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1 (This is a sample cover image for this issue. The actual cover is not yet available at this time.) This is an open access article which appeared in a ournal published by Elsevier. This article is free for everyone to access download and read. Any restrictions on use including any restrictions on further reproduction and distribution selling or licensing copies or posting to personal institutional or third party websites are defined by the user license specified on the article. For more information regarding Elsevier s open access licenses please visit:

2 Theoretical and Applied Mechanics Letters 5 (2015) 1 4 Contents lists available at ScienceDirect Theoretical and Applied Mechanics Letters ournal homepage: Letter Adoint-based optimization of flapping plates hinged with a trailing-edge flap Min Xu a Mingun Wei a Chengyu Li b Haibo Dong b a New Mexico State University Las Cruces NM USA b University of Virginia Charlottesville VA USA a r t i c l e i n f o a b s t r a c t Article history: Received 24 November 2014 Accepted 1 December 2014 Available online 3 February 2015 *This article belongs to the Fluid Mechanics Keywords: Flapping wing Trailing-edge flap Adoint-based optimization It is important to understand the impact of wing-morphing on aerodynamic performance in the study of flapping-wing flight of birds and insects. We use a flapping plate hinged with a trailing-edge flap as a simplified model for flexible/morphing wings in hovering. The trailing-edge flapping motion is optimized by an adoint-based approach. The optimized configuration suggests that the trailing-edge flap can substantially enhance the overall lift. Further analysis indicates that the lift enhancement by the trailingedge flapping is from the change of circulation in two ways: the local circulation change by the rotational motion of the flap and the modification of vortex shedding process by the relative location between the trailing-edge flap and leading-edge main plate The Authors. Published by Elsevier Ltd on behalf of The Chinese Society of Theoretical and Applied Mechanics. This is an open access article under the CC BY-NC-ND license ( creativecommons.org/licenses/by-nc-nd/4.0/). The unsteady aerodynamic phenomena that allows insects to operate efficiently at low-reynolds-number is produced by flapping-wing mechanism. The inherently unsteady nature of flapping-wing kinematics is responsible for the primary force production [1] and also differentiates flapping-wing motion from conventional fixed and rotary wing configurations. In recent years more attention has been paid to the aerodynamics of deformable wings [23]. Results have revealed that the dynamically changed wing surface either actively or passively deformed would potentially provide new aerodynamic mechanisms [4] of force productions over completely rigid wings [15] in flapping flights. The performance of a rigid wing can be significantly improved by adding chord-wise flexibility onto the wing. Vanella et al. [6] used a 2-D two-link model to model the chord-wise flexibility in a flapping plate. It shows that if appropriately chosen the chord-wise flexibility can result in up to 28% enhancement in terms of lift-todrag ratio and 39% increase for lift-to-power ratio comparing to a rigid one. In addition Wan et al. [7] investigated the effect of chordwise flexibility over a range of hovering kinematics parameters using a hinged-plate model. Their results indicated that higher liftto-drag ratio can be obtained by hinging the plate at three-quarter chord from the leading-edge comparing to a fully rigid plate in the same flapping kinematics. Li et al. [8] performed numerical simulations of deformable flapping plate by attaching a trailing-edge Corresponding author. address: mwei@nmsu.edu (M. Wei). flap (TEF) and studied its effects on aerodynamic performance and flow modulation. Results have shown that the optimal lift production can be achieved by tailoring the trailing-edge deflection angle and phase shift. The enhancement of lift production is up to 26% comparing to a completely rigid flapping plate. However if the parameter space of the problem is further increased it is impracticable to find the optimal setting via direct parametric case study. Adoint-based optimization on the other hand offers the same computational efficiency for increased number of control parameters [9 11]. In this work we have developed an adoint-based approach to quickly find the optimal solution of chord-wise wing morphing. The further analysis of the optimal solution allows us to study the effect of wing flexibility through its comparison to rigid and other configuration not optimized for aerodynamic performance. The rest of the paper is arranged in the following manner. The governing equations and numerical simulation details are first introduced. Then the optimization results are discussed. The conclusion is given at the end. Shown in Fig. 1 the configuration of the flapping plates is chosen to be the same as the one used by Li et al. [8] which allows a convenient comparison of performance and accuracy from different approaches. The leading-edge (i.e. main) plate and the trailingedge plate are respectively 0.75 and 0.25 based on the total length. The flapping motion is prescribed by x L (t) = A L 2 (1 + cos 2πft) y L(t) θ L (t) = β L sin(2πft) θ T (t) = β T sin( 2πft + ϕ T ) (1) / 2015 The Authors. Published by Elsevier Ltd on behalf of The Chinese Society of Theoretical and Applied Mechanics. This is an open access article under the CC BY-NC-ND license (

3 2 M. Xu et al. / Theoretical and Applied Mechanics Letters 5 (2015) 1 4 where A L = 3 and β L = 45 are respectively the amplitudes of the leading-edge plate s horizontal and rotational motion and f = 1/(3π) is the flapping frequency. All the variables are nondimensionalised by the total chord-length c and the reference velocity U ref = πfa L. The control parameters are the flapping amplitude β T and the phase delay ϕ T of the trailing-edge flap. The two parameters are optimized to achieve the maximum lift. The Reynolds number Re = U ref c/ν is 100 for all the cases. Here we only consider the periodic state which can be achieved after 8 flapping cycles. The surrounding flow satisfies the incompressible Navier Stokes equations u u i t + u u i + p ν 2 u i x i x 2 with convective boundary conditions at the far field and the Dirichlet condition along the solid wing surface S u i = V i on S (3) where V i is the local velocity of solid. Staggered Cartesian mesh with local refinement through stretching is used to provide both efficiency and numerical stability. Immersed boundary method [1213] is implemented for the description of moving solid boundaries (i.e. hinged plates) in both the forward simulation of fluid flow and the backward simulation of the adoint field. The second-order central difference scheme is used for spatial discretization and the third-order Runge Kutta/ Crank Nicolson scheme is used for time advancement [14 16]. The continuity equation for incompressible flow is enforced by proection method [16]. The computational domain has non-dimensional size A stretched Cartesian mesh is used for an overall Eulerian description of the combined fluid and solid domain. The grid is clustered near solid region with a minimum size of x = y = There are 202 marker points for the Lagrangian description of the moving hinged plates with zero thickness. Among many aerodynamic performance metrics for demonstration lift coefficient C l is picked in this work as the obective function J = C l to achieve the optimal lift performance. The control φ = (β T ϕ T ) has two parameters which are optimized to maximize the cost function. For problems involving moving solid boundaries (or morphing domain) it has been demonstrated that non-cylindrical calculus [1718] has great advantages in efficiency and simplicity in the derivation of an adoint equation in continuous form [1415]. Following the same derivation we get the adoint equation in its continuous form as u u i t + u u i + u + p x i x i with boundary conditions p + ν 2 ui x 2 n and u i = δ i2 on S (5) where δ i is Kronecker s delta. The adoint equation then can be solved backward in time to provide the gradient/sensitivity for the optimization of cost function. Because of its similarity to the forward flow equation the adoint equation is discretized and solved similarly by numerical algorithms for the forward solver as disscussed earlier. The adoint field eventually reaches periodic state by no more than 3 flapping cycles. (2) (4) Fig. 1. Schematic of a flapping plate hinged with a trailing-edge flap. Solving forward the flow equation and backward the adoint equation we get the flow solution (u i p) and the adoint solution (u i p ) which together provide the gradient of the cost function J with respect to the control φ J φ l = where 2 ρu 2 ref ct T0 +T T 0 S V il Z kl u i x k V il = V i Z il = S i φ l φ l ui σ i = pδ i + ν + u x i u σ i = p i δ i + ν + u x i Z kl σ 2 n k σ i n + u u n i dsdt (6) and S i is the location of the solid point n i is the norm direction from solid to fluid and T is the time for one flapping cycle. The control is updated along the gradient to eventually converge to an optimal solution. The accuracy of the gradient computed by adoint approach has been validated in our earlier work and there are also details on derivation [1415]. The optimization is started with an arbitrarily chosen initial control φ 0 = (20 40 ). It is shown in Fig. 2 that after 6 iterations the lift coefficient is improved by 18.7% from to Three cases are compared in Table 1: (a) the rigid plate without trailing-edge flap (b) the initial control and (c) the optimal control which are consistent with the choices of the direct parametric study by Li et al. [8]. The outcome of the adoint-based optimization also matches very well with the direct parametric study. However it is worth noting that the adoint-based optimization (with 6 iterations) has much lower computational cost than the direct approach (with 72 cases). More important the computational cost from the adoint-based approach by its nature should remain at about the same level as the control parameters increase from 2 to a much larger number (e.g. more than 10); on the other hand such increase of control parameters will lead to much larger (or impossible!) computational cost from the direct approach through the increase of cases to cover a much larger parametric space [14]. The optimal control provides much higher lift coefficient than the rigid case with 27% enhancement though the initial control also provides lift enhancement by 7.3% from the rigid case. It is shown in Fig. 3 that during one single flapping circle (8T t < 9T ) the maor lift enhancement happens at t = 8.2T and t = 8.8T. (7)

4 M. Xu et al. / Theoretical and Applied Mechanics Letters 5 (2015) Table 1 The lift coefficients for the rigid flapping plate and hinged flapping plates with initial and optimized controls. Case φ C l Rigid (0 0 ) Initial (20 40 ) Optimized ( ) Fig. 4. The time variation of the translational velocity v xl (solid line) of the leadingedge and the angular velocity ω T = dθ T /dt (dashed line) of the trailing-edge flapping with the optimized control. Two moments t = 8.2T and t = 8.8T are marked by dotted lines for better comparison. Fig. 2. The optimization process showing the change of the cost function (i.e. lift coefficient) by the number of iterations. Fig. 3. Comparison of the lift coefficients for the case of a rigid flapping plate (thick solid line) the case of hinged plates with the initial control (thin solid line) and the case of hinged plates with the optimized control (dashed line). Two moments t = 8.2T and t = 8.8T are marked by dotted lines for better comparison. At these two moments as shown in Fig. 4 the instantaneous translational velocity at the leading edge v xl = dx L /dt reaches its maximum amplitude (positively or negatively). In current study the plates do not interact with their own wake during flapping due to the strong downwash velocity induced by high lift. Therefore despite the unsteadiness quasi-steady model can still be a good approximation [1920] and help to interpret the complex results. It is worth noting that the coefficients of quasisteady model are computed from numerical simulation data. The numerical simulation and optimization still rely on direct numerical simulation approach mentioned above. According to the quasi-steady model the circulation component of the total lift is C l = ργ v xl (8) where Γ is the circulation along the plates and is defined to be positive for anti-clockwise vortices. The formulation (8) implies that the lift is more sensitive to the change of circulation at the moment when the translational velocity v xl has large amplitude. For our case it means the two moments t = 8.2T and t = 8.8T. The trailing-edge flap changes the circulation (and eventually the lift) through two ways: (1) the rotational motion of the flap itself changes directly the total circulation; (2) the angle between the main plate and the trailing-edge plate changes the shedding process of the trailing-edge vortex (TEV) which in turn has a negative impact on the total circulation through the conservation of angular momentum. Both effects can be observed in Fig. 5. At t = 8.2T when v xl < 0 the clockwise flapping motion of the flap brings in additional negative circulation therefore it increases the overall lift; at t = 8.8T when v xl > 0 the additional positive circulation by the anti-clockwise motion of the flap results also the overall lift enhancement. The optimal flow at both time moments also show stronger TEV shedding in comparison to the rigid case and the initial case. The stronger vortex shedding leads to larger lift at these moments. We numerically studied the impact of wing flexibility/morphing towards the aerodynamic performance (i.e. lift coefficient in the current work) of flapping wings in hovering motion. The study is based on a simplified model a flapping plate hinged with a trailing-edge flap. A newly-developed adoint-based approach is used to reach the same optimal setting but much lower computational cost compared to a previous study using direct parametric case study. It is noted that the current extension of using adoint-based approach on moving-boundary problems has only become possible through the introduction of non-cylindrical calculus to handle the boundary derivatives for the derivation of adoint equations. Through the comparison between the rigid (i.e. no hinged flap) case and the optimal trailing-edge flapping case we see a 27% lift enhancement. Further analysis shows that the lift enhancement is the result of circulation changes at the moment when maximum translational speed of the leading edge happens. The implementation of a trailing-edge flap at its optimal rotation and phase delay results in the increase or decrease of the circulation at those critical time moments. The change of circulation the maximum

5 4 M. Xu et al. / Theoretical and Applied Mechanics Letters 5 (2015) 1 4 a b c d e f Fig. 5. (Color Online) The flow field marked by vorticity contours at two time moments t = 8.2T and t = 8.8T : (a) and (b) are the case of a rigid flapping plate; (c) and (d) are the case of hinged flapping plates with the initial control; (e) and (f) are the case of hinged flapping plates with the optimal control. The red and blue colors respectively denote anti-clockwise and clockwise vortices. translational speed of the leading edge and the right sign of both values together result in the lift enhancement in the current study. In other words the impact of the trailing-edge flap to the lift enhancement is through the change of circulation by both the rotational motion and the relative angle of the two plates. Acknowledgment This work was supported by AFOSR (FA ). References [1] M.H. Dickinson F.O. Lehmann S.P. Sane Wing rotation and the aerodynamic basis of insect flight Science 284 (1999) [2] J. Valasek Morphing Aerospace Vehicles and Structures John Wiley & Sons Chichester [3] H. Wang L. Zeng H. Liu C. Yin Measuring wing kinematics flight traectory and body attitude during forward flight and turning maneuvers in dragonflies J. Exp. Biol. 206 (2003) [4] W. Shyy Y. Lian J. Tang D. Viieru H. Liu Aerodynamics of Low Reynolds Number Flyers Cambridge University Press New York [5] M. Sun J. Tang Unsteady aerodynamic force generation by a model fruit fly wing in flapping motion J. Exp. Biol. 205 (2002) [6] M. Vanella T. Fitzgerald S. Preidikman E. Balaras B. Balachandran Influence of flexibility on the aerodynamic performance of a hovering wing J. Exp. Biol. 212 (2009) [7] H. Wan H. Dong G.P. Huang Hovering hinge-connected flapping plate with passive deflection AIAA J. 50 (2012) [8] C. Li H. Dong Y. Ren A numerical study of flapping plates hinged with a trailing-edge flap AIAA Paper (2014). [9] A. Jameson Aerodynamic shape optimization using the adoint method. Lectures at the Von Karman Institute Brussels Febuary 6 (2003). [10] T.R. Bewley P. Moin R. Temam DNS-based predictive control of turbulence: an optimal benchmark for feedback algorithms J. Fluid Mech. 447 (2001) [11] M. Wei J.B. Freund A noise-controlled free shear flow J. Fluid Mech. 546 (2006) [12] R. Mittal H. Dong M. Bozkurttas F. Naar A. Vargas A. VonLoebbecke A versatile sharp interface immersed boundary method for incompressible flows with complex boundaries J. Comput. Phys. 227 (2008) [13] T. Yang M. Wei H. Zhao Numerical study of flexible flapping wing propulsion AIAA J. 48 (2010) [14] M. Xu M. Wei A continuous adoint-based approach for the optimization of wing flapping AIAA Paper (2014). [15] M. Xu M. Wei Using adoint-based approach to study flapping wings AIAA Paper (2013). [16] H. Zhao J.B. Freund R.D. Moser A fixed-mesh method for incompressible flow structure systems with finite solid deformations J. Comput. Phys. 227 (2008) [17] M. Moubachir Moving Shape Analysis and Control: Applications to Fluid Structure Interactions CRC Press Abingdon [18] B. Protas W. Liao Adoint-based optimization of PDEs in moving domains J. Comput. Phys. 227 (2008) [19] A. Andersen U. Pesavento Z. Wang Unsteady aerodynamics of fluttering and tumbling plates J. Fluid Mech. 541 (2005) [20] G.J. Berman Z.J. Wang Energy-minimizing kinematics in hovering insect flight J. Fluid Mech. 582 (2007)