LARGE-EDDY SIMULATIONS OF VISCOELASTIC ISOTROPIC ...fpinho/pdfs/Presentation_APS_2016_v2.pdf ·...

LARGE-EDDY SIMULATIONS OF VISCOELASTIC ISOTROPIC TURBULENCE WITH THE FENE-P FLUID

Fernando T. Pinho1

Pedro O. Ferreira2

Carlos B. da Silva2

1 CEFT, Dept. de Engenharia Mecânica, Faculdade de Engenharia, Universidade do Porto, Porto, Portugal2 LAETA/IDMEC, Instituto Superior Técnico, Universidade de Lisboa, Lisboa, Portugal

webpage: http://www.fe.up.pt/~ceft E-mail: [email protected]://www.fe.up.pt/~fpinho [email protected], [email protected]

69th Annual Meeting of the APS Division of Fluid Dynamics

Portland, Oregon APS | DFD 2016 November, 20-22, 2016

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MOTIVATION & OUTLINE

1) DNS of Homogeneous Isotropic Turbulence (HIT)

2) Development of LES closures: physical arguments and test with a priori analysis of DNS

3) Test with a posteriori analysis

Large-eddy simulations of viscoelastic isotropic turbulence with the FENE-P fluid CEFT/FEUP & LAETA/IDMEC F. T. Pinho, P. O. Ferreira & C. B. Silva

APS | DFD 2016 Portland, November, 20 - 22, 2016

Objective: Develop LES closures for polymer solutions

Starts with HIT (to be extend in the future to inhomogeneous flows: jets and wall flows)

• Continuity:

• Momentum:

• Constitutive equation:

3

INSTANTANEOUS GOVERNING EQUATIONS FOR DNS: FENE-P FLUID

∂ui∂xi

= 0

∂ui∂t

+ uk∂ui∂xk

= − 1ρ∂p∂xi

+ 1ρ∂σ ik

∂xk

σ ij = 2ρν sSij!"#+σ ij ,p

Newtonian solvent Polymer

Sij =12

∂ui∂x j

+∂uj∂xi

⎛

⎝⎜

⎞

⎠⎟

(incompressible fluid)

σ ij ,p =ρν pτ p

f Ckk( )Cij −δ ij( ) Cij =RiRj

R02

f Ckk( ) = L2 − 3L2 −Ckk

Conformation tensorFENE-P

∂Cij∂t

+ uk∂Cij∂xk

= Cjk

∂ui∂xk

+Cik∂uj∂xk

− 1τ p

f Ckk( )Cij −δ ij⎡⎣ ⎤⎦

Evolution equation for the conformation tensor

β = ν s

ν s +ν p

Ratio of viscosities



Peterlin function

• Continuity:

• Momentum:

• Subgrid-scale stress tensor

• Constitutive equation:

4

FILTERED GOVERNING EQUATIONS 1

∂ui∂xi

= 0

∂ui∂t

+ uk∂ui∂xk

= − 1ρ∂p∂xi

+ν s∂2ui

∂x j ∂x j−∂τ ij∂x j

+ 1ρ∂σ ik ,p

∂xk

(incompressible fluid)

σ ij ,p =ρν pτ p

f Ckk( )Cij −δ ij( )FENE-P

∂Cij∂t

+ uk∂Cij∂xk

= Cjk

∂ui∂xk

+Cik∂uj∂xk

− 1τ p

f Ckk( )Cij −δ ij⎡⎣

⎤⎦ −ψ ij − γ ij

Evolution equation for the conformation tensor

τ ij = uiu j − uiu j

filtered polymer dissipation

subgrid-scale conformation advection

subgrid-scale polymer stretchingLarge-eddy simulations of viscoelastic isotropic turbulence with the FENE-P fluid CEFT/FEUP & LAETA/IDMEC

F. T. Pinho, P. O. Ferreira & C. B. Silva APS | DFD 2016 Portland, November, 20 - 22, 2016

5

SUBGRID-SCALE STRESS TENSOR

τ ij ≡ uiu j − uiu j

Classical Smagorinsky model

τ ij = −2νTSij +13τ kkδ ij

νT = CSΔ( )2 S

S = 2SijSij( )1/2Norm of resolved

rate of strain tensor

Δ = Δx ×∆ y ×∆ z( )1/3Filter size

Smagorinsky constant

What is the influence of De on Cs?

CS2 =

−τ ijSij box

2 2Δ2 SijSij( )box3/2

CS =1π

3CK

2⎛⎝⎜

⎞⎠⎟−3/4

= 0.16

Newtonian in HIT (CK=1.6)

CS decreases with De, but we will use 0.16Large-eddy simulations of viscoelastic isotropic turbulence with the FENE-P fluid CEFT/FEUP & LAETA/IDMEC


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VISCOELASTIC TERMS: FIRST MODELLING HYPOTHESIS H1

H1 f Ckk( )Cij ≈ f Ckk( )Cij Justified by analysis of wall flowsMasoudian et al. JNNFM, 202 (2013) 99-111 White & Mungal, ARFM 40 (2008) 235-256

f Ckk( )Cij

f Ckk( )Cij

De=0.38 Δ Δ x = 16

Confirmed in HIT even at high De



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H2 ψ ij ≈ 0 Small magnitude compared with othersThais et al. PoF, 22 (2010) 013103Masoudian et al., JoT 17 (2016) 543-571

VISCOELASTIC TERMS: SECOND MODELLING HYPOTHESIS H2

De=0.38 Δ Δ x = 16

PDF of sub-grid scale advection of Cii and of polymer stretching terms (instantaneous values normalized by their rms)

ψ ii << γ ii when probabilityis high

Confirmed for other De and filter sizes



8

H3 Self-similarity of sub-grid scale polymer stretching

Structurally similar at two different near-sized filter sizes

Liu et al. JFM, 275 (1994) 83-119Bardina et al. AIAA, (1980) paper 80-1357

Δ!Δ = 2Δ

filter size of LES (original)

DISTORTION SIMILARITY MODEL (DSIM): THIRD MODELLING HYPOTHESIS H3



γ ij = Cjk∂ui∂xk

−Cjk∂ui∂xk

⎡

⎣⎢

⎤

⎦⎥ + Cik

∂u j

∂xk−Cik

∂u j

∂xk

⎡

⎣⎢

⎤

⎦⎥

Gij = Cjk∂ui∂xk

!− !Cjk

∂ !ui∂xk

⎡

⎣⎢

⎤

⎦⎥ + Cik

∂u j

∂xk

!− !Cik

∂ !u j

∂xk

⎡

⎣⎢

⎤

⎦⎥

γ ij = CγGij

relies on evolution eq. for Cii

H41Global elastic energy equilibrium

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DSIM- ELASTIC EQUILIBRIUM ASSUMPTION: FOURTH MODELLING HYPOTHESIS

Evolution equation for the trace of the conformation tensor ∂Cii

∂t+ uk

∂Cii

∂xk= 2Cik

∂ui∂xk

− 1τ p

f Ckk( )Cii −δ ii⎡⎣ ⎤⎦



Filtered evolution equation for the trace of the conformation tensor

∂Cii

∂t+ uk

∂Cii

∂xk= 2Cik

∂ui∂xk

− 1τ p

f Ckk( )Cii −δ ii⎡⎣

⎤⎦ −ψ ii + γ ii

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DSIM- ELASTIC EQUILIBRIUM ASSUMPTION: FOURTH MODELLING HYPOTHESIS

2Cik∂ui∂xk box

= 1τ p


⎤⎦

box

Steady HIT

Box averaging

Global elastic equilibrium assumptionH41



polymer stretching polymer relaxation

Filtered evolution equation for the trace of the conformation tensor

∂Cii

∂t+ uk

∂Cii

∂xk= 2Cik

∂ui∂xk

− 1τ p


⎤⎦ −ψ ii + γ ii

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DSIM- ELASTIC EQUILIBRIUM ASSUMPTIONS: FOURTH MODELLING HYPOTHESIS

2Cik∂ui∂xk box

= 1τ p


⎤⎦

box


∂t+ uk

∂Cii

∂xk= 2Cik

∂ui∂xk

− 1τ p


Steady HIT

Box averaging





(H3)

γ ij = CγGij + H1 H41+

f Ckk( )Cij ≈ f Ckk( )Cij

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DSIM- ELASTIC EQUILIBRIUM ASSUMPTIONS: FOURTH MODELLING HYPOTHESIS

2Cik∂ui∂xk box

= 1τ p


⎤⎦

box


∂t+ uk

∂Cii

∂xk= 2Cik

∂ui∂xk

− 1τ p


Steady HIT

Box averaging





Cγ =

12τ p

f Ckk( )Cii −δ ij⎡⎣ ⎤⎦box

− Cjk

∂u j

∂xk box

C jk

∂u j

∂xk

!− !Cjk

∂ !u j

∂xk box

(H3)

γ ij = CγGij + H1 H41+

f Ckk( )Cij ≈ f Ckk( )Cij

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DSIM: A PRIORI TESTS FOR VALIDITY OF H3:

γ ii = Cik∂ui∂xk

−Cik∂ui∂xk

⎡

⎣⎢

⎤

⎦⎥ Gii = Cik

∂ui∂xk

!− !Cik

∂ !ui∂xk

⎡

⎣⎢

⎤

⎦⎥ Cγ 11 =

γ 11G11

;Cγ 22 =γ 22G22

JPDF of Gii and γii

De=0.38 Δ Δ x = 16

PDFs of Gii and γii JPDF of Cγ11 and Cγ22

Cγ is isotropicγ ii &Gii are self-similar

Correlation decreases with De and increasing filter size, but with little impact on a-posteriori resultsLarge-eddy simulations of viscoelastic isotropic turbulence with the FENE-P fluid CEFT/FEUP & LAETA/IDMEC


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DSIM: A PRIORI TESTS FOR CΥ

Cγ =γ jj box

Gii box

Cγ is O(1), increases with filter size, decreases with De Cγ≈ 0 for De=1 consistent with depletion of nonlinear energy cascade



15

Large-Eddy Simulation (LES) a-posteriori analysis

N=483 ; ∆/∆x=8 DNS data explicitly filtered with the same width



16

DSIM: A-POSTERIORI ANALYSIS: SOLVENT DISSIPATION REDUCTION

DR = ε [ p]

ε [s ] + ε [ p]

Incomplete LES

Difference between DNS and LES De and Wi is less than 10% (not shown) Full LES compares very well with filtered DNS Incomplete LES fails when sub-grid scale stretching and nonlinear cascade are relevant Incomplete LES behaves well at large De (absence on nonlinear cascade and direct dissipation by the polymer)



Full

Two LES models: (1) Full LES model; (2) Incomplete LES model (Cγ = 0) DNS data explicitly filtered with the same width

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De=1.23DSIM:A-POSTERIORI ANALYSIS: ENERGY SPECTRUM AND ENERGY BALANCE



Energy balance are rarely shown in LES; Extremely difficult to have perfect matchPerfect match not required for excellent LES performance

Π(k) - Nonlinear cascadeΠp(k) - Polymer-solvent interactionsD(k) - Solvent DissipationF(k) - Forcing

Energy cascade mechanism: scale-by-scale energy budget

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DSIM:A-POSTERIORI ANALYSIS: ENERGY SPECTRUM AND ENERGY BALANCE De=0.38



At high De both LES models behave similarly: nonlinear energy cascade is negligible, no need to model sub-grid scale polymer energy transferMismatch again for k > kcri where filtered DNS carries more energy

• Future work

• Extend model to planar jets

• Extend model to wall turbulence

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FUTURE WORK



• Fundação para a Ciência e a Tecnologia & Feder (COMPETE2020):

PTDC/EME-MFE/122849/2010UID/EMS/50022/2013PTDC/EMS-ENE/6129/2014PTDC/EMS-ENE/2390/2014- POCI-01-0145-FEDER-016669

• Laboratory for Advanced Computing, Universidade de Coimbra

ACKNOWLEDGEMENTS

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