development of large-eddy simulation models for

DEVELOPMENT OF LARGE-EDDY SIMULATION MODELS FOR VISCOELASTIC ISOTROPIC TURBULENCE

Pedro O. Ferreira1

Fernando T. Pinho2

Carlos B. da Silva1

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

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

51ème Congrès Annuel du Groupe Français de Rheologie

Lille, France GFR 2016 October, 25-27, 2016

2

EXISTING MODELS

Development of large-eddy simulation models for viscoelastic isotropic turbulence CEFT/FEUP & LAETA/IDMEC P. O. Ferreira, F. T. Pinho & C. B. Silva

GFR 2016 Lille, October, 25 - 27, 2016

Inherent limitations in unsteady flows, especially with turbulence induced unsteadiness, flows with curvature, rotation

How do they behave in flows other than BL flows?

RANS (Reynolds Average Navier-Stokes) models- Leighton et al., Joint ASME/JSME Fluids Eng. Symp. for DR Viscoelastic Flow (2003)

Honolulu- Pinho et al., JNNFM 154 (2008) 89, JNNFM 181-182 (2012) 51, IJHFF 54 (2015) 220 - Iaccarino et al., JNNFM 165 (2010) 376- Masoudian et al., JNNFM 202 (2013) 99, IJHFF 54 (2015) 220, IJHMT 100 (2016) 332

LES (Large-Eddy Simulation) models

- Thais et al., PoF 22 (2010) 13103- Wang et al., China Phys. B 23 (2014) 34701- Ohta & Miyashita, JNNFM 206 (2014) 29 (inelastic power law fluids)

Complex, expensive, only assessed (calibrated) for wall flows, no info for wall-free flows

Objective: Develop closure for HIT (extend in the future to jets and BL)

Predictive tools are useful and have large scope of application

3

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

4

Direct Numerical Simulation (DNS)



• Continuity:

• Momentum:

• Constitutive equation:

5

INSTANTANEOUS GOVERNING EQUATIONS FOR DNS

∂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 tensor

FENE-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

6

DEFINITIONS



De =τ p Kl

Wi =τ p

τη τη =ν s

ε [s ]Kolmogorov time scale:

Reλ =2K3

λν s

Integral scale:Taylor length scale:

λ = 10ν sKε [s ]

l = π2K

⎛⎝⎜

⎞⎠⎟

E k( )kkmin

kmax

∑

Kinetic energy spectrum E k( ) = 4πk2 12ui!k ,t( )ui*

!k ,t( )

!k

ui!k ,t( ) = FFT ui

!x,t( ){ }with

Turbulent kinetic energy K = E k( )k∑

(*) denotes a complex conjugate

Newtonian solvent dissipation ε [s ] = 2ν s k2E k( )k∑

Average elastic energy dissipated by the polymer ε [ p] = σ ij ,pSijAverage over the computational domain

Wavenumber

7

RESULTS



Newtonian simulation

Viscoelastic simulations

ν s = 0.003 m2s-1All simulations: N = 3843 P = 3.3 m2s-3Collocation points Forcing energy

β = 0.8

Forcing distributed over the 4 lowest wavenumber (Alvelius)Dealiasing at 2/3

Spectral methods used

L2=1002

De=0.38 De=0.91

8

De=0.11



RESULTS: STRUCTURES AS A FUNCTION OF DE

More sheet-like structuresMore worm-like structures

9

Large-Eddy Simulation (LES)



10

FILTERING



φ !x,t( ) = φ !x,t( )+φ ' !x,t( )Filtered/resolvedGridscale(GS)

ResidualSub-gridscale(SGS)

φ !x,t( ) = φ !x ',t( )GΔ!x − !x '( )d!x '

Ω∫

filter property: GΔ!x( )d!x

Ω∫ = 1

Box filter (definition in 1D)

GΔ x( ) =1Δ

if x < Δ2

0 otherwise

⎧⎨⎪

⎩⎪

Local in physical space Nonlocal in Fourier space Formally equivalent to discretisation using FE or FV (more adequate for engineering applications)

• Continuity:

• Momentum:

• Subgrid-scale stress tensor

• Constitutive equation:

11

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 stretching

12



FILTERED GOVERNING EQUATIONS 2

Subgrid-scale conformation advection

Subgrid-scale polymer stretching

ψ ij = uk∂Cij

∂xk− uk

∂Cij

∂xk

γ ij = Cjk∂ui∂xk

−Cjk∂ui∂xk

⎡

⎣⎢

⎤

⎦⎥ + Cik

∂u j

∂xk−Cik

∂u j

∂xk

⎡

⎣⎢

⎤

⎦⎥

Instantaneous 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

13

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 0.16

14

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

15



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

16



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 80-1357Scale similarity for the

sub-grid scale stresses

τ ij = C uiu j! − !ui !u j( )

Δ!Δ = 2Δ

filter size of LES (original)

O 1( )

DISTORTION SIMILARITY MODEL (DSIM): THIRD MODELLING HYPOTHESIS H3

τ ij = uiu j − uiu j

17

DSIM: THE HYPOTHESIS OF SELF-SIMILARITY



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

⎡

⎣⎢

⎤

⎦⎥

H3 Bardina type closure

γ ij = CγGij

relies on evolution eq. for Cii

H41

H42

Global elastic equilibrium

Local elastic equilibrium

18

DSIM- ELASTIC EQUILIBRIUM ASSUMPTIONS: FOURTH MODELLING HYPOTHESIS



2Cik∂ui∂xk box

= 1τ p


⎤⎦

box

H42

Evolution equation for the trace of the conformation tensor

∂Cii

∂t+ uk

∂Cii

∂xk= 2Cik

∂ui∂xk

− 1τ p

f Ckk( )Cii −δ ii⎡⎣ ⎤⎦

Steady HIT

Box averaging

polymer stretching polymer relaxation

Global elastic equilibrium assumption

If 2Cik∂ui∂xk

= 1τ p


⎤⎦ Local elastic equilibrium assumption

H41

19

DSIM: COEFFICIENT OF THE SUB-GRID SCALE POLYMER STRETCHING



(H3)

γ ij = CγGij + H1 H41+

Cγ =

12τ p

f Ckk( )Cii −δ ij⎡⎣ ⎤⎦box

− Cjk

∂u j

∂xk box

C jk

∂u j

∂xk

!− !Cjk

∂ !u j

∂xk box

f Ckk( )Cij ≈ f Ckk( )CijGlobal elastic equilibrium assumption

20

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 results

21



DSIM: A PRIORI TESTS FOR CΥ: GLOBAL VERSUS LOCAL

Cγ =γ jj

Gii box

Cγ =γ jj box

Gii box

Local Global

Local and global variations are similar (H41 and H42 equally valid) Cγ is O(1), increases with filter size, decreases with De Cγ≈ 0 for De=1 consistent with depletion of nonlinear energy cascade

22

Large-Eddy Simulation (LES) a-posteriori analysis



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

23

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)

24

DSIM:A-POSTERIORI ANALYSIS: ENERGY SPECTRUM AND ENERGY BALANCE



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

De=0.38

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

25



De=1.23

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

DSIM:A-POSTERIORI ANALYSIS: ENERGY SPECTRUM AND ENERGY BALANCE

• First LES model developed based on HIT for FENE-P fluids

• Distortion similarity model: closure for sub-grid scale polymer stretching based on scale similarity and global equilibrium of elastic energy

• Local equilibrium of elastic energy assessed for future use in inhomogeneous flows

• Other sub-grid scale terms of evolution equation for Cij are negligible

• Sub-grid scale stress: Smagorinsky model

• a-posteriori testing shows good and consistent results: DR, flow structures, kinetic energy spectra and detailed spectral budgets

• Future work

• Extend model to planar jets

• Extend model to wall turbulence

26

CLOSURE: CONCLUSIONS AND FUTURE WORK



27

ACKNOWLEDGEMENTS

• 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



development of large-eddy simulation models for

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