ANALYSIS AND DESIGN OF WINGS AND WING/WINGLET COMBINATIONS AT LOW SPEEDS

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1 Computational Fluid Dynamics JOURNAL 13(3):76 October 4 (pp.1 99) Special Issue: Physical and Mathematical Foundations of CFD Dedicated to Dr. Isao Imai, Emeritus Professor of University of Tokyo on the Occasion of His 9th Birthday ANALYSIS AND DESIGN OF WINGS AND WING/WINGLET COMBINATIONS AT LOW SPEEDS Jean-Jacques CHATTOT Abstract Numerical treatment in Prandtl lifting-line theory of the nonlinearity associated with a -D lift curve, when the local incidence is larger than the incidence of maximum lift, is proposed. It is shown that the use of an artificial viscosity term makes the solution unique and allows the iterative method to converge to a physically meaningful solution, that is in agreement with the exact solution for the test case. The design and analysis of winglets is presented. The winglets considered are small fences placed upward at the tip of the wing to improve the wing efficiency by decreasing the induced drag. The effect of yaw on a wing equipped with such optimal winglets indicates that they provide weathercock stability. Key Words: 1 INTRODUCTION Prandtl lifting-line theory has been used to predict lift, induced drag and moment of wings of moderate to large aspect ratios very efficiently since its publication [1]. Improvements have followed with the use of -D viscous lift curve to replace the linear relation for the local lift coefficient C l =π(α eff α ) used by Prandtl, where α eff is the effective incidence that includes the induced incidence and α is the angle of zero lift. Anderson et al. [] have shown results obtained with their nonlinear lifting-line method. In the case of a wing of aspect ratio 5.3 with drooped leading-edge, they experience spurious oscillations in the distribution of circulation along the span, at high-α, due to the non-uniqueness problem. Prandtl lifting-line theory can also be used to analyze wing/winglets combinations and, in inverse mode, help find the optimum distribution of circulation to design wings and winglets for practical low speed aircraft applications. Previous papers have been devoted to winglet design since the early work of Munk [3] who investigated the optimum Received on July 14, 4. University of California at Davis, Department of Mechanical and Aeronautical Engineering; One Shields Ave, Davis, California, , USA jjchattot@ucdavis.edu distribution of lifting elements to achieve minimum induced drag, for single dihedral lines of arbitrary shape as well as multiple lifting lines. Whitcomb [4] and Ishimitsu [5] used a combination of theoretical and experimental work to achieve and validate a practical design. A vortex lattice method was used to search for a low induced drag design in both cases, but it is not clear if the procedure provided the optimum solution. Furthermore, the problem complexity was increased by the high subsonic speed range of interest in these studies and the presence of compressibility effects. This paper is in two parts. In the first part, the lifting-line analysis code is described and the details on the handling of the non-uniqueness problem associated with stall are given. The second part is devoted to the aerodynamic optimization of a wing with upward winglets placed at the tip of the wing, using the lifting-line theory [6]. Wing and winglet configurations are analyzed at the design condition and with a 1 deg yaw angle. Contrary to some belief, the optimal winglets do not produce a thrust component at design condition. The % winglet optimum design is also compared with a non-optimal configuration, which underlines some remarkable differences.

2 Jean-Jacques Chattot HIGH-α TREATMENT Prandtl lifting-line method can be associated with a -D viscous polar, experimentally or numerically generated, to account for viscous effects within a strip theory approach. The vortex induced flow in a spanwise plane is matched with the -D polar to give the local lift, viscous and induced drag contributions that are integrated along the span to yield the global coefficients. The -D lift curve, C l (α) is of particular importance. In the useful range of incidence of the wing profile, the lift is an increasing function of incidence until the incidence of maximum lift, (α) Clmax is reached, beyond which the lift will decrease in general, although not always monotonically. If α (α) Clmax everywhere, a solution can be found from the Prandtl integro-differential equation. When α>(α) Clmax at some stations along the wing, the nonlinearity becomes a dominant feature of the equation and the solution needs to be regularized to converge to the correct values. This can be done by introducing an artificial viscosity term in the governing equation. An exact solution has been constructed to test the scheme. The -D lift curve is assumed to be C l (α) = π sin(α), α π. It has a maximum, corresponding to stall, at α = π 4 with (C l) max = π. We consider a wing with an elliptic planform, zero camber and zero twist. The wing aspect ratio is AR = b S = 1.7, where b is the span and S the wing area. The reference dimensional parameters are the semi span b and the incoming flow velocity, V. A Cartesian coordinate system is used with the x-axis aligned with the root chord and oriented in the flow direction and the y-axis in the plane of the horizontal projection of the wing. The z-axis is vertical upward and completes the direct coordinate system. Prandtl integro-differential equation in dimensionless form is given by Γ(y) = 1 c(y)c l[α α (y) + arctan(w ind (y))], where Γ is the circulation, c the local chord, α is the geometric incidence and w ind the downwash at the lifting line w ind (y) = 1 1 Γ (η)dη 4π 1 y η, the integral being taken in principal part. The effective incidence is α eff = α + arctan(w ind (y)). The lift and induced drag coefficients are given respectively by C L = AR C Di = AR Γ(y)dy, Γ(y)w(y)dy. The governing equation is discretized and the values Γ j,c j,c lj,α j and w indj are placed at the nodes y j where y j = cos( j 1 π), j=1,..., jx. jx 1 The integration points are placed between the nodes according to η k = cos( k 1/ π), k=1,..., jx 1, jx 1 ensuring a finite principal value for the downwash as w indj = 1 jx 1 Γ k+1 Γ k,j=,..., jx 1. 4π y j η k k=1 Newton s method is used to linearize the governing equation. One gets and Γ j + Γ j = 1 c j(c lj + dc lj dα α effj), α effj = w indj 1+windj w indj = 1 4π ( 1 1 ) Γ j = a j Γ j. y j η j 1 y j η j Finally, the equation assumes the following iterative form: (1 1 c dc lj j dα a j 1+w indj, ) Γ j ω = 1 c jc lj Γ n j. ω is the relaxation factor. In the equation, the coefficients are evaluated with the latest available values of Γ k as the span is swept in increasing order of the j-index and the circulation is updated as Γ n+1 j = Γ n j + Γ j. Note that a j < and the matrix can be shown to be diagonally dominant when dc lj dα. Over-relaxation can be used with ω = 1.8. However, when dc lj dα < and some part of the wing is stalled,

3 Analysis and Design of Wings and Wing/Winglet Combinations at Low Speeds 3 diagonal dominance is lost and the positivity of the coefficient of Γ j needs to be secured. An artificial viscosity term is added to the right-hand side and a contribution appears in the left-hand side: 3 Cl(α), D CL(α), 3 D Lifting Line (1 1 c dc lj j dα a j 1+w indj +µ) Γ j ω Cl & CL = 1 c jc lj Γ n j + µ(γn j+1 Γn j +Γn+1 j 1 ), where µ is the artificial viscosity coefficient. The iterative method is made stable with some underrelaxation (ω < 1) by choosing µ to verify µ sup( 1 4 c dc lj j dα a j 1+w indj ;) α (deg) Fig.1: Theoretical lift coefficients and lifting-line results With a mesh of jx = 51 points, the iterative method converges in less that 1 iterations. The exact solution is now studied. The circulation is expanded in Fourier series Γ[y(θ)] = 4 A n sin nθ, n=1 Cl & CL 3 1 Cl(Cd), D CL(CD), 3 D Lifting Line where y(θ) = cos θ, θ π. The downwash is found to be w ind [y(θ)] = and the induced drag C Di = πar n=1 na n sin nθ sin θ, na n. n=1 The simplest test case corresponds to elliptic loading with Γ[y(θ)] = 4A 1 sin θ. The downwash is thus w ind = A 1. With a chord distribution c[y(θ)] = c x sin θ, the integro-differential equation reduces to 4A 1 = π c x sin (α arctan(a 1 )). For a given value of α, α π, the above implicit equation can be solved for A 1 (α). A wing with elliptic planform and elliptic loading, no camber and no twist (α = ), has a local lift coefficient which is equal to the global lift coefficient, C L = C l = πar A 1. The induced drag reduces to C Di = πar A 1. The -D and 3-D lift coefficients are shown in Fig. 1. The lifting-line results, for a number of incidences,.1..3 Cd & CD Fig.: Theoretical polars and lifting-line results match the analytic solution very accurately. The - D and 3-D polars (inviscid) are shown in Fig.. As the angle of incidence increases from to π, the polar is swept forward then backward. The wing geometry and the distribution of circulation and downwash for an incidence α =6deg, beyond stall, is shown in Fig. 3. As can be seen, the method yields very accurate results. When the artificial viscosity term is switched-off, with the same -D viscous lift curve, spurious oscillations occur in the solution and the iterative method may not converge. Consider, for example, the case of a rectangular wing of aspect ratio AR = 1 at α = 47deg incidence. With the calculated value for µ the solution is smooth, but for µ =, although the solution is converged, the results are meaningless, and even symmetry is lost, Fig WING AND WINGLETS Winglets are aerodynamic components, placed at the tip of a wing to improve its efficiency during cruise.

4 4 Jean-Jacques Chattot Γ(y) & w(y) α=6 deg wing planform circulation Γ analytical Lifting Line downwash w(y) analytical Lifting Line y Fig.3: Wing planform and distributions of circulation and downwash for α =6deg Γ(y) rectangular wing, α=47 deg µ calculated µ= During the design step, the viscous effects may be neglected as they have a small effect on the optimum distribution of circulation and on the optimum geometry, as was found in the study of optimum wind turbine blades [7]. In the analysis, however, they could be taken into account if the off-design conditions are severe enough. The winglets considered here are perpendicular to the wing span and oriented upward, hence do not contribute to lift, but the method is general and can treat arbitrary dihedral shapes. We make use of the lifting-line theory and assume that the wing is of moderate to high aspect ratio and that the Reynolds number is large. At the design point, away from high lift conditions and stall, a linear -D lift curve is used as C l (α) =π(α α ). Extension to more general lift curve is straightforward [7]. In order to find the wing/winglet combination that corresponds to the minimum induced drag for a given lift, the following objective function F = C D + λc L is defined. C D is the drag coefficient, primarily the induced drag, but could include the viscous drag as given by a -D viscous polar. λ is the Lagrange multiplier that governs the target lift. C L is the lift coefficient and is added to the objective function as a means of enforcing the lift constraint. Let a be the dimensionless winglet height normalized by the semi-span,. The lift and induced drag coefficients are b y C L = AR dy d d, Fig.4: Distribution of circulation for a rectangular wing with and without the artificial viscosity term C Di = AR q n ()d, k Kuhlman et al. present method where (or s) is the curvilinear abscissa along the dihedral shape, and q n is the induced velocity component perpendicular to the dihedral line in the x = plane, with q n = w along the wing, q n = v on the left winglet and q n = v on the right winglet. q n depends linearly on Γ as q n () = a Fig.5: Comparison of efficiency factors for vertical winglets 1 1+a dy dz (y() y(s)) Γ d +(z() z(s)) d (s) 4π (y() y(s)) +(z() z(s)) ds. The minimization equation is obtained by taking the Frechet derivative of the objective function, which gives in the inviscid case F Γ (δγ) = C Di Γ (δγ) + λ C L (δγ) =, δγ. Γ

5 Analysis and Design of Wings and Wing/Winglet Combinations at Low Speeds 5 The boundary conditions are Γ() = Γ(1+a) =. Some mathematical results are derived that show interesting properties of the optimum winglets. The first term in the above minimization equation can be developed as C Di Γ (δγ) = AR [δq n ()+ q n Γ (δγ)]d, where δγ is an arbitrary change of circulation, not necessarily small. The second term in the integral can be transformed, using a sequence of integrations by parts together with the identity 1 8π d d to give q n () = Γ (s) ln[(y() y(s)) +(z() z(s)) ]ds, q n Γ (δγ)d = δq n ()d. This result underlines the antisymmetric property of the kernel of the induced drag integral [7]. The second term in the minimization simply reads C L Γ (δγ) = AR δ dy d d. Altogether, the minimization equation reduces to AR δ(q n () λ dy )d =, δγ. d The above result is actually applicable to an arbitrary dihedral shape. It shows that the normal induced velocity must satisfy q n () = λ dy d, which, in our case, shows that along the wing the downwash is constant and that the normal induced velocity is zero at the winglets. Since C Di is homogeneous of degree two in Γ and C L of degree one, the minimization equation is linear non-homogeneous in Γ when λ. The solution is obtained for a non-zero value of the Lagrange multiplier, say λ = 1, then scaled for target lift. The discrete formulation can be found in [6]. The efficiency of vertical winglets has been estimated by Kuhlman et al. [8] and Lundry et al. [9], from analyses in the Trefftz plane. The efficiency factor k is defined as the ratio of induced drag coefficient for an elliptically loaded planar wing to the induced drag coefficient of the optimal nonplanar wing/winglet configuration having an equal projected span at equal lift coefficient. The present method is compared with the results of [8] in Fig. 5, where a slightly higher efficiency is found, possibly due to a finer mesh in the present calculation. Γ & qn Wing with % winglets Wing w/o winglets Γ(y) qn(y) y & Fig.6: Comparison of the distributions of circulation and normal induced velocity for the wing with % winglets and the wing without winglets Γ & qn Wing with % winglets Wing w/o winglets Γ(y) qn(y) y & Fig.7: Comparison of the distributions of circulation and normal induced velocity for the wing with % winglets and the wing without winglets at same root circulation As a test case, a wing with % vertical winglets designed for a lift coefficient C Ltarget =1.is compared with that of the optimal wing without winglets in Fig. 6. The computation uses a fine mesh of jx = 1 points including the winglets with a cosine

6 6 Jean-Jacques Chattot xle() & xte() Γ & qn Fig.8: Wing and winglets chord distribution Design Analysis Fig.9: Comparison of the design and analysis results CL & e CL(α) e(α) α deg Fig.1: Wing/winglet lift coefficient and Oswald efficiency factor distributions of points on each element. This is required because the normal induced velocity is singular at the wing/winglet junction and small mesh steps are needed there. It can be noted that the winglets do not contribute to the induced drag since the normal induced velocity q n is zero at the winglets, which is in agreement with the theory [3]. Note that the above result proves that the optimal winglets do not produce a thrust, which is a common misconception. The winglets are loaded, although much less than the wing. Their role is to redistribute the loads on the wing to decrease the induced drag by lowering the root circulation and by inceasing it near the wing/winglet junction. The downwash on the wing is again constant, but of smaller absolute value. This effect can be exploited to increase the maximum lift since, as the root circulation is increased to the maximum value possible, the rest of the wing will also experience a higher circulation. In the case of the % winglets, at C L =1., the induced drag is decreased by 18% compared to the elliptically loaded wing. Increasing the root circulation to that of the elliptic wing increases the lift by 9%, Fig. 7. In inviscid flow theory, there is an infinite number of wings and winglets that will produce the optimum loading. They differ by chord, camber and twist. The design step is made simple by having zero camber and zero twist for the wing and winglets (α = ) and having a chord distribution that is proportional to the circulation, i.e. c() =c x Γ max. The wing has an aspect ratio AR = 1 and a maximum root chord c x =.34. In Fig. 8, the wing and winglet geometry is displayed as if the winglets had been rotated to be a continuation of the wing, with the quarter-chord aligned with the y-axis. The winglets are set with a toe-in angle β = 9.1 deg as calculated from the circulation. Indeed, with this choice of chord distribution, the local lift coefficient is constant and equal to the total lift coefficient, C L = C l = Γmax c x. Hence, the winglet will develop the design loading if β = Γmax πc x. The analysis code solves Prandtl integro-differential equation. The design geometry is now analyzed. The design incidence angle is given by α = C l π w ind =1.6 deg. The optimization and analysis codes are in excellent agreement at the design point, Fig. 9. The main difference between the two results lies in the normal induced velocity at the wing/winglet junction, which exhibits a strong singularity in the optimization code and just a rapid transition in five or six points in the analysis code. The induced drag coefficients are for the main wing (C Di ) m =.61 and for the winglets (C Di ) w =.3, this last value giving the accuracy level in the numerics since the result should be zero. The optimal geometry is analyzed for a range of incidences. The lift coefficient and the Oswald efficiency factor e = k = L C πar C Di are shown in Fig. 1.

7 Analysis and Design of Wings and Wing/Winglet Combinations at Low Speeds 7.6 CDi wing winglets wing+winglets CN,om, CN,ow & CN,o wing winglets wing+winglets α deg Fig.11: Component induced drag coefficients a Fig.14: Effect of winglet height on the contributions of wing and winglets to yawing moment at yaw angle of 1 deg & toe in angle=9.1 deg toe in angle= deg &.1.1. Yaw angle= deg Yaw angle=1 deg 1 1 Fig.1: Comparison of distributions of circulation and normal induced velocity for the optimal geometry and the untoed winglets Γ & qn Yaw angle= deg Yaw angle=1 deg Fig.13: Effect of 1 deg yaw on the distributions of circulation and normal induced velocity Fig.15: Effect of 1 deg yaw on the distributions of circulation and normal induced velocity for the untoed geometry e reaches a maximum (e) max =1. at the design incidence. The component induced drag coefficients of the wing, winglets and wing/winglet combination are shown in Fig. 11. At α =deg, the wing has a negative induced drag (thrust) and the winglets a slightly larger positive drag. The induced drag of the winglets changes sign at the design incidence and becomes increasingly negative for larger incidences, whereas the wing induced drag keeps increasing rapidly. These inviscid results need to be taken as indicative when α 15deg since, in general, viscous effects will become significant. In the course of this investigation, a surprising result was found. The optimum geometry was modified by only changing the toe-in angle, from the optimal value of 9.1 deg to zero. The analysis showed that, for the wing and the end plates, the Oswald efficiency factor remains constant for all values of the inci-

8 8 Jean-Jacques Chattot dence angle and equal to (e) toe in= =1.13, although less than (e) max. The same seems to be true for other wings with end plates. The two wing/winglets combinations at incidence α = 1.6 deg are compared in Fig. 1. The winglets of the non optimal configuration actually produce a thrust (C Di ) w =.16, but the price to pay for this is an increased drag of the wing to (C Di ) m =.86. A similar situation is found when designing a wing without winglets with some local thrust-producing upwash. The overall drag will be greater than that of the elliptically loaded wing. The effect of yaw on the optimal wing/winglet combination demonstrates weathercock stability. At a positive 1 deg yaw angle, the flow comes from the right. In the absence of yaw, any optimally designed wing and winglet configuration satisfies the linear Prandtl integro-differential equation =πc()[γ α ()+q n ()], where γ represents the geometric incidence α on the wing and is zero on the winglets (α carries the camber, twist and toe-in angle). The changes due to yaw satisfy on the wing =πc() q n Γ ( Γ) = πc()q n( Γ), and on the right winglet =πc()[yaw + q n Γ ( Γ)] = πc()[yaw + q n ( Γ)]. Here we have used the property of a homogeneous function of degree one. On the left winglet the term yaw is replaced by -yaw. The solution is antisymmetric. The analysis code indicates that the left winglet produces a thrust (C Di ) lw =.73 and the right winglet a drag (C Di ) rw =.367, tending to rotate the wing into the incoming flow. The yawing moment coefficient about the origin, including the contribution of the wing, is given by C N,o = AR q n ()y()d. The integration gives C N,o =.196, a restoring moment. The distributions of circulation and normal induced velocity are shown in Fig. 13. The contributions to the yawing moment of the wing and winglets, at 1deg yaw angle, as function of the winglet height are shown in Fig. 14. It indicates that the optimum winglets of various heights have weathercock stability. The untoed geometry does not provide the weathercock stability. With a yaw angle of 1 deg the left winglet has a drag (C Di ) lw =.113 and the right winglet a drag (C Di ) rw =.3, which contribute with the wing to a total torque C N,o =.63, tending to rotate the wing away from the incoming flow, Fig CONCLUSION Prandtl lifting-line method has been extended to account for nonlinear effects associated with a -D lift curve, when the local incidence is larger than the incidence of maximum lift. An artificial viscosity term has been added to the governing equation that allows the iterative method to converge to the correct solution. An optimization code, based on the same model but with a linear lift curve, has been used to find the optimal distributions of circulation and normal induced velocity for a wing of aspect ratio AR = 1, equipped with % winglets. The results are in agreement with early theoretical work on the subject. A simple wing and winglet design has shown that the analysis and optimization codes are in good agreement. The maximum efficiency is reached at the design incidence. Analysis of the same wing but with winglets set at a zero toe-angle has shown that the Oswald efficiency factor is independent of incidence. End plates parallel to the plane of symmetry have a similar effect on other wings. The effect of yaw has been investigated and the optimal wing/winglet combinations have been found to have weathercock stability. In contrast, the untoed geometry does not have that property. More research is needed to fully understand the way winglets work, and simple theories, such as that used in this effort, are more likely to help, even though validation with experiments and viscous solvers is necessary at design and off-design conditions. REFERENCES [1] Prandtl, L., Tragflachentheorie I und II, Gottinger Nachrichten, p. 451, 1918; p. 17, Reprinted in: Prandtl, L. and Betz, A., Vier Abhandlungen zur Hydro- und Aerodynamik, Gottingen 197. [] Anderson, J.D., Jr., Corda, S. and Van Wie, D. M., Numerical Lifting Line Theory Applied to Drooped-Leading Edge Wings Below and Above

9 Analysis and Design of Wings and Wing/Winglet Combinations at Low Speeds 9 Stall, Journal of Aircraft, Vol. 17, No. 1, December 198, pp [3] Munk, M.M., The Minimum Induced Drag of Aerofoils, NACA Report 11, 191. [4] Whitcomb, R.T., A Design Approach and Selected Wind-Tunnel Results at High Subsonic Speeds for Wing-Tip Mounted Winglets, NASA TN D-86, July [5] Ishimitsu, K.K., Aerodynamic Design and Analysis of Winglets, AIAA Paper No , [6] Chattot, J.-J., Optimization in Applied Aerodynamics, CFD Journal, Vol. 9, No. 3, pp , Special Issue. [7] Chattot, J.-J., Optimization of Wind Turbines Using Helicoidal Vortex Model, AIAA Paper No Also Journal of Solar Energy Engineering, Vol. 15, November 3. [8] Kuhlman, J.M. and Liaw, P., Winglets on Low- Aspect -Ratio Wings, Journal of Aircraft, Vol. 5, No. 1, October 1988, pp [9] Lundry, J.L. and Lissaman, P.B.S., A Numerical Solution for the Minimum Induced Drag of Nonplanar Wings, Journal of Aircraft, Vol. 5, Jan-Feb. 1968, pp