25th AIAA Applied Aerodynamic Conference AIAA Paper # 2006–3160 05–08 June 2006, San Francisco, CA

Induced Drag Minimization with Optimal Scheduling of Virtual Surface Deflection Boundary Conditions

Ernest D. Thompson* and Scott C. Monsch† United States Air Force Research Laboratory, WPAFB, OH, 45433

José A. Camberos‡ United States Air Force Research Laboratory, WPAFB, OH, 45433

and

Franklin E. Eastep§ University of Dayton, Dayton, OH, 45469

This project took an initial step towards our goal of implementing multiple trailing edge “virtual flaps” to control the span-wise lift distribution over a finite wing for minimizing lift–induced drag. The first part of this project compared Computational Fluid Dynamic (CFD) results of an untwisted, finite rectangular wing (NACA 0006, AR = 40/6) using no flap deflections against theoretical results for verification of the methodology. A lifting line code handled the theoretical computations as well as comparison of the results. Dividing the wing into twenty span-wise sections and using a surface integral of pressure at each section provided a method from which to extract a span-wise lift distribution from the CFD solution. A comparison of the numerical and theoretical lift distributions, under flow conditions representing Mach 0.3 – 0.7 subsonic and transonic flows at small angle of attack, shows good agreement with an average error of 2.4% over the wingspan. An important part of the methodology required extracting accurate and robust calculations of induced drag from the CFD solutions. Inaccuracies associated with the (standard) surface integral method of calculating drag prompted the use of a wake integral method. To minimize grid requirements and complexity associated with fully viscous solutions, the CFD solution focused on the Euler equations and only the induced drag obtained. Successful implementation of a wake integral method via a Trefftz-plane analysis provided an approximation of the induced drag, which seemed to prove more accurate than the surface integral method.

Introduction Generally, the wings of a conventional flight vehicle are optimized for specific mission criteria. If the aircraft is a fighter jet, the wings may be designed to provide the aircraft with agility and maneuverability. Likewise, if the aircraft is a commercial jetliner the wings may be designed for a specific cruise condition. Often conventional designs do not have the ability to adapt to deviations in their mission criteria. To address this issue of inflexibility researchers, Kolonay, Eastep, and Sanders used active conformal control surfaces to tailor span-wise lift distribution

*Aerospace Engineering Trainee, Propulsion Directorate, Air Force Research Laboratory / University of Dayton, AIAA Student Member. † Summer Research Assistant, Air Vehicles Directorate, Air Force Research Laboratory / Clemson University, AIAA Student Member. ‡ Research Aerospace Engineer, Air Vehicles Directorate, Air Force Research Laboratory, AIAA Associate Fellow. § Professor Emeritus, Mechanical & Aerospace Engineering, University of Dayton, AIAA Fellow.

24th Applied Aerodynamics Conference5 - 8 June 2006, San Francisco, California

AIAA 2006-3160

This material is declared a work of the U.S. Government and is not subject to copyright protection in the United States.

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of a wing to a desired shape. Their study employed a vortex lattice method coupled with an optimizer to generate an elliptical span-wise lift distribution using trailing edge controls.1 As a result of the author’s of Ref. 1 effort there was desire to elaborate on their work. A new take on their study was developed involving a three dimensional unstructured Euler finite-volume solver and transpiration boundary conditions. The use of an inviscid flow solver is the logical next course in this study because the inviscid solver has a higher fidelity than the vortex lattice method used by Kolonay, Eastep, and Sanders. Unlike a panel method or a potential flow solver, an Euler code does not require knowledge of a wake’s geometry.2 Moreover, an Euler solver serves as a good transition because it has less of a computation overhead than Navier-Stokes flow solver. This is due to not having to compute the viscous fluxes. It can also be shown that lift-induced drag is independent of fluid viscosity. Lift-induced drag is an artifact from the formation of the trailing edge vortices which are by-products of the pressure difference used to generate lift.3 This is why the viscous terms of the governing equations of fluid mechanics can be neglected. The goal behind this project was to use multiple virtual trailing edge control surfaces to change the span-wise lift distribution of a wing inside an Euler solver. The use of virtual control surfaces kept complexity and cost associated with generating physical deflections down. Unlike physical control surfaces, the virtual surfaces do not require grid morphing or grid regeneration when the geometry changes. The manipulation of the lift distribution by the control surfaces will cause a change in lift-induced drag. The end focus of this study is to optimize the lift distribution for minimum induced drag. However later, the techniques developed for this study may be adapted for the minimization of total drag or optimizing for entropy production. This study was conducted following a systematic process. In this process, techniques were developed to extract induced drag data from a Trefftz-plane. A methodology was developed to calculate span-wise lift. A zero-mass-flow boundary condition was created. Lastly, a verification process was developed to validate both the research and the computational fluid dynamics code. Ultimately the larger objective of this project is to achieve a close loop abstract control system by coupling flow solver code with an optimization program. This larger objective will require the transitioning methodologies and techniques developed in post-processing stage of analysis directly into the flow solver. Only then can the flow solver’s link to an optimization routine be constructed.

Lifting Line Theory – Background Thin airfoil theory provides a method to calculate the lift of a two-dimensional airfoil. The assumption being that the span of these airfoils is infinite, which in turn produces a constant lift distribution along the infinite span. Finite wings differ, of course, in that they have a finite span. As the high-pressure flow on the underside of the wing tends to flow outward towards the tip and the low-pressure flow above the wing tends to flow inward towards the root, a trailing vortex is formed as these two flows meet at the trailing edge. This trailing vortex sheet and the tendency for these pressures to equalize induces a downwash velocity in the downward direction, normal to the undisturbed free stream, defined as

( )dzzz

zzwb

b∫

−−

Γ′−=

2/

2/04

1)(π

(1)

where Γ(z) represents the span-wise circulation distribution and b is the total span length. This downwash velocity alters the approach angle of the free stream flow by an amount termed the downwash angle defined as

∞

−∈=U

zw )(tan 1 (2)

which is often simplified to

∞

∈=U

zw )( (3)

where U∞ represents the free stream velocity. The effective angle of attack at a given span-wise location then is defined as

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∈−=ααe (4) where α represents the geometric angle of attack. The downwash velocity increases along the span from root to tip, resulting in a span-wise lift distribution that drops as you approach the wingtip. It is also important to note that since lift acts normal to the freestream velocity direction, the effective lift will act normal to the effective free stream velocity. It follows then that the effective lift has also been altered by the same downwash angle. This effective lift has a force component in the direction of the undisturbed free stream velocity which is termed lift induced drag; the focus of this project. Glauert considered a circulation distribution expressed by a Fourier sine series, the first term of which represents the elliptic distribution. A circulation distribution then can be defined as

( ) ∑∞=ΓN

n nAsU1

sin4 φφ (5)

where s represents the half-span length and the number of terms is determined by the desired number of span-wise locations used to describe the distribution. The physical span-wise coordinate has also been replaced by φ according to the transformation:

φcos−=sz

(6)

Since the span-wise lift distribution represented by the circulation is symmetrical, only the odd terms are used. A derivation is given by Bertin4 that concludes with the governing equation shown here, termed the monoplane equation;

( ) ( )φµφφααµ sinsinsin1

0 +=− ∑ nnAN

nl (7)

where µ is defined as

bcae

4=µ (8)

and ae, the lift curve slope, is assumed to be 2π according to thin airfoil theory. After solving for the Fourier coefficients, lift and drag characteristics can be calculated. The total lift coefficient can be approximated using the equation

ARACL ⋅= π1 (9) where CL is dependent only on the first Fourier coefficient, regardless of the number of terms in the series. The coefficient of induced drag can also be approximated by

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+⋅⋅⋅++++

⋅= 2

1

2

21

27

21

25

21

23

2 7531

AnA

AA

AA

AA

ARCC nL

Di π (10)

which is obviously influenced by the number of terms used. As more terms are added to the Fourier sine series, the induced drag coefficient will more nearly approximate the asymptotic value. The span-wise lift coefficients can also be approximated for a given span-wise unit section by

( ) ( ) ( )cUcU

UCl∞∞∞

∞∞ Γ=

Γ=

φρ

φρφ 22

21 (11)

It is also important to account for compressibility effects, which can be done by applying the Prandtl-Glauert Formula4, which is defined as

21 ∞−

′=

M

CC p

p (12)

At low Mach numbers, just as you would expect, this will not have much affect on the outcome of your calculations, but at higher Mach numbers, it becomes very important. Figure 1 shows the results on the span-wise lift distribution with and without the compressibility effects.

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Figure 1 Compressibility Effects

At M=0.3, compressibility effects alter the lift distribution by about 5%, whereas at M=0.7 the effect is closer to 40%. In an attempt at efficiency, a lifting line code was written using MATLAB to automate the theoretical calculations. Given a set of geometric and flow condition inputs, the code would return a multitude of output variables as shown in Figure 2. Each of these variables is available for manipulation and/or plotting. Efforts were made to write the code in general terms to maintain flexibility across varying input conditions. The code also has the ability to read in post-processed results from numerical cases in order to compare, as well as report the error, when fitting the numerical results over the theoretical.

Figure 2 Lifting Line Code – Inputs/Outputs

Zero-Mass-Flow / Transpiration Boundary Condition The transpiration method was originally developed by Lighthill5 to simulate airfoil thickness is characterized as a near-field boundary with the potential of existing in three states. These states are suction, blowing, or solid wall. The state at which the boundary condition operates is dependent on the transpiration velocity. A transpiration

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boundary condition works by manipulating the normal velocity adjacent to the boundary. This effectively reorients a surface normal; this effect is illustrated in Figure 3. Transpiration boundaries affect velocity profiles of a flow field near the boundary. When the transpiration velocity is less than zero, the boundary creates suction, if it is greater than zero blowing, and if it is zero an impermeable wall.6

Figure 3: Illustration of the transpiration boundary condition

Transpiration allows for small changes in the geometry without having to deform or regenerate a mesh. This has great advantages for accelerating computational times and possibly allowing the efficient use of high fidelity methods in a design environment. Moreover, because the mesh does not have to be regenerated the flow solver can be restarted from a previous solution. This is a significant benefit because a converged solution can be obtained faster than if the flow solver had recomputed a solution from a cold start. The transpiration boundary condition’s design is similar to that of the slip-wall boundary condition. The boundary condition only relies on normal vector component of velocity, the transpiration velocity, and the conservative variables of the neighboring domain cell for its implementation. The tangential flow velocity components at the boundary were computed using the conservative equation. These values combined with transpiration velocity determined the ghost cell velocity.

niontranspiratVnVUU domainghostˆˆ2 ⎟⎠⎞⎜

⎝⎛ +⋅−=rrr

(13)

The transpiration velocity was set by determining the slopes of the deflected control surfaces and then substituting the value the tangent slope relation, equation (14).

dtduu

dtdvv

dxdy

−

−= (14)

Since the flow solver will simulate steady flow conditions, equation (14) reduces to its simplified form;

uv

dxdy

= (15)

The transpiration velocity is calculated by substituting in the u-component of velocity for the neighboring domain cell into equation (15), and solving for the v-component of velocity.

Integral Method for Computing Drag In computational fluid dynamics, there are generally two methods for determining the lift-induced drag of a wing, a surface integration method and a wake integration method. For this study, the latter was used to obtain lift-induced drag. This method was preferred over the surface integration method because the surface integration method relied on measurements of pressure and skin friction over a series of flat surfaces that were used approximated curve surfaces of a three-dimensional wing. This method for determining force was suitable for lift computations because the lift values tend to be one to two orders magnitude large than drag force. The wake integration method referred to as the Trefftz-plane analysis, measures induce drag by extracting data from a cut-plane perpendicular to the free-stream direction. This analysis often takes place in post-processing because it

originaln̂

newn̂

neighborV

iontranspiratV newV

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requires interpolating flow field data to arbitrary planes where the nodes or cell centers of a CFD grid do not necessarily intersect. The Trefftz-plane integration equation is derived from the momentum equation of the governing equations of fluid mechanics. The following equation is set as the objective function in flow visualization software where it is integrated over the domain of the arbitrary plane.

)22(21 wv

dAidD

+∞= ρ (16)

The result is the capturing of the work done by wing on the surrounding fluid. Due to energy conservation, the energy associated with a wing’s lift induced drag can be directly related to kinetic energy generated by the flows perturbation.7

Discussion Results

Geometry and Flow Conditions Two geometric configurations were studied in this investigation, a wing with a NACA 0006 profile, and a wing with the NACA 0012 airfoil shape. Both wings were untwisted rectangular planform with aspect ratios of 6.67. The wing with the NACA 0012 cross-section is presented in Figure 4. The total span is forty feet and the chord length is six feet. Most work was conducted at M=0.3. Although the velocity was varied in the subsonic and transonic flight regimes to gain an understanding of the effects. The geometric angle of attack was selected as five degrees to avoid the complications of high angles of attack while still providing sufficient lift for accurate calculations. The study was also restricted to steady, level flight at sea level conditions.

Figure 4. Wing Geometry

Comparison of Span-wise lift distribution The span-wise lift distribution of the wing was generated to compare numerical data with lifting line theory. The computation of span-wise lift distribution like with the calculation of induced-drag calculation computed as post-processing step. In this computation, a wing was divided into sections. Each section was split at the chord line dividing the sections into their upper and lower surfaces. The static pressure was set as an objective function and the surfaces were integrated over to generate the axial and normal components of force over each section.

dxbpdxbpNTE

LE u

TE

LE l θθ coscos ∫∫ −= (17)

dxbpdxbpATE

LE u

TE

LE l θθ sinsin ∫∫ −= (18)

The normal and axial forces in addition to geometric angle of attack were then used to calculate the average lift of each wing section.3

αα sincos ANL −= (19)

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In the following figure, there is a visual representation of the rectangular wing span-wise lift distribution. The lift is plotted in the half span from wing root to wing tip.

Figure 5. Span-wise Lift Distribution of a Rectangular Wing

The results from the span-wise lift analysis show the numerical results to correlate well with the lifting line theory. The agreement between the two was within 2.4%. A large part of this error can be attributed to tip effects where the flows from the upper and lower surfaces of the wing interact. Rounding of the wing tip would probably improve agreement. The elliptical lift distribution plotted was presented strictly for comparison.

Drag calculations corrections and placement of the Trefftz-plane The results from the Trefftz-plane analysis show quite a bit. Presented in Figure 6 are the coefficients of induced drag results for various grids. Each line in the figure represents a grid. The induced drag coefficient results were depicted with respect to the theoretical value of induced drag obtained from lifting line theory. The information in the legend of the figure indicated the size of each grid presented. As an example, 1.0M line indicated that the grid presented had one million cells. The grids portrayed in the figure were constructed in various ways, some grids possessed a large domain focus while others had small domain focus. The grids with larger focus extended several chord lengths away from wing geometry. The grids that had the smaller focus, the farfield region were only located a few chord lengths away from the wing. This is the reason why some results on map out to seven chord lengths behind the wings trailing edge. The circles that cover the lines indicate different Trefftz-plane surveys taken aft of the wing. The results from the Trefftz-plane analysis contain much detail. Several grids were studied and compared to theoretical values obtained from lifting line theory. Trefftz-plane surveys are taken downstream of the wing; some grids encompassed a large domain while others spanned smaller domains. The plots in the figures below show that the distance of the grid outer-boundary from the wing geometry did not significantly influence the induced drag calculations. This demonstrates that the wing can be modeled with a relatively smaller domain without polluting the solution with reflections from the far-field boundaries. The observed trend from the data indicated that increasing grid density improved the numerical results relative to the value obtained by theory. Another trend observed in the data was that the induced drag decreased, as the Trefftz interrogation plane progressed further downstream of the wing.

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Figure 6. Trefftz Plane Analysis of Various Grids at (M = 0.3) The observed gradual decrease in (numerical) lift-induced drag observed in the plot in Figure 6 is the result spurious drag. This drag is an artificial phenomenon attributed to the relaxation of the grid cells downstream of the wing and the effect of artificial (numerical) viscosity. Grid cell relaxation is used to improve computation times by using coarser cells away from regions of significant flow activity, like the lifting wing geometry. The larger cells reduce the total cell count in a grid. This is a positive trait for a CFD grid when concerning analysis completion times and convergence rates but a negative trait for the Trefftz-plane analysis because it contributes to artificial viscosity. Artificial viscosity is the component of spurious drag that is associated with the formation of the convective fluxes in an inviscid flow solver. Inviscid flow by definition contains no dissipative effects and has no viscosity. The fluid flow solver creates numerical dissipation when the convective fluxes are discretized. Also, numerical damping is used to improve steady-state convergence. To correct the problem different formulations were investigated to improve results. Van der Vooren and Slooff7 developed a near-field correction to the Trefftz-plane analysis, equation (20). This correction included the high-order terms originally neglected in the classical Trefftz-plane lift-induced drag formulation. Using the near field analysis developed by the overall induced drag prediction level off within two-chord lengths of the wing. 7

( )[ ]dAuMwvDA

i ∫∫ ∆−−+= ∞∞2222 1)(

21 ρ (20)

Figure 7 presents two grids where the near field correction was applied. The plots shows the both the classical Trefftz-plane results and the near-field correction.

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Figure 7. Near Field Correction of the Trefftz-plane Results

The near field correction did not completely account for the overall effect of spurious drag as hoped. Since the near-field correction failed to correct the lift-induced drag results, another correction developed by van der Vooren was utilized. The second correction is formulated in equation (21). A third correction provided by Vooren will also be investigated. This third spurious drag corrected lift-induced drag equation involved summing the combination of a volume integral of the divergence of the terms inside the integrand equation(21) and the surface integral presented in equation(21).8

( )[ ]dAwuvuuMwvDA

i ∫∫ ∆−∆−∆−−+= ∞∞**2*222 221)(

21 ρ (21)

⎟⎟⎠

⎞⎜⎜⎝

⎛ ∆−

∆+−=∆

∞∞∞∞ 22

*

uH

RMsuuuu

γ (22)

Figure 8 presents the plot of a single grid’s lift-induced drag coefficient results. Each line in the figure indicates a different formulation of the Trefftz-plane derived lift-induced drag calculation. The classical Trefftz-plane equation relates to the “Apparent

iDC .” The near-field correction relates to “Near Field Correction.” “Correction 2” is the formula presented in equation (21) “Correction 3” is the formula from “Correction 2” with the last two terms found inside the integrand neglected. The figure shows an overall improvement of induced drag at the surveyed cut-plane locations. There was still large amount of error in the near field regions. This was the reason “Correction 3” was developed. However, further downstream of the wing it appears “Correction 2” compares better with lifting-line theory.

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Figure 8. Trefftz-plane Correction Study

Zero Mass Flow Boundary Condition Calculations The total sectional lift was calculated at various locations along the span of the wing and then plotted, Figure 9. The span-wise lift distribution of the symmetric wing at zero angle attack was analyzed to detect changes in lift. The boundary condition was formulated such that velocity injections from the control surface corresponded to the slope of a physical control surface if deflected. Since the wing is symmetric and it is at zero angle attack it is expect that the only lift observed would be from the zero mass flow transpiration boundary condition and numerical dissipation. Figure 9 shows that the zero mass flow boundary condition was indeed effective at changing a wing lift distribution.

-0.005

-0.004

-0.003

-0.002

-0.001

0

0.001

0.002

0.003

0.004

0.005

0 10 20 30 40 50 60 70 80 90 100

Location Along the Semi-Span (% - Half-Span)

Coe

ffici

ent o

f Tot

al S

ectio

nal

Lift

TR-02 TR-01 TR00 TR01 TR02

Figure 9. Comparison of Span-wise Lift Distributions for Wings with Physical and Transpiration Deflecting

Trailing Edge Control, when α=0°, β={±0°,±1°,±2°} The results obtain from the analysis of a wing with zero mass flow boundary condition show good agreement with the results obtained from a wing with physically deflected control surface. Even for a large control surface deflection, β = 20° the deviation between span-wise lift distribution was on average 0.8%.

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00.0050.01

0.0150.02

0.0250.03

0.0350.04

0.045

0 20 40 60 80 100

Location Along the Semi-Span (% - Half-Span)

Coe

ffici

ent o

f Tot

al

Sect

iona

l Lift

0.000.200.400.600.801.001.201.401.601.802.00

%-D

iffer

ence

TR20 CS20 %-Diff

Figure 10 Comparison of Span-wise Lift Distributions for Wings with Physical and Transpiration Deflecting

Trailing Edge Control, when α=0°, β=20°

Conclusions Following the systematic approach all the initial goals were complete toward the idea of using trailing edge deflections to achieve minimum induced drag. A wake integration method was implemented. A technique was developed for extracting span-wise lift distribution. A transpiration boundary condition was tested at the trailing edge of the wing. Lastly, the researcher and code were verified against a theoretical baseline. The next stage in this research project will involve the determination of best location for Trefftz-plane calculation. The result from the position should be independent of location. This will then be followed by the integration of the wake integration method into the flow solver source code. Moreover, the zero mass flow boundary condition will be fully integrated into the follow solver and calibrated. The calibration will involve correlating physical control surface deflections with the virtual deflections made by the zero mass flow boundaries. Lastly, the flow solver will be coupled to optimization software to close control loop. Once this is done the zero mass flow boundary will be used to minimize lift induced drag.

Acknowledgments The authors wish to acknowledge Richard Figliola and Ray Kolonay for their support and direction with project. We thank Greg Brooks, Victor Burnely, Gerald Trummer, and Matthew Grismer for their support using the computing clusters, the AVUS flow solver and post-processing software. Rolf Sondergaard is also thanked for his insights into the intricacies of POSIX command shells and computer aided design tools. Lastly we appreciated the support and resources received from David Moorehouse, Charlie Stevens and Douglas Blake.

References 1 Kolonay R., Eastep F., Sanders, B. “Optimal Scheduling of Control Surfaces on a Flexible Wing to Reduce Induced Drag.” AIAA Paper 2004-4362, September 2004. 2 Bourdin, P. “Numerical Predictions of Wing-tip Effects on Lift-induced Drag.” ONERA –Applied Aerodynamics Department. BP72, 92322 Châtillon Cedex, France. ICAS 2002 CONGRESS. 3 Anderson Jr., John D. Fundamental of Aerodynamics, 2nd ed., Mc Graw-Hill, New York, 1984 Chap 5. 4 Bertin, John J. and Smith, Michael L. Aerodynamics for Engineers, 3 rd ed., Prentice Hall, Upper Saddle River, NJ 1998 Chap 7. 5 Lighthill, M. J. “On displacement thickness.” Journal of Fluid Mechanics, Vol. 4, Part 4, January 1958. 6 White, Frank M. Viscous Fluid Flow, 2nd ed., Mc Graw-Hill, New York,1991. pp. 235-6.

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7 Giles, Michael B. and Cummings, Russell M. “Wake Integration for Three-dimensional Flowfield Computations: Theoretical Development.” Journal of Aircraft, Vol. 36, No. 2, March-April 1999 pp. 357-65. 8 Destarac, D. CFD-based Aircraft Drag Prediction and Reduction. ONERA, 29 Avenue de la Division Leclerc,

92322 Châtillon Cedex, France 03-7 February 2003.