Numerical solution of Prandtl's lifting-line equation · 2018. 3. 1. · 1.1 Lifting Surface Theory L.2 Lifting-line Theory 2.3 Listing Program 2.4 Numerical Results vl vlt vul lx

The LIniversity of Adelaide

NUMERICAL SOLUTION OFPRANDTL'S LIFTING-LINE trQUATION

Boedi Koerniawan

Ir. (Tanjungpura), Grad. Dip. Math. Sc.(Adelaide)

Thesis submitted for the degree of

Master of Sciencein the

University o{ Adelaide

Department of Applied Mathematics

May 1992

This thesis is dedicated to

my rnother and rr. y late father,

whorr-^ l love very much.

The fear of the Lord is the beginning of knowledge

but fools despise wisdom and instruction.

For the Lord giveth wisdom :

out of his mouth cometh knowledge and understanding.

Proverbs I:7 ;2:6

-l '.

Contents

List of Figures

List of Tables

Summary

Signed Statement

Acknowledgements

1 General Introduction

1.1 Lifting Surface Theory

L.2 Lifting-line Theory

2.3 Listing Program

2.4 Numerical Results

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vlt

vul

lx

x

1

1

2 stewartsonts solution of Prandtlts Lifting-line Equation L3

2.L Introduction to Stewa¡tson's Method .

10

13

L7

27

2.2 Numerical Evaluation of Stewartson's Double Integral 15

1V

3 Semi-inflnite Wing with an Asymptotically Constant

Chord

3.1 Numerical Method

3.2 Listing Program


4 Semi-inffnite \ü'ing -

Chord c(y) Proportional to yå ¿s tTends to fnffnity

4.7 Numerical Method

4.2 Listing Program


5 Application of Prandtlts Lifting-line Equation to the Corn-

pliant Layer Problem

5.1 Varley-'Walker Solution .

23

23

27

31

39

39

4L

45

51

51

5.2 Numerical Evaluation of the Varley-Walker Solution 52

5.3 Direct Solution of the Compliant Layer Equation 59

6 Conclusion

Appendices

A DOlAHF - NAG Fortran Routine Summary

B FO4JGF - NAG Fortran Routine Summary

Bibliography

69

72

72

lÐ

79

v

List of Figures

1.1 An aeroplane wing with airfoil section.

L.2 A closed curve C in flow field.

1.3 Uniform flow past a thin wing

4.1 The curves ø : +"(g) are touching the ellipse at one end. 40

5.1 Comparisonbetween our method and Varley-Walker solution

for the compliant layer equation.

2

4

b

68

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List of Tables

2.1 Numerical Results for f (x) Provided by Stewartson .

2,2 Numerical Results for /(r) Obtained from Stewartson's Dou-

ble Integral

3.1 Numerical Results for Prandtl's Lifting-line Equation with

Constant Chord

15

22

34

34

35

Ðt

3.2

3.3

3.4

Optimum Balance between L and n

Numerical Results for /(c) - a Comparison

Numerical Results for Prandtl's Lifting-line Equation with

Chord Defined by (3.2a)


Chord c(fi : yL


Chord Defined by (4.10) . . .

b.r Numerical Results for /(r) Obtained from Varley-Walker So-

lution

5.2 Numerical Results of the Compliant Layer Equation

47

49

58

67

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Surnmary

Prandtl's lifting-line equation is an integro-differential equation which is

used for calculating the span-wise distribution of circulation around wings

which are three-dimensional quasi-planar, with negligible sweep and large

aspect ratio, placed at a small angle of attack to a stream of incompressible

fluid. Since Prandtl introduced this famous equation, there have been many

who have tried to solve the equation. Stewartson (1g00) has solved the

equation analytically for a semi-infinite wing of constant chord, the final

result being a function involving a double integral. To calculate the value of

this function numerically, we need a computer routine which is, in general,

very time consuming. In this thesis, we shall develop an efficient numerical

method for solving Prandtl's lifting-line equation directly, using a non-

uniform grid which is concentrated at the wing-tip. The results agree to

nearly three significant figures with computation, based on Stewartson's

double integral, for the case of semi-infinite wings of constant chord. Our

method is useful for semi-infinite wings in a more general case, not only of

constant chord. The same equation but with negative constant chord, which

applies to a non-aerodynamic problem, e.g. a compliant layer problem, is

also studied. However, our method is less accurate for this problem, because

wave-like behaviour occurs.

vllr

Signed Statement

The contents of this thesis have not been submitted to any university

for the purpose of obtaining any other degree or diploma. Also, to the

best of my knowledge and belief, the thesis contains no material previously

published or written by another person, except where due reference is made

in the text of the thesis.

I give consent to this copy of my thesis, when deposited in the

University Library, being available for loan and photocopying.

Boedi Koerniawan

lx

.A'cknowledgements

I would like to express my sincere thanks to my supervisor, Profes-

sor Ernest Oliver Tuck, for his guidance during my Master's research and

during the preparation of this thesis. I am very much indebted to him for

his interest, ideas and advice during the discussion of all problems.

I am also very much grateful to Professor Ren B. Potts for his guid-

ance prior to my Master's research and during the finishing touches of this

thesis.

I am indebted indeed to the International Development program

of Australian Universities and Colleges (IDP) for its financial supports.

\Mithout the scholarships from IDP, this work could not have been done.

The invaluable help I have received from Dr. David L. clements -the Dean of the Faculty of Mathematical and Computer Sciences, Dr. peter

M. Gill - the Head of the Department of Applied Mathematics, Ms. Lorna

Read - the Secretary of the Department of Applied Mathematics, Ms. Vivien

Hope - overseas Student Adviser and Dr. F. Salzborn are all much ap-

preciated.

x

Chapter 1

General fntroduction

1-.1 Lifting Surface Theory

A lifting surface is a thin streamlined body that supports the flight of birds,

aircraft etc., on which a hydrodynamic lift force is generated when it moves

through the surrounding fluid.

A typical example of lifting surface is a wing of an aeroplane. Let

us consider the wing of an aeroplane as sketched in Figure 1.1. The cross-

sectional shape obtained from the intersection of the wing with a vertical

plane parallel to the centre-line of the aeroplane is called an airfoil or aero-

foil. The most forward and rearward edges of the wing are called lhe leailing

edge and the trailing ed,ge of the wing, respectively. The straight line con-

necting the leading and trailing edges on an airfoil is the chord line of. the

airfoil, and the distance between the leading and trailing edges, measured.

along the chord line, is its choril. The angle between the chord line and the

direction of the free-stream is defined as the geometricøI angle of attøclc, ot

inciilence, of the airfoil. The distance between the two wing-tips is called

lhe span. The mean chord of a wing is the average of the chord of all its

1

I

l

leading trailing edge

T -lI

I

JI

L--J

_II

i.-chordJ

Figure 1.1: An aeropla,ne wing with airfoil section.

section-wise airfoils, namely the wing planform area divided by the span.

The ratio of the span to the mean chord is called tlne aspect ratio of. lhe

wing. For a rectangular wing, the aspect ratio A, is

sA, tc

where s is the span and c is the chord of the wing. For a non-rectangular

wing, the aspect ratio is

Ar:"'.,A,

where s is the span and ,4. is the planform area of the wing. If the aspect

ratio is very large, then two-dimensional theory is valid.

In fluid dynamics, we have a basic equation, which is widely appli-

cable to any fluid, compressible or incompressible, viscous or inviscid, i.e.

lhe conseruation of mass equation, ot continuity equati,on

0oAt + div (Ps-) : g, (1.1)

2

)

where p is the density of fluid and g- is the velocity of fluid particle. For

steady flow, we have

and hence from (1.1) we obtain

div (Pq-) : 9. (t.2)

For an incompressible fluid, where temperature effects are not significant,

the density p remains constant. Hence from (1.2), we obtain

divf:g' (1.3)

In subsonic flow, for instance, the fluid is incompressible.

T}:e uorticity õ at any point in a flow field is defined as

ui : curl g". (1.4)

If al : õ', th" flow is called to be irrotationøl. otherwise, the flow is called.

to be rotational. For irrotational flow, we may define lhe oelocity potential

ó(r,V,z,ú) such that

i: i ó. (1.5)

For an incompressible, irrotational flow, we have

div g'- ¡

and

í:íó,

v.V/ : g

0pat

0

3

and hence

v

curve c

Figure 1.2: A closed cr¡rve C in flow ffeld.

qIntegrationproceeds sothat enclosedarea remainson left

x

or

Y'ë:0. (1.6)

This is the well-kno\¡/n Laplace equation, which is a linear, second-order

partial differential equation.

T}:e circulation is defined as the line integral of the velocity around

any closed curve. Referring to the closed curve C of FigureI.2, the circu-

lation I is given by

I t' íar (1.7)

Kelvin's theorem of the conservation of circulation states that for an invis-

cid fluid acted upon by conservative forces only, the circulation is constant

throughout the fluid.

Now consider an incompressible uniform flow with free-stream veloc-

ity U in the positive r-direction, past a thin wing with small geometrical

angle of attack, as sketched in Figure 1.3. The wing spans along the y-axis.

Since the free-stream velocity is constant, the vorticity there is zero. The

4

I z v

wake

Figure 1.3: Uniform flow past a thin wing

vorticity will remain zero throughout the surrounding fluid, except in the

regions adjacent to wing surface and behind the wing. The region adjacent

to the wing surface, in which non-zero vorticity exists, is called the bounilary

Iayer. The thickness of the boundary layer depends upon the viscosity of

the fluid; thinner layer for smaller viscosity. The existence of the non-zero

vorticity in the boundary layer is due to shear force acted upon the fluid

particles by the surface of the wing, as the effect of viscosity. In most flows

of engineering interest, the viscosity of the fluid is extremely small, and

hence the boundary layer is negligibly thin. The region behind the wing,

where the vorticity is also non-zero, is called lhe walce. Since it forms a

thin sheet behind the trailing edge, it is also called the trailing aorter sheet.

Furthermore, rüe assume the surface of the wing is impermeable. Then

the normal velocity component of the flow field adjacent to wing surface is

equal to that of the wing surface. In any uniform flow, the wing is assumed

to be in the state of rest. So we have

ñ.í6: s (1.8)

x

Ð

on the wing surface, where / is the velocity potential of the flow field and

ñ is the unit vector normal to wing surface.

The velocity g-of the undisturbed free-stream is

where /i" th" unit vector in the ø-direction in Cartesian coordinates system.

This implies

Ó:Uæ

in the free-stream. If Q(r, y,z) is the perturbation velocity potential due

to the wing, then the velocity potential in the neighbourhood of the wing

i: tli,

IS

IS

ó : Ux + Q. (1.9)

Hence

íó: ui +iø. (1.10)

Let z: f+(x,y) and ": l-@,y) be the equations of the upper and lower

surfaces of the wing. The unit vector d on the upper surface " : f+(*,y)

n: !_? - r_*þ,ù) . : ffi, (1.11)l V("-f+@,ù)l W' \¿'r

where i,i and, k- are the unit vectors in the x,y, z-directions of the Cartesian

coordinates, respectively. Substituting (1.10) and (1.11) into (1.g) yields

#:rfftrv+#)+rfftrff>, G12)

which applies on the upper surface , : f+(*,a). A similar boundary

condition applies on the lower surface ,: f-@,y).Since the wing is thin

6

and the geometrical angle of attack is small, the slopes #,W,A{} and,

ff ur. small, except at the leading edge. Hence, as we can see from (1.12),

the vertical perturbation velocity component ff is small.

If both I Í+@,y) | and I f-@,y) | are very small, the exact wing

may be replaced by a cut on the r, gr-plane, with upper surface z : 0+ and

lower surface z :0-. Using Taylor series expansion we have

(ff),=,* : (ff),=o.*#r#),=oa * #r#),=01 * ... = (#),=01.

(1.13)

Similarly, we have.aø. ao( ¿r)"=t-

x ( *)"=o-. (1.14)

The perturbation velocity componentr å3 .rd ffi, which are tangential to

the cut, are negligible since the viscosity of the fluid is small. Hence the

boundary condition (1.12) can be approximated by

Uar+

(1.15)

orL z :0..,", for (*rA) on the cut. Similarly, we have

UaÍ-

(1.16)

oÍt z :0-, for (rry) on the cut.

Another boundary condition, which is known as Kutta-condition,

requires the velocity at the trailing edge to be finite, i.e.

tÕ<oo

at the trailing edge. At infinity, the flow is undisturbed; hence the veloc-

ity at infinity is equal to the free-stream velocity, or in other words, the

0r

ôr

aøOz

aoOz

t

perturbation velocity potential tends to zero at infinity. Since there can be

no pressure jump except through a solid surface, the pressure is continuolls

at trailing edge and throughout the fluid. Consequently, the pressure is

continuous through z :0 within the wake. The pressure coeffi.cient at any

point in the flow field, including the upper and lower wing surfaces z : 0*,

is proportional to ffi. H"n"" we have

.aa, .ôo.(

6*-),=o+ : ( 6*),=o- (1.17)

within the wake.

So, now we have a linear, second-order partial differential equation

V2o :0

throughout the surrounding fluid, subject to the boundary conditions

A6 trãfl-ã-:u+-ozôrlAO ¡rð1-Ø-" ur'VQ < oo,

üÕ --' o,zâOr ¡âÕr\ gs- )z=o1 : \fi )z=o-1

oî z:0..,. for (rra) on the cut,o'n z:0- for (*,a) on the cut,at the trailing edge,at infinity,within the wake.

(1.18)

The perturbation velocity potential Õ(*,A,2) can be decomposed

into even and odd functions of z, i.e.

Q(*, A, z) : Q "(r, y , z) I Q o(n, y, z),

where

(Þ" (", y, z) : e "(*,, u, - z) :

|W {*, a, z) + e (*, y, - ")],

Q o(*, a, z) : - e o(æ, a, - z) : f,[o {r, a, z) - e (r, y, - z))

8

Hence

?Q.(x,A,z) _0z

õQ.(r,A,z) _0z

aÕ v z) 7Q(x,a, -z)ôz 0z

)Q(r,A, -z)0z

+0z

âoir

|r

(r

( at )

Then the boundary conditions on the cut z : O+ are

49 = ++U(%!- eJ:1. ott z : o+3i: =1,û(4'* #j," ;; ; : o;

The even function Q"(*ry,z) represents the perturbation velocity

potential of the flow past a wing which is symmetrical with respect to the

plane z : 0, of thickness /+(ø, y) - l-@,U), at zero geometrical angle of

attack. This potential does not produce lift force and therefore it holds

Iittle interest for the present study. The odd function Qo(r,y,z) repre-

sents the perturbation velocity potential of the flow past curved surface of

zero thickness, with equation z : f,f¡+(r,y) + f-@,y)], namely the n"teo,n-

camber surface of the original wing. This represents the effect of camber

and geometrical angle of attack. This is the important part of the pertur-

bation velocity potential since a lift force will be produced. The lift force

depends merely on the function |lr*@,,ù + r- @,y)l : r(n,,9r). From now

on,\¡/e shall ignore (Þ", writing Õ - Oo.

In order to solve the boundary value problem (1.18), one technique

(see Thwaites, [2] 1960) is to replace the wing and wake by a vortex sheet of

strength l@,y). At any fixed span-wise coordinate y, there is a circulation

l(y) around the chord, as defined by (1.2), and related to 7(ø, y) by

rbf(y): I t@,a)dn, (1.19)Ja

I

where ø and ó are the ø-coordinates of the leading and trailing edges, re-

spectively. After some manipulation, we obtain

r-(ld€dn, (1.20)

@-Ð'+(a-q)'where u(*,A,0) : åÕ( rrU,0) is the vertical component of the perturbation

velocity oÍL z : 0 and the integration is over the wing surface ^9. This

is called the lifting surface equation. The vertical perturbation velocity

component t¿ is induced by the trailing vortex sheet. In general, the induced

velocity is in downward direction and therefore we call it llne downwash

aelocity, and the above boundary condition states that it is a given quantity,

, : UH. Here ff is the local geometrical angle of attack of the mean-

camber surface " : f (*) of the wing. There exists a singularity at rl : y

for all æ ) (, which represents physically the trailing vortex sheet.

L.2 Lifting-line Theory

The first three dimensional-wing theory, which is the most prominent one

in this century for calculating aerodynamic forces acting on the wing is

the lifting- line theory, introduced by L. Prandtl in 1g18. prandtl,s lifting-

line theory leads to a famous linear integro-differential equation which is

singular in the Cauchy principal value sense. \Me call this Prandtl's lifting-

line equation. This equation can be derived from the lifting surface equa-

tion (1.20) by further approximation for very large aspect ratio, using the

method of matched asymptotic expansions (Ashley and Landahl, [B] 1g6b).

Prandtl's lifting-line equation is used for calculating the distribution of cir-

culation round a wing which is three-dimensional quasi-planar, with neg-

w(x,a,,o):¿; lrlffin*

10

ligible s\Meep and large aspect ratio, placed at a small geometrical angle

of attack to a steady, subsonic stream of incompressible fluid. The wing

is considered as a straight line, parallel to its leading edge. An additional

piece of information required for solving Prandtl's lifting-line equation con-

cerns the boundary values at the wing-tips. \Me expect the circulation to

fall to zero al, the wing tips. According to Robinson and Laurmann ([4]

1956), the circulation must drop to zeto at the tips of the wing, otherwise

there will exist trailing vortices of finite strength behind the tips, which

would give rise to infinite downwash velocities at the wing.

Once the distribution of circulation l(y) as defined in (1.19) has

been found, we can subsequently calculate the lift force and induced drag

acting on the wing. Bera ([5] 1991) found that a dragless lifting solution

to the equation exists. However, Bera did not force the circulation at the

wing-tips to be zero.

If we construct a y-axis in the plane of the wing parallel to the leading

edge and the wing spans from y : 0 to A : s, then Prandtl's lifting-line

equation is of the form

f(v) : rc(v)l(J.,s* * l":Hl, 0 ( v { s, (r.2r)

where l(y) is the circulation around the y-axis,, "(v) is the chord of the

wing, U is the velocity of the free-stream and ae is the geometrical angle

of attack of the wing. The integral is a Cauchy principal value.

In this thesis, we emphasize the aerodynamics of the wing-tip. So,

\r¡e are going to use a serni-infinite wing rather than a finite wing. The wing

now spans from y : 0 at the wing-tip to y : oo and the circulation l(y)

11

around the y-axis satisfies (1.21) with s : oo, i.e.

f(y) : nc(y)luas+ * l"*'l!lrl, 0 <y. -., (1.22)

Prandtl's lifting-line equation can also be used for some non-aero-

dynamic applications. For example, if we take c(y) in (1.22) as a negative

constant, say c(y) - -C, where C is apositive constant and define

r(y): -rcuaslr- l@)1,rC

9 - , e)+

(1.23)

then the integro-differential equation (L.22) becomes an integro-differential

equation in /(ø), i.c.

"f("):-!¡*ry, o(r(oo, (1.24)\ / rJo {-nwhich is the integro-differential equation in a compliant layer problem (see

Varley and Walker, [6] 1989). Similar equations can be found in other

physical applications, such as waves near ship bow or stern (Tuck, [z] 1gg1),

\I/ave near a floating dock (Varley and Walker, [6] 1989) and heat conduction

(Varley and Walker, [6] 1939).

Stewartso" ([8] 1960) was able to solve equation (r.22) analytically

for a wing of constant chord. Varley and \Malket (16] 1g8g) have also red-

erived and generalised that solution. A generalised lifting-line theory for

curved and swept wings has been recently introduced by Guermond ([9]

1ee0).

72

Chapter 2

Stewartsonts Solution ofPrandtlts Lifting-line Equation

2.L Introduction to Stewartsonts Method

Stewartso" ([8] 1960) solved Prandtl's lifting-line equation (1.22) for a wing

of constant chord "(A)

: c subject to the boundary conditions

l(0) : 9, l(-) : ¡rc(Jo¿o. (2.1)

He defined a nelv function /(ø) such that

l(y):rcUaslt-f@)1, y:rcr/ . (2.2)

By substituting (2.2) into equation (7.22), he obtained an integro-differential

equation in /(r), i.e.

t(,):+1"*# (,>o), (2.s)

subject to the boundary conditions

/(o) : r, /(-) : o.

13

Instead of solving equation (2.3) directly, Stewartson solved the more gen-

eral integro-differential equation

l(o,r):-i lo* orr(",t)Kr(ol€-*l)ssn((-n) (" >0), (2.4)

subject to the boundary conditions

/(4,0) : 1, /(a, oo) : 6,

where a is real and positive, and .I(r is the Bessel function of order one,

of the second kind and with imaginary argument. In order to solve this

equation, he wrote l@,r) as the inverse Fourier transform of. F¡(u), viz

f(a,r):1r-_t

2r J-- F*(u:)e'i" dt:, (2.5)

where F+(r) is a regular function of ø in the upper half plane, Im c.r ) 0,and

used the \Miener-Hopf technique to determine F+(.). The final result for

the case c : 0 when (2.4) reduces to (2.8) is

1¡æf@):' ¡

7t Jo ""pr-* 1""##ro,e-tt

(1 + úr)(2.6)

once the function /(r) has been found, rve can calculate l(y) using (2.2).

Stewartson did not show how to evaluate the double integral (2.6).

Instead, he provided series expansions of /(z) for large and small values of

r as the approximation of the solutions. For large r,he found that

î,\ 1 I 1 4f @) : ;+ *z*z(z+l"g ")+#12(log r)2 +47log r-1 -i"1*. . ., (2.7)

where 7 is Euler's constant 0.5772...For small r, he found the following

series expansion

r@) : t - zç!¡î - lfiltOosux*, - *74

) + o,(!)*,lf

(2.8)

0.2370.2670.3080.3610.4380.5641r)r7.21.00.80.60.40.20T

Table 2.1: Numerical Reeulte for /(o) Providedby Stewa¡tgon

0.0860.1130.1610.1750.L920.272f(r)4.03.02.01.81.61.4fr

where O¿ means that the order includes an unspecified power of log r. Ste',rr/-

artson also provided a table of the numerical results of (2.6), which is shown

in Table 2.1.

2.2 Numerical Evaluation of Stewartson's Dou-ble Integral

In order to obtain accurate numerical results for small values of r, and to

reduce cpu time, our method for evaluating the double integral (2.6) is as

follows. Let 0 : tan u and hence

larctant

fo*"'^'

Now lettingu: I for the first integralin (2.9) and z : i- i for the second

integral in (2.9), we obtain

l"' '# : T fo'*u"

' log(2 "inf,)a, * i I:-'

arctanú rog(2

"inf,)au: -Tifrarctan t) * o(tr - 2 arctanú)1, (2.10)

where g(d) i. Clausen's integral defined by

s(ó): - loÓ øsçz"inf,)d,, (0 < d < r). (2.11)

log(tan z)du

log(2 sinu)du - log(2 cosu)du. (2.9)

15

According to Abramowitz and Stegun ([t0] L964, page 1005) , Clausen's

integral (2.11) can be expanded as follows

s(ó): -órosó+ó.å1#",rffi: -dlos ó + ó + 0.013888889d3 + 0.000069444ó5 + 0.000000787ó7

+0.000000011de+0(10-10) (0=ø.i), Q.12)

where B2¡ are Bernoulli numbers. Slightly greater accuracy is obtainable

by economising the series (2.72), and we use a set of economised coeffi.-

cients supplied by Prof. J.N. Newman which give at least 14 decimal place

accuracy for g($), 0 < d 1rf2, the economised series being

s(ó) + ór"eöó

A similar economised series is used for the range r12 < þ < r. Hence,the

double integral (2.6) reduces to a single integral

f (*) : + lr* #"*pt#{s( 2arctant)+ s(r -2arctant)}ldt. (2.18)

To evaluate this single integral numerically, we write the whole integral as

Io* o(*,t)dt :

lo' o(*,,t)dt + f,* hçr,,t¡dt, (2.14)

where h(r,t) is the integrand in (2.13). Since h(n,,t) tends Lo zero for 1arge

values of ú , by choosing appropriate values of r, the second integral on the

right hand side of (2.L4) is negligible. The values of r are chosen through

: 1 + 0.0342 6s45s7260fi(+),

+ 0 . 0 0 0 422T 82 5 r2s 6 s f?*f + 0 . 0 0 0 0 1 r B2T BT r s7 s (+f+ 0 . 0 0 0 0 0 0 42 5 5T 8z t 5 f7*f + 0 . 0 0 0 0 0 0 0 1 z s 6 6 6 0, (*)' "

+ 0 . 0 0 0 0 0 0 0 0 0 z b 2 6 6 4 f?*l' + 0 . 0 0 0 0 0 0 0 0 0 0 427 4a (1* )' ^ .

16

experiments to give acceptable results, and r is inversely proportional to

r. The value of r for 6 figure accuracy has been chosen as r: 200f r for

r ) 0 and r :1.441x 105 for z :0. Then the single integral (2.13) can be

evaluated by any integration method.

2.3 Listing Program

The program in Fortran 77 for evaluating the integral (2.13), using the

method described above, is listed below, where DO1AHF is a NAG routine,

available from the computer system, for evaluating the definite integral of

a function FUN over a finite range. For a description of this NAG routine,

see Appendix A.

IMPLICIT REAL*8 (A-H,O-Z)

COMMON X,PI

EXTERNAL FUN

PI : 3. 1415926535897932384626D0

DO 10 J:0,L2

rF (J.LE.10) THEN

X : 0.2*DFLOAT(J)

ELSE

X:DFLOAT(J)-8.0D0

END IF

rF (X.LT.5.0D-7) THEN

R : 1.441d5

T7

i)

'i

,l

ELSE

R : 200.0d0/x

END IF

RESULT : O.ODO

A : 0.0d0

B:REPS : 1.0D-7

NLIMIT: 10000

IFAIL : O

Z : DOL AHF(A,B,EPS,NPTS,REL,FUN,NLIMIT,IFAIL)

RESULT:RESULT +ZIPT

WRITE(7,20)X,RESULT

10 CONTINUE

20 FORMAT(F5.2,5X,F9.6)

PRINT *,"SEE FILE 'FORT.7' FOR THE RESULTS."

STOP

END

,<**{<t(*'F***{<***,t<t(*{<***t<****'F***{<***'t*:ft{<>t*{<t<*{<******{(**{<**

REAL*8 FUNCTTON FUN(T)

IMPLICIT REAL*8 (A_H,O_Z)

REAL*8 B(8),BB(12)

coMMoN x,PI

B(0) : 1.0D0

18

B(1) : 0.034269459726017D0

B(2) : 0.000422782512969D0

B(3) : 0.00001182737r379D0

B(4) : 0.000000425578715D0

B(5) : 0.000000017366602D0

B(6) : 0.000000000752664D0

B(7) : 0.000000000042746D0

BB(0) : 0.693147180559945D0

BB(1) : -0.102808379178034D0

BB(2) : -0.0063 41737696915D0

BB(3) : -0.0007 45724327 445D0

BB(4) : -0.000108523615117D0

BB(5) : -0.0000 77757036264D0

BB(6) : -0.000003726027195D0

BB(7) : -0.000000590910020D0

BB(8) : -0.000000097125621D0

BB(9) : -0.0000 00037 414793D0

BB(10) : 0.000000004587491D0

BB(11) : -0.000000003754394D0

THETA : 2.0D0*DATAN(T)

SUM : 0.0D0

rF (T.LE.1.0) THEN

DO 30 L:0,7

19

30

40

50

60

SUM : SUM + B(L)*(2.0D0*THETA/PI)**(2*L)

CONTINUE

ELSE

DO 40 L:0,11

SUM : SUM + BB(L)+(2,ODO_2.ODO*THETA/PI)**(2*L)

CONTINUE

END IF

rF (T.LE.1.0) THEN

FIRST : SUM*THETA _ THETA*DLOG(THETA)

ELSE

FIRST : SUM*(PI-THETA)

END IF

THETA_PI_THETA

SUM : 0.0D0

rF (T.LE.1.0) THEN

DO 50 L:0,7

SUM : SUM + B(L)*(2.0D0*THETA/PI)**(2*L)

CONTINUE

ELSE

DO 60 L:0,11

SUM : SUM + BB(L)*(2.0D0-2.0D0*THETA/Pr)+*(2*L)

CONTINUE

END IF

rF (T.LE.1.0) THEN

20

SECOND : SUM*THETA - THETA*DLOG(THETA)

ELSE

SECOND : SUM*(PI-THETA)

END IF

ZZ : -0.5D0*(FIRST+SECOND)

FUN : DEXP (-T*X) *DEXP (-ZZ / PI) / ( 1. 0D 0+T* *2) * *0. 7b

RETURN

END


The results obtained from the single integral (2.19) are displayed in Ta-

bre 2.2, compared with the 3-figure table given by Stewartson ([g] 1960).

The Stewartson solution method has been generalised by Varley and

Walker ([6] 1989) to a method for solving the integro-differential equation

u(r) :: [^* :(")^a" (o < , ( -),TtJo s-r

where u(ø) and u(ø) can be expressed in terms of the unknown function

/(ø) and its derivatives as

d" f dn_r fu : d"ñ * an-tã,,-, + ... + crof

and

' : b*# t bn-t#+ ... + óof.

However, this generalisation still does not allow us to treat the case c f con-

stant. Instead, we use a direct method, which we are going to describe in

the following chapters.

27

r)

T¿ble 2.2: Nurnerical Reeults for J(ø) Obtained fro¡n Stewa¡teon'e Double T:rtegral

1.0000.5640.4380.3610.3080.2670.2370.2120.1920.1750.1610.1130.086

1.0000000.5639120.4379150.3610560.3075910.2677980.2368830.2121260.1918420.t749200.1605930.1129380.086303

0.000.200.400.600.801.001.201.401.601.802.003.004.00

from StewartsonTs tableour methodr fr

22

Chapter 3

Serni-infinite Wing with anAsyrnptot ically ConstantChord

3.1- Nurnerical Method

In this chapter, we shall assume that c is a function of g which tends

to a constant coo as y tends to infinity; this includes as a special case

Stewartson's case c : constant. For this class of c(y), the appropriate

boundary conditions on I are

|(0) : g, l(-) : ¡rcao(Jao.

Let us write equation (1.22) as

f(y) : :"c(y)l(rc,s* * 1""1r-

++nJ"r'þt)dnq-a l, (3.1)

where .t is a positive constant. The first integral in (3.1) is the part of

the integral equation which is going to be evaluated numerically, while the

second integral will be evaluated analytically in advance, to give a correction

to the numerical process. We have to choose a value ror L later, to give a

23

certain accuracy. Motivated by Stewartson, let us write

f(y) = rrc*tlas(t - 2) (3.2)a'

as the approximation of l(y) for U ) L, where rc is a constant and is yet

unknown. Hence

f'(y) = nc*Uaorc

.y2 (3'3)

Now the second integral in (3.1) becomes

+ f"r'Ø)dn =! f coouaorcdq. (3.4)4rJt q-y -4lt n,Ø-y)

\Me know that this is equal to f,rc*Uo¡rc times the Hilbert transform of

U-2, so we obtain

* l:W =f,"*u*""(h",st ht -h), v +0,a + L (85)

Using the result in (3.5), we rewrite (3.1) as

f(y) : nc(y)l(ras + * 1""'T44, * I"*u,""(hrcr I h I -hn(3.6)

Since the aerodynamics at the wing-tip is crucial,'we are going to use a non-

uniform grid which is concentrated at the wing-tip to solve the Prandtl's

lifting-line equation. The process will be described below.

Define

y:Lt2, o<¿<1, (B.z)

n:Lu2, o(z(1, (g.s)

and hence

drl :2Ludu.

24

(3.e)

Now if we define

then we obtain the relationships

r(v)

and

h(u):l'(Lu2)2Lu

f,(Lu2)2Ludu

(r)

u

f'

h(

: l,': 1",: 1",

dr¡

)du,

(3.10)

(3.11)

(3.12)

l"

l"

I t'(Lu2)2LuduL(uz - tz)h(u)d,u

L(uz - tz¡

1

By substituting (3.11) and (3.12) into (3.6) we obtain

|o,nçu¡au+f,"{r'*>lI"'m|trcoo(Jag"(#_#'"*l#zlll: rc(Lt2)uao. (9.18)

Now we use a uniform grid in u for equation (3.18), i.e. let

ui : j ln, h¡ : h(u¡), j :0,7)2,... )n. (8.14)

Applying the trapezoidal rule of integration to equation (8.18) with this

grid, is equivalent to applying the trapezoidal rule with a non-uniforrn grid

in 4 to equation (3.6), since q : Lu2. Again we have to choose a value of

n in (3.14) such that, together with choosing ,t in (8.1), we obtain a certain

accuracy. Now if we replace f in (8.13) by

t' t:( 1

-t-i - ,i - ¡)1", (3.15)

25

where 'i : Ir2r3r... , r? respectively, then by applying the numerical method

we have chosen above, we obtain a system of rz linear algebraic equations

with n * 2 unknowns i.e. horhtrhzr. . . ,hn and ¡c. The i-th equation of this

system is of the form

1I-'4n

_c(Lt?-)r ho _ hn _ts 2h¡ .- 8', L@;:6-fçti_,-r- kf@j-q)r1.L1

*]trc*uasc(Lt2¿-)n(øT - ¡ra ,t"g1x-;i x-;l) : rc(Lt?_;)uoo

ó(i) (h¿-, *

1

L-t?,"2

h¿-t I h¿

2

r r?'

(3.16)

where

0 ifi:l#(t , a lr') in i, :2h(no* h¿-t +Di-:r2hi) in i:3,4,5,,

Note that rtye use t¿-r here instead of t¿: if nri:0,1,2r...rn in order to

avoid the singularities.

Using (3.11) we obtain

t(¿) : lo' rrçu¡au. (s.12)

Then using (3.2) and (3.17) we obtain

¡l

Jo' nç"¡a": rcoo(Jaort - i). (3.1S)

Therefore numerically we obtain the extra (n*l)'st equation

*ro, t h* + .i.ro,, * ry3n, : ircoo(J .,o. (s.1e)j=t

So far we have n + 7 equations with n * 2 unknowns.

26

Now using (3.3) and (3.10), we obtain a relationship between rc and

hn as followsLt*: *"*r1^h'' (3'20)

where hn : h(u) l"=t. This reduces the number of unknowns to n*1.

Substituting (3.20) into (3.16) and (3.19) yields

1 .-' h¿-, * h¿,ó(i)+

^(h,-'+ï)Lc(Lt?-r)¡ ho - hn -S 2h¡ t' s" tfç6, - tç,r, - kfej -q)t

+!"e,tl-t)h,( Lt? . Ltl ,t-; x-il"s

I

1 1 1 l): n"(Lt?-¡)U'o,I-t? ,2-;

'i :7r2r3r, . . ,n (3.21)

and1Tr1

b(h" + h* + Lzn¡) * ;n": Tcco(Jo¿o. (3.22);tt, j=t

So we have a system of.nl1 linear algebraic equations with n*1 unknowns

i.e. hsrhtrhrr. . . ,hn which can be solved by any available method.

Once horht,,hrr...rhn have been found, v¡e can calculate

l@o) for

i : 0,1,,2, . .. , n where a¿ : Lt? : L(*)'.This can be done numerically i.e.

from (3.11) we obtain

r(v¿) : 01

2n1

2n

ifi:0(åo*hr) ifi:1(åo * h +D'j--lzhj) if i : 2,,9,4,...,n.

(3.23)

3.2 Listing Program

The program in Fortran 77 for solving Prandtl's lifting-tine equation (I.22)

for asymptotically constant chord, using the method described above, is

27

listed below, taking U : ao: 1, where FO4JGF is a NAG routine, available

from the computer system, for finding the solution of a linear algebraic

equations system. For a description of this NAG routine, see Appendix B.

IMPLTCTT REAL*8 (A-H,O-Z)

PARAMETER (LEN:400)

REAL*8 AA(LEN,LEN),F(LEN),WK(4*LEN)

coMMoN IPAP"I ALPHA,YU

LOGICAL SVD

ALPHA : I.ODO

YU :1.0D0

PI : 3. 1415926535897932384626D0

CINF E ......... (Here is the value of c*)

NRA : LEN

TOL :5.0D-16

LW : 4*LEN

IFAIL : O

PRINT *,"READ FROM FILE 'FORT.l':"

READ(1,*)RL,N

PRINT 'kr)rL:t),RL," N :",N

PRINT *,"'WAIT ......'

IG : N*l

JG:IG

DO 20 I:l,N

28

T : (DFLOAT(r)-0.5D0)/DFLOAT(N)

Y - RL*T>k>I,2

DO 10 J:I,JG

U : DFLOAT(J-1)/DFLOAT(N)

DENOM : RL*(T**2_U**2)

rF (J.EQ.1.OR.J.EQ.Jc) THEN

W : 1.0D0

ELSE

\M : 2.0D0

END IF

AA(I,J) : \M*C(Y)/(8.0D0*DFLOAT(N)*DENOM)

rF (J.EQ.r) THEN

AA(I'J) : AA(I,J) + 3.0D0/(8.0D0*DFLOAT(N))

END IF

rF (J.EQ.r+1) THEN


END IF

rF (r.NE.1.AND.J.EQ.1) THEN


END IF

rF (r.NE.1.AND.J.EQ.r) THEN

AA(I,J) : AA(I,J) + 1.0D0/(2.0D0*DFLOAT(N))

END IF

29

10

20

30

rF (r.GT.2) THEN

rF (J.LT.r.AND.J.GT.1) THEN

AA(I,J) : AA(I,J) + 1.0D0/DFLOAT(N)

END IF

END IF

CONTINUE

TEMP : DLOG(DABS(1.0D0/(1.0D0-T**2))) / (Rr_,*t**+¡

TEMP : 1.0D0/(RL*T**2) - TEMP

AA(I,JG) : AA(I,JG) + C(v¡*TEMP/8.0D0

F(I) : PI*C(Y)*YU*ALPIIA

CONTINUE

AA(IG,1) : 1.0D0/DFLOAT(2*N)

AA(IG,JG) : AA(rG,1)

DO 30 J:2,N

AA(rG,J) - 1.0D0/DFLOAT(N)

CONTINUE

AA(IG,JG) : AA(IG,JG) + 0.bD0

F(IG) : PI*CINF*YU*ALPHA

CALL FO4J GF(IG, J G,AA,NRA,F,TOL,SVD,SIG,IR,\MK,LW,IFAIL)

Y : 0.0D0

GAMMA : O.ODO

WRITE(2,40)Y,GAMMA

DO 50 I:2,IG

30

Y : RL*(DFLOAT(I-1)/DFLOAT(N))**2

GAMMA : GAMMA + (F(I-l)+F(I))/(2.0D0*DFLOAr(N))

WRITE(2,40)Y,cAMMA

40 FORMAT(F7.4,5X,F11.8)

50 CONTINUE

PRINT X,"SEE FILE 'FORT.2' FOR THE RESULTS.''

STOP

END

+'F* * + * * * * * * * * * * + * * * * + * * * * * x {< {< * * + * * {< * * *{< ** * * {< * * * ** * * * * * *

REAL*8 FUNCTTON C(Y)

IMPLICTT REAL*8 (A_H,O_Z)

coMMoN IPAP"/ ALPHA,YU

C: ( Here is the function c: c(y) )

RETURN

END


As a first example, we turn our attention to a wing with constant chord,

as studied by Stewartson ([8]1g60). Let c : fJ : do : 1. If we choose

L - 20rn : 68, for instance, then we obtain the results as shown in

Table 3.1. Further, if we transform the results from l(y) to /(ø) using

(2.2), then we obtain the results as shown in Table 8.3, compared with the

results obtained by Stewartson's method, i.e. from formula (2.1g). From

Table 3.3, we can see that relative error peaks at y/L:0.0002 with error

31

€.1 : 0.00054348, at A /L : 0.0106 with error ez : 0.00053283 and at

AIL:0.2500 with error es: 0.00002712. Relative error increases again

at the upper end of the range 0 < y < L. At y/L: 1: the relative error

is large, which is ea - 0.00362680. In general, if we choose other values

of .t and n, most results have peak errors et¡ê2¡€3 and ea similar those

we mentioned above, with e1,e2 arLd e3 slightly shifting in their positions.

The error e3 is always much less than the others and therefore we shall not

consider it. Both €1 and e2 ãîê sensitive to n, wher€âs €4 is sensitivelo L.

If we keep .t constant, then e1 increases as ?? increases. on the other hand,

e2 decreases as r? increases. Optimum balance between .t and n is when

€1 and €.2 ãîe about equal, and Table 3.2 shows the best such choice of n

for each L,,e.g.rz:68 is the optimal choice for n when L:20. In these

circumtances, maximum relative-errors for all pairs of .t and rL aiÍe equal

to 0.00054, in the range specified in Table 3.2, for L : 10,20,80,40 and

50. For all values of .t and n, maximum relative errors are never less than

0.00054. However, in the case where -t and n are balanced, absolute errors

are less than 0.0005 for all values of y in the whole range 0 < y 1 L.

As a second example, we consider the chord

( 0 if y:Qclv):\ ify>o. G.24)

The solution of equation (L22) is therefore

r(v) : n t*vAgain, we take U : ao : 7. If we choose L : I0 and n - 60, then we

obtain the results as shown in Table 3.4, compared with the exact solution.

v

32

Relative error is large for small value of y. The error decreases as n in-

creases. In order to obtain relative error less than 0.0005 for all values of

y inthe range 01A (.t, wemay chooser¿:60for L:L0, n:85forL :20 and r¿ : 100 far L: 30. For larger value of .t, we need larger value

of n.

33

Table 3.1: Numerical Reeults for Prandtl's Lifting-line Equation with Constant Chord

(L-20,n-68)

Table 3.2: Optimum Bala¡rce between L and n

3.049322583.053336663.057098523.060628323.063944343.067063133.069999713.072767723.075379583.077846603.080179113.082386533.084477463.086459773.088340653.090126643.091823683.093437043.094971243.096429573.097812453.099105503.10033294

9.L5229.55459.9654

10.384910.813111.250011.695572.749772.612513.083913.564014.0528L4.550215.056215.570916.094316.6263L7.L67077.7L6318.274218.840819.416120.0000

2.777798432.804429772.8290L9722.851260302.87L472462.889704t42.906336842.92L488282.935315302.947956362.959533842.970155982.979918602.988906682.9977956L3.004852433.011936783.018501823.024595003.030258753.035531033.040445893.04503391

2.28812.49132.70332.92393.15313.39103.63753.89274.15664.429r4.71025.00005.29845.60555.92L36.24576.57876.92047.27087.62987.99748.37378.7587

0.000000000.260236230.511670150.749567L60.97L547321.776547691.364363061.535358221.690269171.830061001.955825382.068707302.169853542.260377622.34t336952.473719702.478434662.536314882.588112552.634505022.676098702.773434272.74699227

0.00000.00430.01730.03890.06920.10810.15570.21190.27680.35030.43250.52340.62280.73100.84780.97321.10731.25001.40L41.56141.73017.90742.0934

f(v)avIraf(s)v

0.000540.000540.000540.000540.00054

0<alL<0.660<ylL<0.830<ylL<0.910<alL<0.940<alL<0.96

48688395L07

1020304050

Maximum relative errorRangenL

34

Table 3.3: Numerical Resulte for J(o) - a Comparison

(L=2O,n=68)

0.00000.00020.00090.00190.00350.00540.00780.01060.01380.01750.02160.02620.03110.03650.04240.04870.05540.06250.07010.07810.08650.09540.70470.77440.72460.13520.L4620.L5770.16950.18190.19460.20780.22750.2355

0.000000070.000543480.000012520.000262810.000410670.000488190.000523790.000532830.000524330.000503880.000475350.000447440.000404280.000365480.000326300.000287670.000250380.000214930.000181780.000151130.000123320.000098040.000075640.000055870.000038760.000023920.000011470.000001050.000007730.000014290.000019270.000023150.000025470.00002692

0.000000070.000498730.000010480.000200050.000283550.000305210.000296160.000272280.000242100.000210250.000179330.000150690.000125000.000102480.000083090.000066630.000052840.000041400.000032020.000024390.000018270.000013360.000009500.000006480.000004160.000002380.000001060.000000090.000000620.000001070.000001350.000001520.000001570.00000156

1.000000070.917662960.837140810.761205310.690463330.625188030.565413590.511008020.461728510.477263240.377262770.341359330.309189170.280396980.254646270.237622720.211036910.192624500.776146770.161386620.148153060.136273680.725595720.115983800.107318120,099492870.092414600.086000940.080179220.074885320.070062750.065661640.061637920.05795268

1.000000000.977764230.837130330.761405360.690746880.625493240.565709750.511280300.461970610.4L7473490.377447440.341510020.309314170.280499460.254729300.231689350.211089750.192665900.176178190.161411010.148171330.136287040.L25605220.115990280.L07322280.099495250.092415660.086001030.080178600.074884250.070061400.065660120.061636350.05795112

0.00000.00550.02200.04960.08810.73770.19830.26980.35250.44670.55070.66640.79300.93071.0794L.23971.40981.59157.78431.98812.20282.42862.66542.91333.t7273.44793.72284.01.474.3L764.63154.95645.29235.63935.9972

alLrelativeerror

absoluteerrorfrom (2.6)our method

T It

35

See next page -)

Table 3.3 (continued):

0.25000.26490.28030.29610.31230.32890.34600.36350.38150.39990.4t870.43790.4ó760.47770.49830.51920.54070.56250.58480.60750.63060.65420.67820.70260.72750.75280.77850.80470.83130.85840.88580.91370.94200.97081.0000

0.000027120.000026230.000024690.0000224L0.000019300.000015510.000011230.000006440.000000850.000005330.000012110.000019200.000027580.000036310.000046110.000056650.000067980.000080520.000094790.000110930.000128600.000148860.000L72420.000199550.000231070.000269030.000316310.000374940.000450910.000554180.000701300.000930900.001340810.002386620.00362680

0.000001480.000001350.000001200.000001030.000000840.000000640.000000440.000000240.000000030.000000180.000000390.000000590.000000810.000001020.000001240.000001460.000001680.000001910.000002160.000002430.000002710.000003020.000003370.000003760.000004200.000004720.000005360.000006140.000007140.000008490.000010400.000013370.000018660.000032200.00004746

0.054577470.051463800.048602660.045964040.043526610.04L271390.039181470.037241750.035438710.033760280.032195620.030735010.029369670.028091730.026894080.025770290.024714540.02372L570.022786580.021905230.021073570.020287980.019545170.018842140.018176130.017544620.016945280.016376000.015834820.015319920.014829660.014362490.013917010.013491880.01308591

0.054569990.05L462450.048601460.045963010.043525770.041270750.039181030.03724t5r0.035438680.033760460.032196010.030735600.029370480.028092750.026895320.025777750.024776220.023723480.022788740.021907660.021076280.020291000.019548540.018845900.018180330.017549340.016950640.016382140.015841960.015328410.014840060.014375860.013935670.013524080.01313337

6.36626.74627.13727.53927.95228.37638.81139.25749.7L4510.182610.661711.151911.653012.L65212.688313.2225L3.767774.323974.897215.469416.058716.659017.270277.892518.525979.t70219.825520.49L921.169327.857622.557023.267523.988924.727325.4648

alLrelativeerror

absoluteerrorfrom (2.6)our method

fr T

36

0.00000.00030.00110.00250.00440.00690.01000.01360.01780.02250.02780.03360.04000.04690.05440.06250.07110.08030.09000.10030.11110.L2250.73440.14690.16000.17360.18780.20250.27780.2336

0.000000000.000432000.000421860.000405600.000384070.000358380.000329700.000299210.000268050.000237180.000207370.000179220.000153120.000129300.000107880.000088830.000072080.000057480.000044840.000033980.000024690.000016790.000010100.000004440.000000330.000004350.000007730.000010580.000012990.00001504

0.000000000.000071430.000138930.000199000.000248900.000286900.000312300.000325360.000327170.000319340.000303750.000282390.000257120.000229590.000201230.000173080.000145990.000120500.000096950.000075530.000056280.000039150.000024020.000010760.000000810.000010880.000019620.000027200.000033790.00003953

0.000000000.165346980.329328390.490634340.648061080.800551940.947225831.087392311.220553811.346396851.464775r3r.575687251.679251911.775682801.865264881.948333052.025254002.096411082.162L92352.222987462.279750862.331057092.379037732.423409622.464468042.502486632.537717902.570393992.600727752.62891393

0.000000000.165275550.329189460.490435340.647812180.800265040.946913531.08706695L.22022664r.3460775L1.46447r381.575404867.678994797.7754532L1.865063651.948159972.025108012.096290582.L62095402.222905932.279094582.331017942.3790L3772.423398862.464468852.502497572.537737522.570421792.60076t542.62895346

0.00000.00280.01110.02500.04440.06940.10000.13610.17780.22500.27780.33610.40000.46940.54440.62500.71110.80280.90001.00281.11117.22507.34447.46941.60001.73611.87782.02502.L7782.3361

alLrelativeerror

absoluteerrorour exact u

Iv

T¿ble 3.4: Nurnerical Reeulte for Prandtl'e Lifting-line Equation with Chord Defined by (3.2a)

(.L = 1O, zr = 60)

See next page ->

37


0.25000.26690.28440.30250.32110.34030.36000.38030.40110.42250.44440.46690.49000.51360.53780.56250.58780.61360.64000.66690.69440.72250.75110.78030.81000.84030.87110.90250.93440.96691.0000

0.000016780.000018290.000019600.000020750.000021790.000022750.000023630.000024480.000025310.000026130.000026970.000027840.000028750.000029720.000030760.000031890.000033120.000034470.000035960.000037610.000039430.000041460.000043710.000046170.000048840.000051590.000054100.000055410.000052270.000027150.00006045

0.000044560.000049010.000052960.000056520.000059780.000062820.000065670.000068430.000071130.000073820.000076560.000079370.000082320.000085420.000088750.000092310.000096190.000100420.000105060.000110180.000115820.000122080.000128990.000136570.000144750.000153210.000160970.000165170.000156070.000081210.00018107

2.655130402.679539462.702289082.723574052.7433371.82.761870322.779275402.795465262.810704592.825010602.838453792.851098542.863003682.874223072.884805962.894797542.904239272.913168992.92762L822.929629832.937222602.944427362.9ó7269272.95777L522.963955552.969841192.975446792.980789362.985884682.990747382.99539107

2.655174962.679588472.702342042.723570572.743396962.76L933L42.779287072.795533692.810775722.825084422.838530352.851177972.863086002.874308492.884894772.894889852.904335402.9L3269412.92L726882.929740072.937338422.944549442.951398262.957908092.964100302.969994402.975607762.980954532.986040752.990828592.99557274

2.50002.66942.84443.02503.2L773.40283.60003.80284.01114.22504.44444.66944.90005.13615.37785.62505.87786.13616.40006.66946.94447.22507.51117.80288.10008.40288.71119.02509.34449.6694

10.0000

ylLrelativeerror

absoluteerrorexact solutionour method

v I v

38

.1

Chapter 4

Serni-infinite Wing Chord.(V) Proportional to yt as yTends to Infinity

4.L Numerical Method

Another class of c(y), which is of practical importance, is the class where

c(y) is proportion al to yI as y tends to infinity. For example, this is the

case when r4/e are studying flow near the tip of a wing of elliptic planform,

which is very common. In this case, the curves , : t.(A) are touching the

elliptic curvature near the wing-tip as sketched in Figure 4.!.

For this class of c(g), we use the same method as that we have used

in chapter 3, except a different correction term for replacing the second

integral in equation (3.1). Let us write

t(y) x ny* (4.1)

as the approximation of l(y) for a > L, where rc is an unknown constant.

Hence

r'(y) = T"o-r. (4.2)

39

xx:c(y)

x:-c(y)

Figrrre 4.1: The curves o - Ic(y) are touching the ellipse at one end.

Now the second integral in (3.1) can be evaluated analytically, i.e.

Iì/ n)dnn-a

becomes

v

l- /-4" J" = * l:'#f : #^rt'# t (v #o,v * L)

(4.3)

Hence the integro-differential equation for the circulation l(y) becomes

r(y) : nc(y)l(Ias + * 1""';!1, + #"^*l ,," _ r+W ll. @.4)

Performing the same substitutions as those in chapter 3, equation @.a)

lo'nçu¡au+f,"@t ¡tl,'#!h- #".e ri= ll

: rc(Lt2)(Jao. (4.b)

Then from (3.L7) and (4.1) we obtain

lo' nçu¡ar: LL n. (4.6)

Using (3.10) and (4.2), we obtain a relationship between rc and hn asfollows

1

L;

40

hn, (4.7)

where h^: h(u) 1.,=r. Hence using trapezoidal rule with the same grid as

the previous section yields

t h¿-t * h¿,,ó(i)+*(h-a+ 2 )

c( Lt?\ ¿-

8Lt¿-t) 1+Í' 1

lz,, los | å l: ""(Lt?_t)(J ao,

'2i:Lr2r3r...rn (4.8)

and1

(åo * h" + Dzh¡) - hn :0.

4.2 Listing Program

The program in Fortran 77 for solving Prandtl's lifting-line equation (L.22)

when the Chord c(y) is proportional b y* as y tend.s to infinity, using

the method described above, is listed below, taking (J : o,o: 1, where

FO4JGF is a NAG routine, available from the computer system, for finding

the solution of a linear algebraic equations system. For a description of this

NAG routine, see Appendix B.

n-l

2n(4.e)

j=l

Note that

I o if i:rö(i):l *(n"¡h) if i:2

|. #(nr r h¿t +D'j-:zhj) if i :3,4,5,. . . ,n.

Now we have a system of. n 11 linear algebraic equations with n * 1 un-

knowns, which is easily solved. Then we can calculate l(g) bv means of

(3.23) as before.

47


PARAMETER (LEN:400)

REAL*8 AA(LEN,LEN),F(LEN),WK(4*LEN)

LOGICAL SVD

coMMoN IPAP"LI ALPHA,YU

coMMoN /PAP.2/ Pr

ALPHA : I.ODO

YU: 1.0D0

PI : 3.1415926535897932384626D0

NRA : LEN

TOL : 5.0D-16

LW : 4*LEN

IFAIL : O

PRINT *,"READ FROM FILE 'FORT.l':"

READ(1,*)RL,N

PRINT *,ttL :tt,RL," N:",N

PRINT *,''WAIT ......"

IG : NtlJG:IG

DO 20 I:l,Nr - (DFLoAr(r)-o.5Do)/DFLoAr(N)

Y : ¡¡*1**2

DO 10 J:I,JG')

42

U : DFLOAT(J-1)/DFLOAT(N)

DENOM : RL*(T**2_U+*2)

rF (J.EQ. 1.OR.J.EQ.JG) THEN

W : 1.0D0

ELSE

W : 2.0D0

END IF

AA(I,J) : W*C(Y)/(s.0D0*DFLOAT(N)*DENOM)

rF (J.EQ.r) THEN

AA(I,J) : AA(I,J) + 3.ODO/(8.ODO*DFLOAT(N))

END IF

rF (J.EQ.r+1) rHEN


END IF



END IF



END IF

rF (r.GT.2) THEN

rF (J.LT.r.AND.J.cT.1) THEN


43

10

20

30

END IF

END IF

CONTINUE

TEMP : DLOG(DABS((1.ODO+T)/(1.ODO-T)))

TEMP : C(Y)*TEMP/ (8.0D0*RL*T)

AA(r,JG) : AA(I,JG) - TEMP

F(I) : PI*C(Y)*ALPHA*YU

CONTINUE


AA(IG,JG) : AA(IG,1)

DO 30 J:2,N

AA(IG,J) : 1.0D0/DFLOAT(N)

CONTINUE

AA(IG,JG) : AA(rG,Jc) - 1.0D0

F(IG): 0.0D0

C ALL FO4J G F( IG, J G, AA, NRA, F,T O L,SVD, S IG, IR,\ryK, LW, IFAIL )

Y : 0.0D0

GAMMA : O.ODO

WRITE(2,40)Y,cAMMA

DO 50 I:2,IG

Y : RL*(DFLOAT(I_1)/DFLOAT(N))**2

GAMMA : GAMMA + (F(I-1)+F(I))/(2.0D0*DFL9AT(N))

WRITE(2,40)Y,cAMMA

FORMAT(F8.4,5X,F1 2. 8)

44

40

50 CONTINUE

PRINT *,"SEE FILE 'FORT.2' FOR THE RESULTS."

STOP

END

********************x**********************************

REAL*8 FUNCTTON C(Y)


coMMoN /PAR1/ALPHA,YU

coMMoN lPLPú2lPr

C - ............. ( Here is the function c:c(y) )

RETURN

END

4.3 Nurnerical Results

As a first example, we consider

/\1cla) : y' .

The exact solution for equatior' (1.22) is therefore

f(y) : nuasyT

Now let us compare the results obtained from our numerical method with

the exact solution. Let U - do:1. If we choose L:20 and n:70, then

we obtain the numerical results as shown in Table 4.1, compared with the

45

exact solution. As a second example, we consider the chord

.(v) :

The exact solution for (1.22) is

(ol+( taSr[,IooI no!-rl)(t-r)(@

ify:Qif. y :1if0<y<7or1<y<æ.

(4.10)

r(v) :Again, let [/ : oo : 1 and choose L :20rn:70. We obtain the numerical

results as shown in Table 4.2, contpared with the exact solution. In both

these examples, the errors increase monotonously as y increases. The results

are good only for small values of y. However, for a particular value of y,

the error decreases if we take larger values of. n ot L.

46

0.00000.00020.00080.00180.00330.00510.00730.01000.01310.01650.02040.02470.02940.03450.04000.04590.05220.05900.06610.07370.08160.09000.09880.10800.1 1750.72760.13800.14880.16000.17160.18370.19610.20900.22220.2359

0.000000000.000390220.000390370.000390670.000391050.000391540.000392140.000392880.000393710.000394670.000395740.000396940.000398260.000399710.000401300.000403010.000404870.000406870.000409010.000411320.000413780.000416410.0004192L0.000422200.000425380.000428750.000432340.000436150.000440190.000444480.000449040.000453870.000459000.000464440.00047022

0.000000000.000078320.000156700.000235230.000313950.000392930.000472240.000551980.000632170.000712920.000794290.000876370.000959220.001042940.001127610.001213320.001300170.001388250.001477670.001568540.001660990.001755120.001851080.001949000.002049050.002151370.002256150.002363560.002473820.002587150.002703770.002823960.002947990.003076180.00320886

0.000000000.200708990.401417980.602126980.802835971.003544967.204253951.404962951.605671941.806380932.007089922.207798922.408507912.609216902.809925893.010634883.211343883.4\2052873.612761863.813470854.014179854.214888844.475597834.616306824.817015825.0L77248L5.218433805.479742795.619851785.820560786.027269776.227978766.422687756.623396756.824L0574

0.000000000.200630670.401261280.601891750.802522021.00315203r.20378r7L7.4044L0971.605039771.805668012.006295632.206922552.407548692.608173962.808798283.009421563.210043713.410664623.611284193.811902314.012518864.2L3733724.413746754.614357824.874966775.0L5573445.216177655.476779235.617377965.817973636.018566006.219154806.419739766.620320576.82089688

0.00000.00410.01630.03670.06530.10200.14690.20000.26720.33060.40820.49390.58780.68980.80000.91847.04491.17967.32241.4735r.63271.80001.97552.75922.35102.55102.75922.97553.20003.43273.67353.92244.17964.44494.7784

a/Lrelativeerror


v

Table 4.1: Numerical Reeults for Pra¡rdtl's Lifting-line Equation with Chord c({ = gtr

(L=2O,n=70)

47

See next page ->


0.25000.26450.27940.29470.31040.32650.34310.36000.37730.39510.41330.43180.45080.47020.49000.51020.53080.55180.57330.59510.61730.64000.66310.68650.71040.73470.75940.78450.81000.83590.86220.88900.91610.94370.97161.0000

0.000476370.000482900.000489860.000497260.000505150.000513580.000522580.000532210.000542530.000553610.000565520.000578360.000592220.000607220.000623490.000641210.000660550.000681740.000705050.000730790.000759370.000791250.000827050.000867520.000913640.000966680.001028360.001101030.001188020.001294300.001427650.001601230.001840030.002201360.002883090.00333873

0.003346400.003489220.003637780.003792570.003954170.004123190.004300340.004486430.004682330.004889050.005107750.005339760.005586570.005849940.006131900.006434830.006761510.007115250.007500010.007920550.008382650.008893450.009461810.010098920.010819180.011641330.012590490.013701120.015022110.016625790.078625220.0272Lr760.024743840.030044560.039927690.04690797

7.024814737.225523727.426232727.626947777.827650708.028359698.229068688.429777688.630486678.831195669.031904659.2326L3659.433322649.634031639.83474062

10.0354496210.2361586110.4368676010.6375765910.8382855911.0389945811.23970357L7.44047256L7.6411215511.8418305512.04253954t2.2432485312.44395752L2.6446665212.84537ó5713.0460845013.2467934973.4475024913.6482114873.8489204774.04962946

7.021468337.222034507.422594947.623749147.823696538.024236508.224768348.425297258.625804348.826306619.026796909.227273899.427736079.628181699.82860872

I0.0290747910.2293977070.4297523510.6300765810.8303650411.0306119311.2308101211.4309507511.6310226311.8310113712.0308982112.2306580472.43025640L2.62964441t2.8287497213.0274592873.2255823373.4227586513.6181669213.8089927814.00272L49

5.00005.28985.58785.89396.20826.53066.86127.20007.54697.90208.26538.63679.01639.40479.8000

70.204710.616311.036711.465311.902012.346912.800073.267213.730674.208214.693915.187815.689816.200076.7184L7.244977.779678.322418.87351.9.4327

20.0000

ylLrelativeerror


a

48

0.00000.00020.00080.00180.00330.00510.00730.01000.01310.01650.02040.02470.02940.03450.04000.04590.05220.05900.06610.07370.08160.09000.09880.10800.11750.72760.13800.14880.16000.17160.18370.19610.20900.22220.2359

0.000000000.004373740.007545240.002592580.002953630.003110560.003189850.003234520.003262100.003280530.003294110.003304940.003314240.003322780.003331020.003339210.003347550.003356190.003365230.003374700.003384700.003395240.003406360.003418090.003430490.003443560.003457330.003471850.003487170.003503280.003520250.003538100.003556910.003576700.00359752

0.000000000.000016780.000022370.000079920.000153630.000240560.000338830.000447000.000563920.000688630.000820400.000958580.001102630.001252110.001406660.001565950.00L729720.001897780.002069950.002246080.002426090.002609890.002797430.002988680.003183650.003382340.003584790.003791070.004001260.004215430.004433720.004656250.004883200.005L74720.00535103

0.000000000.003836530.014476760.030826410.052013930.077336560.70622L370.138196600.172870280.209914010.249050640.290044590.332694350.376826560.422291240.468957970.576712820.565455830.615098900.665564100.776782270.768691770.82L237520.874370130.928045210.982222721.036866471.091943681.t47424561.203287971.259491181.316029577.372876421.430012697.48742097

0.000000000.003853310.014454390.030746490.051860300.077096000.105882540.137749600.172306360.20922ó380.248230240.289086010.331591720.375574450.420884580.467392020.514983100.563558050.613028950.663318020.714356180.766081880.818440090.871381450.924861560.978840381.033281681.088152617.143423301.19906654L.255057461.311373327.36799322r.424897971.48206988

0.00000.00410.01630.03670.06530.10200.14690.20000.26120.33060.40820.49390.58780.68980.80000.91847.04491.1 7967.3224L.47357.63271.80001.97552.75922.35102.55102.75922.97553.20003.43273.67353.92244.L7964.44494.7784

ylLrelativeerror


a

"a)

Ta,ble 4.2: Numerical Results for Pra¡rdtl's Lifting-Iine Equation with Chord Defined by (+.f O)

(L=2O,n=70)

49

See next page ->


0.25000.26450.27940.29470.31040.32650.34310.36000.37730.39510.41330.43180.45080.47020.49000.51020.53080.55180.57330.59510.61730.64000.66310.68650.71040.73470.75940.78450.81000.83590.86220.88900.91610.94370.97161.0000

0.003619450.003642540.003666860.003692490.003719520.003748020.003778130.003809950.003843610.003879260.003917060.003957210.003999920.004045420.004094010.004146010.004201800.004261820.004326600.004396760.004473080.004556460.004648060.004749290.004862010.004988560.005132120.005296990.005489280.005717990.005997120.006350150.006820830.007506120.008734760.01084271

0.005592360.005838950.006091100.006349120.006613370.006884220.007162140.007447610.00774L170.008043450.008355140.008677040.009010010.009355090.009713440.010086400.010475530.010882650.011309900.011759820.0L2235490.012740580.013279680.013858450.0t4484140.015166030.015916480.076752290.017697050.018785410.020070980.021643090.023667250.026507670.031385310.03962879

r.545084971.60299001L.667122301.719469131.778018731.836760161.895683271.954778602.014037352.073457292.133012752.192714562.252549982.3L2572732.372596892.432796922.493L07672.553524042.614041592.6746559r2.735362902.796158662.857039542.918002062.979042963.040159113.101347573.162605563.2239304L3.285319613.346770773.408281613.469849963.537473763.593151053.65487995

1.539492611.597151061.655031201.713120017.771405361.829875941.888521131.947330992.006296182.065407842.724657672.184037522.243539972.303757642.362883452.4227t0522.482632082.542641392.602731692.662896092.7231274t2.783418082.843759862.904L43672.964558823.024993083.085431093.t45853273.206233363.266534203.326699793.386638523.446L82713.504966093.56L765743.61525116

5.00005.28985.58785.89396.20826.53066.86127.20007.54697.90208.26538.63679.01639.404L9.8000

70.204L10.616311.036711.465311.902012.346912.800073.26L213.730674.208214.693915.187815.689816.200076.718477.244977.779678.322418.8735L9.432720.0000

alLrelativeerror


v vlt

50

ì',,.

'ì :l r\

ìjl)

., .1

¡'. .. ¿l

{

\

Chapter 5

Application of Prandtl'sLiftir,g-line Equation to theCornpliant Layer Problern

5.1- Varley-Walker Solution

Let us now let c(y) in (1.22) be c(y) - -C, where C is a positive constant

Then equation (L.22) becomes

* l'(n)dnq-vf(y): -ncltras**1" I (0(y(oo), (5.1)

The boundary condition is again l(0) :0. Define

f(y):-rcuasll-l(')1, rCu:

-1."4 (5.2)

By substituting (5.2) into equation (5.1), we obtain an integro-differential

equation in /(r), i.e.

f(*): -t7t l"* (, > o), (5.3)

which is the integro-differential equation in a compliant layer problem with

boundary condition /(0) : 1. Varley and Walker ([O] 1g8g) obtained the

51

exact solution for (5.3) as follows

r(*) :,/i cos(æ* ä) - + l"* #"*pt+ 1"" ffioao"

,,E cos(æ * ä)

(5.4)

5.2 Nurnerical Evaluation of the Varley-'WalkerSolution

Our method for evaluating the function (5.a) is similar to that for evaluating

the function (2.6).

\ /ith this method, formula (5.4) can be rewritten as

(r + "')i ""pt-#{ s(2arctans) + sþr - 2arc!,ar.s)}lds,

(5.5)

where g(ó) i" Clausen's integral defined by (2.11). To evaluate (5.b) nu-

merically, we rewrite it as

t@)

f(")

e-"tl,*1

1f

,,E cos(r * ä)

(1 + "')""pt-#{ s(2arctar.s) + oQr - 2arctans)}lds

SEI7f

I7f

e

l.*e-",

""pt-#{ s(2arctans) + oQr - 2arctans)}lds.

(5.6)

By choosing appropriate values of r, the second integral on the right hand

side of (5.6) is negligible. The values of r are chosen through experiments

to give acceptable results, and r is inversely proportional to z. The value

of r for 6 figure accuracy has been chosen as about r : r00 I r for all values

52

of æ, except for x : 0. Then (5.6) can be evaluated by any integration

method.

The program in Fortran 77 Íor evaluating (5.6), using the method

described in section 2.2, is listed below, where DO1AHF is a NAG routine,

available from the computer system, for evaluating the definite integral of

a function FUN over a finite range. For a description of this NAG routine,

see Appendix A.


coMMoN x,PI

EXTERNAL FUN

PI : 3. 1415926535897932384626D0

PRINT *,"READ FROM FILE 'FOFùT.l':"

READ(1,*)RL,N

PRINT '*r))L :)),RL," N :",N

PRINT *,"WAIT ..."

DO 10 J:O,N

x : RL*(DFLOAT(J)/DFLOAT(N))**2

rF (X.LT.5.0D-7) THEN

R : 1.0d5

ELSE

R: 1.0D2/X

END IF

RESULT : DSQRJT(2.ODO)*DCOS(X+PI/8.ODO)

oó

A : 0.0D0

EPS: 1.0D-7

NLIMIT : 10000

IFAIL : O

Z : DOl AHF(A,B,EPS,NPTS,REL,FUN,NLIMIT,IFAIL)

RESULT: RESULT _ZIPI

\4/RITE( 7, 20 )X, RES ULT

10 CONTINUE

20 FORMAT(f8.4,5X,F10.6)

PRINT *,"SEE FILE 'FORT.7'FOR THE RESULTS."

STOP

END

*>ß*rk*********+>t>k*****'1.>lc*,ß**{<>t<*****{<rt<>ß**{<***rkrF{<*t<>t<***rkrßrt<*

REAL*8 FUNCTTON FUN(S)


REAL*8 B(8),BB(12)

COMMON X,PI

B(0) : 1.0D0

B(1) : 0.034269459726077D0

B(2) : 0.000422782512969D0

B(3) : 0.00001 1827371'379D0

B(4) : 0.000000425578715D0

B:R

54

i-t-:1

.I

I

1.l

'l

I1l

I

30

B(5) : 0.000000017366602D0

B(6) : 0.0000000007õ2664D0

B(7) : 0.000000000042746D0

BB(0) : 0.693147180559945D0

BB(1) : -0.102808379178034D0

BB(2) : -0.0063 4L737696915D0

BB(3) : -0.0007 45L24327 445D0

BB(4) : -0.000108523615117D0

BB(5) : -0.0000t7757036264D0

88(6) : -0.0000 03L26027 L95D0

BB(7) : -0.000000590910020D0

BB(8) : -0.000000097125621D0

BB(9) : -0.000000037414793D0

BB(10) : 0.000000004587491D0

BB(11) : -0.000000003754394D0

THETA : 2.0D0*DATAN(S)

SUM : 0.0D0

rF (s.LE.1.0) THEN

DO 30 L:0,7

SUM : SUM + B(L)*(2.0D0*THETA/PI)** (2*L)

CONTINUE

ELSE

DO 40 L:0,11

bb

40

50

60

SUM : SUM + BB(L)*(2.0D0-2.0D0*THETA/PI)**(2*L)

CONTINUE

ENDIF

rF (s.r8.1.0) THEN

FIRST : SUM*THETA _ THETA*DLOG(THETA)

ELSE

FIRST : SUM*(PI-THETA)

END IF

THETA:PI_THETA

SUM : 0.0D0

rF (s.LE.1.0) THEN

DO 50 L:0,7

suM : suM + B(L)+(2.0D0*THETA/PI)** (2*L)

CONTINUE

ELSE

DO 60 L:0,11

SUM - SUM + BB(L)*(2.0D0-2.0D0*THETA/er)**(2*L)

CONTINUE

END IF

rF (s.LE.l.o) THEN

SECOND : SUM*THETA - THETA*DLOG(THETA)

ELSE

SECOND : SUM*(PI-THETA)

56

END IF

ZZ : -0.5D0*(FIRST+SECOND)

FUN : DEXP (-X*S) *DEXP (ZZ I PD I (1. 0D0+ S * *2)** 7.25

RETURN

END

The numerical results obtained from the above program for r :L(*)', L :20, n :70, j : 0,I,2,. .. , n is shown in Table 5.1.

57

Table 5,1: Numerical Results for J(ø) Obtained from Varley-Walker Solution

1.1169131.3023951.3687891.3000007.0926240.7587280.327093

-0.157670-0.638565-1.052301-7.337677-1.445358-r.347640-1.046378-0.577158-0.007995

0.5684251.0488091.3380301.369516]-7230230.635102

-0.001723-0.653165-1.169869-L.422L35-1.334999-0.914159-0.2537740.4794321.0853511.3851971.2774370.7772990.024136

5.28985.58785.89396.20826.53066.86127.20007.54697.90208.26538.63679.01639.40479.8000

10.204110.616311.036711.465311.9020L2.346912.8000L3.267213.730674.208214.693915.187815.689816.200016.718477.244977.779678.322418.873519.432720.0000

1.0000250.9998240.9984250.9945790.9869440.9740770.9544290.9263560.8881460.8380580.7743830.6955260.6001090.4870950.3559430.2067700.040531

-0.L40797-0.334045-0.534718-0.736901-0.933258-1.115133-7.272789-1.395809-1.473674-7.496525-1.456087-7.346772-1.166480-0.918239-0.610456-0.2577220.1192580.4943520.837477

0.00000.00410.01630.03670.06530.10200.14690.20000.26720.33060.40820.49390.58780.68980.80000.91847.04491.17967.3224t.47351.63271.80001.97552.L5922.35102.55102.75922.97553.20003.43273.67353.92244.r7964.44494.71845.0000

r@)tr)ftr

58

5.3 Direct Solution of the Cornpliant LayerEquation

Our numerical method for solving the integro-differential equation (5.3)

directly is again using a non-uniform grid which is concentrated at the

wing-tip, as described in chapter 3. We rewrite equation (5.3) as

l@): _! [" f'G)de _! f f'G)d€ (, > o), (5.2)rJo (-æ nJt (-xwhere ,t is a positive constant. For ø > L, f @) is approximated by

f @) * tÆcos(x+ ii - að' rrr(5.8)

Hence

t: l'G)dc(-r

: t/llcos(n + f,).i{z - r) + sin(ø + f,)ci{z -')l.Ï,#bet*t-*t, (5.e)

where

and

si(ø) : - l,*sin ú

dt (5.10)t

ci(r) : - l: Yor. (5.11)

Now equation (5.7) becomes

r@) ++ 1"" # : -fl.o"(*+ f)si(r - u) + sin(z r ä)ci(¿ - r)l

å[år"s l*l-hl. (5.12)

The functions si(ø) and Ci(ø) can be evaluated using series expan-

sions (Abramowitz and Stegun, [10] L964, pages 2J2-2JJ), i.e.

59

si(r) Ën=O (2n * 1)(2n + 1)! 2

(-t¡"*2"+r 7f

Ci(r)

si(r)

ci(r)

7rrog'+Ëffia r -1¡"*zn7*ros'+tffi,

-f @) cos 0 - 9(r) sin c,

/(o) sin r - g(r) cos 0,

for r ( 1, with 12 figure accuracy,

for u ( 1, with 14 figure accuracy,

foræ)1,

forr)1,

40.02L433322.624977570.236280L57.705423

48.796927482.485984

1114.978885449.690326

where 7 is Euler's constant 0.5772156649. . . and

r@) :38.027264

26õ.187033335.67732038.102495

Iæ

h:b2:bs:b4:

A1 :

Q2:c$:A4:

h:b2:h:b4:

A1 :A2:A3:A4:

42.242855302.757865352.01849821.821899

So equation (5.12) together with the boundary condition /(0) : 1

can be solved using the method described in chapter B. Unfortunately, the

method is less accurate for this equation. This is a more diffi.cult problem

which needs furthur work in order to obtain better results. The problem is

due to the wave-like behaviour that occurs in the solution.

60

)

The programin Fortran 77 fot solving (5.3) directly, using the method

described above, is listed below, where FO4JGF is a NAG routine, avail-

able from the computer system, for ffnding the solution of a linear algebraic

equations system. For a description of this NAG routine, see Appendix B.


PARAMETER (LEN:400)

REAL+8 AA(LEN,LEN),F(LEN),WK(4+LEN)

LOGICAL SVD

COMMON PI

PI : 3. 1415926535897932384626D0

NRA : LEN

TOL :5.0D-32

LW : 4*LEN

IFAIL : O

PRINT *,"READ FROM FORT.l:''

READ(1,*)RL,N

PRINT *,"L:",RL," N :",N

PRINT *,"WAIT ......"

IG : Nf1

JG:IG

DO 20 I:l,NT : (DFLOAT(r)-0.5D0)/DFLOAT(N)

X - RL*T¡F{<2

61

DO 10 J:I,JG

u : DFLOAT(J-1)/DFLOAT(N)

DENOM : RL*(U**2_T**2)

rF (J.EQ.1.OR.J.EQ.Jc) THEN

W : 1.0D0

ELSE

W : 2.0D0

END IF

AA(I,J) : W / (2.0D0*DFLOAT(N)*DENOM*Pr)

rF (J.EQ.r) THEN

AA(I,J) : AA(I,J) + 3.ODO/(8.ODO*DFLOAT(N))

END IF

IF (J.EQ.I+l) THEN


END IF


AA(r,J) - AA(r,J) + 1.0D0/(2.0D0*DFLOAT(N))

END IF



END IF

rF (r.cT.2) THEN

rF (J.LT.r.AND.J.cT.1) THEN

62

')

10

20

30


END IF

END IF

CONTINUE

P : DLOG(DABS(RL/(RL-X)))/X**2 - 1.0D0/(RL*X)

R:X+PI/8.ODO

Z:RL_XF(r) : -DSQRT(2.0D0)*(DCOS(R)*Sr(z)+DSrN(R)*Cr(z))/pr

1 - 1.0D0

F(I) : F(I)-PIPI**2

CONTINUE


AA(IG,JG) : AA(rG,1)

DO 30 J:2,N

AA(IG,J) : 1.0D0/DFLOAT(N)

CONTINUE

PL : PI*RL

F(IG) : DSQRiT(2.0D0)*Dcos(Rl-.,+prl8.0D0) - 1.0D0 - 1.0D0/pL

CALL FO4JGF(IG,JG,AA,NRA,F,TOL,SVD,SIG,IR,\A/K,L'W,IFAIL)

X : 0.0D0

FX : 1.0D0

WRITE(2,50)X,FX

DO 40 I-2,IG

x : RL*(DFLOAT(I-1)/DFLOAT(N))**2

63

FX : FX + (F(r-l)+F(r))/(2.0D0*DFLOAT(N))

WRITE(2,50)X,FX

40 CONTINUE

50 FORMAT(F8.4,5X,F10.6)

PRINT *,"SEE FILE 'FORT.2'FOR THE RESULTS."

STOP

END

{<******rF*rF***t<{<*<{<***t<>t<****>t(*t<*****'t<t<*(*{<*rk*t<*<***rk{<{<t<'Èt r<*

REAL*8 FUNCTTON Sr(Z)

IMPLICIT REAL*8 (A-H,O_Z)

COMMON PI

rF (2.LE.1.0) THEN

SI : g.¡¡g

DO 60 N:0,7

D : DFLOAT(2*N+1)

SI : SI + (-1.0D0)*<'r,N:tz,t<*(2*N+1)/(D*H(D))

60 CONTINUE

SI:SI-PI/2.0D0

ELSE

SI : -DCOS(Z)*FF (z) - DSIN(z)*c(z)

END IF

RETURN

END

* * * * * * * {< * + * * t<,k *,F * * *,t< {< * * * * * * * * * * * * * * *,k * * * {< * * {< t( )t< *,Fr< X X *

64

REAL*8 FUNCTTON CI(Z)


rF (2.LE,7.0) THEN

cI : 0.5772756649 + DLOG(Z)

DO 70 N:1,8

D - DFLOAT(2*N)

CI : CI + (-1.0D0)**19*7**(2*N)/(D*H(D))

70 CONTINUE

ELSE

CI : DSIN(Z)*FF(Z) - Dcos(z)*c(z)

END IF

RETURN

END

,<***+>k*>1.+*{<*t **'ßt**{<****t<rt(***r<********'F*>t<**<*'ß*t *{<***

REAL*8 FUNCTTON FF(Z)

IMPLICIT REAL*8 (A_H,O_Z)

L - Z**8 + 38.027264*Z**6 + 265.L87099*Z**4

A : A + 335.677320*Z**2 + 39.102495

B :7**3 + 40.021433*Z**6 + 322.62497L*fZ**4

B : B + 570.236290*Z**2 + r57.L05423

FF : A/(B*Z)

RETURN

END

65

REAL*S FUNCTTON G(Z)


A - Z**8 + 42.24285ó*Z**6 + 302.757865*Z**4

A : A + 352.018498*2**2 + 21.821899

B : 2**3 + 48.196927*Z**6 + 482.485984*Z**4

B : B + 11L4.978885*Z**2 + 449.690326

G : A/(B*Z**2)

RETURN

END

*,k * >k * * * * * * *< * * * + * * * * X >k * * t< * * * *,F * * {< {< * X * * {< * * * {< * * r. * * * t< * >k *

REAL*8 FUNCTTON H(D)


H : 1.0D0

N : INT(D)

DO 80 I:2,N

H : H*DFLOAT(I)

80 CONTINUE

RETURN

END

Table 5.2 shows the numerical results, using our method with L :20and r¿ - 70, compared with that obtained from Varley-Walker solution; and

Figure 5.1 shows the results in graphs.

** t< * * * * * * ** * * * * * * ** ** * * * * * ** * * * * * * * * * ** * * * * t< * * * ** * r. *

66

1.1 169131.3023951.3687891.300000r.0926240.7587280.327093

-0.157670-0.638565-1.052301-L.337677-1.445358-7.347640-1.046378-0.577158-0.007995

0.5684251.0488091.3380301.369516L.7230230.635102

-0.001723-0.653165-1.169869-1.422735-1.334999-0.914159-0.2537740.4794321.0853511.3851977.2774370.7772990.024136

1.241600t.3620541.358165r.2206500.9534470.5756260.1.21644

-0.360516-0.813585-r.L77529-1.397836-L.434597-L.270927-0.919189-0.4237440.1450700.6938351.7262451.3578181.3346977.0484790.543672

-0.085381-0.709439-1.189788-1.409809-7.305746-0.888480-0.2488770.4576281.0464061.3520937.2780070.8317630.023187

5.28985.58785.89396.20826.53066.86127.20007.54697.90208.26538.63679.01639.404L9.8000

70.204L10.616311.036711.465311.902072.346912.800073.267213.730614.208214.693915.187815.689816.200076.778417.244977.779678.322418.873579.432720.0000

1.0000250.9998240.9984250.9945790.9869440.9740770.9544290.9263560.8881460.8380580.7743830.6955260.6001090.4870950.3559430.2067700.040531

-0.140797-0.334045-0.534718-0.736901-0.933258-1.115133-1.272789-1.395809-r.473674-7.496525-1.456087-7.346772-1.166480-0.918239-0.610456-0.2577220.1192580.4943520.837477

1.0000000.9862520.9709430.9526070.9297790.9009540.8645810.8190770.7628570.6943880.6L22480.5L52280.4024280.2733960.728269

-0.032072-0.205836-0.390179-0.581069-0.773208-0.960012-1.133684-7.285407-1.405669-r.484746-1.513352-7.483429-1.389073-7.227573-1.000083-0.713059-0.378244-0.013137

0.3594510.7724461.016316

0.00000.00410.01630.03670.06530.10200.14690.20000.26120.33060.40820.49390.58780.68980.80000.91841.04491.1796t.3224L.47357.63271.80001.97552.75922.35102.55102.75922.97553.20003.43273.67353.92244.77964.44494.77845.0000

from(5.4)our methodt

5.4our methodr ftr

Table 5.2: Numerical Results of the Compliant Layer Equation

(L=2o'n-70)

67

r(")our methodVarley-\Malker sulution

2.O

1.5

1.0

0.5

.0"9OO

c

c

o

o5 x-o.5

-1.O

-1.5

-2,O

.o 10

o¿

15 20

'?9o.0å

Figure 5.1: Comparison between ou¡ method and Varley-Walker solution for the cornpliant layer

equation.

68

Chapter 6

Conclusron

The lifting-line equation was developed by Prandtl (1918) in order to calcu-

late the span-wise distribution of circulation around wings of large aspect

ratio. This is an aerodynamic problem, but with some modification, the

lifting-line equation can also be used in other applications, such as the com-

pliant layer problem. Solving Prandtl's lifting-line equation is a difficult

task, since it is a singular integro-differential equation. In the aerodynamic

problem, the degree of difficulty depends on the wing chord distribution,

i.e. the chord function c(y).

An analytic solution of Prandtl's lifting-line equation for a semi-

infinite wing of constant chord has been found by Stewartson (1g60) and

confirmed by Varley and Walker (1989). However, it is still difficult to

obtain accurate numerical results from the Stewartson's solution, because

of the double integral and semi-infinite range involved in the function ap-

pearing in the solution. It is therefore convenient to simplify the function

before evaluating it numerically. This can be done by some manipulation of

the inner integral of Stewartson's double integral, leading to the so-called

Clausen's integral. By evaluating Clausen's integral beforehand, it becomes

69

easier to calculate the whole function of Stewartson's solution, which now

reduces to a single integral. The same technique can also apply to the

Varley-\Malker solution of the compliant layer equation.

Direct numerical solution of Prandtl's lifting-line equation for a wing

of finite span 0 I A 1 s can be obtained by transforming the integro-

differential equation into an equivalent integral equation, and then using

an appropriate grid and integration algorithm. A suitable grid for the

equation is a non-uniform grid which is concentrated at the wing tips, the

grid spacing increasing quadratically as the distance from the wing tips

increases. The integration algorithm used is the trapezoidal rule. The

use of this grid and integration method is especially accurate for a wing

of constant chord, where the circulation l(y) near the wing tip g : 0 is

approximately proportional b y*.

The numerical task becomes more difficult for the case treated in

this thesis - a semi-infinite wing as s tends to infinity, because rü/e norv

need to devise a satisfactory procedure for truncating the infinite range

of integration. This can be achieved by separating the whole integral in

Prandtl's lifting-line equation into two parts, one orì. a finite range and

the other on a semi-infinite range. The unknown function on the semi-

infinite range is approximated with a predicted function which is valid for

large values of the span-wise coordinate. It is not easy to predict such a

function. It has to be treated case by case, as we vary the chord function

c(y). In other words, we have to choose different functions in the truncation

correction for different classes of c(y).

For the compliant layer equation, the source of difficulty is the tvr¡ave-

70

like behaviour that occurs in the solution of the equation. In my opinion,

in order to obtain more accurate results in solving this equation in future

work, we should consider other grids and integration methods. The results

are very sensitive to the angular frequency and the phase of the wave-

like function which we choose for the truncation correction. Therefore the

angular frequency and the phase should be correctly chosen, as well as the

actual form of this function.

7t

Appendix A

DO]-AHF - NAG Fortrarr.Routine Summary

Important: For a complete specification of the use of this routine see the

NAG FORTRAN Library Manual. Terms marked ll...ll may be imple-

mentation dependent.

A. Purpose:

I /D}LAHF I / cornputes a definite integral over a finite range to a

specified relative accuracy.

B. Speciffcation:

I lreall I FUNCTION //D01AHF I I (4, B, EPSR, NPTS, RELERR,

F, NLIMIT, IFAIL)

C IîITEGER NPTS, NLIMIT, IFAIL

C l/reaIl/ A, B, EPSR, RELERR, F

C EXTERNAL F

C. Parameters:

A,- llrcatl/ .

on entr¡ A must specify the lower limit of integration. unchanged

72

on exit.

B-llrcatl l.On entry, B must specify the upper limit of integration. Unchanged

on exit.

EPSR - l/rcarll.On entry, EPSR must specify the relative accuracy required. Un-

changed on exit.

NPTS _ INTEGER.

On exit, NPTS contains the number of function evaluations used in

the calculation of the integral.

RELERR-llrcarll.

On exit, RELERR contains a rough estimate of the relative error

achieved.

F - l/rcal// FUNCTION, supplied by the user.

It is called by I ID}LLHF / I to evaluate the integrand at the point

X. Its specification is:

llrcatll FUNCTION F(x)

I lrcatll xx- llreat/l .

on entry, x specifies the point at which the value of the integrand

is required by I /D}LAHF / / . X must not be reset by F.

F must be declared as EXTERNAL in the (sub)program from which

//D01AHF // is called.

NLIMIT _ INTEGER.

73

On entry, NLIMIT must specify a limit to the number of function

evaluations. If NLIMIT.LE.O, the routine uses a default limit of 10,000.

Unchanged on exit.

IFAIL _ INTEGER.

Before entry, IFAIL must be assigned a value. For users not familiar

with this parameter, the recommended value is 0. Unless the routine detects

an error (see next section), IFAIL contains 0 on exit.

D. Error fndicators and 'Warnings:

Errors detected by the routine:

IFAIL: 1

The integral has not converged to the accuracy requested. It may

be worthwhile to try increasing NLIMIT.

IFAIL : 2

Too many unsuccessful levels of subdivision have been invoked.

IFAIL : 3

Invalid accuracy request (i.e. on entry EPSR.LE.0.0).

when IFAIL : 1 or 2 a result is obtained by continuing without

further subdivision, but this is likely to be *+inaccurate** .

74

Appendix B

FO4JGF NAG FortranRoutine Surnrnary

Important: For a complete specification of the use of this routine see the

NAG FORTRAN Library Manual. Terms marked /1...1/ may be imple-

mentation dependent.

A. Purpose:

I |î} JGF / I frnds the solution of a linear least squares problem,

Ax:b, where A is a real m*n (m.GE.n) matrix and b is an m element

vector. If the matrix of observations is not of full rank, then the minimal

least squares solution is returned.

B. Specification:

suBRourINE //F04JGF// (M, N, A, NRA, B, TOL, SVD,

SIGMA, IRANK, WORK, L\ /ORK, IFAIL)

C LOGICAL SVD

C INTEGER M, N, NRA, IRANK, L\MORK, IFAIL

C llrealll A(NRA,N), B(M), TOL, STGMA, \MORK(L\^/ORK)

tÐ

C. Parameters:

M _ INTEGER.

On entry, M must specify the number of rows of A, where M.GE.N.

Unchanged on exit.

N _ INTEGER.

On entry, N must specify the number of columns of A, where

1.LE.N.LE.M. Unchanged on exit.

A - llrcalf f anay of DIMENSION (NRA,t), where t.GE.N. Before entry,

the leading M*N part of A must contain the matrix to be factorised.

On successful exit, if SVD is returned as .FALSE., the leading M*N

part of A, together with the first N elements of the vector WORK, contains

details of the Householder QU factorisation of A. See NAG Library routine

llF}zwDFll for further details. If SVD is returned as .TRUE., then the

top N*N part of A contains the right hand singular vectors, stored byt<'Frows*{< . The rest of the first N columns of A is used for workspace.

NRA _ INTEGER.

on entry, NRA must specify the first dimension of A as declared in

the calling (sub)program, where NRA.GE.M. Unchanged on exit.

B - I lrealf f artay of DIMENSION at least (M).

Before entry, B must contain the M element vector b. on successful

exit, the first N elements of B contain the minimal least squares solution

vector The remaining M - N elements are used for workspace.

TOL- //reaIl l.on entry, ToL must specify a relative tolerance to be used to de-

to

termine the rank of A. TOL should be chosen as approximately the largest

relative error in the elements of A. For example, if the elements of A are

correct to about 4 significant figures then TOL should be set to about

5*19**(-4). See Section 11 of the Library Manual routine document for

a description of how TOL is used to determine rank. If TOL is outside

the range ( "pr,

L.0), where eps is the machine accuracy (see NAG Library

routine I /XlzAAF I / ), then the value eps is used in place of TOL. For

most problems this is unreasonably small. Unchanged on exit.

SVD _ LOGICAL.

on successful exit, SVD is returned as .FALSE. if the least squares

solution has been obtained from the QU factorisation of A. In this case A

is of full rank. SVD is returned as .TRUE. if the least squares solution has

been obtained from the singular value decomposition of A.

SIGMA - l/reat/l .

on successful exit, SIGMA returns the value seRT(r(transpose)r/(M-

IRANK)) when M.GT.IRANK, and returns the value zero when M -IRANK.

IRANK _ INTEGER.

on successful exit, IRANK returns the rank of the matrix A. Itshould be noted that it is possible for IRANK to be returned as N and.

SVD to be returned as .TRUE.. This means that the matrix U only just

failed the test for non-singularity.

WORK - I lrcal// array of DIMENSION (L\ /ORK).

on successful exit, if svD is returned as .FALSE., then the first

77

N elements of WORK contain information on the QU factorisation of A,

(see parameter A above and NAG Library routine I |F}2WDF I I ) and

WORK(Nf 1) contains the condition number norm(U)*norm(U(inverse)) ,

where norm denotes the Euclidean norm, of the upper triangular matrix U.

If SVD is returned as .TRUE., then the first N elements of \MORK

contain the singular values of A arranged in descending order and

\MORK(N*l) contains the total number of iterations taken by the QR

algorithm. Otherwise WORK is used as workspace.

L\MORK _ INTEGER.

On entry, LWORK must specify the dimension of the array WORK

as declared in the calling (sub)program, where L\MORK.GE.4 *N. Un-

changed on exit.

IFAIL _ INTEGER.

Before entry, IFAIL must be assigned a value. For users not familiar

with this parameter, the recommended value is 0. Unless the routine detects

an error (see next section), IFAIL contains 0 on exit.

D. Error Indicators and Warnings:

Errors detected by the routine:

IFAIL : 1

On entry, N.LT.l, or M.LT.N, or NRA.LT.M, or LWORK.LT.4*N.

IFAIL : 2

The QR algorithm has failed to converge to the singular values in

50*N iterations. This failure can only happen when the singular value

decomposition is employed, but even then it is not likely to occur.

78

Bibliography

[1] Prandtl, L. 1918 "tagflügeltheorie" Nachr. Ges. Wiss. Göttingen,

Math. Phys. K1., 1st part, 451-477;2nd part, 107-137.

[2] Thwaites, B. 1960 "Incompressible Aerodynamics" (Oxford University

Press, Oxford).

[3] Ashley, H and Landahl, M. 1965 "Aerodynamics of \Mings and Bodies',

(Addison-\Mesley, Massachusetts).

[4] Robinson, A and Laurmann, J.A. 1g56 "\Ming Theory,, (Cambridge

University Press, Cambridge).

[5] Bera, R.K. 1991 "The Lifting Line Equation - revisited", fnterna-

tional Journal of Mathematical Educøtion in Science and Technology

22,341-349.

[6] Varley, E. and \Malker, J.D.A. 1g8g "A Method for Solving Singular

Integrodifferential Equations", IMA Journal of Apptied, Mathematics

43,77-45.

[7] Tuck, E.o. 1991 "Ship-Hydrodynamic Free-Surface problems \Mithout

Waves" ,Journal of Ship Research, 95, 2ZT-287.

79

I

I

I

I

I

1

I

II

I

,]

ì:J

J

I

I

.;

t,l

'.tl

'lI.i

[8] Stewartson, K. 1960 "A Note on Lifting-line Theory" , Quørterly Jour-

nal of Mecha,ni,cs ønd, Applieil MøthematicsL3,49-56.

[9] Guermond, J.-L. 1990 "A Generalized Lifting-line Theory for Curved

and Swept \Mings", Journal of Fluid Mechanics zLL,497-6L3.

[10] Abramowilz, M. and Stegun, I. A. 1964 "Handbook of Mathematical

Functions with Formulas, Graphs, and Mathematical Tables".

jl

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Numerical solution of Prandtl's lifting-line equation · 2018. 3. 1. · 1.1 Lifting Surface Theory L.2 Lifting-line Theory 2.3 Listing Program 2.4 Numerical Results vl vlt vul lx

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