A k-! LOW REYNOLDS NUMBER TURBULENCE

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1 A - LOW REYOLDS UMBER TURBULECE MODEL FOR TURBULET CHAEL FLOW OF FEE-P FLUIDS P. R. Resende Centro de Estudos de Fenómenos de Transporte, Universidade do Porto, Portugal F. T. Pinho Centro de Estudos de Fenómenos de Transporte, Universidade do Porto, Portugal B. A. Younis Dep. Civil and Environmental Engineering, University of California, Davis, USA K. Kim Dep. Mechanical Engineering,Hanbat ational University, Daejeon, South Korea R. Sureshumar Dep. Biomedical and Chemical Engineering, Syracyse University, Syracuse, Y, USA VI th Annual European Rheology Conference 7 th -9 th April 21 Göteborg, Sweden

2 Drag reduction: motivation Drag reduction in fully-developed channel flow 3 u u + = 11.7 ln y We DR [%] u + = y + u + = 2.5 ln y Existing models (1 st order) -": Pinho et al, JFM 154 (28) 89 -" improved:pinho et al (21) in prep -"-v 2 -f: Iaccarino et al,165(21)376 Can a - model improve on -"? Advantages: Valid across all BL (no damping) Better in BL with adverse pres. grad y + Disadvantages: Too sensitive to in free stream CEFT-FEUP Centro de Estudos de Fenómenos de Transporte AERC 21, Göteborg, Sweden 2

3 DS cases: channel flow u Fully-developed channel flow 2,y 1,x 2h We = "u 2 # Re = hu " DS test/calibration cases Re = 395," =.9, L 2 = 9 Low Drag Reduction We = 25, DR = 18% High Drag Reduction We = 1, DR = 37% CEFT-FEUP Centro de Estudos de Fenómenos de Transporte AERC 21, Göteborg, Sweden 3

4 ew model: Governing Equations Continuity: U i = x i Momentum balance: Reynolds decomposition: ˆB = B + b Overbar upper-case: time-averaged quantities Lower-case: fluctuating quantities "U i "t "U + U i = # "p " 2 U +$ i "x "x s # " (u i "x "x "x i u ) + "% i, p "x Rheological constitutive equation: FEE-P ij = 2" 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 LT ij Independent of turbulence model Closures required CEFT-FEUP Centro de Estudos de Fenómenos de Transporte AERC 21, Göteborg, Sweden 4

5 Conformation (RACE) equation " + #c ij % #u C ij + u $ c i #u j - j + c i #x,- (. * ) / = $ +, f ( C )C ij $ f ( L)1 ij. / $ f ( C + c )c ij M ij CT ij LT ij Model for LT ij essentially identical to that for -", except in some coefficients/ functions f ( C mm ) LT ij = f ( C mm ) *, + f 1 C ij -, f ( C mm ) $ #U " f 2 C i #U j., j + C i )/ % (, $ C + f n $ #U j #U 3 u i u m #U m + u j u i #U m m ) 1 2S pq S pq % #x n #x + 1 $ #U #U m C jn u i u m + C in u j u m % n ( 1 2S pq S pq % #x n $ #U " f 4 C #U i #U jn + C #U j 2 #U j #U in + C i n + #U #U i j #x n #x n 3 4 #x n #x n 6 7 ) + f 5 C mm : ij % ( s ( ) )) (() f i = f (We, y + ) CEFT-FEUP Centro de Estudos de Fenómenos de Transporte AERC 21, Göteborg, Sweden 5

6 The specific dissipation rate: "u i u j = 2µ T S ij 2 3 "# ij Prandtl- Kolmogorov closure for Reynolds Stress µ T = l Transport equation How to determine l? Generally difficult Various alternatives 1) Estimate of dissipation (large scale) 3 2 l [] = length2 time 3 Chou (1945) µ T 2 " 2) Specific dissipation rate: [ ] = 1 time " Kolmogorov (1942) is better behaved near walls, but more sensitive far from walls µ T " " 2# C $ y 2 CEFT-FEUP Centro de Estudos de Fenómenos de Transporte AERC 21, Göteborg, Sweden 6

7 Reynolds stress closure: eddy viscosity model (-" -) Prandtl-Kolmogorov model Pinho et al (21): -" u i u j = 2" T S ij 2 3 # ij T = T " T P T = C µ f µ 2 " T P = C µ f µ C µ P f µ P C 2 " = " ew model with ote: C = C µ C T = f µ P " T = f µ C P µ f P µ C " CEFT-FEUP Centro de Estudos de Fenómenos de Transporte AERC 21, Göteborg, Sweden 7

8 Transport equation for D Dt = "u i u D V = p " ( p " P exact # $ C i f ( C mm + c mm )u i + c i f ( C mm + c mm )u i % # $ ) f C mm % #U i # " u i " # pu i # 2 #u + $ s " $ i #u i s + #% i, pu i " % i, p dx i #x i #x i #x i ( ) C i ( FU ) i + ( CU ) ij 2 D T Unchanged (ewtonian) * D exact #" $ D V Previous Model " = C # Essentially unchanged Coefficients functions #u i " V Previous Model C i u ( FU ) i f FU C i u i n x n f FU = f FU ( We) f ( C mm )CU ij ( ) % $C j $C = " f #1 u i u m + u j u i ( m $x m $x m ) * " f f C # mm 7 f 1, f 7 = f ( We) + ± u 2 j C i ± u 2 i C j,-. / CEFT-FEUP Centro de Estudos de Fenómenos de Transporte AERC 21, Göteborg, Sweden 8

9 Viscoelastic stress wor: " V V " 1 # $ i, p %u i p ) + %x #( c f C + c i mm mm * ( ) %u i %x,. - u f c i u i " f x # V $ f ( C mm )c i i x LTii f V = f V ( We) Same model as in -" V = f V " p #$ f ( C mm ) LT ii 2 Unchanged CEFT-FEUP Centro de Estudos de Fenómenos de Transporte AERC 21, Göteborg, Sweden 9

10 Transport equation of : final modeled form = d + % p + s + " f T# T - dy, $ Based on ewtonian model of agano Hishida (1984) ( ) * d dy. + P 1 "C 2 + p / 3 d dy + -, f ( C mm ) C n ( FU ) n + CU nny 2. 1 p / f ( C mm ) 3 LT nn 2 ew form = 1.1 f T = exp " R T 15 # ( ) 2 Variable Prandtl numbers: agano Shimada (1993), Par and Sung (1995) $ % CEFT-FEUP Centro de Estudos de Fenómenos de Transporte AERC 21, Göteborg, Sweden 1

11 Specific rate of deformation: transport equation D Dt T = P " # + $ + D Production Destruction Redistribution Turbulent diffusion + D Molecular diffusion Viscoelastic interaction V + E D Dt = P " + # + D T + D + D V " V = " C µ D Dt = 1 C µ D" Dt # D Dt D" Dt D Dt T = P " # + $ + D + D V + E " = C "1 P + #.( $ s + $ p + % + T #" 1 #x i ) *, C "2 " 2 + C ( " $ s / #x i 2 + % + # #" V T ) *, - + E #x i #x " i Viscous cross-diffusion (Bredberg et al. 22) CEFT-FEUP Centro de Estudos de Fenómenos de Transporte AERC 21, Göteborg, Sweden 11

12 Viscoelastic contribution to : model Definition and model V E = 1 C µ E V " # D V + "V Slide 9 Slide 1 # E V " 2# u p i s $ ( L 2 % 3) x m x +, -x m ( f ( C nn ) f Ĉ pp ( )c qq C i. ) */ Model of V E E V 2 " # f DR f DR = f DR $ V ( C F1 L2 # 3 % ( We,", L 2 ) ( ) ( ) 2 + C F 2 C ii f C $% 2 ) improved version relative to -", it also incorporates effects of % L 2 CEFT-FEUP Centro de Estudos de Fenómenos de Transporte AERC 21, Göteborg, Sweden 12

13 Mean velocity 1: Re " = 395; %=.9, L 2 =9 3 u We= We= 25 }-" We= 1 DS- We= 25 DS- We= 1 We= } We= 25 - We= 1 u + = 11.7 ln y u + = 2.5 ln y u + = y y + CEFT-FEUP Centro de Estudos de Fenómenos de Transporte AERC 21, Göteborg, Sweden 13

14 Turbulent inetic energy: Re " = 395; %=.9, L 2 = DS- Mansour (We= ) DS- We= 25 DS- We= 1 We= We= 25 -" We= 1 We= We= 25 - We= 1 } } T = C µ f µ 2 ( ) " 1# C P µ f P µ C y + CEFT-FEUP Centro de Estudos de Fenómenos de Transporte AERC 21, Göteborg, Sweden 14

15 Dissipation of by solvent: Re " = 395; %=.9, L 2 = DS- Mansour (We=) DS- We= 25 DS- We= 1 We= } We= 25 -" We= 1 We= We= 25 } - We= y + CEFT-FEUP Centro de Estudos de Fenómenos de Transporte AERC 21, Göteborg, Sweden 15

16 LT ii : Re " = 395; %=.9, L 2 =9 5 LT ii * 4 DS- We= 25 DS- We= 1 We= } We= 25 -" We= 1 We= We= 25 } - We= y + CEFT-FEUP Centro de Estudos de Fenómenos de Transporte AERC 21, Göteborg, Sweden 16

17 Conclusions, Future Wor and Acnowledgments - - model developed, it wors well at Low DR and High DR (5%) - Closure for elastic terms: similar to corresponding in -" - Slightly better than -" - More stable (easier convergence) - eed for 2 nd order Reynolds stress closures - eed to extend models to Maximum DR, % L 2 Acnowledgments - Funding Fundação para a Ciência e Tecnologia Project PTDC/EQU-FTT/7727/26 CEFT-FEUP Centro de Estudos de Fenómenos de Transporte AERC 21, Göteborg, Sweden 17