F. T. Pinho. Centro de Estudos de Fenómenos de Transporte, Universidade do Porto & Universidade do. Minho, Portugal

Transcription

1 F. T. Pinho Centro de Estudos de Fenómenos de Transporte, Universidade do Porto Universidade do Minho, Portugal Colaboradores P. R. Resende Centro de Estudos de Fenómenos de Transporte, Universidade do Porto, Portugal B. A. Younis Dep. Civil and Environmental Engineering, University of California, Davis, USA R. Sureshumar Dep. Energy, Environmental and Chemical Engineering, Washington University of St. Louis, St Louis, MO, USA K. Kim Korea Institute of Energy Research,Daejeon, Republic of Korea II Conferência Nacional de Métodos Numéricos em Mecânica de Fluidos e Termodinâmica 8 th -9 th May 8 Universidade de Aveiro, Portugal

2 Drag reduction: some characteristics Applications: drilling of oil an gas wells; district heating cooling sys. 1-1 f Blasius eq. 6 u PAA Re=49.5 CMC Re= CMC Re=43.9/.9 CMC/XG Re=453 4 u + =11.7 ln PAA. PAA.5 CMC. XG.9/.9 CMC/XG Virs MDRA asymptote Re !Reduction of shear Reynolds stress DR Reynolds stress deficit -!Increase of normal streamwise Reynolds stress -!Dampening of normal radial and tangential Reynolds stress No universal law of the wall 1 u + =.5 ln +5.5 Escudier et al JNNFM 1999 Pinho, Resende, Younis, Sureshumar Kim

3 Governing equations for non-newtonian fluids Reynolds decomposition ˆB = B + b where b = ^- instantaneous quantities Overbar or upper-case letters- time-averaged quantities or lower-case letters- fluctuating quantities Continuity incompressible: Momentum:! "Ûi "t!ûi!x i = +!Û "Ûi "x = # "ˆp "x i + " ˆ$ i, fluid "x Rheological constitutive equation: Purely viscous: GNF ˆ! ij = µ I Sij, II Sij, III Sij Ŝij Power law fluid µ = K! n"1! = S pq S pq Viscoelastic: ˆ! ij, fluid = " s Ŝ ij + ˆ! ij, p Upper Convective Maxwell ˆ! ij, p + " ˆ! ij, p # = $p Ŝ ij Pinho, Resende, Younis, Sureshumar Kim 3

4 Existing models: equations - Mizushina and Usui Cruz Mixing length with Van Driest damping Stoe s nd problem! "U i "t "U +!U i = # "p + " $ "U i "x "x i "x "x Mizushina Usui 1977 PoF v, S1 f µ = 1! exp! y+ #!" + " +1 A + $ 1.34! t! = 3.76 #!U "18 $ D! T = f µ " y du dy 1/ 1/! = 5 * # "!u "x i u * +,!uv = " T du dy Turbulent relaxation time " s " [ ] Mc T! = " t # u $ A + * Rouse relaxation time Power law- purely viscous Cruz et al HME v, 1 " " f µ = * n 1! 1+ 1! n * # $ # $ 1+ n A +! 1+n 1!n Pinho, Resende, Younis, Sureshumar Kim + - -, 4

5 Existing models: equations- Malin inelastic fluids! "U i "t "U +!U i = # "p + " $ "U i "x "x i "x "x * # "!u "x i u Malin 1997 ICHMT v4,977!u i u j = " T S ij! 3 # ij! T = C µ f µ " Viscosity: Power law No elasticity: No polymer stress Malin: Lam-Bremhorst model low Re -! :! : =!. " + " T!. U =!. # + # T!. U" #$! + P + -, $ " f µ = " # 1! exp!.165re n 1/4 *!". + " / c 1 " f 1 P 1 c " f " f 1 = 1+.5 f µ $ 1+ 5 Ret 3 Pinho, Resende, Younis, Sureshumar Kim 5

6 Existing models: Pinho et al 3 to 6! ij = µs ij n1 Modified GNF: µ = K v "#! $! K e "#! p1 $! "U i "t "U +!U i = # "p + " $ µ "U i "x "x i "x "x + " $ µ "u i "x "x # "!u "x i u µ = f v µ h + 1! f v " v µ h = C µ! 3m m"1 A 8+3m m"1 A 4 m m"1 A 8+3m m"1 A 6 m m"1 A! v = K v #$ "! n1 8+3m m"1 A # [ 8"3 m"1 A ]m 8+3m m"1 A B 8 8+3m m"1 A µ!u i =!C K K v e p#1!x j A " $C µ f µ µ!" S mn S mn * p+n# C µ f µ!" + 1 L c + S ij S pq S pq Low Reynolds number anisotropic -! model!"u i u j = # t S ij! 3 $ ij! f! 1n S * i S * j! 1 $ 6 S* ij * +! f n f! 3n 3 W * i S * j! S * * i W j Pinho 3, Cruz Pinho 3, Cruz et al 4 JNNFM; Resende et al, 6 Int. J. Heat Fluid Flow Pinho, Resende, Younis, Sureshumar Kim 6

7 Constitutive model for dilute polymer solutions: FENE-P ˆ! ij = " s Ŝ ij + " p # f Ĉ Ĉij $ f L ij f with! Ĉ ij " Ĉ Ĉij +! Ĉij = "Ĉij "t = # ij +U "Ĉij "x # Ĉ j Molecular conformation "U i "x # Ĉi "U j "x 1 # v 1-1! p =.1 Pa.s! s =.1 Pa.s! = " s " s + " p L =9 L =1 f Ĉ =!!" L L! Ĉ Couette flow f L = L L! 3 =! p C!" +! s : analytical solution C! !" 1 5 Pinho, Resende, Younis, Sureshumar Kim 7

8 Time-averaged governing equations: RANS and RACE Continuity:!U i!x i = Momentum balance:! "U i "t "U +!U i = # "p " U +$ i "x "x s # "!u i "x "x "x i u + " i, p "x Rheological constitutive equation: FENE-P! ij = " s S ij +! ij, p RACE! ij, p = " p # f C C ij $ f L ij + " p # f C + c c ij! "c ij $ "u C ij + u # c i "u j j + c i "x "x "x = # * ij, p + p M ij CT ij NLT ij Closures required Pinho, Resende, Younis, Sureshumar Kim 8

9 Existing models for FENE-P: Li et al 6- equation model! "U i "t "U +!U i = # "p " U +$ i "x "x s # "!u i "x "x "x i u + " i, p "x Reynolds stress!uv = " T,v du dy! T,v = "! T,N! T,N = "u # y b DR! = "# a DR $ equation model shear stress only Re! "! xy, p dy = " M xy d NLT xy dy Re! Re! " I M xy = a+ b DR + c DR I NLTxy = a"+ b"dr + c"dr / 1 DR = 8 1! exp +!.5 We "! * # $ Re " 15!.5, 31. 4* 1! exp!.75l,- -51 Pinho, Resende, Younis, Sureshumar Kim 9

10 Existing models for FENE-P: unpublished FENE-P and based on DNS Leighton, Waler and Stephens APS meeting - Reynolds stress transport model -!Slow pressure-strain redistribution term is modified by polymer limits energy redistribution -!New term in RS equation: interaction of " p,ij turbulence -!New term in C ij equation NLT ij -!Additional diffusive flux terms not modeled Shaqfeh 6 AIChE Conference - -! v -f extension of Durbin s model !Simplified model: " p,ij proportional to mean strain elongation -!Coefficient has laminar and turbulent contribution -!Laminar part proportional to!u/!y -!Turbulent part proportional to ; -!Modifies pressure strain v equation -!One transport equation for C ; Pinho, Resende, Younis, Sureshumar Kim 1

11 DNS cases: channel flow u Fully-developed channel flow,y 1,x h We! = "u! # Re! = hu! " Re! = 395," =.9, L = 9 Low Drag Reduction We! = 5, DR = 18 High Drag Reduction We! = 1, DR = 37!7 models Pinho et al JNNFM 8 unpublished - Only LDR!8 model under develop.- Recalculated DNS + LDR HDR!Closures valid for 1 st higher order turbulence models Pinho, Resende, Younis, Sureshumar Kim 11

12 Time- average polymer stress! ij, p = " p # f C C ij $ f L ij + " p # f C + c c ij. -5 fc 1 /fc C 1.15 We=5; DR=18 We=1; DR= fc C 1 fc 1 fc C 1 fc 1 We= 5 DR= 18 We= 1 DR= f C C 1 >> f c 1 Can be neglected! Pinho, Resende, Younis, Sureshumar Kim 1

13 Function f C Function: f 8 Ĉ = L! 3 L! C + c 1 fc 1 7 C c We= 5 DR= 18 6 C c We= 1 DR= C > c C 3 f Ĉ b lm d i! f C b lm d i 1 f Ĉ b ij! f C b ij = This will be used frequently Pinho, Resende, Younis, Sureshumar Kim 13

14 Time-average evolution equation for the conformation: RACE " + #c ij #u! C ij +! u $ c i #u j. - j + c i #x #x #x * = $ +, f C C ij $ f L1 ij,- /! "c ij $ "u C ij + u # c i "u j j + c i "x "x "x M ij CT ij NLT ij + D ij = # * ij, p + p DNS: Housiadas et al 5 Phys Fluids 17, 3516, Li et al 6 a JNNFM. / Added for stability Should be negligible Oldroyd derivative Mean flow distortion Exact and large : turbulent distortion originates in distortion of Oldroyd derivative- not negligible Must be modeled : originates in advective term, negligible no need for modeling Pinho, Resende, Younis, Sureshumar Kim 14

15 Approximate equation for NLT ij!u f Ĉ i!u mm c j + f j Ĉ!x mm c i + C!x j f Ĉ mm!u i + C!x i f Ĉ mm!u # j + "!u!c i j!x!x!t $ # +"!C j!u u i!x n +!C!u i j u n!x!x n +! U c n j!u i +! U c n i!u j!c j!u + u i!c n!x!x n!x!x n!x n + u i!u j!x n!x n!x n!x $ # "!U *!u c i!u - j!x jn + c n!x in +,!x. / +!U j!u c i!x n +!U!u * i j!u j!u c n!x!x n + C i n!x n +!u!u - i j +,!x n!x!x n!x. / $ #!u " C!u i!u jn + C!u j!u!x n!x in + c!u i!u!x n!x jn + c!u j!u j!u!x n!x in + c i!u!x n!x n + c i!u j!x n!x n!x n!x $ = +!u j!c i!x!t + Pinho, Resende, Younis, Sureshumar Kim 15

16 First generation model for NLT ij st model First model Exact equation is too complex. Alternative model based on: 1! Identification of possible dependencies from inspection of exact equation Simplicity, but capturing main features 3 We=5 DR=18 f C mm NLT ij! $ = function S ij,w ij,c ij," N ij, #u u i j, #C ij, #NLT ij, M ij,u i u j #x #x #x n f C mm NLT ij! C E3 u " = f µ1 C u i u j + C $ 14 u i u W n C nj + u j u W n C ni + u u i W jn C n # # * coefficients 1 damping function C E3 =.35;C!14 =.15 f µ1 = 1! exp! 6.5 Pinho, Li, Younis, Sureshumar 8 JNNFM, in press Pinho, Resende, Younis, Sureshumar Kim 16

17 Second generation model for NLT ij- - 7 nd model Based on: 1 Modeling of exact equation: Ver Resende- 6ª feira, 11h15 HERE We=5 DR=18 ONLY f C mm NLT ij! = f C mm! - u C ". N1 C ij f C mm / #$ s C +C n U j N3 u i u m $ S pq S pq x U f N1 C N4 C jn x n U i x 4 : N +C N5 f N C mm ; ij 15 #$ s C N C j U i U m x n + C in U x n x + u j u m U i x + C i U j x U m x n U j 4 U j + C n x 5 6 x n *, + U i x * 1, U i x n U j x 7 * 8 9, + 4 coefficients: damping functions: C N1 =.164;C N =.3;C N3 =.4;C N4 = 1.11;C N5 =.45 " f N1 = 1!.8exp! y+ + * # $ 3 -, Pinho, Resende, Younis, Sureshumar Kim " f N = 1!.8exp! y+ + * # $ 5 -, 4 17

18 7 models for NLT ij : First versus second model 4 3 fc*nltxy fc*nltyy fc*nltii fc*nltxy fc*nltyy fc*nltii fc*nltxy fc*nltyy fc*nltii DNS First Second We=5, DR= Pinho, Resende, Younis, Sureshumar Kim 18

19 Modeling the Reynolds stress Major issue: what model to use for Reynolds stresses? 1 Reynolds stresses: Prandtl-Kolmogorov model -! closure Dissipation of turbulent inetic energy:! N "u! i " $ # Du i s "x m "x m Dt!u i u j = " T S ij! 3 # ij! T = C µ f µ +! s "u i "x m " with $ "U #u i "x +! s m "x!" N +" V "u i "x m " $ # "u u i "x m "x "u +! i " #"p "u s *! i " #" u i "u "x m "x m $ "x s! i " #"+ i, p i "x m "x m $ "x s = "x m "x m $ "x New term will be considered in the future As for Newtonian fluids, most terms in! N are approximated Next slide Pinho, Resende, Younis, Sureshumar Kim 19

20 Transport equation for turbulent inetic energy! D Dt = "!u #U i # iu "!u i " # pu i # #u + $ s " $ i #u i s + # i, pu i " i, p #x dx i #x i #x i #x i #x #x #x P Q N D N! #$ Q V #u i #x!" V When We increases DR P decreases! N decreases Q v increases in buffer l., but remains small! V increases in inertial l. -.5 We=5 DR=18 We=1 DR=37 Need to model well! V ! V P Q N Q V D N -! N P Q N Q V D N -! N -! V Q v is small Need to modify model of! N Pinho, Resende, Younis, Sureshumar Kim

21 Zoom of balance of : inertial sub-layer We=5 DR=18 P Q N Q V D N -! N -! V We=1 DR=37 P Q N Q V D N -! N -! V Pinho, Resende, Younis, Sureshumar Kim 1

22 Assumptions for viscoelastic stress wor:! V! V " 1 # $ i, p u i = p * x # C f C + c i mm mm u i u i x + c i f C mm + c mm x + -, 5 We=5; DR= 18 C i f C mm + c mm!u i << c i f C mm + c mm!u i!x 1 We=1; DR= 37!x C i f!u i /!x fc i!u i /!x SUM 8 C i f!u i /!x fc i!u i /!x SUM Except in viscous sublayer and buffer, but here ! V is not important Pinho, Resende, Younis, Sureshumar Kim

23 Further assumptions for viscoelastic stress wor:! V 1 8 fc mm c i!u i /!x fc mm c i!u i /!x fc mm c i!u i /!x fc mm c i!u i /!x We=5 DR=18 We=1 DR=37!u f c i!u i " C!x # V $ f C mm c i i!x 6 4 C! V! O 1 at We! = 5 but larger as DR increases This is NLT ii Pinho, Resende, Younis, Sureshumar Kim 3

24 Viscoelastic stress wor model! V " # p $ C u f C! V mm c i i x = C! V * We 5 +, - n.1 # p $ f C mm NLT ii Modeled 4 We! = 5 " 1.7 We! = 1 " 1.56 C! V = 1.7 n = ! V Re " We=5 Model DR=18! V Re We=1 " Model DR=37! V + Re " C! V versus Re " 1# $ We " * f C ii NLT jj 1 Previous model: We " = 5 only C! V = 1.76 Pinho, Li, Younis, Sureshumar 8 JNNFM, in press Pinho, Resende, Younis, Sureshumar Kim 4

25 Viscoelastic turbulent transport: Q V Q V! "# i, pu i = $ p "x f C mm CU ij! # 5 = f µ $ We ".53 f µ = 1! exp! 6.5 C!1 = 1.1;C!7 =.3 " C i f C mm + c mm u i + c i f C mm + c mm u i "x CFU ii, # +C j +C C *1 u i u m + u j u i m $ +x m +x m C * 7.! f C mm -, -. CU ii ± u j C i ± u i C j 15 1 CFU ii +fc mm CU ii Model CFU ii +fc mm CU ii Model // 1 1 We=5 DR=18 We=1 DR=37 CFU ii = C i f C mm + c mm u i! C FU " We # $u f C mm C i u i n $x n 5 Closure development followed similar procedures as that for NLT ij Pinho, Resende, Younis, Sureshumar Kim 5

26 and! transport equations: modified Nagano Hishida = d +! s + " f T# T - dy, $ Based on Newtonian model of Nagano Hishida d. d + f C * + P 1 "! N 1 "D N +! mm p - dy / dy, CU nny. / 1! f C mm p 3 " d!! N =! N + D N D N = 1.1 =! s $ # dy f T = exp "! R T 15 $ # Variable Prandtl numbers: Nagano Shimada 1993, Par and Sung 1995 NLT nn = d,! s + " f T# T. dy - $! " = 1.3 * + d! N dy /! N 1 + " f 1 C 1 " " f C N P f 1 = 1 f = 1!.3exp!R T + "E + E 3 p C!1 = 1.45 C! = 1.9 E =! s " # T 1$ f µ d U dy * E! p = Pinho, Resende, Younis, Sureshumar Kim 6

27 Predictions U + : Re! = 395; We! = 5; =.9, L =9 3 u + 5 u + = 11.7 ln -17. DNS Newtonian- Variable Pr 7 First- Variable Pr 7 Second- constant Pr 7 Second- Variable Pr 7 1 st model 7 nd model 7 nd model 15 1 u + = u + =.5 ln Pinho, Resende, Younis, Sureshumar Kim 7

28 Predictions! N : Re! = 395; We! = 5; =.9, L =9 +! N DNS 7 First- Variable Pr 7 Second- Constant Pr 7 Second- Variable Pr.! N+.15 DNS 7 First- Variable Pr 7 Second- Variable Pr 7 Second- Constant Pr Pinho, Resende, Younis, Sureshumar Kim 8

29 Predictions u v " p,xy : Re! = 395; We! = 5; =.9, L =9 uv + +! p,xy zoomed 1.3 uv +.8 DNS 7 First- Variable Pr 7 Second- Constant Pr 7 Second- Variable Pr! p,xy +.5 DNS 7 First- Variable Pr 7 Second- Constant Pr 7 Second- Variable Pr y * y* 1 Pinho, Resende, Younis, Sureshumar Kim 9

30 Conclusions and Acnowledgments - Closures for Low DR and High DR Resende, 6ª feira, 11h15 -!Closures for NLT ij,! V and Q V in fact for! V ij and Q ijv -!Developed simple low Reynolds -! model wors reasonably well -!Need to incorporate with better Reynolds stress closures: -, modified -! or -, Menter s SST or Durbin s v-f or RS transport deficiencies in base model are imp. -!Need to extend models to Maximum DR, L -!DNS in other canonical flows required for extension of turb. models Acnowledgments - Funding Fundação Calouste Gulbenian: Project 759 Fundação para a Ciência e Tecnologia Projects SFRH/BSAB/57/5 ; SFRH/BD/18475/4 Project Turbopol POCI/EQU/5634/4 Pinho, Resende, Younis, Sureshumar Kim 3