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

Document Type: 
Collection: 
Document Number (FOIA) /ESDN (CREST): 
CIA-RDP81-01043R001200180003-2
Release Decision: 
RIPPUB
Original Classification: 
U
Document Page Count: 
405
Document Creation Date: 
December 22, 2016
Document Release Date: 
September 25, 2012
Sequence Number: 
3
Case Number: 
Publication Date: 
September 19, 1957
Content Type: 
REPORT
File: 
AttachmentSize
PDF icon CIA-RDP81-01043R001200180003-2.pdf181.29 MB
Body: 
Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 R STAT Next 1 Page(s) In Document Denied Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 L.I.Sedov C FHOI3 AND D1)iSIONS 14ECHANICS State Publishing ikruee for Technical-Theoretical Literature Koacoll 19 54 STAT4, Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 -'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 Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 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. Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 ....,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 _ '??? - Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 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 Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 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 Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 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, Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 :;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, Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 , . , ' , ? ? -. . .,.' ?''.,' . ?,, ?,.. ,. 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'-',"''',-- - Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 _ 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 Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 . , . . . . . . , . ? , . -,,, ,..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 Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 g xt,iork STAT Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 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. Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 r I (4.0 d aro z at no int Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 IrrrI Taft Prvr, t 11'1 C 7_27 -;!:. Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 , ottlottIsto illterrtra Itte a,11. . I " ZO (.1 x6) after t44.13 140 obtai.r. *"' (, tit )S - ( SO '''''' I s ? _., -.,.,..?.: ' Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 ' }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. Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 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- Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 ';',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 Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 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?,,,,,, ; Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 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 Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 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. Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 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 _ -- Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 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. Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP8 1-01043R001200180003-2 Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 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. Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 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 Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 , . ,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. Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 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 Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 A Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 + , ''` 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 Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 rr'',..?=4,rrrmw,7,1virT-?St ??, _ ate -4111ita. ? ? ? ? Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 ? , . _ ? _ ? ? ??? ? . 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. Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 STAT ? ?.4 Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 .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. Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 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) Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP8 1-01043R001200180003-2 Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 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 Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 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 Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 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). Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 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.- Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 - _ ??, Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 _ ? ?? .? ,?? ? ? ?.,? ??? ? ?. . ? ? ? ?? ???? ? ? ? ,.? ? ? -??? ? ?. ? ..? : ,? ? .. .........., . : ? 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 ? . . : , ? ............., -- ?. ? Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 STAT Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 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 Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 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 Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 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 Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 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 Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 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 Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 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 Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 - Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 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 Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 esoaiitiNOWkil '',1,(44new 4 sQ. 'o:gliakI;;Viqikot$FAT1204,?... Point C* consequ n of sy, e and uniqueness of igorously proven. lindr44's41 and plane syTCtry By using the solution of equt ions (7.10) and (7.11) we ea 5I].7 f sPherical sytnetry .9) accordingt? formaa (7.14) from equa and Ras fune ions of V. In the ease of Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 7 , Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 p6,1 STAT Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 riff VeloeitY Diatributioa behind the 3he'zk 3eri-Crk-1 Crd'r-S8 C-Iirdricll c-ise rlane case 441, ?Density Distribution behind the Shock Wave Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 Declassified in Part - Sanitized Copy Approved for Release.2012/09/25 : CIA-RDP81-01043R001200180003-2 'OA* ' 01 ature distribution. behind the Shock Wave Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 Val 54 scw th Wt1 go 56 stows ttarceril,ture 1t t(fla e unbroken 'tine ;r;1ve tA r1cL ?_ CIA, 0 'IS A ``)Ildthe Pwivrt, th 4 ? v`1.-? -t ? tr.,rrr6ditl. ,',211a,P.ntitie$ 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 Declassified in in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 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 Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 " 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 Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 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. Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 Mk. Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 ...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 Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 4 Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 0 feet 01 t.o Constant y on the behind a Shock irlave Front for Spherical 3nncry Ornnt......9.1.0 Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 ?????? Owe. Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 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 _ Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 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 Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 ,--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 Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 Declassified in Part - Sanitized Copy Approved for Release 2012/09/25: CIA-RDP81-01043R001200180003-2 .. . ? . , , - , . . .. ? ' '',.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.,