CONFERENCE ON COMPLEX FLOWS OF COMPLEX FLUIDS University of Liverpool, UK, March 17-19, 2008 SOME THOUGHTS ON DIFFERENTIAL VISCOELASTIC MODELS AND THEIR NUMERICAL SOLUTION Paulo J. Oliveira Universidade da Beira Interior Dep. Eng. Electromecânica UBI
TOPICS Derivation of a generalized Oldroyd-B model The traceless stress tensor formulation Law-of-the-wall for viscoelastic flows
DERIVATION OF GENERALISED MODEL First TOPIC
GOVERNING EQUATIONS iu= 0 Du ρ Dt τ η 2 = p+ i τtot = p+ s u+ iτ
OLDROYD-B EQUATIONS Differential formulation Original: τ λ τ η D λ D τ + ( τ ) = 2 + tot tot 0 r Alternative: = + = 2 D+ τ τ τ η tot s p s τ + λ ( τ ) = 2η D p τ With: η0 = ηs+ ηp ηs λr β = = η λ 0
OLDROYD-B: molecular formulation ( ) A= A I η p λ τ = ( A I) λ (1) (2) ( ) λ η A p η I τ = A I = p + 2D λ τ λτ = ηp + 2D η λ τ + τ = 2η pd p ( λ and ηp are constants) with: I = 2D
Equations with conservative property Idea based on conservativeness property (at continuum level) Convected variables should appear inside derivative: D λ τ ( ) ( ) Dt λ τ = + t div( λuτ ) so that balances of fluxes (out-in): ( ) n+ 1 ( ) n V P λτ λτ P P + λu iaτ λu iaτ = t flux e flux w e W w P e w E... x
Example of Conservative stress models FENE-MCR: not conservative FENE-CR: conservative PTT: not conservative (apparently) λ τ + ( τ ) = 2η p D f λ τ + τ = 2η p D f f τ + λ τ += η D ( ) 2 p FENE-P: original equation also apparently not conservative D lnz λ( τ ) + Z τ λ + ( 1 εb) nktλ = 2( 1 εb) nktλ Dt τ I D
Example of Conservative stress models: FENE-P (cont) with function: Z 3 tr( τ ) = 1+ ( 1 εb) + 2 b 3nkT ε = bb ( + 2) But re-arrangement shows conservativeness: λ ( η / ) p aη D a f p τ + τ = 2 D I f f Dt with: f Z a b+ 5 b = ηp = nktλ b + 2 b + 5
CONCLUSION So, in general all previous equations can be cast in terms of variable relaxation times and viscosity coefficients: as: η f p λ λ ηp τ + τ = 2 D+... f f f with different functions for the various models:
TYPICAL STRESS FUNCTIONS: FENE-CR: f [ τ] = L 2 + ( λ / η ) Tr( τ ) 0 p0 L 2 3 FENE-P: f 3 λ = 1+ 1 + Tr( ) b [ τ] 0 τ + 2 3aηp 0 PTT: f λε = 1 + Tr( τ ) [ τ] 0 η p0
GENERALISATION: Now, both viscosity and relaxation time are variable, functions of invariants of τ : η = η ( τ ) p p λ = λ( τ ) We shall use for the constant zero-shear rate values: In general: ( fa) η λ p0 0 Df = A+ f A Dt
DERIVATION OF GENERALISED MODEL: Assume as still valid: ( ) A= A I η p λ τ = ( A I) λ (1) Equilibrium between rotation/stretching and relaxation (2) Kramers expression Upper convected derivative of Eq. (2) gives: Dη Dη λτ ηpa ηpi A+ ηpa I+ ηpd Dt Dt ( ) ( ) ( ) p p = = 2
DERIVATION OF GENERALISED EQUATION Substituting Eq. (1): Dη ( ) p λτ = 2η ( ) pd+ A I ηp Dt Eq. (2) again: λ Dη ( ) p λτ + τ = 2ηpD+ τ η Dt Finally: Dη λ + p = η Dt p ( λ τ ) τ 1 2 p η p A I ( ) D λ
GENERALISED EQUATION SPECIAL CASE Recall: η p0 λ η = λ= 0 p f f assume same function f τ for η and λ. Dη λ [ ] + p = η Dt p ( λ τ ) τ 1 2 Gives: p η Dλ λ Dη λ + + = p D 0 p τ τ τ 1 2ηpD Dt ηp0 Dt
MANIPULATION FOR SPECIAL CASE: ( λ / ) D( η / ) p f λ D f λ η τ τ τ D f Dt ηp0 Dt f 0 0 0 0 p0 τ+ τ + τ 1 = 2 τ fτ 2η 0 p 0 λ + = D the PTT equation!!
COMPARISON WITH FENE-P: FENE-P under conservative form: ( λ τ ) τ 2 p D η τ τ η D p I + = + = Dt or in a more compact and conservative form: ( λ τ ) τ ( η Ι ) p with: λ = λ 0 /f η = aη f p p0 / Present model: + = ( λ τ ) τ 2 η p D λτ + η p D η Dt p Advantage: permits to control functions separately
COMPARISON WITH FENE-P: material functions Stress growth upon inception of shear flow: viscosity 1 FENE-P and FENE-M, L 2 =100 2 L b = + 3 η + (t,γ)/η p. 0.1 0.01 We=1.0 We=10 We=100 0 1 2 3 4 5 t/λ FENE-P (solid lines) goes beyond the linear viscoelastic limit envelope; cf. DeAguiar (1983)
COMPARISON WITH FENE-P: material functions Stress growth upon inception of shear flow: viscosity 1 FENE-P, L 2 =10 1 FENE-M, L 2 =10 0.1 0.1 η + (t,γ)/η p. 0.01 We=1.0 We=10 We=100 η + (t,γ)/η p. 0.01 We=1.0 We=10 We=100 0.001 0.001 0 1 2 3 4 5 t/λ 0 1 2 3 4 5 t/λ Same as before, for lower extensibility, L2=10
COMPARISON WITH FENE-P: material functions Similar for start up of first normal stress coefficient 10 FENE-P, L 2 =100 10 FENE-M, L 2 =100 1 1 Ψ 1 + 0.1 Ψ 1 + (t,γ) 0.1 0.01 0.001 We=0.1 We=1.0 We=10 We=100 0.01 0.001 We=0.1 We=1.0 We=10 We=100 0 1 2 3 4 5 t/λ 0 1 2 3 4 5 t/λ
COMPARISON WITH FENE-P: material functions Start-up of uniaxial elongational flow 200 FENE-P and FENE-M, We=5 η E 160 120 L2=25 L2=50 L2=100 80 40 0 0 1 2 3 4 5 t Modified model allows a less steep rise of extensional viscosity (more realistic?? cf. contrast FENE FENE-P: Keunings 97; van Heel et al 98; Herrchen &Ottinger 97)
COMPARISON WITH FENE-P: material functions Modified model in uniaxial elongational flow 50 FENE-M, We=5, L 2 =25 40 30 η E 20 10 L2vis=25 L2vis=10 L2vis=50 0 0 1 2 3 4 5 t Variation of L2 in viscosity funtion permits some control
TRACELESS STRESS TENSOR FORMULATION Second TOPIC
TRACELESS STRESS TENSOR FORMULATION P-correction solver Standard equations of motion: u x j j = ρu ρuu p i + j i = + t x x x 0 τ j i j ij Coupling term (1 β )De Traceless tensor: = 1τ δ ij ij 3 kk ij τ τ so that: ( τ ) = = 0 Tr τ kk
TRACELESS MOMENTUM EQ. Substitution in original eqs. gives: ( 1 ρ u ρuu p τ ) τ i j i 3 kk + = + t x x x j i j ρ u ρuu i j i p τ ij + = + t x x x j i j ij with: p = p 1τ 3 kk (modified pressure)
TRACELESS STRESS: OLDROYD-B Oldroyd-B equation, indicial notation: Dτ u u τ + λ = η + + λg ij i j ij p ij Dt xj x i with generation tensor term: G u = τ + τ j ij ik jk xk Using the TST we get: u x Dτ ij u u i j τ ij+ λ = η p + + λ ij 3 δij Dt xj x i i k ( 2 G G )
TRACELESS STRESS TENSOR: increased viscosity with: and: uj ui G ij = τ ik + τ jk xk xk ul G ll = 2τ lk 2G x l (in turbulence modelling, G= τ is the generation rate) lk xk u k An increased viscosity arises: = + 1λτ p p 3 kk η η very high, where it matters! (link with AVSS )
Evolution of the trace Relations for the evolution of the trace of the stress: Contracting indices: G= G + Dτkk uk τkk + λ = 2η p + 2λG Dt x k = 0 1 3 τ kk u xj = 0 enabling calculation of the trace j
TRACELESS STRESS TENSOR for OLDROYD-B Resumé in tensor notation: τ + λ ( τ ) = 2η D p 1 τ = τ Tr( τ ) Ι 3 D = D 1Tr( D) Ι 3 η = η + 1λ τ and the extended definition: Dτ ( τ ) = τ + τ 3 τ : Dt Tr p p 3 ( ) { T 2 i u u i ( u) I}
TRACELESS STRESS TENSOR: RESULTS (1) UCM in channel flow, 20x20, L=10, H=1 Total number of iterations to solve pressure equation: We Outer iterations Total number of inner iterations for pressure Standard method Traceless method Ratio 0.3 265 3434 2193 1.6 0.6 459 6129 2428 2.5 0.9 653 8768 2316 3.8 1.2 824 10957 2163 5.1 1.5 975 12609 2080 6.1
TRACELESS STRESS TENSOR: RESULTS (1a) 16000 UCM: 2D channel flow, 20x20 pressure iterations 12000 8000 4000 Standard Traceless 0 total iterations 1000 800 600 400 200 0 0.4 0.8 1.2 1.6 We 0 0.4 0.8 1.2 1.6 We
TRACELESS STRESS TENSOR: RESULTS (2) UCM, 4:1 plane contraction flow (mesh 2960 CV) Total number of iterations to solve pressure equation: We 0.0 Outer iterations 798 Total number of inner iterations for pressure Standard method 8641 Traceless method -- Ratio 0.3 651 2442 1859 1.3 0.6 664 4074 3008 1.4 1.2 725 6935 3711 1.9 Ratio proportional to relative CPU times
TRACELESS STRESS TENSOR: RESULTS (2) UCM, 4:1 plane contraction flow (mesh 2960 CV) pressure iterations 8000 6000 4000 2000 0 UCM: 4:1 plane contraction Standard Traceless 0.2 0.4 0.6 0.8 1 1.2 We (ratio proportional to relative CPU times)
TRACELESS STRESS TENSOR: CONCLUSIONS Advantages: TST offers better coupling and much less CPU Number of pressure iterations approximately constant Problems: Creation of strong normal stress normal to a wall Difficulty to implement such BCs (oscillations) Standard: TST: e.g. UCM, channel aligned with x: τ xx xx ληγ τ = 2 = 2 yy 0 τ = ληγ 4 3 2 τ 2 2 yy = 3ληγ xy τxy = ηγ τ = ηγ Problems are particular to our FVM (collocated) (?) Would like to see other attempts with FEM
LAW-OF-WALL FOR VISCOELASTIC FLOWS Third TOPIC
LAW-OF-WALL FOR VISCOELASTIC FLOWS At present, 2D meshes are so fine that any attempt to extend to 3D is out of question (ex: cylinder) NC=44280; even a coarser mesh NC=11040x100=1.104.000
LAW-OF-WALL FOR TURBULENT, FLOWS Flow depends on: Dimensional analysis gives: and a wall layer: u u τ y τw ρ µ τ u w τ ρ y + ρyu µ u u + = f1( y ) f ( ) uv = τ + + 2 y = f 2 3( y ) y y u τ τ = 0 τ = τl + τt const. y Small acceleration implies: (constant-stress layer) Experiments show, for large y+: f3 1 uτ = uv f2 const. = 1/ Κ Κ= 0.41 τ u u = τ y Κy u u 1 = ln( Ey ) Κ + + u τ (the log-law)
LAW-OF-WALL FOR VISCOELASTIC FLOWS Carry same ideas for viscoleastic flows near walls List of dependent variables includes relaxation time: y p x ρ η λ Giving the characteristic scales: tc λ y = η = c c 2 λp λ x u = η p Instead of a logarithmic variation, power law attempts were followed in the next fittings of shear flows (following Zagarola et al., Phy Fluids 1997) x
LAW-OF-WALL FOR VISCOELASTIC FLOWS Pipe flow: Linear SPTT
LAW-OF-WALL FOR VISCOELASTIC FLOWS Pipe flow: Exponential SPTT
LAW-OF-WALL FOR VISCOELASTIC FLOWS Channel flow: Leonov fluid
LAW-OF-WALL FOR VISCOELASTIC FLOWS Lot to be done. In future attempts, a wall law should be sought for the stresses (and not velocity), based on the notion of local equilibrium
CONCLUSIONS Theoretical (empirical): a naive derivation led to a modified FENE-P equation with some advantages for time-dependent flows. It is the PTT when functions are the same. Numerics: the traceless approach allows tighter coupling of equations in decoupled methods. Unsolved problems with normal stresses at wall. Pratical applications (3D): proposal for a loglaw -like treatment to bridge the near wall BLs.
ACKNOWLEDGMENTS: Prof. Manuel A. Alves (Univ. of Porto, FEUP) (law-of-wall) Project POCI/EME/58657/2004