Aspects of Wing Design - Stanford Universityaero-comlab.stanford.edu/Papers/stanford-aa294.pdf ·...

Aspects of Wing Design

Antony Jameson

Department of Aeronautics and AstronauticsStanford University, Stanford, CA

Stanford UniversityStanford, CAMay 2, 2007

c© A. Jameson 2007Stanford University, Stanford, CA

1/100 Aspects of Wing Design

+ Outline

ã Wings

ã Economics

ã Basic Equations and Scaling Laws

ã Optimum Design via Control Theory

ã Shock Free Airfoils

ã Wings for wide-body Jets

ã P51

ã Supersonic Biz Jets



+ Albatross



+ Raptor



+ Wright Flyer



+ Spitfire



+ P51



+ Me 262



+ P 188



+ JU 132



+ JU 287



+ Horten 9



+ Lockheed F104



+ Lockheed U2



+ Lockheed SR-71



+ Dassault Mirage 3



+ Northrop F18



+ Boeing 737



+ Global Flyer



+ Global Hawk



+ X45-B



+ X47-A



+ Overall Design Process

15−30 engineers1.5 years$6−12 million

$60−120 million

6000 engineers

Weight, performancePreliminary sizingDefines Mission

$3−12 billion5 years

2.5 years

Final Design

100−300 engineers

DesignPreliminary

DesignConceptual

Figure 1: The Overall Design Process



+ Cash flow

−12 b

400 aircraft

80 b sales

Year

Economic Projection (Jumbo Jet)

Preliminary Design

9 15

(if atleast 100 orders)

Launch

Conceptual Design

−300 m

Decisions here decide

final cost and performance

Leads to performance guarantees

Detailed Design

and certification

−12

−2

−4

−6

−8

−10

4

Cash Flow

$ billion

1.5



+ Environmental Impact



+ C02 Emission of a Car

C02 generated per gallon of gasoline : 22 pounds

Suppose you drive 15000 miles a year at 25 mpg. Annual consumption : 600gallons

100 gallons produces 1 ton of C02

Annual Emission of CO2 : 6 tons



+ CO2 emission of Jumbo Jet

CO2 emission per gallon of gasoline : 22 pounds

TO weight 880000 pounds, 400 tons

Fuel Burn : 280000 pounds

CO2 emissions : 227× 280000 = 880000 pounds = TO weight, 400 tons



+ Breguet Range Formula

For a jet aircraft,

dW

dt= V

dW

dx(1)

= −sfcT= −sfc W

L/D

So,

dx = − V

sfc

L

D

dW

W

Flying at fixed V L/D,

R =V

sfc

L

Dlog

W1

W2



+ Drag

CD = CD0 +C2L

επAR

Then,D = D0 +Dv

where

D0 =1

2ρSCD0V

2

Dv =L2

1/2επρb2V 2

L/D is maximized when

d

dCL

CDCL

= −CD0

C2L

+1

επAR= 0

or

Dv = D0 =1

2D

Then,CL =

√επARCD0

andL

D=

1

2

√√√√√√√επAR

CD0



+ How fast should you fly? (1)

If L/D is independent of V (no compressibility effect), one should fly at CLwhich maximizes L/D,

Then,

W = L =1

2ρSCLV

2

or

V =

√√√√√√√2W

ρSCL=

4

επARCD0

14

√√√√√√√W

ρS

and

VL

D=VW

2D0=

1

CD0

W

ρSV=

1

CD0

επARCD0

4

14

√√√√√√√W

ρS

Thus Range is maximized by flying higher where the air density, ρ, is reduced,and hence faster.

This is limited

ã by reaching a Mach number at which drag rise begins.

ã the engines reaching their thrust limit as ρ is reduced.



+ How fast should you fly? (2)

At a fixed altitude

dR = − V

sfc

L

D

dW

W

where

D = D0 +Dv (2)

D0 = k1V2, k1 =

1

2CD0ρS

Dv =k2

V 2, k2 =

L2

1/2επρb2

At any instant L = W and one should minimize

D

V= k1V +

k2

V 3

Then,

d

dV

D

V= k1 − 3k2

V 4



which vanishes when

Dv =1

3D0 =

1

4D

Then,

CL =

√√√√√√πARCD0

3,L

D=

3CL4CD0

=

√3

4

√√√√√√√επAR

CD0

V =3k2

k1

14

= 314VmaxL/D =

12

επARCD0

14

√√√√√√√W

ρS

and

V L/D =3

4

VW

D0=

3

2CD0

W

ρSV=

1

CD0

27επARCD0

64

14

√√√√√√√W

ρS



+ Wing Weight vs. Loading : Tennekes

W

S= 0.38V 2



+ Engine Sizing

Consider the alternatives of flying at max L/D at a higher altitude with airdensity ρ∗ and drag D∗ and at max V L/D with density ρ, drag D. If V L/D iskept equal then

ρ∗

ρ=

√√√√√√16

27,D∗

D=

√√√√√√3

4

The engine size depends on t/ρ, where

T ∗

ρ∗/T

ρ=D∗

D

ρ

ρ∗=

√√√√√√3

4

√√√√√√27

64=

√√√√√√81

64



+ Effect of Drag Rise

0 0.5 1 1.5 26

8

10

12

14

16

18

20Variation of L/D vs. M

M

L/D

0 0.5 1 1.5 20

2

4

6

8

10

12

14

16

18

20Variation of M L/D vs. M

M

M L

/D



+ Effect of Increasing Span

Consider a configuration for which

CD = CD0 +C2L

επAR

D =1

2ρCD0SV

2 +L2

1/2επρb2V 2

and

maxL

D=

1

2

√√√√√√√επAR

CD0

with Dv = D0

If one doubles b2 at fixed S, the new drag is

D′ = D0 +Dv

2=

3

4D

However, the new chord is c/√

2, and the new aspect ratio is 2AR.

Thus, if one reoptimizes CL, the new lift drag ratio isL′

D′ =√

2L

Dand with L′ = L = W ,

D′ =1√2D



+ Effect of Doubling L/D on fuel Weight Wf

Denote new fuel weight by W ′f , Then,

R =V

sfc

L

Dlog

W0 +Wf

W0=

2V

sfc

L

Dlog

W0 +W ′f

W0

so,

log

1 +

W ′f

W0

=

1

2log

1 +

Wf

W0

W ′f

W0=

√√√√√√√1 +Wf

W0− 1

If,Wf

W0= α,

W ′f

Wf= β, this gives, β =

√1+α−1α

For small α, β = 1/2 and dβdα < 0, so β decreases as α increases,

α β0 1/21 .4143 1/3



+ How Far should you fly?

Consider a trip of length R. Then

R =V

sfc

L

Dlog(1 + α)

where the fuel load isWf = αW0

Suppose the trip is divided into N segments each with fuel load,

WfN = αNW0

Then,

log(1 + αN) =1

Nlog(1 + α)

αN = (1 + α)1/N − 1

and the total fuel burn in

NαN = N[(1 + α)1/N − 1

]

If α = 1, N = 2,NαN = 2(

√2− 1) = 0.818



+ Representative Drag breakdown of jumbo jet

CL = 0.5 (Wing = 0.425, Fuselage = 0.675)

CD wing vortex 105 counts (0.0120)shock 15friction 45

fuselage 50nacelles 20tail 20

10—265 (0.0265)

L/D = 18.8



+ Representative Weight breakdown of Jumbo Jet

Fraction of TO weight

Empty 45 percentPayload 10Fuel 45

Structure Weight 25 percent of TO weightincluding wing 10 percent

fuselage 7 percent



+ Attained L/D Ratios

DC3 13B717 14

B747-400 19.4B777 19.8B787 20.5

Global Flyer 37

Best SailPlanes 60

2D Airfoilwith laminar flow 500



+ Scaling Laws



+ Wing Structure Weight

As a first approximation, the wing weight is propotional to the weight of itsmain box

a

σ t

d

M = σtdc

Wper unit span = ρtc

Thus if d is doubled, t is halved and W is halved.



+ Wing Scaling with fixed AR, fixed d/c, fixed lift

At corresponding span stations,

M ′ = rM

where with skin thickness, t, stress σ,

M = dctσ,M ′ = r2cdc′σ

so,t′ = t/r

The areas areS = bc, S ′r = r2bc

so the weight scales asW ′ = rW

Also, with fixed reference area, Sref , the skin friction of the wing scales as

C ′Df

= r2CDf

For fixed weight, d′ = r2d, t′ = t/r2,

d′

c′= r

d

c



+ Wing Scaling with fixed area, fixed lift

At corresponding span stations, M ′ = rM ,

Also, with fixed skin thickness, t, stress, σ,

M = cdtσ,M ′ = c/rd′tσ

so,

d′ = r2d,d′

c′= r3d

cwhile,

AR = b/c, AR′ = r2b/c = r2AR

Thus,

d′

c′/d

c=

AR′

AR

32

Alternatively, with fixed d/c, d′ = d/r,

M ′ =cd

r2t′σ = rM ; t′ = r3t

while the area is fixed. So,W ′ = r3W



+ Effect of Chord Variation with fixed span

With skin thickness t, stress σ, fixed L, fixed bending moment, M ,

M = σtcd = σt′r2cd

Thus,

t′ =t

r2

and the weight scales as

c′t′ =ct

r

or

W ′ =W

r

So the weight decreases as the chord increases until the wing becomes so thinkthat it is subject to buckling.

The skin friction scales as

D′f = rDf



+ Fuselage Scaling

r d

∆ p p∆

d

Assuming skin thickness is sized by pressurization load, hoop stress is

σ =∆pd

2t=

∆pr0d

2t′

so t′ = rt,

Also surface area is S ′ = rS

Thus the weight scales as W ′ = r2W

while the friction drag scales as D′f = rDf



+ N-Body Aircraft

add picture here



+ Weight Tradeoffs

a

σ t

d

The bending moment M is carried largely by the upper and lower skin of thewing structure box. Thus

M = σ t d a

For a given stress σ, the required skin thickness varies inversely as the wingdepth d. Thus weight can be reduced by increasing the thickness to chord ratio.But this will increase shock drag in the transonic region.



+ Vision

Effective Simulation

Simulation−based Design



+ Aerodynamic Design Process

{PropulsionNoiseStabilityControlLoadsStructuresFabrication

Conceptual Design

CADDefinition

MeshGeneration

CFDAnalysis

Visualization

Performance Evaluation

Multi−Disciplinary Evaluation

Wind Tunnel Testing

Central Database

Detailed Final Design

Release toManufacturing

ModelFabrication

Ou

ter

Loo

pM

ajo

r D

esi

gn

Cy

cle

Inn

er

Lo

op

Figure 2: The Aerodynamic Design Process



+ Automatic Design Based on Control Theory

ã Regard the wing as a device to generate lift (with minimum drag) by controllingthe flow

ã Apply theory of optimal control of systems governed by PDEs (Lions) withboundary control (the wing shape)

ã Merge control theory and CFD

ã Find the Frechet derivative (infinite dimensional gradient) of a cost function(performance measure) with respect to the shape by solving the adjoint equationin addition to the flow equation

ã Modify the shape in the sense defined by the smoothed gradient

ã Repeat until the performance value approaches an optimum



+ Aerodynamic Shape Optimization: Gradient Calculation

For the class of aerodynamic optimization problems under consideration, thedesign space is essentially infinitely dimensional. Suppose that the performanceof a system design can be measured by a cost function I which depends on afunction F(x) that describes the shape,where under a variation of the designδF(x), the variation of the cost is δI . Now suppose that δI can be expressed tofirst order as

δI =∫ G(x)δF(x)dx

where G(x) is the gradient. Then by setting

δF(x) = −λG(x)

one obtains an improvement

δI = −λ ∫ G2(x)dx

unless G(x) = 0. Thus the vanishing of the gradient is a necessary condition fora local minimum.



+ Aerodynamic Shape Optimization: Gradient Calculation

Computing the gradient of a cost function for a complex system can be anumerically intensive task, especially if the number of design parameters is largeand the cost function is an expensive evaluation. The simplest approach tooptimization is to define the geometry through a set of design parameters, whichmay, for example, be the weights αi applied to a set of shape functions Bi(x) sothat the shape is represented as

F(x) =∑αiBi(x).

Then a cost function I is selected which might be the drag coefficient or the liftto drag ratio; I is regarded as a function of the parameters αi. The sensitivities∂I∂αi

may now be estimated by making a small variation δαi in each designparameter in turn and recalculating the flow to obtain the change in I . Then

∂I

∂αi≈ I(αi + δαi)− I(αi)

δαi.



+ Symbolic Development of the Adjoint Method

Let I be the cost (or objective) function

I = I(w,F)

where

w = flow field variables

F = grid variables

The first variation of the cost function is

δI =∂I

∂w

T

δw +∂I

∂FT

δF

The flow field equation and its first variation are

R(w,F) = 0

δR = 0 =∂R

∂w

δw +

∂R

∂F δF



+ Symbolic Development of the Adjoint Method (cont.)

Introducing a Lagrange Multiplier, ψ, and using the flow field equation asa constraint

δI =∂I

∂w

T

δw +∂I

∂FT

δF − ψT

∂R

∂w

δw +

∂R

∂F δF

=

∂I

∂w

T

− ψT∂R

∂w

δw +

∂I

∂FT

− ψT∂R

∂F

δF

By choosing ψ such that it satisfies the adjoint equation∂R

∂w

T

ψ =∂I

∂w,

we have

δI =

∂I

∂FT

− ψT∂R

∂F

δF

This reduces the gradient calculation for an arbitrarily large number of designvariables at a single design point to

+ One Flow Solution + One Adjoint Solution



+ Design using the Euler Equations

In a fixed computational domain with coordinates, ξ, the Euler equations are

J∂w

∂t+R (w) = 0 (3)

where J is the Jacobian (cell volume),

R (w) =∂

∂ξi(Sijfj) =

∂Fi∂ξi

. (4)

and Sij are the metric coefficients (face normals in a finite volume scheme) Wecan write the fluxes in terms of the scaled contravariant velocity components

Ui = Sijuj

as

Fi = Sijfj =

ρUiρUiu1 + Si1pρUiu2 + Si2pρUiu3 + Si3p

ρUiH

.

where

p = (γ − 1)ρ(E − 12u

2i ), ρH = ρE + p.




A variation in the geometry now appears as a variation δSij in the metriccoefficients. The variation in the residual is

δR =∂

∂ξi(δSijfj) +

∂

∂ξi(Sij

∂fj∂w

δw) (5)

and the variation in the cost δI is augmented as

δI − ∫

D ψTδRdξ (6)

which is integrated by parts to yield

δI − ∫

B ψTniδFidξB +

∫

D∂ψT

∂ξ(δSijfj)dξ +

∫

D∂ψT

∂ξiSij∂fj∂w

δwdξ




For simplicity, it will be assumed that the portion of the boundary thatundergoes shape modifications is restricted to the coordinate surface ξ2 = 0.Then equations for the variation of the cost function and the adjoint boundaryconditions may be simplified by incorporating the conditions

n1 = n3 = 0, n2 = 1, dBξ = dξ1dξ3,

so that only the variation δF2 needs to be considered at the wall boundary. Thecondition that there is no flow through the wall boundary at ξ2 = 0 is equivalentto

U2 = 0, so that δU2 = 0

when the boundary shape is modified. Consequently the variation of the inviscidflux at the boundary reduces to

δF2 = δp

0

S21

S22

S23

0

+ p

0

δS21

δS22

δS23

0

. (7)



+ Design using the Euler EquationsIn order to design a shape which will lead to a desired pressure distribution, pd, anatural choice is to set

I =1

2

∫

B (p− pd)2 dS

and the integral is evaluated over the actual surface area.

Then under a shape modification

δI =∫

B(p− pd)δpdS +1

2

∫

B(p− pd)2dδξ

In the computational domain the adjoint equation assumes the form

CTi

∂ψ

∂ξi= 0 (8)

where

Ci = Sij∂fj∂w

.

To cancel the dependence of the boundary integral on δp, the adjoint boundarycondition reduces to

ψjnj = p− pd (9)

where nj are the components of the surface normal

nj =S2j

|S2|.




This amounts to a transpiration boundary condition on the co-state variablescorresponding to the momentum components. Note that it imposes norestriction on the tangential component of ψ at the boundary.

We find finally that

δI = − ∫

D∂ψT

∂ξiδSijfjdD

− ∫∫

BW(δS21ψ2 + δS22ψ3 + δS23ψ4) p dξ1dξ3. (10)

Here the expression for the cost variation depends on the mesh variationsthroughout the domain which appear in the field integral. However,the true gradient for a shape variation should not depend on the way in whichthe mesh is deformed, but only on the true flow solution. The author hasshown how the field integral can be eliminated to produce a reduced gradientformula which depends only on the boundary movement.



+ The Need for a Sobolev Inner Product in the Definition of theGradient

Another key issue for successful implementation of the continuous adjointmethod is the choice of an appropriate inner product for the definition of thegradient. It turns out that there is an enormous benefit from the use of amodified Sobolev gradient, which enables the generation of a sequence ofsmooth shapes. This can be illustrated by considering the simplest case of aproblem in the calculus of variations.

Suppose that we wish to find the path y(x) which minimizes

I =b∫

aF (y, y

′)dx

with fixed end points y(a) and y(b). Under a variation δy(x),

δI =b∫

a

∂F

∂yδy +

∂F

∂y′δy

′ dx

=b∫

a

∂F

∂y− d

dx

∂F

∂y′

δydx




Thus defining the gradient as

g =∂F

∂y− d

dx

∂F

∂y′

and the inner product as

(u, v) =b∫

auvdx

we find that

δI = (g, δy).

If we now set

δy = −λg, λ > 0

we obtain a improvement

δI = −λ(g, g) ≤ 0

unless g = 0, the necessary condition for a minimum.




Note that g is a function of y, y′, y

′′,

g = g(y, y′, y

′′)

In the well known case of the Brachistrone problem, for example, which calls forthe determination of the path of quickest descent between two laterallyseparated points when a particle falls under gravity,

F (y, y′) =

√√√√√√√1 + y′2

y

and

g = − 1 + y′2 + 2yy

′′

2(y(1 + y′2)

)3/2

It can be seen that each step

yn+1 = yn − λngn

reduces the smoothness of y by two classes. Thus the computed trajectorybecomes less and less smooth, leading to instability.




In order to prevent this we can introduce a weighted Sobolev inner product

〈u, v〉 =∫(uv + εu

′v′)dx

where ε is a parameter that controls the weight of the derivatives. We nowdefine a gradient g such that

δI = 〈g, δy〉Then we have

δI =∫(gδy + εg

′δy

′)dx

=∫(g − ∂

∂xε∂g

∂x)δydx

= (g, δy)

where

g − ∂

∂xε∂g

∂x= g

and g = 0 at the end points. Thus g can be obtained from g by a smoothingequation. Now the step

yn+1 = yn − λngn

gives an improvementδI = −λn〈gn, gn〉

but yn+1 has the same smoothness as yn, resulting in a stable process.



+ Outline of the Design Process

The design procedure can finally be summarized as follows:

1. Solve the flow equations for ρ, u1, u2, u3, p.

2. Solve the adjoint equations for ψ subject to appropriate boundary conditions.

3. Evaluate G and calculate the corresponding Sobolev gradient G.

4. Project G into an allowable subspace that satisfies any geometric constraints.

5. Update the shape based on the direction of steepest descent.

6. Return to 1 until convergence is reached.

Sobolev Gradient

Gradient Calculation

Flow Solution

Adjoint Solution

Shape & Grid

Repeat the Design Cycleuntil Convergence

Modification

Figure 3: Design cycle



+ Constraints

ã Fixed CL.

ã Fixed span load distribution to present too large CL on the outboard wingwhich can lower the buffet margin.

ã Fixed wing thickness to prevent an increase in structure weight.

- Design changes can be limited to a specific spanwise range of the wing.

- Section changes can be limited to a specific chordwise range.

ã Smooth curvature variations via the use of Sobolev gradient.



+ Application of Thickness Constraint

ã Prevent shape change penetrating a specified skeleton (colored in Red).

ã Separate thickness and camber to allow free camber variations.

ã Minimal user input required.



+ Shock free airfoil design

ã The search for profiles which give shock free transonic flows was the subject ofintensive study in the 1965-70 period.

ã Morawetz’ theorem (1954) states that a shock free transonic flow is an isolatedpoint. Any small perturbation in Mach number, angle of attack, or shape causesa shock to appear in the flow.

ã Nieuwland generated shock free profiles by developing solutions in the hodographplane. The most successful method was that developed by Garabedian and hisco-workers. This used complex characteristics to develop solution in thehodograph plane, which was then mapped to the physical plane. It was hard tofind hodograph solutions which mapped to physical realizable closed profiles. Itgenerally took one or two months to produce an acceptable solution.

ã By using shape optimization to minimize the drag coefficient at a fixed lift,shock free solutions can be found in less than one minute.



+ Two dimensional studies of transonic airfoil design(cont’d)

Pressure distribution and Mach contours for the GAW airfoil

Before the redesign After the redesign



+ Two dimensional studies of transonic airfoil design

Attainable shock-free solutions for various shape optimized airfoils

0.7 0.71 0.72 0.73 0.74 0.75 0.76 0.77 0.78 0.79 0.8 0.81 0.82 0.83 0.84 0.850.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

Mach number

CL

KornDLBA−243RAE 2822J78−06−10G78−06−10G70−10−13W100W110Cast7GAWG79−06−10



+ Performance of the optimized RAE 2822 airfoils for 3 Mach numbersat CL = 0.6

Mach 0.76 Mach 0.77 Mach 0.78



+

Inverse Design :

Recovering of ONERA M6 Wingfrom its pressure distribution

A. Jameson, 2003-2004



+ NACA 0012 WING TO ONERA M6 TARGETNACA 0012 WING TO ONERA M6 TARGET Mach: 0.840 Alpha: 2.954 CL: 0.305 CD: 0.02024 CM:-0.1916 Design: 0 Residual: 0.1136E+01 Grid: 193X 33X 49

Cl: 0.285 Cd: 0.05599 Cm:-0.1137 Root Section: 6.2% Semi-Span

Cp = -2.0

Cl: 0.326 Cd: 0.01486 Cm:-0.0887 Mid Section: 49.2% Semi-Span

Cp = -2.0

Cl: 0.256 Cd:-0.00660 Cm:-0.0546 Tip Section: 92.3% Semi-Span

Cp = -2.0NACA 0012 WING TO ONERA M6 TARGET Mach: 0.840 Alpha: 3.137 CL: 0.304 CD: 0.01390 CM:-0.1841 Design: 100 Residual: 0.7604E-01 Grid: 193X 33X 49


Cp = -2.0


Cp = -2.0


Cp = -2.0

Starting wing: NACA 0012 Target wing: ONERA M6

This is a difficult problem because of the presence of the shock wave in thetarget pressure and because the profile to be recovered is symmetric while thetarget pressure is not.



+ Pressure Profiles at 48% span

Starting Cp Target Cp

The pressure distribution of the final design match the specified target,even inside the shock.



+

Planform and Aero-Structural Optimization

K. Leoviriyakit and A. Jameson, 2003-2005



+ Planform and Aero-Structural Optimization

The shape changes in the section needed to improve the transonic wing designare quite small. However, in order to obtain a true optimum design larger scalechanges such as changes in the wing planform (sweepback, span, chord, andtaper) should be considered. Because these directly affect the structure weight,a meaningful result can only be obtained by considering a cost function thattakes account of both the aerodynamic characteristics and the weight.

Consider a cost function is defined as

I = α1CD + α21

2

∫

B(p− pd)2dS + α3CW

Maximizing the range of an aircraft provides a guide to the values for α1 and α3.



+ Choice of Weighting ConstantsThe simplified Breguet range equation can be expressed as

R =V

C

L

Dlog

W1

W2

where W2 is the empty weight of the aircraft.

With fixed VC , W1, and L, the variation of R can be stated as

δR

R= −

δCDCD

+1

logW1W2

δW2

W2

= −

δCDCD

+1

logCW1CW2

δCW2

CW2

.

Therefore minimizingI = CD + αCW ,

by choosing

α =CD

CW2logCW1CW2

, (11)

corresponds to maximizing the range of the aircraft.



+ Boeing 747 Euler Planform Results: Pareto Front

Test case: Boeing 747 wing-fuselage and modified geometries at the followingflow conditions M∞ = 0.87, CL = 0.42 (fixed), multiple α3

α1.

80 85 90 95 100 105 1100.038

0.040

0.042

0.044

0.046

0.048

0.050

0.052

CD (counts)

Cw

Pareto front

baseline

optimized section with fixed planform

X = optimized section and planform

maximized range



+ Boeing 747 Euler Planform Results: Sweepback, Span, Chord, and Section

Variations to Maximize Range

Baseline geometry —Optimized geometry —

Geometry Baseline Optimized Variation(%)Sweep (deg) 42.1 38.8 - 7.8Span (ft) 212.4 226.7 + 6.7Croot (ft) 48.1 48.6 + 1.0Cmid (ft) 30.6 30.8 + 0.7Ctip (ft) 10.78 10.75 + 0.3troot (in) 58.2 62.4 + 7.2tmid (in) 23.7 23.8 + 0.4ttip (in) 12.98 12.8 - 0.8

SYMBOL

SOURCE SYN88 DESIGN 19SYN88 DESIGN 0

ALPHA 1.930 2.189

CD 0.00872 0.01077

COMPARISON OF CHORDWISE PRESSURE DISTRIBUTIONSBOEING 747 WING-BODY

REN = 0.00 , MACH = 0.870 , CL = 0.420

Kasidit Leoviriyakit20:48 Thu

1 Jan 04COMPPLOT

JCV 1.13COMPPLOT

JCV 1.13COMPPLOT

JCV 1.13COMPPLOT

JCV 1.13COMPPLOT

JCV 1.13COMPPLOT

JCV 1.13COMPPLOT

JCV 1.13COMPPLOT

JCV 1.13COMPPLOT

JCV 1.13COMPPLOT

JCV 1.13

Solution 1 Upper-Surface Isobars

( Contours at 0.05 Cp )

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

Cp

X / C 12.1% Span

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

Cp

X / C 33.7% Span

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

Cp

X / C 53.7% Span

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

Cp

X / C 74.7% Span

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

Cp

X / C 95.8% Span

ã CD is reduced from 107.7 drag counts to 87.2 drag counts (19.0%).

ã CW is reduced from 0.0455 (69,970 lbs) to 0.0450 (69,201 lbs) (1.1%).



+ Super B747 V.6

ã Design a new wing for the Boeing 747

ã Strategy

- Use the Boeing 747 fuselage

- Use a new planform (from the Planform Optimization result)

- Use new airfoil section (AJ airfoils)

- Optimized for fixed lift coefficient at four Mach numbers:

.78, .85, .86, and .87



+ Super B747 at Mach .78 and 0.85: (Solid line = Redesigned Config.), (Dashed line = Initial Config.)

B747 WING-BODY Mach: 0.780 Alpha: 2.677 CL: 0.448 CD: 0.01110 CM:-0.1594 Design: 30 Residual: 0.7768E-02 Grid: 257X 65X 49


Cp = -2.0


Cp = -2.0


Cp = -2.0B747 WING-BODY Mach: 0.850 Alpha: 2.235 CL: 0.448 CD: 0.01146 CM:-0.1717 Design: 30 Residual: 0.1814E-02 Grid: 257X 65X 49


Cp = -2.0

Cl: 0.540 Cd:-0.00275 Cm:-0.2478 Mid Section: 50.5% Semi-Span

Cp = -2.0


Cp = -2.0



+ Super B747 at Mach .86 and 0.87: (Solid line = Redesigned Config.), (Dashed line = Initial Config.)

B747 WING-BODY Mach: 0.860 Alpha: 2.142 CL: 0.448 CD: 0.01156 CM:-0.1743 Design: 30 Residual: 0.1151E-02 Grid: 257X 65X 49


Cp = -2.0


Cp = -2.0


Cp = -2.0B747 WING-BODY Mach: 0.870 Alpha: 2.023 CL: 0.448 CD: 0.01182 CM:-0.1796 Design: 30 Residual: 0.7744E-03 Grid: 257X 65X 49


Cp = -2.0


Cp = -2.0


Cp = -2.0



+ Drag Rise and Wing LD of Super B747 V.6

Drag Rise Wing LD

0.78 0.8 0.82 0.84 0.86 0.88 0.9 0.9215

20

25

30

35

40

45

Mach

Win

g L

/D

Wing L/D vs. Mach at fixed CL .45

BaselineRedesign

0.78 0.8 0.82 0.84 0.86 0.88 0.9 0.92100

120

140

160

180

200

220

240

260

280

Mach

CD

(co

un

ts)

Drag rise at fixed CL .45

BaselineRedesign

- - - : Baseline

—– : Redesign



+ Comparison between Boeing 747 and Super B747 V.9

CL CD CWcounts counts

Boeing 747 .45 141.3 499(107.0 pressure, 34.3 viscous) (82,550 lbs)

Super B747 .50 138.3 475( 99.2 pressure, 39.1 viscous) (78,640 lbs)

“Lower drag and lighter wing weight at higher CL”



+ P51 Racer

ã Aircraft competing in the Reno Air Races reach speeds above 500 MPH,en-counting compressibility drag due to the appearance of shock waves.

ã Objective is to delay drag rise without altering the wing structure. Hence tryadding a bump on the wing surface.



+ Partial Redesign

ã Allow only outward movement.

ã Limited changes to front part of the chordwise range.

P51 Mach: 0.780 Alpha:-0.963 CL: 0.100 CD: 0.01336 CM:-0.0450 Design: 0 Residual: 0.3278E-03 Grid: 257X 65X 49


Cp = -2.0


Cp = -2.0

Cl: 0.048 Cd: 0.00544 Cm:-0.0663 Tip Section: 92.3% Semi-Span

Cp = -2.0P51 Mach: 0.780 Alpha:-1.093 CL: 0.100 CD: 0.01036 CM:-0.0408 Design: 50 Residual: 0.1889E-03 Grid: 257X 65X 49


Cp = -2.0


Cp = -2.0


Cp = -2.0

Before the redesign After the redesign

P51 Mach: 0.780 Alpha:-1.093 CL: 0.100 CD: 0.01036 CM:-0.0408 Design: 50 Residual: 0.1889E-03 Grid: 257X 65X 49


Cp = -2.0


Cp = -2.0


Cp = -2.0

Zoom



+ Flight at Mach One

A viable alternative for long range business jets ?



+ Flight at Mach One

It appears possible to design a wing with very low drag at Mach 1, as indicatedin the table below :

CL CDpres CDfriction CDwing

(counts) (counts) (counts)0.300 47.6 41.3 88.90.330 65.6 40.8 106.5

The data is for a wing-fuselage combination, with engines mounted on the rearfuselage simulated by bumps.

The wing has 50 degrees of sweep at the leading edge, and the thickness tochord ratio varies from 10 percent at the root to 7 percent at the tip.

To delay drag rise to Mach one requires fuselage shaping in conjunction withwing optimization.



+ X Jet : Model D, Mesh at side of body

X-JET : Model D GRID 256 X 32 X 48

K = 1



+ X Jet : Model E, Pressure Distribution on the wing at Mach 1.1



+ X Jet : Model E, Pressure Distribution on the Upper Surface



+ X Jet : Model E, Pressure Distribution on the Lower Surface



+ X Jet : Model E, Wing Section at 40% of Span



+ X Jet : Model E, Drag Rise

X-JET : Model E

CL 0.330 CD0 0.000

GRID 256X64X48

0.85 0.90 0.95 1.00 1.05 1.10 1.15

Mach

0.0E

+00

0.5E

+02

0.1E

+03

0.2E

+03

0.2E

+03

0.2E

+03

0.3E

+03

0.4E

+03

0.4E

+03

CD

(cou

nts)

0.0E

+00

0.8E

+01

0.2E

+02

0.2E

+02

0.3E

+02

0.4E

+02

0.5E

+02

0.6E

+02

0.6E

+02

L/D



+ X Jet : Model E, Estimated Range in Nautical Miles

RANGE ACCORDING TO THE BREGUET EQUATION ANTONY JAMESON

X-JET : Model E

MACH NO CL WING CD WING CL EXT CD EXT SFC C0

1.1000 0.3300 142.0000 0.0450 155.0000 0.7250 968.0000

W EMPTY W LOAD W FUEL W RESERVE R RESERVE

100000.0000 2000.0000 42500.0000 3000.0000 0.0000

V L/D LOG(W1/W2) RANGE

968.0000 12.84 0.3689 5518.0



+ Boeing Sonic Cruiser



+ Conclusions

ã An important conclusion of both the two- and the three-dimensional designstudies is that the wing sections needed to reduce shock strength or produceshock-free flow do not need to resemble the familiar flat-topped and aft-loadedsuper-critical profiles.

ã The section of almost any of the aircraft flying today, such as the Boeing 747 orMcDonnell-Douglas MD 11, can be adjusted to produce shock-free flow at achosen design point.



+ Conclusions

ã The accumulated experience of the last decade suggests that most existingaircraft which cruise at transonic speeds are amenable to a drag reduction of theorder of 3 to 5 percent, or an increase in the drag rise Mach number of at least.02.

ã These improvements can be achieved by very small shape modifications, whichare too subtle to allow their determination by trial and error methods.

ã When larger scale modifications such as planform variations or new wingsections are allowed, larger gains in the range of 5-10 percent are attainable.



+ Conclusions

ã The potential economic benefits are substantial, considering the fuel costs of theentire airline fleet.

ã Moreover, if one were to take full advantage of the increase in the lift to dragratio during the design process, a smaller aircraft could be designed to performthe same task, with consequent further cost reductions.

ã It seems inevitable that some method of this type will provide a basis foraerodynamic designs of the future.



Aspects of Wing Design - Stanford Universityaero-comlab.stanford.edu/Papers/stanford-aa294.pdf ·...

Documents

Transcript of Aspects of Wing Design - Stanford Universityaero-comlab.stanford.edu/Papers/stanford-aa294.pdf ·...