Numerical tests on some viscoelastic flows-multiscale approach

Bangwei She, Mária Lukáčová
Institute of Mathematics, JGU-Mainz

June 18, 2013
The 8th Japanese-German International Workshop on
Mathematical Fluid Dynamics, Tokyo

Outline

Introduction

Multiscale approach

Numeric

B. She, M. Lukáčová (JGU-Mainz)

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Introduction
viscoelastic flow

Governing equations
Re( ∂u∂t + u · ∇u) = −∇p + α∆u + ∇ · σ∇·u =0(1)u, p, σ, α, Re = ρ UL
µ are velocity, pressure, elastic stress, andportion of Newtonian viscosity in total viscosity, Reynolds number.

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Introduction
viscoelastic flow

Governing equations
Re( ∂u∂t + u · ∇u) = −∇p + α∆u + ∇ · σ∇·u =0(1)u, p, σ, α, Re = ρ UL
µ are velocity, pressure, elastic stress, andportion of Newtonian viscosity in total viscosity, Reynolds number.
Two approaches describing the elastic stress
Macro: constitutive law (Oldroyd-B model):∂C1+ (u · ∇)C − ∇u · C − C · (∇u)T =(I − C)∂tWe(2)U
where C = 1−αWe (σ − I), We = λ L is Weissenberg number, λ isrelaxation time.
Results do not converge for high We.
Micro: molecular theory

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Multiscale approach
molecular theory

End to end dumbbell
Assumption:a chain of beads and spring,
dilute,zero-mass.

1 H. C. Öttinger, Stochastic Processes in Polymeric Fluids: Tools and Examples forDeveloping Simulation Algorithms.
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Multiscale approach
molecular theory

End to end dumbbell
Assumption:a chain of beads and spring,dilute,zero-mass.

Spring force: F(R) = R (Hooke law)
√Stochastic force1 : Bi = 2kT ζdWi /dt
Friction force: f = ζ(ṙ − v(r, t))k Boltzmann constant, T absolute temperature, ζ = 6πµs a friction
coefficient, µs solvent viscosity, a radius of bead.

1 H. C. Öttinger, Stochastic Processes in Polymeric Fluids: Tools and Examples forDeveloping Simulation Algorithms.
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Miltisacle approach
Fokker-Planck equation

Newton’s second law−ζ(ṙ1 − v(r1 , t)) + F(R) + B1 = 0,−ζ(ṙ2 − v(r2 , t)) − F(R) + B2 = 0.
2Ṙ = ∇v · R − F(R) +ζ

B. She, M. Lukáčová (JGU-Mainz)

s4kT dWt.
ζ dt

Numerical tests on some viscoelastic flows

(3)
(4)
(5)

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5 / 12Miltisacle approach
Fokker-Planck equation

Newton’s second law−ζ(ṙ1 − v(r1 , t)) + F(R) + B1 = 0,−ζ(ṙ2 − v(r2 , t)) − F(R) + B2 = 0.
2Ṙ = ∇v · R − F(R) +ζs

4kT dWt.
ζ dt

(3)
(4)
(5)

Probability distribution function ψ(x, R, t):
at a position x and time t, the probability of a dumbbell vector
that stays between R and R + dR.
ZZ
ψ(x, R, t)dR = 1.

B. She, M. Lukáčová (JGU-Mainz)

Numerical tests on some viscoelastic flows

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5 / 12Miltisacle approach
Fokker-Planck equation

Newton’s second law−ζ(ṙ1 − v(r1 , t)) + F(R) + B1 = 0,−ζ(ṙ2 − v(r2 , t)) − F(R) + B2 = 0.
2Ṙ = ∇v · R − F(R) +ζs

4kT dWt.
ζ dt

(3)
(4)
(5)

Probability distribution function ψ(x, R, t):
at a position x and time t, the probability of a dumbbell vector
that stays between R and R + dR.
ZZ
ψ(x, R, t)dR = 1.
Fokker-Planck equation∂ψ
11+ u · ∇ψ = ∇R · ((−∇u · R +F(R))ψ) +∆R ψ (6)∂t2We2We
B. She, M. Lukáčová (JGU-Mainz)

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Multiscale approach
Relation between micro and macroZZ
1−α(−I +R ⊗ RψdR).
σ=We
ZZ
R ⊗ R × (6)dR ⇒ (2).
The micro approach is equivalent to the Oldroyd-B model!

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Multiscale approach
Relation between micro and macro
ZZ1−α(−I +R ⊗ RψdR).
σ=WeZZR ⊗ R × (6)dR ⇒ (2).
The micro approach is equivalent to the Oldroyd-B model!
Multiscale systemRe( ∂u∂t + u · ∇u) = −∇p + α∆u + ∇ · σ∇ · uRR=01−ασ=(−I+R ⊗ RψdR)
We ∂ψ
11+u·∇ψ=∇·((−∇u·R+ 2WeR)ψ) + 2We∆R ψR∂t

B. She, M. Lukáčová (JGU-Mainz)

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6 / 12Numericnumerical method

1. Navier-Stokes

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Numericnumerical method

1. Navier-Stokes
2. Fokker-Planck: space splitting∂ψ1
1∂t + u · ∇ψ = ∇R · ((−∇u · R + 2We F(R))ψ) + 2We ∆R ψ
In physical space x ∈ Ω(geometry) we have∂ψ
+ u · ∇ψ = 0∂t(8)Upwind in physical space.
In configuration space R ∈ (−∞, +∞) × (−∞, +∞), we use animplicit schemeψ∗ − ψn1
1= ∇R · ((−∇u · R +F(R))ψ ∗ ) +∆R ψ ∗ (9)∆t
2We2WeB. She, M. Lukáčová (JGU-Mainz)

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Numeric
numerical method

The configuration space is infinite!
FENE, A. Lozinski, C. Chauviere, F =R
2 , |R| ∈ (0, R0 ).
1− |R|2R

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Numeric
numerical method

The configuration space is infinite!
FENE, A. Lozinski, C. Chauviere, F =R2 , |R| ∈ (0, R0 ).
1− |R|2R

0Idea:
polar coordinates (ρ, θ) = (|R|, arctan RR12 ),1.
infinite plane to unit circle r = ρ+1∂φ∂φ= b0 (κ, θ)L0 φ − b1 (κ, θ)+ L1 φ,∂t∂θwhere L0 and L1 are linear operators,

(10)

21
b0 = κ11 cos 2θ + κ12 +κ
sin 2θ, L0 φ = −4r (1 − r )2 [−s(1 − η)−1 φ + ∂φ],2∂ηκ21 −κ12κ12 +κ21cos
2θ+,b1 = −κ11 sin 2θ +2222L1 φ = c1 φ + c2 ∂φ+ c3 ∂∂ηφ2 + c4 ∂∂θφ2∂η1c1 = We [1 + 2s(1 − η)−1 (−3r 4 + 2r 3 − r ) + 8r 4 (r − 1)2 s(s − 1)(1 − η)−2 ],216 4c2 = We
(3r 4 − 2r 2 + r ) − Wer (r − 1)2 s(1 − η)−1 ,21r8 4)2 , κ = ∇u,c3 = We r (r − 1) , c4 = 2We ( 1−rsψ(t, x, R) = (1 − η) φ(t, x, η, θ), s = 2, η = 2(1 − r )2 − 1.
B. She, M. Lukáčová (JGU-Mainz)

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Numeric
Pseudo-spectral method

Pseudo-spectral method
We look for an approximate solution to the Eq.(10) of thefollowing form
φ(t, x, η, θ) =Nη
Nθ X
1 X
X

φikl hk (η)Φil (θ),(11)

i =0 l=i k=1

where Φil (θ) = (1 − i) cos(2lθ) + i sin(2lθ),Nθ , Nη number of discretization points,hk (η) Lagrange interpolating polynomial,ηm (m = 1, · · · , Nη ) Gauss-Legendre points η ∈ (−1, 1).
n∗nwhere φ̄
tn = n∆t.

φ̄ = [I − ∆t(M0 + M1 + M2 )]−1 φ̄ ,(12)is the vector of the expansion coefficients φikl at time
M0 , M1 , M2 · · ·B. She, M. Lukáčová (JGU-Mainz)

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test for Peterlin modelRe( ∂u∂t + u · ∇u) = −∇p + α∆u + ∇ · σ∇·u=0σ= (tr C)C ∂C11T+(u·∇)C−∇u·C−C·(∇u)= We(tr C)I − We(tr C)2 C + ǫ∆C∂tand ∂C∂n = 0 on the boundary.
Re = 1, We = 5

Re = 1, We = 50.02

160.018140.016
12100.012
trace

kinetic energy

0.014

0.010.008

860.006
40.00420.002005

10

B. She, M. Lukáčová (JGU-Mainz)

15time

20

2530

005

Numerical tests on some viscoelastic flows

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test for Peterlin model

Slight modification tr C −→ max(tr C).
Re = 1, We = 5

Re = 1, We = 5

0.02

3.6

0.018

3.40.016

3.230.012
trace

kinetic energy

0.014

0.01

2.8
2.60.008
2.40.006

2.20.004

20.0020051015time

2025301.8051015time

202530

Table: L2-error of σmesh points
3264
128

||σ1 − σ1 (256)||L20.0636
0.05590.0355EOC
0.18580.6538||σ3 − σ3 (256)||L2
0.4466
0.31650.1643EOC
0.49690.9457

Convergent!
B. She, M. Lukáčová (JGU-Mainz)

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Thank you for your attention!

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