ihcHf^-m^ 
 
 ?*t. 
 
 >^AilQM 
 
 < 
 
 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 
 
 GOVERl^nNG UNSTEADY FLOW IN A GAS 
 
 By Lorenzo Poggi 
 
 Translation of "Estensione ai Casi di Simmetria Centrale Bi-e Tri- 
 
 Dimensionale di Due Particolari Soliizioni delle Equazioni del Moto 
 
 Gassoso Non Permanente" in "Niimero Speciale in Onore di 
 
 Modesto Panetti" published by L'Aerotecnica, Associazione 
 
 Tecnica Automobile, and La Termotecnica, Turin, Italy 
 
 November 25, 1950. 
 
 Washington 
 June 1952 
 
 UNIVERSITY OF FLORIDA 
 
 Pa^So',f^^^csuB4Rv 
 
 GAINESViaE.FL 32611-7011 JSA 
 
\^L ^f7 6 3^7rr/i 
 
 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 synmetry in two- 
 and three-dimensions. 
 
 SOLUTIONS 
 
 As in a previous study (ref. 1), the equations of motion describing 
 unsteady one-dimensional flow for a gas obeying the adiabatic law of 
 expansion may be cast into the form: 
 
 Su Su 
 
 ^- + U :— = ■ 
 
 St Sx 
 
 2 Sa 
 
 7 - 1 ^ Sx 
 
 Sa Sa 
 ^- + u ^ = ■ 
 St Sx 
 
 7-1 Su 
 " 2 ^ Sx 
 
 (l-one) 
 
 (2-one) 
 
 -^Original Italian Report appeared as Est ens lone ai Casi dl 
 Simmetria Centrale Bi-e Tri-Dimensionale di Due Particolari Soluzioni 
 delle Equazioni del Moto Gassoso Non Permanente , in the anniversay 
 volume entitled Numero Speciale in Onore di Modesto Panetti , published 
 by L'Aerotecnica^ Associazione Tecnica Automobile^ and La Termotecnica, 
 Tirrin, Italy, 25 November 1950. 
 
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 press\ire 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: 
 
 u 
 
 t + t. 
 
 > 
 
 and 
 
 (C-one) 
 
 u 
 
 7 + 1 t + tf 
 
 1 + 
 
 -2(7-1) 
 7+1 
 
 7 - 1 
 7 + 1 
 
 — 1 (- (D-one) 
 2 
 
 ao2(t + to)' 
 
 where 
 
 and 
 
 jlq qh^ oq 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) . 
 
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) 
 
 Su Su 
 St hx 
 
 2 Sa 
 
 7-1 Sx 
 
 (1-two) 
 
 Sa , Sa 7-1 1 S(-ux) /„ . x 
 
 ^ + u ^— = — a ^^ — '- (2-two) 
 
 dt dx 2 X Sx 
 
 For the case of three-dimensional flow one then obtains in an 
 analogous way that: 
 
 Su , Su 2 Sa / -, J.-U \ 
 
 ■TT + u T-- = - ;- a -r— (1-three) 
 
 dt ox 7 - 1 ox ^ ' 
 
 Sa , Sa 7 - 1 a S(ux2) / ^ 
 
 3t ^ ^ 5^ = - ^ -2 ^^ (2-three) 
 
 r2i '. 
 
 ■^For the derivation of equations ( 1-two) and ( 1-three), which 
 are identical to equation (l-one), it suffices simply to 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 encoiintered; that is, from 
 
 Bp 5(xup) , . , 
 
 X -vr = - — N (two-dimensional) 
 
 and 
 
 X 
 
 2 |p _ ^(fup) (three-dimensional) 
 ot Sx 
 
 wherein substitution is made for the density p and its derivative, 
 when expressed as functions of a, in accordance with the usual practice. 
 These equacions then become equations (2-two) and (2-three), respectively. 
 
h ■ MCA TM 1332 
 
 The system of equations (l-two) and (2-two) possess as particxilar 
 solutions the following two expressions for u and a: 
 
 and 
 
 u = 
 
 7(t + to) 
 
 u = 
 
 a = 
 
 t + t. 
 
 ^°ri 
 
 a = a^ 
 
 -2(7-1) 
 
 r-o ' 
 
 2\ 7 
 
 — 1 
 2 
 
 ao^Ct + to)^ 
 
 (C-two) 
 
 (D-two) 
 
 This result may be quickly checked by insertion in the differential 
 equations. 
 
 The system of equations (l-three) and (2-three) also possess as 
 particular solutions the following two analogous expressions for u 
 and a: 
 
 u = 
 
 t + tr 
 
 3(7-1) 
 
 EL — 3l, 
 
 °'-t 
 
 and 
 
 (C -three) 
 
 u = 
 
 37 - 1 t + t, 
 
 a = a. 
 
 1 + T^ 
 
 6(7-1) 
 
 37-1 
 
 + 31 
 
 7-1 ' 
 
 37 - ly 
 
 ao2(t + to)2 
 
 ~ -, I (D-three) 
 2 
 
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 (l) 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 u^ (which is propor- 
 tional to its kinetic energy) . 
 
 In fact, throiigh use of quite obvious manipulations, it appears 
 that the various D-type solutions lead to relationships of the general 
 form: 
 
 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 
 
NACA TM 1332 
 
 REFERENCE 
 
 1. Poggi, L. : Contributo alia Pulsoreazlone. Part 1, L'Aerotecnica, 
 Vol. 6, I9k9. 
 
 NACA-Langley - 6-12-52 - 1000 
 
U] 
 
 Hs 
 
 
 l/l 
 
 
 o c^ i? n 
 
 01 
 
 u 
 
 S"ll 
 
 a, 
 
 3 
 
 (" -• 2 ^ 
 
 o 
 
 
 
 
 
 .-r< <D ti 
 
 ■< 
 
 
 M>y O C 
 
 
 ho 
 
 
 C 
 
 o < c a 
 
 u, 
 
 b 
 
 a 2; O Q. 
 
 CM KH S S 
 
 
 
 J 
 
 
 
 < 
 
 
 
 z 
 
 
 
 o 
 
 
 
 
 
 
 (/J 
 
 
 CO 
 
 ;^. 
 
 
 
 w 
 
 
 3 
 
 § 
 
 
 ffl 
 
 
 
 C 
 
 u 
 
 
 o 
 
 o 
 
 
 01 
 
 < 
 
 is 
 
 
 i^ 
 
 
 
 n 
 
 Pk 
 
 
 
 O 
 
 
 
 
 
 
 
 
 
 
 
 F 
 
 < 
 
 
 E 
 
 U 
 
 
 o 
 
 W 
 
 
 U 
 
 a 
 
 
 >> 
 
 f- 
 
 
 o 
 
 o 
 
 CO 
 
 > 
 
 •n 
 
 H 
 
 z 
 
 ?: 
 
 < 
 
 o 
 
 H 
 
 « 
 
 z 
 
 < 
 
 
 w 
 
 O 
 
 
 H 
 
 < 
 
 rt 
 
 X 
 
 z z w 
 
 HgO 
 b< O < 
 
 °h2 
 
 o-i < " 
 
 ocyo 
 
 J w J 
 ^ w ^ 
 
 U M Jx 
 
 S^9 
 
 W H ^ 
 S CO m 
 
 gz z 
 go a 
 
 """HO 
 >< 3 Z 
 J J s 
 
 -JO 5 
 "5 > 
 
 K < O 
 
 a jj O 
 
 E -^ ►, 
 
 
 CO « 
 
 "52 
 
 U: 
 
 c c 
 o o 
 
 m 
 
 m 
 
 c 
 
 c 
 
 cu 
 
 ai 
 
 
 
 m 
 
 c 
 
 W 
 
 r> 
 
 c < Z 
 
 2 o - 
 
 E < 2 
 
 3 z D 
 
 2 UP o 
 
 o .2 
 
 O lO o 
 
 u) as ^ 
 
 => cu 
 
 "= t. 
 
 O 
 
 • c 
 
 -^ 'bbO 
 
 a» bD r- 
 T3 O C 
 
 .- 0. o 
 
 ^ r- i: 
 
 o o ^ 
 
 .- N . 
 
 N C W -^ 
 
 3 1-. « S 
 
 o 
 
 IS 
 
 M D O " = 
 
 S °.S o 
 
 S m ^ •-; 
 
 ■" c c 3 
 
 ■a 2 o -s 
 
 <u r^ o o 
 
 u a ui '" 
 
 !" 3 2 0) 
 
 >, o 
 
 ^ s 
 
 .2^ cH 
 
 5 u 
 
 i"5 
 
 ti S 
 
 CTJ 
 
 
 C dj -(_) 
 
 5 .is ho c i! i; 
 
 3 3 C ,S _ 0) 
 
 « S S -a ^ S 
 
 ^ t: t <D ^ s 
 
 £ m o c S >. 
 
 < 
 u 
 < 
 z 
 
 3 ^^ 
 
 CO 
 
 
 o 
 
 eg 
 
 (1) 
 
 w 
 
 
 
 
 o 
 
 U) 
 
 (A 
 
 O eg 
 
 
 
 
 N CO 
 
 n 
 
 OJ 
 
 a 
 
 3 
 
 
 
 
 e 
 
 o 
 U 
 
 
 3^ 
 
 
 2 
 
 
 
 — '~< 
 
 OJ 
 
 ^ 
 
 pT 
 
 
 bDr V 
 
 
 ttj 
 
 
 bo 
 
 o 
 
 c 
 
 
 c: 
 
 o < 
 
 c: 
 
 rt 
 
 a. z o a ::^ 
 
 CM I-, ;=? s 
 
 ►J 
 < 
 Z 
 
 o 
 
 . CO 
 
 m 2 
 
 c Q 
 
 2o 
 
 ^ H 
 
 o (^ 
 
 •" O 
 
 a! CO 
 
 ti W 
 
 ■- CO 
 
 E < 
 
 O td 
 
 uac 
 
 >.H 
 
 CM O O 
 
 - T3 Z 
 5 < 2 
 
 '^ « z 
 
 < § w 
 
 (-> :2 f- 
 
 < ra X 
 Z Z W 
 
 o o to 'e: 
 
 C " O 
 
 &< O < .A = 
 
 op;zm-S 
 
 CO <; " <p .„ 
 
 o a o ii -2 
 
 J w J n 9 
 ^ w * o o 
 
 O K >■ ~, CO 
 a H Q .2 — 
 H O ^ 
 
 CO d, 
 
 C f-i > 
 
 £ "^ o 
 
 c <; Z 
 
 dj Z (u 
 
 c ■ Q. 
 
 W H 
 
 r, -.-» ra -7 CO O 
 
 S CO fe g .ii 
 
 S 2 2 .S t; 
 >H O a CO n 
 
 CO p ^3 ^ a 
 !« D z „ s; 
 
 J ^ s ■ - = 
 
 a: 
 
 M 
 
 > 
 o 
 
 in CM ^ 
 
 JO 
 
 < CO 
 
 ^K 
 
 E< „ 
 
 U J O ^ ^ 
 
 =s H z -2 .2 
 
 a o z s s 
 
 "i E s c c 
 
 < Q, 2 Cl Q 
 
 Q^ 
 
 - 3)0 
 
 •O O C 
 
 N C U) — • 
 
 5-0 i;s^ 
 
 M 01 O ''' ? 
 
 c -a « 5 4^ 
 •" c >> o 
 
 ^ E "' "■ S 
 
 ■- - ^ w ra 
 
 S w C ■- S" 
 
 •" c c 3 £ 
 
 n f-, -r i-* 
 
 !n 2 "1 
 
 > E 
 
 ra 
 
 o a> 
 
 -o *^ — 
 >'2 ^ ^ 
 
 o >, 
 
 
 c 5 i; 
 ■2 > c 
 
 £ — hc c i; ii 
 
 ■t=; 3 c C n» 
 
 H ci w o 
 
 0) 
 
 1^ 
 
 T3 
 
 s 
 
 ni 
 
 
 
 
Digitized by tlie Internet Archive 
 
 *in 2011 witli funding from 
 
 University of Florida, George A. Smathers Libraries with support from LYRASIS and the Sloan Foundation 
 
 http://www.archive.org/details/extensiontocasesOOunit 
 
o 
 U 
 
 = " ^ J* 
 
 01 — I o > 
 o g '^Z 
 '-' ^ ^ ._- 
 
 ■r-T < OJ ti 
 
 a 2 o a. 2 
 
 
 
 ^ 
 
 
 < 
 
 
 Z 
 
 
 
 
 
 
 
 . w 
 
 
 " Z 
 
 
 .y w 
 
 
 ^S 
 
 
 c3 ■:? 
 
 
 c Q 
 
 
 2o 
 
 
 H 
 
 
 
 
 b. 
 
 
 ■" 
 
 
 OJ 
 
 
 
 
 ti CO 
 
 
 f< 
 
 
 S^ 
 
 
 w 
 
 
 UK 
 
 
 >,H 
 
 
 So 
 
 CO 
 
 i/j H 
 
 
 •^z 
 
 ? 
 
 <y 
 
 f- 
 
 S z; 
 
 < 
 
 ^ w 
 
 I) 
 
 •S H 
 
 < 
 
 ra X 
 
 z z w 
 
 &< O < 
 
 CTj <■ •-' 
 
 o cj-o 
 
 ►J W J 
 
 u a >< 
 ^° w 
 
 W ^ H 
 
 S Cfi C/j 
 
 SZ z 
 
 """HO 
 ►J J 5. 
 
 CQ 4i^ ^ 
 "J -a -g 
 
 0) .- 0, 
 
 ™ o 
 U o " : 
 
 GO a 
 
 S 5! 
 
 O _r 
 
 < s 
 
 
 .- Q. 
 -a „, 
 
 O 
 
 JO 
 
 « < o 
 = ^z 
 
 z < 
 < a, S 
 
 K (J . 
 
 -SO) 
 
 Oi d -a 
 
 = c _ 
 
 o o c 
 
 en m 2 
 
 2 2" 
 
 en H 
 '^ ^ 
 
 (1> -- 
 
 o 
 ■ c 
 
 - O 
 
 I — < b£) 
 
 ^ 2 
 
 o — 
 
 N . 
 
 3h2 
 
 , « X i; 
 1 i o ■" = 
 
 c >> o 
 
 c e. 
 
 ' -. s 2 "-^s 
 
 o 
 ? w 
 
 •" c 
 
 T3 O 
 
 01 S 
 Lh d 
 
 > 13" 
 
 O OJ 
 
 m ^ 
 >.2 
 
 2 .s >, o 
 g 3£-S 
 
 ^ o g g 
 
 « > s 
 
 n] a; tu .3 
 
 bD w *e -o 
 
 O) o 
 
 . o S 
 
 3 a ^ u, ^ 
 ■3 01 g a) ■" 
 
 
 
 H a. tn 
 
 < 
 u 
 < 
 z 
 
 m 
 
 
 CM m 
 
 
 
 N CO « 
 
 
 
 C CO ^ 
 
 Q. 
 
 3 
 
 0) rt 
 
 a 
 
 Q, 
 
 Ss^ 
 
 
 
 m 
 
 -> H -3 
 
 
 
 
 
 c 
 
 — < 01 
 
 p£ 
 
 hu 
 
 boO 
 
 
 
 < C 
 
 Ix, 
 
 PU 
 
 a 2 
 
 a, ^ 
 
 CM M s ;=; 
 
 So?; 
 
 §5 
 
 o- 
 
 < ■ 
 
 z ; 
 
 I L, m a 
 
 1 Z (^ O < .i =! ~ 
 iwOpgCQ^S 
 
 ; '-' o a o ^ -2 " 
 . o J W J c 2 c 
 
 ; P ** W * U o K 
 
 H u B >- w 0" 
 
 ; (x< 2 H Q .2 - c 
 
 w S c« E I .ii g 
 S Z Z S r o 
 
 (_, S O D 1^ I S 
 ^ c/: C n .- Q, a 
 
 H-. 'i > ra Qj 'T' 
 
 55 ffi p 5, .0 O c 
 Z "^ y r! w w 2 
 
 X z -3 o « E g. 
 w < a SSq w 
 
 " -2 
 S a; 
 
 <z 
 
 — c 
 
 M 0; O ■" C 
 
 .S •a " S g 
 
 a> 
 
 2IS 
 
 .S ^-o 
 
 o o ^ 
 
 0> 
 
 >> o 
 
 ^ o 
 
 6 5^ 
 o 
 
 S "1 " 
 
 3 ffl CD 
 VT bD w 
 
 O OJ 
 
 i^ 
 
 2s 
 
 0, ;- 
 
 o 
 ■ c 
 so 
 
 ho ^ 
 
 o c 
 
 Q, O 
 
 o — 
 
 eg . 
 
 c tn ^ 
 
 jh2 
 
 2 ■" 
 ■3 " 
 
 
 >-2 ^ ^ g *' 
 
 w „ ° u S -a 
 
 .2 >.^£ " 
 
 OJ 3 n] « ;, ? 
 
 L — * OJ 5 tu *-> 
 
 a o -K .2 > c: 
 
 r* " g m o .5 
 
 3 3 C .5 ^ 01 
 
 « .a 3 ? ^ E 
 
 77 ra w c 
 
 H a. w o 
 
 <: 
 o 
 
 
 <u 
 
 1-5 
 
 
 
 OJ 
 
 CM M 
 
 
 N CO Oj 
 
 
 C CO ^ 
 
 3 
 
 0) -. 
 
 a 
 
 S^^ 
 
 OJ 
 
 >Jh^ 
 
 •-''< ■" 
 
 
 hoy 
 
 ho 
 
 c 
 
 < c 
 
 U4 
 
 Q. 2 
 
 a x; 
 
 CM I— I »=t S 
 
 < 
 
 z 
 o 
 
 . m 
 
 2o 
 
 O fci 
 
 •" O 
 <u 
 
 0) c« 
 
 t; W 
 
 - K) 
 
 s < 
 oS 
 
 >,H 
 
 CM O O 
 
 CO M H 
 ^ -a 2 
 3 < 2 
 
 "^ « 2 
 
 < § W 
 O s ^ 
 
 < « X 
 Z 2 W 
 
 55^, 
 HgO 
 
 (1. O < 
 
 °hS 
 
 CO <■ " 
 &D5 
 O O'O 
 ►J b3 J 
 
 O 33 >H 
 W H g 
 
 S CO CO 
 
 S 2 2 
 
 So 3 
 
 "■"HO 
 Jx D 2 
 J hJ S 
 -JO K 
 
 y 5 > 
 
 K< o 
 
 "h2 
 
 Q K ft 
 2<: o 
 
 < a § 
 
 CQ ^ i^ 
 ►*4 -n w 
 
 O O a* 
 
 ^ « ° , 
 
 <z 
 
 o _r 
 
 < 2 
 
 2 0> 
 
 ^ Ot 
 
 CO ra 
 
 .- a 
 
 -:; OJ 
 o> ra T3 
 c c „ 
 00? 
 
 I ra X 
 ho QJ o "^ 
 
 C T3 ■" J 
 
 5 °.S § 
 
 ? w c; vr 
 
 ■" c c 3 
 
 ■a o o ^ 
 
 OJ r; cj o 
 
 t. ra 1^3 w 
 
 ^ = ra OJ 
 
 ^ ST ho w 
 
 >. o 
 
 t. c 
 
 O OJ 
 
 ^ s 
 
 OJ OJ 
 
 w E 
 
 cy 
 
 Q W 
 
 ' i" 6 
 ) a> ,s 
 1 rt S 
 
 I c ^ 
 
 > 3 Q 
 
 ; ^ t, 
 , hoO 
 
 bo r^ 
 O S 
 
 o — 
 
 N . 
 
 C CO '^ 
 
 jh2 
 
 ^ ra ji: c OJ 
 ^ c H 0; 2; 
 
 •S >^ g- 
 
 
 o 
 
 S SB 
 
 .2 > c 
 
 S S o 
 c ^ {^ 
 
 S S .s I ^ 
 
 
 < 
 
 < 
 z 
 
 — rout: 
 
 H a w o 
 
■^ 
 
 'JNIVERSITY OF FLORIDA 
 
 1 XDCUMENTS DEPARTMENT 
 
 ■,20 MARSTON SCIENCE LIBRARY 
 
 PO. BOX 117011 
 GAINESVILLE.FL 32611-7011 USA