XIIIth International Workshop on Numerical Methods for non-newtonian Flows Hôtel de la Paix, Lausanne, 4-7 June 23 PROGRAMME Wednesday Afternoon, 4 June 23... 2 Novel Numerical Methods I (Chairman : Roland Keunings)... 2 High-Order Numerical Methods (Chairman : Antony Beris)... 2 Thursday Morning, 5 June 23... 3 Novel Numerical Methods II (Chairman : Tim Phillips)... 3 Concentrated Solutions and Melts (Chairman : Mike Graham )... 3 Thursday Afternoon, 5 June 23... 4 Viscoplastic/Inelastic Flows (Chairman : Ian Frigaard)... 4 Suspensions and Mixing (Chairman : Gareth McKinley)... 4 Friday Morning, 6 June 23... 5 Bead-Spring Modelling and Brownian Dynamics (Chairman : Eric Shaqfeh)... 5 Turbulent Flows (Chairman : Radhakrishna Sureshkumar)... 5 Friday Afternoon, 6 June 23... 6 Free Surface Flows (Chairman : Bamin Khomami)... 6 Stability and Nonlinear Dynamics (Chairman : Peter Monkewitz)... 6 Saturday Morning, 7 June 23... 7 Macroscopic Constitutive Modelling and Benchmark Problems (Chairman : Jay Schieber)... 7 Theoretical Developments (Chairman : Raj Huilgol)... 7 1
Wednesday Afternoon, 4 June 23 Registration 1: AM 12: PM Lunch 12: PM 1:3 PM Introduction 1:45 PM 2: PM Lectures 2: PM 6:3 PM Novel Numerical Methods I (Chairman : Roland Keunings) 2: PM 2:2 PM M. Laso and J. Ramírez. Implicit micro-macro methods. 2:25 PM 2:45 PM M. Ellero, M. Kröger and S. Hess. A hybrid method for efficient CONNFFESSIT simulations of fully uncorrelated ensembles of polymers. 2:5 PM 3:1 PM C. Le Bris, B. Jourdain and T. Lelièvre. On variance reduction issues in the micro-macro simulations of polymeric fluids. 3:15 PM 3:35 PM C. Chauvière and A. Lozinski. Simulation of dilute polymer solutions using a Fokker-Planck equation (comparison between 2D and 3D FENE models). 3:4 PM 4: PM G. Pan and C. Manke. Simulation of polymer solutions by dissipative particle dynamics. Break 4:5 PM 4:25 PM High-Order Numerical Methods (Chairman : Antony Beris) 4:25 PM 4:45 PM M. I. Gerritsma. Least-squares spectral element methods for non-newtonian flow. 4:5 PM 5:1 PM N. Fiétier. Simulation of viscoelastic fluid flows through contractions and constrictions with spectral and mortar element methods. 5:15 PM 5:35 PM R. G. M. van Os and T. N. Phillips. The prediction of complex flows of polymer melts using spectral elements. 5:4 PM 6: PM T. N. Phillips and K. D. Smith. A spectral element approach to the simulation of viscoelastic flows using Brownian configuration fields. 6:5 PM 6:25 PM X. Ma, V. Symeonidis and G.E. Karniadakis. A spectral vanishing viscosity method for stabilizing viscoelastic flows. Dinner 7: PM 2
Thursday Morning, 5 June 23 Breakfast 7: AM 8:3 AM Lectures 8:3 AM - 12:1 AM Novel Numerical Methods II (Chairman : Tim Phillips) 8:3 AM 8:5 AM I. J. Keshtiban, F. Belblidia and M. F. Webster. Simulating weakly compressible non-newtonian flows. 8:55 AM 9:15 AM H. K. Rasmussen. The 3D Lagrangian integral method. 9:2 AM 9:4 AM A. Lozinski and R. G. Owens. Modelling highly non-homogeneous flows of dilute polymeric solutions using Fokker-Planck-based numerical methods. Concentrated Solutions and Melts (Chairman : Mike Graham ) 9:45 AM 1:5 AM J. van Meerveld and H. C. Öttinger. Molecular-based description of polydisperse polymeric liquids. Break 1:1 AM 1:3 AM 1:3 AM 1:5 AM P. Wapperom and R. Keunings. Impact of decoupling approximation between tube stretch and orientation in rheometrical and complex flow simulation of entangled linear polymers. 1:55 AM 11:15 AM J. Fang, A. Lozinski and R. G. Owens. More realistic kinetic models for concentrated solutions and melts. 11:2 AM 11:4 AM J. Schieber. Solving a full chain temporary network model with sliplinks, contour-length fluctuations, chain stretching, and constraint release using Brownian dynamics. 11:45 AM 12:5 PM P. K. Bhattacharjee, J. Ravi Prakash and T. Sridhar. Stress relaxation after step extensional strain in an entangled polymer solution. 12:1 PM 12:3 PM T.M. Nicholson. Measurement and modelling of polymer melt flow and extrudate swell. 3
Thursday Afternoon, 5 June 23 Lunch 12:4 PM 1:45 PM Lectures 2: PM 6:1 PM Viscoplastic/Inelastic Flows (Chairman : Ian Frigaard) 2: PM 2:2 PM E. Mitsoulis. Flow of viscoplastic fluids through expansions and contractions. 2:25 PM 2:45 PM E. Mitsoulis and R. Huilgol. Finite stopping times in Couette and Poiseuille flows of viscoplastic fluids. 2:5 PM 3:1 PM M. A. Moyers-Gonzalez and I. A. Frigaard. Accurate numerical solution of multiple visco-plastic fluids in ducts. 3:15 PM 3:35 PM S. Alexandrov. Frictional effects in viscoplastic flows. 3:4 PM 4: PM D. Vola. On a numerical strategy to compute non-newtonian fluids gravity currents. Break 4:5 PM 4:25 PM 4:25 PM 4:5 PM S. Miladinova and G. Lebon. Thin-film flow of a power-law liquid down an inclined plate. Suspensions and Mixing (Chairman : Gareth McKinley) 4:55 PM 5:15 PM V. Legat. Micro-macro modelling of black carbon mixing. 5:2 PM 5:4 PM W. R. Hwang, M. A. Hulsen, H. E. H. Meijer. Direct simulations of particle suspensions in viscoelastic fluids in Lees-Edwards sliding bi-periodic frames. 5:45 PM 6:5 PM V. Valtsifer and N. Zvereva, Computer simulation and experimental investigation of rheological behaviour of nanoparticles in suspension. Reception 6:45 PM Bus departs 7:45 PM Lake cruise with buffet dinner on the Henry Dunant from Ouchy 8: PM 1:3 PM 4
Friday Morning, 6 June 23 Breakfast 7: AM 8:3 AM Lectures 8:3 AM - 12:1 AM Bead-Spring Modelling and Brownian Dynamics (Chairman : Eric Shaqfeh) 8:3 AM 8:5 PM R. Prabhakar and J. Ravi Prakash. Superposition of finite extensibility, hydrodynamic interaction and excluded volume effects in bead-spring chain models for dilute polymer solutions. 8:55 AM 9:15 AM P. T. Underhill and P. S. Doyle. On the coarse-graining of polymers into bead-spring chains. 9:2 AM 9:4 AM R. Akhavan, Q. Zhou. A multi-mode FENE bead-spring chain model for dilute polymer solutions. 9:45 AM 1:5 AM R. M. Jendrejack, J. J. de Pablo and M. D. Graham. DNA dynamics in a microchannel: relaxation, diffusion and cross-stream migration during flow Break 1:1 AM 1:3 AM Turbulent Flows (Chairman : Radhakrishna Sureshkumar) 1:3 AM 1:5 AM V. K. Gupta, R. Sureshkumar and B. Khomami. Numerical simulation of polymer chain dynamics in turbulent channel flow. 1:55 AM 11:15 AM K. D. Housiadas and A. N. Beris. Direct numerical simulations of polymerinduced drag reduction in turbulent channel flows. 11:2 AM 11:4 AM M. Manhart. A coupled DNS/Monte-Carlo solver for dilute suspensions of small fibres in a Newtonian solvent. 11:45 AM 12:5 PM D. O. A. Cruz and F. T. Pinho. A low Reynolds number k model for drag reducing fluids. 5
Friday Afternoon, 6 June 23 Lunch 12:15 PM 1:45 PM Lectures 2: PM 5:4 PM Free Surface Flows (Chairman : Bamin Khomami) 2: PM 2:2 PM G. McKinley. Free surface flows of viscoelastic fluids 2:25 PM 2:45 PM G. Bhatara, E. S. G. Shaqfeh and B. Khomami. A study of a free surface viscoelastic Hele-Shaw cell flow using the finite element method 2:5 PM 3:1 PM A. Bonito, M. Laso and M. Picasso. Numerical simulation of 3D non- Newtonian flows with free surfaces. 3:15 PM 3:35 PM K. Foteinopoulou, V. Mavrantzas and J. Tsamopoulos. Numerical simulation of bubble growth during filament stretching of pressure-sensitive adhesive materials. 3:4 PM 4: PM Y. Dimakopoulos and J. Tsamopoulos. Gas-penetration in straight tubes partially or completely occupied by a viscoelastic fluid. Break 4:5 PM 4:25 PM Stability and Nonlinear Dynamics (Chairman : Peter Monkewitz) 4:25 PM 4:45 PM K. Atalik and R. Keunings. On the occurrence of even harmonics in large amplitude oscillatory shear experiments. 4:5 PM 5:1 PM B. Sadanandan, K. Arora and R. Sureshkumar. Stability analysis of nonviscometric viscoelastic flows. 5:15 PM 5:35 PM M. Sahin and R. G. Owens. Linear stability analysis of the non-newtonian flow past a confined circular cylinder in a channel. Workshop Banquet. Bus Departs 6:3 PM 6
Saturday Morning, 7 June 23 Breakfast 7: AM 8:55 AM Lectures 8:55 AM 12:1 AM Macroscopic Constitutive Modelling and Benchmark Problems (Chairman : Jay Schieber) 8:55 AM 9:15 AM G. Mompean, L. Thais and L. Helin. Numerical simulation of viscoelastic flows using algebraic extra-stress models based on differential constitutive equations. 9:2 AM 9:4 AM Y. Fan. Boundary layers in the viscoelastic flow around a confined cylinder. 9:45 AM 1:5 AM M. A. Alves, P. J. Oliveira and F. T. Pinho. Flow of PTT fluids through contractions effect of contraction ratio. Break 1:1 AM 1:3 AM Theoretical Developments (Chairman : Raj Huilgol) 1:3 AM 1:5 AM X. Xie and M. Pasquali. A convenient way of imposing inflow boundary conditions in two- and three-dimensional viscoelastic flows. 1:55 AM 11:15 AM B. Caswell, G. E. Karniadakis and V. Symeonidis. The hole-pressure due to a tube on one wall of a plane channel. 11:2 AM 11:4 AM A. R. Davies. Transient decay rates in some common constitutive models of differential and integral type. 11:45 AM 12:5 PM M. Renardy. Jet breakup of a Giesekus fluid with inertia. Lunch 12:15 PM 1:45 PM 7
A LOW REYNOLDS NUMBER k-! MODEL FOR DRAG REDUCING FLUIDS D. O. A. Cruz Departamento de Engenharia Mecânica, Universidade Federal do Pará, Belém, Brasil, doac@ufpa.br F. T. Pinho Centro de Estudos de Fenómenos de Transporte, Faculdade de Engenharia, Universidade do Porto, Portugal fpinho@fe.up.pt,http://www.fe.up.pt/~fpinho XIIIth International Workshop on Numerical 4-7th June 23 Hôtel de la Paix, Lausanne, Switzerland
Outline Introduction Proposed model Results Conclusions Future developments
Introduction Fluid rheology affects hydrodynamic behaviour Viscoelastic effects on turbulence are largely unpredicted Phenomenological models exist WE WANT Fluid rheology: measured properties Flow rate or pressure gradient and geometry Turbulent flow characteristics Turbulence modelling of viscoelastic engineering flows remains a challenge
Introduction - Objective Objective: Development of a coupled turbulenceconstitutive equation closure Selection of (simple) constitutive equation Reynolds averaged equations Modelling new terms and modification of transport equations Damping functions for low Reynolds number effects Comparisons for parameterisation and behaviour assessment
Proposed model - Rheological constitutive model Route 1 Viscoelastic model: shear-thinning, viscoelastic, DNS simulations FENE-P: $ f ( A kk )A ij +! "A ij & % "t [ ]! ij = " p # f ( A kk )A kk $% ij + u k "A ij "x k # A jk "u i "x k # A ik "u j "x k ' ) = * ij ( with f ( A kk ) = L 2 L 2! A kk TOO COMPLEX at this stage: double, triple & quadruple (?) correlations
Proposed model - Rheological constitutive model Route 2 Simple model: shear-thinning, with assumed relevant features Reduced modifications relative to Newtonian turbulence model Drag reduction: relevance of strain-hardening extensional viscosity for constant shear viscosity Fluids are usually shear-thinning Strain/shear- hardening Trouton ratio MODIFIED GENERALISED NEWTONIAN FLUID
Proposed model - Modified GNF! ij = 2µS ij µ =! v " K [ e # 2 ] p$1 2 % µ = K [ v & 2 ] n$1 2 K [ e # 2 ] p$1 2 Shear viscosity contribution Extensional viscosity contribution via Trouton ratio n <1 p >1 1! e " 3! v # ( ) ( ) = K e [" 2 ] p$1 2 Effect of! only in turbulent flow Effect of! conditions reduced under low Reynolds number
Proposed model - Transport equations 1 Momentum Modified stress New stress! "U i "t ( ) "U +!U i k = # "p + " 2µ S ik + 2µ's ik #!u i u k "x k "x i "x k Turbulent kinetic energy Closure by 2- equation model: k-!! Dk Dt = " #u j p " # $ 1 #x j #x j 2!u ' iu j u i " 2µu i s ij " 2µ'u i S ij " 2µ'u i s % & ij ( ) " 2µs ij 2 2 " 2µ's ij "!2µ's ij S ij! "u i u j S ij New term Dissipation:!
Proposed model - Transport equations 2 Rate of dissipation 1) Very complex: as with Newtonian fluids, all terms are modelled 2) There are new terms originating from advection (order of mag.) 3) Weakness of any turbulence model Two models tested: Model A: modified stress and without new stress Model B: modified stress and new stress (preliminary res.)
Proposed model - Average viscosity (Models A & B) µ = f v µ h + ( 1! f v )" v high Reynolds number contribution.. (effect of fluctuating " and! ) pure shear viscosity at walls! v = K [ v " 2 ] n#1 2 µ h = ( C µ!) 3m( m"1) A2 8+3m( m"1) A 2 2 4 m( m"1) A 2 8+3m( m"1) A 2 k 6m( m"1) A 2 [ ] m 8+3m( m"1) A 2 # 8"3( m"1) A 2 8 8+3m( m"1) A 2 8+3m( m"1) A B 2 (derived from order of magnitude and pdf arguments) Rheological measurements: B, m
Proposed model - New stress (Model B) 1 2µ' s ij? µ'! K v K e "' p#1 $ ' n#1!' " # ' A! s ij s ij A! = S A! 2µ' s ij! K v K e A " p#1 S p+n#2 s ij In boundary layers: P k =!" #!uv $U $y s ij! "u i # u iu j "x j L % 2µ S2 Inviscid estimate of dissipation of k
Proposed model - New stress (Model B) 2 2µ' s ij = {[ 1 + C ] p+n!2!1} K v K e p!1 A ", &%U).#$ T ( +. ' %y *.. 2µ -. 2 / 1 1 1 1 1 p +n!2 2 $ T 1 L C du dy du dy with 1 =! L 3 c u and u 2 k R = ) R # exp! k &, + % 2(!1 * $ u " '. - 1 / to match low and high Reynolds number behaviour 1 L c =! u " 3 1 L c =! k 3/ 2
Proposed models A & B - Pipe flow Momentum: Reynolds stress: k equation: 1 r! equation: 1 r 1 r # k d ) dr r # % µ du $ dr! "uv + 2 µ' s & xr( * + ' d * dr r $ µ! + " ' T & ) dk -, / uv 1U + % ( dr. 1r 2µ' s xr d + dr r % µ +!" ( T ' & ) * d $. $ - +!f, dr 1 C $1 / k P 1!f 2 C $2 # $!"uv = "# T $U $r, -.! dp dx = Model B %! "uv = "C µ f µ k 2 $ 2 k +" " T 1 1 f µ 1U 1r $ 2 + 2" & % Model B ( ) 22 U ' 1 k 1r Parameters and functions: Nagano - Hishida (except f! ) % ' & 2r 2 ( * ) 2 & ' ) ( 2 $U $r = " + C T d $ dµ $3 # E " dr dr =
Proposed model - Damping function Van Driest s (1956) philosophy: Stokes second problem Newtonian: f µ = [ 1! exp (! y + A + )] " [ 1! exp (! y + A + )] Shear-thinning and strain-hardening contributions ( * " f µ = 1! $ 1 + 1! n 1 + n y + % ) ' # & + * Van Driest s parameter! 1+n 1!n A + to quantify, ( * * " - / ) 1! $ 1 + p!1. * 3! p y + C * # $ + 1! p 2! p (wall viscosity) viscometric contribution extensional contribution % ' &'! 3! p p!1 A+, * - *.
Results - Experimental (Escudier et al, 1999) 1 1-1 f 6 u + 5.125% PAA Re=429.25% CMC Re= 166.3% CMC Re=43.9/.9% CMC/XG Re=453 4 1-2 3 1-3.125% PAA.2% PAA.25% CMC.3% CMC.2% XG.9/.9% XG/CMC 1 3 1 4 Re 1 5 2 1 1 1 1 1 2 1 3 y +
Results - Experimental (Escudier et al, 1999) 2
Results - Experimental (Escudier et al, 1999) 3 N 1 data not used
Results model A- Determination of C -.125% PAA 1-1 f f =.316Re!.25 inelastic shear-thinning Two formulations of f µ y + + y w M1: in f! M2: in f! f = 64 / Re 1-2 1-3 Exp data C=5; M1 C=7; M1 C=9; M1 C=5; M2 C=7; M2 C=9; M2 M1; p=1 M2; p=1 MDRA-Virk 1 3 1 4 Re 1 5 w Best damping function Formulation M2 C= 9
Results model A - Friction factor - Limiting cases 1-1 f Inelastic shear-thinning n<1;p=1 1-1 f Strain-hardening n= 1; p>1 K e =2 K e =.5 Dodge & Metzner, n=1 1-2 n 1..8.6.4 1-2 f=64/re p 1. 1.2 1.4 1.6 Virk's MDRA 1.8 1-3 1 3 1 4 Re 1 5 g 1-3 1 3 1 4 Re 1 5 w
Results model A - Friction factor - Other drag reducing fluids 1.2% XG.25% CMC 1-1 1-1 f f =.316Re!.25 Dodge & Metzner (1959) f f =.316Re!.25 Dodge & Metzner (1959) f=64/re f=64/re 1-2 1-2 MDRA- Virk MDRA- Virk Exp data Pred M1 Pred M2 Exp data Pred M1 Pred M2 1-3 1 3 1 4 1 5 Re w 1-3 1 3 1 4 Re 1 5 w
Results model A - Friction factor - Other drag reducing fluids 2.3% CMC.9% CMC/ XG 1-1 f 1-1 f f =.316Re!.25 Dodge & Metzner (1959) f =.316Re!.25 Dodge & Metzner (1959) 1-2 f=64/re 1-2 f=64/re Exp data Pred M1 Pred M2 MDRA- Virk Exp data Pred M1 Pred M2 MDRA-Virk 1-3 1 3 1 4 Re 1 5 w 1-3 1 3 1 4 1 5 Re w
Results model A - velocity profile - Limiting cases 4 u + 35 3 Inelastic shear-thinning n<1;p=1 Dodge & Metzner log law n=.4.6 8 u + 7 6 Strain-hardening n= 1; p>1 p=1 p=1.2 p=1.4 p=1.6 p=1.8 Virk's MDRA 25 u + + = [ y DM ] 1/ n.8 5 2 1. 4 15 3 1 2 5 1 Newtonian log law 1-1 1 1 1 1 2 1 3 y + DM 1 1 1 1 2 1 3 y + w
Results model A - velocity profile - drag reducing fluids 1 35 u + 3 25.9% CMC/ XG (Re=45 3) Exp data Pred M1 Pred M2 Pred Newt u + =2.5lny + w +5.5 u + =11.7lny + w -17. 35 u + 3 25.25% CMC (Re=16 5) Exp data Pred M1 Pred M2 u + =2.5lny + w +5.5 u + =11.7lny + w -17. u + =y + w u + =y + w 2 2 15 15 1 1 5 5 1 1 1 1 y + w 1 1 1 1 y + w
Results model A - velocity profile - drag reducing fluids 2.3% CMC (Re=4 3).125% PAA (Re=42 9) 35 u + 3 Exp data Pred M1 Pred M2 u + =2.5lny + w +5.5 35 u + 3 Exp data Pred M1 Pred M2 u + =2.5lny + w +5.5 25 u + =11.7lny + w -17. u+=y + w 25 u + =11.7lny + w -17. u+=y + w 2 2 15 15 1 1 5 5 1 1 1 1 y + w 1 1 1 1 y + w
Results model A - Turbulent kinetic energy.125% PAA (Re= 429) 14 k + 12 Exp data Pred Newt Pred M1 Pred M2.14 k/u 2.12 Exp data Pred Newt Pred M1 Pred M2 1.1 8.8 6.6 4.4 2.2. 1 1 1 1..2.4.6.8 r/r 1. y + w
Results model A - Shear stress & damping function.125% PAA (Re= 429) 1. 1. uv/u t 2.8 Pred Newt Pred M1 Pred M2 f µ.8 Newt Model 1 Model 2.6.6.4.4.2.2...2.4.6.8 r/r 1.. 1 1 1 1 y + w
Results model B - 1.125% PAA.125% PAA - Re!4 1-1 f 6 u + 5 Exp. data C=9 C = C=9 C =-.6 C=5 C =-.9 4 1-2.125%PAA p=1 Exp data C=9 C = C=9 C =-.6 C=7 C =-.6 C=7 C =-.9 C=5 C =-.9 3 2 1 1-3 1 3 1 4 Re 1 5 w 1-1 1 1 1 1 2 1 3 y + w
Results model B - 2 12 k + 1 Exp. data C=9 C = C=9 C =-.6 C=5 C =-.9.125% PAA - Re!4 1.!/"U! 2.8 C=9 C =-.6 2µS ij -uv 2µ's ij 8.6 6.4 4 2.2 1 1 1 1 2 1 3 y + w...2.4.6.8 r/r 1.
Conclusions Coupled turbulence-rheological model derived. Only requires fluid properties as input No previous tuning required Predicts intense drag reduction (DR) for large p Predicts well DR for inelastic shear-thinning fluid Predictions of DR for elastic fluids is fair Improvements required in u + - y + & k New term in momentum and k helps, but more inv. required Different formulation?
Future developments SHORT TERM Wall consistent damping functions Non-linear k-! for Reynolds stress anisotropy Reynolds stress model for Reynolds stress anisotropy MEDIUM TERM Adoption of true viscoelastic rheological equation: Oldroyd-B/FENE-CR and FENE-P Development of turbulence closure. Use of DNS results k-!, non-linear k-! and Reynolds stress models
A LOW REYNOLDS NUMBER k-! MODEL FOR DRAG REDUCING FLUIDS D. O. A. Cruz Departamento de Engenharia Mecânica, Universidade Federal do Pará, Belém, Brasil, doac@ufpa.br F. T. Pinho Centro de Estudos de Fenómenos de Transporte, Faculdade de Engenharia, Universidade do Porto, Portugal fpinho@fe.up.pt,http://www.fe.up.pt/~fpinho ACKNOWLEDGMENTS 1) FCT (Portugal): Proj. POCTI/EME/37711/21; POCTI/EQU/37699/21 2) ICCTI (Portugal)- CNPq (Brasil) XIIIth International Workshop on Numerical 4-7th June 23 Hôtel de la Paix, Lausanne, Switzerland