Modelling of turbulent flows: RANS and LES Turbulenzmodelle in der - - PowerPoint PPT Presentation

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Introduction Algebraic stress models Non-linear eddy viscosity models

Modelling of turbulent flows: RANS and LES

Turbulenzmodelle in der Str¨

• mungsmechanik: RANS und LES

Markus Uhlmann

Institut f¨ ur Hydromechanik Karlsruher Institut f¨ ur Technologie www.ifh.kit.edu

SS 2012 Lecture 7

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Introduction Algebraic stress models Non-linear eddy viscosity models

LECTURE 7 Algebraic stress models

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Introduction Algebraic stress models Non-linear eddy viscosity models

Questions to be answered in the present lecture

How can the linear eddy viscosity assumption be avoided without the need for solving transport equations?

◮ algebraic stress models ◮ nonlinear eddy viscosity models

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Introduction Algebraic stress models Non-linear eddy viscosity models

Intermediate between RSM and Boussinesq approximation

Reynolds stress transport models

◮ naturally incorporate transport effects ◮ describe stress production exactly

BUT: high computational cost (equations for 6 components)

Standard eddy-viscosity models (Boussinesq approximation)

(A) local relation between Reynolds stress and mean strain (B) linear relation between Reynolds stress and mean strain (A) is inevitable ⇒ (B) can be changed

⇒ non-linear Reynolds stress/mean strain relationships

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Introduction Algebraic stress models Non-linear eddy viscosity models

Algebraic stress models

Reynolds stress equations for transport model: ¯ Du′

iu′ j

¯ Dt + (Tkij),k

• ≡Dij (transport)

= Pij + Rij − 2 3 ˜ εδij

Basic idea of algebraic stress models (ASM):

◮ approximating transport terms Dij by local expressions

→ resulting model is free from derivatives: 6 algebraic equations relating u′

iu′ j, k, ˜

ε, ui,j ⇒ approach benefits from known models for pressure-strain Rij

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Introduction Algebraic stress models Non-linear eddy viscosity models

Algebraic stress models – equilibrium assumption

Reynolds stress equations for transport model: ¯ Du′

iu′ j

¯ Dt + (Tkij),k

• ≡Dij (transport)

= Pij + Rij − 2 3 ˜ εδij

Simplest local equilibrium assumption:

◮ neglect the transport term altogether:

Dij = 0 ⇒ implies for the turbulent energy:

1 2Dll = P − ˜

ε = 0 problem: equality P = ˜ ε not verified in general!

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Introduction Algebraic stress models Non-linear eddy viscosity models

Algebraic stress models – weak equilibrium assumption

Reynolds stress equations for transport model: ¯ Du′

iu′ j

¯ Dt + (Tkij),k

• ≡Dij (transport)

= Pij + Rij − 2 3 ˜ εδij

Weak equilibrium assumption (Rodi 1972)

◮ rewriting Reynolds stress in terms of anisotropy and TKE:

u′

iu′ j = 2k bij + 2 3k δij ◮ neglecting transport of anisotropy: ¯

Du′

iu′ j

¯

Dt =

u′

iu′ j

k ¯

Dk

¯

Dt +✘✘✘✘✘✘

✘

k

¯

D

¯

Dt u′

iu′ j

k

• ≈

u′

iu′ j

k ¯

Dk

¯

Dt

◮ applying the approximation to the entire transport term:

Dij ≈

u′

iu′ j

k

· (transport of k) =

u′

iu′ j

k 1 2Dll = u′

iu′ j

k

(P − ˜ ε) ⇒ final model:

u′

iu′ j

k

(P − ˜ ε) = Pij + Rij − 2

3 ˜

εδij

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Introduction Algebraic stress models Non-linear eddy viscosity models

ASM predictions for homogeneous shear flow

LRR-IP pressure-strain model

Rij = −CR2˜ εbij − C2(Pij − 2

3Pδij) ◮ corresponding ASM:

bij =

1 2 (1−C2)

CR−1+P/˜ ε · Pij− 2

3 δijP

˜ ε ◮ in homogeneos shear flow:

bij has finite limit for P

˜ ε → ∞

b11 → 4

15

b22 → − 2

15

b12 → − 1

5

⇒ stress remains realizable

1 2 3 4

• 0.2
• 0.1

0.0 0.1 0.2 0.3 b12 b12(k-ε) b22=b33 b11

P/ε

bij P/˜ ε — —, ASM predictions; – – – –, k-ε model (from Pope “Turbulent Flows”, 2000) 8 / 16

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Introduction Algebraic stress models Non-linear eddy viscosity models

Stress/mean strain relation implied by ASM

LRR-IP pressure-strain model

◮ define:

−u′v′ = Cµ k2

˜ ε u,y

with unknown function Cµ

◮ substituting ASM:

Cµ =

2 3 (1−C2)(CR−1+C2P/˜

ε) (CR−1+P/˜ ε)2

⇒ Cµ decreases with P/˜ ε

1 2 3 4 0.00 0.10 0.20 0.30

P/ε Cµ

P/˜ ε — —, ASM predictions (from Pope “Turbulent Flows”, 2000) 9 / 16

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Introduction Algebraic stress models Non-linear eddy viscosity models

Assessing the ASM approach

Achievements of algebraic stress models

◮ partial differential equations reduced to algebraic equations ◮ physics of pressure-strain model is carried over

Problems of the ASM approach

◮ implicit system of equations ◮ dependence is in general non-linear ◮ system can have multiple solutions ◮ numerical stiffness

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Introduction Algebraic stress models Non-linear eddy viscosity models Performance in free shear flow Performance in flow with system rotation

Explicit ASM or non-linear eddy viscosity models

Explicit ASM (EASM)

◮ explicit expressions for the stress components are numerically

desirable

◮ there are two routes (viewpoints) to achieve this:

• 1. construct an implicit ASM (as above):

bij = fi(bij, k

˜ ε ui,j)

then derive equivalent explicit form analytically ⇒ bij = fe( k

˜ ε ui,j)

• 2. construct an explicit expression for the Reynolds stresses:

⇒ bij = fe′( k

˜ ε ui,j)

⇒ both approaches have been realized ⇒ results also known as “non-linear eddy viscosity models”

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Introduction Algebraic stress models Non-linear eddy viscosity models Performance in free shear flow Performance in flow with system rotation

Deriving explicit algebraic stress models

◮ ansatz:

bij = Bij

• S,

Ω

• where normalized mean rate of strain/rotation are defined:
• Sij ≡ k

2˜ ε (ui,j + uj,i),

• Ωij ≡ k

2˜ ε (ui,j − uj,i) ◮ most general consistent expression (Pope 1975):

Bij

• S,

Ω

• = 10

n=1G(n)

T (n)

ij ◮ with independent, symmetric, deviatoric functions:

• T (1)

ij

= S

• T (2)

ij

= S Ω − Ω S

• T (3)

ij

= S2 − 1

3 trace(

S2)I

• T (4)

ij

= Ω2 − 1

3 trace(

Ω2)I

• T (5)

ij

= Ω S2 − S2 Ω

• T (6)

ij

= Ω2 S + S Ω2 − 2

3 trace(

S Ω2)I

• T (7)

ij

= Ω S Ω2 − Ω2 S Ω

• T (8)

ij

= S Ω S2 − S2 Ω S

• T (9)

ij

= Ω2 S2 + S2 Ω2 − 2

3 trace(

S2 Ω2)I

• T (10)

ij

= Ω S2 Ω2 − Ω2 S2 Ω

and undetermined scalar coefficients G(n)

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Introduction Algebraic stress models Non-linear eddy viscosity models Performance in free shear flow Performance in flow with system rotation

Examples of EASM

Linear case – Boussinesq hypothesis

◮ G(1) = −Cµ; G(n) = 0 for n ≥ 2

→ bij = −Cµ Sij

Statistically two-dimensional flow (Pope, 1975)

◮ sum contains only three terms

(G(n) = 0 for n ≥ 4)

General three-dimensional flow

◮ all 10 terms are non-zero

• 1. ASM approach:

Gatski & Speziale (1993), based on linear pressure-strain

• 2. direct approach:

Shih, Zhu & Lumley (1995), based on realizability

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Introduction Algebraic stress models Non-linear eddy viscosity models Performance in free shear flow Performance in flow with system rotation

Performance of EASM in mixing layer flow

Self-similar mixing layer

◮ spreading rate dδ/dx: exp. EASM (SZL) k-ε 0.019 0.014 0.016

EASM by Shih et al. not well calibrated for free shear flows

◮ anisotropy well predicted

(normal stresses)

y δ

• ✁✄✂✆☎
• ✁

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b11

(experiment of Bell & Mehta, 1990) 14 / 16

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Introduction Algebraic stress models Non-linear eddy viscosity models Performance in free shear flow Performance in flow with system rotation

Performance of EASM for rotating pipe flow

Rotation in axial direction

◮ rotation has stabilizing

effect (production term) ⇒ EASM predictions are reasonable linear eddy-viscosity fails

u

1.2 0.8 0.2 0.6 0.4 0.4 0.8 1.0 1.6

r/R

lines: EASM of Wallin & Johansson (2000) symbols: experiment of Imao et al. (1996) no rotation; • medium; strong 15 / 16

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Introduction Algebraic stress models Non-linear eddy viscosity models Performance in free shear flow Performance in flow with system rotation

Performance of EASM for rotating channel flow

Rotation in spanwise direction

◮ rotation causes

non-symmetric profiles ⇒ EASM predictions are comparable to full transport model

u y/H

— —, EASM of Gatski & Speziale (1993) experiment of Johnston et al. (1972) 16 / 16

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Conclusion Further reading

Summary of today’s lecture

How can the linear eddy viscosity assumption be avoided without the need for solving transport equations?

◮ transport terms eliminated by weak equilibrium assumption

• 1. algebraic stress models (ASM)

◮ inherit properties of pressure-strain model

• ften numerical difficulties
• 2. nonlinear eddy viscosity models (EASM)

◮ provide general explicit expressions for Reynolds stresses

⇒ allow for prediction of complex straining fields

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Conclusion Further reading

Summary (2): A hierarchy of RANS models

(a) elliptic relaxation RSM (b) standard RSM (c) ASM (with k-ε equations) (d) nonlinear eddy-viscosity model (with k-ε) (e) standard (isotropic eddy-viscosity) k-ε model (f) one-equation k-model (with ℓm) (g) mixing-length model

Which assumption is added when stepping from (a) to (g)?

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Conclusion Further reading

Further reading

◮ S. Pope, Turbulent flows, 2000

→ chapter 11

◮ D.C. Wilcox, Turbulence modeling for CFD, 2006

→ chapter 6

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