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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 onedimensional duct. These solutions are now extended to cover the more noteworthy cases of central symmetry in two and threedimensions. SOLUTIONS As in a previous study (ref. i), the equations of motion describing unsteady onedimensional 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 (1one) (2one) *Original Italian Report appeared as Estensione ai Cast di Simmetria Centrale Bie TriDimensionale 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 flowvelocity, 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(y1) t Y7+1 +o) 71 + 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 typeC 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 onedimensional case. The object of the rest of this paper is to point out what the analogous solutions are for the cases of two and threedimensional centrally symmetric flow (which will be denoted by Ctwo, Cthree, and Dtwo, Dthree, respectively). (Cone) (Done) 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 twodimensional 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 (1two) (2two) For the case of threedimensional 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 (1three) (2three) (2)For the derivation of equations (1two) and (1three), which are identical to equation (1one), 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 (twodimensional) (threedimensional) 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 (2two) and (2three), respectively. 4 NACA TM 1332 The system of equations (1two) and (2two) 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(71) v + 21)2 s[>2 +I This result may be quickly checked by insertion in the differential equations. The system of equations (1three) and (2three) 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(71)  3 to +3 2 37 1) and (Ctwo) (Dtwo) and (Cthree) a=a 1 o 1 (Dthree) 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 onedimensional 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 onedimensional 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 Ctype 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 xcoordinate, it is readily found that the velocity of these par ticles turns out to be timeindependent. This result follows immediately from the fact that the lefthand member of equation (1) represents the total derivative of the velocity in each one of the cases. When the various Ctype 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 Dtype 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 Dtype 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 threedimensional flow. Their specific values are derived with little trouble directly from the three Dtype 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. A NACALangley 61252 1000 I I ^ j<. ci N L. En 00 c 5 4) E! a a ~ E 00 Z u U CT., 4' Zr m N  700 'i .cm z L. 0 000 L 4  o., E0 oo 0 0 . 01 P P 2 C C .. S"U:E 0 m zz< U0 3 0  is8 o 1r ZZW<0..SQU3 J2H  I. o= CU 0 4 Z c >' 0 c w E  C cz 3 0 0   ci 4' 4 c co 6 = ooeaj ZT:u >5 i0 . S O g H" a. u a a U CM (0 4 LA N CO i ,1 U, 4I ,4 oc N  4 42 U2 Nr 4 K ta 0 L 2 ai G C m . 3 0 8 Q. '4 ^,. u i ;;^ * u~. lu o ;0 E 00 c (uo .0. cu 0 o oyu o r a C 0 4 wZ' wL Ca C o 0 Sz crzc 0 n O CiU 0 0C U = ^ pr c 4 tri p 0 if zu C 4' 0 Ec c C u 2 0 L S_? o , cr ma 1 cuEu ^tv s " >50  C C 5 . E V 004' 144 3 .0 C  c nV 0 3 a * 0~~ L.E3 E Sa, E ScJ a o in gp O 0 >, > ~ E OC Lj rl a 4, H ac1i0 U Digitized by Ihe Inleinel Archive inl 2011 Will lIundnI g Iroin University of Florida, George A. Smaillers Libraries with support Irom LYRASIS and the Sloan Foundaiion http://www.archive.org details e: Cu 1 " 2^s . 0 0 U S fgi6  D. 3 0 N ~ _] En Z we mn*: CuCiOCA.*~ 2 5 a, a E <. & z ^j^ iu . 1w 0Ld 0 Ef 2 u LN lz E z Lr, in in n i  g22:zz o% 0 ~,k m .c .]~ 2 g. 'n= *0 .5 H c.Q . 0> Z 1% 0 Q c c 0 0 z 0 o L4 Z 0a o~ l~iza ssai < En 4a~i , EI.u c' Cx 0 R a ZZ I "3 Cu 41, 0 M C Im Cu Ul a EnE~ w oC C 3uU '0 C2 C 03 0 f U Cac, 111 M V5a E ~a,~ a to o V" g C 0s C L *a ,0 5 C ' s~Cu O5j r z m'! g.JI Z~ N I a z u L. '1' 0 0 w0O r 2. 0. 0 L l4 o Go 0 it di 3 P zo azo2 G I E 0 O i N W 9 c Lf'" V, > u F E' a 0 " Eu, Ufv,) Ui D z0< W )~ E 0c o .afio~za^w0,0 u21 co o g C. = QU, Z O oc ao W o' 0% cc x cc vCu a z O!:ggg.2.2 1'C  C d < M N W 1 C Cu au5 3 o V u z < sH2 11g S a Ho1sna...u 0 Sor u a 'a a, W S 5. E E O 0 o in im o I I" C Ii. [u 0 01 a  ' 0 0 5. :6 C < u M 0. Z 0 0.  N S o ." U o o Z cu  '0 o "0 0 E' 0 0E z0 2w L. OOWu 2 L4 iDFi. r S~.3^o" c o o i:S. ^^ ^llh h~lgpip5 IsO~z^ss s^d~lU 0 33mar J ^gillli^ i Mo  c u r S0 0 t 0 0 r F :"W 0 a .. S S . o C ka g.. mu2 mm l? oo E : L UNIVc1Iai I ur rLUNIU 3 1262 08106 650 