Extension to the cases of two dimensional and spherically symmetric flows of two particular solutions to the equations o...

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Material Information

Title:
Extension to the cases of two dimensional and spherically symmetric flows of two particular solutions to the equations of motion governing unsteady flow in a gas
Series Title:
NACA TM
Physical Description:
6 p. : ; 27 cm.
Language:
English
Creator:
Poggi, Lorenzo
United States -- National Advisory Committee for Aeronautics
Publisher:
NACA
Place of Publication:
Washington, D.C
Publication Date:

Subjects

Subjects / Keywords:
Aerodynamics   ( lcsh )
Airplanes -- Motors   ( lcsh )
Gas flow -- Research   ( lcsh )
Genre:
federal government publication   ( marcgt )
bibliography   ( marcgt )
technical report   ( marcgt )
non-fiction   ( marcgt )

Notes

Abstract:
The author previously discovered two interesting particular solutions to the equations of motion describing unsteady flow in a gas confined solely to a one-dimensional duct. These solutions are now extended to cover the more noteworthy cases of central symmetry in two and three dimensional.
Bibliography:
Includes bibliographic references (p. 6).
Funding:
Sponsored by National Advisory Committee for Aeronautics
Statement of Responsibility:
by Lorenzo Poggi.
General Note:
"Report date June 1952."
General Note:
"Translation of "Estensione ai Casi di Simmetria Centrale Bi-e Tri-Dimensionale di Due Particolari soluzioni delle equazioni del moto gassoso non permanente" in "Numbero Speciale in Onore di Modesto Panetti" published by L'Aerotecnica, Associazione Tecnica Automobile, and La Termotecnica, Turin, Italy November 25, 1950."

Record Information

Source Institution:
University of Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 003875517
oclc - 156997661
sobekcm - AA00006195_00001
System ID:
AA00006195:00001

Full Text
fPk r 33-











NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS


TECHNICAL MEMORANDUM 1332


EXTENSION TO THE CASES OF TWO

DIMENSIONAL AND SPHERICALLY SYMMETRIC FLOWS OF TWO

PARTICULAR SOLUTIONS TO THE EQUATIONS OF MOTION

GOVERNING UNSTEADY FLOW IN A GAS*

By Lorenzo Poggi


SUMMARY


The author previously discovered two interesting particular solu-
tions to the equations of motion describing unsteady flow in a gas
confined solely to a one-dimensional duct. These solutions are now
extended to cover the more noteworthy cases of central symmetry in two-
and three-dimensions.


SOLUTIONS


As in a previous study (ref. i), the equations of motion describing
unsteady one-dimensional flow for a gas obeying the adiabatic law of
expansion may be cast into the form:


bu au 2 da
- + u = ---- a-
6t ux 7 1 cx


a da 7 1 5u
t- + u a-
)t ox 2 ax


(1-one)




(2-one)


*Original Italian Report appeared as Estensione ai Cast di
Simmetria Centrale Bi-e Tri-Dimensionale di Due Particolari Soluzioni
delle Equazioni del Moto Gassoso Non Permanente, in the anniversary
volume entitled Numero Speciale in Onore di Modesto Panetti, published
by L'Aerotecnica, Associazione Tecnica Automobile, and La Termotecnica,
Turin, Italy, 25 November 1950.






2 NACA TM 1332


where u is the flow-velocity, a is the local velocity of sound
corresponding to the state of the fluid, x is the distance along the
abscissa or duct axis, t is the time, and 7 is the ratio of the
specific heat at constant pressure to that at constant volume. The
first of these equations constitutes the "equation of motion" (arising
from the equilibrium condition on the dynamic forces) while the second
arises from the "condition of continuity" applied to the fluid masses.

It was shown in reference 1 that equations such as the ones
set down here possess, among others, the following two particular
solutions:


X
xt
t + t


a = ao(l


2 x
u t + to
7 + 1 t+to


a = ao


-2(y-1)
t Y7+1
+o)


7-1
+ t 2-


where ao and to are constants having the dimensions of velocity and
time, respectively. These two solutions are here distinguished from
each other by naming them type-C and -D, as was done in the cited
reference, and they are further differentiated from those to follow by
appending the suffix "one" in order to indicate that they refer to the
one-dimensional case.

The object of the rest of this paper is to point out what the
analogous solutions are for the cases of two- and three-dimensional
centrally symmetric flow (which will be denoted by C-two, C-three,
and D-two, D-three, respectively).


(C-one)












(D-one)






NACA TM 1332


Retaining the same system of notation as used previously, it is
easy to verify that in these cases the equation of motion remains
unaltered, provided simply that x now represent the radial distance
from the axis or center of symmetry; the continuity equation, however,
undergoes a slight change. Indeed, one finds in the two-dimensional
case that(2):


6u 6u 2 6a
+ u a
at ax i 1 ox


6a -a -1 (ux)
a + 7 a -
ot ox 2 ox


(1-two)




(2-two)


For the case of three-dimensional flow one then obtains in an
analogous way that:


bu Ou 2 6a
o- + u ox = a x
dt Ox a- Zx


6a 3a ? 1 a 6(ux2)
w-+ u 2 2 ox
ox 2 x2 6x


(1-three)


(2-three)


(2)For the derivation of equations (1-two) and (1-three), which
are identical to equation (1-one), it suffices simply td refer back to
what was stated in reference 1. In the cases of two- and three-
dimensional flow, the equation of continuity can be derived from the
forms in which they are more commonly encountered; that is, from


-p (xup)
x =t ax




x2 p (x2up)
at 6x


(two-dimensional)




(three-dimensional)


wherein substitution is made for the density p and its derivative,
when expressed as functions of a, in accordance with the usual practice.
These equations then become equations (2-two) and (2-three), respectively.






4 NACA TM 1332

The system of equations (1-two) and (2-two) possess as particular
solutions the following two expressions for u and a:

u = x
t+to 1


a = ao ( +


x
u = 7(t + to)


a = ao +
t"/


-2(7-1)
v


+ 2-1)2
s[>2
+I


This result may be quickly checked by insertion in the differential
equations.

The system of equations (1-three) and (2-three) also possess as
particular solutions the following two analogous expressions for u
and a:

x
U =t+ to


a = ao(l +


t 3(?-1)
-Tt 2
to)


2 x
37 1 t + to
37 It + to


6(7-1)
-- 3-
to


+3 2
37 -1)


and


(C-two)







(D-two)


and


(C-three)


a=a 1
o


1 (D-three)







NACA TM 1332


The two sets of solutions exhibited here have a rather interesting
physical significance, which may not have been brought out with suf-
ficient clarity when the one-dimensional case was examined in
reference 1.

To be more exact, when state parameters of the fluid are initially
uniform through the flow field, while the velocity is proportional to
the abscissa coordinate (in the case of one-dimensional flow this
latter relationship may be directly interpreted as the fact that the
velocity varies linearly with respect to the location coordinate along
the duct axis), the various C-type solutions show the following features:
In successive intervals of time this type of wave setup is maintained
unaltered, except that a uniformly applied alteration of the fluid
properties is seen to occur, and likewise the coefficient of propor-
tionality between the velocity and the abscissa coordinate is changed.
If one now examines the motion of the individual particles of the fluid,
rather than considering the velocity to be a function of the time and
the x-coordinate, it is readily found that the velocity of these par-
ticles turns out to be time-independent.

This result follows immediately from the fact that the left-hand
member of equation (1) represents the total derivative of the velocity
in each one of the cases. When the various C-type solutions are
inserted in this same left hand side the result should be zero, because
so is the right hand side, inasmuch as "a" is solely a function of time.

As regards the several D-type solutions, we may observe that, from
the initial moment on through the subsequent flow process, they are
characterized by a linear variation of a2 (which is proportional to
the internal energy of the gas) with respect to u2 (which is propor-
tional to its kinetic energy).

In fact, through use of quite obvious manipulations, it appears
that the various D-type solutions lead to relationships of the general
form:


a2 ao + t ) + Cu2


where Z and C are constants with different values in the three
cases of one-, two-, and three-dimensional flow. Their specific values
are derived with little trouble directly from the three D-type solutions
already presented explicitly.


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







6 NACA TM 1332


REFERENCE


1. Poggi, L.: Contributo alla Pulsoreazione. Part I, L'Aerotecnica,
Vol. 6, 1949.
















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