerofoil camber line projected onto the Chebychev polynomials and expressed as Glauert expansions $w=\sum \limits _{i=0}^n {{\mathit {w}}_i \cos ({i\varphi })}$ and $\eta _c =\sum \limits _{i=0}^n {\eta _i \cos (i\varphi )}$, respectively, the generalised forces acting on the aerofoil are (Peters 2007)

$$\begin{array}{@{}rcl@{}} F_0^a &=&\pi \rho_\infty b^2\left( {\dot{\mathit{w}}_0 -\frac{\dot{\mathit{w}}_2}{2}} \right)+2\pi \rho_\infty bV_\infty \left( {\mathit{w}_0 -\lambda_0 +\frac{\mathit{w}_1 }{2}} \right), \\ F_1^a &=&\frac{\pi }{8}\rho_\infty b^2\left( {\dot{\mathit{w}}_1 -\dot{\mathit{w}}_3 } \right)-\pi \rho_\infty bV_\infty \left( {\mathit{w}_0 -\lambda_0 -\frac{\mathit{w}_2 }{2}} \right),\\ F_2^a &=&-\frac{\pi }{2}\rho_\infty b^2\left( {\dot{\mathit{w}}_0 -\frac{2}{3}\dot{\mathit{w}}_2 +\frac{\dot{\mathit{w}}_4 }{6}} \right)-\frac{\pi }{2}\rho_\infty bV_\infty \left( {\mathit{w}_1 -\mathit{w}_3 } \right), \\ F_i^a &=&\frac{\pi }{4}\rho_\infty b^2\left[ {\frac{1}{i+1}\left( {\dot{\mathit{w}}_i -\dot{\mathit{w}}_{i+2} } \right)-\frac{1}{i-1}\left( {\dot{\mathit{w}}_{i-2} -\dot{\mathit{w}}_i } \right)} \right]\\ &&-\frac{\pi }{2}\rho_\infty bV_\infty \left( {\mathit{w}_{i-1} -\mathit{w}_{i+1} } \right), \end{array} $$

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where the influence of the unsteady wake on the aerofoil load is given by the inflow velocity (Berci 2011)

$$ \lambda_0 =\left[ {1-\mathrm{C} (k)} \right]\left( {\mathit{w}_0 +\frac{\mathit{w}_1 }{2}} \right)-\int\limits_0^t {\frac{dV_G }{d\tau}\Psi (t-\tau)d\tau}, $$

(27)

and, due to the non-penetration boundary condition, the coefficients of the flow velocity due to the aerofoil motion are given by (Peters 2007)

$$ {\begin{array}{*{20}c} {\mathit{w}_0 {}={}{}-{}\dot{\eta}_0 {}-{}V_\infty \sum\limits_{i=1,3}^n {\frac{i\eta_i }{b}} ,} & & {\mathit{w}_i {}={}{}-{}\dot{\eta}_i {}-{}2V_\infty \sum\limits_{j=i+1,i+3}^n {\frac{j\eta_j }{b}}.} \end{array} } $$

(28)

Finally, by employing suitable approximations (Jones 1940; Zaide and Raveh 2006; Venkatesan and Friedmann 1986; Leishman 2002) of both the Theodorsen function C(k) in the reduced frequency domain and the Kussner Ψ(t) functions in the time domain, the aerofoil aerodynamic load can entirely be written in the time domain in a state-space form (Bielawa 2006), where two appropriate added aerodynamic states are defined and introduced via Laplace transformation (Poirel 2001). Note that a lift-curve slope C _L/α =2π has been assumed implicitly, which is strictly valid for a two-dimensional flat plate only; nevertheless, this restriction is lifted by employing a three-dimensional lift-curve slope which accounts for the effect of the wing tip vortices and the Prandtl-Glauert correction for subsonic compressible flow (Anderson 2007).

1.2 High-fidelity model: Xfoil nonlinear CFD

Xfoil is an MIT code (Drela and Youngren 2001) for the viscous or inviscid analysis of subsonic isolated aerofoils, allowing for the Karman-Tsien compressibility correction, forced or free transition of the boundary layer (including transitional separation bubbles) and limited trailing edge separation; lift and drag can be predicted just beyond stall.

The inviscid formulation is a 2D panel method based on linear-vorticity stream function, where a finite trailing edge thickness is modelled with a source panel; the flow equations are then closed with an explicit Kutta condition. Due to the theoretical foundation of the Karman-Tsien correction, accuracy rapidly degrades as soon as the transonic regime is entered and shocks cannot be predicted.

In the viscous formulation, the boundary layer and wake are described with a two-equation lagged dissipation integral formulation and an envelope e ^N turbulence transition criterion (Luchini and Quadrio 2004); the viscous solution is strongly coupled with the incompressible potential flow via the surface transpiration model, which permits proper calculation of limited separation regions. The aerofoil drag is determined from the wake momentum thickness far downstream, a special treatment being used for blunt trailing edges in order to calculate the pressure drag properly. The total velocity at each point on the aerofoil surface and wake, with contributions from the freestream, the aerofoil surface vorticity and the equivalent viscous source distribution (accounting for the boundary layer thickness), is obtained from the panel solution, with the Karman-Tsien correction added; this is incorporated into the viscous equations, yielding a nonlinear elliptic system which is readily solved by a Newton-Rapson method. If an angle of attack is specified, the wake trajectory is taken from an inviscid solution at that given angle of attack: this is not strictly correct, since in general viscous effects decrease lift and change the trajectory; however, the effect of this approximation on the overall accuracy is small and felt near or past stall only. Turbulence transition can be either left free or forced,however the e ^N method is always active and free transition can occur upstream of the forced transition. Note that such method is only appropriate for predicting transition when the growth of 2D Tollmien-Schlichting waves (Luchini and Quadrio 2004) via linear instability is the dominant transition mechanism; however, this is typically the case in aerofoil aerodynamics, in the presence of adverse pressure gradients.

Having calculated the local pressure coefficient, the lift and moment coefficients are calculated by direct pressure integration along the aerofoil surface, while the drag coefficient is obtained by applying the Squire-Young formula at the last point in the wake and, if the aerodynamic flow is separated at the trailing edge, much of the drag contribution (i.e., energy dissipation due to viscous shear and turbulent mixing) occurs in the aerofoil wake. Note that the Squire-Young formula extrapolates the momentum thickness to downstream infinity and assumes that the wake behaves in an asymptotic manner downstream of the point of application: this assumption is strongly violated in the near-wake behind an aerofoil with trailing edge separation but always reasonable some distance far behind the aerofoil. In addition to calculating the total viscous drag from the wake momentum thickness, the friction and pressure drag are also determined.

Appendix B: Typical section static aeroelastic response

A very thin symmetric aerofoil with chord c = 1 m is considered and a wood-like material with uniform density ρ=1,000 kg/m ³ and Young’s modulus E=10⁹ Pa is assumed for its idealised structure, with h _c =0.03 m. The elastic axis of the wing is placed at 37.5 % of its chord (i.e., half-way between the CG and AC), with vertical spring constant k _η =10⁵ N/m and concen- trated mass M _η =4 kg; several values of the torsional constant k _𝜗 are then considered as fractions of the vertical one (Kim and Lee 1996). A free-stream flow of $V_{\infty } = 100$ m/s with an angle of attack α _r =4° is chosen at sea level. Fifteen Chebychev modes were employed in the aeroelastic analysis and sufficient to achieve modal convergence for both static and dynamic responses.

Figure 18 compares low- and high-fidelity calculations of the aerofoil’s static aeroelastic response, using Xfoil in inviscid (left) and viscous (right) mode, for validation purposes: the high-fidelity results are almost identical to the low-fidelity ones in the former case, whereas the effect of the boundary layer turbulent transition and the viscous wake separation can be clearly identified in the latter case, where a lower local aerodynamic load corresponds to a lower structural deformation.

Fig. 18

Multiobjective optimisation results: Pareto front (left) and non-inferior optimum points (right)

Note that when a nonlinear CFD code (Chung 2002) is employed for calculating the static aerodynamic load on a flexible aerofoil and modally coupled to a linear structural code for calculating its deformation, the resulting FSI equations are nonlinear and have to be solved numerically. In particular, when a staggered approach is used, this translates into solving a first-order discrete aeroelastic equation of the form

$$ \mathbf{K}_{\mathrm{s}} \mathbf{x}^i=\textit{\textbf{F}}_{\textit{g}} +\textit{\textbf{F}}_{\textit{a}}^i $$

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up to potential convergence x ⁱ⁺¹≈x ⁱ of the aerofoil displacement, where i is the iteration number, K _s the structural stiffness matrix, F _g the gravity load vector and F _a the aerodynamic load vector, respectively. The above equation is quasi-linear and valid for small displacements only; however, calculating the aerodynamic load via CFD is always a nonlinear problem in its own right, which must be solved numerically within each aeroelastic iteration. Moreover, the aerofoil structure cannot be too thin and soft, thus causing the structural stiffness matrix to be close to singular and the aeroelastic problem ill-conditioned (Quarteroni et al. 2000). Such a staggered approach was implemented in Matlab within the modal framework of the aeroelastic equation for a flexible Typical Section: the static inviscid calculations required less than five iterations to converge, whereas the viscous ones required up to thirty iterations; of course, convergence can be guaranteed under certain conditions only (Bindolino et al. 2000).