- Email: [email protected]
Adjoint-based optimization of flapping plates hinged with a trailing-edge flap
Accepted Manuscript Adjoint-based optimization of flapping plates hinged with a trailing-edge flap Min Xu, Mingjun Wei, Chengyu Li, Haibo Dong PII: DOI: Reference:
S2095-0349(14)00006-3 http://dx.doi.org/10.1016/j.taml.2014.12.005 TAML 5
To appear in:
Theoretical and Applied Mechanics Letters
Received date: 24 November 2014 Accepted date: 1 December 2014 Please cite this article as: M. Xu, M. Wei, C. Li, H. Dong, Adjoint-based optimization of flapping plates hinged with a trailing-edge flap, Theoretical and Applied Mechanics Letters (2015), http://dx.doi.org/10.1016/j.taml.2014.12.005 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Manuscript submitted to the Theoretical & Applied Mechanics Letters
Adjoint-Based Optimization of Flapping Plates Hinged with a Trailing-Edge Flap Min Xu,1 Mingjun Wei,1, ∗ Chengyu Li,2 and Haibo Dong2 1 New
Mexico State University, Las Cruces, NM 88003, USA
2 University
of Virginia, Charlottesville, VA 22904, USA (Dated: December 23, 2014)
Abstract 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 adjoint-based approach. The optimized configuration suggests that the trailingedge flap can substantially enhance the overall lift. Further analysis indicates that the lift enhancement by the trailing-edge 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. Keywords: flapping wing, trailing-edge flap, adjoint-based optimization.
∗
Email: [email protected]
2 I.
INTRODUCTION
The unsteady aerodynamic phenomena that allows insects to operate efficiently at lowReynolds-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 [2, 3]. 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 [1, 5] in flapping flights. The performance of a rigid wing can be significantly improved by adding chord-wise flexibility onto the wing. Vanella et al. used a 2-D two-link model to model the chord-wise flexibility in a flapping plate [6]. It shows that if appropriately chosen, the chord-wise flexibility can result in up to 28% enhancement in terms of lift-to-drag ratio and 39% increase for lift-to-power ratio comparing to a rigid one. In addition, Wan et al. investigated the effect of chord-wise flexibility over a range of hovering kinematics parameters using a hinged-plate model [7]. Their results indicated that higher lift-to-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. performed numerical simulations of deformable flapping plate by attaching a trailingedge flap (TEF) and studied its effects on aerodynamic performance and flow modulation [8]. 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. Adjoint-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 adjoint-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 introduced in § II. The optimization results are discussed in § III. Finally, the conclusion
3 is in § IV. II. A.
NUMERICAL METHOD Configuration and Kinematics
Fig. 1: Schematic of a flapping plate hinged with a trailing-edge flap. Shown in figure 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 trailing-edge plate are respectively 0.75 and 0.25 based on the total length. The flapping motion is prescribed by xL (t) =
AL (1 + cos 2πf t), yL (t) = 0, 2
θL (t) = βL sin(2πf t),
(1)
θT (t) = βT sin(−2πf t + φT ), where AL = 3 and βL = 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 non-dimensionaized by the total chord-length c and the reference velocity Uref = πf AL . The control parameters are the flapping amplitude βT and the phase delay φT of the trailingedge flap. The two parameters are optimized to achieve the maximum lift. The Reynolds number, Re = Uref c/ν, is 100 for all the cases. Here, we only consider the periodic state, which can be achieved after 8 flapping cycles.
4 B.
Flow Equations
The surrounding flow satisfies the incompressible Navier-Stokes equations, ∂uj = 0, ∂xj ∂ui ∂uj ui ∂p ∂ 2 ui + + − ν 2 = 0, ∂t ∂xj ∂xi ∂xj
(2)
with convective boundary conditions at the far field, and the Dirichlet condition along the solid wing surface S: ui = Vi on S,
(3)
where Vi 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 [12, 16] 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 adjoint field. The second-order central difference scheme is used for spatial discretization and the third-order Runge-Kutta/CrankNicolson scheme is used for time advancement [13–15]. The continuity equation for incompressible flow is enforced by projection method [15]. The computational domain has non-dimensional size 30 × 30. A stretched 290 × 320 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 = 0.02. There are 202 marker points for the Lagrangian description of the moving hinged plates with zero thickness.
C.
Adjoint Equations
Among many aerodynamic performance metrics, for demonstration, lift coefficient Cl is picked in this work as the objective function, J = Cl , to achieve the optimal lift performance. The control ϕ = (βT , φT ) 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 [17, 18] has great advantages in efficiency and simplicity in the derivation of an adjoint equation in continuous form [13, 14]. Following
5 the same derivation, we get the adjoint equation in its continuous form as: ∂u∗j = 0, ∂xj ( ∗ ) ∂u∗j ∂u∗i ∂ui ∂p∗ ∂ 2 u∗ + uj + + + ν 2i = 0, ∂t ∂xj ∂xi ∂xi ∂xj
(4)
with boundary conditions, ∂p∗ /∂n = 0, and u∗i = −δi2 on S,
(5)
where δij is Kronecker’s delta. The adjoint 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 adjoint equation is discretized and solved similarly by numerical algorithms for the forward solver in § II B. The adjoint field eventually reaches periodic state by no more than 3 flapping cycles.
D.
Gradient Calculation and Optimization Algorithms
Solving forward the flow equation and backward the adjoint equation, we get the flow solution ui and the adjoint solution u∗i , which together provide the gradient of the cost function J with respect to the control ϕ: ( ) ] ∫ T0 +T ∫ [ ) ∂ui ( ∗ ∂J 2 ∂σ2j ∗ ˙ = nk − Vi,l − Zk,l σij nj + uj uj ni dsdt, (6) Zk,l 2 ∂ϕl ρUref cT T0 ∂xj ∂xk S where ∂Vi ∂Si V˙ i,l = , Zi,l = , ∂ϕl ∂ϕl ) ( ∂ui ∂uj + , σij = −pδij + ν ∂xj ∂xi ( ∗ ) ∂u∗j ∂ui ∗ ∗ σij = p δij + ν + , ∂xj ∂xi
(7)
and Si is the location of the solid point, ni 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 adjoint approach has been validated in our earlier work, and there are also details on derivation [13, 14].
6
1
0.95
J
0.9
0.85
0.8
0.75
0.7
0
1
2
3
4
5
6
Niteration
Fig. 2: The optimization process showing the change of the cost function (i.e. lift coefficient) by the number of iterations.
Table 1: The lift coefficients for the rigid flapping plate and hinged flapping plates with initial and optimized controls. Case
ϕ
Cl
Rigid
(0◦ , 0◦ )
0.739
Initial
(20◦ , −40◦ )
0.793
Optimized
(57.0◦ , 53.4◦ )
0.941
III.
SIMULATION AND OPTIMIZATION RESULTS AND DISCUSSION
The optimization is started with an arbitrarily chosen initial control, ϕ0 = (20◦ , −40◦ ). It is shown in figure 2 that, after 6 iterations, the lift coefficient is improved by 18.7%, from 0.793 to 0.941. 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 adjoint-based optimization also matches very well with the direct parametric study. However, it is worth noting that the adjoint-based optimization (with 6 iterations) has much lower computational cost than the direct approach (with 72 cases). More important, the computational cost from the adjoint-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
7 cost from the direct approach through the increase of cases to cover a much larger parametric space [13].
4
Cl
3
2
1
0
-1
8
8.5
9
9.5
10
t/T
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.
10 8 6 4
vxl , ωT
2 0 -2 -4 -6 -8 -10
8
8.5
9
9.5
10
t/T
Fig. 4: The time variation of the translational velocity vxL (solid line) of the leading-edge and the angular velocity ωT = dθT /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. 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 figure 3 that, during one single flapping circle (8T ≤ t < 9T ), the major lift enhancement happens at t = 8.2T and t = 8.8T . At these two moments, as
8 shown in figure 4, the instantaneous translational velocity at the leading edge, vxL = dxL /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 [20, 21] and help to interpret the complex results. It is worth noting that the coefficients of quasi-steady 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 = ρΓvxL ,
(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 vxL 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 figure 5. At t = 8.2T , when vxL < 0, the clockwise flapping motion of the flap brings in additional negative circulation, therefore, it increases the overall lift; at t = 8.8T , when vxL > 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.
IV.
CONCLUSION
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
9
(a) t = 8.2T
(b) t = 8.8T
(c) t = 8.2T
(d) t = 8.8T
(e) t = 8.2T
(f) t = 8.8T
Fig. 5: 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.
newly-developed adjoint-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 adjoint-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 adjoint equations.
10 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 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. This work was supported by AFOSR under the contract FA9550-12-1-0071
[1] Dickinson, M. H., Lehmann, F. O., and Sane, S. P., Wing rotation and the aerodynamic basis of insect flight. Science 284, 1954–1960, (1999). [2] Valasek, J., Morphing aerospace vehicles and structures. John Wiley & Sons, Chichester, (2012). [3] Wang, H., Zeng, L., Liu, H., and Yin, C., Measuring wing kinematics, flight trajectory and body attitude during forward flight and turning maneuvers in dragonflies. J. Exp. Biol. 206, 745–757, (2003). [4] Shyy, W., Lian, Y., Tang, J., Viieru, D., and Liu, H., Aerodynamics of low Reynolds number flyers. Cambridge University Press New York, (2008). [5] Sun, M., and Tang, J., Unsteady aerodynamic force generation by a model fruit fly wing in flapping motion. J. Exp. Biol. 205, 55–70, (2002). [6] Vanella, M., Fitzgerald, T., Preidikman, S., Balaras, E., and Balachandran, B., Influence of flexibility on the aerodynamic performance of a hovering wing. J. Exp. Biol. 212, 95–105, (2009). [7] Wan, H., Dong, H. and Huang, G. P., Hovering Hinge-Connected Flapping Plate with Passive Deflection. AIAA J. 50, 2020–2027, (2012). [8] Li, C., Dong, H., and Ren, Y., A Numerical Study of Flapping Plates Hinged with a Trailing-
11 Edge Flap. AIAA Paper 2014-2049, (2014). [9] Jameson, A., Aerodynamic shape optimization using the adjoint method. Lectures at the Von Karman Institute, Brussels, Febuary 6 (2003). [10] Bewley, T. R., Moin, P. and Temam, R., DNS-based predictive control of turbulence: an optimal benchmark for feedback algorithms. J. Fluid Mech 447, 179-225, (2001). [11] Wei, M. and Freund, J. B., A noise-controlled free shear flow. J. Fluid Mech. 546, 123–152, (2006). [12] Yang, T., Wei, M., and Zhao, H., Numerical study of flexible flapping wing propulsion, AIAA J., 48, No. 12, 2909–2915, (2010). [13] Xu, M. and Wei, M., A continuous adjoint-based approach for the optimization of wing flapping. AIAA Paper 2014-2048, (2014). [14] Xu, M. and Wei, M., Using adjoint-based approach to study flapping wings. AIAA Paper 2013-0839, (2013). [15] Zhao, H., Freund, J. B., and Moser, R. D., A fixed-mesh method for incompressible flow– structure systems with finite solid deformations. J. Comput. Phys. 227,3114–3140, (2008). [16] Mittal, R., Dong, H., Bozkurttas, M., Najjar, F., Vargas, A., and Von Loebbecke, A., A versatile sharp interface immersed boundary method for incompressible flows with complex boundaries. J. Comput. Phys. 227, 4825–4852, (2008). [17] Moubachir, M., Moving Shape Analysis and Control: Applications to Fluid Structure Interactions. Abingdon: CRC Press, (2006). [18] Protas, B. and Liao, W., Adjoint-based optimization of PDEs in moving domains. J. Comput. Phys. 227, 2707-2723, (2008). [19] Press, W. H., Flannery, B. P., Teukolsky, S. A., and Vetterling, W. T., Numerical Recipes. Cambridge university press, (1986). [20] Andersen, A., Pesavento, U., and Wang, Z., Unsteady aerodynamics of fluttering and tumbling plates. J. Fluid Mech. 541, 65–90, (2005). [21] Berman, G. J. and Wang, Z. J., Energy-minimizing kinematics in hovering insect flight. J. Fluid Mech. 582, 153–168, (2007).
S2095-0349(14)00006-3 http://dx.doi.org/10.1016/j.taml.2014.12.005 TAML 5
To appear in:
Theoretical and Applied Mechanics Letters
Received date: 24 November 2014 Accepted date: 1 December 2014 Please cite this article as: M. Xu, M. Wei, C. Li, H. Dong, Adjoint-based optimization of flapping plates hinged with a trailing-edge flap, Theoretical and Applied Mechanics Letters (2015), http://dx.doi.org/10.1016/j.taml.2014.12.005 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Manuscript submitted to the Theoretical & Applied Mechanics Letters
Adjoint-Based Optimization of Flapping Plates Hinged with a Trailing-Edge Flap Min Xu,1 Mingjun Wei,1, ∗ Chengyu Li,2 and Haibo Dong2 1 New
Mexico State University, Las Cruces, NM 88003, USA
2 University
of Virginia, Charlottesville, VA 22904, USA (Dated: December 23, 2014)
Abstract 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 adjoint-based approach. The optimized configuration suggests that the trailingedge flap can substantially enhance the overall lift. Further analysis indicates that the lift enhancement by the trailing-edge 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. Keywords: flapping wing, trailing-edge flap, adjoint-based optimization.
∗
Email: [email protected]
2 I.
INTRODUCTION
The unsteady aerodynamic phenomena that allows insects to operate efficiently at lowReynolds-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 [2, 3]. 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 [1, 5] in flapping flights. The performance of a rigid wing can be significantly improved by adding chord-wise flexibility onto the wing. Vanella et al. used a 2-D two-link model to model the chord-wise flexibility in a flapping plate [6]. It shows that if appropriately chosen, the chord-wise flexibility can result in up to 28% enhancement in terms of lift-to-drag ratio and 39% increase for lift-to-power ratio comparing to a rigid one. In addition, Wan et al. investigated the effect of chord-wise flexibility over a range of hovering kinematics parameters using a hinged-plate model [7]. Their results indicated that higher lift-to-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. performed numerical simulations of deformable flapping plate by attaching a trailingedge flap (TEF) and studied its effects on aerodynamic performance and flow modulation [8]. 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. Adjoint-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 adjoint-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 introduced in § II. The optimization results are discussed in § III. Finally, the conclusion
3 is in § IV. II. A.
NUMERICAL METHOD Configuration and Kinematics
Fig. 1: Schematic of a flapping plate hinged with a trailing-edge flap. Shown in figure 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 trailing-edge plate are respectively 0.75 and 0.25 based on the total length. The flapping motion is prescribed by xL (t) =
AL (1 + cos 2πf t), yL (t) = 0, 2
θL (t) = βL sin(2πf t),
(1)
θT (t) = βT sin(−2πf t + φT ), where AL = 3 and βL = 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 non-dimensionaized by the total chord-length c and the reference velocity Uref = πf AL . The control parameters are the flapping amplitude βT and the phase delay φT of the trailingedge flap. The two parameters are optimized to achieve the maximum lift. The Reynolds number, Re = Uref c/ν, is 100 for all the cases. Here, we only consider the periodic state, which can be achieved after 8 flapping cycles.
4 B.
Flow Equations
The surrounding flow satisfies the incompressible Navier-Stokes equations, ∂uj = 0, ∂xj ∂ui ∂uj ui ∂p ∂ 2 ui + + − ν 2 = 0, ∂t ∂xj ∂xi ∂xj
(2)
with convective boundary conditions at the far field, and the Dirichlet condition along the solid wing surface S: ui = Vi on S,
(3)
where Vi 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 [12, 16] 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 adjoint field. The second-order central difference scheme is used for spatial discretization and the third-order Runge-Kutta/CrankNicolson scheme is used for time advancement [13–15]. The continuity equation for incompressible flow is enforced by projection method [15]. The computational domain has non-dimensional size 30 × 30. A stretched 290 × 320 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 = 0.02. There are 202 marker points for the Lagrangian description of the moving hinged plates with zero thickness.
C.
Adjoint Equations
Among many aerodynamic performance metrics, for demonstration, lift coefficient Cl is picked in this work as the objective function, J = Cl , to achieve the optimal lift performance. The control ϕ = (βT , φT ) 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 [17, 18] has great advantages in efficiency and simplicity in the derivation of an adjoint equation in continuous form [13, 14]. Following
5 the same derivation, we get the adjoint equation in its continuous form as: ∂u∗j = 0, ∂xj ( ∗ ) ∂u∗j ∂u∗i ∂ui ∂p∗ ∂ 2 u∗ + uj + + + ν 2i = 0, ∂t ∂xj ∂xi ∂xi ∂xj
(4)
with boundary conditions, ∂p∗ /∂n = 0, and u∗i = −δi2 on S,
(5)
where δij is Kronecker’s delta. The adjoint 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 adjoint equation is discretized and solved similarly by numerical algorithms for the forward solver in § II B. The adjoint field eventually reaches periodic state by no more than 3 flapping cycles.
D.
Gradient Calculation and Optimization Algorithms
Solving forward the flow equation and backward the adjoint equation, we get the flow solution ui and the adjoint solution u∗i , which together provide the gradient of the cost function J with respect to the control ϕ: ( ) ] ∫ T0 +T ∫ [ ) ∂ui ( ∗ ∂J 2 ∂σ2j ∗ ˙ = nk − Vi,l − Zk,l σij nj + uj uj ni dsdt, (6) Zk,l 2 ∂ϕl ρUref cT T0 ∂xj ∂xk S where ∂Vi ∂Si V˙ i,l = , Zi,l = , ∂ϕl ∂ϕl ) ( ∂ui ∂uj + , σij = −pδij + ν ∂xj ∂xi ( ∗ ) ∂u∗j ∂ui ∗ ∗ σij = p δij + ν + , ∂xj ∂xi
(7)
and Si is the location of the solid point, ni 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 adjoint approach has been validated in our earlier work, and there are also details on derivation [13, 14].
6
1
0.95
J
0.9
0.85
0.8
0.75
0.7
0
1
2
3
4
5
6
Niteration
Fig. 2: The optimization process showing the change of the cost function (i.e. lift coefficient) by the number of iterations.
Table 1: The lift coefficients for the rigid flapping plate and hinged flapping plates with initial and optimized controls. Case
ϕ
Cl
Rigid
(0◦ , 0◦ )
0.739
Initial
(20◦ , −40◦ )
0.793
Optimized
(57.0◦ , 53.4◦ )
0.941
III.
SIMULATION AND OPTIMIZATION RESULTS AND DISCUSSION
The optimization is started with an arbitrarily chosen initial control, ϕ0 = (20◦ , −40◦ ). It is shown in figure 2 that, after 6 iterations, the lift coefficient is improved by 18.7%, from 0.793 to 0.941. 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 adjoint-based optimization also matches very well with the direct parametric study. However, it is worth noting that the adjoint-based optimization (with 6 iterations) has much lower computational cost than the direct approach (with 72 cases). More important, the computational cost from the adjoint-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
7 cost from the direct approach through the increase of cases to cover a much larger parametric space [13].
4
Cl
3
2
1
0
-1
8
8.5
9
9.5
10
t/T
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.
10 8 6 4
vxl , ωT
2 0 -2 -4 -6 -8 -10
8
8.5
9
9.5
10
t/T
Fig. 4: The time variation of the translational velocity vxL (solid line) of the leading-edge and the angular velocity ωT = dθT /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. 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 figure 3 that, during one single flapping circle (8T ≤ t < 9T ), the major lift enhancement happens at t = 8.2T and t = 8.8T . At these two moments, as
8 shown in figure 4, the instantaneous translational velocity at the leading edge, vxL = dxL /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 [20, 21] and help to interpret the complex results. It is worth noting that the coefficients of quasi-steady 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 = ρΓvxL ,
(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 vxL 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 figure 5. At t = 8.2T , when vxL < 0, the clockwise flapping motion of the flap brings in additional negative circulation, therefore, it increases the overall lift; at t = 8.8T , when vxL > 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.
IV.
CONCLUSION
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
9
(a) t = 8.2T
(b) t = 8.8T
(c) t = 8.2T
(d) t = 8.8T
(e) t = 8.2T
(f) t = 8.8T
Fig. 5: 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.
newly-developed adjoint-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 adjoint-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 adjoint equations.
10 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 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. This work was supported by AFOSR under the contract FA9550-12-1-0071
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