Elasto-inertial turbulence 1 4 2

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1 Elato-ertal turbulece 4 Eergy trafer ad drag reducto elato-ertal turbulece lade wth elogated cotravarat ad covarat polymer E-mal:mauda..af@m.ttech.ac.jp E-mal:khorut@ me.ttech.ac.jp E-mal:uzuk.a.ap@m.ttech.ac.jp Shu Suzuk Tokyo Ittute of Techology -- Ookayama Meguro-ku Tokyo Kyo Horut Tokyo Ittute of Techology -- Ookayama Meguro-ku Tokyo Ao Suzuk Tokyo Ittute of Techology -- Ookayama Meguro-ku Tokyo We carry out umercal tudy to reveal the prmtve proce of elogato of the polymer releaed the Newtoa olvet ad how exchage th eergy betwee polymer ad olvet. We ue BDS-DNS method whch combe Browa dyamc mulato BDS wth DNS forced homogeeou otropc turbulece. Compared wth the complete affe cae α=. cotravarat more dratc DR acheved whe o-affty maxmum α=. covarat. We try kd of Weeberg umber whch defed by relaxato tme of. The hgher Weeberg umber the more drag reducto we could get. We focu o the τ j whch calculate by. We revealed the dfferece of cotravarat ad covarat eergy flow betwee polymer ad olvet.. Newto Tom Newto Lumley 969 de Gee 986 DNA 3 Vrk maxmum drag reducto MDR aymptote Tom col tretch 7 rod SIS Shear Iduced Structure affe affe DNS affe 8 affe affe affe Drect Numercal Smulato BDS-DNS Wataabe et al 9 affe BDS-DNS affe affe affe P.C.Valete Weeberg W Weeberg W = 5. W = Weeberg. BDS-DNS b. Dumbbell. BDS-DNS bead Lagrage 9 x x bead affe Fg. coector vector R x x = vector Fg. Cofgurato of elatc model

2 9 e E l rg = x R x.. rg f 9 dr = u x u x F dt req W W τ = u x u x d rg dt req 8τ t = W t W Wα F m α. β j = δαβ δ j δ mδ t affe.5 affe lp velocty ug affe flexble α α =. 8tep ν = Re = 3 Wray 3 Ruge-Kutta alag τ τη Weeberg 5 τ =.4 W = 5. Weeberg W = Kolomogorov Newto.5 Δt =. 3/ W = τ τη col-tretch Kolomogorov S bead 83.4 u = ug u g R Rg α{s R Rg }. < α <..3. <α>=<β>=<α >=<β >= π.3.4. < > α k β k. W {k α k β k } 6πk 4 Δt #c. k.5 E f k = " f! otherwe Δt c f =.5. E f k f k t = W W u x Whte Gaua. f DNS req = LMAX 5 LMAX = η =. η a =.85 Nt = 9 lp parameter α =. rgd τ j ~ τ j = 3. 3πa a Nt {R R j reqδ j } τ Re = Bead FENE.6 3 k FENE dampg R $ "R F = kf " & rmax # f z = z.. 8 α =. α =. f 4 BDS-DNS Newto 3..7 Weeberg Weeberg. m.7 u b W = a W = 5. Fg 3. Tme evoluto of eergy dpato b Naver-Stoke.8.9 τ j u u u j p u = f t x j x Re x j x x j.8 u = x.9 x = 3 u Re b W = a W = 5. Fg 3. Tme evoluto of legth

3 a b Fg 3.3 P.D.F. of legth Fg 3. Fg 3.3 p.d.f. : probablty of dtrbuto fuctoa b affe Q Q = 3. S k S k Ω k Ω k 3. 5 A A j j = Sk Ωkj Ωk A j [ j 3. A ] 3 Fg 3.4 Fg 3.4aFg 3.4b A ] [ A ] [ A a a [ j j j ] a a a W = 5. W = a 8 BDS-DNS α =.. Taylor 3.3 dr u = R j R 3.3 dt x τ j acotravarat α =. bcovarat α =. Fg 3.4 Vualzato of vortex heet ad tube W = 5. a α =. b α =. Fg 3.5 Cofgurato of A ] egevector o the vortex heet ad 3.5 dr u j = R j R 3.5 dt x τ [ j S Fg d u O-XYZ A j a a a Dumbbell a R a R a R R jot-p.d.f. R R jotp.d.f. Fg3.5 R R = R R A j Fg 3.7 a R 3 a α =. b α =. Fg 3.6 Jot-p.d.f. ad R R

4 3 R R 4.4σ σ σ 4.3S k 4.3 P e Fg 3.7 Cofgurato of A j egevector o the vortex heet ad hghly tretched 4. u τ j u P.C.Valete τ j Joho-Segalma 6 α =. α =. Oldroyd-B Oldroyd-A Dτ j! u = α τ j k u $! u # τ kj &α τ k k u $ k Dt " x k x # τ kj k " x j x & τ j 4. 4.t = τ j = Brd τ j t ν β t e t S j d ν β t r α dr d e t S k u j r u r * & x k x k ν β t r α dr d e t S k u k r u k r & x j x * 4. Horut 8 τ j ν β S j ν β { αs k S k Ω kj Ω k } 4.3 τ j 3 A j E σ σ σ S j ω P e = α τ j S j 4.5 P e ν β α [ S j S j 4 α S k S j 8 α S k S kl S lj S j A j A j ] 4.6 Ω k Ω kj S j 4.8S k S j S j S j P e = τ j S j 4ν αs k S j 4.7 D! Dt S $ # js j & = S k S j Ω k Ω kj S j 4.8 " 4.9a =. Pe P ε S k S j Valete et al. a =. Pe P ε S k S j u p.d.f. p.d.f. 5 # σ σ σ & E T τ j E ν β σ σ σ $ σ # σ σ & σ σ σ ν β α σ σ σ σ σ σ $ # σ σ ω & ν β $ σ σ ω 4.4 Fg 4. Dvo of the legth to eve rage 4

5 a α =. bα =. Fg 4. P.D.F. producto codtoal amplg by the legth of Fg 4.5 Dvo of the dervatve kewe to eve zoe a R R S b R R S Fg 4.3 P.D.F. decompoto of elatc eergy producto term α =. codtoal amplg by legth of a α =. b α =. Fg 4.6 P.D.F. decompoto of elatc eergy producto term codtoal amplg by S j S k a R R S b R R S Fg 4.4 P.D.F. decompoto of elatc eergy producto term α =. codtoal amplg by legth of Fg 4. Fg.4. p.d.f. Pe S j R R j 3.3 R R S Fg 4.3Fg 4.4 p.d.f. dervatve kewe S j S k p.d.f. Fg 4.5 S j S k p.d.f. Fg a α =. b α =. Fg 4.7 P.D.F. decompoto of dpato rate codtoal amplg by S j S k S j S k p.d.f. Fg4.7 p.d.f.s j S k S j S k S j S k zoe4 S j S k zoe 5. Weeberg BDS-DNS

6 de Gee Tom BA.Some obervato o the flow of lear polymer oluto through traght tube at large Reyold umber I Proc.t Itl Cogr.Rheol Cho H. J. Lm P. -Y.S.T. ad Cha C. K. Turbulet Drag Reducto ad Degradato of DNAPhy.Rev.Lett VrkP.S.Drag reducto fudametalaichej J. L. Zak J. M. ad Chara Z.New lmtg drag reducto ad velocty profle aymptote for opolymetrc addtve ytem AICHEJ Taybor M. ad Gee P. G. D.A Cacade Theory of Drag Reducto Europhy. Lett W. K. Lee R. C. Vaelek ad A. B. Metzer Turbulet Drag Reducto polymerc Soluto Cotag Supeded Fber AIChEJ K. Horut K. Matumoto K. Fujawa Phy. of Flud T. Wataabe T. Gotoh. Hybrd Eulera - Lagraga mulato for polymer-turbulece teracto. J. Flud Mech 77 3 : BDS-DNS 7 C-3 3 P.C. Valete et al. The effect of vcoelatcty o the turbulet ketc eegy cacade J.Flud Mech.76 4 R. G. Gordo ad Schowalter J. Rheology Brd R. B. Curt C.F. Armtrog R. C. ad Haager O. Dyamc of polymer Lqud d ed. Wley New York H.R. Warer Jr. Ph. D. The Uverty of Wco Mado 97 Id. Eg. Chem. Fudametal T.Gotoh D. F. ad Nakao T. Velocty feld tattc homogeeou teady turbulece obtaed ug a hgh-reoluto drect umercal mulatophy.of Flud K. Horut ad Y. Takag Idetfcato method for vortex heet tructure turbulet flow Phy. of Flud M.W. Joho Jr. ad D. Segalma A model for vcoelatc flud behavor whch allow oaffe deformato J.No-Newtoa Flud Mech.997 6