Effect of the acceleration of enlongated bodies of revolution upon the resistance in a compressible flow

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Material Information

Title:
Effect of the acceleration of enlongated bodies of revolution upon the resistance in a compressible flow
Series Title:
Technical memorandum ;
Uniform Title:
Prikladnaya matematika i mekhanika
Physical Description:
8, 2 p. : ill. ; 28 cm.
Language:
English
Creator:
Frankl, F. I
Langley Aeronautical Laboratory
Publisher:
NACA
Place of Publication:
Washington, D.C
Publication Date:

Subjects

Subjects / Keywords:
Body of revolution   ( lcsh )
Two-body problem   ( lcsh )
Aerodynamics   ( lcsh )
Unsteady flow (Aerodynamics)   ( lcsh )
Genre:
federal government publication   ( marcgt )
bibliography   ( marcgt )
technical report   ( marcgt )
non-fiction   ( marcgt )

Notes

Abstract:
The effect of acceleration of enlongated bodies of revolution upon the resistance in a compressible gas is investigated. It is assumed that the body produces only small disturbances in the flow so that the velocity potential satisfies the simple wave equation. The results of the analysis show that for accelerations not higher than 1000 meters per second² and velocities comparable with the velocity of sound, the added pressures arising from the acceleration are negligibly small.
Bibliography:
Includes bibliographical reference (p. 7).
Original Version:
Originally published: Moskva : Nauka, 1946 : Prikladnaya matematika i mekhanika.. Vol. 10, no. 4, 1946
Statement of Responsibility:
by F.I. Frankl.
General Note:
This work is part of the library's "Parachute History Collection" donated by Sandia National Laboratories through the American Institute of Aeronautics and Astronautics Aerodynamic Deceleration Systems Technical Committee.
General Note:
Caption title.
General Note:
At head of title: National Advisory Committee for Aeronautics.
General Note:
"Technical Memorandum 1230."--T.p.
General Note:
"Translation of 'O vliianii uskoreniya na soprotivlenie pri dvizhenii prodologovatykh tel vrascheniya v gazakh.' Prikladnaya Matermatika i Mekhanika, Vol 10, no. 4, 1946."--Cover.

Record Information

Source Institution:
University of Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 003759947
oclc - 71365487
sobekcm - AA00006228_00001
System ID:
AA00006228:00001

Full Text
^-A T 12r 10







7i?'


NATIONAL ADVISORY COMMITfEE FOR AERONAUTICS

TECHNICAL MEMORANDUM 1230


EFFECT OF THE ACCELERATION OF ELOIGATED BODIES OF REVOLUTION

UPON THE RESISTANCE IN A COMPRESSIBLE GAS

By F. I. Frankly


The problem of the motion of an elongated body of revolution in
an incompressible fluid may, as is known, be solved approximately with
the aid of the distribution of sources along the axis of the body. In
determining the velocity field, the question of whether the body moves
uniformly or with an acceleration is no factor in the problem. The
presence of acceleration must be taken into account in determining the
pressures acting on the body. The resistance of the body arising from
the accelerated motion may be computed either directly on the basis of
these pressures or with the aid of the so-called associated masses
(inertia coefficients). A different condition holds in the case of the
motion of bodies in a compressible gas. In this case the finite
velocity of sound must be taken into account. If it is assumed that
the body produces in the flow only small disturbances, the velocity
potential p satisfies the wave equation:


a2,p = 2 (1)
't2

where a is the velocity of sound. The method of sources still remains
applicable but in computing the effect of the sources it is necessary
to make use of the retarded potential.

We introduce two systems of coordinates, one a fixed system xr,
the other moving with the body tr, where x and t are, respectively,
the coordinates along the axis and r is the distance from the axis.
The coordinate 9 is computed from the nose of the body.

Let r = r(g) be the equation of the body of revolution (0 < E < 1).
The position of the body is characterized by the inequalities
-f(t) < x < I -f(t), where f(t) is a given function of the time
characterizing the motion of the body. The coordinates x and t are
connected by the equation x = E f(t).


*"0 Vliianii uskoreniya na soprotivlenie pri dvizhenii prodolgovatykh
tel vrashcheniya v gazakh." Prikladnaya Matematika i Mekhanika, Vol. 10,
no. 4, 1946, pp. 521-524.






NACA TM 1230


Let q = q(x',t') be the value of the linear density of the source
at the point x' at the instant of time t' and R = (x x')2 + r2
be the distance between the source and the point considered. The
potential of the disturbance velocities is then


(xt) = q (x',t dx

where the integral is taken between the limits determined by the
inequalities


-f (t-

In order to establish the range of integration, we must consider
in the plane x't' a strip -f(t') < x' < I f(t') between the
lines of motion of the front and rear points of the body (fig. 1) and
the part of the semihyperbola in this strip.

t' = t = t 1/(x -x)2 +2
a a

The abscissas of the part of the semihyperbola considered form the
range of integration (fig. 1). As may be seen from the figure, the
range of integration may consist of one or several segments. The ends
of the corresponding arcs of the hyperbola may lie either on the line
of motion of the nose or on the line of motion of the base. In
particular, in the motion of the body with supersonic velocity it is
possible that within the strip considered there lies an entire arc of
the hyperbola, both ends of which lie on the line of motion of the
nose (as will be the case, for example, for a uniform supersonic motion
or for a motion approximating this type).

The function q(x',t') must be determined from the integral
equation expressing the fact that the normal component of the velocity
of the gas on the surface of the body is equal to the normal component
of the motion of the body:

f ( t l- d./d f'(t) (2)
on 41r 6n afr + ['( a

where 6/6n denotes differentiation along the outer normal and
S = x + f(t). In what follows we corbider the derivative 2&/P as
small and shall give an approximate solution of equation (2). For the
assumption made, 86p/n may be replaced by cqp/or. We assume for






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simplicity that the range of integration reduces to the segment (xl, 2).
Equation (2) then assumes the form


SX2

4v 6r xl


Sa] r=F


12
dxv = 1 r
41r x(

RlR
Xlt T ) =


+-
3 a2 t'=


f.(t) dr
dt


- q(,2t -~x- +
a/ dr KI


If r/1 and dr/dI are considered small quantities of the first order
and their higher degrees or derivatives are neglected, an approximate
solution of equation (3) will be'


q(x,t) = 2rf'(t)r(S) d-
dt


We shall show that the error in substituting this expression in
equation (3) is a magnitude of the second order of smallness. For this
prupose it is convenient to write er* for F where r* = r*(E) is a
variable quantity of the order of unity relative to I and e is a
small constant number. The degree of E in each expression then shows
its order of smallness. We consider the order of the terms of the left
side of equation (3). The terms outside the integral on the left of
equation (3) containing q as a factor for finite terms (in the general
case) will be of the second order of smallness. In the same way the
expression


is of second-order smallness. We write the remaining term in the form


Sx2
dx' = rq(x,t)
JC1 2


dxR
S


+ r [q(x',t)
41t


- q(x,t)] dx'
R


4 2
+ r 2 [q ',
41 tIZ L v


-E -


q(x',t) d -
]JR E3


r X2
rf
Jx,


1 1,
-q x',t
pR3 \


'This solution (4) was first used by Karan (reference 2) in the case
of a uniform motion (in particular, for obtaining the approximate value of
the wave resistance).






NACA TM 1230


As a result of the computation of the first integral of this expression
we obtain

r 2
r f x2 I [2 + O(e2]
1x1 3 er*() j


For the estimates of the differences in the brackets in
and third integral of expression (6) we have, respectively


q(x',t) q(x,t) = 0(E2)R,


the second


q(x',t) q(x',t R/a) = 0(c2)R


Hence the estimates of integrals are of the form

nx2
o(c2)r x < 0n(E2)
X1 R2
so that these integrals are likewise magnitudes of the second order of
smallness. Thus


[q (x't a)r


dx' = f'(t) d + 0(2)


(10)


as was required to be proven.

The expression for the velocity potential after substituting in
equation (4) becomes


cp(,t) =-' t t ( ) dx'
2JR a)


(7(Q)


For computing the field of pressures we have the generalized formula
of Bernoulli


p P = _w2 _
p 2 6t


(11)


(12)


where p is the pressure at the given point, p the pressure in the
undisturbed region, p the density and, w the modulus of the velocity.
Neglecting magnitudes of the second order w2/2 and taking account of
the equations


-' =x' + f (t )


(13)


1 a 2
'^^Jxi


= ri


- = ft (t R
it a)






NACA TM 1230


we obtain


p p
P


-2 t 2 x7 f= d
_ rLt1 f (t. a)] 62
L 2R a T2- t


(14)


In the particular case where the pressure is required on the surface
of a projectile, the motion of which approximates to uniform supersonic
motion, the point x2 lies on the line of motion of the nose of the
projectile and the last term on the right of equation (14) is equal
to zero. We obtain


(15)


Sf t R ()] ) + f" t- )

where x, and x2 are determined from the equation


Xl,2 = f 1,2


(16)


In the case of uniform motion x = vt + c and equations (15)
and (16) become


P-
P


x1 *


dx'
(


1.2 = v (x.2* x)2 + r2 )+ c


(17)

The additional pressure produced by the acceleration is therefore given by
the equation


x2 l2 -', ( ,)
dxl ( [a! ) ] I-


+ [f' j 2 L0]'(Y '(*)] ) '+


x Lf(t)]2 I*(-') R
X-l


+ 2 [ rf'(t)]2 7(') -


fit E7( ) -- +


5p = 2






fACA TM 1230


where *' is determined from the equation


x' = k*' f'(t)t' + f'(t)t f(t)


(19)


From the structure of equation (18) it is easily seen that the
relative increase of the pressure arising from the acceleration has,for
velocities comparable with the velocity of sound, an order of magni-
tude bl/v2 where b = f"(t) is the acceleration. Hence for rocket
projectiles of the usual dimensions, for accelerations not higher
than 1000 meters per second2 and velocities comparable with the velocity
of sound, the added pressures arising from the acceleration are
negligibly small.

This is confirmed also by the following illustrative computation.
The length of the war head portion of the shell is equal to I = 0.25 meter
and the maximum radius r = 0.07 meter. The generator of the head is an
arc cf a parabola which smoothly goes over into the cylindrical part so
that it is given by the equation ? = 0.56i 1.122 (scale in meters).
The motion is one of uniform acceleration with an acceleration
of 1000 meters per second2 so that the line of motion of the
projectile x = -5000t2 (t > 0), x = 0 (t < 0).


The pressure distribution over the war head is found at the
of time t = 0.5 second such that the velocity is equal to v =
per second. The air density p = 0.125 kilograms-per second2.
were obtained for the pressure distributions (with account taken
acceleration) and the additional pressures bp produced by the
acceleration the following values:


t = 0.05 0.10 0.15 0.20


instant
500 meters
There
of the


= 0.05 0.10 0.15 0.20 (meters)


p = 8550,7448,5970,3588; bp = -5.0 -5.5 -11.0 -16.0 (kilograms per meter21


The wave resistance and the corresponding added resistance were then
obtained as


Q = 2 r dr (p p) dt = 100,
Jo us


= o d=


(kilograms)
(20)







HACA TM 1230


The added pressures 6p arising from the acceleration were computed from
the approximate equation (18) which for the example given has the form


R7(') x'
R


7'(t') dx' +bt 2
]La)


J


7" (e') R dx'


/t228x2
+ 'Y'(O) -
R2


where


02\a
5xI = xI xl*
bt IX --
a R-1
a RI


3\a/
&x2 = x2 x2 b '2 -
t x x2
a R2


Translated by S. Reiss
National Advisory Committee
for Aeronautics


REFERENCES

1. Lamb, H.: Hydrodynamics. Cambridge, 6th edition, 1932.

2. v. Kgrmin, Th.: Problem of Resistance in a Compressible Fluid.


1935.


- 2
a k


5p = pb
2 x
6P=1 /








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