An analytical procedure is described for designing smooth transition surfaces for blended wing-body configurations. Starting from two specified cross section shapes, the procedure generates a gradual transition from one cross section shape to the other as an analytic blend of the two shapes. The method utilizes a conformal mapping, with subsequent translation and scaling, to transform the specified and shapes to curves that can be combined more smoothly. A sample calculation is applied to a blended wing-body missile type configuration with a top mounted inlet

Barger, R. L.

English

NASA Technical Reports Server

NASA 
Technical 
Paper , 
2012 
May 1982 
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m= An Analytical Procedure for -dzzEE ~ 
fn= Comtwting. Smooth Transitions = I U 
Between Two Specified 
Cross Sections With 
Applications to Blended 
Wing-Body Configurations 
Raymond L. Barger 
I .  
https://ntrs.nasa.gov/search.jsp?R=19820017317 2020-03-21T07:27:05+00:00Z
TECH LIBRARY KAFB, NM 
NASA 
Technical 
Paper 
2012 
1982 
3 
National Aeronautics 
and Space Administration 
Scientific and Technical 
Information Branch 
An Analytical Procedure for 
Computing Smooth Transitions 
Between Two Specified 
Cross Sections With 
Applications to Blended 
Wing-Body Configurations 
Raymond L. Barger 
Langley Research Center 
Hampton ,  Virginia 
SUMMARY 
An analytical pxocedure is described fox designing smooth transition surfaces 
fox blended wing-body configurations. Starting from two specified cross-section 
shapes, the procedure genexates a gradual transition from one cross-section shape to 
the other as an analytic blend of the two shapes. The method utilizes a conformal 
mapping, with subsequent translation and scaling, to transform the specified end 
shapes to curves that can be combined more smoothly. A sample calculation is applied 
to a blended wing-body missile-type configuration with a top-mounted inlet. 
f 
I 
INTRODUCTION 
The construction of aerodynamic configurations requires a detailed numerical 
description in the form of position coordinates of the surface points. Often these 
coordinates are read from drawings of the configuration cross sections on graph 
paper. AS a general rule, certain of the cross-section shapes are closely determined 
by the requirement for packaging internal equipment of a given shape, or by the pres- 
ence of a jet inlet or exhaust, or by some similar condition. The intermediate 
cross-section shapes, between those that are thus determined, are usually obtained by 
making a smooth transition between the predetermined cross sections. When this tran- 
sition is performed by hand lofting, it becomes tedious and time-consuming. 
Fox some types of configurations or configuration components, however, a com- 
puter can be utilized to compute the transition shapes. Reference 1 describes a 
procedure for computing transition cross-section shapes for forebodies. The method 
of reference 1, however, treats only cross-section shapes that are single valued when 
expressed in polar coordinates; thus, it is not applicable to more general design 
problems. 
The present procedure, however, treats the problem of computing cross-section 
shape transitions for a class of wing-body configurations. It is not intended to 
perform the same function as a general geometry program, such as that described in 
reference 2. Such a general geometry program utilizes point-to-point fairing, which 
is usually by means of low-order polynomials. In essence, it can be used to approxi- 
mate the geometry of a given configuration, and to display rapidly the geometric 
effects of altering individual point locations. The present procedure, on the other 
hand, is intended for application primarily in the design process to generate rapidly 
a large section of a wing-body configuration in an analytic or semianalytic form. 
The method uses function averaging rather than point-to-point fairing. Thus, it 
produces transition surfaces between two given cross-section shapes that repxesent at 
each station a true analytic blend of the end shapes without the slope and curvature 
excursions that are inherent in polynomial fairing. The resulting surfaces, for 
smooth input cross sections, should have a degree of smoothness (continuous second 
derivatives) sufficient for accurate calculation of flow properties and radar cross- 
section parameters. 
I 
SYMBOLS 
t ransformation parameter (eqs. ( 1 )  and ( 4 ) )  
exponent func t ion  def ined by equation ( 3 )  
t r a n s i t i o n a l  transformed shape func t ion  a t  x, def ined by equat ion ( 2 )  
single-valued func t ions  of C determined by transforming the i n i t i a l  and 
f i n a l  c r o s s  sec t ions ,  r e s p e c t i v e l y  
parameter i n  Kgrkn-Treff t z  c a l c u l a t i o n  
complex variable y + i z  
long i tud ina l ,  lateral ,  and ver t ical  coordinates ,  r e spec t ive ly  
exponent func t ion  i n  equat ion ( 3 )  
l a t e ra l  and ver t ical  coordinates ,  r e s p e c t i v e l y ,  i n  transformed plane 
i n t e r i o r  angle  a t  corner  p o i n t  
complex v a r i a b l e  i n  txansformed plane,  17 + i Z  
Subscr ipts :  
1 i n i t i a l  s t a t i o n  
2 f i n a l  s t a t i o n  
i ma 11 est  value 
m maximum value 
s t  s c a l e d  and t r a n s l a t e d  value 
ANALY SI S 
The cross-sect ion shapes a t  two x-s ta t ions  are assumed t o  be given, and they are 
assumed t o  be symmetric about t he  ver t ical  x,z-plane. Each of the  symmetric halves  
may be represented as  a single-valued funct ion y of t h e  vertical  coordinate  z (as 
shown i n  the example of f i g .  l ( a ) ) ,  or it may .be represented by a more complex shape, 
such as  t h a t  of f i g u r e  l ( b ) ,  which might describe a s e c t i o n  of a blended wing-body 
configurat ion.  More extreme cases, such as a s e c t i o n  through a fuse l age  and wing 
with nacel le ,  could only be treated with much d i f f i c u l t y ;  t he re fo re ,  they are n o t  
considered herein.  . .  
I f  both of t he  shapes can be represented as single-valued func t ions  of z ,  then 
a smooth t r a n s i t i o n  between them can be computed by the method of reference 1. How- 
ever,  i f  a t  least  one of t he  shapes cannot be represented i n  such a manner (as that  
shown i n  f i g .  l ( b ) ) ,  then the method of reference 1 f a i l s  and a new approach i s  
required.  
2 
Ill ' 
This approach involves transforming each of the two end shapes by a conformal 
mapping into single-valued functions of a similar nature so that they can be easily 
combined. 
First, the general point (y,z) on each of the end shapes is written as a complex 
variable s = y + iz. For this purpose, a proper choice of axes is important. Where 
a corner exists, as at the wing tip in figure l(b), the y-axis is taken through that 
point. For a shape like that of figure l(a), no corner point exists, but there is a 
point representing the maximum width, and consequently, the y-axis is taken through 
that point. 
Various choices of the complex mapping function could be made. 
Trefftz transformation (including the Joukowski transformation as a special case) is 
normally used because of its versatility and its familiarity as a mapping to open up 
and round out a complicated shape. (See ref. 3.) The transformation equation is 
The Kgrmgn- 
cr- c \ s  - nc) 
where n and c are parameters to be assigned in accordance with the particular 
shapes to be transformed. This transformation is described in detail in reference 4 
(sec. 7.32). For a shape with a corner, like that of figure l(b), the parameter c 
is determined by the location of the corner. (Since fnc are singular points of the 
transformation, s = nc or s = -nc is a corner location.) The parameter n is 
determined by the interior angle A at the corner in the physical plane in accor- 
dance with the relation: 
For a shape like that of figure l(a), no singularity point exists on the boundary; 
therefore, the singularity of the transformation is taken inside the boundary. A n  
appropriate location is midway between the point of maximum width and the center of 
curvature at that point. For smooth shapes the value of c does not have to be 
determined with great precision. The value of n for this cross section (fig. l(a)) 
was assumed to be the same as that for the cross section of figure 1 (b) . 
When the transformation is applied to both of the end shapes, the resulting 
curves can be represented as single-valued functions of the vertical coordinate c .  
Figure 2 shows, for example, the transformed curves for the cross-section shapes of 
figure 1. For this example, the similarity of the transformed curves is quite marked 
despite the distinct dissimilarities of the original shapes. In the event that 
essent,ial differences in the form of the transformed shapes exist, then a second 
Ka/rman-Tref f tz transformation could be applied. An example of applying successive 
mappings is shown in figure 3 where the initial shape is similar to that in fig- 
ure l(a) but with a discrete concave corner. A n  initial transformation rounds out 
the lower part of the curve, and a second transformation is applied to straighten out 
the corner. 
In order for the transformed curves to be combined, they must be written as 
functions having the same direction on the c-axis. For this purpose, one of the 
curves can be translated and scaled as follows. If the domain of the initial func- 
tion is 
then a new independent variable for the second curve is defined by 
cli < c < clm and the domain of the other end function is s2i c c c s2m 
3 
Denote the  r e s u l t i n g  func t ions  of C as f , ( C )  and f 2 ( C ) .  Various methods can be 
used t o  combine these two func t ions .  The simplest of these  methods is  to form a kind 
of arithmetic average of the  func t ions .  However, experience demonstrates t h a t  a 
smoother, more aerodynamically d e s i r a b l e  surf  ace is  obta ined  when geometric averaging 
is  used. 
Thus, the  combination curve a t  t he  x-s ta t ion  is  given i n  terms of the  end curves 
f l  and f 2  by the  formula 
where 
and a i s  an a r b i t r a r y  parameter which may be assigned as a cons t an t  o r  as a smooth 
func t ion  of x. This  parameter provides  a means of a d j u s t i n g  the  rate a t  which the  
shape changes. If a i s  a small f r a c t i o n ,  E i s  approximately one, except  very 
near  x2. Consequently, the  f a c t o r  conta in ing  f l  i s  dominant except  near  x2. 
Thus, the  shape changes slowly near x1 from the  i n i t i a l  shape ( f l ) .  Conversely, 
when a i s  r e l a t i v e l y  l a rge  ( > 2 )  t he  t r a n s i t i o n  i s  r ap id  a t  f i r s t  with l i t t l e  of t he  
change occurr ing near  x2. 
Values of t he  t ransformation parameters n and c are requi red  a t  each 
x-s ta t ion .  They are obtained from the  va lues  a t  the  t w o  end s t a t i o n s  by a formula 
s i m i l a r  t o  equat ion ( 2 ) .  
Figure 4 ( a )  shows an example of a t r a n s i t i o n  curve computed from the shapes 
shown i n  f i g u r e  2, toge ther  with the  transforms of t he  two end shapes, f o r  compari- 
son. For each such t r a n s i t i o n  curve, t he  corresponding cross -sec t ion  shape i s  
obtained by transforming the  coord ina tes  back t o  the  phys ica l  plane by the  i n v e r s e  
t ransformation,  which is  
L n s + nc 
s - nc = (=) ( 4 )  
For the  transformed c ross  s e c t i o n s  of f i g u r e  4 ( a ) ,  t he  corresponding phys ica l  
plane shapes are shown i n  f i g u r e  4 ( h ) .  
Afte r  t he  requi red  number of t r a n s i t i o n  c ros s  s e c t i o n s  have been computed, the  
r e s u l t s  can be displayed as a u n i t ,  a s  i l l u s t r a t e d  i n  f i g u r e  5. I f  some t a i l o r i n g  of 
the  ca l cu la t ed  t r a n s i t i o n  shape i s  required,  modi f ica t ions  can be accomplished by 
varying the  parameter a i n  equat ion ( 3 ) .  The e f f e c t  of varying Q: on the  o v e r a l l  
sur face  is  i l l u s t r a t e d ,  f o r  t he  sample t r a n s i t i o n ,  i n  f i g u r e  6. The e f f e c t  on a 
s p e c i f i c  c r o s s  sec t ion ,  taken a t  the  middle a x i a l  s t a t i o n ,  i s  shown i n  f i g u r e  7. 
4 
Once the  t r a n s i t i o n  shapes have a l l  been ca l cu la t ed ,  they can be independently 
scaled f o r  the purpose of a t t a i n i n g  a s p e c i f i e d  a x i a l  area d i s t r i b u t i o n  o r  f o r  smooth 
matching with the rest of t he  configurat ion.  
Whether t he  rest of t h e  configurat ion has  been designed by a s i m i l a r  method o r  
whether it has been independently spec i f i ed ,  the c o n t i n u i t y  a t  the j o i n i n g  c r o s s  
sec t ion  i s  assured by specifying t h a t  t he  f i n a l  c r o s s  sec t ion  of one sec t ion  match 
the i n i t i a l  c r o s s  sec t ion  of the next. However, t he  problem of ob ta in ing  a smooth 
j o i n i n g  involves  s p e c i f i c  mathematical requirements on t h e  a x i a l  s c a l i n g  and on a, 
which may be t r e a t e d  as a func t ion  of x. Th i s  problem is  not  d i f f i c u l t  b u t  i s  out- 
s i d e  the scope of t he  p r e s e n t  paper. 
An example of a blended wing-body missile-type conf igu ra t ion  with a topmounted 
i n l e t  is  shown, i n  p l an  v i e w ,  i n  f i g u r e  8. An af terbody determined by the end c r o s s  
s e c t i o n s  of f i g u r e  1 w a s  f i t t e d  onto a forebody ahead of t he  i n l e t .  The forebody w a s  
spec i f i ed  t o  have the i n i t i a l  shape of an e l l i p s e ,  with a terminal  shape l i k e  t h a t  of 
f i g u r e  1 ( a )  b u t  without  the superimposed i n l e t .  
It  i s  r e l a t i v e l y  simple t o  superimpose a s p e c i f i e d  camber l i n e  on the configura- 
t ion.  Since t h e  camber i s  given as a func t ion  of x, t he  value of t he  func t ion  a t  
each x - s t a t ion  i s  simply added t o  t h e  z-coordinates of t he  t r a n s i t i o n  cross-sect ion 
. shapes a t  t h a t  s t a t i o n .  
CONCLUDING REMARKS 
An a n a l y t i c a l  procedure has  been descr ibed f o r  generat ing smooth t r a n s i t i o n  
su r faces  f o r  blended wing-body configurat ions.  S t a r t i n g  from two s p e c i f i e d  cross- 
s ec t ion  shapes, t he  procedure generates  a gradual t r a n s i t i o n  from one cross-sect ion 
shape t o  t h e  o t h e r  a s  an a n a l y t i c  blend of t he  two shapes. 
The method used a conformal mapping, with subsequent t r a n s l a t i o n  and sca l ing ,  t o  
transform the  end shapes t o  curves t h a t  could be e a s i l y  combined. The example 
described w a s  a blended wing-body missile-type conf igu ra t ion  with a top-mounted 
i n l e t .  
Langley Research Center 
Nat ional  Aeronautics and Space Administration 
Hampton, VA 23665 
A p r i l  28, 1982 
5 
RFFEReNCES 
1. Barqer, Raymond L.: A Procedure €or Designing Forebodies With Cons t r a in t s  on 
Cross-Section Shapes and Axial Area Dis t r ibu t ion .  NASA TP-1881, 1981. 
2. Stack, Sharon H.; Edwards, Clyde L. W.; and Small, W i l l i a m  J.: GEMPAK: An A r b i -  
t rary A i r c r a f t  Geometry Generator. NASA TP-1022, 1977. 
3. Moretti ,  Gino: Grid Generation Using Classical Techniques. Numerical Grid Gener- 
a t i o n  Techniques, NASA CP-2166, 1980, pp. 1-35. 
4. Milne-Thomson, L. M.: Theore t i ca l  Hydrodynamics, Second ed. Macmillan Co., 1950. 
L 
6 
z 
(a) Half of symmetric cross section representable as single- 
valued function y = y ( z ) .  
z 
t 
1- - 
(b) Cross section for which y is not representable as single- 
valued function of z .  
Figure 1.- Configuration cross-section examples. 
I 
7 
I 
( a )  Transform of f i g u r e  1 (a). ( b )  Transform of f igu re  l ( b ) .  
Figure 2 .- Transforms of cross-sect ion curves of f i g u r e  1. 
(a) Cross section in physical plane. (b) Result of initial (c) Result of second 
transformation. transformation. 
Figure 3. - Effect of applying successive Kgrmgn-Tref f tz mappings. 
Final Initial Intermediate 
(a) Transformed cross sections. 
Figure 4.- Transformed cross sections and corresponding shapes in physical plane. 
I n i t i a l  F ina l  Intermediate 
(b) Physical plane shapes corresponding t o  transforms of f igure  4 ( a ) .  
Figure 4. - Concluded. 
Figure 5.- Perspective view of surface generated as a transition between the 
cross-section shapes of figure 1. 
i 
(a) a = 1.5. 
Figure 6.- Influence on configuration shape of varying parameter a. 
(b) a = 2.5. 
Figure 6 .- Continued. 
( c )  a = 3.5. 
Figure 6.- Concluded. 
(a) a = 1.0. 
Figure 7.- Effect of varying a on midstation cross-section shape. 
Figure 8.- Transition section smoothly fitted on to forebody. 
17 

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An analytical procedure for computing smooth transitions between two specified cross sections with applications to blended wing body configuration

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