On rotational conical flow


Material Information

On rotational conical flow
Series Title:
Physical Description:
12 p. : ill. ; 27 cm.
Ferrari, Carlo
United States -- National Advisory Committee for Aeronautics
Place of Publication:
Washington, D.C
Publication Date:


Subjects / Keywords:
Aerodynamics   ( lcsh )
Rotational motion   ( lcsh )
federal government publication   ( marcgt )
bibliography   ( marcgt )
technical report   ( marcgt )
non-fiction   ( marcgt )


Some general properties of isoenergetic rotational conical fields are determined. For such fields, provided the physical parameters of the fluid flow are known on a conical reference surface, it being understood that they satisfy certain imposed conditions, it is shown how to construct the hodographs in the various meridional semiplanes, as the envelope of either the tangents to the hodographs or of the osculatory circles.
Includes bibliographic references (p. 11).
Sponsored by National Advisory Committee for Aeronautics
Statement of Responsibility:
by Carlo Ferrari.
General Note:
"Report date February 1952."
General Note:
"Translation of "Sui moti conici rotazionali" in "Onore di Modesto Panetti" published by L̕ Aerotecnica, Associazione Tecnica Automobile, and La Termotecnica, Turin, Italy, November 25, 1950."

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Source Institution:
University of Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 003779475
oclc - 86175742
sobekcm - AA00006187_00001
System ID:

Full Text

c, / 7 r/ L 2. 2.. ;.




By Carlo Ferrari


The author determines some general properties of isoenergetic rota-
tional conical fields. For such fields, provided the physical parameters
of the fluid flow are known on a conical reference surface E, it being
understood that they satisfy certain imposed conditions, it is shown how
to construct the hodographs in the various meridional semiplanes, as
the envelope of either the tangents to the hodographs or of the osculatory


1. A method for determining the field of flow about a cone of revo-
lution, the axis of which is aligned with the direction of the impinging
supersonic stream, which is taken to have a uniform velocity distribution
sufficiently far ahead of the body before the conical field is created,
was developed by Busemann (reference 1) at an early date. Several years
later the present author (reference 2) extended Busemann's procedure to
cover the case of a cone of any shape whatsoever situated in the flow,
so that its axis was at any arbitrary finite angle of attack; that inves-
tigation was confined, however, solely to irrotational conditions. With
proper alterations, nevertheless, this treatment of the yawed arbitrary-
shaped conical surface can be applied to the case of rotational flows.
The purpose of the present investigation is just to give the relationships
which permit one to draw the hodographs for the flow, in reference to
the various meridional planes through the body axis, in the case where
the motion is defined by a rotational field of conical flow with arbitrary
specification of the cone shape.

2. Upon the mere assumption of isoenergetic flow in a perfect fluid,
one may write the equations of motion in the form (reference 3):

*Original Italian Report appeared as Sui Moti Conici Rotazionali in
Onore di Modesto Panetti, published by L'Aerotecnica, Associazione Tecnica
Automobile, and La Termotecnica, Turin, Italy, November 25, 1950.

NACA TM 1333

4 --> C2
curl V x V = grad S


div V2 v2 = 0 (1)

wherein R is the universal gas constant, 7 is the adiabatic exponent,
V2 is the limiting velocity obtained when the flow expands to a vacuum,
C is the velocity of sound, S is the entropy, and V is the fluid

Upon employment of a spherical coordinate system (r,9,(), as depicted
curl V
in figure 1, the components of l V are taken as wr, Wp, and we;
where wr is the radial component, w(p is the component lying in the
meridional plane and normal to the radius vector, while we is the com-
ponent that is perpendicular to the meridional plane. Likewise, the
corresponding components of V/V1 are denoted by vr, vm, and ve.
Between these components there subsist the following relationships

1 'v O(vo sin cp)
-= rwr
sin -p 0 =T

6(rvo) 1 vr (2)
6r sin (p 0 rw

&vr ( rvqp)
r = rve

Based on the assumption that the flow is conical, the following
scalar equations are derived in a straightforward way from the first of
the equations (1):

NACA TM 1333

wevp Wqv = 0

c2 1 as
- Vr + W(Vr = 7R r sinq) e

1 c2 s
VoWr VrW r yR a

wherein c2 = C2/V2.

The set of equations (3) are not independent of each other as is
evident from consideration of equation (1) directly, but they are inter-
related through the expression

V grad S = 0 (-)
V x grad S = 0 (4)

which, for a conical field, becomes

s vye 1 as
vcp ) V sin cp e


From the second of equations (1), upon use of the hypothesis
the flow is conical, and by taking into account the relationships
by equations (2) and (4'), one then obtains that

c2 (

v2 )

2 V(PV qdV _V
sin cot 9 v
sin c c2 0 v

+ 1
sin (p

V~P ds
-YR d'V

ao /




This expression differs only by the presence of the rotationality terms
from the analogous relationship derived in reference 2 previously mentioned.

NACA TM 1333

3. By means of equations (3), (4), and (4') one may deduce some
interesting properties of conical fields. Let it be assumed that. one of
the stream surfaces of the flow is conical (this will be the case for a
field of flow arising by the action of a uniform supersonic stream
impinging from any direction whatsoever upon a conical-shaped obstacle);
it will be convenient to designate this surface as Ec. Let the versor
of any arbitrary general on eof the generatrices of the conical sur-
face Zc be denoted by then grad S x r = 0. On the other hand, if
the versor of the tangent to any streamline whatsoever that is traced
upon the surface of the cone Ec is denoted by r then it is true, in
addition, that grad S x T = 0. It is evident, therefore, that grad S
is perpendicular to the surface Lc; that is, the above-described conical
surface is a surface of constant entropy.

Besides, let it be assumed that the conical flow is symmetric with
respect to the meridional plane e = +900 (this will be the case already
mentioned for the field of flow about a conical-shaped obstacle). At
all the points of this plane it is true that vg = 0. One then deduces,
upon the basis of the first of equations (3), that we = 0 provided
that v~, is not zero everywhere. On account of this, and through
utilizing the third of equations (3) it follows that _S = 0. Thus even

the meridional plane of symmetry for the conical field is itself a con-
stant entropy surface, and at this plane the flow is irrotational as is
easily deduced upon taking cognizance of equations (2).

In conjunction with the result obtained above one can derive from
this latter fact that, for the case of flow about a conical obstacle, the
shock wave in the two semiplanes e = 900 and 0 = -900 must produce
the same change in direction of the stream velocities; and so the tangents
to the trace of the shock wave in these semiplanes are symmetrically
inclined with respect to the undisturbed stream velocity vector. Now
let us consider an obstacle in the form of a right circular cone. Upon
the surface of this cone it is true that __S = vqp = 0, and therefore one
gets that vw = 0. Thus the following relationship results

ve = -- r (6)
sin p 6@

NACA TM 1333 5

On the cone one can always express the vr values as a periodic
function of 0, and thus vr = Bo + L Bn sin nO. It follows that on
the cone's surface the peripherial velocity component is given by

vQ = 1 n Bnn cos ne
sin cp

just as in the case of irrotational flow.

Now, if we let the angle of incidence of the axis of the cone be
denoted by P, then the expression for vr becomes simply

vr = Bo + B1 sin 8

if only terms of the order of magnitude of 0 are taken into account.
Since this is true, then because B1 is proportional to 0, the Bn
coefficients have to be at least as small as 32.

The relationship given by equation (6) may be generalized for the
case of a cone of any shape whatsoever. It is assumed for this purpose
that the cone's surface is divided up by a network of orthogonal coordi-
nate lines 0a and a2 (r = const. and c = const., respectively).
The former of which are the intersections of the spheres with radius r
upon the cone under consideration, while the latter are the generatrices
of the cone itself. At an arbitrary general point P on the cone the
length of the linear element dol can be written as: dol = rh1 (E) dO
while the length of the linear element do2 along the line a2 is
given by: do2 = dr.

The component, in the direction of the normal to the cone at the
point P, of the curl is

1 v \' 1 3V2
wn = (vih-r) -
rhl 6r Y lh1

where v1 and v2 are now the velocity components in the direction
of al and 02, respectively. On the other hand, upon referring to
the first of equations (1), it is still true that wn = 0, and on account
of this it is evident that:

NACA TM 1333


If the component of velocity in the
expressed as a periodic function of
equation (6') immediately furnishes
spending expression for vl.

0, as is
the means

of the radius vector is
still permissible, then
of obtaining the corre-

4. It is now easy to determine how to continue the construction of
the flow field downstream of a given conical reference surface, E, upon
which the physical conditions of the flow are assumed known. Let the
equation of the conical reference surface, Z, be given by

(P = (p(e)

From the relationship

/dS\ = 6S ~S
\d0^ 60 ac



one obtains

( dS)
-- = ve s
CCP 4vJ vp sin q

provided p is expressed as a function of 0
above relationship allows one to calculate LS

surface E.

In like manner, by use of the equation

as in equation (7). The
at the points of the

36( vp\
l60 ,

1 v2
vl = h1i e
hl 60

+ CP

NACA TM 1333

and by setting

S+Vr = RI

it is found that


\de /z

- pR1 + vr = A1 RR1


wherein the Al are calculated at the points of the conical surface
through means of the relation

A = ( v


Finally, it is found (the intermediate steps are omitted, and just
the final result presented) that:

9 ?

in Ri
sin (p

OVr V82 vs 1
r-e = R2 Vp =) s An
byc V9 p sin 'P

where the quantities A2 and A3 are the expressions:

\de v\
A2 = WE
V( E9G7

sn ve cos \
sin \

c2 1 as
+ Al sin cp
7R ve p

ve sin c +




NACA TM 1333

T- (2 + vy e)
I sin CP

Vq( sin q( ( ve

and their values are known on the reference conical surface,
so also is the value of R2.

E, and thus

By means of equation (5), therefore, one obtains


sin rp

s2 V c2V
sin p c2

c2 Av
c2 A1 -


sin2 cp

v -2 /
1 + 2) cot (P -

] R1 =

v 2)


v{ 3S VcVr
- c2- rwe
RR5 c2 2


Thus it is possible to calculate the values of R1 at the points of
the reference conical surface, Z. This formula is the natural extension
of the analogous relationship already derived in reference 2 in the case
of an irrotational flow.

The complete solution of the problem as to how to continue constructing
the flow field downstream of the reference surface Z is thus presented
by the formulations given as equations (8), (11), (12), and the additional
equation (13), since it is also evident that

1 i A c2 1
-v A Rt I vO cos ( -
6p sin Cp 7R v

sin (]

vv wk

A3 =



NACA TM 1333

5. The graph of the hodograph corresponding to an arbitrary gen-
eral semiplane, 0 = const., is represented in figure 2. Let P be
any point whatsoever on this hodograph, and let R signify the vec-
@ -I
tor R= Rlr R2T. In addition let r again denote the versor that
has the sense and direction of the radius vector in the semiplane in
question, and let T now be the versor normal to r in the semiplane
and oriented in the sense of an increasing cp. With these conventions
it follows that R x = 0 and therefore it becomes clear that the
tangent to the hodograph at the point P is perpendicular to the above-
defined vector R. Thus this tangent is inclined to the direction of
r by an angle

-1 R2
X = tan (14)

If the linear element of length along the hodograph is denoted by ds
then the absolute value of ds/dcp is given by

H s= R12 + R22

Thanks to the formulae developed in the preceding sections the
drawing in of the tangent to the hodograph at the point P, situated on
this hodograph, is therefore made possible, since everything is known
about the point P. If the element of length along the hodograph is
calculated as

ds = /R2 + R22 dp

then it is easy to find another neighboring point P'. When used repe-
titiously in the various semiplanes, this procedure allows one to determine
the respective hodographs as the envelope of their tangent lines.

The angle of intersection da with respect to the direction of the
linear element of the hodograph ds is thus determined by

NACA TM 1333

( dR2 dR
/ Rd W1 ~ -
da = d (1 = dp 1 2-
S12 + R2 /

The radius of curvature of the hodograph at the point P just
mentioned turns out to be consequently

( 12 + 3/2
R12 + R22
Rc = (15)
/ dR2\ / dRi1
R (3 + R2 R2 + -
dp dp

When the point P' is determined in the manner that was described
above, in consequence of having available all information about the known
point P, it follows that R1 and R2 are also known at the point P',
dR1 dR2
and thus the values of -- and are calculable. Through means of
dip dcp
equation (15) the radius of curvature Rc is then determined, using
dR1 dR2
these values of -- and -. The direction of the principal normal
d4p dCp
as defined by equation (14) is also known, and thus the hodograph can
be obtained in every meridional semiplane as the envelope of the respec-
tive osculatory circles.

6. The results obtained here for the determination of the conical
field of flow in the case of rotational motion are employed in an exactly
analogous way as described in reference 2 when making a numerical appli-
cation. In general it will be convenient to assume as the reference
surface, Z, upon which the initial values are taken to be known, the
conical surface of the shock wave. The required information about the
shock wave may be obtained in first approximation by utilization of the
hypothesis that the flow is irrotational.

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

NACA TM 1333


1. Busemann, A.: Drucke auf kegelf6rmige Spitzen bei Bewegung mit
Uberschallgeschwindigkeit. Z.f.a.M.M., Bd. 9, Heft 6, Dec. 1929,
pp. 496-498.

2. Ferrari, C.: Atti R. Accad. Sci. Torino, vol. 72, Nov.-Dec. 1936,
pp. 140-163. (Available as R.T.P. Translation No. 1105, British
Ministry of Aircraft Production)


Ferrari, Carlo: Interference between Wing and Body at Supersonic
Speeds Analysis by the Method of Characteristics. Jour. Aero.
Sci., vol. 16, no. 7, July 1949, pp. 411-434.

3. Crocco, L.: Una nuova funzione di corrente per lo studio del moto
rotazionale dei gas. Rend. della R. Accad. dei Lincei. Vol. XXIII,
ser. 6, fasc. 2, Feb. 1936.

NACA TM 1333


Figure 1.- Coordinate and vectorial orientation in Lhe spherical coordinate

Figure 2.-

Construction in the hodograph plane for finding
P' when they are known at P.

velocities at

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