ABSTRACTOne of the aerospace design engineer’s goals aims to reduce drag for increased aircraft performance, in terms of range, endurance, or speed in the various flight regimes. To accomplish this, the designer must have rapid and accurate techniques for computing drag. At subsonic Mach numbers drag is primarily a sum of lift-induced drag and zero-lift drag. While lift-induced drag is easily and efficiently determined by a far field method, using the Trefftz plane analysis, the same cannot be said of zero-lift drag. Zero-lift drag (CD,0) usually requires consideration of the Navier-Stokes equations, the solution of which is as yet unknown except by using approximate numerical techniques with computational fluid dynamics (CFD). The approximate calculation of zero-lift drag from CFD is normally computed with so-called near-field techniques, which can be inaccurate and too time consuming for consideration in the design environment. This paper presents a technique to calculate zero-lift and boundary-layer drag in the subsonic regime that includes aeroelastic effects and is suitable for the design environment. The technique loosely

The AeronAuTicAl JournAl november 2015 volume 119 no 1221 1451

Paper No. 4205. Manuscript received 6 June 2014, revised version received 17 December 2014, accepted 10 July 2015.This is an adapted version of a paper first presented at The 2014 Royal Aeronautical Society Biennial

Applied Aerodynamics Research Conference, Advanced Aero Concepts, Design and Operations.

An efficient method for predicting zero-lift or boundary-layer drag including aeroelastic effects for the design environmentJ. A. CamberosR. M. Kolonay US Air Force Research Laboratory Dayton, Ohio USA

F. E. Eastep University of Dayton Dayton, Ohio USA

R. F. Taylor Wright State University Dayton, Ohio USA

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couples a two-dimensional aerofoil boundary-layer model with a 3D aeroelastic solver to compute zero-lift drag. We show results for a rectangular wing (baseline), a swept wing, and a tapered wing. Then compare with a rectangular wing with variable thickness and camber, thinning out from the root to tip (spanwise direction), thus demonstrating the practicality of the technique and its utility for rapid conceptual design.

1.0 INTRODUCTIONAn aerospace design engineer often seeks to reduce drag in order to increase aircraft performance, such as improved range/endurance or speed in the various flight regimes. To accomplish this, one must have rapid and accurate techniques for computing total drag. For subsonic Mach numbers, drag is primarily the sum of lift-induced drag and zero-lift drag. While lift-induced drag is easily and efficiently calculated by far field methods using the Trefftz plane technique(1) the same cannot be said of zero-lift drag (the drag that exists when the wing is not generating lift). For conceptual design studies, zero-lift drag can be approximated from historical data or from ‘heuristically’ determined formulas for skin friction, as in a compu ter code like FRICTION(2). However, in the preliminary design environment a more accurate computational method with viscous and aeroelastic effects is needed.

Zero-lift drag usually requires consideration of the Navier-Strokes equations, solutions of which are as yet unobtainable in general except by using approximate numerical techniques with computational fluid dynamics (CFD). The approximate calculation of drag from CFD is normally computed by so-called near field techniques (discrete integration of the pressure field over the body surface), but has been shown to be inaccurate and too time consuming for the conceptual or preliminary design environment. This research focuses on a simple technique to calculate zero-lift drag in the subsonic regime that includes aeroelastic effects and is suitable for the design environment. The technique loosely couples a two-dimensional aerofoil boundary-layer model with a 3D aeroelastic solver to obtain zero-lift drag.

The technique itself was inspired from a paper by Jepson, et al(3) who investigated automated drag reduction on a wing with multiple trailing-edge flaps. In the present investigation we use a similar technique to computationally determine zero-lift drag, with aeroelastic effects included, on a generic wing ‘flying’ at design conditions.

2.0 TECHNICAL APPROACH – CALCULATION OF WING CD,0 OR BOUNDARY LAYER DRAG ON LIFTING WING

Zero-lift drag in the subsonic regime may be calculated in a simplified manner. For a specified configuration and flight conditions, the wing is divided into a number of span-wise strips and the wing is placed at angle-of-attack so that the total wing lift equals zero. The span-wise lift distribution of lift can then be determined by efficient codes such as a vortex lattice approach in the ASTROS software(4). Further, at each wing strip, the sectional drag polar is calculated for a specified sectional profile using the computational code PABLO(5).

PABLO allows one to compute the sectional zero-lift drag for both the laminar and turbulent portions of the boundary layer with a prediction of the sectional drag polar. The sectional drag polars are then over-laid on curves representing lift variation at various span locations on a selected number of spanwise strips (see for example Fig. 12 in the discussion below). The intersection of the sectional drag polar with lift distribution for a wing under zero-lift conditions yields the

KolonAy et al An efficienT meThod for predicTing zero-lifT or boundAry lAyer 1453

zero-lift drag for that interval. Finally, the computed sections boundary-layer zero-lift data is summed over all the selected spanwise strips to yield a simple total zero-lift drag coefficient for a selected wing. The same procedure can be used to determine the boundary-layer drag on a lifting, aeroelastic wing.

2.1 Sectional drag polar

The simplified technique of calculating zero-lift drag, CD,0, first requires consideration of sectional drag on a selected two-dimensional aerofoil. A MATLAB-based boundary-layer code called PABLO(5) allows one to compute the sectional drag for both the laminar and turbulent portions of the boundary layer. Additionally, the lift versus angle-of-attack can be obtained and the drag polar plots generated for a particular aerofoil profile. For example, the selected aerofoil used for this investigation is the NACA 24126 aerofoil shown in Fig. 1. The aerofoil has a 12% chord thickness with a maximum camber of 2% chord located at 40% of the aerofoil chord.

PABLO is a pedagogical, low-speed aerofoil analysis program written in MATLAB. It uses a ‘one-way coupled’ inviscid plus boundary-layer model. The inviscid flow is solved using a panel method(7) with options of three different kinds of singularity distributions. The boundary-layer equations use the inviscid flow velocity-field provided by the panel method, but the effect of the boundary layer on the inviscid flow is not taken into account, as in PANDA(8). The boundary-layer model is described in greater detail by Moran(9).

With flow conditions specified by a Mach number of 0·7 (with Prandtl-Glauert compressibility correction) and a Reynolds number of 5·7 × 106, the drag polar curve shown in Fig. 2 was calcu-lated from the PABLO based code for NACA 2412 aerofoil.

Figure 1. Aerofoil geometry for NACA–2412.

Figure 2. Sectional drag polar for a NACA 2412 aerofoil at M → 0 and M = 0·7 (using P-G compressibility correction).

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calculated from the PABLO based code for NACA 2412 airfoil.

Prof

ile, y

/c

Chord, x/c

Figure 1: Airfoil geometry for NACA–2412

B. Zero-Lift Condition for Cruise of Flexible Wing

As mentioned above, consider a simple rectangular flexible wing “flying” at Mach number 0.7 and dynamic pressure of 5 psi. The wing structural model consisted of three spars, five ribs and the NACA 2412 airfoil with constant skin thickness. The finite element model of this wing is shown in Figure (3) and represents a wing structure with uniform stiffness in the span direction. The computational code ASTROS4 produced the span variation of lift shown in Figure (4) for the cruise conditions at zero-lift, including (static) aeroelastic and inertial effects. To determine the span variation of Cl when the total wing lift is zero, the angle of attack is varied until the total lift is approximately zero for the wing including aeroelastic and inertial effects. Total lift of zero occurs at an angle of attack of about −1.7 deg. for this planform and flight conditions; the span variation of Cl is shown in Figure (4).

Cl

Figure 2: Sectional drag polar for a NACA 2412 airfoil at M 0 and M = 0.7 (using P-G compressibility correction).

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calculated from the PABLO based code for NACA 2412 airfoil.

Prof

ile, y

/c

Chord, x/c

Figure 1: Airfoil geometry for NACA–2412

B. Zero-Lift Condition for Cruise of Flexible Wing

As mentioned above, consider a simple rectangular flexible wing “flying” at Mach number 0.7 and dynamic pressure of 5 psi. The wing structural model consisted of three spars, five ribs and the NACA 2412 airfoil with constant skin thickness. The finite element model of this wing is shown in Figure (3) and represents a wing structure with uniform stiffness in the span direction. The computational code ASTROS4 produced the span variation of lift shown in Figure (4) for the cruise conditions at zero-lift, including (static) aeroelastic and inertial effects. To determine the span variation of Cl when the total wing lift is zero, the angle of attack is varied until the total lift is approximately zero for the wing including aeroelastic and inertial effects. Total lift of zero occurs at an angle of attack of about −1.7 deg. for this planform and flight conditions; the span variation of Cl is shown in Figure (4).

Cl

Figure 2: Sectional drag polar for a NACA 2412 airfoil at M 0 and M = 0.7 (using P-G compressibility correction).

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2.2 Zero-lift condition for cruise of flexible wing

As mentioned above, consider a simple rectangular flexible wing ‘flying’ at Mach number 0·7 and dynamic pressure of 5psi. The wing structural model consisted of three spars, five ribs and the NACA 2412 aerofoil with constant skin thickness. The finite element model of this wing is shown in Fig. 3 and represents a wing structure with uniform stiffness in the span direction. The computational code ASTROS(4) produced the span variation of lift shown in Fig. 4 for the cruise conditions at zero-lift, including (static) aeroelastic and inertial effects. To determine the span variation of Cl when the total wing lift is zero, the angle-of-attack is varied until the total lift is approximately zero for the wing including aeroelastic and inertial effects. Total lift of zero occurs at an angle-of-attack of about −1·7°. for this planform and flight conditions; the span variation of Cl is shown in Fig. 4.

2.3 Wing total drag for zero-lift condition

To determine wing drag for the zero lift condition the wing is divided into N strips (eight shown for display purposes). At the mid-span location of each strip we overlay the sectional drag polar from Fig. 2 as shown in Fig. 4. The intersection of the Cl versus span curve with the sectional drag polar curve determines the sectional drag coefficient associated with each strip. Sectional drag coefficient cd,0 is assumed to be constant over each strip and the strip cd,0 can be obtained by multiplying cd,0 by the strip span length. It is then a simple matter of summing each strip cd,0 to obtain the wing drag coefficient CD,0 for a wing in a zero-lift condition, according to the summation equation (1):

. . . (1)

where cd(yi) is determined from Fig. (4) and ∆yi is the interval length.

Figure 3. Structural wing model dimensions (blue lines) for rectangular planform with aerodynamic shape shown overlaid in green.

CS

c y c y dys

c y c y yD d d i ii

N

i 1 101 ( ) ( ) ( ) ( )

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2.4 Boundary-layer drag for wing in lifting condition

In the previous section, a technique was described to calculate zero-lift drag. This required a determination of the wing angle-of-attack such that the total lift on a flexible wing was zero. The same technique can be used to determine boundary-layer drag on a lifting wing without adjusting the wing angle-of-attack. Here we select an appropriate flight condition and calculate the wing angle-of-attack and the coefficient of lift distribution along with span from an aeroelastic trim analysis within ASTROS. The lift distribution along the span is shown in Fig. 6 for a straight, rectangular wing for selected aerofoil profile (NACA 2412) and flight conditions described previously. The spanwise twist angle has the same shape as shown in Fig. 5, which is the zero lift twist distribution, but a different vertical scale. We next overlay the sectional drag polar over a number of intervals (eight intervals shown for display purposes). The intersection of the drag polar with the Cl distribution will allow one to determine the sectional boundary-layer drag coefficient

Figure 4. Spanwise lift distribution for rectangular, flexible wing at zero-lift condition with wing angle-of-attack of –1·731°.

Figure 5. Spanwise twist angle distribution.

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for that interval. Next the total wing boundary-layer drag can be determined from Equation (1). It was determined that the selected aerofoil profile (NACA 2412) that the boundary-layer drag for a lifting wing was approximately equal to the zero-lift drag case. Hence for this case, computing the viscous drag at the trimmed lift condition or at the zero lift condition yielded the same results. This should not be expected for other aerofoil profiles or when the aerofoil profiles vary along the span.

3.0 PLANFORM VARIATIONS AND RESULTSIn addition to the rectangular wing used above to describe the simple procedure for determining CD,0, other planforms were considered. These included a swept wing 30°. sweep angle, untapered) as show in Fig. 7. The wing angle-of-attack resulting in zero-lift was found to be −1·56°. and the lift coefficient variation is displayed in Fig. 8. Next, a tapered wing planform (taper ratio of 0·4) with no sweep was considered as shown in Fig. 9. The wing angle-of-attack resulting in zero lift was determined to be −2·01°. and the lift coefficient variation is shown in Fig. 10.

The above results were obtained when the aerofoil profile was uniform in the spanwise direction. In general, a wing has aerofoil sections which are similar but the thickness and camber have some variation from root to tip. A study of the effect of aerofoil section on CD,0 was conducted for the rectangular wing. At the wing root the aerofoil was again the NACA 2412 section. The thickness and camber at the tip was reduced in a linear manner to 50% of root values, as shown in Fig. 11. The wing angle-of-attack resulting in zero lift is −1·24°. and the sectional lift coefficient variation is shown in Fig. 12. The sectional drag polars are overlaid at five intervals as shown in Fig. 13. Notice that the sectional drag polar curves change from interval to interval.

4.0 DISCUSSION OF RESULTSFollowing the computational method described above, the wing zero-lift drag coefficient for an unswept wing with no taper was determined as CD,0 = 0·00602. The wing with the same aerofoil was then swept through an angle of 30° and CD,0 calculated. Next, an unswept wing but with the

Figure 6. Sectional drag coefficient overlaid across spanwise lift distribution for rectangular, aeroelastic trim condition. M = 0·7, q = 5psi, and aeroelastic trimmed angle-of-attack of –1·37°, rigid trimmed angle-of-attack = –2·16°.

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chord taper ratio of 0·4 and the same aerofoil profile was considered. There was only a slight change in the CD,0 calculated for the swept and tapered wing when compared to the value of CD,0 for the rectangular wing (both were about 0·00602). Hence, for the configurations studied, the value of CD,0 is insensitive to sweep angle and taper ratio. For the cases studied, a good approximation for the value of wing CD,0 can be obtained from the sectional drag polar curve at a sectional lift coefficient of zero if the drag polar curves are constant along the span. We next removed this simplifying restriction and consider a wing with decreasing thickness and camber along the span. The value of CD,0 was found to be 0·00577 in this case. In conclusion, the simple computational procedure as described allows for the calculation of zero-lift drag in the subsonic flight regime, including the effects of viscosity and static aeroelasticity. The technique loosely couples a two-dimensional boundary-layer model with a 3D aeroelastic solver to compute zero-lift drag. The technique is accurate and efficient for use in the preliminary design environment, where the engineer seeks to reduce flight vehicle drag.

5.0 FUTURE INVESTIGATIONSCurrently, the procedure described in this paper will allow one to determine zero-lift drag in the transonic Mach regime. However, the span-wise lift variation must be calculated from a numerical

Figure 7. Wing planform geometry with 30°. sweep, untapered.

Figure 8. Spanwise lift distribution for 30° sweep, untapered, at wing angle-of-attack of –1·56°.

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Figure 9. Tapered wing geometry, with a taper ratio of 0·4 used.

Figure 10. Spanwise lift distribution for tapered wing, no sweep, wing angle-of-attack at –2·01°.

Figure 11. Variable thickness wing geometry (50% thinned wing from root-to-tip).

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solution of the Euler equations to account for the nonlinear effects that are not present in standard panel codes for the subsonic regime. It also will require that the sectional drag polar be determined numerically or experimentally. An efficient numerical solution of the two-dimensional Navier-Stokes equations could be effective and accurate for capturing the surface shock wave effects for the sectional drag polar. With this modified procedure as described, the zero-lift drag can be determined up to the drag divergence Mach number in the transonic Mach regime. Additionally, the reduction of zero-lift and induced drag can be considered for vehicles flying at off-design Mach numbers. It is also proposed to investigate the settings of external control surfaces to reduce total drag coefficients at those off-design conditions.

REFERENCES1. KolonAy, R. and eAsTep, E. Optimal scheduling of control surfaces on flexible wings to reduce induced

drag, J Aircr, November-Decemter 2006, 43, (06), pp 1655-1661.2. mAson, W.H. FRICTION: From the Virginia Tech Aerodynamics and Design Software Collection,

Website, 2011, www.aoe.vt.edu/~mason/Mason f/friction.f.3. Jepson, J.K. and gopAlArAThnAm, A. Computational study of automated adaption of a wing with multiple

trailing- edge flaps, AIAA Aerospace Sciences Meeting & Exhibition, January 2005.

Figure 12. Spanwise lift coefficient for 50% thinned wing from root-to-tip, wing angle-of-attack –1·2395°.

Figure 13. Sectional drag polar overlaid with spanwise lift distribution for 50% thinned wing, root-to-tip.

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4. neill, D.J. and herendeen, D.L. ASTROS User’s Manual, Agency: Universal Analytics, WL-TR-96–3004, Tech rep, Universal Analytics, May 1995.

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