III J II
Hi
l i l i l l i l l
O
a O
^ O CD 0>vjn o
OD 00 VJ1 00QUASI HOMOGENEOUS APPROXIMATIONS
FOR THE CALCULATION OF WINGS WITH
CURVED SUBSONIC LEADING EDGES
FLYING AT SUPERSONIC SPEEDS
PROEFSCHRIFT
TER VERKRIJGING VAN DE GRAAD VAN DOCTOR
IN DE TECHNISCHE WETENSCHAPPEN AAN DE
TECHNISCHE HOGESCHOOL DELFT, OP GEZAG VAN
DE RECTOR MAGNIFICUS IR. H. R. VAN NAUTA LEMKE,
HOOGLERAAR IN DE AFDELING DER
ELEKTROTECH-NIEK, VOOR EEN COMMISSIE AANGEWEZEN DOOR
HET COLLEGE VAN DEKANEN TE VERDEDIGEN OP
WOENSDAG 28 MAART 1973 TE 14.00 UUR.
door
/8^S' 955»^
RENÉ COENE
vliegtuigbouwkundig ingenieur
geboren te Amsterdam
'OU 3 2^61
- * /
BIBLIOTHEEK TU Delft
P 1888 4284
C 1040863Dit proefscinrift is goedgekeurd door de promotor prof. dr. ir. J. A. Steketee
Abstract
In this thesis a method is developed for the calculation of supersonic wings with planforms characterized by curved subsonic leading edges. The method extends the range of applicability of Germain's and Fenain's homogeneous flow theory which is valid for supersonic wings with
straight leading edges. With boundary conditions at the wing surface and leading edges of polynomial form the boundary va.lue problems can be reduced to algebraic problems which permit a systematic treatment.
L i s t o f Symbols V I I I
Summary XIV Chapter I
Introduction
1.1. Outline of the thesis • 1
1.2. The governing equation 3
1.3. The boundary conditions k
^.h. The four types of problems 5
1.5. Integral representations 6 1.6. Methods based on Eward's principle 8
1.7. Expansion with respect to a slenderness parameter 12
1.8. Straight subsonic leading edges 13 1.9. The quasi-homogeneous approximation 11+
Chapter II
Review of homogeneous flow theory
2.1. Introduction l6 2.2. Definition of homogeneous flows 17
2.3. Statement of the problem 18 2.1». The n derivatives 19 2.5- Two conformal mappings 22 2.6. The compatibility relations 23 2.7. Euler's relation for homogeneous functions 26
2.8. Elementary flows 28 2.9. The boundary conditions on 5 for elementary flows 28
2.10. The functions 4J^io in an elementary D.P. 31 2.11. The analytic functions related to the boundary conditions in
the X -plane 3U 2.12. Determination of the coefficients introduced in the n
VI
2.13. Reduction of the elementary D.L.P. to aji algebraic problem 37
2. lit. Unified expression of the solutions. Ul
Chapter III
Wings with slightly curved leading edges
3.1. Introductory remarks kh
3.2. The transformation U5 3.3. The transformed equation (1.2) 1*7
3.'t. The transformed boundary conditions k9
3.5. The D.L.P. 5l* 3.6. The D.T.P. 58 3.7. The I.L.P. 62 3.8. The I.T.P. 65 3.9. The D.L.P. 67 3.10. Discussion of the results. 68
Chapter IV
Wings with leading edges of algebraic form
h.^. Introductory remarks 71
U.2. The transformations 73 't.3. The transformed differential equation 82
k.h. The construction of solutions for the perturbation potential
in the 3Cj-space B^t ^ . 5 . A special case 86 U.6. The general case 93 't.7. Expansion of solutions in the x,(y,2.-space in the variables Jc.^
97
' t . 8 . R e l a t i o n between t h e e x p r e s s i o n s ('t.76) and Ct.SO) 102
Chapter V
Evaluation o f some results - Examples of applications
5.1. Preliminary considerations 107 5.2. Flat plate with leading edges iii,|= "r jc + £a^oc^ 108
5.3. Flat plate with leading edges itpi = Ïjc + <£ o^ oc"* 11't 5.'t. Wings with parabolic camber and leading edges \ii.\=XXi + e.Q.^yJ' 1l8
5.5. Flat plate with parabolic leading edges 122 5.6. Wings with parabolic leading edges at ideal angle of attack 129
5.7' Concluding remarks lltl
Table 1 Ilt2 Table 2 llt3 Cited references I't't
Appendix A 150 Appendix B 15't Appendix C I58 Appendix D 163
VIII LIST OF SYMBOLS d parameter introduced in ('t. 20).
Q speed of sound in the unperturbed flow. a.^ coefficients defined in (3-2).
Qj coefficients defined in ('t.7). Q.^ coefficients defined in ('t.61t).
^ , ^ o coefficients introduced in I.T.P. (2.311). - V i -
(-%'%
c<-i,_/-o Q coefficients introduced in I.L.P. (2.36).
'^Li.A coefficients introduced in the transformation ('t.2). Q, . , coefficients introduced in the trsmsformation
('t.5)-R = - T ^ a-'\! 1--^"'
-jr half the wing span (see Fig. I.io), parameter introduced in (11.20).
"^^-/-t Y coefficients introduced in I.T.P. (2.3't).
"^t-V
coefficients introduced in I.L.P. (2.36).
coefficients appearing in the inverse transformation (11.12). coefficients appearing in the special inverse transformation (C.lta).
•^c,;^ coefficients introduced in the transformation ('t.2), 4- a coefficients introduced in the transformation ('t.5)'
C^_^_^ coefficients introduced in D.T.P. (2.33). ^ti-ss coefficients introduced in D.L.P. (2.35). C* coefficients introduced in D.L.P. (2.37).
•h-Z-i.S
c,^^
coefficients introduced in the transformation ('+.2).r I, coefficients introduced in the transformation ('t.5)> C* , coefficients introduced in (5-113a)
E.(p) the integral part of the r e a l n\imber p . '1
•jr function defined in ('t.66).
-§. . coefficients defined in (C.5).
-flpc)=2^(Xiyi , polynomial function introduced in ( 3 . 2 ) . 'J^(j^j= Ji—— ,polynomial function defined in ( 3 . 9 ) .
"Ti (i- -U.)
f (a:,|u|) function of polynomial form in .3c and |a| ; see C t . l ) , T"(Z) complex function of the form (2.'t6).
1^ (^,^) functions defined by (2.79a) for the D.T.P.
'Y*C^) functions defined by (2.79b) for the D.L.P.
a* function defined in ('t.66). ^i.sj, coefficients defined in
(0.5)-<J(a,(^) the wing surface i s given by 1 = Q,(X,U.)
Q ^'iij=Z-fc('.^J , polynomial function introduced in ( ' t . l 6 ) .
f^
functions introduced in ('t.3't); see also appendix C.Q,, . coefficients introduced in (C.5)<
9 =
v T ^
-C (^,ij functions defined by (2.79a) for the D.T.P. 1^* (f) functions defined by (2.79b) for the D.L.P. ^t^.''^-' functions defined by (2.79e) for the D.L.P.
-H function defined in ('t.66). •"c.ti coefficients defined in (C.5).
-k function defined in ('t.78) for section 't.7. '^s,it coefficients defined in ('t.78).
-Ac functions defined in Ct.ltl) and ('t.'t2). -k^ ^f_ coefficients defined in ('t.87).
H.^^{f) functions defined by (2.79d) for the I.T.P. ^wd--*:) functions defined by (2.79c) for the I.L.P.
X
j(I)
first reflexion integral; see Fig. 1.8. and equation (I.I5).
^I-1)IU L.i.S....{l^-t). Q.j)!l= l.H. (>.... (1^).
K ' '= 'V. u( , f u n c t i o n which s a t i s f i e s ( 2 . 7 8 ) ; s e e a l s o a p p e n d i x B.
-t- l e n g t h of t h e w i n g .
•iy^ c o e f f i c i e n t s d e f i n e d i n ( 2 . 5 6 ) f o r t h e I . T . P . M.= — Mach niimber of t h e u n p e r t u r b e d flow.
Pi^. c o e f f i c i e n t s d e f i n e d by ( A . ' t ) . M,, c o e f f i c i e n t s d e f i n e d by ( A . 9 ) . <r Ta d e g r e e of homogeneity M^i c o e f f i c i e n t s d e f i n e d i n ( A . ' t ) . A4( c o e f f i c i e n t s d e f i n e d i n ( A . 9 ) . to p r e s s u r e of t h e a i r . ' ^ ' B , c o e f f i c i e n t s d e t e r m i n e d i n t h e r e s u l t s of c h a p t e r V. Vsit c o e f f i c i e n t s d e f i n e d i n ( 5 . 7 1 ) and ( 5 . 7 2 ) . P ^ ( ó S 3 f u n c t i o n s i n t r o d u c e d i n ( 2 . 7 6 ) . I n a D . P . Q^^ ( JcJ f u n c t i o n s i n t r o d u c e d i n ( 2 . 6 8 ) Q f u n c t i o n s d e f i n e d i n ( 2 . 7 1 a ) . t = U^ (3c~ Au) , c h a r a c t e r i s t i c c o o r d i n a t e u s e d i n s e c t i o n 1.6. 7? r e m a i n d e r of t h e p l a n e x = o , o u t s i d e of S, ( s e e F i g . I . l t ) , ( a l s o u s e d t o d e n o t e t h e r e a l p a r t of a complex f u n c t i o n ) . "RCJci) r e m a i n d e r of t h e p l a n e 3Cj=o, o u t s i d e of S(oc^^).
R-H^-^) f u n c t i o n s i n t r o d u c e d i n ( 2 . 7 6 ) . 7^.^^ f u n c t i o n s d e f i n e d i n ( 2 . 7 9 b ) .
S= Lls'(^:z + fhu) , c h a r a c t e r i s t i c c o o r d i n a t e u s e d i n s e c t i o n 1.6. S wing p l a n f o r m ( s e e F i g . I . ' t ) .
P^^(óc) functions introduced in (2.76).
^ functions introduced in (2.79c). In an I.P. 5 = T T - .5
Lf speed of the oncoming unperturbed flow. U|V-,uj- perturbation velocity components.
U
V TI A ^^
Ö3C,- ^2.
X,C(,t trirectangular coordinates in the physical space. cc.,,Xj,DCj trirectangular coordinates in the transformed Jt^ space. ^ =3é-i-i'u,= ^ - ; see section 2.5.
angle of incidence; see Fig. 1.3.
l o c a l angle of incidence
parameters defined in ('t.25).
coefficients defined in (2.3't) for the I.T.P. functions defined in ('t.39).
coefficients defined in ('t.'tO).
coefficients defined in ('t.'tO ) on page 88. functions defined in ('t.62).
coefficients defined in (2.63), (2.66) and (A.7). coefficients defined in (2.65) and (A.8).
/3 =.
VFc^.
1^ coefficients used in section 't.8.; ('t.96). r Mach cone JC^_/3''(^ uV 4^) = 0 , J C ^ O . iJs—^:-—,parameter defined in (5.1). £ parameter defined in (3.l). o^iac.u) c<,,°^z ^.-t,t
M'
<^
i,is,2.fcU''^'
<
^l
XII
e = 4<,^~' i » .
Av^ c o e f f i c i e n t s i n t r o d u c e d i n ( 2 . 5 2 ) f o r t h e D . T . P . A.W., c o e f f i c i e n t s i n t r o d u c e d i n ( 2 . 5 3 ) f o r t h e D . L . P . X"*'^' f u n c t i o n d e f i n e d i n ( ' t . 7 ' * ) . ^i,tt c o e f f i c i e n t s d e f i n e d i n ( ' t . 7 5 ) and ( D . ' t ) . ^j. c o e f f i c i e n t s d e f i n e d i n ( A . l ) and ( A . 2 ) .A d e g r e e o f a p o l y n o m i a l ; s e e pages 139 and I'tO. V> d e g r e e of h o m o g e n e i t y ; s e e a p p e n d i x B. V; c o e f f i c i e n t s d e f i n e d i n ( A . l ) ajid ( A . 2 ) . v' p a r a m e t e r d e n o t i n g o r d e r s of a p p r o x i m a t i o n ; s e e page 111 V c o e f f i c i e n t s i n t r o d u c e d i n ( 2 . 5 7 ) f o r t h e I . L . P . /3 d e n s i t y o f t h e a i r . X t a n g e n t of t h e apex a n g l e of t h e w i n g . ip p e r t u r b a t i o n v e l o c i t y p o t e n t i a l . (p v e l o c i t y p o t e n t i a l \jJ complex f u n c t i o n : K. (j)^ = W^ y. s o l u t i o n of t h e wave e q u a t i o n u s e d i n s e c t i o n ' t , 7 .
ABBREVIATIONS
D . P . d i r e c t p r o b l e m .
I.P. inverse problem. T.P. thickness problem. L.P. lifting problem.
D.T.P. direct thickness problem. D.L.P. direct lifting problem.
I.T.P. inverse thickness problem. I.L.P. inverse lifting problem .
D.L.P. direct problem in which 09 is odd in u and X.
SUBSCRIPTS AND SUPERSCRIPTS
In general the first subscript indicates the degree of homogeneity of a function. The second subscript of a function indicates that a differen-tiation must be carried out. We write, for instance,
The function U) (oc,,i:^,Jfc,)is homogeneous of degree yi in Jc,,Jc^ and Jc^ and is differentiated with respect to Jt, . Thus in (J) the first subscript re-fers to the degree of homogeneity of II) and not to the degree of homoge-neity of Ü) , which is yi-I .
In chapter II, equations (2.6) and (2.7), a special notation is used in which the differentiations are indicated in subscripts as well as super-scripts. To avoid confusion the use of this notation is restricted to chapter II.
p , ~T accents denote a perturbation, or as in equations (3.1't) and (3.17), a differentiation.
D^ 00 denotes the situation at infinity where no perturbations exist.
i- top and bottomsides of the wing surface. (J), (J denotes orders in £.
(/), J denotes the degree of homogeneity of (M . (oJ.nOI
V
XIV SUMMARY
In this thesis a method is developed for the calculation of supersonic wings with a planform characterized by curved subsonic leading edges. The method represents a natural extension of Germain's and Fenain's ho-mogeneous flow theory which is valid for supersonic wings with straight leading edges.
In chapter I a review is given of the steady linearized potential flow theory which is available for wings with subsonic leading edges. Homoge-neous flow theory is discussed in chapter II.
In chapter III general formulae are derived for the solutions of bounda-ry value problems for wings with leading edges which are only slightly curved. A systematic treatment of wings with delta-like planforms is possible. The planform may be considered as variable. The boundary value problems are formulated in a space where the leading edges are straight lines and where the Mach cone being the envelope of the disturbances in the air, remains a straight circular cone. After expansion of the solu-tions with respect to a small parameter, chosen as a measure for the de-viation from straight of the leading edges, the first terms in the ex-pansion can be determined.
In chapter IV the transformed boundary value problem is solved after arranging the terms in the solutions with respect to ascending degrees of homogeneity. These terms can be calculated successively. When the boun-dary conditions at the wing surface and the leading edges are of poly-nomial form the boundary value problems can be reduced to algebraic pro-blems, which permit a standardized treatment.
In chapter V some illustrative calculations are carried out. A compari-son is made between the present approximations and those based on other methods. The parameters defining the leading edges and those defining the boundary conditions at the wing surface for the warp, the thickness distribution or the pressure distribution can be given the same priority in the calculations.
C h a p t e r I INTRODUCTION
1.1. Outline of the t h e s i s
In t h i s t h e s i s some methods are developed t o c a l c u l a t e t h e flow around supersonic wings with curved subsonic l e a d i n g edges. Some of t h e planforms we have in mind are i n d i c a t e d below.
Fig. 1.1.
The velocity component of the oncoming flow in the direction perpendi-cular to the leading edges is, in every point of these edges, less than the speed of sound. The possibility then exists for perturbations
created on the top side of the wing to travel around the leading edges and to make their influence felt at the bottom side. It follows that the flow at the upper- and lower sides of the wing are not independent. It will be assumed that the trailing edges of the wings are supersonic. In that case their form has some interest for the properties of the wing as a whole, but they have no influence on the flow field near the wing ahead of them.
2
Wings of the type indicated can be calculated in a satisfactory way only in some special cases:
(i) If, at the Mach numbers considered, the wing lies well within the envelope of the disturbances generated by the wing, the slender body theory and its extensions present a natural approach.
(ii) If, on the other hand, the leading edges lie close to the envelope of the disturbances, are "nearly sonic", the approximate methods based on Eward's principle yield acceptable results.
(iii) If the leading edges are straight the homogeneous flow theory yields, within the scope of the linearized theory, a large class of exact solutions, which remain valid in the "slender" as well as the "sonic" limit.
(iv) If the leading edges are curved the quasi-homogeneous approxima-tions developed in this thesis permit a systematic approach, within the linearized theory, to wings with delta-like planforms (Fig.1.1). These methods are based on homogeneous flow theory and extend the applications of this theory to wings with curved sub-sonic leading edges. The answers obtained in this way remain valid as in case (iii) both in the "slender" and in the "sonic" limit. An interesting feature of the methods presented is that the parameters defining the leading edges and those defining the boundary conditions at the wing surface appear in an equivalent manner. This offers some new possibilities for design- and optimization problems, where so far the planform of the wing had to remain fixed.
In the remainder of the first chapter we summarize some basic information on steady linearized flow theory for wings with subsonic lead.ing edges.
In chapter II a review is given of homogeneous flow theory.
In chapter III formulae are derived for the calculation of wings with slightly curved leading edges.
In chapter IV some methods for wings with not so slightly curved leading edges are discussed.
In chapter V some examples of applications are given.
1.2. The governing equation
For p r a c t i c a l purposes one i s i n t e r e s t e d in a wing t r a v e l l i n g at
constant supersonic speed Lf through a homogeneous atmosphere at r e s t .
This problem i s equivalent t o the case when a wing at r e s t i s placed in
a uniform flow with speed U . Omitting v i s c o s i t y , heat conduction, the
effect of body forces and assuming
t h a t the a i r i s an ideal gas with
constant s p e c i f i c heats i t follows
t h a t Kelvin's theorem applies and the
flow w i l l be free of v o r t i c i t y . A
v e l o c i t y p o t e n t i a l 0 of t h e form
Fig. 1.2
<p
=
Uü
+ (p,
( 1 . 1 )can then be i n t r o d u c e d , where W i s the p e r t u r b a t i o n v e l o c i t y p o t e n t i a l .
Linearizing t h e equations of motion one finds t h a t (D has t o s a t i s f y
t h e equation.
where S •= M^,— I, A/ = — , £Z is the speed of sound in the
unper-t u r b e d flow. The s u b s c r i p unper-t s denounper-te d e r i v a unper-t i v e s wiunper-th respecunper-t unper-t o .2r,u,,X.
Other r e l a t i o n s , obtained from the l i n e a r i z e d equations of motion are
and
(1.3)
( I . ' t )
The accents denote the perturbations
f D = b — p),
Equation (i.'t) is the linearized Bernoulli law.1*.
Equation ( 1 . 2 ) i s a l i n e a r second order p a r t i a l d i f f e r e n t i a l
equation of hyperbolic type with constant c o e f f i c i e n t s . This equation
occurs in mathematical physics and has been studied extensively. I f one
i n t e r p r e t s J t as a time variable one e s t a b l i s h e s t h e analogy with the
two-dimensional wave equation. On t h e other hand by putting JC = t ^
one e s t a b l i s h e s a formal analogy with t h e three-dimensional '
Laplace equation (L = — l) . The i n i t i a l value and boundary value
problems, however, are specific for supersonic wing theory.
1.3. The boundary conditions
I f t h e (x,iL,X) system has i t s o r i g i n at t h e apex of the wing, the
envelope of t h e disturbances in upstream d i r e c t i o n i s given by t h e Mach
cone / :
Jt^-^3V^+X'>=0. •^>0 ( 1 . 5 )
since ahead of / no p e r t u r b a t i o n s e x i s t , t h e boundary condition on /
is
(f=0.
The boundary condition at t h e wing surface follows from the requirement
t h a t t h e flow must be t a n g e n t i a l t o t h e wing s u r f a c e . The angle between
the normal t o t h e wing surface and t h e X - a x i s i s assumed t o be small.
The geometry of wings i s e s s e n t i a l l y p l a n a r . Near round leading edges,
t h i s assumption i s obviously violated but t h e region in which t h i s
happens w i l l , in g e n e r a l , be r e l a t i v e l y small. I f t h e perturbation
v e l o c i t y i s small with respect t o Lc , t h e boundary condition can be put
in t h e form
a r = ( ^ ^ « _ [Zo4(:t,^j, (1.6)
ci(:i:,U,) being t h e local angle
of %¥ts incidence.
The position of the wing surface is near the plane X = 0.
The relation between the wing surface, defined by t = aL.(7:,U'} and Oi. (X, U) is given as
(1.7) A superscript will be used to indicate the flowfield near the upper-side of the wing surface, for instance, CLT . A superscript indicates the lower side. As usual, the boundary condition (l.6) will be applied on the projection S of the wing surface on the plane "X = O .
p can be referred to as the planform. The remainder of the plane "X = O will usuaJ.ly be indicated as R . It should be noted that (2)^ on p is
uniquely determined if the wing geometry X=qr(x,il)is given. Conver-sely if Ul has been determined, the geometry of the wing is only determi-ned up to a function of tt and an arbitrary constant. The arbitrary part must be chosen in such a way that the wing remains planar and admits zero thickness at the edges.
F i g . I.'t.
(a=o)
1 . It. The four types of problems
The pressure p e r t u r b a t i o n i s r e l a t e d to t h e p e r t u r b a t i o n v e l o c i t y
p o t e n t i a l by the l i n e a r i z e d Bernoulli equation:
p'=-r^^f:t>
:i.8) /0<j, being the density of the air in the unperturbed flow. Equations (1.8) and (1.6) suggest two types of problems.If the wing slope is given, one knows the upwash ur= (Ü on S •
6
wing is called the direct problem, (D .P. ). If the pressure distribution is prescribed and one is asked to find the geometry of the wing
gene-rating this pressure distribution, one calls it the inverse problem,(l .P. ). The perturbation potential (p(Jc,U,ji.] of a flow around a planar wing which lies near the plane 2=0 can be considered as the sum of an even
and an odd part:
(f)^ If^'-t- f
(1.9a)
The first part is even in X and is associated with obstacles symmetric with respect to 'Ji=,0. Boundary value problems for this part of the solu-tion are referred to as thickness problems, (T.P.)
The second part is odd in X and is associated with wings without thick-ness. Boundary value problems for this part of the solution are refer-red to as lifting problems, (L.P.)
One is thus led to four types of problems: (i) The direct thickness problem (D.T.P.), (ii) The direct lifting problem (D.L.P.), (iii) The inverse thickness problem (I.T.P.), (iv) The inverse lifting problem (I.L.P.). 1.5. Integral representations
Volterra [33] derived an integral representation for the solution of the general boundary value problem associated with equation (1.2). This integral representation is formally equivalent to the representa-tion of a harmonic funcrepresenta-tion as the result of distriburepresenta-tions of sources and doublets, obtained by means of Greens' theorem. The result for a planar wing can be expressed as follows:
z
dZ
; i . i o )In this integral 2. is the part of P within the forecone of the point
Xyx^
while
d
Z^=
dx^aij^,
X-Xo
The f i r s t part on the r i g h t hand side of (I.IO) can be recognized as
an i n t e g r a l over a d i s t r i b u t i o n of d o u b l e t s . The second part i s an
i n t e g r a l over a d i s t r i b u t i o n of sources and s i n k s . The f i r s t part i s
odd in Ï. and i s associated with the warp of the wing; i t corresponds
t o t h e L.P. The second p a r t i s even in 'X and i s associated with the
t h i c k n e s s d i s t r i b u t i o n ; i t corresponds to the T.P. In g e n e r a l , one i s
i n t e r e s t e d in t h e solutions on the wing surface. Seperating (1.10) in
t h e even and odd p a r t s and after some manipulations one obtains from
t h e second p a r t :
(T.P.) m (x,u.,o) = 1Vfr-^°)'^*(^'>r^J
^^of^^^
{Lx-xS-f^\^-^.ff'
(1.11a) The first part can be written in the form
8
The symbol J j i n d i c a t e s t h a t the f i n i t e part of the i n t e g r a l , as defined
by Hadamard £30j , has t o be taken. The expressions ( l . l l ) can be thrown
i n t o several a l t e r n a t i v e forms [38] . For given 2- , one may d i s c e r n two
classes of problems. The f i r s t class of problems a r i s e s when t h e known
function occurs in the integrand of the r i g h t hand sides of ( I . I I ) .
The problem then c o n s i s t s of the evaluation of t h e double i n t e g r a l s .
The second class of problems a r i s e s when the functions on t h e l e f t hand
sides are given. In t h a t case a double i n t e g r a l equation has t o be
solved. In a p a r t of the l i t e r a t u r e (for instance [.38] and 0+8] ) , the
f i r s t c l a s s of problems i s referred to as d i r e c t problems and the
second class i s referred t o as inverse problems. I f the region X
i s bounded by two subsonic leading edges, the second class of problems
i s more d i f f i c u l t t o solve than the f i r s t .
It should be noted t h a t the meaning of the terms d i r e c t and inverse in
the l i t e r a t u r e quoted i s d i f f e r e n t from the meaning adopted i n section
( i . ' t ) which i s used in the French l i t e r a t u r e . In what follows the
d e f i n i t i o n s of section ( i . ' t ) w i l l be used.
A considerable amount of l i t e r a t u r e i s devoted t o the construction of
approximate s o l u t i o n s of equations equivalent t o ( I . I I ) . The techniques
range from purely numerical t o almost e n t i r e l y a n a l y t i c a l . No attempt
w i l l be made t o give an exhaustive survey of t h i s l i t e r a t u r e , but a
number of relevant points w i l l be presented.
1.6. Methods based on E w a r d ' s p r i n c i p l e
In a L.P. the p e r t u r b a t i o n p o t e n t i a l can be expressed as an i n t e
-g r a l over a source and sink d i s t r i b u t i o n in the plane X= O . On p
the strength of t h e d i s t r i b u t i o n i s known in a D.P. but o u t s i d e of S ,
on R , the strength w i l l not be known a p r i o r i .
After introduction of c h a r a c t e r i s t i c coordinates X and 5 by
S,A,
Fig. 1.6.
^
one may write
(p%,s]=:I ff UJ(U.S,)d'uclS. (1.12a)
For b r e v i t y t h i s w i l l be w r i t t e n as
^ » = a ( z ^ j ,
= a(r)+Jarj+J{z2z)-^Jrir;.
(1.12b)
Expressing the fact t h a t the p e r t u r b a
-I , ^ N .
.
.
.
/ s,s„ t i o n p o t e n t i a l i s zero m the part of
Zl t h a t belongs t o R , e s p e c i a l l y on
the l i n e s t = c o n s t . and S = c o n s t .
through X , leads t o
Xïïh^imUo and 3(E[)-i-3(ll)=0,
so that (1.12b) reduces to
(1.13)
and employing (1.12b) and (1.13)
10
N
CJ(lj i s r e f e r r e d t o as t h e f i r s t r e f l e x i o n i n t e g r a l and can be used as
a f i r s t approximation t o t h e inversion of the i n t e g r a l equation (1.11b), Etkin and Woodward [lit] proposed a second approximation:
ipt?)^ W - T O . ( 1 . 1 5 )
For some special cases (1.15) can be shown to give an improvement with respect to J U j but, in general this is not necessarily the case. The relation (l.15) can be extended to yield a converging series but the resulting (J) ('2) does not converge to the exact inversion of (1.11b).
/ <
<
X
\N
- ^ ^
'^ /
/ s \ \\ / ^
x\ -^
5M
\L r
X41
N
, X
Fig. 1.If the leading edges in region LZ are sonic, the upwash in region JJT is known. The regions TZ and Y then coincide and the expression (1.1't) is an inversion of (1.11b) and the expression (1.15) is exact.
In general, one can expect that the first and the second approxi-mation improve when the leading edges become more nearly sonic. Then the regions lY and 2" become relatively smaller and more distant from P , so that 3(lZ) and j(F/ can be expected to differ by a smaller amount. For slender wings the method is not satisfactory.
A different treatment of tl{ZI'jwas given by Stewartson [_5lJ . The upwashfield in region TT can be expressed as an integral of doublets over the part of p that belongs to JZ . Substituting this expression into 3(I3rJ of equation (1.1't) and inverting the order of integration leads to:
^=|'n
Fig. 1.9.
The expression (I.16) is not a
general inversion of equation (1.11b) but is interesting for several
reasons.
(i) If (p is prescribed on region VI and the wing warp on region I , Stewartson's expression (I.16) gives the solution W in a region
down-stream.
(ii) If (the envelope of) the part of the leading edges in region M are straight lines, a D.L.P. can be solved for the forward part of the wing by homogeneous flow theory (see section 1.8). In such a case the integral equation (1.11b) has in effect been inverted.
(iii) The integral equation (1.11b), which is of the first kind is re-placed by an integral equation (I.I6) of the second kind. The first term on the right hand side in (I.I6) is 3(1/ and can be used to start an iteration process.
Only the possibility mentioned in (ii) seems to have been investi-gated systematically L26J.
If an argument, similar to the one used by Stewartson is applied to the expression (1.11a) one finds
(1.17)
JT^\/flp)}(s-jhjj-n
12
Calculations based on the methods discussed in t h i s section are at t h e i r
best for wings with nearly sonic leading edges. The f i r s t r e f l e x i o n
i n t e g r a l s can be i n t e r p r e t e d as the nearly sonic counterpart of the
slender wing s o l u t i o n s .
1.7- Expansion with respect t o a slenderness parameter
The solutions of equation (1.2) can
be expanded with respect t o a
slender-ness parameter <£ = P-y .
• • I
I f the wing l i e s '^
well within / , the parameter £ i s
small and the wing is c a l l e d "slender'.'
The f i r s t term in the expansion i s
considered in t h e well known slender
wing theory. Especially Jones [3'tJ
and Ward [53] have studied t h i s term
e x t e n s i v e l y .
Adams and Sears [ l ] have shown how the slender wing theory can be
extended and improved by including a second term in the expansion i n v o l
-ving gauge functions of the form £ and £ ^ ° 9 ^ • This approximation
i s c a l l e d the "not-so-slender body t h e o r y " . More r e c e n t l y , Fenain [27j
has given a systematic treatment of the expanded solutions i n such a
way t h a t as many terms as desired can be determined successively.
The expansions are given for the unsteady flow p a s t harmonically
o s c i l l a t i n g wings in which the steady motion i s considered as a p a r t i
-cular case where the frequency i s zero. The gauge functions involved
can be determined completely and the solutions can in many cases be
c a l c u l a t e d by a systematic a n a l y t i c a l t r e a t m e n t . Some numerical a p p l i
-cations for the steady case including t h r e e or four terms i n the
expan-sion i n d i c a t e t h a t considerable improvement can be obtained with respect
to the slender body solutions [27]. The approach i s natural for small
values of <£ . For values of <£ near u n i t y , the leading edges are
nearly sonic and the approach is artificial. Many terms may be required to obtain an acceptable approximation of the exact linearized solutions or there may occur divergence.
The results obtained by the methods indicated in section (1.6) and by the method indicated in this section can be compared with exact homogeneous flow solutions for those cases where the leading edges are straight.
1.8. Straight subsonic leading edges
In the next chapter, on homogeneous flow theory, the case of wings with straight subsonic leading edges is discussed in detail. In this section we consider some general features of the methods which can be applied in this case in relation to the methods discussed in sections
(1.6) and (1.7).
If the leading edges are straight, a large class of boundary value problems permits a systematic analytic treatment. They can be reduced to the construction of two dimensional harmonic functions. The solu-tions can be expressed in terms of known funcsolu-tions multiplied by
coefficients that are uniquely related to the boundary conditions, Especially Fenain's synthesis [25] , based on Germain's theory [28,29] is very elegant and reduces a large class of problems to completely algebraic ones. They are particularly well suited for a systematic approach to design problems for wings with straight lead.ing edges.
Comparisons can be made between the exact homogeneous flow solu-tions and the corresponding results based on the approximasolu-tions dis-cussed in sections (1.6) and (1.7). ïhis comparison indicates that neither of the two approaches is satisfactory in the whole range of
(5 (O ^ (5 ^ // .To cover the whole range of £ it is necessary to shift from approach (i) (see section I.I) for not too large values of
£ to approach (ii) for values of £• not too far from unity, or conversely.
For a given wing with fixed geometry, the number of terms that is required in both approximations to obtain a certain accuracy depends on
1U
the Mach number. Moreover, t h i s number of terms depends also on the
type of boundary conditions on S•
The exact homogeneous flow theory permits a unified treatment for
the whole range of <£ , and a large c l a s s of boundary conditions on S •
Therefore i t i s a t t r a c t i v e to base the treatment of the more general
problem, for wings with curved subsonic leading edges, on the methods
of homogeneous flow theory.
1.9. The quasi-homogeneous approximation
A l a r g e class of wings of p r a c t i c a l i n t e r e s t has a d e l t a - l i k e
planform with s l i g h t l y curved subsonic leading edges. We c a l l such
planforms quasi-conical (see Fig.1.1) The solutions can be expected t o
d i f f e r l i t t l e from the homogeneous flow s o l u t i o n s . In chapters I I I , IV
and V i t w i l l be shown t h a t for quasi-conical planforms, i t i s possible
t o construct quasi-homogeneous approximations which can be c a l c u l a t e d
in a systematic manner.
I f the leading edges are given by a r a t i o n a l function in X and lU-l
and the boundary conditions on S are given by a polynomial in X and |U.|,
the boundary value problems can be reduced t o completely algebraic ones.
Many applications based on homogeneous flow theory can be extended in
t h i s manner t o the more general problem of wings with curved leading
edges. If the leading edges are strongly curved, the approach i s a r t i
-f i c i a l ; too many terms may be required t o obtain an acceptable
approxi-mation or divergence may occur. Generally speaking, however, the number
of terms included can be associated with orders of approximation. In
many cases of p r a c t i c a l i n t e r e s t , a reasonably small number of terms
w i l l be s u f f i c i e n t .
The d i f f i c u l t i e s associated with curved leading edges and v a r i a
-t i o n s in -the planform are due -to -the fac-t -t h a -t d i s c o n -t i n u i -t i e s and
s i n g u l a r i t i e s occur at the leading edges. I t i s therefore impossible t o
write the solution in a form
with y)^X,U.,5t.) representing the solution for a delta wing with straight leading edges, in some way "close" to the actual leading edges, and <S as a small parameter representing a measure for the "deviation from straight". A solution of that type would possess singularities and dis-continuities not on, but only in the neighbourhood of the actual leading edges.
A possibility to circumvent these difficulties is the introduction of a transformation shifting all points of the curved leading edges to straight lines through the origin. Moreover it will be required that the Mach cone / remains a straight circular cone. The surfaces where boundary conditions have to be imposed, are then of the same conical nature as those occurring in homogeneous flow theory.
The transformed differential equation and the transformed boundary conditions can be satisfied in terms of functions which are solutions of homogeneous flow problems.
Before proceeding to the development of this method it is
16
Chapter i l
REVIEW OF HOMOGENEOUS FLOW THEORY
2 . 1 . I n t r o d u c t i o n
I n some problems of s u p e r s o n i c flow t h e p e r t u r b a t i o n v e l o c i t y i s c o n s t a n t on s t r a i g h t l i n e s t h r o u g h a f i x e d p o i n t , c a l l e d t h e v e r t e x of t h e flow f i e l d . With t h i s v e r t e x as t h e o r i g i n o f a system o f c a r t e s i a n c o o r d i n a t e s .3r, , X^ , Xj , t h e p e r t u r b a t i o n v e l o c i t y components U.I^'.UT a r e homogeneous o f o r d e r z e r o i n Jr,,JC^and .3:^ . These v e l o c i t y f i e l d s a r e c a l l e d c o n i c a l f i e l d s . They were f i r s t d e s c r i b e d by Busemann [3] who developed a l i n e a r i z e d t h e o r y f o r t h e s e flows ['t] . The t h e o r y was l a t e r g e n e r a l i z e d by Germain i n t o t h e t h e o r y of homogeneous flows [28] , [29] . S u b s e q u e n t l y , o t h e r a u t h o r s c o n t r i b u t e d t o t h e t h e o r y by u s i n g d i f f e r e n t m e t h o d s .
Robinson ['to] i n t r o d u c e d h y p e r b o l o i d o - c o n a l c o o r d i n a t e s and o b t a i n e d s o l u t i o n s by employing a method which i s a c o u n t e r p a r t of t h e t r e a t m e n t of L a p l a c e ' s e q u a t i o n by systems of o r t h o g o n a l c o o r d i n a t e s . The s o l u t i o n s i n v o l v e Lame's f u n c t i o n s . S e v e r a l a p p l i c a t i o n s of p r a c t i c a l i n t e -r e s t we-re made by Rope-r ['t2 - lt6] .
Lomax and H e a s l e t d e r i v e d s i n g l e i n t e g r a l e q u a t i o n s which r e l a t e e i t h e r t h e t h i c k n e s s of a s y m m e t r i c a l wing t o t h e p r e s s u r e d i s t r i b u t i o n or t h e l o a d i n g of a l i f t i n g wing t o i t s s h a p e . They a r e a p p l i c a b l e t o t r i a n g u -l a r p -l a n f o r m s w i t h s u b s o n i c -l e a d i n g edges [ 3 7 ] .
G e r m a i n ' s t h e o r y and i t s s y s t e m a t i c a p p l i c a t i o n s t o s u p e r s o n i c w i n g s , i n p a r t i c u l a r t h e a p p l i c a t i o n s by Fenain and V a l l e e , ^l6 - 26] , seem a s u i t a b l e p o i n t of r e f e r e n c e f o r t h e f o r m u l a t i o n and s o l u t i o n of our p r o b l e m . I n [ 2 5 ] F e n a i n s y n t h e s i z e d t h e t h e o r y i n a very e l e g a n t form.
A large class of problems is reduced to completely algebraic ones.'The functions that occur in the solutions are simple and can be determined once and for all. A rational approach to the design of delta wings is made possible in this way. The discussion in this chapter will, there-fore, be based on Fenain's theory.
2.2. Definition of homogeneous flows
Germain [28] defined a homogeneous flow of order 71 as a flow in which the perturbation potential ip^(x,,X2,,Xj) is a homogeneous function of degree 11 in the variables Jfc,,3:^ and J^ . This potential satisfies
the equation
J(X^„\X^)KX^)^XJCX„X^,X^). (2.1
0^,,X^,X_^) CXoc,.)iX^^\t \ D i f f e r e n t i a t i n g with respect t o A
and p u t t i n g A = / , one obtains „ E u l e r ' s r e l a t i o n '.
Fig. 2 . 1 .
Equations (2.1) and (2.2) are e q u i v a l e n t ; the one can be derived from the o t h e r . For n a t u r a l numbers IT , i t follows t h a t a l l tl d e r i v a t i v e s with respect t o 3:,,Jhj^and X^ are homogeneous functions of degree z e r o . They are constants on s t r a i g h t l i n e s through the o r i g i n . The o r i g i n i s c a l l e d t h e v e r t e x of the flow f i e l d .
18
2 . 3 . Statement of the problem
a
.^1/_ —
^ ^ - ^
\ x \ ^\ \ \
\ \
\
-/
- ^
•V /s
-^
} — „ ^4
/^
/
—
/\
^
/
r
\
1~r^
f i
1 ' Fig. 2.2aThe general considerations given in chapter I are valid if one replaces the variables Jc.ü and X by Jt, , Jc and Xj respectively.
The perturbation potential must satis-fy
/SV-,-.-/'...-W"- 1^-3)
The Mach cone / is given by
[X,^ö)
Fig. 2.2b:t;'-/3(^:-h^s)=o, ^.>°> (2.
' t )and the boundary condition on / is
The boundary conditions at the wing surface are applied on 2,(^_i=o) The subsonic leading edges are straight lines through the origin.
They are given by
| : t j = XXi , (2.5)
the modulus bars indicating that there is symmetry with respect to X^=0. The trailing edges are supersonic.
In the T.P. the perturbation potential (p(jc,,X2_,X^)aiid. its derivatives (A and U) are even in Xj . U)^ is odd in Jt^ and equal to zero on R .
In the L . P . , the p e r t u r b a t i o n p o t e n t i a l (Z) , (J) , and iD^ are
odd in J t . (J) i s even in X, . Since on R , W i s continuous and odd
in X, , (p and (P are equal t o zero on R .
The boundary c o n d i t i o n s i n the D.L.P. and the I . T . P . in the plane
Jt = 0 are of mixed type because one knows (p in one p a r t of the plane
and (p. in t h e remainder. In the D.T.P. and the I . L . P . the boundary
conditions in - ^ = 0 are simpler because only U) or 0. are involved.
In an I . L . P . , U= (D i s prescribed in the plane .3:^=0 . This
determines (D in the plane J:, = 0 up t o a function of X^ . This
function must be chosen in such a way t h a t U) i s equal t o zero at t h e
l e a d i n g edges. In t h i s c a s e , the sidewash V==(p i s also uniquely
determined.
In the I . T . P . , (D and (D for ^-^O follow in the same way from
(D-j^ up to an a r b i t r a r y function of X. • This implies t h a t a c e r t a i n
p r e s s u r e d i s t r i b u t i o n in a T.P. does not uniquely determine the c o r r e s
-ponding upwash XiXzzi CD , at the wing surface. Another indeterminacy
a r i s e s upon i n t e g r a t i n g CD. with respect t o X, in order t o obtain the
t h i c k n e s s d i s t r i b u t i o n . In p r a c t i c e , of course, the f i n a l s o l u t i o n
should admit only the edges of the planform as l i n e s of zero t h i c k n e s s .
The planforms considered are symmetric with respect t o (jX, and the
boundary conditions o n p a r e , in g e n e r a l , even in X]_ . The case of
boundary conditions t h a t are antisymmetric in -^2 , which makes sense
in t h e D.L.P. o n l y , i s indicated as D.L.P. Such problems a r i s e in the
study of r o l l i n g wings, in the cases of non-symmetric gusts as studied
by Lance [35,36] or in cases of antisymmetric deformations.
2. 't. The 71 d e r i v a t i v e s
I f 03 i s a solution of equation ( 2 . 3 ) , a l l d e r i v a t i v e s of U)
with respect t o X,Xj. and X are s o l u t i o n s of equation ( 2 . 3 ) . This
ap-p l i e s in ap-p a r t i c u l a r t o the Ti d e r i v a t i v e s :
20
If U) is homogeneous of degree 71 in a:,,Xj_and Oc^, the ?i derivatives u) ^ "•* ^'"''•'are homogeneous of degree zero and constants on straight lines through the origin.
The boundary conditions will be specified at A:_j= O , usually for the first derivatives of C^.^ , and in most cases one is primarily in-terested in the solutions at the wing surface. Therefore,the 7i derivatives directly related to the solutions and the boundary condi-tions at the wing surface deserve special attention. These are defined by the notation:
f ^.^- (f^
4
C-H-J, J,e>J V-H-/ U .i a . ; - ' - i ^ P c , ï '
V-.. = fn
('M-/-'L,t+/.oJ èi^-'f^ ^^. •n-i-xèx^
on.
^.. = f.
i'^-t.t.o}
H - / - 5 •/ •vt^xi
=
_±_^
^ j c . ^ - ' ^ J c J= a
•He ( 2 . 7 )Where U l ^ and UT^ are the f i r s t d e r i v a t i v e s of Ü3 with respect t o
X, ., X^ and JCj r e s p e c t i v e l y .
I t i s usual t o introduce the coordinates t , X and 6 by
x,= /3xx.
-t^ = ^ Ccrt 0 ,
. J t , = X \i/n. 6 .
X > O
7 r < e < TT
( 2 . 8 )In t h i s way a one-one r e l a t i o n s h i p i s e s t a b l i s h e d between a p a i r of
values X . Ö and a s t r a i g h t l i n e through the o r i g i n . The Mach cone /
corresponds t o X = I • "X. y I corresponds t o the i n t e r i o r of /"" . The 'M
d e r i v a t i v e s of ^ depend on "X and 0 only. The transformed equation
( 2 . 3 ) , s a t i s f i e d by the tl d e r i v a t i v e s , i s no longer dependent on t
and s i m p l i f i e s t o
C':'-')/,, + Vx+/«.-"•
With
•)d=aaAy
one obtains
jyy + fee
= o.
( 2 . 9 )
( 2 . 1 0 )
. V * .
and the problem for the 71 d e r i v a t i v e s i s reduced t o the c o n s t r u c t i o n
of harmonic functions. In order to a s s i s t in the construction of t h e s e
functions, the e n t i r e theory of complex functions i s a v a i l a b l e . Taking
in f i r s t instance O + LTj/ as complex v a r i a b l e we obtain the following
s i t u a t i o n :
tV|
A/
I
N
L
I
-IT TTFig. 2.3a.
Fig. 2.3b.
Ö
I i s represented by the & axis ( T</=Oy
The boundary conditions in the Tj/, Q plane are :
^71.— (3-9 , p , 9 )( i ) For y = O i - 7 r < 0 <7r one has (p = 0 and e s p e c i a l l y a l l
Uy, = O and a l l UJy,^ = 0 .
( i i ) At the wing s u r f a c e , MN , t1N~, ^N and A?A/ a l l Cü^^ are known
in a D.P. and a l l U.^„ are known in an I . P .
( i i i ) Outside of the wing, at Lh and LM , one has in a L.P. a l l
a = 0 and in a T.P. a l l <^y,^=0.
22
More s u i t a b l e p l a n e s can be found by c o n f o r m a l mappings. 2 . 5 . Two conformal mappinj^s
By p u t t i n g
- y te
LB( 2 . 1 1 )
the interior of the Mach cone is mapped on the interior of the unit circle in the Z plane. The Z. plane can be used with advantage for the construction of the form of the solutions.
F i g . 2 . ' t .
r
"ï
-I. -I
®
'Wvm ^ / F i g . 2 . 5 .=L^3i
A p l a n e which i s more p r o f i t a b l e f o r a c t u a l c a l c u l a t i o n s i s found by
^ 1+2. ( 2 , 1 2 )
The i n t e r i o r o f t h e Mach cone now c o r r e s p o n d s t o t h e whole X p l a n e . i s mapped on t h e r e a l a x i s Ixl^l and t h e p o i n t a t i n f i n i t y . For JC. = O , one h a s
± _ /3JC^ _
X^ Ortit-y / - ^ p ' = - 3 C , ( 2 . 1 3 ) which implies that the boundary conditions and the solutions at the wing surface (Xj = 0) are related to the X plane in a particularly simple manner. If the leading edges are given by iXj^l=~C X, , the leading edges are mapped on the real axis ( a, = 0) x z=.'t M , with .*r = /3~r.
2 . 6 . The c o m p a t i b i l i t y r e l a t i o n s
The 71 d e r i v a t i v e s (M a r e harmonic f u n c t i o n s i n ' V ' and
0 which can b e c o n s i d e r e d as t h e r e a l p a r t s o f a n a l y t i c f u n t i o n s
jr (^7)-/.-i,p,iJ . . ^
\U of t h e v a r i a b l e W-f-iU o r of one o f t h e v a r i a b l e s /L or X . A l l t h e U) a r e d e r i v a t i v e s of t h e same id and a r e t h e r e f o r e n o t , i n g e n e r a l , i n d e p e n d e n t . To f i n d t h e r e l a t i o n s b e -tween t h e d e r i v a t i v e s we may p r o c e e d a s f o l l o w s . I n t r o d u c e a f u n c t i o n (J) , homogeneous o f d e g r e e one i n Jc,, X,^ and JCj , byr = % (2.1U)
/ / I * • The t o t a l d i f f e r e n t i a l o f W i scLf^ (f^dx, + (f^^lx^-^ (flcLx^. (2.15)
On t h e o t h e r h a n d , E u l e r ' s r e l a t i o n g i v e s : and t h e t o t a l d i f f e r e n t i a l J^ a l s o g i v e n b y : From ( 2 . 1 5 ) and ( 2 . l 6 ) i t f o l l o w s^,^fl, + ^AfZ^^Afl~^- (2.17)
-f-(2.16)2't
^.^^CZ.2.),
( 2 . 1 8 ) Introducing t h e functions (2.17) can be w r i t t e n in t h e form ^ . 2/0 (7i-p-3.,p,3.) ^employing ( 2 . l 8 ) , t h e r e l a t i o n ( 2 . 1 9 )The functions between the brackets are functions of L. only and since Z. and Z are to be considered as independent variables, these expres-sions must vanish identically. One easily deduces
l-hZ
i-T
( 2 . 2 0 )These r e l a t i o n s can be considered as recurrence r e l a t i o n s . For p= ^ = 0 one has
• ('>',o,o) (•n-1,1,0)
-IficZJc^$f.-''°'^2.21a)
For p = l>^=0 :
For
p = ö,g;»/
^z6.-/,0,/;_ , ^ ^^7,-z.,,;^ .0y<-l,O.l) (2.21c)
^
Proceeding in the same way one obtains expressions that can be substi-tuted successively into equation (2.21a) and one establishes the equi-valence of the expressions
In the X plane, the compatibility relations (2.22) can be expressed as :
^r""'=^'J?|J''V^f "^ !fi"''"""
:2.23)The r e l a t i o n s (2.23) can be thrown i n t o a more useful form, by introducing t h e v e l o c i t y f i e l d more e x p l i c i t l y :
:2.2it)
one may w r i t e :
\ / 3 ^ oLS- ' /3^ dx ^ (3' [s/iZfi dx '
( 2 . 2 5 ;The relations (2.25) express the relations between the pressure pertur-bation, the sidewash and the upwash. It should be noticed that knowing one of the 7i derivatives, all the 71 derivatives can be found from these compatibility relations.
26
For i n s t a n c e :
2 . 7 . E u l e r ' s r e l a t i o n for homogeneous functions
E u l e r ' s r e l a t i o n (2.2) can be i n t e r p r e t e d as a recurrence r e l a t i o n .
From
^ %= "^^ fr.., + ^.%^^ -t- ^.f.^, , (2.27)
one deduces by d i f f e r e n t i a t i o n with respect t o X,j X^ and X^ :
( ^ - ' ) fn:.. = -^. %j.,x, -f- ^^fvx,^. + j^x.:t3 " (2-28)
S u b s t i t u t i o n in (2.27) gives
Repeating t h i s process one obtains
This shows t h a t Wy. can be obtained from i t s 71 derivatives without
carrying out any i n t e g r a t i o n s .
In the same way one obtains
'TL-I
For J ^ = 0 the relations (2.30) simplify and with (2.7) one finds:
30)
\
>,_/ • n - / - 3 . ^a = Z — —
^%
>
" ito (-n-.-i)/ i /
^=o {n-\-x}lxl
( 2 . 3 1 a ) • t t - / - S .s V. L C T , — Z _ - 7 7^ • 5 = 0 (It - I -s)l 3/ One may adjoin (x = o) •/3
^"^ to (yi-t}J tf
( 2 . 3 1 b )
The f u n c t i o n s ü.^_ , l/^,^ e t c , a r e c o n s t a n t s on s t r a i g h t l i n e s t h r o u g h L/, hence only f u n c t i o n s o f 2£j-= .^ . /Xi=Oj
One can a l s o w r i t e :
«-/
U ^ = Jc
IT
rz
a
! ! l _j.=o (-vi-z-i)/2./ 7^
(If'
~ > t .
to (n-.-s)/^./ ^/3^
( 2 . 3 2 )
28
A further simplification arises when one of the functions U»,p , V'.n^^ , etc. is constant over the entire wing. The corresponding func-tion U„ , V'.^ , etc. is then a polynomial in X and X^ of degree (T»-/).
2.8. Elementary flows
If the first derivatives of U) which determine the boundary con-ditions on S are homogeneous polynomials in X, and 3:^ of degree (TI-O , the homogeneous flow is called "elementary". This implies that for elementary flows the (71-/) order derivatives of one of the functions
U^ , V.fi or UJl^ , which are given by '^•,0»^^°^ "^s respectively, are constants on S • If the wing surface is of polynomial form in JC, and Xj^ and contains an arbitrary function of X , one has to do with ele-mentary D.P.'s.
Prescription of U^ in polynomial form (elementary I.P.), implies that U) a.t S will be of polynomial form up to an arbitrary function of
Xj^ . It is natural to take this function of the form C X^ . in the I.T.P. the coefficient C is arbitrary. In the I.L.P. the coefficient
C is determined by the requirement that (U is equal to zero at the leading edges. It follows that in an elementary I.P. the sidewash 1/^ on S will be of polynomial form and dependent on C .
The functions U^ »''^Y ''*^I ^^"^ T t introduced in (2.7) can be lated to the coefficients specifying the boundary conditions. The re-lations are given in the next section.
2.9. The boundary conditions on P for elementary flows
For — <^ T , "the boundary conditions for the different types of I ^ ; I
elementary flows can be w r i t t e n as follows:
"^-l -n-i-s . . . .S
(D.T.P.) ur:=L ^...,-s.s *. If
.s = o
( I . T . P . )
u^=-zZ
a J c ,- i
hcSlf.=-rZ
^ 11=0 •n (2.3U) 'H-t.fc ( D . L . P . ) • n - / ^-'-s.j^,s S = 0I r l
(2.35) •n-/f < = - ^ Z
( I . L . P . ) -K-;- 5_ ft^^^>,= +
l^J
7 i X
^ T « - ^ ^ o (71-2.) L ' f ^ ' ' ^ 1 ) 2- IJC.(2.36)
(D.L.P.) c j ^ i ^ J y ' c * j c ' - ' - ' i ^ l - " (2.37:
•*-i s=o ' *- ' The c o e f f i c i e n t s i n t r o d u c e d i n ( 2 . 3 ' t ) f o r t h e I . T . P . a r e n o t i n d e p e n -dent . One e a s i l y v e r i f i e s :{
For the coefficients in (2.36) for the I.L.P. one has:
J(n-/--.}i^_,_^_^=-(t+/j a^_^_^^^^^ .(o<t<7^-z;
n-1 :^30
Comparison with equations (2.31) gives the r e l a t i o n s between the
c o e f f i c i e n t s introduced in t h i s section and the ^ ^ ^ , <^.^5 ^^^s ^^'^
CD. . The r e s u l t i s 4-*^'^'
(ü.i.f.) ^.^^ x^~Sc^ '-h-'-s,s •
( I . T . P . ) "3- - J - 2 - ' ^ 2 . ^^-i-%'% '<
[$
I I <'>•+'U ^ ( 7 1 - / - X) Xl I^J
' ^ t = - t-f T, tT " - ' X^
o ( 4 , - 6 ; t ( 2 . 3 8 ) ( 2 . 3 9 ) ( D . L . P . )or =
'Via_ in-i-s]! sl\x,f ^*
•T--S j^S ^-'h->-S,.S ( 2 . ' t o ) ( I . L . P . )r^
.'*;
( 2 , ^ i ; ( D , L . P .ur
•Mi(^1~l-s)! sl IxJ
r'
- ;
i + /- - C
H - / - i j ^ ( 2 . ' t 2 )The remaining question i s now, how t o proceed from these conditions at
the wing surface t o the complete solution for the flow f i e l d in the
X^,X^,Xj space. The f i r s t step in t h i s process i s the construction of
2.10. The functions
dU
ctZ
Ü? in an elementary D.P.
For the c o n s t r u c t i o n of the complex functions introduced in
equa-t i o n s (2,2'equa-t) equa-t h e Z plane i s well s u i equa-t e d . The argumenequa-ts w i l l be
presen-ted in d e t a i l for D . P . ' s , the arguments for the I . P . ' s being analogous,
From the c o m p a t i b i l i t y r e l a t i o n s i t i s c l e a r t h a t i t i s s u f f i c i e n t
to operate on one 71 d e r i v a t i v e of ffl only, say ïv!^ • The c o m p a t i b i l i
-t y r e l a -t i o n s g i v e :
2dZ ^M. = / 3 / / - z ' j " ^
«p ( 2 , U 3 )The function LT has to be regular a t Z = + / and (2.lt3) then implies that ^LIXÏO must be devisible b y ( / - Z j .
dZ y
/ ®
- a
\r
^
^y^Q \
<x
y
On / one has:
l^=o, | Z | = / , Z = e ,
G and/
/
d Z dQ
Fig. 2 .
Since UJ,uj i s zero on the Mach
cone, W.y^ has a zero r e a l p a r t on
/ and one may introduce a function
TLz)
dXvTo
Z^-/
dZ
'
:2.u't)with t h e following p r o p e r t i e s :
32
On the two edges of the cut (—a, a.) ,t(Z)ha.s a zero real part in an elementary D.P.
To avoid singularities at the origin, 7^(Zj must be divisible by Z . The only admissible singularities for |Z| < / in TCZ) will occur at
Z = it CL . In the neighbourhood of Z = a one may put
T7z)- Z l c ^ f z - a } !
.* (2.'*5)
where X and B are defined in t h e
f i g u r e .
Fig. 2 . 7 .
Inspection shows t h a t only two p o s s i b i l i t i e s of i n t e r e s t e x i s t :
( i ) (2^ imaginary; m an i n t e g e r .
( i i ) C ^ r e a l ; 7n an i n t e g e r -*• y •
A s i m i l a r argument can be given for Z = - a so t h a t riZ) can be w r i t t e n
in t h e form:
\[^:z^' ^'-''^
in which 0 ( Z J and "WiZj admit poles at Z = i C L as only s i n g u l a r i t i e s .
I f the flow f i e l d i s symmetric with respect t o t h e plane J t , = 0 , the
r e a l p a r t of W^,^ must be even i n X •
This implies t h a t r ( Z j i s purely imaginary on C7 / and one can w r i t e
For the elementary D.T.P. one obtains functions r(Z.) t h a t s a t i s f y a l l
requirements by t a k i n g \ll(Z.j=-0 in (2.116). One may write
-T'f-z) = 7 '- i^P ^
with 71 r e a l c o e f f i c i e n t s ha • The s i n g u l a r i t i e s at Z = ± <1 are r e
-f l e c t e d in the u n i t c i r c l e / Z l = / , t o s a t i s -f y the boundary conditions
on the Mach cone. F o r | Z | = / the expression (2,lt8) i s purely imaginary.
For y = O , ~F(Zj i s purely imaginary.
For the i n s i d e of the unit c i r c l e the points Z = ± o, are poles
of order p . The upper l i m i t i s 71 i f one admits logarithmic s i n g u l a
-r i t i e s at most in t h e p e -r t u -r b a t i o n v e l o c i t i e s . S t -r i c t l y speaking t h i s
contradicts the assumptions of l i n e a r i z e d theory and one should take
(yi-l) as the upper l i m i t for p . However, t o obtain a n o n - t r i v i a l
solution for 71= ƒ a l s o , and t o obtain solutions t h a t depend on Tl
parameters n t, , j u s t as the boundary conditions depend on TL
coefficients C.y,_i_^ 5 (in equation ( 2 . 3 3 ) ) , one admits 7l as t h e
upper l i m i t . The pressure then remains i n t e g r a b l e . The edge forces
remain f i n i t e and one supposes t h a t the regions in which the assumptions
of l i n e a r i z a t i o n are v i o l a t e d are s u f f i c i e n t l y small.
For the D.L.P. one takes Ch{Z\=0 and one may write
-Tfv] - T ^ B , Z ' ' ( Z - / j
in which the c o e f f i c i e n t s 3 p are r e a l . F o r | Z | = / , r ^ Z j i s purely
imaginary. At the c u t ^ d , a),"7^(Zj is also purely imaginary. The upper
l i m i t in (2.1t9) i s 71. , which implies t h a t square root s i n g u l a r i t i e s
for the perturbation v e l o c i t i e s are admitted at the leading edges. They
remain integrable and the edge forces remain f i n i t e . In t h i s case t h e r e
are also 71 c o e f f i c i e n t s which can be determined in such a way t h a t t h e
boundary conditions at the wing surface are s a t i s f i e d . One i s thus led
to adopt the following s o l u t i o n s :
3't if!-2 ^
(D.T.P.): ^ _ ^ = r "'^fr ^ (^Zll^ , ( 2 . 5 0 )
a 7 p4-/ {Ca.'-Z')(l-a'Z')}^
4U°= f ^-B^z'^-Vz^-zj'
dZ p4, {(a^_Z^j(/-a^Z";}'
( D . L . P . ) : dM;o^ Y LBP Z (Z - I) _ (2.51)
2.11. The analytic functions r e l a t e d t o the boundary conditions in
the X plane
In the 2 p l a n e , the expressions (2.50) and (2.51) can be put
in the form (JU = ^^ ] :
^ l-hO^J
(D.T.P.): ° L M p ^ _ l i r Akp -L (2.52)
(D,L,P,): c^W^ _ i t \jl-'i^ y A j i t i (2,53)
J ? TT VjL^y^ Z 7 ? t F ) f
From the compatibility r e l a t i o n s (2,26) one obtains
(D.T,P.): 4J:lJl} = - ii fi:^]^ r ^^ ^ , (2.5't)
d-i -rr ^icl ^, iV-i.^}^
(D.L.P.): "i^^-ii. (z;^f\fJZX' f )LJI\ (2.55)
(I.T.P.)
4JLl=
lil
/zêf-]/^y''LiLj^ ,
(2.56)
( I . L . P . :
LU,. _ lil / - ^ ) ^ 7
^y., ^
. (2.5^)
[7
77 I T ^ p^_, ^£^_4^)P
The precise fonn of the coefficients appearing in these solutions is not essential but a matter of convenience in view of the calculations to be carried out in the next sections. The preceding expressions have been derived for subsonic leading edges but the expressions (2.5't) and
(2.57) can also be used for wings with supersonic leading edges. One may also notice for instance that (2.55) simplifies to (2.5't) for
-K = / , i.e. for sonic leading edges, as it should. By analogy one is led to adjoin:
(D.L.P.) iM^=^^i ±(-J}]"\n^ yjL^^'. (2.58)
m
2.12. Determination of t h e c o e f f i c i e n t s introduced in t h e 7i
d e r i v a t i v e s
The expressions (2.5't) t o (2.58) contain 7i or 71+/ a r b i t r a r y
real constant c o e f f i c i e n t s which have t o be r e l a t e d t o the c o e f f i c i e n t s
appearing in the boundary conditions.
( i ) In a D.T.P. a l l ^ , j are constants on the cut (--U^Xc) and zero on
t h e Jc - a x i s for (£|>-/!c . I n t e g r a t i o n of (2.5't) gives (^^^- '^TTl^^
i n which •*.„, i s the residue of dVJ^s at x = ^ .
Evaluation of the residues and
36
K^^^li.,r(sz^Ks,^3)^s + l ^
A^=^-//(v>-/-^;.'../c_.^,. (2.59)
p=s- (-ip-i),'.' ^
With ( 2 p - i ) .'.' = J2.^.6. C-^-p-X), and O^Siyi-i. The solution of this system can be given explicitly [29J in the form:
\ =Y c T (~'f~ Cp-o!(yi+it-i)! _ (2.60)
5 = 0
^ ( (zb-HS-i) 'cp-'=)lth-i)JCit-z)l
( i i ) In a D.L.P. one finds from (2.55) that for odd S = 2yn-I , the constants ' ^ j ^ ^ .are easily calculated. A suitable path of inte-gration i s the imaginary axis and a quarter circle around the origin. Comparison with (2.'t0) then yields
T '^ u r , - Ci-yn-ljUn-Z-m)! C* , , , ^^.61)
and
Z (-0'"' ^^ Z ' {it)\L^^^^'^-^^-'^)l p\ ^^'^ .
= llyn-\)l(n-Xyn)\ C^, , , . (2.62)
For even 5=Zrti(hi>/J the situation i s more complicated. The i n -tegrations t o be carried out are more d i f f i c u l t . The details are given in the appendix. The equations can be summarized as follows:
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