On the determination of certain basic types of supersonic flow fields


Material Information

On the determination of certain basic types of supersonic flow fields
Series Title:
Physical Description:
17 p. : ill. ; 28 cm.
Ferrari, Carlo
United States -- National Advisory Committee for Aeronautics
Place of Publication:
Washington, D.C
Publication Date:


Subjects / Keywords:
Aerodynamics, Supersonic   ( lcsh )
Aeronautics -- Research   ( lcsh )
federal government publication   ( marcgt )
bibliography   ( marcgt )
technical report   ( marcgt )
non-fiction   ( marcgt )


Abstract: A discussion is given of the application of Fourier series techniques to the problems of linearized supersonic flow. The formulation presented is an extension of the doublet type of "fundamental solution" to higher order types of singularity. The equations developed have application to wing theory but are primarily of importance in wing-body interaction problems. A specific example of a wing-body interference problem is discussed in light of the presented methods.
Includes bibliographic references (p. 15).
Statement of Responsibility:
by Carlo Ferrari.
General Note:
"Translation of Sulla determinazione di alcuni tipi di campi di corrente ipersonora," from Rendiconti dell'Accademia Nazionale dei Lincei, Classe di Scienze fisiche, matematiche e naturali, serie VIII, vol. VII, no. 6; read at the meeting held on December 10, 1949."
General Note:
"Report date November 1954."

Record Information

Source Institution:
University of Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 003807561
oclc - 127125473
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Full Text
I N A 1 \-13i8





By Carlo Ferrari


A quite universal mode of attack on problems which arise in super-
sonic flow, whether connected with flow over wings or over bodies of
revolution, is explained, first, in great generality, and then in more
detail, as specific applications to concrete cases are illustrated. The
method depends on the use of Fourier series in the formal definition of
the potential governing the flow and in the setting up of the boundary
conditions. This new formulation of the many problems met in supersonic
flow is really an extension of the doublet type of "fundamental solution"
to higher order types of singularity. The limitations and, in contrast,
the wide field of applicability of such a means of handling these prob-
lems with complex boundary conditions is discussed in some detail, and
a specific example of a wing-body interference problem is cited as proof
of the versatility of the method, because the -esults obtained by applying
the techniques expounded herein agree well with experimentally determined
data, even for the quite complex configurrtior .srd to exemplify the kind
of problem amenable to such treatment.


For purposes of analytic treatment of the flow problem to be con-
sidered here the usual rectangular Cartesian coordinate system is employed
with the x-axis taken to lie in the direction of and having the same sense
as the uniform (undisturbed) free-stream velocity, V,. This free-stream
velocity, V,,, is taken to be supersonic in the discussion that follows;
i.e., V. > C, where C, denotes the velocity of sound in the undis-
turbed stream. The flow of the gaseous fluid to be investigated is to
"Sulla determinazione di alcuni tipi di campi di corrente iper-
sonora," Rendiconti dell'Accademia Nazionale dei Lincei, Classe di
Scienze fisiche, matematiche e natural, series VIII, vol. VII, no. 6;
read at the meeting held on December 10, 1949.

NACA TM 1581

be considered as resulting from the superposition upon the free-stream
velocity of a nonuniform flow, having velocity components that are des-
ignated as Vx, Vy, and Vz, and lying in the direction of the respec-
tive axes (x,y,z) of the coordinate system. This nonuniform super-
imposed flow is supposed to be small enough, in comparison with the
speed of sound, C,, that it is permissible to neglect the ratios Vx/Coo
V/ C), etc. in the equations governing the flow.

It is taken for granted that, under the conditions stated above,
there exists a velocity potential describing the flow in question, and
in practically all cases which are of any interest for actual designs
it will really be true that this assumption can be made legitimately.
If it is then agreed that the nonuniform superimposed part of the flow
is to be denoted by the potential 0, it will be recognized that this
potential will have to satisfy the relationship:

( M2 =O (1)
1x2 ay2 0z2

where the free-stream Mach number, MN, is defined as M, = Vg/C and,
of course, the potential 0 will also have to obey the boundary condi-
tions which are peculiar to each stated problem.

A way of handling the determination of the function 0, so that it
will satisfy equation (1) and so that it will obey the imposed boundary
conditions, will now be explained, and its usefulness illustrated by
consideration of problems which can be attacked by this means, both in
the case of lifting surfaces (that is, wings) as well as in the case of
bodies of revolution. The proposed method is based on the use of Fourier
series. Although this technique does not afford complete universality
in treatment of all the posed problems, as will be more clearly pointed
out in what follows, it can be used to fine advantage in a.goodly number
of situations by replacing the procedures which are based on the Fourier
or the Laplace transforms (which, for that matter, have just as restricted
limits of applicability as the analogous ones which arise in connection
with the approach being discussed herein) or by being substituted in
place of the techniques which stem from use of the "fundamental" (source,
sink, doublet) solutions to equation (1), or from use of transformations
carried out in the complex plane.


As usual, the wings are imagined to be very slender and so placed
that the wing span lies along the y-axis; i.e., the long dimension is

NACA TM 1381

out the y-axis (see fig. 1). Let the equations which define the ventral
and dorsal surfaces of the wing surface be, in fact, given in the form:

z = Zv(x,y) and z = zd(x,y)

and then the slenderness of the wing is supposed to be slight enough
that the above-defined values of z will be so small at all locations
on these surfaces as to make it possible to accept the fact that the
derivative is, to all intents and purposes, equal to the direction
cosine, with respect to the free-stream x-axis, of the normal to the
surface. It is further assumed that the wing is immersed in a stream
of supersonic flow which has a constant value for its component lying
in the direction of the x-axis, of magnitude Vo. The component in the
direction of the z-axis, meanwhile, is assumed to be known, but of rela-
tively small size in comparison with the V, velocity, and it may take
on various values, which will be denoted by Vz'. If, now, the potential
describing the flow perturbed by the wing is denoted by 0 this poten-
tial will have to satisfy equation (1), and it will also have to conform
to the conditions which are imposed at the boundaries. These further
(boundary) conditions may be stated as follows:

(2) Upstream of a certain surface, which may be immediately defined
just as soon as the wing-like body is specified which is to invest the
impinging stream, the value of is zero; i.e., the basic condition is

-= 0 (2)

(5) On the wing surface, it must be true that

= -V 8 Vz cos (n,z) = H(x,y) (5)

wherein the value of z to be employed is either the zv or zd quanti-
ties, depending on whether one is concerned with a point which is lying
on the under ventral surface or on the upper dorsal surface, respectively.
The notation cos (n,) signifies the cosine of the angle between the
z-axis and the unit vector taken in the direction of the exterior normal
to the wing surface in question; i.e., this vector is represented by the
vector n, and under the present hypothesis cos ,z = il.

NACA TM 1581

It is convenient to distinguish between two basic types of problem
which come under this kind of analysis, and to make the differentiation
on the basis of the sort of boundary conditions met with in each type;
that is,

Symmetric Types of Configuration

In this case, the boundary conditions to be satisfied on the wing
may be expressed in the form

=( H(x,y) (5')


\z=O= -H(x,y)

Asymmetric Types of Configuration

In this case, the boundary conditions are expressed as

()z=o+ z=0 ,y) (")

The first type of problem corresponds to a configuration for which
the wing has a zero angle of attack with respect to the free-stream
undisturbed flow, V,, and which possesses a symmetric profile. The sec-
ond type of problem corresponds to a configuration for which the wing
is a flat plate, but which has any local angle of attack whatsoever,
with respect to the free-stream vector, V,, so long as it is small.


In this case it will suffice to examine the flow solely in the
upper half-plane, where z > 0. If 0(l)(x,y,z) stands for the flow
which takes place in this upper region, and if 0(2)(x,y,z) represents
the flow in the nether region, then, of course,

(2)(x.yz) = #()(x,y,-z)

NACA TM 1581 5

The boundary conditions in this case are composed of equations (5'),
together with the restriction that

( = 0 (for locations lying beyond the region
\ /z /z=0 occupied by the wing surface) (2')

Now let the definition of the function describing the velocity com-
ponent at the wing surface, and also the potential function itself, be
cast into the convenient forms

H*(x,y) = V Hm (1) cos y = V. Hm() cos mi
m m


0(l)(x,y,z) = O(l) = V. b m(e ,) cos I m1 (4)

wherein = x/b, y = y/b, and = z/b, while b is a suitable length
used for purposes of nondimensionalization. The value used for b will
be equal to the semispan of the wing in the case where the leading edge
of the wing is supersonic everywhere, and provided that the wing tips
are cut off in such a way that the wing surface remains outside of the
tip Mach cones emanating from either one of the wing-tip extremities out
at the farthest reaches of the wing span. The value used for b will
be larger than this semispan just defined, if, in contrast, these geo-
metrical relationships do not hold; the magnitude employed for b in
this latter case is illustrated in figure 2.

Finally, it should be observed that H* is a periodic function
of y, which is equal to the values taken on by the function H at the
wing's surface and it is zero for points lying out of this region, and
this definition is to hold throughout the spanwise interval for which
-b < y < b.

The fact that it is possible to write H*(x,y) in the form given
as equation (4) (i.e., the possibility of expressing the component-
velocity field describing the normal velocities to the wing surface by
means of a Fourier series instead of in terms of a Fourier integral)
stems from the property already noted to the effect that the perturba-
tions, which are created at any arbitrary point P(x,y) whatsoever, do
not make themselves felt anywhere outside of the Mach cone emanating
from P. As a result of this situation, therefore, as far as the

NACA TM 1581

determination of the field of flow about the given wing is concerned, it
makes no difference to this flow whether one considers the wing to be
operating by itself as an isolated entity within the impinging stream or
whether, instead, one imagines it to be accompanied by an infinite number
of reflections of this primary wing in the planes y = tmb.

If one now inserts the second of the expressions given as equa-
tion (4) into equation (1), it will be seen that this differential equa-
tion reduces to

- B2 2 = k2m

wherein B2 = Ma2 1 and where k replaces the constant 1M.
Meanwhile, it is also evident that, on the basis of the first of
the formal developments given as equation (4), the boundary condition
reduces to


The expression given as equation (5) above is formally analogous to
the so-called "telegraph equation," and its solution, which is suitable
for applying the type of boundary condition exemplified in equation (6),

SB -B0

hm(') Jol (l7) ETB2tj dti

where JO is the cylindrical Bessel function of zeroth order.

Consequently, the vertical derivative turns out to be

a0M = (6-B +) j -B

h(') 1 BE, B22')2
B~ Q --g 2);-B2

= Bm(S)

NACA TM 1581

and, because of the boundary condition (6), it follows that

so that th sought potential must have the frm
so that the sought potential must have the form




The possibility of being able to find solutions to such asymmetric
problems by means of the method being propounded here is restricted in
this case to those configurations for which the leading edge as well as
the trailing edge of the wing are supersonic, and where the wing tips
are cut off in such a way that the wing surface lies outside of the Mach
cone emanating from the very tip of the leading edge where the maximum
span occurs.

Under these circumstances the boundary conditions are constituted
from the restrictions given as equations (3"), and of equation (2')
once again. If one then follows the same procedure as was utilized in
section 5, it follows that the expression for the sought potential is
formally given as (ref. 1)

0M = nhmE B + k f -B I

1 ^ )2 B2 (gg)2-B22
m\~ ''L~-t t

where h, is, a priori, an undetermined function, and where it should
be recognized that the + sign is to be employed for the lower half-plane
where t < 0, and where the sign is to be employed for the upper half-
plane where t > 0. It is evident, therefore, that the derivative of m
with respect to will be continuous along the plane 0 = 0, but the

Bl(S ')Jo dq'

NACA TM 1581

derivative of m with respect to t will be discontinuous, and the
"jump" will be of such size that

holds true.

It is clearly permissible here again to concentrate attention solely
upon the disturbed flow in the upper half-plane where t > 0, therefore,
because the observation just made above will tell one how to compute what
the flow will be in the other lower half-plane, once the former is

The boundary conditions in this instance may now be recast into the

IBh. () +

k2 /F
it T

h(t ') Jo

nk (t') Jil k
'-'0 I

B-L) d' -

S--') d' = Gm(d)

provided, as in the previous section, one sets up the convenient con-
vention that G*(x,y) is to represent a periodic function in y that
is to be equal to the values taken on by the function G(x,y) at the
wing's surface, and it is to be zero for points lying out of this region.
This definition is to hold throughout the spanwise interval for which
-b S y < b. In addition, the form of G*(x,y) is to be assumed, specif-
ically, to have the appearance

G*(x,y) = Vc G~ m() cos mT

while it has also been assumed that the derivative of a function by the
sole parameter upon which it depends is to be denoted by a dot over the
function, that is,




/-H -
W6t /t~o-

NACA TM 1581

The integro-differential equation defining bm may also be immedi-
ately simplified to the compressed expression

B(hm() + j hm( ') (Jo + J2)d' = Gm( ) (9)

Now apply a Laplace transformation to this integro-differential
equation (i.e., multiply through by the factor e-PE and integrate
from 0 to m). Thus, one obtains

2 +2
k2 1 p2 +2
CBfmp + Gm
B p 2 2 k2

where a bar over a symbol serves to indicate that this quantity stands
for the Laplace transform of the function so designated.

Standard tables of Laplace transforms could be consulted to check
these results, which may now be simplified by noting that

1 + 2 k2 +
1 4(5B2-
p + k2 k2 +2 k2
SB2 B2_ TB2

k2 + p2 k2 p2 2p /p2 + k2

k2 p B2
B2 T B2

= 2 B p2 + p

Thus the Laplace transforms of equation (9) simplifies to

tBpbm + xB p2 + p m = Om

or the explicit expression for the Laplace transform of the unknown func-
tion hm is given in the form

2 k2
p B2


NACA T' 1381

so that finally one may invert the transformation to obtain


GCm(')Jo ( ')] d'

Once having obtained the value of hm, it is easy to write down the
expression for the component of velocity lying in the x-direction and
located at the wing-surface, because one has simply that this component
is given by the partial derivative of the potential 0, taken with respect
to t, and evaluated at the plane of the wing; i.e., one finds that


cos m2 = -V Y ]m(w) cos m

Furthermore, the formula giving the lift on the wing is just

L = 8pV. 2b2

S(-1) 2 hm/b)m


for m = 1,,5,5,. ., etc.

where the symbol, 2, is used to denote the distance along the x-axis
measured back from the leading edge of the root chord to the projection
into the plane of symmetry of the trailing edge of the wing-tip profile.
In regard to the moment taken about the y-axis, it is apparent that
it may be computed from the relation:

My = 2npV.2b5

= 2zpm b5V2

f cos (on
W-l )

()(-l) 2

for m = 1,3,5,. ., etc.


dI f/b

bm )


hm() d)


a ( =0


NACA TM 1381


The procedure discussed in the preceding sections can be extended at
once to apply also to the solution of problems which are concerned with
the flow over bodies of revolution.

For this purpose let a cylindrical coordinate system (x,Y,8) be
set up, and then the equation which governs the potential, 0, being
sought will have the form

20 -L B2 02 (13)
6y2 aY y2 6e2 ,x2

Now let R = R(x) be the equation of the meridian line of the body,
and let it be assumed that R is sufficiently small at all locations
along the body so that the direction cosine of the normal to this merid-
ian line, measured from the Y-axis, may be taken to be equal to unity.

Furthermore, let Vnf' stand for the component, taken in a direc-
tion perpendicular to the circular cross-section of the body, which
arises from the impinging flow which invests the body. Then the boundary
condition which must be satisfied at the surface of the body may be
expressed mathematically by the relation

-= -Vnf

For sake of simplicity, it is also now assumed that the treatment
to be developed is to be restricted to the case where symmetry with
respect to the semiplanes 8 = exists in the incident flow. Under
this hypothesis it is convenient to write the normal velocity components
and the potential being sought in the following explicit formulations:


S= Vm Fm () sin m o

0 = V. m(x,Y)Ym sin m a

12 NACA TM 1581

If one now inserts the second of the expressions given as equa-
tion (14) into the differential equation governing the flow (13), it
will be found that the defining equation for the potential will have
the form


+ 2m + 1 m
Y Ny-


= B2 m

and the boundary condition turns out to be


= -F


A suitable solution to equation (15), which can be made to satisfy
the boundary condition being imposed as equation (15'), will be found
to be

~m (lm 0
Y Y) 00aY

are ~



fo(x-BY cosh u)du


NACA TM 1581

Thus, the successive individual potentials are given by the expressionsI

Yo1 = -B f

Y02 = B2 f

cosh --

cosh XA

fl(x-BY cosh u) cosh u du

f2(x-BY cosh u) cosh2 u du


Upon imposition of the requirement that the boundary condition (15')
is to be satisfied, one obtains a set of integral equations which serve

iTranslator's note: It was pointed out on page 650 of an article
by R. H. Cramer in the Journal of the Aeronautical Sciences, vol. 18,
no. 9, September 1951, entitled "Interference Between Wing and Body at
Supersonic Speeds Theoretical and Experimental Determination of
Pressures on the Body," that the result given here for O, for m > 1,
is incorrect; the correct formula is, for m = 2,

Y2 = B2 r x f2(x-BY cosh u)(2 cosh2 u 1) du
arc cosh By

while, in general, the use of hyperbolic functions of multiples of the
argument u gives a more compact form, which is easy to work with; i.e.,
in general it is true that

Y = Bm()m

Sco fm(x-BY cosh u) (cosh m u) du
are cosh


NACA TM 1381

to determine the arbitrary functions fm, which are a priori unknown.
Thus, applying these conditions, one finds that2

Bm+l (_-)m+l f fm(x-BR cosh u) (cosh u)m+l du = -Fm (18)
arc cosh x

The determination of the values of the fm's appearing in formula (18)
may be carried out by using a step-by-step procedure which is entirely
analogous to the one employed by Von KArmAn in his work on determining
the flow about a body of revolution at zero angle of attack.

It is important to point out that if one only has in mind to calcu-
late the force distribution along the axis of the body and the corre-
sponding moment, and if one is not interested in knowing the local veloci-
ties or pressures around the body, then it is merely necessary to
calculate 0 and 02.


The procedure that has been propounded above has been applied (ref. 2)
to the situation arising in the study of the question of wing-body inter-
ference. The wing-body configuration considered in this particular appli-
cation of the method is depicted in the appended figure 5.

The wing used in this configuration is a flat plate, whose semispan
is equal to 4R0, where RO is the radius of the circular cross-section
taken through the body at the location where the body is widest. The
leading edge of this wing is located 5RO downstream from the tip of the
nose of the body. The free-stream flow is impinging on the body at a
speed which is twice the speed of sound in the undisturbed stream.

2Translator's note: In view of the correction pointed out in Note 1
above, it will be seen that this formula for determining the fm functions
is also incorrect, except for m = 1; for higher integral values of m,
the correct formula is:

Bm+1 (-i)m+l1 m(x-BR cosh u) [cosh m u cosh u] du = -Fm
arc cosh

NACA TM 1381

The curves shown in figure 5 give the value of the pressure coeffi-
cient, P ---P, out along the span of the wing, in the mid-chord loca-

tion (i.e., along the wing axis), for points on the upper (dorsal) side
of the wing. These coefficients have been calculated by the method out-
lined in section 4, and there are shown results for various angles of
attack, which apply to such points on the upper side of the wing pro-
files at their mid-chord positions.

In addition, some experimental test points obtained by R. H. Cramer
(see ref. 2) are also plotted on these curves. These results were
obtained from experiments carried out in the supersonic tunnel of the
Daingerfield Aeronautical Laboratory.

The agreement between the computed and experimentally determined
results is very good from a qualitative viewpoint. In regard to the
more precise details of the quantitative comparison between the results
it is worthy of note that the experimental results exhibit a certain
amount of dissymmetry as one passes from positive angles of attack to
negative angles of attack. Such a dissymmetry cannot be predicted, or
should not be expected, from the type of theoretical treatment being
considered here.

In order to bring about a more valid comparison of these results,
it would appear logical, in face of such evident dissymmetry, to take
for the representative experimental value, at a given value of the angle
of attack, 0, the one which is obtained from averaging the result
obtained for an angle of attack equal to +p with the result obtained
at -0. Such average values have been computed and are designated in
the plots of figure 3 by means of solid circles. These adjusted values
lie much closer to the theoretically derived curves at almost all

Translated by R. H. Cramer
Cornell Aeronautical Laboratory, Inc.
Buffalo, New York


1. Ferrari, C.: Interference Between Wing and Body at Supersonic Speeds -
Theory and Numerical Application. Jour. Aero. Sci., vol. 15, no. 6,
June 1948, pp. 317-556.

2. Ferrari, C.: Interference Between Wing and Body at Supersonic Speeds -
Note on Wind Tunnel Results and Addendum to Calculations. Jour.
Aero. Sci., vol. 16, no. 9, Sept. 1949, pp. 542-546.

NACA TM 1381


Figure 1.- Orientation of coordinate axes and location of typical wing
plan form therein.






Figure 2.- Definition of the interval of periodicity required for application
of the Fourier series technique when leading edges are subsonic.

NACA TM 1581

Cp- Poo 2 0.00--
1/2p, Voo 0.00 e
Upper 0.04
surface 4
0.08 0 =-40

o I X 1 --I
0.160 4Ro=b

Q20 2-R

QuWing juncture
0. .10 .20 .30 .40 .50 .60 .70 .80 .90 100
77 y/b

Figure 3.- Pressure distribution along the wing axis: Comparison of
experimental results with predictions based on the method expounded
in section 4.

NACA-Langley 11-3-54 1000


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