TRANSLATION: THE METHODS OF SIMILARITY AND DIMENSIONAL ANALYSIS IN MECHANICS, (METODY PODOBRYA I RASMERNOSTI V MEKHANIKO)

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• Original Classification: U
• Document Page Count: 405
• Document Creation Date: December 22, 2016
• Document Release Date: September 25, 2012
• Sequence Number: 3
• Publication Date: September 19, 1957
• Content Type: REPORT

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R
STAT
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L.I.Sedov
C FHOI3 AND D1)iSIONS
14ECHANICS
State Publishing ikruee
for Technical-Theoretical Literature
Koacoll 19 54
STAT4,
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-'14t4,1V 15,
#40,
. ?many defissiencies. These queetiots are toeehed even only seellsille and in Peseine.
The basic concepts, even the concepte of d4men ional and dieensioniese quante
the problem of the number of fundamental units of measurement, etc., are not explain-
clear manner. Moreover, befuddled , d ituitive representatione of
the ,concept of dimensions often serve as the :starting point for the
the cen-
..of viewpoints in which the foreallae for dimensions are ascrleed a serteln myetis or
eret significance. In certain instances such confusion has led te
eeich serve as an el:elect of perplexity. We will analyse in detail one eeemple
of such a
misunderstandine in connection with "Relefen on1 on onernini7
heat exchange of a body in a liquid stream. In presenting the theory of s2.esilari-
ties, relationships and eathematical devices which are not eseentially connected
with this theory are
tions of the theory
frequently introduced.
of dimensions and similarities, as is eenerally the case edts
any theory, esee the a.4:ia
of methods and basic premises appropriate to the essence of
the theory. Such a construction makes it possible to clearly outline the lieits and
pessibilities of the theory. This is especially necessary in the case of the theory
of dimensions and similarities since one often encounters extre, opinions: on the
one hand concerning the omnipotence of this theory and on the other, its tr4v4elity.
Neither opinion can be considered correct.
It should be noted, however, that most realistic and useful results an be ob-
tained by combining the theorems of the theory of dimensions with the propositions
of general physics, which, in itself, yields interesting conclusions. Therefore,
order to illustrate various applications more completely we will consider a whole
Series of mechanical problems and examples of the combination of dimensional methods
of various types with other qualitative mechanical and mathematical theorems.
This has likewise impelled us to concern ourselves with the problems of the
turbulent movements of a liquid in more detail. In the theory of turbulence, methods
of similarity are the basic working theoretical methods since, in this field, we
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3ti1l do not have 3 (710,Se t-Jy5tem. of 1,1-.,0c. At possible to re.lce the
mechanical probleme to nat,hemiAticit\ hir
In the set ion on t' ?::ent
or a liqu.?d rtw results art pre ti wHQ.h nupplemont an Clarify several problems
involved.tri the theory of turhaewe.
in addition to the enamrle:5 of the N.1)3e of the methods of dimensions an)i similar-
itits, we have ttttonriteto 9hOd some rj::nt on the re i: of a 7.;:,1m-ter
'roLiOO1,:vH - ;:tr*-.1 %-ery of re
()f
reL.ttionsir
in ;40,$44 what more [!!rtail on an exazdnn of the e:77?laions of
express Newton' cn.4law. point cf view which we wi11 present is t':Ierefore
not new, however, it in consi,..ier%bly different from the treatment of this hasin prob-
lem in mechanics as it appears in 7ertain wiely use:.1 text-ooks on theoreic
mechanics.
The nurber of known applications of the theory of .iinsiow; and sirrities
in mechanics is very 7,7reat and we have not touched upon m.%ny of tbem. The author
hopes that the hook
til give the
reader In a,J.ea of the rrocedures and possibilities
or these methods and will help in the analysis of new predens and in the forrila-
tion and execution of new experiments.
No special preparation in requirel for readin,' the 7re%ter part of the book.
In order to understand the material presented in the se;:ond half of the book, it is
necessary to have a general knowledge of hydromechanics.
Moscow, 1943
Preface to the Second Edition
This book is an extended and revised version of the book Methods of the Theory
of Dimensions and the Theory of Similitude in Mechanics published in 1944.
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....,m,w,WMCW10111,10AWAROWLA,
:
rn the s. ond
iition, in addition to certain corrections an0 minor improv
manta,. supplemental information has been introduced in which the theoremBof the
.theory of dimensi,no are utilized to determine a sere s of precise deri?ations in the
theory of a wave on the surface of a heavy ideal liquid, in the theory of the move-
ment of a vi-ous fluid, and in the theory of one Aimen ional nonSteady otates of
motion of a gas.y an analog
the dimeti3ion15 zoerMient of 1;71orl 0.
the
state .of weirM:t:
,are as cletrnining.
'e
kg
we.ght
-;;-
Ea:- of the nart,..s -f
are -distribute0, 17),-..! a - inite
Let the MaglitUdeofteie bo t n t. e
41.
of def4?aing -amete.rs will be t',..e
E, U. og, P ?
in this case we hwe n -2;
.. .
-.1..a . _. a,,,,?
eaJ..'..: .....ar states of elastic emilibrium wi7.7.1. be the fella ,
ng k.ep 1.,."(i, 3s' the case of .the th.ree-diirtens.1..ona),, problep . an(.. ? ..
:. the Shape? r'"1. r thitt rlat w-ef-1,7.e of irAfillite .. span .:14.1 the case of the- 'p341re. (probllein are -:
- , 4, . . ? .. .. ,
? . . ,
, .. . .
inte..7'est.4?47,?, ii.tlIzIt .t.II#.a:i.rk sl-,,tt'f-a.ce- 1,.'s .f i....?4641 cete1y- ' by the cyrle r,eq,-11.irezierl-t,
. . . .
.. . .
. . .. . .
? 1?,-"?:-., :.-,-...,,- :metricalsil,4.11'1.r-4, ty., .ibk.- -41.e.p.",itra.1,,e o',..., geor,-;e4,--.r...1..'?,:is., ,;,,, ? -;---:,1.1....Ltar? ,f,....onesco,..me ,o c24ne .5:-.
?7he. surface of the .con.s and this surface or the flat wedge are ?,letextrined
, . _ . .. .. . .
._
..??..;. completel.y . dies3.on1css geometrical irttla,ntitles.
_
-:',:?-? .?
We will sine tl'..14. *I- e rlu3, d recu-ies '1r' e'"4 ".1.**() im-ferace bound,ed., by
.. ? . _
h tontal plane and that we can neglect the weight property ar."1 viscosity of the
ru1d. Therefore we will ectsiier that the fluid i incomPressible, -0,71?C071?115, 3-111
ideal.
1e n J.
o
,,,, g
.. 4,41-dell r,fa.4....n.l.ter, the contact by the ,
Lot t ,-, 0 be the tit ii tne,Yraml ,...,,,e
boy 'with tht5 fluid at rest, no body,submergin& into the 4.1711.12,d? achieves fonictrd
4, .
velocity v Constant 11 -value and d:irrection..
? .
. ? . . . . r .
e -4-11 ass.grv.,0 ,thr,t on, the free surface, Pressure. .1) has a constant value pe..
the lne0mpre5..5i.bi3.1t,yof the .tIui;c it follerkts that the va.7.ue po on t
free
-."--leurfa("te cannot influence the tltri:)ulen,t moti an. of the fluid. ,Instead or pressure p
e can comeidel- the difference p po; in this case the parameter po is tvnaterlal.
Therefore the me)chtlnical properties of the fluid are detertvined by a single para-
-imeter by density P.' ?I'M that I have 34,4do ,it follows that all. the mechan'tcza
'"---101.giamacte7ri5tS.ca of the motion of the f13.114 at each point are deterrai.ried by the tzaan-
,-,-1
?" -1 tities
II" t s, Vo CIL 3 Xo Z
? e ?
---4-torhere a and fi are angles deterr,...iviirlgthi, .direction of velocity relative to the body, .
'and x, - and z Are eooroxiatea of the point under consideration), either in a
STAT
_
'???
-
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oin
? .C,O,
oidcr ri1 tirrni Os* Qv'
,he fluid i3 none tato ;,ion,
tion of the fi4.1 retucos to
function of four indcpnd.t var'iable
nd
he ?oi1a of the .foJ.th form is vana:
if
For the velodty of parttcle of the fluid
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The constants
angles of attack rf the piae o s
relative, to the nonto ant level. of he free slIrfacFr the pne prb1. e.
re are appro te theoretical
? ertical-vibate and-- for the teabzersico of a plate
of the body th
t1he for of r-
for bodie of
he wed upon the
itY of the wedge
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4'luid e c4 'a. coripment,
"
4"1. ? ., 177 5 :4*
azi beteiin th
2 the 1IL1
4
tenda
ratea;
,52, 1936
Le .AtheraLics n? Mechari.CS
Institute, ilo.2520 1936).
the t';aue Poissonwave Theory. Trudy Materaaticheekoso
tuta imeni VJi.Stek1ova 0:t3 of the Mathematios Institute :uerii 1r
Y ? Jim
eklov Vol 9, 1935.
Thory of Waves on the urface of an Incolwressible Fluid.
Univ. ro.U, 1948,
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:;P!#hr40
ort " , 4 'which 1.4 .4 f nstrnt $ fr
4 4
IVY 1 Y ce " ici
0314,1k 'WA UT 1 to
iif 43,
v S ip %,"
,
1,1n as u ... 4, A 7-f,........, ..?
..V4.1:13
*,wiaere ' (xlt) epe.zet t heLght of pit; or-race .bcve theuncils-
e and p t1 eit 40 T.,ne
The taring ,,;1CflZ3 can b 1atJ a folic: iher4 L 0, we have:
vi.irt tritthe 3 eXCePt ariab
d n cr:zir drii otats. ince the oru1ated in kine,
tit craantitie it i evident that therecar be no oe thitwo dime sional con-
, tants l.rith indeper4, ?ensions i he functions
atisfy the Laplace equations if 10.e assl
(13.6)
x i3 4L charac+eri3tic ?tnction 04t* the LIQW of the 1icuid,
8,331.41= that the ..characteristfunction w z) i tale, fthlte,
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, . , ' , ? ? -. . .,.' ?''.,' . ?,,
?,..
,. rosa 4:,),1,6 c..01?1r1, tic!,r)!1 (13,2) ii:, :C0114)14.13 t,1t?, t,I.le.;.'ie:r,''i?.'(-,trii'43,4Nro ^;',-,1Z,- Ii PP 1,,,Ivall
,
ne on ax b z-ertrerbnter,1
11/4.,t
. '
?? 4
0 ?
?nka,.3,,r,?-as . ? ,11.v.33b,.. 44?, to, ,;(4,t,.?;iIr?Lb. 11.1.
i
.upper- serdpane? :If) t.,11.at ,t, i?on t ?-? ?
?., ..? - ? , , . ? ? . ? . ? ? ?
??-? 132Arte .0f i. iz?
1:11e. ..gOtter641..1 ftr,ov-ement.. ,t11.71.t? ?
4L1 lie cn.the. ?;:.ea
IAt us soak- a s ,01,t1..011 4' Z .?en.e411,,,25
? ,
,42,4 e ey tt pp1rrta74.1, tir
. _
?deterie 'kho.pteioz the -tte11,71..14 ,fe 141,11 Ilot, be ab_ to gt\re e t
-
roAnt to these- cotioris.
" tk 4'...11 4 .0., 4* eicen't,..; cori?sider.
7."rom the linearit,y prooler,
c,as,e3 Ithere ,?-2ave c1j one ti .7, s t
1" ant 8 tar., be
w(zi .41.epen,45 lear v (the
0; ,,,,e,,,erit,.&?/"1. at,
4" 4 tr. nara?...
,
Front the ftsTami)tions maicle we ;lee that a cor4pletue
meters cart be repree/).4,,,?ef.21, 1-?r7 t e ta121t,-?;-on
? X iy 1, '4, a.
,
. .
...iii.,,,,,,- :..--!:Lot :us. now - asstu.te '? ' .
., y. .. .
...... -...-._ P: if. ....1 ..
.? :_,......., ., .. . ... .. ? . .. ?... . .
?4:5...,..,ir., .? ?. ..:,.. ..j.'. .: .... , :-. ., , .,..,: ... .. .
- '''''.:?-i'''.?':h' e ? expO. ' ' tiellt a ..'?at.'..',-iari a?:----01.--..-h-,4- -'v'? 0:: been..., s.e.16-e, .t. 04...1::. '9"? t...h4t. ?...Y. ..s, , an: abstra'ct qu.. tit ? .8 . ce ? ,...
k:t?ii.,) ',?-',,'LP'rq 0 1.,t,',?,,,,i.._ .,..,,,,,?,,,,.4,Aeri+.,,,...' ,,,,,,?. '4,or, : "1:s.:::, to ' b,-e..s, 7.,;,.....,._.,,
' ? --- ???? - ..: ,I1-4-.4...+ ,1 1.1,.1.,-;,,,, 1...."73
.....:L__....:,,,...,..,:,... - , . ....
..-.. ? . .. ...,...',5'2.,.....1:-.-?H? ? ' r ' ' .. .
from' . hi .h' .r.'' ?-." ? - - . ' - ? ? ? - ?-? - ? - -? ? .: ? : -
.: - ??? - ?-? ,, ??? -:':-- ?: - ? ????? ? ',. - -? ? ' .
,. ..,.., . .., . ?_...... ? ., , , . ?
''.. , I.: ...'.. '...:?.,. . .; . . ' .. . :!..,. '. . ': ....:: : ...' ...' :-..' .:: . . .. : . ':.L?-A-.t.'-'7' :?'''''''''''''..c'-',"''',--
-
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_
fol
Since
CO
point i thi3 fant, cien 3 in t series
re pare
Let
on a freeurfac3 if we take a
s singlaari es only n tile re
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. ,
. .
. .
. . , .
? , . -,,, ,..I.,,,ti a,. ',c,fn Aitt,i,;11't'' r It. I.,"11 1 1 1.'1'.' 0 IV$.114::, .4:11.1 Z1 1:41tv, r...4
,
' ? ' 't ? '4 4,". ? 44 VT .; not 1c
lete1411e , c.),,,A~A,( tert 5 to .1713114 ,vr?
noce$sa/-t.y, to' illt?egrz.tte #.,:rle U fent (13 4}
; t t*. *)3?, tio4iono wih 1rlart i sfroe Lof
ai Prk,
f3tate that .0.rhon t Q? the, of r qt1 thPric'e
.rogtzlar f r.ori. the. . e 3 : of 0, ?
, .; ? ? ,
tr'.!...1 on GI. ( A) ??:.? C!'cl sa i t '?;)
? ?
tt -"I ?
V 4-? 1?;??,:rx, .4 et ?pc, 4. 4 11-4
:74 A 444 1;1 .4444 4.4 Z . ? / Nt...,.14 4 L. :4. 34,4' 4 4
. . ,
? ?, ,.4?1?1?1?4.,t2 4.!-`4. .44
?:1 ? ? 4 es
to. , -
s
this eqtation. a in zulabiary- n.arztTt 1 3 ""'Ot .;"
olex, the solbation tjc17.,,i2) al:30 -rielei wrtve notion.
141,te
basin- s 4; 4 ctrt. ? 4-fr 4 ?4;:. ???,;?e (74.
4.. aation (13i2) after qtab14. it.,12("L in p he 'variable
- e soll'Itit)ri. of ettation (23,13.) is expresseti ir; terms of the c nflt.tent., hzrpergeo?
metric funetionS. 14(1c, Y X) -1- satisfy the dii`ferential equation'''. ?
. xj?E -(?X) ().
A meta soluti,on of eiratItion, (.13.13) trAttes the form
land, F.1W,00 Table, of Funet?onil With Pozmuthe and Curves4 GTI, 19490
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g xt,iork
STAT
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41?-ta c*e'"Itlit ?Sotrlf" iorv-fr t4., Alerir4C- PrIl 11 n
, 4 A. 41- 4 ) -
equ.,11, ctn. ' n4.1.1 7, ":""i *
1411.13 k x) -1 :07.itta? (1 ?
,
rot,ion. l'';111ft,r1 the fc:),L,1
The f t.)-r.1.3 j:
thAt t 0 7 (x. / t.r?) (/3-0 431i-
-
.4%
f t?.? 4 A. / 4
\ pu .1 0,1%, tr. 4 et4 4 the..1,41 .4, on c.A.. 4.; ?0115 take
. ?
- d the fn 41 is 4
? A. , ?,?
(", riitt 1,44Set!t
- 4". t; ipt...Z -tt 2.1-4 C.2.41 ?C )2
Frotri this it follora that if 1%,. ) is ,*tal then tile (41,,,r4 -kir initial contii tions
4 $
are obtained:
? ihmt n d y 0 41)2 ?'1i 0,041.
= F (x).
This
54 easo Ifas analyzed by il.re.Kochirl. It us now explain the ol rae ter or the
cowittions
in. a general case where aL.
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r
I (4.0 d aro
z
at no int
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IrrrI
Taft
Prvr, t 11'1 C 7_27 -;!:.
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,
ottlottIsto illterrtra Itte a,11. .
I " ZO (.1 x6)
after t44.13 140 obtai.r.
*"'
(, tit )S -
( SO ''''''
I
s
?
_.,
-.,.,..?.:
'
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'
}I ill
,VPLICATIONS TO ?-foTrov! OF A VISCOUS
THAolty OF
C.I`rortsi(,,let-artions of the ..the or of dintensi.ons can render great, assistance in: the
tical solution of certain p1y8ical problems. Here and in the following para-
:r?s we shall , present examples of such an application of the theory of d mensioris
Lot ua consider' f-he pr^1,1 nf the AiffnAinT1 or vorti.eeim in a vistous
...gft41,114 fizid under the assumption that the motion of the fluid is plane parallel
, ,and the fluid occuPies the entire plane. The muLluq under consici2ratiorl is non-
-"
!steady. Let, at the i.niti.14.1 .mottent of
vihere, with the exception of the pole
?t's utotiorl of an infinite rectilinear
tip t 0, the fluid move potentially eery-
0 which represents the trace on the plane of
concentrated vortex with circulation r
Let us assume that 44he possesses aLdal synnietry. Let us desi.gnate by
ti.te.Kochin, I.A.Kibelt and N.V.Rozes Teoreticheskaya GidrotagAdwiika (Theo-.
retical ilyttrodIrgurics), Part 11, State Technical Press, 1948. This book
sontairls a detailed description and solution of this problem utilizing the con-
siderations concerning dimeneic)nal ana,11sis.
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v i h fi.int v......0 1.4,4.. . cosit
-- q
he t r rtami_tio n a t ixas a function
From that
th a p b it o
nt a. t
The dinonsionles combination mat be expres a function or the sing e in -
,2
? dependent dimensionless quantity' -- h can 1--- .. c of the dimensional
v t
paramet
ersv ? i, t. cQn5eqUflti1 w have,
r. formal* (I s evident that the equation in Partial derivatives (1.3) for
fun_ on of t with two independent variables r an reduces to an ordinary dif-
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';',11?75;?.
1ttvt,,b1,4 e5..31'?1' to
roapiowin
(4) 1.14' () 4
The rIt is ocival to zero for the
Litgratiflg the collation
c143.
c re1ft of radiue
For t s. 0 for aryr-radiw4,R greater than
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Let U3 designate by v (r.0 t) tt-tev1oit theTiartic?il..es of the fluid.
ion of the ?..ta 1,.ossesea 1.;.ence the veloci.ty of the particles
e t1?d if$ 4iiredted perrto:1,11 c.,?11,ar i,Pr to tho radius vector ,1ra.41 tr the foiLt ur-lex.
c:onsid.,eration fror t?he
? ,
Talting t) co:15,3,2.1.era',,,,ion the ',1,i,ree .t.1143ioi. we obtai,ri the .
foilfsif1r.4 relati.on between
1.'41*.
, ' . ? '. s? ' ,... . ..N? ?4? ori,,? I. r ,f, n , '.41 50 ,7-4 1;', ' .46a0c.f ? 'A. 4..k.0 V ,,,,,,..',4,,...,,,,t, .- .- .
. . ' . ? ' ? ' . .. ' ,. . ' . ' . . . ' ' ' . . ' ' ? '
?, 0 Cit.:/' along raglitats..r..arti:1 -1.:1?, ttr-e .-t: :... y . .. . ., . .
;:.....,...... ,. .. ,,.. . . . . . . . . ??? ??
. ? .1, . - .-...,.
?,.: , .. r n ' . . ' . n . 1 ,
b...... ._,- 1 ?1,
.? For t 0 there is obtait-ed thi) law goliparnitig the distri?bu,tion
,
that corresp)rvis to a poir4t trcir.tex In an ideal, fiuJ. For r > 0 an t, 0 the
tiort of the fitti4 is 1:,-otentiall vorees are abserlit; for r > 0 arid. t, > 0 the
notion is of the fluid is vortical at each. vi.?,()irlt of the fllaid. Me eq. (1.7) gi'res
the ialt*t ct'terriiriff, theca?o;alzato:1 .e. d'4,,I,145,1.011 7 Of
icates that. the of the vortax. at each point irIcreases the C0r,Se
of
time fro -Pero up to a xiur equal to 1-2? .ami thereafter tertiris to zex'd.
2nra6-e
,te t:,,he eq. (1.3) is linear,
then '111"oceetliTIZ fro:; the obtee.irlal, solutio74
the lagl) eitiorl 0 a, int vortex we can construct b the riot-1'10d of superposition
P:il. the ixti?n Or the ., , .
'-`''.-1 l'i:r3- Preb1641 11).' theeirsYllullet'ri Pl'?t1?/.1 rcili all'Ir Init''Ial distribution
.
or vol.*oi ties. . -
. or
th
i
, 1 Sao-tic:III 2, ,*xaot Solutionst.ion5 of -$ tion of a Viscc)uat I,n0o_1r
.....,,. ertlaib....,1?e?,,,,,, ;
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ght for quantitit3 will t- the pro
=tic press d o: the plane and
us 5tu:y the 3o1.jon of he
r*
of S 0constant
LA
where p and q arc cc st
In the of 11 on it
ions eons of h introduCei quatiti'3will he functions of only three ab-
stract parazet3T5?
en the sought for functiOfls can be ropreSefltGd in the foll form
0), (1c
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egss. (2
?
The rattober or independent variable,si reduzea ifW4.1; 3,:',Isurae- that
P ?
?,... . _ ? ...... .
??
This dO,ndit.iOn ignifi Airtat '''the-t:Iiension Of. ?the' o-Onstant -A is represellt
a ertaill, power of the :,:itrionsion or thfa ? cot!: ficiont,' Of 1.'"?irtfrtr4ati- Nri.sooSit
r ? ? ? ' ?
?
BeSi4e3.,.'th14..a3WaritiAiOnt flIrthPr' 4:1SqMI that the motion under 'study. poSsesses
?
axial syttastry., so that the! variable X is norwss,mtial.
'From the aSsumption5 madf.1 it. fo11ow.3 that for the sought for 'quan ities the
?
foUowir.cft'or.,v:Las
r -(2,2
,-tm-- /0) ? ' '.. r .., A :-.-.-- ,---, % ,i, (0) II (0)?
.. ...
? r
. ,ThOSe i40,141Dttaat i ,k, ' eititabliah adi'...Apeild elle a of 'the vetloc 4 4' - tio'ld as-lci pressure f iela
. .
.upon the variabl.t, .. -
in this-oas, we obtain from the egs. (2.1) for the 'four fun tiOnS f
a syteta of ordinary, nonlinear ..d,ifferen-ti,41 equations
.. . ,
.,
?t1:; etg 0 + etg --
-
I tp (tg --
,
After 11.?tlation of the function Far etr-tn ip1 ransf()I'ma ions we ob-
ressions:
+et g 0) -1- (ft etg 0)' 'c-21.1' +2/1= 0,
ci (';i1 (`,1g 0)",
= tp utg 0).
.}3efore stuckving f-be solutiorits of the system of eqs. (2.4) wi3 note certain gen-
.
priosii)erties of the cortsiderf)ct motions of a viscous flitid.
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on; o th ii of t1u'4.. upon.
th
r 4,11.
. teiiznat by th vo1z disch
mentuxr throi a c1o?d urf L-0 0 that
.1)
Zr: he ? of contract- of tho r.,a,,,? S toward the poi0 into a Paint we obtain'
tht fact that uheq iti3s . cr depend only upon the coeffici and uo-
on the constant A., he inion 0 ih is expressed by the dimensions of v.
Since the dirensiOn3 of Q anti v ari, Lidpefldeflt,. then it is evident that the dis-
charge CI i equal to either zero o init)'. The dimensionality of J is expressed
lx, the dimensiOnality of he coefficient v therefore the quantity J can be finite.
Another situation holds true in the case of plane parallel motions If for
_
lane pt%rallel motions the velocity field in the n.tirc Plane ,4ePende only upon the
coordinates of the Point and upon constants which Possess a dimens--- nalitY that is
_ --
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del 1 0 o th 0,t
i 1441(511 th e d
in polar'
coc>rirtatOo hk, Qrrt. ?oz'u1awill be
(In,
t4hi3 ca34 r is the raciiw3 votor in the pl,a,ne
t cons df,'1, ' 1 char . e
C n . the ' nmotions
di5eh e rrapondi . . a be equai toro or
tv.
TbI cir nce .it azt anti a rrbr of other aifthora in the Problm
act solu-
o the 'o, of i ? ?. two p to obtain the
,tion o thc aviertoke quation3 by wq oi their educ,i n to ordin
ontialequationsLet us noi pass .
over to he coziicration of the solutio to the ystti of eqs.
The first of the
s. be represente in the -o A
?
These so1ution are expounded in clitail in the book: N.Ye Kochins 1.A. and
N.V4?--Roz Veoreticheskaya Gidrom;ekhanika (Theoreticea. Hydrodynamic), Part
State Technical Pre, 1948.
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sin 0, 241, + et 4)
s (2.
i th constantof in on.
The fu 'Pn'i(0) efin)? he , r A.1, on the ponerts of v1ocitiea p,
mdicx1ar ,,o th& pin of th-1 ,
rid&i.
pal
n t ,lifficult to that. 0 or 1 , conat th?t1 ". Of
4 4 AVOS or -la ' o o tho Ict - ancl ,
. e dot.; i
elatio
n
C
io ( s
The con ongives field for vx corr nda
r the o
to a rectilintar vortex h coincUea'4th thztxi of a
one hequations 01 're atif led, to any '.'e1ocLty
of he type undir cons 1 rat or we add "JrAtb.1,14.4 .ol
. c rom th rtiLinc3ar
,
sin coos 29)
be
hen the eq. gratei thr tii, after whi.h the solu ion of the
rob ?e ruo to the integration of h following quation
2 0 Sin 0 OS 0 M C? COS 0 + 1.? 2 41?
been obtained in another war by N.A.S1eskin. ehen77e ZaPialek
Ta;..1(Scientific ,10tell of Moscow State Univers 3r P No.2, 1934.
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k., .
It i..e net dirr'..1..ott1t to r. t for thil3 3? Gn '''"rithe ca a() or i.,AIfr3* ttli?, ,
ieharg,; Of. i'aui,4 through a s't,t,rfa.,e eyl,closini; the. origin of coovi..i.nates ils secrt.l.al to?,.' ..
. .
._, ,... .?.
' ,'' ' ? , . ' . ,
? t , 1 ,, zero, but, ir,1 ,t,h,elt case. [Al q +,,,h.e ?,,.:iia.ch4rge,:. ,
.
'' . 113 ?4:1111.t1 to
ill, it,y. , , . . ,, . , ?
I For thet flolii of the pr.o,,j,...ct.i,.011? o.f
- ? _ .. ,. .... ,
. r ,,,4,co*Z)
..i.', -i:tin.'4-14
4 2 thfl momentum upon the a,xis syrzactry
,
througlI zi.ny phere 'en??th i:::enter at the
,':::oor.....tinates the rollolfing forzu-
7 s tAirtla 1)
}t )
? Fig.20. The lines .of flux for a
?
of zero power and for finite
. ?
.3,4-1 -6-c,,:ixtts
-point ,at the origin of coordinates.
For this current the eluati,ons of
r
tj
7
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, .
,t),r.' h? tQU of th* lin of rittic: .:i 1nii t in
Th i
h .ca otfar roa1 to thg :Li 3f fiw dU .
t f.11 motion along ? 'et ?h ,. 3 0,lin of 11) . riu vito
injwr va1u6for a crtain viu or ; Id OY he fOnow r,Ig
be ir tie wirctto
n of thacL 41th fin1e i (ioztWe ca .
td=1V'point to a Idlole 3erI of o1utionsof
0) -lett c'- ,64,-.4..
partithir val.- O tho - nstant '
21 ,,,,,,,? ,', . _ ,
, $ ,
,have 'oflowing soiution
See V.I.ratyev, Zhurnal acaperilitenta Teoreticheskay Fiz?11.4
arid Theoretical Fhpsice) 11,1 1950, page 1031.
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gr"r"r?,','w
?
' ? 0:- aq' ' . . 0 rtxtucedrto th(1 fi)1.:144 11 0 -
ioR
e.,qs . (3 . 6) an!I - the boundary ..:-.ort-itt tot-1,-$ . , c.,, an be con3i*der-e-ui' a,,,L. f or -
ImiaLion of tho --,rol_er of a 1--?-iounia 7.1z)rer iik I?t4 raen,tio-,1-?','-s. f ot.r. ale 3 c,lut ion of
- t.`; i ?.
this p,.*oblert (,,.arulot, ,lepenti upon thY:z.k, quantit?y --..---.= ,, t vala.,,ch fi o
res in n way in
., thAt eq., (3.6) and il, 't,hebottn
these partlal caz3C3, ani conseq'uefltl.Y also in the ger;eral ease the lagoverniric w
the dwtp1x deper15 in an e3efltiLLi: way woon- the .,properties of the .ini?
al- die wrb-ances Thereforo, in order to obtain the asymptotic laws .governing
amping :with the aid of the considered solutions, it is still necessary to employ
:
either add.ittOrA1? Iwpotheses of a mechanical character or experimental. data.
6.The problem of turbulent motion ,in an aerodynamiC pipe. As we, have already
jr4icated, the investigation of isotropiC turbui,ence is connected with the study of
1,-6
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A
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+ , ''` ono c U. 1 , Ile:, '4tit t'., ,L1 * e al ,Ivies , n royaLLc,ttie:a r.4, fi,. ? .
: I o r the , b' en of 1 d :4, or.aae/ t ur ' en totion 1 ? I-
7
presslble fluid 'Jae: at th :a. -t gi, rii A" 4,:r ` SiV With . constant, V02,0 itY
S ior sake 4 or Ilpa. ci , ,,,,, ?h tl, ta s 1
Iatt.e 4 r
. formed of o Ongiu. t - 1 c 1 each rel-
..'ativ to the othersa ..
r , .ra...
perr au: to the . axis.
io the 2; Ices ..6,th. ::.,.,. IteraiI for:..
Th.
0
e, ::.'.otion: of ,. ,;.1,. 4
., r. ?,,,,. , , r .
tme. x .,..-4,3
.the oi1owi1, v 5 4: k., n oi .
on.I ss . coritic of the.-,otiot 'por1.. upon two ?ret
pill
ssue that for ufficiant1y iare 1..ta., of it
r.
hetuebulent rot1or is s
he x azis the develoPt;ent of
n this ease the caract,ersti
renplanes perperaic-olar to
s iiiferert. only in r-A-zase.
bnn
?efficient of correlation f n1 h eper upon
e
a consequence of the assumption that for vurficien ly
or this Paramet r becomes nonessentia
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rr'',..?=4,rrrmw,7,1virT-?St
??,
_
ate
-4111ita.
?
? ? ?
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? , .
_
?
_
? ? ??? ? .
sett. lug 0 * we obtain the following:
Ttle es 4.2-#:?) 8.ive t'he law govert-lini; the tktrI)uletlt, 1?)talsat.i.ons al
the axis, of the
t1Ie basis of .a ,iat.a obta4,..,ned. ta,eroil:rn.a..-dc
rwopo8e\-3. he ex4)irica1 forzault1 of the follow-ing, form
(4.27)
er A ati41 a are con
For a,.eeraent of the ass. 4.2L) a1is riecessa:ry to set Tor
, .
we
have: Ay
ut
ulent motiona with larile Pul$ationas If the Parameter is not esse-
tia.1 then the coeffi.cieritrs of rreiatior rinair constant v-alue for >:
v t
zust. The influence of tine is roluced to the oha.f.-1.1,,,Ye of the scale for r.
In the solutions under eatasiderati.on 'the change in the scale is determined by
he following relation
? (44.28)
0 )
or* ?
From the eq. (4. it follows that forr x const., .e. at a fixed point relative-
.
to the, lattice, the scale 1 is c-onstant ir time Besides this, the eq. (4.28)
mdi-
cate8 that this scale depends 'upon the velocity U.
In the cited 14vrk of Taylor is'iresentai r..ertain entperimental data which does
Taylor, Fr ce Soc ol; 1510 1935 and A0 vol. 1560 1936.
16A
-
we.
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STAT
?
?.4
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.r,
itot
Ionb I ha
ery cab to the ce of iar LatiOfl3.3
value 1ergQer an t ran be-
h in these proCe33C th amn role
te 0 :Tir,kd by ,
'h proprt, of iner.iA or the fiuii.4
111 rope
Tr 1- ,
t On 1 41-4 :I: in r.,:.?eath ,
11 .
rer;
. , o.
s ,,,,t,-
.4.4r,,,,,i
e II W 4 We a $ WS& t hat ror ufficj(t.
, and
lit or
cess of I$3
1
o correlation
50 thatbe uence of upon the oeffi.cie
reducedto a chanZe in the scale for the
ice 1. The propert:, of viscosi
exertsty
an influen e qiatUtie5 b ar11, upon the henor
enOfl
To the quafltitY 1 we shall riot. attribute any concrete oi.etri or mothanical
sijriificance. The de tion of I can. be considered a an additional hypot.hesis.
there exists a
nar4.icular1 if one assuzes that the Invariant A 7( 0, 00) i.e.
,r Further conclusions rely essentially upon the assturiptions included ir the eqs.
(4.29) and (4.30) ApplicabilItPof these assumptions for the description of the
actual phenomena require ditional clarification.
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0.41
where
con
nal
crt. If e f ther ate th,
ol or). tor t
a ,
? The dimensionless quantities kc
from he gnerai pro r he lunation f(Y) t fo11.ows that he quantities
.1ta ane -0 are con to-.
Integrating he eqs. 4) ai (4.35) we obtain the. fo11oirgrelatioru
170
STAT
di
dt 4.34)
b di
i di
(4?35)
o riot aepc pon From the eq. (4.33)
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The d ion
the vojty of = tLtrb
o t,ire.
at.
A.N.Kolmo 0 0)347
Slt 101. MI, tlo 6 , 1941
-
- P t. A el etel amrkr CNC l'Atie.leri
5SZ-574'1 :VaPtart., 4,
30) and (4 31
ained which
vero, in paragraph 8 it will be shown that the a lair. iCOS ( 4 29
are traitetory o; but if h /0 hen s are o
ifferent fro 57
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Tt ,ws oVerflU,. ,
t
L
?RAV?
ther w 4n,
(i14) iuri
resert,
,
thiCh
arc ttdedth
ox
,
_
rs ? anc
ertdr:artitol: 1:17 thec orz:
f ,
,.
,.
, cn, sider ,I, means
tt, 1-kr ad 5 ::,ol,at, rn.5 01
'3 'Lae *1-,,
tions of he
ere I aroci b are oerta
In consequence ofofthese asmutptione eq. (4.14) 1eai to he relation
r aetall more exact solut,ions or his equa?
or conatant
us rn'
.consider
-ion Differ iating the eq. (4.32) with r
he ?follo!ring
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ctio with raticm (4 3t1) wo ahafl tutio t.b foiiirOa
(N)
arc 1itr1yr nden.
i1 me ,ndepcndent linear 'c1at ior withco a tut coer
t
there 4,4 a er ,
onwit.h cota coefficients?
Cir,c r (..:1 t.herthese rela car
In the firstcas all e coefi ciotthe roiation(L.38)
ere k, c and are cert. 4 con5tant liw;ber. 14.1e systez of .e1a4.4ona (4.40) gives:
4-
/ rnv I
v
ti-to 2 ?
Cor3entir there 4 obtained corr e ely-definite law for the ariatio of
ctio Is of the coincides with - law (4.37)forthe
and b a ,....
J1.11
'a this c-abe only- on equation is obtainea for th tro functions r(x)
It .3 i1eit he in this case the quantity/i cannot be finite and a.. ,Irivarian
equal to zerof since in the contrary case the equality (.4.31) would contradict
th,equality (4,'41).
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Cie( 11,
aow izf
f3
.uu st
r
? be
i$ 0 '14?0,4' ,ation
high. Snat.1:1021:5 Zr1 ?
3,
on. N. but, for the cae w Obt. O which 14044'11 ?
is
,
the ?
'
ino?3 . ? It.? ion 4;.
Since the functio xf ? ai f ar 1 e ? inde ,it is eiient that
pre ns ir the square brackets becor, ; hrce it follow :
Id!v
=
dt y dt
-ere p and q are cor at of rtLrat.io.
Substituting the. relatiOns (4.43) in' (4.32) wad keeping in mind the equat4o
4.1.1? ?,,,L. find the aril li_Or4 for the deterldnation of h(V) in the followi .or
-it WV' t -...".2.-.--.. - -
bLa
d
2.0n of the funct? on -.7. h ir
- --' 2
b ers o
herefore i foilow3 from the equa.-
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-
_
??,
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_
? ?? .? ,?? ? ? ?.,? ??? ? ?. .
? ? ? ?? ???? ? ? ? ,.? ? ? -??? ? ?. ? ..? : ,? ? .. .........., .
: ? r. ? ..r.? ? .?.? ? ? .?
:4* L ',t1*.i.at. q,
? ' - ---, -
ti).1.3,13 wi,?haVe, ro13.1-4 ixy the case ,A1.71..i.er .conSidel,a.t;.i.on..the co,I.tr?;)-)e't,efott,444i of eqs.
r ?
r
* ) 0 ,( 4. 4;133 ) WI,.( 4 .44 ) ,for.. the de,4?..>.e,,r1d.r4.ation.. oi. h and f as flinct.,-1..c)ris (..1.f the var.- L
.. ,.. .
i 41i;11, 8, : , and . fo,', the. ;,:leterna.tion ? of the cluti.eit...1.ties 1 and b as fil.nttiorlf sof the '
. . .... . . . . ..
- ,... .
These ecptions contain three abstract const,ant 4)t a?,), :t is not- diff-
cult to discern that one of these coT.Istaxts is nonesset,ti,al. In fact, the transfor-
wl.,ere X is a cert.a.i.n constant, rti1t-tr).1, cat' o-- of the
,
, .
it'ill'Ini,(.1.eter.rfirine,4_,- 'Scale. 1 by the ,..is
t:ors
? . ?
t...r.'Isto.rmation. in the f..,49s. (.4.42.) ' l'*1) (+.4A) 41)tair' the:- Sar''''3. 7 ?
'
* ' r.** .1-rt- 11. f`i)110vi r?
,but trzmsformea ,talues ar.4, ar4 ?-allic.,h 3.re -Je.-..ertane..
, .
?
?FLY:in.c of a definJ.te value for
lection of a definite scale for 1.
On the basis of the second eq.. (4.143) it follows from the condition for the
=pine of turbulent pulsations (b --C for t oo) that a ',O.
or for al /C is equivtlent to thf.,
b -O.
.43 eqs. (4..43) can. be integrated, arter which we arrive at the following
r"urther we also assume as a p. sical condition that 1-- + for t oo or for
'
relation
7777-
2p41,1,,.*17-1;
(e --Zs ? PCU-
where e is the constant or imtegration? here 2a, al. From this relation, from the
J- ?
ineualiti
L.> 0 and from the condition 1-ioo for b -0 follows that 0.
Let us consider also the case ai ft 0, For the case al ft 0 the eq. (4.42) pos-
0580 a unique solution s?fyin the condition f(0) - 1.
175
? . . : , ?
.............,
--
?. ?
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The 61.)t i- rePztort 6 tO Ilat
n eh iet IT f.
w
,..,tiie c? .. ::.
?,,? ,? se ,.
,. , th
an the moments 0. the
v hav 1 / 1-77t :1.,aus 1 ng Ito
bri
order exerts an tnfluerice upon -? e coeffioien. o orre1a4on , .
Ugh tion-
.he eperdcte of he linear s.aie 1 upon 4 .
The funo o h() on the basi of the eq. 4) ( 0). s eaSilY eXp eSSed
eh the fnrictior fi Integratinz. the eq. (4.44) arid ke and . he condition
-for Y'' '0.;-' we obtain ?the'.-.Cp.31
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0
thee it 1 cv,Lent that the h( Y') vanishesfor
therefore th oment of fo..th or(ler b/1---i)oss sse
1.
rorhav A 400,
or Ly 0 lk
Sof te
:llffrtfr s4ero
are
oents of the thiDd or4er are e u . to zero1
Thus, if zervoz.ents of th th order ar
hypotheses that are ee1 the equali
,,integral Aequai zero or i1riity are coequent1y .neonvenient for the leter-
dnation the ie corre rere with the equality(4.31), o (ii) for
he quantity 4 varies with the oure of tire and therefore caznot be con.;
51 ered as the haracteri5tic constant
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l'SRI
where C is the 1.-"on ant of inte,gation.
For p / 0 aril ,,,he general aolutior of th 4. e repre-
sentedn the followirci form
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the letters :b4 arid t4 desi?lnatct the :klonstant of J-nt; grat-iono which .I.)osdess the
mension. of the squalve of velocity and' i;ime .(14 posar:u*es the digiension of "l'(.t.ngth).
,
, ,It, is rlot diffitlult t-o see 'that, the .law governing the degeneration of the tur.-
bvtlent motion is determd,r.Aed' essentiallY onlY by one,' c'onst".dnt 'ince b* 4111 1* p
lay
, the role of constculstta,ad th constant t* clerxtr.siis only upon t),,,e of
time rec'honing.
- Let uS define the Re:Tr:4.014s mbo accorair to the
ollovrt
it foiolet3 44
1.
_fience, oi z then fl-0 for w -0 or t oo; 1f. then fol- w 0 we have:
1.e. the:t111.r.,ber nosiA .,
sses a fi4+' e 14 .0 .4.1 value.
evident
re disrezard the moments of the third order the behavior of the
b Iry t forl 0.1) possesses the same charaiter
The laws obtained by lis for the degenel.ation 013 isotropic turbulence (4.50)
, and (4.51) are 1:,:he consequences of the assur/Ptfions (4.29) and (4.30). For the deter-
.
rination of the constant (t we need additional hypotheses or experimental data.
? .
"i7x*osl izhe 'eggs. (4.50), (4.51.) and (4.52) it, follows that for oc>0.1 and when
00 Lne asyrTtotie. Ne.1,4
-? which indicate that the' laws ux1' by us for the variation of b and 1 as functions
of the time, taking into consideration the moments of the third order, tend aymp-
totically toward the laws obtained by !Carman and Howarth when the moments of the
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however,, lir ..in otin
1 ca e trot,44,on
z14. fiuii in a
p and ix hnel, e erictowt-r turbulent motions for ,hjc 4:lie air god otioxi
..".t a - h'riow are called oaiy tintio1.
Let us conjter th ""Oblel,D the, toad:Jr:rbJ.ent C/1'. e,>7 sl le
velcit1ezi jt,ei ao the azi n p.111.1ri_
vr..0
e arbi :tuci s :lot ocna
e ''. 4 . 'v'et . .?
. 0-nZ ,, % 4 1 0 cr'
. oorlinat -4-. 2,..on -
-4 4.
), 4 po.1 v..
he i velocity able p YIS 7,41011 e - ,..ance r -
der tzonsi. era e center of the pipe.
there a is the radius of the rasan value of
4
icradient of presu.4.e
along the pipe, and is he tre5s of 4" I ion on the
of te pe. 1l the
quantities are functions of the foliowir two parameters:
5
voap .2)
epresenta the so callt velocity of tar.,cntiai &tress on the walls
s onless quantities character r the properties of the ro-
;ion in the l.rge do not depend upon the variable r and consequently are deteriinei
niy by some Ronolds nunber.
Let us designate he rgnitude of the velocity of the averagtyl tion by the
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tter Uat he veloc..
lejor1 it follow thz
Frn tho.the
Th quart,?,
o ve1L,
of the
the14.,p
ity ISexplained
otIA or le-v-eli ve1octie of the observed rotIa. and leads to
"transforta:., on of the ie tic eoriof the observ moti the energy of hea
ew of the kineti ther or mact . tc proper
aotic molecular motion which enable the
the vesence
gover 11, the conservation of ener-- is expressed in the constant7 of
the wwr. of mechanical -en of the observed r;otior and energ.r of molecular notion.
Both forms of energy can be considered: as component*,uf difforent fr, m51 elf. rAchAn-
al entir-tr. If one disregards the intra-molecit
eosit7 is detorLined bythe mean kinematic characteriSties of the state of the mo-
eeular rotion and IV the property of the inertia of the mlecules of the finid.
orderless turbulent mixing in relation to the averaged motion is analogous
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ors or the Itolecular
artia
(,otoAy-Tu ffc?
:the-..th6.,r*z.?1 ; pro
rolwzies Qf the fliti,,,
J*t of es .
t ioz., a o v7,7 s aLi .41t
of th averae
. o rii,:;.' on .i.,', 1 .1, ..k..t3
. .
Since the quantity ET is- con,stant, then the relzitionship
In the caae of cylindrical symmetry the factor 41,,,r2 is replaced by 21'4,44 and for
plane waves, it is a constant The relp..tionship (7.13) y1e3_d3 the integral being
sought; it is valid for equations of notion of a gas which are more general than the
egliationt (1.2)
For an ideal gas s (y 1) p By placing this expression in formula (7.13)0
_
- replacing v p and p in it with Vo R and P according to formulas (1.1) and irAt
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esoaiitiNOWkil
'',1,(44new
4
sQ.
'o:gliakI;;Viqikot$FAT1204,?...
Point C* consequ n
of sy,
e and uniqueness of
igorously proven.
lindr44's41 and plane syTCtry
By using the solution of equt
ions (7.10) and (7.11) we ea 5I].7 f
sPherical sytnetry
.9) accordingt? formaa (7.14) from equa
and Ras fune ions of V. In the ease of
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7
,
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p6,1
STAT
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riff
VeloeitY Diatributioa behind the 3he'zk
3eri-Crk-1 Crd'r-S8
C-Iirdricll c-ise
rlane case
441,
?Density Distribution behind the Shock Wave
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'OA*
'
01
ature distribution. behind the Shock Wave
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Val
54 scw th Wt1
go 56 stows ttarceril,ture
1t t(fla
e unbroken 'tine ;r;1ve tA r1cL
?_
CIA, 0 'IS A ``)Ildthe Pwivrt, th 4 ?
v`1.-? -t ? tr.,rrr6ditl. ,',211a,P.ntitie$ t)ehiti?,1
,
varlakos.on ot V P p0114 $-.,7 ?-oa`CVI'l wtl 4th? r 5 41::t
by the i',-,rmalas (i74',1) whe..e in pl;..c#:,!. theyve?, t.)1kr ,C 1,1.1e it
ton in t e ririz or 4, ',Tod t t,ttd3. I ',. tb :t; t,
COnne ion bet,tieen. :And t *at! clbtf,kilt:
w
presoure di,:at,ribution
* r I .272' .1
t v
(4._ ,-.....-,....? ?,,,.....
81'' r
...-...=,- .-----... .,. ...._
4 -tv 2t'rt , .0
''''''fff) 1 p i i' 1) ' 1
i
For th terriperat,xe be1iP.,3 tb hock wave, we?
P2
102
From the forriUsa which give tile it is ? 7-matter to Le.....ae
as ntotic fomalas for v arnd p red t-em-,erture T close to the center of tile
exp1osi' in. the cse 0, r
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The re4Ationship of the con tants k1,. k2
Near the center ot yietry the velocity
constant. Under conditions o1 pproa h to the center o.4. the explosion, the density
very rapidly tends toward zero and the temperature towards infinity. It is easily
seen that entropy also tends toward s infinity, Close to the center of the explosion
'Large temPereture gradients arise; to this fact the property of heat conthictiv-
ity becomes very imPortent, If heat conductivity is considered then the tempera -
um, obtained in the center of the explosion is terminal.
The mass of gas scatters from the center of the explosion at r = Owe have
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"
i ,,Iff',44A
44 illi,t'4,'
;'.2P4Mnf:',4g'AA
pressure in the center i finite but ten46 toward z.c?
the ct':?,se of ,,,,.strong explosion in a gas
'=
the Formulas (7.1,
", kyj e
r 3, 40 ) ? 4 44- '4 4, '4 "
A.,
which Give As,rmptotic Values for Denit, Pressnre and, Tempert.are
Close to the Center of tht. cpiosion
a) Spherical symmetry,', i
Cylndrical symmetry; c,1 ,T-4,ane waves
in chi) prior to the explosion there is finite pressure, thereimust Occur a motion
f the gas back towards the center of the explosion. We observe Well, this effect
in the case-of underwater explosions where repeated pulsations of the gas bubble
occur.
ks. it of the scitt-tering of the gas rr m.the center and .the. drop inpres?
aura, an Archimedean lArt force set,s in Which causes a floating of the region of
..
turbulent motion or the as in the surreunding.atirkosPhere. Aeoonling, to data in
_ .
the photographs of an atomir,-, bong) explosion Ne 14' Mexico the vertical rate of rise
of the luminous nucleus was of the order of 3, sec.
The effect or the constant Y on the distribution of. characteristics of the
motion behind the shock wave front rOr spherical sYmmetrY is presented in Fig058,
-
STAT
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41..?
t )1.4: tUt tenc14
* that in tho 1,1$0, r 3t ng cxpLcton in gi71.s
7, ,I--
T)-... as:.
e .,....?. 4.
*. Vir:Ir
4a, ..` a., i. ...P..1"..? 1 4, *...i...:717 eratre.
Ch.....GiVe-'.
, ,,..w.I o the Certr ? ion-.
,. ? 1
3th0.?.,,,
try; "ictr? es H-.....
.4:, ,z.
of the gas back twrd the . the 1 'n. etc observe we... tiis effe t
in the or undr4tore ons 'where repeated Latior of t gs bubble
,As a ruit of the catt ring of the fr n .he center 3nd th drop in pres
e, an Archim n lift forceset5 in which causes f1oting of e region of
turbulent motion of the gas in the surrauntLng atuo5phere. According to data in
e photographs of an atomic bomb explosion in New Mexico the vertical rate of ris
of the luminou nucleus was of t.e orir of 35 co
The aft.. of the constant y on the distribution of characteristics of the
tion behind the shook wave front for sPherical symmetry is presented in Fig.58,
59 arid 60.
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Mk.
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...r.,,IJ''..0114,0it?ho in ter of the chirge cnergy
?
enters into the zboveli'ived
on' The zion?frenjona
curve's iz!ependent of the
tiozial t it.
.,..teriotics of
or the tota erst7e,y we hv the formuIa
2 a2
4r,r2dr in the phricai CP se,
r2 py2
2 n 2 dr ir the cylind
first term the -kinetic energy of he gas; the second term is the he .t
the . By introducing non?dimensional quantities we rind
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4
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0
feet 01 t.o Constant y on the
behind a Shock irlave Front for Spherical 3nncry
Ornnt......9.1.0
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??????
Owe.
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4%04
AWAILk4Ma
? (r.,-L r rl:',;?Wif
rtios heat conductivitv can ex
dC) not
hen the cefficiet cf r3ccity nnd the.
the,equations of gasHmoti
be noted that the coeffieidnt of heat cond-activity enters
sereeor .emperat
In ev,11.uating thetot
eary to keep in mind th
on radiation.
energ7 re1eaed man, atomic bomb exp1oion it is neces-
considerihlA part of EhR 4
AnPrmi
.1:2
_
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n tt the tolt11-,1.-ortat;Ire T witlt the- telpc..!,?,?n of.,11,1,z1tion?f(forf$trk.to
?
siop325.cittr
W &11assl,00th to be KlApayroil T)1, there will, enr ?
'tic)
The r?,??!....,. ?-?:,11. i" ,ction of the
b) syr try sirimetry
1.
c\ z 11 arie
me t X7

x
co fi 4.,enttotx aetwaly, but he rt.ltio (It beitig? the -gas constant,. : ?fts .
? ell.. 'V3101414 the:. COtirti.ierlt0= ti ._ . 704, dePEtna on the tersPerzture. . Or'dinarilY theY
. . .
re ..t A as, Ir.of).prti0nl....a. to the telip(Irtit-.).1.re T in. some .power (mpst , of ten they
?
. taken rortiorlaL to a the$rf 4.1.re constants ).. We shall.asstme- that,
., , . as..p . . ., . . . . or ? ... . ? ? .
following ne dimensional. eons alltS enter into the
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,--ensions are egtvill
tt
411d 1-1'ir,,"?),
[.?1. !.-tr"2,
In
?'' that l'1',e Ir,t-iti.tr t),e se;Lf rn 1/36.
STAT
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.. . ? . , ,
- , . . .. ?
' '',.talt: cOnstant C.::1.s?deterr.ti..ncd, 'tv t'he riph . side o
', . 011 .1:110 ..s11001*, .waye. (()?17).?
tv,erywhe.re?-In .what. f.ollovs. w shal.,.. pot 1 c. .7,. !.. and s' 7., -Z. . 170:1,62r,i,..pri the :In- ,
.. .. . ,
?
.
.: .i3O'cir .2) (e. arld (t) . ''' ,..,:-.. :-....,.. ? ? . .. ... ... ???,. .. .....
")8) ,'. 29) ' we eatAillt trxryross P. P and 'F' by: an of the . .
.......;-...t.yirailIt ....tomitia's In V. .
? .
:11.13st :i. tut o.n or. these: ,,,J;:cp re ti s '1, on s a. nto 0.,