IOS Journal of Mahmacs IOS-JM ISSN: 78-578. olum 3 Issu Sp-Oc. PP 37-53 Mof-Zro-Fourh-Orr-Cumulan-xpanson pproxmaon Mho for Invsgaon of urbulnc wh acng an Mxng Chmcal lmns M.C.Mshram an Kr Sahu Dparmn of Mahmacs Laxmnarayan Insu of chnology ashrasan uaoj Maharaj Nagpur Unvrsy Nagpur-4433Ina Dparmn of Mahmacs S. ncn Pallo Collg of ngnrng an chnology Nagpur-448 Ina SC: h govrnng quaons for h scrpon of urbulnc wh racng an mxng n-chmcal lmns scrb by. S. roy an.. O rn ar consr. For convnnc w consr a urbuln racon mxur wh wo racng an mxng chmcal lmns an. h racons ar rrvrsbl sohrmc scon orr an h yp of racons s + Prouc. brf accoun of rvaon of h quaons scrbng h ynamcs of h nrgy spcrum nsy funcons for h vlocy fl an h concnraon fls an rv on usng Lws-Krachnan 3 spac-m vrson of h Hopf 4 funconal formalsm an mulpl-scal-cumulan-xpanson mho s gvn. h quaons for spcrum funcons of h vlocy fl an h concnraon fls rv on mployng mulpl-scal-cumulan-xpanson mho by Josh N.. an Mshram M. C. 5 ar consr. h quaons for spcrum funcons ar rwrn by usng a mof-zro-fourh-orr-cumulan-xpanson MZFOC approxmaon mho whch ar frs wrn n mnsonlss form an hn ngra numrcally. h numrcal valus of h nrgy of vlocy fl as wll as ha of concnraon fls ar oban. hs valus ar us o valua h sascal quans scrbng h urbulnc wh racng an mxng chmcal lmns for larg ynols numbrs up o = 6.h quans nclu nrgy ransfr funcons of concnraon fls nsropy of h concnraon fls swnss of concnraon fls sspaon nrgy of concnraon fls an aylor s mcro-scals for concnraon fls. W prsn hs n h form of graphs for h rprsnav valus of ynols numbr = 4 an = 6.h analyss of h numrcal valus of sascal paramrs hus oban s carr ou an h laws govrnng hs quans ar nvsga. lso w scuss h mrs an scop of h prsn closur schm for suyng smlar yps of urbuln flows. Ky Wors: Hopf funconal formalsm Cumulans Mulpl scals urbulnc wh racng an mxng chmcal lmns I. Inroucon I s accp ha h urbulnc slf s vw as las unsolv problm of classcal physcs an h aon of h chmcal racon amplfs h ffculs normously. frs glanc o h unna h subjc of urbuln racng flows woul appar o b rlavly smpl. In h basc govrnng prncpls can b ruc o a samn of consrvaon of chmcal spcs an nrgy an a samn of consrvaon of flu momnum. Howvr anyon who has amp o masr hs subjc wll ll you ha s n fac qu complca subjc. On h on han n orr o unrsan how h flu flow affcs h chmsry on mus hav an xclln unrsanng of urbuln flows an of urbuln mxng. On h ohr han gvn s paramoun mporanc n h rmnaon of h yps an quans of chmcal spcs form an qually goo unrsanng of chmsry s rqur. vn a cursory rvw of h lraur n any of hs aras wll qucly rval h complxy of h as. In gvn h normous rsarch proucon n hs aras urng h wnh cnury woul b saf o conclu ha no on coul smulanously masr all aspcs of urbulnc mxng an chmsry. Nowhsanng h nllcual challngs pos by h subjc h man mpus bhn h vlopmn of compuaonal mols for urbuln racng flows has bn h ncrasng awarnss of h mpac of such flows on h nvronmn. For xampl ncompl combuson of hyrocarbons n nrnal combuson ngns s a major sourc of ar polluon. Lws n h chmcal procss an pharmacucal nusrs naqua conrol of prouc yls an slcvs can prouc a hos of unsrabl by-proucs. vn f such by-proucs coul all b succssfully spara ou an ra so ha hy ar no rlas no h nvronmn h conomc cos of ong so s ofn prohbv. Hnc hr s an vr-ncrasng ncnv o mprov nusral procsss an vcs n orr for hm o rman compv n h marplac. 37 Pag
Mof-Zro-Fourh-Orr-Cumulan-xpanson pproxmaon Mho for Invsgaon of Gvn hr complxy an praccal mporanc shoul b no surprs ha ffrn approachs for alng wh urbuln racng flows hav vlop ovr h las sxy yars. On h on han h chmcal-raconngnrng C approach cam from h applcaon of chmcal ncs o h suy of chmcal racor sgn. In hs approach h als of h flu flow ar of nrs only n as much as hy affc h prouc yl an slcvy of h racor. In many cass hs ffc s of sconary mporanc an hus n h C approach grar anon has bn pa o ohr facors ha rcly affc h chmsry. On h ohr han h flu-mchancal FM approach vlop as a naural xnson of h sascal scrpon of urbuln flows. In hs approach h mphass has bn prmarly on how h flu flow affcs h ra of chmcal racons. In parcular hs approach has bn wly mploy n h suy of combuson. brf ls of such conrbuon nclu osnr 6 ; Prs 7 ; Ponso an ynan 8 ; ynan an rvsch 9 Mshram an Junghar 4 Mshram 3 Mshram an Sahu 4. In hnsgh h prmary facor n rmnng whch approach s mos applcabl o a parcular racng flow s h characrsc m scals of h chmcal racons rlav o h urbulnc m scals. In h arly applcaons of h C approach h chmcal m scals wr largr han h urbulnc m scals. In hs cas on can safly gnor h als of h flow. Lws n arly applcaons of h FM approach o combuson all chmcal m scals wr assum o b much smallr han h urbulnc m scals. In hs cas h als of h chmcal ncs ar of no mporanc an on s fr o concnra on how h ha rlas by h racons nracs wh h urbuln flow. Mor rcnly h shorcomngs of ach of hs approachs hav bcom apparn whn appl o sysms whrn som of h chmcal m scals ovrlap wh h urbulnc m scals. In hs cas an accura scrpon of boh h urbuln flow an h chmsry s rqur o prc prouc yls an slcvs accuraly. Wh hs obsrvaons n mn Josh N.. an Mshram M.C. 5 mploy h mulpl-scalcumulan-xpanson mho for h nvsgaon of urbuln racon mxur wh scon orr rrvrsbl sohrmc racons for a racon of yp + Prouc. W hav noc ha h mulpl-scal-cumulanxpanson mho for chmcal racans nabls us o unrsan hr rol o g an a abou analyss of varous sascal quans. Such an analyss sms o smula a nw vlopmn n h suy of h physcal mchansm of urbulnc wh racng an mxng chmcal lmns. hr rmans howvr an mporan quson concrnng h physcal bass an h mahmacal accuracy of h prsn approxmaon a larg ynols numbr. Frsch U. Lusur M. an Schrzr D. 5 hav pon ou ha h rsuls oban on mployng mulpl-scal-cumulan xpanson mho by asum. Ka S. an Mzushma J. 6 o no complly scrb h sascal faurs of ncomprssbl soropc urbulnc a larg ynols numbrs. hr obsrvaon was bas on rsuls hy oban by numrcal compuaons.. asum an S. Ka 7 nrouc h mof-zro-fourh-orr-cumulan xpanson approxmaon mho for soropc urbulnc n rspons o h fcs pon ou by Frsch U. Lusur M. an Schrzr D. 5. y nroucon of hs mho for ncomprssbl soropc urbulnc hy succssfully rcf h fcs of arlr mho. h mho nrouc s capabl o complly scrb h sascal faurs of ncomprssbl soropc urbulnc for ynols numbrs as larg as 6. On xnson of h mulpl-scal-cumulan xpanson mho of asum. Ka S. an Mzushma J. 6 o urbulnc wh racng an mxng chmcal lmns of h yp + Prouc w coul carry ou our nvsgaons for ynols numbr up o 6.hs suggss h mprovmn n h closur schm. Hnc our man naural choc o mprov h fnngs s o mploy h mof-zro-fourh-orr-cumulan-xpanson mho for h nvsgaon of urbulnc wh racng an mxng chmcal lmns of h yp + Prouc. Josh N.. an Mshram M. C. 5 oban h ynamcal quaons for spcrum funcons of h vlocy fls an h concnraon fls on mployng mulpl-scalcumulan-xpanson mho. brf accoun of hs quaons s prsn n scon. In scon 3 h quaons for nrgy spcrum funcons on usng mof-zro-fourh-orr-cumulan-xpanson mho ar gvn. quaons for nrgy spcrum funcons n mnsonlss form ar gvn n scon 4. hs quaons ar ngra numrcally an h sascal quans scrbng h urbuln racon mxur wh scon orr racon ar valua an al analyss s carr ou n scon 5. h prsnaon conclus wh scon 6 whr h summary of h wor s gvn an rsuls oban ar scuss. hs s follow by scop of h mho for suyng h smlar urbuln flows. II. quaons For Spcrum Funcons Of locy Fl n Concnraon Fls h hory of urbulnc wh racng an mxng chmcal lmns wh a scon orr rrvrsbl sohrmc racons may b vlop n rms of jon probably srbuon P n h spac of h vcor fl Ux an h n -concnraon scalar fls x = 3... n. whch sasfy h followng sysm of quaons: 38 Pag
Mof-Zro-Fourh-Orr-Cumulan-xpanson pproxmaon Mho for Invsgaon of U U j U x j ρ P x U γ x x P P P U D K Pq. x x x U.3 Whr U = Ux = vlocy of h flow fl p = px = concnraon of racan p D = molcular ffusvy γ = nmac vscosy an K = racon ra. In orr o sola h ra of small-scal flucuaon from ohr ffcs h sascal homogny of h concnraon fls p s assum. Smlarly h urbulnc s assum o b soropc homognous an axsymmrc. W gnor h ynamc an chmcal rol of h prouc. o nsur smplcy h racon ra K s slc as consan an h sochomrc coffcns q as uny. vn hough h hory can b vlop wh hs quaons for n-concnraon fls w rsrc ourslvs o wo racans an png fuur convnnc n mn. Smlar compuaons can b conuc for a praccal suy nvolvng mor han wo racans. hus for a urbuln mxur wh racans an h sysm of smulanous quaons.. an.3 ar rprsn by followng quaons: U U P U U j γ.4 x j ρ x x jx j U x U U D x x x D x x x j j K K.7 Hr rpa lowr ncs mply summaon. h abov quaons ar scrb n h wor of roy.s. an O rn... h govrnng quaons for scrpon of urbuln mxur wh racans an for an soropc homognous an axsymmrc urbulnc ar hn ransform no a Fourr spac by Josh N.. an Mshram M.C. 5. hy hav fn h characrsc funconal for jon probably srbuon of vlocy an concnraon fls. hy rv h quaons for urbulnc unr suy o h funconal ffrnal quaons an hrafr hy fn h scon characrsc funconal. hy ransla h quaons n characrsc funconal n rms of scon characrsc funconal an oban h quaons for h cumulans of varous orr. hs quaons scrb h ynamcs of varous orr cumulans an also rflc h characrsc ffculy of h urbulnc hory ha mans a hypohss o oban a clos s of quaons for cumulans. y consrng h followng hypohss of mulpl scals rspcvly for lngh an m as n D n 3... n.8 n n D n3....9 hy succssfully oban a clos s of quaons for cumulans whou any ahoc assumpon. h assumpon of mulpl scals for h urbulnc slf srvs h sr purpos. vn hough h quaons for hghr orr cumulan ar avalabl hy consr quaons for scon orr cumulans an hr orr cumulan whch also nvolvs h fourh orr cumulan. On applcaon of mulplscal-cumulan-xpanson mho hy oban a clos s of quaon for scon orr cumulan snc scon orr cumulans ar rla o spcral nsy funcons. hy rv h followng quaons scrbng h..5.6 39 Pag
Mof-Zro-Fourh-Orr-Cumulan-xpanson pproxmaon Mho for Invsgaon of ynamcs of spcral nsy funcon for vlocy fl spcral nsy funcon for h racan. an h spcral nsy funcon for racan 4 3 } D 4 } D 3 D. q D 4 } D q D 3.. hs quaons ar ngra numrcally by Mshram M. C an Sahu K. 8.hy oban h sascal quans scrbng urbulnc wh mxng an racng chmcal lmns for ynols numbrs as larg as 6.h rsuls so oban o no complly scrb h sascal faurs of urbulnc a larg ynols numbrs. Hnc w nvsga h urbulnc wh mxng an racng chmcal lmns usng mof-cumulan-xpanson mho. hs mho s scrb n h scon ha follows. III. Mof-Zro-Fourh-Orr-Cumulan-xpanson-pproxmaon Mho h numrcal rsuls of Frsch U. Lusur M. an Schrzr D. 5 rval ha h nvsgaon of homognous soropc urbulnc by asum. Ka S. an Mzushma J. 6 on mployng mulpl-scalcumulan-xpanson mho o no complly scrb h sascal faurs of ncomprssbl soropc urbulnc a larg ynols numbrs. o xamn h suaon an consr h problm ras by Frsch U. Lusur M. an Schrzr D. 5 ; asum. Ka S. an Mzushma J. 6 mploy mof-zro-fourh-orrcumulan-xpanson-approxmaon mho. hy go succss n scrbng h sascal faurs of ncomprssbl soropc urbulnc a larg ynols numbr up o 6. Hnc w fn worhy o nrouc mof-zro-fourh-orr-cumulan-xpanson-approxmaon mho o suy h sascal srucur of urbulnc wh racng an mxng chmcal lmns. In h prsn wor w nvsga urbulnc wh racng an mxng chmcal lmns for ynols numbr up o 6. Howvr w prfr o prsn h 4 6 rsuls whch ar rprsnav for an.hs rsuls rval svral faurs of urbulnc a hs ynols numbrs whch wr no clarly c by usng mulpl-scal-cumulanxpanson mho n arlr nvsgaon. h quaons for nrgy spcrum funcon of h vlocy fl nrgy spcrum funcon of racan an nrgy spcrum funcon of racan of urbulnc wh racng an mxng chmcal lmns n symbolc form can b wrn as: 4 Pag
Mof-Zro-Fourh-Orr-Cumulan-xpanson pproxmaon Mho for Invsgaon of 4 Pag 3. 3. 3.3 whr an ar nrgy spcrum funcon for vlocy fl nrgy spcrum funcon of racan nrgy spcrum funcon of racan an nrgy ransfr funcon for vlocy fl nrgy ransfr funcon of racan nrgy ransfr funcon of racan rspcvly s h wavnumbr s h m. Unr h mof-zro-fourh-orr-cumulan-xpanson-approxmaon mho h nrgy ransfr funcon of vlocy fl nrgy ransfr funcon of racan an nrgy ransfr funcon of racan ar xprss n rms of h rspcv nrgy spcrum funcon of vlocy fl nrgy spcrum funcon of racan an nrgy spcrum funcon of racan as follows: } 3.4 } 3.5 } 3.6 whr μ an h nal conons an hav bn assum for smplcy. h s of quaon 3.4 3.5 an 3.6 consus a clos s of quaons for h nrgy spcrum funcon of vlocy fl nrgy spcrum funcon of racan an nrgy spcrum funcon of racan. quaons for nrgy spcrum funcon of vlocy fl ; nrgy spcrum funcon of racan an nrgy spcrum funcon of racan usng mof-zro-fourh-orr-cumulan-xpanson-approxmaon mho bcom as follows: } 3.7
Mof-Zro-Fourh-Orr-Cumulan-xpanson pproxmaon Mho for Invsgaon of 4 Pag } 3.8 } 3.9 whr I. quaons For nrgy Spcrum Funcons In Dmnsonlss Form For h fuur convnnc of numrcal compuaons all varabls ar ma non-mnsonal wh rspc o a rprsnav valu of h nrgy spcrum an a rprsnav valu of h wav numbr as funcons of h non-mnsonal wav numbr m an ynols numbr whch rspcvly ar gvn as follows : Wav numbr ; m 3 ; ynols numbr / ; hus h quaon 3.7 for nrgy spcrum funcon for vlocy fl h quaon 3.8 for nrgy spcrum funcon for h racan an h quaon 3.9 for h nrgy spcrum funcon for racan ar rspcvly wrn n h non-mnsonal form as follows: / 4. / 4. / 4.3
Mof-Zro-Fourh-Orr-Cumulan-xpanson pproxmaon Mho for Invsgaon of 43 Pag whr. W mus mphasz ha h prsn suy rlas o h nvsgaon of urbulnc wh racng an mxng chmcal lmns an. Hr an ar no chosn as any spcfc racans as w wsh o prsn h nvsgaons n h gnral form. For h suy of spcfc racans hr corrsponng nal valus for spcral nsy funcons may b slc an h nvsgaon b carr ou corrsponng o hos spcfc racans. lso for h prsn suy w hav slc Schm numbr Sc D Sc as uny snc no much nformaon s los by hs assumpon. h Schm numbr rprsns h rao of h nmac vscosy of h flu an h molcular ffusvy of parcular spc. W hav also assum qual ffusvs for all spcs. W slc h nal conons for spcral nsy funcons of vlocy fls racan an h racan as follows xp 4* p 4.4a xp 4* p 4.4b xp 4 * p 4.4c Smlarly w slc h nal conons for h nrgy spcrum funcon for vlocy fl nrgy spcrum funcon for h racan an h nrgy spcrum funcon for h racan as gvn blow by quaons 4.5a 4.5b an 4.5c rspcvly. xp 4.5a xp 4.5b xp 4.5c h ngrals on h rgh han s of quaons 4. o 4.3 ar calcula numrcally by usng appropra sum ruls a scr valus of an an rmnang h nfn ngrals wh rspc o a a suffcnly larg valu of. vry larg ynols numbr h ngran changs raply a small an os no vansh vn a larg. In orr o valua such an ngral accuraly an ffcnly w mploy a non unform msh for whos sz s an ncrasng funcon of.o m hs rqurmn w nrouc h followng varabls an : log ;. so ha. 4.6 In rms of hs varabls quaons 4. 4. an 4.3 as h form as gvn blow: / xp 3 4.7
Mof-Zro-Fourh-Orr-Cumulan-xpanson pproxmaon Mho for Invsgaon of 3 3 xp xp / 4.8 /. valuaon Of Sascal Quans n Invsgaon Of Govrnng Laws h numrcal ngraon of quaons 4.7 4.8 an 4.9 s carr ou for 4 5 6 an usng h sam numrcal mho as mploy for mulpl-scal-cumulan-xpansonmho bu mployng h largr cuoff wav numbr for h ngral an fnr sps for h m N ngraon. h cuoff wavnumbr s an as.. wh 6 for 4 5 N 7 for 5 6 8 for h m sp s an as.. hus h quaons 4.7 4.8 an 4.9 ar solv numrcally for h nal conons4.5a 4.5b an 4.5c; h ynols numbr = 4 o = 6 ; an h nrgy spcrum funcons for racan an nrgy spcrum funcons for racan ar oban as funcons of h wav numbr an m. Imporan sascal quans such as nrgy of concnraon fls an ransfr funcons of concnraon fls an nsropy of concnraon fls an sspaon nrgy of concnraon fls D an 4 4.9 swnss of concnraon fls S an S D an aylor s mcro-scals for concnraon fls an ; characrzng h urbulnc wh racng an mxng chmcal lmns of h yp + Prouc ar rv from numrcal valus of nrgy spcrum funcons. W prfr o prsn only rprsnav of hs corrsponng o = 4 an = 6. In h nr scusson blow h nrgy spcrum funcon shall no b scuss as all rlaons govrnng ar no ffrn from hos oban by. asum.al. 7 an o no fch any aonal nformaon for h vlocy fl usng mofcumulan-xpanson mho. Of all h numrcal aa concrnng h voluon of nrgy spcrum funcons an of concnraon fls an rspcvly for = 4 an = 6 ar chosn as rprsnav aa an shown graphcally n fgur 5. an fgur 5. rspcvly.. asum.al. 7 on mployng mof-cumulanxpanson-approxmaon mho for h vlocy fl oban posv nrgy spcrum funcon a all ms for all h ynols numbrs. On xnng hs mho o h urbulnc wh racng an mxng chmcal 44 Pag
Mof-Zro-Fourh-Orr-Cumulan-xpanson pproxmaon Mho for Invsgaon of lmns of h yp + Prouc w also oban posv nrgy spcrum funcon for boh concnraon fls an a all ms for all h ynols numbrs as larg as 6. h spcrum of h concnraon fls xns much furhr no h hgh wav numbr rang han ha of h vlocy fl. h form of h spcrum funcon an of h racan an spcally s srucur a hghr wav numbrs ar splay mor clarly on logarhmc scal as shown n fgur 5. an fgur 5.. hr ar som rmarabl faurs n h voluon of h spcrum funcons shown n hs fgurs. Frs h posv slops of an a vry low wav numbrs whch ar almos nvaran n m. hus h spcrum funcons n h vry low wav numbr rang ar nrly fx by h nal conons rflcng h wll nown prmannc of h larg scal componn of urbulnc. Scon n h smlary pro h hghr wav numbr par of h spcrum ar no nflunc apprcably by h ffrnc bwn h cass corrsponng o ynols numbrs 4 an 6 a vry low wav numbrs bu has a narly ncal form for boh h cass. h mor al srucur of h spcrum n h smlary pro s pnn upon h ynols numbr so ha wll b scrb blow for small an larg ynols numbrs n urn. closr nspcon of h spcrum rvals ha h pnnc of h spcrum funcons on h wav numbr s no prcsly ha of xp - as woul b xpc from h lnarz form of 4.4. an 4.3 an h nal conons 4.5a 4.5b an 4.5c bu s mor closly rprsn by.5 xp 5..5 xp 5. as whr σ s a consan pnn upon an. h spcrum funcons n h unvrsal wav numbr rang ar compos of hr pars havng ffrn funconal forms. In h lows subrang of h unvrsal wav numbr rang h nrgy spcrum funcons follow h laws whch ar xprss as follows: 5 3 5.3 5 3 5.4 5 h nrgy spcrum funcons of racan an ha of racan s proporonal o h ngav 3 powr of h wav numbr.w noc ha h abov law for h nrgy spcrum funcons of racan an racan has xacly h sam powr as Kolomogorov s nral-subrang spcrum. hs rsul s n accoranc wh h Kolomogorov s scon hypohss whch assums h xsnc of h nral subrang a larg ynols numbrs. h par of h subrang n whch Kolomogorov s scon hypohss ar val 5 ncrass nfnly wh ynols numbr. h ngav 3 powr for spcrum funcons hav bn obsrv by many xprmns n varous urbuln flows wh xrmly larg ynols numbr hr n naur or n laboraory. Nx o prvous subrang hr xss a subrang n whch h spcrum funcons follow h laws as follows: h nrgy spcrum funcons of h racan an h racan ar nvrsly proporonal o a wav numbr. Symbolcally h laws ar wrn as follows: 5.5 5.6 s h ynols numbr ncrass hs subrang s shf o hghr wav numbr whou changng s wh on logarhmc scal bu bcoms wr on lnar scal. sll hghr wav numbr boh h spcrum funcons an cays xponnally. hus h rsuls oban by h applcaon of mof-cumulan-xpanson- approxmaon mho suppor h asympoc cay of h spcrum funcons for vry hgh wav numbr rrspcv of ynols numbrs. 45 Pag
Mof-Zro-Fourh-Orr-Cumulan-xpanson pproxmaon Mho for Invsgaon of h nrgy ransfr funcon of h racan an of h racan rspcvly hos for = 4 an = 6 ar shown graphcally n fgur 5.3 an fgur 5.4. From hs fgurs w noc ha h nrgy ransfr s from lowr o hghr wav numbrs. h naur of h ransfr spcra of boh h chmcal racans an s analogous. h ffrnc obsrv n h graphs for an a ffrn ms s manly u o h ffrnc n h choc of hr rspcv nal valus. oh ransfr spcra ar posv n h frs shll. h prsn closur schm mploy for nvsgaon las o h rsul ha for a gvn ynols numbr n h nghbourhoo of h wav numbr boh h nrgy ransfr funcons chang hr sgn an sasfy h followng rqurmn W furhr noc ha as h ynols numbr as h largr valu h wavnumbr a whch vansh s shf o h lowr wavnumbr. h nsropy of h concnraon fl an nsropy of h concnraon fl ar rspcvly fn by h followng ngrals: 4 5.9 4 5. hs quans mosly rflc h shap of nrgy spcrum funcons an n h nrgy conanng rang of h wav numbr. h quaons 4.7 4.8 an 4.9 ar ngra numrcally an oban h numrcal valus of h nrgy spcrum funcons of h vlocy fl nrgy spcrum funcons of h racan an h racan. h valus of nrgy spcrum funcons hus oban ar us o fn h numrcal valus of nsropy of h racan an an h plos of hs valus ar shown n fgur 5.5 an fgur 5.6 rspcvly. W nfr from hs graphs of nsropy for racan an ha hy ar analogous o ach ohr. h shap of ach of hs graphs can b v no hr pars pnng upon hr bhavour wh rspc o m. h frs par bng from = o =6 h scon par from =6 o =35 an hr par >35. h nsropy for boh h racans an ncrass monooncally wh rspc o m n h nal pro from = o =6. hrafr n h scon pro from =6 o =35 h curvs aan h Gaussan shap. In h fnal pro corrsponng o >35 h nsropy of boh h racans ncrass monooncally. h ffrnc n nsropy of concnraon fls an concnraon fl s solly u o h choc of nal conons. W also obsrv from hs fgurs ha as h ynols numbr ncrass hr s ncras n h m an o aan h maxmum nsropy n cas of boh h racans an. For boh h racans hr nsrophy grows vry raply n h nal pro. h nsropy of h racan aans a qu larg maxmum valu.5 for = 4 an 7.5 corrsponng o = 6. Smlarly h corrsponng maxmum valus for h racan ar 6.5 an 3.5 rspcvly. hs rvals ha hr s a fall n h maxmum valu of nsropy wh rs n h ynols numbr. In h nal pro bfor aanng h maxmum valu h nsropy of h spc s proporonal o scon powr of m an analogus o hs nsropy of h spc s proporonal o.5 powr of m.symbolcally hs laws ar xprss as follows: 5. 5.7.5 5. 5.8 an 46 Pag
Mof-Zro-Fourh-Orr-Cumulan-xpanson pproxmaon Mho for Invsgaon of Nx o prvous pro jus afr aanng h maxmum valu for = 4 an = 6 h valu of nsropy of boh h racans crass an hs cras s govrn by h followng laws: h nsropy of h concnraon fl s proporonal o h ngav. powr of m an h nsropy of h concnraon fl s proporonal o h ngav.3 powr of m. Symbolcally hs laws ar wrn as follows:. 5.3.3 5.4 hs laws agr qu wll wh h xprmnal rsuls. In h las pro nsropy of h racan an ha of vars wh rspc o m as pr h followng laws: h nsropy of h concnraon fl s proporonal o h.3 powr of m whras h nsropy of h concnraon fl s proporonal o h.3 powr of m. hs laws symbolcally ar wrn as follows:.3 5.5..3 5.6 h swnss of h srbuon of h m rvav of h racan S an racan s fn an rla o h corrsponng nrgy spcrum funcons an h nrgy ransfr funcons as follows: 3 3 S 3 4 5.7 S S 3 3 4 3 5.8 quaons 5.7 an 5.8 for swnss of h racan S an of h racan S rval hr pnnc on nrgy spcrum funcons an nrgy ransfr funcons spcrum funcons. h swnss of boh h racans s manly rmn by h forms of nrgy an nrgy ransfr funcons n h hghr wavnumbr rang so ha rflcs h shap of h nrgy spcrum byon h nrgy-conanng rang of h wav numbrs. h swnss of boh h racans s valua by subsung numrcal valus of an no 5. 7 an 5.8. h numrcal valus of swnss of boh h racans ar valua an prsn n h form of graphs n fgurs 5.7 for ynols numbr = 4 an n fgur 5.8 for ynols numbr = 6.h nal valu of snss s ncally zro for boh h racans u o our choc of h nal conons. h chmcal racon has a ramac ffc upon h voluon of h swnss coffcn. h swnss S an S fn by 5.7 an 5.8 gvs a non-mnsonal masur of h srngh of h vorcy proucon. Snc S an S s xprss as an ngral of an rspcvly wgh owars hghr wav numbrs rflcs h bhavour of h spcrum n h unvrsal rang or h small-scal srucur of urbulnc wh racon mxur. lso from hs fgurs w noc ha h swnss of boh h racans s a masur of h parur from Gaussany an nca how ransfr funcons vary 47 Pag
Mof-Zro-Fourh-Orr-Cumulan-xpanson pproxmaon Mho for Invsgaon of wh h m a hgh wav numbr. h swnss of boh h racans crass subsanally afr havng rach hr sharp maxmum an hn ncrass monooncally wh rspc o m. Hnc pnng upon hr bhavour wh rspc o m h shap of ach of hs graphs can b v no hr pars. h frs par bng from = o =6 h scon par from =6 o =35 an hr par >35. h swnss for boh h racans S an S ncras monooncally wh rspc o m n h nal pro from = o =6. hrafr n h scon pro from =6 o =35 h curvs assums h Gaussan shap. In h fnal pro corrssponng o >35 h swnss of boh h racans ncrass monooncally. h maxmum valu of h swnss aan by h racan for = 4 s.5 an ha for h racan s narly 6.5. Smlarly for = 6 h maxmum valus of h swnss aan by h racan an racan rspcvly ar 7.5 an 3.5 approxmaly. hs rvals ha hr s a fall n h maxmum valu of swnss wh rs n h ynols numbr. In h nal pro bfor aanng h maxmum valu h swnss of h spc s proporonal o scon powr of m an analogus o hs swnss of h spc s proporonal o.5 powr of m.hs laws symbolcally ar wrn as follows: S 5.9.5 S 5. Nx o prvous pro jus afr aanng h maxmum valu for = 4 an = 6 h valu of swnss of boh h racans crass an hs cras s govrn by h followng laws: h swnss of h concnraon fl s proporonal o h ngav. powr of m an hs law for h swnss of h concnraon fl s proporonal o h ngav.3 powr of m. hs laws symbolcally ar wrn as follows:. S 5..3 S 5. In h las pro swnss of h racan an vars wh rspc o m as pr h followng laws: h swnss of h racan s rcly proporonal o.3 powr of m an smlarly h swnss of h racan s rcly proporonal o.3 powr of m. hs laws symbolcally ar wrn as follows:.3 S 5.3..3 S 5.4 h nrgy sspaon funcon for concnraon fl an concnraon fl rspcvly ar fn blow as: D 5.5 D 5.6 h numrcal valus of nrgy spcrum funcons of boh h racans ar us o valua hr sspaon funcons.h valus so oban ar plo vrss wavnumbrs a m = an =3 for ynols numbrs = 4 n fgur 5.9 an = 6 n fgur 5.. hs graphs clarly rflc h characrsc of nrgy spcra.i s obsrv ha for larg ynols numbr n h nrgy conanng rang an h nral subrang h nrgy sspaon s much bggr n magnu han h nrgy ransfr. Fnally s obsrv ha for larg ynols numbr h - spcrum always occurs n h nghbourhoo of wavnumbr a whch nrgy ransfr changs s sgn. 5.7 5.8 Lasly h aylor s mcro-scal for h racans xprss n rms of h nrgy spcrum funcons as gvn blow: an for h racan ar fn an 48 Pag
Mof-Zro-Fourh-Orr-Cumulan-xpanson pproxmaon Mho for Invsgaon of 5.9 5 5.3 5 h valus of aylors mcro-scal of racan an ha of racan oban by subsung numrcal valus of an no 5.9 an 5.3 an h numrcal valus hus oban ar plo n fgur 5. for ynols numbr = 4 an n fgur 5. for ynols numbr = 6. h aylor s mcro-scal fn by 5.9 an 5.3 gvs anohr paramr rprsnng h small scal srucur of urbulnc wh racng an mxng chmcal lmns an. larg ynols numbrs h mcro-scal for h racan an ha of h racan cras vry raply n h nal sag. hn hrafr for = 4 h aylor s mcro-scal of racan aans a mnmum aroun =. an ha of h racan aans a mnmum aroun =.5 an hs valus for = 6 rspcvly ar =.8 an =.4. h varaon of h aylor s mcro-scal of racan an ha of h racan wh m s govrn by h laws ha h aylor s mcroscal of boh h racans vars proporonally o ngav.5 powr of h m. Symbolcally hs laws ar wrn as follows:.5 5.3.5 5.3 h varaon of h aylor s mcro-scal of racan an ha of h racan wh m confrms h asympoc cras of aylor s mcro-scal wh m. 49 Pag
Mof-Zro-Fourh-Orr-Cumulan-xpanson pproxmaon Mho for Invsgaon of Fgur 5.5 nsropy of h racan sol ln an h racan ash ln for = 4 Fgur 5.6 nsropy of h racan sol ln an h racan ash ln for ynol numbr = 6 5 Pag
Mof-Zro-Fourh-Orr-Cumulan-xpanson pproxmaon Mho for Invsgaon of 5 Pag
Mof-Zro-Fourh-Orr-Cumulan-xpanson pproxmaon Mho for Invsgaon of I. Summry n Dscusson W frs wr h quaons for urbuln mul-componn mxur n-racans wh scon orr rrvrsbl sohrmc racon. In h prsn suy w rsrc ourslvs o wo racans an. Josh N.. an Mshram M. C. 5 nvsga urbulnc wh racng an mxng chmcal lmns of h yp + Prouc on mployng h mulpl-scal-cumulan-xpanson mho. h choc of mulpl m scal an mulpl lngh scal of Josh N.. an Mshram M. C. 5 s ran n h prsn wor. In orr o mprov h rawbacs of mulpl-scal-cumulan-xpanson mho asum. an Ka S. 7 hav nrouc h mof-zro-fourh-orr-cumulan-xpanson mho for nvsgaon of homognous soropc urbulnc. W hav mploy hs mho for h nvsgaon of urbulnc wh racng an mxng chmcal lmns of h yp + Prouc. W consr h quaons for h nrgy spcrum funcon for h spc an ha for h spc rv by Josh N.. an Mshram M. C. 5 on mployng hs mho. hs quaons ar hn ngra numrcally an varous sascal quans scrbng h urbulnc wh racng an mxng chmcal lmns of h yp + Prouc such as nrgy of concnraon fls nsropy of h concnraon fls swnss of concnraon fls sspaon nrgy of concnraon fls ransfr funcons of concnraon fls an aylor s mcro-scals for concnraon fls an ar valua an rprsnav valus for ynols numbrs = 4 an = 6 ar prsn n h form of h graphs. h characrscs of urbulnc wh racng an mxng lmns rflc n hs graphs ar su. h mporan faurs of h sascal quans scrbng h urbulnc unr suy rval h followng:. h nrgy spcrum funcon for boh h racan an h racan ar foun o b posv fn for all valus of ynols numbrs.. h spcrum of h concnraon fls xns much furhr no h hgh wav numbr rang han ha of h vlocy fls. 3. h nrgy spcrum funcons of racan an ha of s proporonal o h ngav 5/3 powr of h wav numbr for all h m n h nral subrang whos xn ncrass nfnly wh ynols numbr. hs rsul s n accoranc wh h Kolomogorov s scon hypohss. 4. h unvrsal spcrums ar compos of hr ffrn subspcra whch ar n orr of ncrasng wavnumbr h -5/3 spcrum h - spcrum an h xponnal spcrum. 5. h naur of h ransfr spcra of boh h chmcal racans an s analogous. s h ynols numbr as h largr valu h wavnumbr a whch an vansh s shf o h lowr wavnumbr. 5 Pag
Mof-Zro-Fourh-Orr-Cumulan-xpanson pproxmaon Mho for Invsgaon of 6. h graphs of h nsropy for racan an ar analogous o ach ohr. s h ynols numbr ncrass hr s ncras n h m an o aan h maxmum nsropy n cas of boh h racans an. 7. nsrophy of boh h racans grows vry raply n h nal pro an aans a qu larg maxmum valu.5 for = 4 an 6.5 corrsponng o = 6 for h racan rspcvly. Smlarly h corrsponng maxmum valus for h racan ar 7.5 an 3.5 rspcvly. 8. In h nal pro h nsropy of h spc s proporonal o scon powr of m an analogus o hs nsropy of h spc s proporonal o.5 powr of m. In h nrma pro h nsropy of h concnraon fl s proporonal o h ngav. powr of m an hs law for h swnss of h concnraon fl s proporonal o h ngav.3 powr of m whras n h hr pro h nsropy of h concnraon fl s proporonal o h.3 powr of m an h nsropy of h concnraon fl s proporonal o h.3 powr of m. 9. h nal valu of snss s ncally zro for boh h racans.. h maxmum valu of h swnss aan by h racan for = 4 s.5 an ha for h racan s narly 6.5. Smlarly for = 6 h maxmum valu of h swnss aan by h racan an racan s 7.5 an 3.5 approxmaly..h graph of h swnss for boh h racans s v no hr ffrn m pro. In h nal pro h swnss of h spc s proporonal o scon powr of m an analogus o hs swnss of h spc s proporonal o.5 powr of m. In h nrma pro h swnss of boh h racan assums h Gaussan shap an h swnss of h concnraon fl s proporonal o h ngav. powr of m an h swnss of h concnraon fl s proporonal o h ngav.3 powr of m an fnally n h hr pro swnss of h concnraon fl s proporonal o h.3 powr of m an h swnss of h concnraon fl s proporonal o h.3 powr of m..in h nrgy conanng rang an h nral subrang as ynols numbr ncrass h nrgy sspaon crasss n magnu han h nrgy ransfr. 3.n mnn faur oban rla o sspaon funcon s ha for larg ynols numbr h - spcrum always occurs n h nghbourhoo of wavnumbr a whch nrgy ransfr changs s sgn. 4. s h ynols numbr ncrass h m an for obanng zro aylor s mcro-scal for boh h concnraon fls an crass. 5. h aylor s mcro-scal for boh h racans an s nvrsly proporonal o half powr of m. h prsn nvsgaons for urbulnc wh racng an mxng chmcal lmns + Prouc usng mof-zrofourh-orr-cumulan-xpanson-approxmaon mho has prov sgnfcan srucur of urbulnc unr suy. hrfor w fors ha hs mho whn mploy for h nvsgaon of smlar urbuln flows may prov sgnfcan nformaon n gnral an n h suy of sascal nvsgaon of urbulnc wh racng an mxng chmcal lmns + Prouc; amosphrc urbulnc wh racng an mxng chmcal lmns n parcular. h wor n such aras s n progrss an upa shall b prsn n u cours of m. frncs [] roy. S. 975 urbulnc n Mxng Opraons: hory an pplcaons o Mxng an acon Nw Yor: camc Prss. [] O rn.. 97 urbuln Mxng of wo aply Chmcal Spcs Phys. Flus 4 pp. 36-33. [3] Lws. M. an Krachnan. H. 96 Spac-m Funconal Formalsm for urbulnc Comm. Pur ppl. Mah. 5pp.397-4 [4] Hopf. 95 Sascal Hyroynamcs an Funconal Calculus J. a.mch. nal. pp. 87-3. [5] Josh N.. an Mshram M. C. 988 h spac-m funconal formalsm for urbuln mulcomponn mxur wh scon orr racons J.Mah. Phy.Sc. 6 pp.7-737. [6] osnr D. ranspor Procsss n Chmcally acng Flow Dovr Nw Yor. [7] Prs H. an ohors. horcal an Numrcal Combuson.. vrs Inc.. [8] Ponso an ynan horcal an Numrcal Combuson.. vrs Inc.. [9] ynan an rvsch urbuln Combuson Mollng Progrssng ngg. Sc. 8 pp. 93-66. [] Mshram M.C. an Junghar J.. 4 Invsgaon of amosphrc urbulnc wh mxng an racng lmns of h yp [] + Prouc usng wo-pon closur mho Nagpur unvrsy Scnc Journalpp.4 [] Mshram M.C. an Junghar J.. 4 Invsgaon of amosphrc urbulnc wh mxng an racng lmns of h yp [3] + Prouc usng wo-pon closur mho Nagpur unvrsy Scnc Journalpp.9 [4] Mshram M.C. wo-pon Closur Mho for urbulnc wh acng an Mxng Chmcal lmns of yp + C [5] Phys. Scr.4 pp.-8. [6] Mshram M.C. Facors Cumulan xpanson pproxmaon Mho for urbulnc wh acng an Mxng Chmcal lmns of yp + Prouc Phys. Scr.o appar. [7] Mshram M.C. an Sahu K. Invsgaon of urbulnc wh racng an mxng chmcal lmns of h yp + Prouc scrb by spac-m funconal formalsm In.Jour.Mah.Sc. an ngg.ppl.o appar [8] Frsch U. Lsur M. Schrzr D. 98 Commns on h Quas-Normal Marovan pproxmaon for fully-vlop urbulnc J. Flu Mch. 97 p. 8-9. [9] asum. Ka S. an Mzushma J. 978 h Mulpl-Scal -Cumulan- xpanson for Isoropc urbulnc J. flu Mch. 85 pp. 97-4. [] asum. an Ka S. 98 h Mof Cumulan xpanson forisoropc urbulnc a Larg ynols Numbrs J. Phys.Soc. Japan 49 pp. 4-5. [] Mshram M. C an Sahu K. Invsgaon of Magnohyroynamc urbulnc Dscrb by h Spac-m Funconal [] Formalsm Physcs Scrpa o appar. 53 Pag