OPTIMUM TRANSONIC WING DESIGN USING CONTROL THEORY

Transcription

1 OPTIMUM TRANSONIC WING DESIGN USING CONTROL THEORY Thomas V. Jones Professor of Engineering, Deartment of Aeronautics and Astronautics Stanford University, Stanford, CA jamesonbaboon.stanford.edu 1. Introduction While aerodynamic rediction methods based CFD are now well established, and quite accurate and robust, the ultimate need in the design rocess is to find the otimum shae which maximies the aerodynamic erformance. One way to aroach this objective is to view it as a control roblem, in which the wing is treated as a device which controls the flow to roduce lift with minimum drag, while meeting other requirements such as low structure weight, sufficient fuel volume, and stability and control constrains. Here we aly the theory of otimal control of systems governed by artial differential equations with boundary control, in this case through changing the shae of the boundary. Using this theory, we can find the Frechet derivative (infinitely dimensional gradient) of the cost function with resect to the shae by solving an adjoint roblem, and then we can make an imrovement by making a modification in a descent direction. For examle, the cost function might be the drag coefficient at a fixed lift, or the lift to drag ratio. During the last decade, this method has been intensively develoed, and has roved to be very effective for imroving wing section shaes for fixed wing lanform [3, 4, 8 11, 13, 14]. In the resent work a continuous adjoint formulation has been used to derive the adjoint system of equations, in which the adjoint equations are derived directly from the governing equations and then discretied. This aroach has the advantage over the discrete adjoint formulation in that the resulting adjoint equations are indeendent of the form of discretied flow equations. The adjoint system of equations has a similar form to the governing equations of the flow, and hence the numerical methods develoed for the flow equations [1, 2, 5] can be reused for the adjoint equations. Moreover, the gradient can be derived directly from the adjoint solution and the surface motion, indeendent of the mesh modification.

2 2 In order to accelerate the convergence of the descent rocess the gradient is then smoothed imlicitly via a second order differential equation. This is equivalent to redefining the gradient in a Sobolve sace. The resulting rocedure is very efficient, often yielding the otimum in 1-2 design cycles. 2. The general formulation of the Adjoint Aroach to Otimal Design The aerodynamic roerties which define the cost function are functions of the flow-field variables,, and the hysical location of the boundary, which may be reresented by the function,, say. Then and a change in results in a change (1) in the cost function. Using control theory, the governing equations of the flow field are introduced as a constraint in such a way that the final exression for the gradient does not require re-evaluation of the flow-field. In order to achieve this, must be eliminated from equation (1). Suose that the governing equation which exresses the deendence of and within the flow field domain can be written as Then is determined from the equation Next, introducing a Lagrange Multilier, we have which can be rearranged as )+*! #, %$ & * Choosing to satisfy the adjoint equation -! ', )( (2) (3) (4)

3 6 Otimum Transonic Wing DesignUsing Control Theory 3 the first term is eliminated and we find that. where./! (5) (6) In this way the gradient with resect to the shae is obtained at the cost of one flow and one adjoint solution. After taking a ste in the negative gradient direction, the gradient is recalculated and the rocess is reeated to follow the ath of steeest descent until a minimum is reached. In order to avoid violating constraints, such as the minimum accetable wing thickness, the gradient can be rojected into an allowable subsace within which the constraints are satisfied. In this way one can devise rocedures which must necessarily converge at least to a local minimum and which can be accelerated by the use of more sohisticated descent methods such as conjugate gradient or quasi-newton algorithms. There is a ossibility of more than one local minimum, but in any case this method will lead to an imrovement over the original design. 3. Adjoint and Gradient formulations for the equations of transonic flow In alying the adjoint method one may aly the above rocedure directly to the artial differential equations to derive a continuous adjoint equation, which must then be discretied to obtain a numerical solution. Alternatively one may derive a discrete adjoint equation directly after first discretiing the flow equations. In this work the first rocedure has been adoted because it allows more flexibility in the formulation of the gradient. The rocedure is illustrated here for the Euler equations. These are reresented in transformed coordinates 21 on a fixed comutational domain. Let %8:9 where Then the transformed equations are 1 ; 3 1 > 4? BADCD E6F 1<; >G 1H;2ID; J D1 D1 Consider the case of an inverse roblem where one wishes to find the shae which brings the ressure as close as ossible to the secified target ressure, K>L. Hence we try to minimie the cost function

4 M 4 M ON PRQTS K K L VUD 3 over the design surface W, which for convenience is assumed to be the surface U X consequently. Now a shae modification induces a change K in the ressure and Q S K K L K 3 Also the change in the solution is given by Here the flux changes are where 1 3 \ 1 N P Q S YZ 3 1<;[ID; \ 1 1<; I ; Consequently one can augment the cost variation by Q^] 1 Q^S`_ 1 Now choose to satisfy the adjoint equation \ 1 1 with the boundary condition U'ab dc ae gf a!h S K K>L U 3 QT] K K>L where ab ae a!h are the comonents of the surface normal. Then the boundary integrals involving K and the field integral involving the gradient is reduced to N PRQTS K K L VUD 3 QiQ^S 3 U'9 U 3 UYU c 3 U c f K are eliminated and where tyically the first term is negligible and can be droed. The evaluation 3 of the field integral requires the evaluation of the metric variations 1<; throughout the domain. However, the true gradient should not deend on the way the mesh is modified. Consider the case of a mesh variation with a fixed boundary. Then ) 9 c Q^] 3 1<; I ; V

5 3 S Otimum Transonic Wing DesignUsing Control Theory 5 but there is a variation in the transformed flux 3 1<;[ID; Here the true solution is unchanged, so the variation is actually the variation kj due to the mesh movement at fixed. Therefore and since it follows that or Q^] Q ] j l ; 3 1<;^ID; 1 3 1<;2ID; 1 i 1<; Q^] ID; 3 Q ] \ 1 ; 1H; > ; ID; ; j A similar relationshi can be verified in the general case with boundary movement. Now, Q ] Q ] \ 1 j V 1 Q] \ 1 j V D1 Q S \ 1 j V (7) Hence on the wall boundary \ U G U 3 U ;^ID; Thus by choosing to satisfy the adjoint equation and the adjoint boundary condition, we have the following exression for the reduced gradient: ) QmQ S 3 QmQ S U ;2ID; \ U j V 9 nc 3 U'9 U 3 UYU dc 3 U cogf K 9 nc (8) It has been confirmed in numerical exeriments that these alternate formulations yield comuted values of the gradient which are in close agreement,

6 Qs r G Qs 6 and that the otimiation rocedure converges to essentially the same result whichever is used. On a structured mesh one can exlicitly define mesh deformations which allow the field terms to be evaluated easily. On an unstructured mesh this is not the case and the reduction to a boundary integral yields large savings in the comutational cost. The discrete adjoint does not rovide for such a transformation. The need for a Sobolev inner roduct in the definition of the gradient Another key issue for successful imlementation of the continuous adjoint method is the choice of an aroriate inner roduct for the definition of the gradient. It turns out that there is an enormous benefit from the use of a modified Sobolev gradient, which enables the generation of a sequence of smooth shaes. This can be illustrated by considering the simlest case of a roblem in calculus of variations. Choose to minimie q V ut with fixed end oints Evw and yxd. Under a variation, r Qs r Qs Thus defining the gradient as and the inner roduct as $ G $ G G { Ÿ} r we find that )~ Then if we set i` { } G t ( ut >G ( t G t

7 9 Q Q Q t Otimum Transonic Wing DesignUsing Control Theory 7 we obtain an imrovement unless of t t t, Now each ste )i` ƒ, the necessary condition for a minimum. Note that B T 9 t t t B % is a function reduces the smoothness of by two classes. Thus the comuted trajectory becomes less and less smooth, leading to instability. In order to revent this we can introduce a modified Sobolev inner roduct [18] { Ÿ} ˆ { }R Š { t } V t where Š is a arameter that controls the weight of the derivatives. If we define a gradient such that Then we have where i ) Œ l by a smoothing equa- and at the end oints. Thus tion. Now the ste gives an imrovement but T B T 9 )i` ˆ Š V t Œ Š > > Š > is obtained from B % ˆ has the same smoothness as, resulting in a stable rocess. In alying control theory for aerodynamic shae otimiation, the use of a Sobolev gradient is equally imortant for the reservation of the smoothness class of the redesigned surface, and it has been emloyed to obtain all the results in the next section.

8 8 4. Redesign of the Boeing 747 wing Over the last decade the adjoint method has been successfully used to refine a variety of designs for flight at both transonic and suersonic cruising seeds. In the case of transonic flight, it is often ossible to roduce a shock free flow which eliminates the shock drag by making very small changes, tyically no larger than the boundary layer dislacement thickness. Consequently viscous effects need to be considered in order to realie the full benefits of the otimiation. Here the otimiation of the wing of the Boeing is resented to illustrate the kind of benefits that can be obtained. In these calculations the flow was modeled by the Reynolds Averaged Navier-Stokes equations. A Baldwin Lomax turbulence model was considered sufficient, since the otimiation is for the cruise condition with attached flow. The calculation were erformed to minimie the drag coefficient at a fixed lift coefficient, subject to the additional constraints that the san loading should not be altered and the thickness should not be reduced. It might be ossible to reduce the induced drag by modifying the san loading to an ellitic distribution, but this would increase the root bending moment, and consequently require an increase in the skin thickness and structure weight. A reduction in wing thickness would not only reduce the fuel volume, but it would also require an increase in skin thickness to suort the bending moment. Thus these constraints assure that there will be no enalty in either structure weight or fuel volume. Figure 1 dislays the result of an otimiation at a Mach number of.86, which is roughly the maximum cruising Mach number attainable by the existing design before the onset of significant drag rise. The lift coefficient of.42 is the contribution of the exosed wing. Allowing for the fuselage to total lift coefficient is about.47. It can be seen that the redesigned wing is essentially shock free, and the drag coefficient is reduced from 1269 (127 counts) to 1136 (114 counts). The total drag coefficient of the aircraft at this lift coefficient is around 27 counts, so this would reresent a drag reduction of the order of 5 ercent. Figure 2 dislays the result of an otimiation at Mach.9. In this case the shock waves are not eliminated, but their strength is significantly weakened, while the drag coefficient is reduced from 1819 (182 counts) to 1293 (129 counts). Thus the redesigned wing has essentially the same drag at Mach.9 as the original wing at Mach.86. The Boeing 747 wing could aarently be modified to allow such an increase in the cruising Mach number because it has a higher swee-back than later designs, and a rather thin wing section with a thickness to chord ratio of 8 ercent. Figures 3 and 4 verify that the san loading and thickness were not changed by the redesign, while figures 5

9 Otimum Transonic Wing DesignUsing Control Theory 9 and 6 indicate the required section changes at 42 ercent and 68 ercent san stations. 5. Conclusions The accumulated exerience of the last decade suggests that most existing aircraft which cruise at transonic seeds are amenable to a drag reduction of the order of 3 to 5 ercent, or an increase in the drag rise Mach number of at least.2. These imrovements can be achieved by very small shae modifications, which are too subtle to allow their determination by trial and error methods. The otential economic benefits are substantial, considering the fuel costs of the entire airline fleet. Moreover, if one were to take full advantage of the increase in the lift to drag ratio during the design rocess, a smaller aircraft could be designed to erform the same task, with consequent further cost reductions. It seems inevitable that some method of this tye will rovide a basis for aerodynamic designs of the future. 6. Acknowledgement This work has benefited greatly from the suort of the Air Force Office of Science Research under grant No. AF F References [1] A. Jameson, W. Schmidt, and E. Turkel, Numerical Solution of the Euler equations by finite volume methods using Runger-Kutta time steing schemes, AIAA Paer , June, [2] A. Jameson and T.J. Baker, Imrovements to the Aircraft Euler Method, AIAA Paer , 25 Ž AIAA Aerosace Sciences Meeting, Reno, January, [3] A. Jameson, Aerodynamic Design via Control Theory, Princeton University Reort MAE 1824, ICASE Reort No , November, 1988, also J. of Scientific Comuting, Vol. 3, , [4] A. Jameson, Comutational Aerodynamics for Aircraft Design, Science, Vol. 245, , [5] T.J. Barth, Aects of unstructured grids and finite volume solvers for the Euler and Navier- Stokes equations, AIAA Paer , 29 Ž AIAA Aerosace Sciences Meeting, Reno, January, [6] J. Elliot and J. Peraire, Aerodynamic design using unstructured meshes, AIAA Paer , 33 y AIAA Aerosace Sciences Meeting, Reno, January, [7] K. Anderson and V. Venkatakrishnan, Aerodynamic Design Otimiation on Unstructured grids using a continuous adjoint formulation, AIAA Paer , 34 Ž AIAA Aerosace Sciences Meeting, Reno, January, [8] A. Jameson, L. Martinelli, and N. Pierce, Otimum Aerodynamic Design Using the Navier- Stokes Equations, Theoret. Comut. Fluid Dynamics, 1, , 1998.

10 1 [9] A. Jameson, A Persective on Comutational Algorithms for Aerodynamic Shae Analysis and Design, Sixth Taiwan National Conference on Comutational Fluid Dynamics, Taitung, Taiwan ROC, August, 1999, Progress in Aerosace Sciences, Elsvier, 21. [1] A. Jameson and L. Martinelli, Aerodynamic Shae Otimiation Techniques Based on Control Theory, CIME (International Mathematical Summer Center), Martina Franca, Italy, [11] J. C. Vassberg and A. Jameson, Comutational Fluid Dynamics for Aerodynamic Design: Its Current and Future Imact, AIAA , 39th AIAA Aerosace Sciences Meeting & Exhibit, Reno, NV, January, 21. [12] S. E. Cliff, S.D. Thomas, T. J. Baker, A. Jameson, and R. M. Hicks, Aerodynamic Shae otimiation using unstructured grid method, AIAA Paer 2-555, 9 Ž AIAA Symosium on Multidiscilinary Analysis and Otimiation, Atlanta, Setember, 22. [13] J. C. Vassberg and A. Jameson, Aerodynamic Shae Otimiation of a Reno Race Plane, International Journal of Vehicle Design, vol.28 no.4, , 22. [14] S. Kim, J.J. Alonso, and A. Jameson, Design Otimiation of High-Lift Configurations Using a Viscous Continuous Adjoint Method, AIAA , 4th AIAA Aerosace Sciences Meeting & Exhibit, Reno, NV, January, 22. [15] O. Pironneau, Otimal Shae Design for Ellitic Systems, Sringer-Verlag, New York, [16] J.L. Lions, Otimal Control of Systems Governed by Partial Differential Equations, Sringer-Verlag, New York, 1971, Translated by S.K. Mitter. [17] A. Jameson, Otimum Aerodynamic Design Using Control Theory, Comutational Fluid Dynamics Review 1995, Wiley, [18] A. Jameson, L. Martinelli, and J. Vassberg, Using CFD for Aerodynamics - A critical Assesment, Proceedings of ICASE 22, Toronto, Canada, Setember 8-13, 22. [19] A. Jameson and S. Kim, Reduction of the Adjoint Gradient Formula in the Continuous Limit, AIAA Paer, 41 E AIAA Aerosace Sciences Meeting, Reno January, 23.

11 Otimum Transonic Wing DesignUsing Control Theory 11 COMPARISON OF CHORDWISE PRESSURE DISTRIBUTIONS B747 WING-BODY REN 1, MACH.86, CL SYMBOL SOURCE SYN17 DESIGN 5 SYN17 DESIGN ALPHA CD % San % San - Solution 1 Uer-Surface Isobars ( Contours at 5 ) % San % San - - COMPPLOT JCV % San Figure 1. MCDONNELL DOUGLAS Redesigned Boeing 747 wing at Mach.86, distributions % San 14:4 Tue 28 May 2 COMPARISON OF CHORDWISE PRESSURE DISTRIBUTIONS B747 WING-BODY REN 1, MACH.9, CL SYMBOL SOURCE SYN17 DESIGN 5 SYN17 DESIGN ALPHA CD % San % San - Solution 1 Uer-Surface Isobars ( Contours at 5 ) % San % San - - COMPPLOT JCV % San Figure 2. MCDONNELL DOUGLAS Redesigned Boeing 747 wing at Mach.9, distributions % San 18:59 Sun 2 Jun 2

12 12 COMPARISON OF SPANLOAD DISTRIBUTIONS B747 WING-BODY REN 1, MACH.9, CL.421 FOILDAGN Sanwise Thickness Distributions C*CL/CREF SYMBOL SOURCE ALPHA SYN17 DESIGN SYN17 DESIGN CD SYMBOL AIRFOIL Wing 3 Wing 2.3 SPANLOAD & SECT CL CL Half-Thickness Maximum Thickness Average Thickness PERCENT SEMISPAN COMPPLOT MCDONNELL DOUGLAS 18:59 Sun JCV Jun Percent Semi-San FOILDAGN MCDONNELL DOUGLAS 19:2 Sun JCV.94 2 Jun 2 Figure 3. San loading, Redesigned Boeing 747 wing at Mach.9 Figure 4. Sanwise thickness distribution, Redesigned Boeing 747 wing at Mach FOILDAGN Airfoil Geometry -- Camber & Thickness Distributions SYMBOL AIRFOIL Wing 13 Wing 2.13 ETA R-LE Tavg Tmax X :2 Sun 2 Jun FOILDAGN Airfoil Geometry -- Camber & Thickness Distributions SYMBOL AIRFOIL Wing 22 Wing 2.22 ETA R-LE Tavg Tmax X :2 Sun 2 Jun Half-Thickness Half-Thickness Camber Camber Airfoil Airfoil Percent Chord Percent Chord MCDONNELL DOUGLAS FOILDAGN JCV.94 MCDONNELL DOUGLAS FOILDAGN JCV.94 Figure 5. Section geometry at.42, redesigned Boeing 747 wing at Mach.9 Figure 6. Section geometry at.68, redesigned Boeing 747 wing at Mach.9