SMS-618, Particle Dynamics, Fall 2003 (E. Boss, last updated: 10/8/2003) Conservation equations in fluids

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1 SMS-68 Parcle Dnamcs Fall 3 (E. Boss las daed: /8/3) onseraon eqaons n flds onces e need: ensor (Sress) ecors (e.g. oson eloc) and scalars (e.g. S O). Prode means o descrbe conseraon las h comac noaon We need o defne a coordnae ssem () h n normal ecor ( j k ) and an (nfnesmal) elemen of olme δδδδ. Fgre : coordnae ssem and nfnesmal olme. he Lagrangan frameork s he frameork n hch he las of classcal mechancs are ofen saed. Assme a arcle nall a oson (). he coordnaes () descrbe he rajecor of hs arcles. he dens ma change along he rajecor (()). he change of dens (or oher scalar) along he rajecor s dered sng he chan rle (n ecor noaon): d d d d cons cons d( ( ) ) D ( ) d D he rae of change along he rajecor (Lagrangan frame) eqals he local rae of change ls he adecon of gradens (Eleran frame). he Lagrangan derae (D/D) need no be ero e.g. f here s a sorce or snk. Eamle: Le s assme ha e are n a rer ha feeds on glacal mel. Le s assme ha he aer arms a a consan rae ha s a fncon of dsance from he sorce ().e. here s a sorce of hea (he ar). If e drf don rer (A la Hckleberr Fn ) he emerare ncreases h me (D/D>). A one on along he rer hoeer e see no change

2 n emerare h me (/) as he aer arrng here s alas a he same emerare. he hea fl s adece (/>). Mass conseraon a. Eleran dfferenal aroach: Acconng for he change n mass (M) nsde a fed consan-olme olme: (- / / /) ( / / /) (- /- / /) A A A A A ( / /- /) A (- /- /- /) ( /- /- /) Fgre. A fed olme. hange of mass hn he olme are de o dfferences of fles beeen ha comes n and ha goes o: ( ) A ( ) A ( ) A ( ) Sbsng for he areas and dedng b : / / / / / / ( ) ( ) ( ) / / / / / akng he lms and n ecor noaon form: ( ) ( ) ( ) ( ) / b. Eleran negral aroach: Acconng for he change n mass nsde a fed consan-olme olme ( ) sng he dergence heorem ( denoes he srface area enclosng he olme): d d d nds d d

3 c. omarson of Lagrangan and Eleran: he Lagrangan mass conseraon s sml: D D d Where he olme alloed o change along he rajecor (). he Eleran mass conseraon s: d d ( )d d d. onn eqaon and non-dergence: he mass conseraon can be ren as: r ( ) Whch s eqalen o: D r D he second erm s he dergence of he flo (rae of oflo of olme er n olme). In he absence of forcng hs can be nonero onl for comressble flds. I s he rae of loss of dens de o eanson. For boh aer and ar e can assme ha n erms of her dnamcs. For some rocesses (e.g. sond roagaon) comressbl canno be negleced. onseraon of mass of a scalar: he mass of a scalar s here denoes he concenraon (e.g. n mol or mg er Kg fld). Addng he ossbl for a dffse fl (Fck s la) and sbracng D()/D: d ( K ) Usng he dergence heorem: r K d r nds And snce he olme s arbrar r ( ) r ( ) ( K)

4 Momenm balance (he Naer-Sokes eqaons): Neon s nd la of moon saes ha he me rae of change of momenm of a arcle s eqal o he force acng on. hs la s Lagrangan he me rae of change s h resec o a reference ssem follong he arcle. hs: d d ( ) d ( ) gd ( ) ds Where g s he bod force er n mass (e.g. gra) and s he srface force er n srface area bondng. If he olme s small enogh ha he negrands can be aken o of he negral: d d d d d ( ) ( δ ) d d d d δ δ δ d d d d d ( ) ( ) here conseraon of mass along he ah elmnaed a erm. he bod force s smlarl reaed: gd gδ ( ) We ge: D g Τ D Srface forces (sresses): For an nscd fld he srface force eered b he srrondng fld s normal o he srface.e. -n and s called he ressre force. In general scos sress force σ s also resen so for scos flds: -nσ. B defnon n and e no hae -IΣ here σ Σ n and I s he den ensor. Noe ha he ressre s soroc a an gen on. For Neonan ncomressble flds (see belo) Σ And he reslan Naer-Sokes eqaons are (see belo): D g D Defnng a sress ensor: Τ n and alng he dergence heorem: ds Τ d Τδ ( ) ( )

5 Some characerscs of he sress ensor : Fgre 3: he drecon and noaon of he elemen of he sress ensor. he sress ensor s smmerc: Pressre s relaed o he dagonal elemens of he sress ensor: In Neonan flds he sress s lnearl relaed o he shear and he rooronal consan s he dnamc scos. Sokes 845:. Σj lnear fncon of eloc gradens.. Σj shold ansh f here s no deformaon of fld elemens. 3. Relaonsh beeen sress and shear shold be soroc. Derng he RHS of he Naer-Sokes eqaon: ( ) 3 j j j δ ( ) ( )

6 he Naer-Sokes eqaon (ecor and non-ecor noaon): he Bossnesq aromaon: Searae balance of fld a res (denoed b ero) from mong fld (denoed b rme). he rmar balance s he hdrosac balance: he ne order balance s he modfed (Bossnesq) N-S eqaons: g g D D g D D g

7 Renolds decomoson of he N-S eqaons: Assme a errbed (e.g. rblen) flo. A an gen on n sace e searae he mean flo (mean can be n me sace or ensemble) and deaon from he mean sch ha: Sbsng no he conn eqaon (lnear): Sbsng no -momenm Naer-Sokes eqaon: he eolon of he mean s forced b correlaons of flcang roeres. he correlaon erms me he dens are he Renolds sresses. hese erms domnae oer he moleclar sresses. he ne sress ensor s: Sbsng no a scalar conseraon eqaon: ec ; ( ) ( ) ( ) ( ) ( ) ( ) K

8 he closre roblem: o deelo eqaons for he eolon of he Renolds sresses hemseles hgher order correlaon are needed (e.g. ) and so on. For hs reason heores hae been desed o descrbe n erms of he mean flo. One solon o he closre roblem s o lnk he Renolds sress o mean-flo Qanes. For eamle: K edd { } { } edd hs e of formlaon s aealng becase : a. Prode for don-graden fl. b. Is remnscen of moleclar dffson and scos. c. Prode closre o he eqaons of he mean felds. hs e of formlaon s roblemac becase: a. K edd s a roer of he flo and no he fld. b. K edd s lkel o ar h drecon (e.g. ercal edd dffs s smaller han horonal edd dffs de o gra) nlke moleclar rocesses. Ho s K edd relaed o he rblence? Assme a graden n a mean roer (momenm hea sole ec. Remember: no mean graden no fl). Assme a flcang eloc feld: Fgre 4: change of oson of fld arcels n he resence of a mean graden n a scalar Ψ resls n a fl of roeres. l s he dsance a arcel raels before loses s den. he rae of ard ercal rblen ransfer of <Ψ> s don he mean graden: ψ ψ ψ l l K edd ψ

9 Ho s K edd relaed o he rblence? K edd { } { } edd ennekes and Lmle (97) aroach hs roblem from dmensonal analss based on assmng a sngle lengh scale-l and a sngle eloc scale ω< > /. cω ; c ~ O() he eddes noled n momenm ransfer hae orces ω/l; hs orc s mananed b he mean shear (l s he lengh scales of he eddes e.g. he decorrelaon scale). U ω / l c ; c ~ O() I follos ha: e dd ~ lω ~ l U In analog h momenm fl for a sole e hae: K edd I s mos commonl assmed and erfed ha K edd edd. Edd-dffson: ersece from a de ach (fgres from lecre noes b Bll Yong USD): Fgre 5a: De ach << domnan scale of eddes. Dashed crcle denoes nal oson and se of racer ach.

10 Fgre 5b: De ach ~ domnan scale of eddes. Dashed crcle denoes nal oson and se of racer ach. Fgre 5c: De ach >> domnan scale of eddes. Usefl references: Acheson D. J. 99 Elemenar fld dnamcs. Oford U. ress. ennekes H. and J. L. Lmle 97 A frs orse n rblence MI ress. Wlkes J. O. 999 Fld Mechancs for hemcal Engneers. Prence Hall.

11 Aend: onece derae: Graden of a scalar (s a ecor): Dergence of a ecor (s a scalar): Dergence of a ensor (s a ecor): Lalacan of a ecor (s a ecor): D D k j φ φ φ φ k j ( ) k j