SOME THOUGHTS ON DIFFERENTIAL VISCOELASTIC MODELS AND THEIR NUMERICAL SOLUTION

Transcription

1 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

2 TOPICS Derivation of a generalized Oldroyd-B model The traceless stress tensor formulation Law-of-the-wall for viscoelastic flows

3 DERIVATION OF GENERALISED MODEL First TOPIC

4 GOVERNING EQUATIONS iu= 0 Du ρ Dt τ η 2 = p+ i τtot = p+ s u+ iτ

5 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

6 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

7 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

8 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

9 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

10 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:

11 TYPICAL STRESS FUNCTIONS: FENE-CR: f [ τ] = L 2 + ( λ / η ) Tr( τ ) 0 p0 L 2 3 FENE-P: f 3 λ = Tr( ) b [ τ] 0 τ + 2 3aηp 0 PTT: f λε = 1 + Tr( τ ) [ τ] 0 η p0

12 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

13 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

14 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

15 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

16 MANIPULATION FOR SPECIAL CASE: ( λ / ) D( η / ) p f λ D f λ η τ τ τ D f Dt ηp0 Dt f p0 τ+ τ + τ 1 = 2 τ fτ 2η 0 p 0 λ + = D the PTT equation!!

17 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

18 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 We=1.0 We=10 We= t/λ FENE-P (solid lines) goes beyond the linear viscoelastic limit envelope; cf. DeAguiar (1983)

19 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 = η + (t,γ)/η p We=1.0 We=10 We=100 η + (t,γ)/η p We=1.0 We=10 We= t/λ t/λ Same as before, for lower extensibility, L2=10

20 COMPARISON WITH FENE-P: material functions Similar for start up of first normal stress coefficient 10 FENE-P, L 2 = FENE-M, L 2 = Ψ Ψ 1 + (t,γ) We=0.1 We=1.0 We=10 We= We=0.1 We=1.0 We=10 We= t/λ t/λ

21 COMPARISON WITH FENE-P: material functions Start-up of uniaxial elongational flow 200 FENE-P and FENE-M, We=5 η E L2=25 L2=50 L2= 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)

22 COMPARISON WITH FENE-P: material functions Modified model in uniaxial elongational flow 50 FENE-M, We=5, L 2 = η E L2vis=25 L2vis=10 L2vis= t Variation of L2 in viscosity funtion permits some control

23 TRACELESS STRESS TENSOR FORMULATION Second TOPIC

24 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

25 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)

26 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 )

27 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 )

28 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 = τ kk u xj = 0 enabling calculation of the trace j

29 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}

30 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

31 TRACELESS STRESS TENSOR: RESULTS (1a) UCM: 2D channel flow, 20x20 pressure iterations Standard Traceless 0 total iterations We We

32 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 Ratio proportional to relative CPU times

33 TRACELESS STRESS TENSOR: RESULTS (2) UCM, 4:1 plane contraction flow (mesh 2960 CV) pressure iterations UCM: 4:1 plane contraction Standard Traceless We (ratio proportional to relative CPU times)

34 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 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

35 LAW-OF-WALL FOR VISCOELASTIC FLOWS Third TOPIC

36 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=

37 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 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)

38 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

39 LAW-OF-WALL FOR VISCOELASTIC FLOWS Pipe flow: Linear SPTT

40 LAW-OF-WALL FOR VISCOELASTIC FLOWS Pipe flow: Exponential SPTT

41 LAW-OF-WALL FOR VISCOELASTIC FLOWS Channel flow: Leonov fluid

42 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

43 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.

44 ACKNOWLEDGMENTS: Prof. Manuel A. Alves (Univ. of Porto, FEUP) (law-of-wall) Project POCI/EME/58657/2004