P AD-A094 775 AIR FORCE INST OF TECH WRIOHT-PATTEOSON.AFB OH SCHOO--ETC F/9 20/4*

-AD 77 AHiGHER-ORDER TRAPEZOIDAL VECTOR VORTEX PANEL FOR SUS NC PLO--ETC (U)

UNCLASSIFIED AFIT/GAEtAA/80ODIAMI

UNITED STATES AIR FORCEAIR UNIVERSITY

AIR FORCE INSTITUTE OF TECHNOLOGYWrig ht-Patterson Air Force SaseOhi.

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4 IIGIE- RDR RAPEZOIDAL VEjCTOR FVORTEX PANEL FOR SUBSONIC FLOW"

Thesis

,1AFIT/GAE/AA/8,GD-14 Ronald lU/LutherCapt USAF

Approved for public release; distribution unlimited.

AFIT/GAE/AA/80D-14

I

A HIGHER-ORDER TRAPEZOIDAL VECTOR

VORTEX PANEL FOR SUBSONIC FLOW

THESIS

Presented to the Faculty of the School of Engineering

of the Air Force Institute of Technology

Air University

In Partial Fulfillment of the

Requirements for the Degree of

Master of Science

by

Ronald E. Luther, B.S.

Capt USAF

Graduate Aeronautical Engineering

December 1980

Approved for public release; distribution unlimited

Preface

I wish to thank my advisor, Major Stephen Koob, for

his constant aid and guidance and also tolerance to my

efforts. To my family, Freda and Amanda, I can only hope

that in the future some reward for their tremendous sacrifice

will be granted.

ii

Contents

Preface..................... . .. ... . .. .. .. .... i

List of Figures ...................... v

List of Symbols. .................... vi

Abstract ....................... viii

I. Introduction.....................

Background....................1Approach.....................3

Ii. Basic Problem Development ... ............ 6

Vorticity Distribution Function. .......... 6Basic Panel Geometry. .............. 7Interpolation Functions .. ............ 7Network Assembly ................. 10Inter-Panel Continuity. ............ 12Intra-Panel Continuity. ............ 12

III. Solution Process .................. 14

Planform Edge Boundary Conditions .. ....... 14Leading, Tip and Trailing Edges .. ........ 15Root Edge. ................... 17Kinematic Flow Condition. ........... 19Intra-Panel Continuity Condition. ........ 22Matrix Formulation ................ 24Compressibility Correction. .......... 27

IV. Aerodynamic Data .................. 28

V. Computer Code. .................. 30

VI. Results and Discussion ............... 31

VII. Conclusions and Recommendations .. ........ 38

Bibliography ...................... 40

APPENDIX A: Interpolating Function Equations .. ..... 41

Page

APPENDIX B: Evaluation of Integrals. ......... 43

APPENDIX C: Computer Code ................ 46

Vita..........................83

iv

List of Figures

Figure Page

I Basic Panel Geometry ....... .............. 8

2 Panel Subregions With Assigned Nodal Values 9

3 Typical Panel Network With Nodal Unknowns . 11

4A Lift Distribution ..... ............... .. 16

4B Vortex Pattern - Straight Wing ........... ... 16

4C Vortex Pattern - Swept Wing ... .......... . 16

SA Leading Edge Root Chord Panel of Swept Wing 18

5B Boundary Conditions Applied to Leading EdgeRoot Chord Panel of Swept Wing .. ........ .. 18

Nine Panel Network With Planform BoundaryConditions Applied ..... .............. 20

7 Control Point Reflection .... ............ . 26

8A Lift Distribution Versus Span Station forVarious Control Point Locations .......... .. 32

8B Control Point Locations .... ............ . 32

9 Distribution of Local Lift Coefficient Overthe Span of a Rectangular Wing; AR=5, c=5 ° 33

10A Vortex Pattern - Single Panel Model ....... .. 34

10B Vortex Pattern - Four Panel Model ........ .. 34

11 XCP Versus Span Station for Rectangular Wing;AR=5, a=5 ° , Single Panel Model......... ... 36

12 Comparison of y Distribution Along Root Chord . 37

V

List of Symbols

a angle of attack

6(x,y),y(x,y) components of vorticity vector

6i yi nodal values of vortex vector

A difference symbol

w vorticity vector

set of planform nodal values of vorticity

[A. ] coefficient arrayiJ

A,B,C,D,E,F coefficients for 6 distribution

G,H,I,J,K,L coefficients for y distribution

u,v,w, perturbation velocities in x, y and z directions

i,j unit vectors in x and y directionTkTLk integral of specific region

R region of integration

U free stream velocity

moo free stream Mach number

c chordlength

C pressure coefficient

CL local lift coefficient

Cm local moment coefficient

k ratio of specific heats

M number of chordwise panels

N number of spanwise panels

{ } vector

[ ] matrix

vi

Subscripts and Superscripts

L,T Leading or trailing triangle

i Node number

vii

AFIT/GAE/AA/80D-14

Abstract

A higher-order trapezoidal vector vortex panel method

is developed for application to linearized subsonic potential

flow. Each panel is subdivided into two triargular subregions

on which a quadratic vorticity strength distribution is pre-

scribed for both the spanwise and chordwise components of

the vorticity vector. The vorticity strength distribution

is expressed as a function of the components of the vorticity

vector at selected nodes oil the boundary of each triangular

subregion. Nodal values on the shared boundary of the sub-

regions are made equal, assuring continuity of the vorticity

distribution function throughout the trapezoidal panel. A

lifting surface of no thickness is modeled with a network of

the trapezoidal panels. Again, nodal values on the common

panel boundaries are matched to achieve complete continuity

of the vorticity distribution throughout the lifting surface.

Aerodynamic data for several wing planforms is obtained with

the flow model. Results from this method are compared to

those from other computational and theoretical methods.

viii

A HIGHER ORDER TRAPEZOIDAL VECTOR

VORTEX PANEL FOR SUBSONIC FLOW

I. Introduction

Background

The concept of modeling the flow over a lifting surface

by replacing that surface with a distribution of vorticity

began in the early part of this century and continues to be

expanded and explored. The first quantitative results were

achieved by Prandtl in 1919 with his lifting line theory

(Ref 9: 112-123) in which the entire surface was represented

by a single bound vortex and two infinitely long, free vortices.

Despite the many simplifying hypotheses involved in Prandtl's

theory, results obtained with it are sufficiently accurate

for many purposes and the theory provided the foundation for

many subsequent analytical analyses of the lifting surface

problem. In 1925 Blenk (Ref 1) extended Prandtl's idea by

representing the surface with not one but a distribution of

bound vortices over the surface. Blenk's method was an

improvement over the single lifting line approach, but still

had limitations and the computations involved were very

lengthy considering the absence of electronic computers at

that time.

A more recent development has been the vortex-lattice

theory (Ref 6). This method covers the surface with a grid

1

of horseshoe vortices and has produced very useful results.

Yet another approach, and the one pursued in this report, is

to subdivide the surface into a network of panels, each with a

discrete vorticity distribution. Such an approach is termed

a vortex paneling method.

An accurate flow model via the vortex panel technique

is achieved as follows. The vorticity distribution is found

such that, at as many points as possible on the surface, the

normal component of velocity is zero. This is the so-called

kinematic flow condition (Ref 9: 126). With the surface

vorticity distribution known, the perturbation velocity at

any point on the surface is easily determined. The velocity

field and Bernoulli's theorem are then used to compute the

pressure distribution and subsequently the aerodynamic lift

and moment coefficients.

Ideally, the vorticity vector distribution used to

model the lifting surface should be one that resembles the

observed physical distribution of vorticity on a finite wing.

That is, the vorticity vectors at the wing root lie predom-

inantly in the spanwise direction while those in the region

of the wing tip lie predominantly in the chordwise direction.

Thus the chosen vorticity distribution function must permit

the vorticity to vary in direction. Such a scheme has been

proposed by Sparks (Ref 10). His results were flawed, however,

until a computer program logic fault was detected by this

author. Subsequent results have been encouraging. Sparks'

vorcicity distribution has two undesirable features. First,

2

continuity in vorticity distribution is not enforced through-

out the surface (Ref 10: 9). This lack of continuity across

panel boundaries permits the vorticity distribution at the

boundaries to violate Helmholtz's second theorem regarding

the continuity of vorticity (Ref 8: 168). Second, the vorti-

city distribution is linear. It has been shown (Ref 4)

that increasing the number of terms in the polynomial expres-

sion representing the vorticity distribution has several

advantages. The higher order representation reduces surface

veloc. y errors and gives significantly improved accuracy as

the number of panels used to model the surface is increased.

The present report develops a vortex panel having a

quadratic vorticity distribution. The panel is derived spe-

cifically to provide a continuous vorticity distribution over

the surface while identically satisfying the second Helmholtz

condition at every point on the surface.

Approach

The basic problem is developed in Section II. First,

the quadratic vorticity distribution function and the

importance of the properties of the function are discussed.

The panel geometry is presented and unknown values of the

components of the vorticity vector are assigned to specified

nodes on the panel boundary. This permits the vorticity

distribution function to be expressed in terms of the unknown

nodal values and the panel geometry. The method of joining

panels to form a network to model a wing planform is then

3

~ j

described. This global network has a certain number of nodes

and, consequently, a set number of global unknown nodal values

to be determined.

Section III describes the solution process. The first

step is the application of specific boundary conditions to

the leading, tip, trailing and root edges of the planform.

This process reduces the total number of unknown nodal values

which must be explicitly solved for. The solution is obtained

by generating an equal number of linear equations to be solved

simultaneously.

Two conditions are satisfied on each panel to generate

the required equations. First, the kinematic flow condition

is enforced at two control points on each panel. To accomplish

this, the normal velocity at each control point must be found.

As the normal velocity at any point is affected by the vor-

ticity distribution on the entire planform, each panel's

individual contribution to the velocity at any point must be

determined. The Biot-Savart Law is applied to the vorticity

distribution function of each panel to calculate the induced

velocity caused by that panel's vorticity on any desired

point in the plane of the panel. The velocities induced by

each panel on the desired control point are then summed.

This summation yields one equation representing the total

induced velocity at one control point. The second condition

to be satisfied involves intra-panel continuity of the vor-

ticity distribution. As will be explained in Section I,

inter-panel continuity is achieved by the commonality of

4

nodal values at shared panel boundaries. Such commonality

will be shown to not exist, however, on the intra-panel

boundary, resulting in a discontinuity along that boundary.

This problem is resolved by forcing commonality of suffici-

ent nodal values along the boundary to ensure continuity.

Satisfaction of the kinematic flow condition and intra-panel

continuity condition results in a system of linear equations.

The solution of these gives the nodal values for the vorticity

distribution.

Section IV describes how the nodal values are used to

calculate the components of the vorticity vector at any point

on the planform. From the fully described vorticity distri-

bution, the velocity distribution on the surface of the plan-

form is readily determined and, subsequently, the pressure

distribution and aerodynamic coefficients can be computed.

The computer code used to evaluate the theory is out-

lined in Section V. Section VI presents the results achievcd.

Comparisons are made to other theories for rectangular wings.

Section VII draws conclusions from the results of this

method and offers recommendations for further improvement.

5

II. Basic Problem Development

Vorticity DistributionFunction

The vorticity distribution is a vector function in the

x-y plane:

U7(x,y) = 6(x,y)i + y(x,y)j (2.1)

Each component of the vorticity vector is allowed to vary

quadratically throughout the plane, thus expressions for the

components are:

6(x,y) = A + Bx + Cy + Dxy + Fy 2 + Fx 2 (2.2)

y(x,y) = G + Hx + Iy + Jxy + Ky2 + Lx 2 (2.3)

It has been shown (Ref 8: 168) that vorticity is solenoidal

and so:

V * = 0 (2.4)

This statement of continuity is known as Helmholtz's second

theorem (Ref 8: 168) and also as the condition of source-free

vortex distribution (Ref 9: 124).

Equation (2.4) can be satisfied by replacing Eq (2.2)

by

6(xy) = A - Ix + Cy - 2Kxy + Ey2 jX2 (2.5)

6

Basic Panel Geometry

Figure 1 shows the geometry of the basic trapezoidal

panel. This panel is subdivided into a leading triangle and

a trailing triangle (hereafter denoted by the subscripts L

and T respectively). The basic panel has nine nodes labeled

as shown. Two features of the nodal coordinates should be

noted. The x-coordinates of nodes 5-9 can be expressed in1

terms of the x-coordinates of nodes 1-4 (e.g. x5 = l(Xl+X

Only two y-coordinates need be specified in advance since

=1Y3" = (Yl+y2) .

Interpolation Functions

It will be shown later that it is advantageous to repre-

sent the expressions for 6 and y as functions of nodal values

and nodal coordinates. To accomplish this, each triangular

subregion of the panel is treated separately. Figure 2 shows

the panel split into its subregions with each subregion assigned

nine nodal values. Since Eqs (2.5) and (2.3) involve nine

coefficients, only nine nodal values are required to uniquely

define 6 and y within the subregion. Substituting the nine

nodal values of the leading triangle into Eqs (2.3) and (2.5)

gives

{IF } = [f.(xj,yj)]{Fi} (2.6)L 1 L

where

{rj= t 1 63 66 67 1 3 4 Y6 Y7 1T(27L L L L L {L L L L 1 (2.7)

{FL} = [AL CL EL GL tL IL JL KL L]T (2.3)

7

yi y 3 y2

((x 3 'p2'

xx Y

Fig) 1. BscPnlGoer

L83

y 7

7 ~ y 3

67Y

y -Y a

x

6

Fig 2. Panel Subregions With Assigned

Nodal Values

Similarly for the trailing triangle:

{rJ}= [gi(x],yj]{F T } (2.9)

T 1j yj T

where

{r [1 62 6 64 1 2 8T (2.10)T T 65TT 68125, T

F i ( iKLT (.1T (AT CT ET GT HT 'T aT KT LTI (2.11)

Eqs (2.6) and (2.9) are solved for in the FL and FT, respec-

tively. The method used for obtaining the solutions is

explained in Appendix A.

The vorticity vector components can now be written in

the following form:

6LT(x,y) = fT (xy; xj ,yj)]rJ (2.12)LTL Tj j L ,T

YL,T(x,y) = [gLT(xy; xj,yj)]{F , T (2.13)

In Eqs (2.12) and (2.13) the functions f, and g , are the11, T ar the

interpolating functions for the components 6L,T and 1LT'

Network Assembly

Figure 3 shows how the panels are connected to model

the semi-span of a wing planform. While the planform showni

has straight leading and trailing edges, the methd also

permits analysis of cranked leading and/or trailiig edges.

The root edge, tip edge and all chordwise boundaries are

parallel to the x-axis.

10

'0 0

U) 0.0

Ct . H

0 c0

0 r0

:30

00

u -%

0)

4-4

-- 4

Lt')

-4

mL

U

r-q 11

Inter-Panel Continuity

Continuity of the vorticity function is assured across

inter-panel boundaries for the following reason. The line

forming a common boundary between two panels is described by

y = mx + b (2.14)

Let A and B be the two panels whose common boundary is des-

cribed by Eq (2.14) and examine the value of y along the

boundary for each panel. The value of y for each panel along

the boundary is given by substituting Eq (2.14) into Eq (2.13):

YAJBoundary C 1 + C 2x + C3 x2 (2.15)

YBIBoundary C 1 + C 2x +C 3x 2 (2.16)

The three constants in either Eq (2.15) or Eq (2.16) are

determined if three values of y are specified on the boundary.

If the same three values and locations are specified for both

Eqs (2.15) and (2.16), then the constants must be identical

and y is continuous along the boundary. The argument can be

repeated for the 6 function showing it too to be continuous

along the boundary.

Intra-Panel Continuity

The same conditions that assure continuity between

panels are applied to the shared boundary of the two triangu-

lar subregions of the basic panel. Figure 2 shows that

continuity is not assured along this boundary since the only

nodal values common to both subregions are the 5 and y

12

components at node 1. Two more common values for both com-

ponents are required to establish continuity. This can be

achieved by enforcing the following conditions. The 6 com-

ponent value at node 4 of the leading triangle is made equal

to 64. The y component value at node 4 of the trailing

4triangle is made equal to y . Finally, the 6 and y component

values at node 9 of the leading triangle are made equal to

the 6 and y component values at the same node of the trailing

triangle. The enforcement of the above conditions assures

three common values and locations of both vorticity vector

components along the boundary and guarantees intra-panel

continuity. The method of incorporating these conditions

into the solution process is explained in Section III.

13

III. Solution Process

Figure 3 shows the problem at hand. A quadratic vor-

ticity vector distribution has been prescribed on the surface

of a wing planform of no thickness. The vorticity distribu-

tion has been discretized by representing the wing as a

network of vortex panels without sacrificing continuity of

vorticity anywhere on the wing. By expressing the vorticity

distribution in terms of interpolating functions and nodal

values, the value of the vorticity vector is determined any-

where on the wing once the nodal values are obtained. This

section details the method of solving for the nodal values.

Planform Edge Boundary

Conditions

As stated in the introduction, the vorticity distri-

bution has certain observed physical characteristics. These

characteristics can be assigned to the model being developed

and will serve to simplify the solution process by reducing

the number of nodal values that must be determined in order

to completely specify the wing's vorticity distribution.

The method used by Cohen (Ref 2) to develop vortex

patterns on elliptic wings both with and without sweep is

the basis for the boundary conditions that will be imposed

here. These boundary conditions are not unique to Cohen's

work, with Kuchemann (Ref S: 140) suggesting similar vortex

patterns.

14

Leading, Tip and Trailing

Figures 4A and 4B illustrate Cohen's straightforward

method for deriving the vortex pattern of a tapered wing in

straight flight. Figure 4A shows an arbitrary distribution

of lift assumed for the wing. Cohen shows the relationship

that exists between the pressure distribution and the vortex

pattern which yields Fig 4B. The contour lines in Fig 4B

correspond to the vorticity pattern of a continuous vortex

sheet. Based on these results, the boundary conditions for

the leading, tip and trailing edges are formulated.

On the leading edge the vorticity vector is tangent

to the leading edge. The slope of the leading edge is given

by Ay/Ax, so the vorticity tangency requirement necessitates

the 6 and y components at any point on the leading edge be

related by the following

yXA

5 Ax

or alternatively

= (= ( )y (3.1)

In the solution process, therefore, the 6 nodal values on

the leading edge can be expressed as functions of the y com-

ponents and, consequently, need not be solved for simultaneously.

At the trailing edge, the Kutta condition that no pres-

sure difference exists at the trailing edge requires that the

Y nodal values on the trailing edge are identically zero.

15

Fig 4A. Lift Distribution (Ref 2)

Fig 4B. Vortex Pattern Straight Wing (Ref 2)

Fig 4C. Vortex Pattern - Swept Wing (Ref 2)

16

This is because the lift at any point on the surface is pro-

portional to the cross-stream, or y, component of the vorticity

(Ref 2: 544), so requiring no load implies that y be zero.

Similarly, since the pressure differences between the upper

and lower wing surfaces decrease to zero toward the wing

tips, the y component of vorticity must also decrease to

zero at the tip. This results in the y nodal values being

zero along the tip edge.

Root Edge

Treatment of the root edge is not as straightforward

as the leading, tip, and trailing edges. This is especially

true for swept wings with pointed apices. Figure 4C shows

Cohen's results for the vortex pattern on a sweptback wing.

The vortex lines cross the root chord without a discontinuity

in slope. It would, therefore, indicate that a boundary con-

dition requiring the 6 component of the vorticity vector to

be zero along the root chord would be desirable. A conflict

develops near the apex, however, where the leading edge bound-

ary condition required that the 6 component be non-zero to assure

tangency at the leading edge. This dilemma is dealt with as

follows. Figure 5A shows the leading edge, root chord, panel

or a network modeling a swept wing (node 1 is the apex of the

wing). The boundary condition of leading edge tangency is

enforced at the apex. At node 2, however, the 6 component

is made equal to zero, permitting the vorticity vector to

cross the root chord without discontinuity in slope. At

17

4, y

61

x

Fig 5A. Leading Edge Root Chord

Panel of Swept Wing

y

x

Fig SA. Boundary Conditions Applied to

Leading Edge Root Chord Panel of Swept Wing

18

node S, the 6 component is peiiritted to be finite, but is

restricted such that the slope, y/6, of the vorticity vector

at node 5 is twice that of the vorticity vector at node 1.

Figure 5B shows the nature of the vorticity vector, -, on the

leading, root chord panel. Because the 6 component at node

5 is proportional to the y component at that node, it does

not have to be solved for explicitly. Any other nodes on

the root chord due to other panels are treated similarly to

node 2.

Applying all edge boundary conditions to the planform

of Fig 3 reduces the number of nodal values which must be

found to determine the vorticity at any point on the planform.

Figure 6 shows the same planform with only the unspecified

nodal values numbered. The number of nodal values which

must be determined for any network arrangement is 6MIN, where

M is the number of chordwise panels and N is the number of

spanwise panels.

Kinematic Flow Condition

Ideally, the vortex distribution on the lifting surface

would result in it being a stream surface such that the normal

component of velocity were zero everywhere on that surface.

The paneling method as developed here permits the enforcement

of only a finite number of boundary conditions so the kine-

matic flow condition can be satisfied only at certain control

points on the surface. In words, the kinematic flow condition

states that when the induced velocity at a point on the

19

A4)

r- 00

*00

r- 40--x 00 C

00 C-C>el

0

-4C:

00

-4-

C14C

200

surface, w(x,y), caused by a vortex distribution, w(x,y),

on the surface is added to the normal component of velocity

at the point caused by a free stream velocity U incident to

the surface at some angle of attack, a, the resultant velocity

is zero. Stated mathematically, the kinematic flow condition

isU sin a + w(x,y) = 0 (3.2)

The Biot-Savart Law serves to uniquely define the induced

velocity coexistent with a given vorticity field (Ref 8: 170).

Sparks (Ref 10: 10) has used the Biot-Savart Law and shown

that the normal velocity component induced at the origin of

an x-y plane, when that plane has a vorticity distribution

of the form (2.1), is given by

w(o,o) = I FR1(xy - Y6 ) / ( x 2 ) 3 / 2 d R (3.3)

Substituting Eqs (2.5) and (2.3) for 6 and y in (3.3) yields:

w - fRf(Gx + Hx2 + 2Ixy - Ay - Cy 2 + 3Kxy 2

41T

Ey 3 +Lx 3 + Jx 2y)/(x2 + y2 )3/2dR (3.4)

The induced velocity caused by one quadrilateral panel is

the sum of the velocities induced by each of its two triangu-

lar subregions. The coefficients in Eq (3.4) are constants

over a subregion, they being functions of the nodal values

and panel geometry. Eq (3.4) is rewritten as:

21

w (GLT I + 1 "' H T ±L LL + Lb 3_ A T4 C T5 + 3K T 6

-ET 7 + L T8 + 3 JT9 + GT1 + ItT2 + 21IT3 (3.5)L L LTL f LTL 'T T ITTT T T

A AT 4 _ C T 3K T6 E T7 LTT 8 + T3)T9TT T'FT T T T T+ TT"

where

TLk T f RxiyJ/(x2 + y233/2dR (3.6)

and

k= 1 2 3 4 5 6 7 8 9

i= 1 2 1 0 0 1 0 3 2

j= 0 0 1 1 2 2 3 0 1

The evaluation of the first five integrals given by Eq (3.6)

is found in Sparks (Ref 10: 47-53). The last four were evalu-

ated in a similar manner; the results are given in Appendix B.

With the integrals evaluated in terms of the panel geometry,

Eq (3.5) is expressed in the form:

w (fi 6 + gi') (3.7)i=l1 1

where the fi and gi are expressions involving only the panel

geometry.

Eq (3.7) gives the normal velocity induced at the

origin of the x-y plane by a vorticity vector distribution

over a trapezoidal panel in the plane.

Intra-Panel Continuity

Condition

Intra-panel continuity is achieved by enforcing the

22

four conditions set forth in Section II. The first of the-.e

is

6 6 L(x4,y2) (3.8)

Substituting Eq (2.12) for the right-hand side of Eq (3.8)

gives an expression in terms of only nodal values and panel

geometry. The second condition involves matching the I values

at node 4 and yields:

Y4 = YT(x 4,y2) (3.9)

The right-hand side of Eq (3.9) is replaced by Eq (2.13),

giving an expression in terms of nodal values and panel

geometry.

The final two conditions match the 6 and y values

at node 9:

6T(x 9 ,y3) 6L (x9 y3 ) (3.10)

YT(x9,Y3) YL(x9,y3) (3.11)

Eqs (2.12) and (2.13) are substituted into the left- and

right-hand sides, respectively, of Eqs (3.10) and (3.11).

Eqs (3.8) through (3.11) involve only nodal values and

panel geometry, and satisfying these four expressions assures

intra-panel continuity of the vorticity distribution function.

Eq (3.9) cannot be applied to the trailing edge root

chord panel (panel 3 in Fig 6) under all conditions. If

the trailing edge of this panel is parallel to the y-axis,

Eq (3.9) is ,;rivial. The reason for this can be seen by

examining Figs 2 and 6. Suppose the panel in Fig 2 is the

23

trailing root chord panel. Then, by the planform edge

boundary conditions, the following are all identically zero:

61, 65 62, Y , Y , and y4 . The value of y on the trailing

edge is

YT = C1 + C2y + C2Y2 (3.12)

since x is constant on the edge. Two values of Y, (Y2 and

Y8 ), are specified. Helmholtz's second theorem, Eq (2.4),

says that:

af + 3 o = 0 (3.13)aNx ay

1ic I 2 an 5 96Since 6, 6 and 5 are all zero, - at node 2 is zero,

therefore 2- must also be zero at that node. Eq (3.12) is

completely specified if the three conditions

Y2 = Y8 = -y = 0 (3.14)Y

are dictated. The solution is that y is zero along the line.

Consequelitly, trying to apply Eq (3.9), recalling that y is

also zero, is redundant. This problem is avoided by always

insuring that the trailing edge of this panel (root-trailing

edge) has a non-zero slope so that the edge is never parallel

to the y-axis.

Matrix Formulation

Eq (3.7) is the induced velocity at a point caused by

one panel. The total induced velocity at a point is found

by summing the effects of all panels in the planform. The

effect of panels on the planform to the left of the x-axis

24

is accounted for by reflecting the control point about the

x-axis and applying Eq (3.7) to that point. Figure 7 illus-

trates this procedure. The induced velocity at point A

caused by panels 1 and 2 is desired. The effect of panel 1

is found by straightforward application of Eq (3.7). From

symmetry, the induced velocity at point A caused by panel 2

is identical to the effect of panel 1 on point A'. So, the

induced velocity at point A due to panels 1 and 2 is the sum

of the effect of panel 1 on A and the effect of panel 1 on

A'. For a selected control point, the induced velocity is

given by:

6MNZ A. . = w..1)

j=l iJ (.)

In Eq (3.15) the coefficients A.. are functions of panel1J

geometry only, and the 4j are the unknown nodal values of

the vorticity vector distribution. The right-hand side of

Eq (3.12) is the normal component of velocity such that

Eq (3.2) is satisfied:

w. = -UO sin a (3.16)

Two control points are selected per panel so Eq (3.15) gives

2MN equations for the 6MN unknown nodal values.

The remaining 4MN equations come from the four intra-

panel continuity conditions being applied to each panel.

The resulting system of equations has the form:

25

y

Fig 7. Control Point Reflection

26

[Ai] 6M = sin 2N (3.17)

The solution to this system is:

P [A..] 1 (3.18)

Compressibility Correction

The Prandtl-Glauert Rule is used to account for the

effect of compressibility as the Mach number is increased

(Ref 8: 276). In practice, the adjustment for compressibility

is made by multiplying all x-coordinates in the A matrix of

Eq (3.18) by the factor

= / 1 (3.19)

27

IV. Aerodynamic Data

The Cj, as determined from Eq (3.15), are used in

Eqs (2.12) and (2.13) to compute the components of the vor-

ticity vector at any point on the planform. The perturbation

velocities on the surface due to the vorticity distribution

are:

i 2Y 1 (4.1)

v 16j (4.2)

where the upper and lower signs correspond to the upper and

lower sides of the surface (Ref 9: 124).

Following Sparks, the perturbation velocities are

used to determine the pressure coefficient by one of two

methods. The exact isentropic expression is (Refl: 433):

Cp = 2[l+([k-llM2/2) (1- [(tJ *-u)

+V 2 ]/U 2 ) k / ( k - ).I]/K 2 (4.3)

while the second order approximation is (Refll: 433):

Cp = -2[2u/U.+(l-N 2 )u 2 /U 2 + v 2 / U1 ] (4.4)

Let the difference in the pressure coefficients of the lower

and upper surfaces be defined by:

AC = C - C (4.5)

28

The local coefficients of lift and moment are then given by

(Ref 9: 30):

C C AC dx (4.6)L Co p

and

CM - fo (AC )x dx (4.7)

29

V. Computer Code

The theory set forth in Sections II, III and IV has

been incorporated into FORTRAN code. The program, WING2,

takes its basic structure from the program WING, developed

in Ref 10. WING2's mainline and two of its subroutines,

MESH and LOADS2, are the result of only minor modifications

to their counterparts in WING. Each subroutine is briefly

described below and a complete listing of WING2 is in Appen-

dix C.

WING2: Reads data describing the planform to be

analyzed. Calls all subroutines.

MESH: Computes the x-y coordinates of the nodes and

control points for the input planform.

BIOT: Generates the 2MN equations which result from

enforcing the kinematic flow condition at each control point.

INT: Evaluates the nine integrals given by Eq (3.6).

STRIP: Is called by INT and contains expressions for

evaluating the integrals.

CONT: Generated 4MN equations which result from

assuring intra-panel continuity.

COEFF: Evaluates the coefficients required to define

the interpolating functions.

LOADS2: Uses the nodal values of the vorticity dis-

tribution to determine the pressure distribution and aero-

dynamic coefficients.

30

VI. Results and Discussion

The theory was evaluated by obtaining aerodynamic

data for a rectangular wing of aspect ratio 5 at an angle of

attack of 50* Two panel networks were used. First, the

wing semispan was modeled with one panel. Second, four

panels of equal size were used.

Figure 8A shows the distribution of the local lift

coefficient, CL, over the span using the one panel model

and varying the control point location. Figure 8B shows the

control point set locations on the semispan. The results

shown for these locations illustrate the sensitivity of the

solution to the control point placement. Analysis of results

for the four control point locations shown and others indi-

cated that control point set A was the most desirable.

Figure 9 shows the CL distribution for control point

set A compared to the distribution predicted by Truckenbrodt's

lifting surface theory (Ref 9: 164). Although WING2 consis-

tently underpredicted CL values, the distribution is smooth

and exhibits the desired rate of decay from a maximum at the

root to zero at the tip.

Figure 1OA illustrates the value of the vorticity

vector at various points on the semispan, again for the single

panel model. Also plotted on the figure are the two control

point locations. The nature of the vortex pattern agrees

31

AA

.C3-

.232

o Truckenbrodt's Lifting SurfaceTheory

3 WING2, Single Panel Model

.5

.4

u .3

,.-4

44

0U

'J

.8.2

014

U

0.1 .2 .3 .4 .5 .6 .7 .8 .9 1.0

y/s

Fig 9. Lift Distribution Versus Span Station

for Rectangular Wing; AR=5, a=S °

33

j

.2Sy Control Point

.2-6

+

Fig 10A. Vortex Pattern Single

Panel Model

.2Y + Control Point

.2- _ ____ _ _

6 4.+

-4 4

+

+ ."t.

Fig 108. Vortex Panel -Four Panel Model

34

with the desired pattern, Fig 4B.

Figure 11 plots the center of pressure versus span

station. The X shifts slightly forward towards the tip

instead of slightly rearward as desired (Ref 9: 159).

The desirable features exhibited by the one panel

model all but vanish when four panels are used. Figure 10B

graphically shows the altered nature of the vortex pattern.

The prevalence of negative values of both 6 and y is not

physically reasonable for this low a case. A further illus-

tration of the departure of the four panel solution is shown

in Fig 12. Here the chordwise distribution of Y is given

along the root chord. Three distributions are shown; the

one panel model, four panel model, and the exact answer for

two-dimensional flow about a flat plate (Ref 5: 62):

XL -xY(x) = 2a(x-x ) (6.1)

where x L and xT are the values of the leading and trailing

edges respectively. For the single panel model, WING2 is

more correct in the midchord region than in either the lead-

ing or trailing edge regions.

35

.4

0

.3 0 0

0~00 00

;2

.1

0 I I I I I I I I II

.1 .2 .3 .4 .5 .6 .7 .8 .9 1.0

y/S

Fig 11. XCp Versus Span Station for

Rectangular Ving; AR=S, t=5 °

36

1. (

Exact 2-D Solution

- - WING2, Single Panel

.8 - - WING2, Four Panels

.7-

.6-

.4-

.3-

.1 2 31 . 5 6 7 -- 8 . .

-. 4

Fig 12. Comparison of y Distribution

Along Root Chort

37

VII. Conclusions and Recommendations

The linear discontinuous vorticity vector method of

Sparks (Ref 10) has been extended to a quadratic, continuous

vector vortex panel method. The desirable features of the

higher-order, continuous vorticity distribution are apparent

from the results of the single panel model. The spanwise and

chordwise lift distributions, while consistenly low in value,

do exhibit the characteristics of the desired solutions.

The method breaks down, however, if the planform is modeled

with four panels.

No reason is readily apparent for the drastic change

in the nature of the vortex pattern for the four panel solu-

tion. It is suspected that the vorticity distribution func-

tion must be further restricted so that the solution converges

toward a physically reasonable result.

The following are recommendations which may improve

the results of the method.

a. Sacrifice some control point equations to provide

continuity in the first derivative of y with respect to y

across panel boundaries.

b. Incorporate analytical results into the solution

process. Davies (Ref 3) has demonstrated an accurate closed

form approximation to the behavior of the lift distribution

near the wing apex. Using the analytical result will insure

38

a more correct result in the apex area and may then improve

the overall lift distribution.

c. Adopt an iterative procedure. Starting with the

single panel solution, use selected values of 6 and y as

given quantities in the four panel solution process.

39

Bibliography

1. Blenk, Hermann. "The Monoplane as a Lifting VortexSurface," National Advisory Committee for Aeronautics,Technical Memorandum No. 1111. Washington, D.C., 1947.

2. Cohen, Doris. "A Method for Determining the Camber andTwist of a Surface to Support a Given Distribution ofLift, With Applications to the Load Over a SweptbackWing," National Advisory Committee for Aeronautics,Technical Report No. 826. Washington, D.C., 1945.

3. Davies, Patricia J. "The Load Near the Apex of a LiftingSwept Wing in Linearized Subsonic Flow," Royal AircraftEstablishment Technical Report 72031, January 1972.

4. DeJongh, J.E. and Kooh, S.J. "An Evaluation of HigherOrder Aerodynamic Paneling Methods for Two-DimensionalIncompressible Flow," AIAA Paper 78-14, January 1978.

S. Kuchemann, D. The Aerodynamic Design of Aircraft.London: Pergamon Press, 1978.

6. Langley Research Council. Vortex-Lattice Utilization.National Aeronautics and Space Administration, SP-405,1975.

7. The Mathlab Group. MACSYMA Reference Manual. Laboratoryfor Computer Science, Massachusetts Institute of Tech-nology, 1977.

8. Milne-Thompson, L.M. Theoretical Aerodynamics. NewYork: Dover Publications Inc., 1973.

9. Schlichting, Hermann and Truckenbrodt, Erich. Aerodyna-mics of the Airplane. New York: McGraw-Hill Book Co.,1979.

10. Sparks, John C. "Development and Application of a SubsonicTriangular Vortex Panel." Unpublished MS) thesis (A}I'T/GAE/AE/80J-l). Air Force Institute of Technology, Wright-Patterson AFB, Ohio, June 1980.

11. Yuan, S.W. Foundations of Fluid Mechanics. EnglewoodCliffs, N.J.: Prentice-Hall, Inc., 1967.

40

APPENDIX A

Interpolating Function Equations

41

Interpolating Function Equations

Eqs (2.5) and (2.3) are transformed into functions of

nodal values and nodal coordinates. The solutions to Eqs (2.7)

and (2.10) are

{FL} = [fi(x,y-)]I {fj} (A.)L 1jjL(A)

and

1F = [gi(xj,yj)] " I {rj } (A.2)

The fi and gi matrices were symbolically inverted using the

computer program MA CSYMA (Ref 7). With the symbolic inversion

the F are readily expressed as linear polynomials in theL ,T

nodal values. For example,

FL = AL = (ALDl)(6 1) + (ALD3)(6 3 ) + (ALD6)( 6 )

71 3 A3

+ (ALD7)(6L7) + (ALGI)(YL) + (ALG3)(y3) (A.3)

(ALG4)(- 4 AG6( + (ALG7)( 7

where the ALDX and ALGX represent the appropriate element of

the inverted f. matrix. The complete algebraic expressions1

of the inverted fi and gi matrices are found in the subroutine

COEFF listed in Appcndix C.

42

APPENDIX B

Evaluation of Integrals

43

Evaluation of Integrals

The procedure for evaluating the four integrals in

Eqs (3.11) - (3.14) is given in detail in Sparks (Ref 10:

47-53). Only the results are given here.

Following Sparks' notation, the STRIP functions for

the four additional integrals are:

S6 [(x0 ,y0),(x l,y l)] = mY + 3bi62(m 2+1) m 2(2+ 2)

S(m2y + m2 x2 )1/2 ( my 0 + 3bm2(m 2+1) 2(m 2+1)

S(m2Y2 + m2 2 /2 + 3bui2 - m(m 2+1)b20 2m2+1 5/2

(2+l 1/2 2 2 /2

(m }1) (y2+x 1 ) +MY +x(

LN( 2 1/2 2 2 1/2 (B.1)(m +1) (y 0 +X0 ) +my 0 +X0

2b-yl 3bS7 [(x 0 ,y 0 ),(x 1 ,y l )] = {( 2 2 2

2(m +1) 2(m +1)

2b+Yo 3b

(my + m x) ( +2(m 2+1) 2(m 2+1)

• m~ 2 3m 2 b 2

my0 0) 2(2-)/2(m2+1) g /

(m 2 + I ) 1/2 (y2+x2) 1/2+my +x 1LN( 1/2 2 2 1/2 ) (B.2)

(m 2+1) (y 2x 0 ) +mY 0 +x 0

44

02 2

2(m +1)

" (x2+2 )1/2 + y rnx0(m 2+1)-rn2 b+2b

1 0 2(m 2 +1 2

"(2 +y2 )1/2 + 3mb 2

( 0-- 0) 2(m2 +1)5/

(in2 +1) 1/2 (X2+Y 1/ 2 +nY +xLN(- 20/ ( 04-0 0 (B.3)

(in .1) (x 1+Y,) 1 /2 mY1 4xl

3rnb+m 4 x +n 2 xiSg[(O.-Y).-Xl.,j~l2(m2+1)2

~2+X21/2 3mb+m 4x 0 '+m 2 x0(YI 1)2(m 2+1)

2

( +2 ) 1/ + OILN(+(x 2 + 2 )/ 2

2l 292112 2 2 b 2 -b 2

2 LN~xl(xl~y,)2(rn2 +1) /

(r 2+1)1/ 2(y 2 +x2 ) /2+nY 0+x 0LN( (m2+ 1/22+ 2 1/2 (B.4)

(in 1)( 1 x1) +my14-x1

The expressions for the four integrals are then

T= S i[(x,,Yl),(x 3 ,y2)] - Si[(xl1 ,),(x 4,y2 )I (B.5)

T= S i[(xlyl),(x4,y2 )] - SilKx 2,yl),(x 4,y 2 )] (B.6)

45

V

APPENDIX C

Computer Code

46

.. I qL , L( r L 1 ,- .

)" ')~~~L i Ii~L;I I'l -/r LL , X

11~~I~ f N 7,L -K I LI

Ali., ,-N T ?', . . s , : ,.''n L VL/

1 L 9 , L ~ L

3L3,L 1,iL,

*. <L * <4 - , . , _ : ; , .. , L

L I 7LI. 9 p J1 L L,L L

C] r r' =T, . , r-. . . ) ,.. - ,:Tr , -, S 'T ~o 7 JT , Ij. I JJ I , I ,-

vr <.' Lig ', -,H~ L 7'. ., '3 ,LTG Ld'pt. , I L) Ll; ,i.~~r

2 L;,7 ~L r 3 Lv, C- '.- L, L- , .-." . -.T T

"I : J-.': t'C", v, , I J 'JI(T1) 1

v r , <

r~l T r Y7 ;' , S -)T

H -. L 7-) T-7L L P L r;.

LLL

1 Tr I , I 3 1 5" , 9

1 j E:S:- . 3 s ,'3,.,,i, ,r , :, ,?, - ,rF~nl 1

SL 7-A ii T

3 , :G.' I 7PU Gj -'l-, - "- T;I rlT71

ZAD SD, 14 DATA.

t , ' -, -t, S'AT C. /Aq K , L IUX, . )

"' C( ,."

"3 r( ) _ '. ,Wi7T C,? 3 )

$- = -,, t/ 3 ) : ->J, S I ;' -I ). r I: (

T 2

I V

r I: s

2 **IT)' (,,:. ) I !

,r:T ('.,:'

. 1 Y. , V

" . r o .

-V I C ;', ?k J1 ',.S,

'r T <'7 T- )

W I ..

J'E T :1

(/ a", 31 - 7 - 7 31

,-,) y , ?

" W' T C ,L)

II

Z. 4-., : Lt~ "I +3 i-L'."JL4 ., w, r . , " S :5 t _? E, t,"- .

CA CJL T Nfl4-I DA..'I5 1".",.VFr;= 1;- j

-LL '+' ( 3)

T fl'*= *

. <'' - . L 1 7U

7 " L L L r 3 4

48

w

49

TNrS 'rO IT Ci~ i 14 A~ rK S 7-4 0 t N L 4' 74GC'A IS. IiL C, U'! IDJ~ 'S 1 E U7TV7

-CONJ 1 %L OnIJT 1" 73 j .)IIT t 4tNST

IL L,Il

CLCL7 ST.") C7) 3,-Y)N7

f-L. STF~1P~i,YI,XZ,Y?, I)* ~CtLr~i,%TE rT'-Z rJ T -L '!'Ul- ~ ~ 1?) tN IAGNN~AL 4I1T-4 ~ PIT ITS (X,,fi.

CLL M ITF Y 1 Y l2,p1)C AL C UL IE S TRT r' F'YJr 11~ ).4 D Ti H

jJ IXe Y

I V AL'J-1r P AN--L I NTZ ZL 3

r L I Q I 1 2

ii

so

5 Ll - 73 [ U.1LT~.~ 3 CL J I T Cr

T TK'OF PIJ T: -,.Lt(r II T J I

-j FijT~ LLP T~ 3,1 7) i

flT 1 ',c-o SO~ T) '.

Y -Y I2 (D ')-~f,

'.1+1

P.I ! ILL '('

911* r)/(. -I , 3'

pj'r- '.'rI* U" *'

~- +

* (,I' ty j) 2 )

21 f( I I I -

1 y 5I

3'JI 7 zALz-G(C'J-.J)

Socr(r ) (y . r -- ,4nji 1 f(

WR-T m " PF'(L L1 1

P 7")1 LTP.. e1(

I Ut 0' I*4C)1- %/ 'L P. 3 -13: 'N 14 11|I . / - 9',LTC T 1 1 , . r

I C-1

r) I - 3 (2) ( (",?)-' (,'=!

2 v~.T.-2,t (K+l)

y3 ( ')l )no) 2' 1T- 1 l

Y ('9), f): C(! , I )+: ( ,) ( ( ) n ]

Y (1) '3 ) ( C , S , 1) / J )

'CC, C)=. (.C ,:) * X ''4) "(J !)-1)/-_

C'J3, r ,y() - )

3 ( 4-' ") ( ( 21 E( ? .) -

, 'I) j y 1 7 + )

V ' 1 ( D -1))

Y { 3 "):t=A (y , ' *u 2-'..-) ) / (YP(VC,)-C' (!.y 111 I, (' - j ,:, :

? ) L" FC3 A L CULA T -- CO0 iJ. L 4 ['T .

y : I = y ., -

Y( .~7 =CC, Y , I , C) I tV ', , L

Y( I 5r'I T , T "

: " 'o T ['' ; T :

L :L53C

tf 4T K, IZ

y L9 1

C --' C'L L L7

I IL L )

5d

FRI--- *'

1, (.Y T*~ j*.1 T J 3 4

,-. ,1 Clv!4

7;~, 41

;1, 4 t

,' 41

rl G1 4'4 %

n'. T17 42

55

L , ,' "

,L ,-) -L , '-Cl ,L , L ,2 c LI. 7 Lj L' L

JLr IL" JL-I J. , . J JL-,

L I <

7 -L D]It ' 1 L 3u, 4:L3: '.. -L,'

4 LLC ,5 L I L. L

1 3 T I

C I I' I

C J T JYJT' ,

T CT' ,;

H LT 1,L T, LT

3 LLF, , L L ,,

JT 'l, J I I , J I

VT D w IL r2,. 7 .'7r

+ , 4 Y Y '-V'-?7 - (Y I-2-," (Y+ '-'') )'. -)

" '? - ( 0', , '4< ^. )-v2 _' ( ( - - -. ' t)•

V .. v ' 4 <t'~', .Vi I"'o)'Y: Y •

"Y . ° + > " * I ' .)" 1, ' ?)/ (Y " -- f+

y L2 Y y - - y + ( I y 2 y3 I t 2)

T V = y (y,, 1. 'Y.'*? 1 'YV )Y '7AT - --. .5

Y 4( X

1 '

X ' i

(' " I X ) ' I y 1-1 1 f2 #"(( Z 4 Y?* '

2 Y24 , IV 2 , ;- ? ' -4') 2-. $( I' y 4 ',2- 4'" ' "-

X Y1 ,lyf~t y ,~ ct2Y14?Y:'I'I' V

4- y 1 4K)

2 1" Y- c -3Y( " I (" ' " '"s " 2' 3 4 yy" 2-6')-' 'Z ("6 Yy. ~ fv - *] -y 2- y'-'( K y y'' I X Y. ' ) Y '

J 1 IVI~ ( -y ,-. . Y . 2 ' -- " ' . / . , - } , .

2 ' 2 / (Y " .. - yl- " ." 1 "v"'' : "("C { " '' [ "X ;"-"Y ' "

x 4 1( - 1. , ") . yt- - . f -. , €+ . I -f . ] v:

(7 1" 1 1+ ,

56

2 'XTi X I Y 2 (Y 4e -- ' '

2 ?

1 2s 21P + , Q 2+-'~( < f~y) (I; >2 1 V''LZ "E 2

2Y +- ?-, - 2s 2: x?- y

x __2 Y, Y, ~ ?

2 % e y IVt'7(X1' 23 s.4 Y I Vyy y

T '0 34 Y Z Y I..'. Y' 3- +? ViL -3*r 3

I Y1'K<

2 x'~ " ? 3)+

2 ??<X4 22"V2) 12)

Z/Y2 C?21'2't * 2

2 x'Vs .. ,t7

4 X ( 4.-'-~ .Vj 7rr2)( Y2 '' iV2''

P ~ ~ ~ ~ ~ ~ y V:M

2 ( -2' ?- X r'u%' V> 2.('

I P Y- 'YeV: ?'yC~ ? 2 u y

2 G Y - W ?'3) V1fV2( VC2~* 1 .+ ~ 1 '5) -'yj' v' r '

2 ~ ~ ~ ~ *' OY254)0<I ';

1 + Y <2 * 24 I

I I? )' ?,- ++V4 V3Y ?f I I)

57

2 )'YI' 2)

Yi~ 'Y?+X1-4) *-- I + 7Yj < (X ?-? IyfI Yil9()t.< ':2 'V +X: ,)v )

T 2)Y <)- k Y -/ (V ''2 +~''

2 X) y2' 4- Y- V I'7 -* -K' +2'-

2 *< V4'-2') 2 Y, 7 2

2~ 2 Yi- V . 2+

2 yt.2

leT-,- J ' V+ 2f.' . Y V YV - '(? A;:

2 , Y2 2 C 2Y !t 7: V2

T- . V?'yi.V2#(y . y IyL' , , X -

1 yV7 +L)2V: ')Y(I2 -?'+Xi 2 Y >2 I +~1 2

2'V 2+'1 ''- + *yq.

I - U *. B ) - X

+,. 2 Y:2*'v y+. r X / V +

2L 2 ,1+y YIy,+y 'F' 2

3C 7)'±''' 1 U *CX'*iUNSV~'3

T Y Y 1 +8

1 . 2 Y."' 3 2)YI T , V +V

1 Y21" )/( +'2, ' *C 3 V? (- .',+' .C

NLS' I.'j Vn/ (V '' X -y , - + (Y!?Y

2 2~2'+ y ' - x I.2('- .) - Y

2 ~ ~ ~ ~ . ( rj ,' -t' )9'4

y

I 3

Yi tV (.A.Y) ya . S 2)y

It z v3? y: I)fY -< I ' 3

L -X Y.: I' ?+ - f " - XY'' 2) Y-*2 '1 t." +* y~" - Y 3-) '1 2+ .'' -: I* *V -3) 'V'

2 ,( ' ~ Y ', f'~ V 2)

2 3 )13,3

.Y:X- x ~Y'< * ,~. * * ~ t/( Y Y::"

4K V. 3' Y5 '< 13 +' 2Y +(/ '-2 I -'U 2.f

-- I - ''- '' . *.- -, *

1~ ~ I j ?4 'e ,? Y2 ( -'5 2 YY'V. <; +'- , %

I.~l Y- +X,~ +?~ y 2)-. *'

:P'''-. '

'-31 y I -Y+9

EL3 = CO .- -* I-t~'. I-1I.4

Y 4-Z> + a X. 12) U' IX. Y 7S . 2-2-', Y, +X I ) Y-

2 )

11 '-- v+ -. 11 1 y -

3 -Y5'(-Y '-1-' ' ) '"1:

'LS. ~ ~ ~ ~ ~ ~ ~ ~ (w (03 - ': 4 . 7 V- y-*lf tY,4

2 'X Y 3J' . 3- Y, ,- :.Yr -- : I'~i 2 ''V -V4 y', >.X'2 jV~. V')( .- Y) *

IL)L

ILI Y' I (X: 2CL'%y.-))1 , I5

ILS +Y Y y I>1 Y:21 1*1.7''1+ + (?'V?* 3''

L I-'C- +j' X~'f/( 7 -'X Vt -*19 2 fpir

1 L ' " + X Y)V't ,? y ~ '(' x' '> v: 7 ~ ~ ' '' y +'<'+

IL IX 3-=--((C Y"' Yt''- + I ''+''' Y -' I '.r2 '2f

2 24'X2 Lv y 2- 2 Y - *1 y ,*4 /' ')

1 (Z '3 YC) 1 Y> X.-' f1 f

-1 fCC -'-'W YjY'C

X ~ Xi + ! 2- (V I (Y v) 2' ,

JL SI

I I -V 2Y)

ICLn 4'i y <) 2- X. / f 2 + Y-. 2 -' 3' Y 's- t y

A y '? V:

WL'3y7 1)j Vd

60

2 2)

I y 47-yx +- X A j2y

32 2rL4 =, +14V .tVx7V)I'I2(-)-Y 4O y'" V Y 1

2 - ) y A9 Y X * ) V 2 ( ~ ~ 'V*

,L 13C2)2 - .

GL/U.' -nt. Y. IB '1 Yy

1I ?'N2'( X 4 Xt < ' B ?'iY&

j -Y4 .('. %2X-X'' ) ) " '.

1 -+ -'!

2 LS _It If2-2?A' , V, N~

STY ?YrtJ)r4 - 2) f -Y L+

(

2 Y:61

S;l' I, U'l !E r IT ( j.p ',i~

SCLEFF C- f' : FU - . J%i "1 i - JLl N'

F;", ,.0'' Lf 1 ilT

" ' -'4l k/ ;" .. U - w ,/ L - L )-)- f" , vv/ , 2 -7.

L4 k" vu L v L I - L L ,

r L I L L L L

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82

Vita

Ronald E. Luther was born in Waukegan, Illinois on

11 June 1948. He was graduated from Chillicothe High School,

Chillicothe, Ohio in 1966. He graduated from Purdue Univer-

sity with a Bachelor of Science degree (Aeronautical Engineer-

ing) in June 1970 and was commissioned into the Air Force at

that same time. After completing Undergraduate Pilot Training

he was assigned to the F-111 weapons system and was stationed

at Nellis AFB, Nevada, Taklhi Royal Thai AFB, Thailand, and

RAF Upper Heyford, England. In 1979 he entered the Air Force

Institute of Technology.

Permanent Address: 4801 Sparrow DrDayton, Oh 45424

83

UNCLASSIFIED

SECURITY CLASSIFICATION OF THIS PAGF (Whe.n )0.tm Fnt-r.d

REPORT DOCUMENTATION PAGE READ INSTRUCTIONSDCMN O BEFORE COMPI.I'TINC, F(I<M

I. REPORT NUMBER .2. GOVT ACCESSION NO. 3. RECIPIENT'S CATALOG NUMBER

AFIT/GAE/AA/80D- 14 ,4. A9! ' __"__ _.. . .4. TITLE (and Subtitle) S. TYPE OF REPORT & PERIOD COVERED

A HIGHER-ORDER TRAPEZOIDAL VECTOR VORTEX MS ThesisPANEL FOR SUBSONIC FLOW 6. PERFORMING O1G. REPORT NUMBER

7. AUTHOR(s) 8. CONTRACT OR GRANT NUMBER(s)

Ronald E. LutherCaptain, USAF

9. PERFORMING ORGANIZATION NAME AND ADDRESS 10. PROGRAM ELEMENT. PROJEC, TASK

AREA & WORK UNIT NUMBERS

Air Force Institute of Technology (AFIT/E.N)Wright-Patterson AFB, Ohio 45433

1 . CONTROLLING OFFICE NAME AND ADDRESS 12. REPORT DATE

12 December 198013. NUMBER OF PAGES

9314. MONITORING AGENCY NAME & ADDRESSif different from Controlling Office) 1S. SECURITY ZLASS. (of th,s report)

UNCLASSIFIED

ISa. DECLASSIFICATION DOWNGRADINGSCHEDULE

16. DISTRIBUTION STATEMENT (of this Report)

Approved for public release; distribution unlimited.

17. DISTRIBUTION STATEMENT tof the ahst,.¢c entereJ in f,, ,ck 20, if different from Report)

IS, SUPPLEMENTARY NOTES Approved for public release; lAW AFR 190-17

Fredric C. Lynch, Major, USAF

Director of Public Affairs O0 DEC 98019. KEY WORDS (Continue on reverse ii-ei nre .... rv arid tder,tlv 6y block numttter)

Airloads Perturbation MethodAerodynamic Loading Paneling MethodPotential Flow Vortex PanelSubsonic Flow Singularity Method

20. ABSTRACT (Continue tn reverse -side it necessary end identity hv bloc-k rt-thor)

A higher-order trapezoidal vector vortex panel methodis developed for application to linearized subsonic potentialflow. Each panel is subdivided into two tria-,tilar subregionson which a quadratic vorticity strength distribution is pre-scribed for both the spanwise and chordwise Loponcnts of thevorticity vector. The vorticity strength distril, ilion is

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expressed as a function of the components of the vorticityvector at selected nodes on the boundary of each triangularsubregion. Nodal values on the shared boundary of the sub-regions are made equal, assuring continuity of the vorticitydistribution function throughout the trapezoidal panel. Alifting surface of no thickness is modeled with a network ofthe trapezoidal panels. Again, nodal values on the commonpanel boundaries are matched to achieve complete continuityof the vorticity distribution throughout the lifting surface.Aerodynamic data for several wing planforms is obtained withthe flow model. Results from this method are compared tothose from other computational and theoretical methods.

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