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Planform Load Distribution Mapping for Straight Tapered Wings with Linear Twist

Matthew Brown1, Balaji Kaushik2 and Willem A.J. Anemaat3 Design, Analysis and Research Corporation, Lawrence, KS 66049

Load distributions used for preliminary airplane structural analysis are generally based on simplistic load distributions as outlined in the Federal Aviation Regulations or based on Computational Fluid Dynamics (CFD) simulations. The simplistic load distributions are very rudimentary and are used only when data from reliable methods are unavailable. Computational Fluid Dynamics simulations are very time consuming. The model/mesh definition is very important in getting good CFD results and often lead to tedious and time consuming model discretization. The Planform Load Distribution Mapping (PLDM) method is a Quasi Panel Code developed specifically to aid load generation for structural analysis. The computation times for the PLDM method are a fraction of the computation times from CFD simulations. The model discretization times are minimal as the input is tied in with the parameter definition of the Advanced Aircraft Analysis (AAA) software.

Nomenclature

wb = wing span

c = wing local chord

ac = sectional axial force coefficient

dc = sectional drag coefficient

flc = unscaled airfoil lower surface skin friction coefficient

fl rc = root airfoil lower surface skin friction coefficient

fl tc = tip airfoil lower surface skin friction coefficient

fl wc = wing lower surface skin friction coefficient

fl wc = unscaled wing lower surface skin friction coefficient

fuc = unscaled airfoil upper surface skin friction coefficient

fu rc = root airfoil upper surface skin friction coefficient

fu tc = tip airfoil upper surface skin friction coefficient

fu wc = wing upper surface skin friction coefficient

fu wc = unscaled wing upper surface skin friction coefficient

lc = sectional lift coefficient

1 Aerospace Engineer, Design, Analysis and Research Corporation, Lawrence, KS 66049, AIAA Member. 2 Senior Aerospace Engineer, Design, Analysis and Research Corporation, Lawrence, KS 66049, AIAA Member. 3 President, Design, Analysis and Research Corporation, Lawrence, KS 66049, AIAA Associate Fellow.

50th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition09 - 12 January 2012, Nashville, Tennessee

AIAA 2012-0396

Copyright © 2012 by Design, Analysis and Research Corporation. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

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nc = airfoil normal force coefficient

plc = unscaled airfoil lower surface pressure distribution

plrc = root airfoil lower surface pressure distribution

pltc = tip airfoil lower surface pressure distribution

plwc = wing lower surface pressure distribution

plwc = unscaled wing lower surface pressure distribution

puc = unscaled airfoil upwer surface pressure distribution

purc = root airfoil upper surface pressure distribution

putc = tip airfoil upper surface pressure distribution

puwc = wing upper surface pressure distribution

puwc = unscaled wing upper surface pressure distribution

x = X-location of any point on the wing planform

ly = lower surface Y-location of any point on the wing planform

uy = upper surface Y-location of any point on the wing planform

0 = zero lift angle of attack

= local twist angle

= non-dimensional rectangular grid coordinate

= pressure scaling function

= viscosity scaling function

I. Introduction oad distributions from methods recommended in Federal Aviation Regulations often end up being very conservative and will lead to an overdesigned, heavy structure. Typical load distribution, from the FAR, on the

horizontal tail or vertical tail are as shown in Figure 1.

Figure 1. Surface Load Distribution from FAR

L

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Higher fidelity data from Comptational Fluid Dynamics (CFD) simulations can be obtained but the process is very time consuming. CFD simulations often require a very good model discretization to obtain accurate results. As a result, CFD simulations are not suited for airplane preliminary design.

The purpose of this paper is to present a method to scale the chordwise pressure distributions based on the spanwise sectional force and moment coefficients obtained from lifting line theory from Advanced Aircraft Analysis (AAA) software (ref. 1). The root and tip pressure distributions corresponding to the local angle of attack and Reynolds number are obtained from 2d vortex panel codes like XFOIL (Ref. 2) or JAVAFOIL (Ref. 3) or wind tunnel data. The spanwise distributions based on equations described in reports from NASA (Ref. 4-7) are applicable to straight tapered wings with linear twist distribution in the subsonic flow regime.

II. Planform Load Distribution Mapping To simplify calculations, the three dimensional coordinates of the lifting surface panel chord plane in free space

can be transformed into a non-dimensional rectangular grid identified by two variables ,x

c

, where is defined

as follows: 2

w

y

b (1)

Figure 1 shows a schematic of the parameter definitions.

Figure 1. Geometry Parameter Definitions

The unscaled planform pressure distribution is given in terms of the two dimensional pressure distributions for the root and tip airfoils and assumes a linear variation between root and tip. The same variation is assumed for airfoil geometry if airfoil sections differ from root to tip. The two dimensional pressure coefficients are obtained from Refs. 2, 3 or wind tunnel data. This unscaled pressure map is then scaled to match the lift distribution obtained from Refs. 1-5. Figure 2 shows an example pressure coefficient variation on upper and lower surfaces of NACA 4412 airfoil.

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Figure 2. Example Airfoil Pressure Coefficient Variation

For wings with large twist angles or a number of different airfoils the linear variation assumption may not be acceptable in which case the unscaled planform pressure distribution should take on a piecewise form. The endpoints of the piecewise equations should be determined from inspection of the airfoil pressure distributions. Equations 2, 3, 4 and 5 show the unscaled pressure and shear distribution for the upper and lower airfoil surfaces.

,p p p pu u u ut r r

x x x xc c c c

c c c c

(3)

,p p p pl l l lt r r

x x x xc c c c

c c c c

(4)

,f f f fu u u ut r r

x x x xc c c c

c c c c

(5)

,f f f fl l l lt r r

x x x xc c c c

c c c c

(6)

The spanwise normal and axial force distributions are obtained from the lift and drag distributions as shown in

Equations 7 and 8.

0 0cos sinn l dc c c (7)

0 0cos sina d lc c c (8)

dc is the spanwise drag distribution obtained from experimental data. Figure 3 shows an example

spanwise drag distribution from Ref. 9.

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Figure 3. Example Spanwise Drag Distribution

lc is the spanwise lift distribution function obtained from Refs. 1-5. An example spanwise lift distribution

is as shown in Figure 4.

Spanwise Station, %

Lift Coefficient

cl

0.0 10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0 90.0 100.0

2.00

1.50

1.00

0.50

0.00

-0.50

Additional Lift Basic Lift Total Lift Max. Airfoil Lift

CLw cln p.off

= 0.7051

clrw

= 6.0501 deg-1

cltw

= 6.0501 deg-1

c/4w

= 0.6 deg

ARw = 7.40

Sw = 175.15 ft2

aw

= -3.0 deg

w = 0.59

M1 = 0.106

Figure 4. Example Spanwise Lift Distribution

The scaling functions needed to match the unscaled sectional results to the spanwise load distributions are shown in Equations 9 and 10. These equations constitute a system of 2 equations with 2 unknowns (namely the scaling functions) which must be solved at each point along the span to yield the two local values of the scaling functions,

and .

1 1

0 0, , , ,x x x u x l

n p p f fl u u lc c c c

dy dyc c c dx c c dx

dx dx (9)

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1 1

0 0, , , ,x u x l x x

a p p f fu l u lc c c c

dy dyc c c dx c c dx

dx dx (10)

Once the values of the scaling functions are obtained at each spanwise location, the scaled spanwise pressure and

shear distributions can be obtained by applying the scaling functions to the unscaled pressure and shear distributions given by Equations 11, 12, 13 and 14.

, ,p pu uw w

x xc c

c c

(11)

, ,p pl lw w

x xc c

c c

(12)

, ,f fu uw w

x xc c

c c

(13)

, ,f fl lw w

x xc c

c c

(14)

Figure 5 shows the computed pressure distribution for several stations along the wing span at a Reynolds

Number of 1.5 million and an angle of attack of 4 degreees for a wing with a NACA 0015 airfoil at the root and tip.

Figure 5. Wing Pressure Distribution

Figure 6 shows the static pressure contours on the wing upper surface with a NACA 0015 airfoil at 4 degrees

angle of attack from the Planform Load Distribution Method.

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Figure 5. Pressure Distribution Contours on Wing Upper Surface

Figure 6. Pressure Distribution Contours on Wing Lower Surface

III. Validation A validation study was conducted comparing the results against CFD (Ref. 10) and Ref. 9. No experimental data

for shear distribution on a lifting surface has been found as of yet thus no validation of the predicted shear distribution is possible. Figure 7, 8 and 9 show pressure distribution comparisons.

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Figure 7. Lifting Surface Pressure Distribution at Re=1.5 Million and = 4 deg from NASA TP 3151 ,

CFD and Planform Force Distribution Mapping

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Figure 8. Lifting Surface Pressure Distribution at Re=1.5 Million and = 8 deg from NASA TP 3151 ,

CFD and Planform Force Distribution Mapping

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Figure 9. Lifting Surface Pressure Distribution at Re=1.5 Million and = 12 deg from NASA TP 3151 ,

CFD and Planform Force Distribution Mapping

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IV. Conclusion A method has been developed to map chordwise and spanwise load distributions. This method can be used in a

variety of applications such as wind turbine blade loading to aircraft wing pressure and shear distributions. Research is underway to extend the theory to cranked lifting surface planforms and non-planar planforms.

It is seen from the validation studies that Planform Load Distribution Mapping estimation of the wing spanwise and chordwise pressure distribution shows excellent correlation with measured wind tunnel data.

The results at 12 degrees show less correlation as this is beyond the linear range. It does, however, offer a reasonable approximation. The discrepancies shown toward the tip region are in large part due to the failure of the lift distribution method to predict effects of a blunt tip. The resulting difference in local lift distribution manifest themselves as mismatched pressure distributions.

The CFD simulation for one angle of attack for a 3 dimensional half-wing model with a NACA0015 airfoil takes approximately 15 hours with a node count of 725,000 on a Dell Precision T7500 with a dual quad core Intel Xenon processor. The Planform Load Distribution Method takes less than 15 seconds to run. This clearly demonstrates that the Planform Load Distribution Mapping provides a quick, reliable and easy way of obtaining lifting surface loads for preliminary and detailed finite element analysis of lifting surfaces. The results are valid over the linear range with decreasing accuracy in the nonlinear lift range.

Further development is under way to account for the non-linear lift range. Methods to account for control surfaces and flaps are also being developed.

References 1AAA, Advanced Aircraft Analysis, Software Package, Ver 3.3, Design, Analysis and Research Corporation, Lawrence, KS,

2011. 2XFOIL, Ver 6.96, Mark Drela and Harold Youngren, 2005. 3JAVAFOIL, Ver 2.07, Martin Hepperle, 2010. 4DeYoung, J., and Harper, C.W., “Theoretical Symmetric Span Loading at Subsonic Speeds for Wings having Arbitrary Plan

Form,” NACA TR-921, 1948. 5DeYoung, J.,“Spanwise Loading for Wings and Control Surfaces of Low Aspect Ratio,” NACA TN-2011, 1950. 6Blackwell, Jr., J., “A Finite-Step Method for Calculation of Theoretical Load Distributions for Arbitrary Lifting-Surface

Arrangements at Subsonic Speeds,” NASA TN D-5335, 1969. 7DeYoung, J., “Nonplanar Wing Load-Line and Slender Wing Theory,” NASA CR-2864, 1977. 8Anderson, J.D., Fundamentals of Aerodynamics, 3rd edition, McGraw-Hill, New York, NY, 2001. 9McAlister, K.W., and Takahashi, R.K.,“NACA 0015 Wing Pressure and Trailing Vortex Measurements,” NASA TP-3151,

1991. 10CFDesign, Software Package, Ver. 2010, Blue Ridge Numerics, 2010.