Background

This spreadsheet uses lifting line theory to calculate the optimum planform shape and twist to minimize induced drag and maximize lift. It can also analyze the lift and drag of an optimum shape or any other planform.

Lifting line theory is the simplest way to estimate the lift and drag of a three-dimensional wing or sail. This is the same theory that says an elliptical wing is the optimum choice for a single planar planform away from the ground.

Any lifting surface deflects the fluid flowing by it, leaving a wake which trails behind. This wake in turn has an effect on the surface, inducing a cross flow which modifies the local angle of attack (the angle between the surface and the free stream.

In the case of an aircraft in level flight this "downwash" causes the aircraft to behave as though it were flying through a downdraft. In the case of a sail, it is as though the sailboat had experienced an unfavorable wind shift - a "header".

The lift at a given station acts perpendicular to the local flow angle, and this change in the local angle of attack causes the lift vector to tilt backward, resulting in a force component that is parallel to the free stream wind.

This acts to slow the vehicle down, and is called, "lift induced drag," or simply, "induced drag." This induced drag varies with the square of the lift, and is in addition to the drag caused by the influence of the boundary layer.

Lifing line theory assumes that the wake trails straight back from the trailing edge. The calculations are done at a fictious plane far downstream, and the results are related to back to the location of the lifting surface.

Many significant effects are not included in this theory. For example, real wakes roll up into two large trailing vortices, and this wake rollup is not modeled. The freestream wind is also considered to be uniform in this version.

Lifting line theory is not capable of handling planforms with sweep or rake. For such planforms, the lift distribution along the span will be the same, but the planform shape and twist distribution required to achieve it will be different.

However, lifting line theory has been found to give results which match experimental results to an accuracy which is sufficient for many practical engineering applications. For example, most aircraft designed before the Korean War were based on this theo

The current version of this spreadsheet is also restricted to a single wing or sail surface, although a sloop rig (main and jib) could be considered to be one surface and the jib/main interaction handled as a slotted airfoil problem.

In this case, the twist distribution will have to be changed or reinterpreted in the light of the combined geometry.

Lifting line theory is a linear theory. Nonlinear effects, such as stall, are not modeled. This limits its applicablility to low to moderate angles of attack, say 10 degrees or less.

Surface Effects

When a wing or sail is operating in proximity to a surface, such as the ground or sea, the flowfield is influenced by the surface. For sails, the water is assumed to be a flat, solid surface like the ground.

A wing or sail near such a surface behaves as though it had a virtual twin, mirrored in the surface. This virtual image makes the surface a plane of symmetry, enforcing the condition that there can be no flow through the surface.

For hydrofoils, such as keels, rudders, and hydrofoil boats, the free surface of the water is more complex. At very low speeds and at high speeds, the water surface is approximately flat, and this theory is applicable.

At very low speeds, the gravity forces are large compared to the pressures exerted by the hydrofoil, and the water's surface acts like a solid surface, and the same kind of virtual image is used. This is the "zero Froude number" condition.

The high speed free surface condition acts as though the surface were located half way between the wings of a biplane, with the one wing formed by the hydrofoil, and the other a virtual wing.

The geometry of this virtual wing is the same as the virtual image used for the solid surface, but the lift is directed in the opposite direction. This enforces the condition that the pressure at the surface is constant.

This is the "infinite Froude number condition. For hydrofoils operating at approximately one chord length in depth, this is good approximation for Froude numbers greater than four.

Optimal Planforms

In 1920, Max Munk used lifting line theory to prove that the minimum induced drag for any planform is obtained when the downwash is uniform along the span, and proportional to the cosine of the dihedral angle.

Thirty years later, Robert T. Jones showed that the minimum induced drag for a given bending moment is obtained when the downwash tapers linearly along the span. For sails, this corresponds to the minimum induced drag for a given heeling moment.

So by specifying the downwash distribution, it is possible to calculate the spanwise lift distribution to which it corresponds. Once the lift distribution is known, the optimal planform shape can be calculated.

At any given station, the nondimensional lift per unit span is given by the lift coefficient, Cl, times the chord. Since the lift is established by the downwash, the optimal planform shape (chord) will depend on the twist distribution along the span.

When a uniform downwash distribution is applied to a planar wing far from the ground, the result is the elliptical lift distribution and an elliptical planform shape. However, for any other condition, the optimum is not elliptical.

In the case of sail rigs, the right height is generally restricted by the stability of the boat. A tapered downwash distribution is best for this case because it results in a greater span for the same moment.

Once the optimum planform is found for one condition, it still remains to see how it operates in other conditions. And the designer will also want to know what the effect will be of modifying the shape for practical construction.

Both these capabilities are included in this workbook. The "Design" sheet is used when the downwash distribution is to be specified and the shape computed. The "Analysis" sheet is used when the planform shape is known and the lift, drag, and downwash.

Using the Design Sheet

This woorkbook is intended to be a working tool, modified and added to as necessary. The two key sheets are named "Design" and "Analysis". Both are organized similarly, with Input, Constraints, and Results sections.

The axes convention is Z up, Y to the right. The ground or free surface, if any, is at Z=0, and the moment reference center is at the origin. The separate force and moment contributions can be used to refer the moments to any other location.

The span size and orientation is specified by giving the coordinates of one end of the span (Yfoot, Gap), and the coordinates of the other end (Yhead, Gap+Height), as shown in the cartoon.

Airfoil characteristics are given for each end, and are linearly interpolated along the span. These consist of the angle or attack at which the airfoil produces zero lift, and the slope of the lift curve.

If desired, a maximum lift coefficient can be supplied as well (also interpolated linearly along the span). This is used for reference, and any excursions above this value noted in the Constraints section.

For the planform design problem, the loading on the wing or sail is specified by the downwash distribution. For convenience, this is given as an angle, in degrees, at the head and foot, and linearly interpolated in between to give the optimal conditions.

The wind speed is only used to calculate the dynamic pressure, qbar, and thus the dimensional forces. The nondimensional coefficients are not affected by the choice of wind speed or fluid density.

Although English units are shown in the labels, any consistent set of units may be used.

The surface type is used to model ground effects or free surface effects, and should be set to 1 or -1, as indicated. A value of 0 may also be used to indicate no surface effects.

Automatic calculation has been turned off to speed up the input process. Once the input values are entered, press F9 to recalculate the results.

The twist at the head and foot have been placed in the Results section, because they are currently calculated automatically so as to produce a constant lift coefficient across the span. These formulae can be replaced if a different twist is to be used.

All of the cells except the input and twist values are locked to prevent being inadvertently changed. Protection can be removed by selecting the Tools/Protection/Unprotect menu option.

Using the Analysis Sheet

The Analysis sheet is used to analyze the characteristics of a given planform. The default values for all the inputs are taken from the Design sheet. However these formulae may be replaced with any desired values.

The span geometry input follows the same convention as for the Design sheet. This time, the twist distribution is given as an input, rather than being calculated as an Analysis sheet result.

The remaining four optional geometry variables may be used to calculate the planform shape. For example, a linearly tapered planform can be calculated from the chord lengths at the head and foot.

The airfoil characteristics are specified in the same way as on the Design sheet. The aerodynamic loading is provided by specifying the angle of attack in this case. The downwash is a result to be calcuated for this problem.

The big difference in the Analysis sheet input is that the chord distribution is an input. A straight tapered planform can be specified by replacing the formula in cell I11 with: "= $G11*$D$14 + (1-$G11)*$D$15" and filling down.

As before, once the input values are entered, press F9 to recalculate the results. These are plotted in the sheets following the Analysis sheet, and the results from the Design sheet are shown for comparison.

Input ConstraintsGeometry GeometryLuff Height (vetical, foot to head), ft Height 15 Minimum Chord, ft Min_chord #VALUE!Lateral Offset of Head, ft Yhead 0Lateral Offset of Foot, ft Yfoot 0 AirfoilGap Between Foot and Surface, ft Gap 0 Actual Max Lift Coefficient Cl_max #VALUE! Area, ft^2 Area 49 Actual Min Lift Coefficient Cl_min #VALUE!

Sum Error^2 of Cl>Cl_max E2_Clmax #VALUE!AirfoilZero Lift Angle of Attack @ Head, deg alpha0_head 0 Note: Automatic calculation has been turned off.Zero Lift Angle of Attack @ Foot, deg alpha0_foot 0 Press F9 to update results.Lift Curve Slope @ Head, /deg a_head 0.1096622711Lift Curve Slope @ Foot, /deg a_foot 0.1096622711Max Lift Coefficient @ Head Cl_max_head 1Max Lift Coefficient @ Foot Cl_max_foot 1

Wind and LoadingDownwash Angle @ Head, deg Whead 3Downwash Angle @ Foot, deg Wfoot 3Wind Speed, ft/sec V_fps 20Air Density, slug/ft^3 rho 0.002378

Surface Type 1 = Solid surface (0 Froude number) Surf_factor 1 0 = None-1 = Free surface (infinite Froude number)

Z

Y

Yfoot

GapSurface

Yhead

Height

Wfoot

DownwashDistribution

Sail or Wing

Whead

View Looking Upstream

Lift Distribution

ResultsGeometry Span DistributionsLuff Length, ft Length 15 Index Span. Dist. Chord, ft Nondim. Lift Lift CoefficientTwist @ Foot, deg twist_head 0 I, ,j Si chord Cl*c/cbar ClTwist @ Head, deg twist_foot 0 0Aspect Ratio AR 4.591836735 1 0.02312 #VALUE! #VALUE! #VALUE!Mean Chord cbar 3.266666667 2 0.2072256 #VALUE! #VALUE! #VALUE!

3 0.57090351 #VALUE! #VALUE! #VALUE!Planform 4 1.10519877 #VALUE! #VALUE! #VALUE!Angle of Attack, deg Alpha #VALUE! 5 1.79695526 #VALUE! #VALUE! #VALUE!Range of Lift Coefficients (max-min) Cl_range #VALUE! 6 2.62913964 #VALUE! #VALUE! #VALUE!Lift coefficient CL #VALUE! 7 3.58126076 #VALUE! #VALUE! #VALUE!Induced Drag Coefficient CDi #VALUE! 8 4.62987426 #VALUE! #VALUE! #VALUE!Oswald Efficiency Factor e #VALUE! 9 5.74915977 #VALUE! #VALUE! #VALUE!Lift Curve Slope a3D #VALUE! 10 6.91155678 #VALUE! #VALUE! #VALUE!Lift/Drag Ratio CL/CDi #VALUE! 11 8.08844322 #VALUE! #VALUE! #VALUE!Lateral Center of Effort, ft Yce #VALUE! 12 9.25084023 #VALUE! #VALUE! #VALUE!Vertical Center of Effort, ft Zce #VALUE! 13 10.3701257 #VALUE! #VALUE! #VALUE!

14 11.4187392 #VALUE! #VALUE! #VALUE!Forces 15 12.3708604 #VALUE! #VALUE! #VALUE!Total Lift, lb Lift #VALUE! 16 13.2030447 #VALUE! #VALUE! #VALUE!Induced Drag, lb Drag #VALUE! 17 13.8948012 #VALUE! #VALUE! #VALUE!Total Moment, ft-lb Moment #VALUE! 18 14.4290965 #VALUE! #VALUE! #VALUE!Horizontal Force, lb Fy #VALUE! 19 14.7927744 #VALUE! #VALUE! #VALUE!Vertical Force, lb Fz #VALUE! 20 14.97688 #VALUE! #VALUE! #VALUE!Moment Due to Horizontal Force, ft-lb Mxy #VALUE!Moment Due toVertical Force, ft-lb Mxz #VALUE!Dynamic Pressure, lb/ft^2 qbar 0.4756

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

2

4

6

8

10

12

Lift Distribution

Nondimensional Spanwise Distance Along Panel, eta

Nond

imen

sion

al L

ift p

er U

nit L

engt

h, C

l*c/c

bar

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

Planform Shape

Nondimensional Spanwise Distance Along Panel, eta

Nond

imen

sion

al C

hord

, c/L

engt

h

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

2

4

6

8

10

12

Local Lift Coefficient Distribution

Nondimensional Spanwise Distance Along Panel, eta

Sect

ion

Lift

Coef

ficie

nt, C

l

Single Wing Planform Analysis

This spreadsheet calculates the induced drag and lift distribution of a single wing (or sail), given the geometry of the planform and its apparent wind.along the span. The spreadsheet also calculates the angle of attack and the chord distribution which provides the calculated lift distribution.

Input ConstraintsGeometry Planform Shape AirfoilLuff Height (vetical, foot to head), ft Height_a 15 Index Span. Dist. Chord, ft Actual Max Lift CoefficientLateral Offset of Head, ft Yhead_a 0 I, ,j Eta_a Si_a chord_a Actual Min Lift CoefficientLateral Offset of Foot, ft Yfoot_a 0 0 Sum Error^2 of Cl>Cl_maxGap Between Foot and Surface, ft Gap_a 0 1 0.001541 0.02312 #VALUE!Twist @ Head, deg twist_head_a 0 2 0.013815 0.2072256 #VALUE!Twist @ Foot, deg twist_foot_a 0 3 0.03806 0.5709035 #VALUE!Optional input variable 1 Chead 1.866667 4 0.07368 1.1051988 #VALUE!Optional input variable 2 Cfoot 4.666667 5 0.119797 1.7969553 #VALUE! Note: Automatic calculation has been turned off.Optional input variable 3 6 0.175276 2.6291396 #VALUE! Press F9 to update results.Optional input variable 4 7 0.238751 3.5812608 #VALUE!

8 0.308658 4.6298743 #VALUE!Airfoil 9 0.383277 5.7491598 #VALUE!Zero Lift Angle of Attack @ Head, deg alpha0_head_a 0 10 0.46077 6.9115568 #VALUE!Zero Lift Angle of Attack @ Foot, deg alpha0_foot_a 0 11 0.53923 8.0884432 #VALUE!Lift Curve Slope @ Head, /deg a_head_a 0.109662 12 0.616723 9.2508402 #VALUE!Lift Curve Slope @ Foot, /deg a_foot_a 0.109662 13 0.691342 10.370126 #VALUE!Max Lift Coefficient @ Head Cl_max_head_a 1 14 0.761249 11.418739 #VALUE!Max Lift Coefficient @ Foot Cl_max_foot_a 1 15 0.824724 12.37086 #VALUE!

16 0.880203 13.203045 #VALUE!Wind and Loading 17 0.92632 13.894801 #VALUE!Angle of Attack, deg Alpha_a #VALUE! 18 0.96194 14.429096 #VALUE!Wind Speed, fps V_fps_a 20 19 0.986185 14.792774 #VALUE!Air Density, slug/ft^3 rho_a 0.002378 20 0.998459 14.97688 #VALUE!

Surface Type Note: Put your formulas or values in the chord 1 = Solid surface (0 Froude number) Surf_factor_a 1 distribution, using the optional input variables.

Z

Y

Yfoot

GapSurface

Yhead

Height

Wfoot

DownwashDistribution

Sail or Wing

Whead

View Looking Upstream

Lift Distribution

0 = None-1 = Free surface (infinite Froude number)

Z

Y

Yfoot

GapSurface

Yhead

Height

Wfoot

DownwashDistribution

Sail or Wing

Whead

View Looking Upstream

Lift Distribution

ResultsGeometry Span Distributions

Cl_max #VALUE! Luff Length, ft Length_a 15 Index Span. Dist. DownwashNondim. Lift Lift CoefficientCl_min #VALUE! Area, ft^2 Area_a #VALUE! I, ,j Eta_a Si Wi Cl*c/cbar ClE2_Clmax #VALUE! Aspect Ratio AR_a #VALUE! 0

Mean Chord cbar_a #VALUE! 1 0.0015 0.02312 #VALUE! #VALUE! #VALUE!2 0.0138 0.2072256 #VALUE! #VALUE! #VALUE!3 0.0381 0.5709035 #VALUE! #VALUE! #VALUE!

Planform 4 0.0737 1.1051988 #VALUE! #VALUE! #VALUE!Note: Automatic calculation has been turned off. Lift coefficient CL_a #VALUE! 5 0.1198 1.7969553 #VALUE! #VALUE! #VALUE! Press F9 to update results. Induced Drag Coefficient CDi_a #VALUE! 6 0.1753 2.6291396 #VALUE! #VALUE! #VALUE!

Oswald Efficiency Factor e_a #VALUE! 7 0.2388 3.5812608 #VALUE! #VALUE! #VALUE!Range of Lift Coefficients (max-min) Cl_range_a #VALUE! 8 0.3087 4.6298743 #VALUE! #VALUE! #VALUE!Lift Curve Slope a3D_a #VALUE! 9 0.3833 5.7491598 #VALUE! #VALUE! #VALUE!Lift/Drag Ratio CL/Cdi #VALUE! 10 0.4608 6.9115568 #VALUE! #VALUE! #VALUE!Lateral Center of Effort, ft Yce_a #VALUE! 11 0.5392 8.0884432 #VALUE! #VALUE! #VALUE!Vertical Center of Effort, ft Zce_a #VALUE! 12 0.6167 9.2508402 #VALUE! #VALUE! #VALUE!

13 0.6913 10.370126 #VALUE! #VALUE! #VALUE!Forces 14 0.7612 11.418739 #VALUE! #VALUE! #VALUE!Total Lift, lb Lift_a #VALUE! 15 0.8247 12.37086 #VALUE! #VALUE! #VALUE!Induced Drag, lb Drag_a #VALUE! 16 0.8802 13.203045 #VALUE! #VALUE! #VALUE!Total Moment, ft-lb Moment_a #VALUE! 17 0.9263 13.894801 #VALUE! #VALUE! #VALUE!Horizontal Force, lb Fy_a #VALUE! 18 0.9619 14.429096 #VALUE! #VALUE! #VALUE!Vertical Force, lb Fz_a #VALUE! 19 0.9862 14.792774 #VALUE! #VALUE! #VALUE!Moment Due to Horizontal Force, ft-lb Mxy_a #VALUE! 20 0.9985 14.97688 #VALUE! #VALUE! #VALUE!Moment Due toVertical Force, ft-lb Mxz_a #VALUE!Dynamic Pressure, lb/ft^2 qbar_a 0.4756

Z

Y

Yfoot

GapSurface

Yhead

Height

Wfoot

DownwashDistribution

Sail or Wing

Whead

View Looking Upstream

Lift Distribution

Z

Y

Yfoot

GapSurface

Yhead

Height

Wfoot

DownwashDistribution

Sail or Wing

Whead

View Looking Upstream

Lift Distribution

Matrix Sizei j Alpha_rad_a20 20 20 #VALUE!

Geometry Aero paramI, j Chord/Len Yj_a Zj_a Yi_a Zi_a dYi_a dZi_a dSi_a dA_a Cl_Maxi_a a_a

0 0 01 #VALUE! 0 0.092337 0 0.02312 0 0.092337 0.092337 #VALUE! 1 6.2831852 #VALUE! 0 0.367076 0 0.207226 0 0.274739 0.274739 #VALUE! 1 6.2831853 #VALUE! 0 0.817451 0 0.570904 0 0.450375 0.450375 #VALUE! 1 6.2831854 #VALUE! 0 1.432373 0 1.105199 0 0.614921 0.614921 #VALUE! 1 6.2831855 #VALUE! 0 2.196699 0 1.796955 0 0.764327 0.764327 #VALUE! 1 6.2831856 #VALUE! 0 3.091611 0 2.62914 0 0.894911 0.894911 #VALUE! 1 6.2831857 #VALUE! 0 4.095071 0 3.581261 0 1.003461 1.003461 #VALUE! 1 6.2831858 #VALUE! 0 5.182373 0 4.629874 0 1.087301 1.087301 #VALUE! 1 6.2831859 #VALUE! 0 6.326742 0 5.74916 0 1.144369 1.144369 #VALUE! 1 6.283185

10 #VALUE! 0 7.5 0 6.911557 0 1.173258 1.173258 #VALUE! 1 6.28318511 #VALUE! 0 8.673258 0 8.088443 0 1.173258 1.173258 #VALUE! 1 6.28318512 #VALUE! 0 9.817627 0 9.25084 0 1.144369 1.144369 #VALUE! 1 6.28318513 #VALUE! 0 10.90493 0 10.37013 0 1.087301 1.087301 #VALUE! 1 6.28318514 #VALUE! 0 11.90839 0 11.41874 0 1.003461 1.003461 #VALUE! 1 6.28318515 #VALUE! 0 12.8033 0 12.37086 0 0.894911 0.894911 #VALUE! 1 6.28318516 #VALUE! 0 13.56763 0 13.20304 0 0.764327 0.764327 #VALUE! 1 6.28318517 #VALUE! 0 14.18255 0 13.8948 0 0.614921 0.614921 #VALUE! 1 6.28318518 #VALUE! 0 14.63292 0 14.4291 0 0.450375 0.450375 #VALUE! 1 6.28318519 #VALUE! 0 14.90766 0 14.79277 0 0.274739 0.274739 #VALUE! 1 6.28318520 #VALUE! 0 15 0 14.97688 0 0.092337 0.092337 #VALUE! 1 6.28318521

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InvM_a #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!

Resultsalpha0_a twist_a Known_a gamma_a Wi_a dLift_a dDrag_a dFz/qbar dFy/qbar dMxz/qbar dMxy/qbar dCl_max_error

0 0 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!0 0 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!0 0 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!0 0 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!0 0 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!0 0 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!0 0 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!0 0 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!0 0 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!0 0 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!0 0 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!0 0 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!0 0 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!0 0 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!0 0 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!0 0 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!0 0 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!0 0 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!0 0 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!0 0 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!

#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!

#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!

0 0.2 0.4 0.6 0.8 1 1.20

0.1

0.2

0.3

0.4

0.5

Planform Shape

Nondimensional Spanwise Distance, eta

Nond

imen

sion

al C

hord

Dis

tribu

tion,

c/L

engt

h

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

2

4

6

8

10

12

Lift Distribution

AnalysisDesign

Nondimensional Spanwise Distance, eta

Nond

imen

sion

al L

ift D

istri

butio

n, C

l*c/c

bar

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

2

4

6

8

10

12

Lift Coefficient Distribution

DesignAnalysis

Nondimensional Spanwise Distance, eta

Sect

ion

Lift

Coef

ficie

nt, C

l

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

2

2.5

3

3.5

Downwash Distribution

AnalysisDesign

Nondimensional Spanwise Distance, eta

Dow

nwas

h An

gle,

w, d

eg