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RANS modeling The turbulent viscosity assumption

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 3

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RANS modeling The turbulent viscosity assumption

LECTURE 3 Introduction to RANS modelling

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RANS modeling The turbulent viscosity assumption

Questions to be answered in the present lecture

How can the Reynolds-averaged equations be closed? What are the different types of models commonly used? Do simple eddy viscosity models allow for acceptable predictions?

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RANS modeling The turbulent viscosity assumption

The challenge of turbulence

Recap of the salient features of turbulent flows

◮ 3D, time-dependent, random flow field ◮ largest scales are comparable to characteristic flow size

→ geometry-dependent, not universal

◮ wide range of scales: τη/T ∼ Re−1/2, η/L ∼ Re−3/4 ◮ wall flows: energetic motions scale with viscous units

δν/h ∼ Re−0.88

◮ non-linear & non-local dynamics

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RANS modeling The turbulent viscosity assumption

General criteria for assessing turbulence models

Level of description

◮ how much information can be extracted from the results?

Computational requirements & development time

◮ how much effort needs to be invested in the solution?

Accuracy

◮ how precise and trustworthy are the results?

Range of applicability

◮ how general is the model?

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RANS modeling The turbulent viscosity assumption

Possible discrepancies between computation & experiment

(adapted from Pope “Turbulent flows”) 6 / 24

RANS modeling The turbulent viscosity assumption

Reynolds averaging procedure – need for modeling

◮ decompose velocity field into mean and fluctuation:

u(x, t) = u(x, t) + u′(x, t)

◮ average continuity & momentum equations:

ui,i = ∂tui + (uiuj),j + 1 ρp,i = νui,jj − u′

iu′ j,j ◮ task of RANS models:

→ supply the unclosed Reynolds stresses u′

iu′ j

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RANS modeling The turbulent viscosity assumption

Reynolds averaging – the closure problem

Averaging always introduces more unknowns than equations

◮ transport equation for the nth moment

→ contains (n + 1)th moment . . . and so on ⇒ requires closure at some level

◮ the higher the level, the more terms need modeling

Most successful closures:

◮ n = 1: turbulent viscosity models ◮ n = 2: Reynolds stress models

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RANS modeling The turbulent viscosity assumption

Common types of RANS models

Models based on the turbulent viscosity hypothesis

u′

iu′ j = −νT (ui,j + uj,i) + 2 3δij k ◮ turbulent viscosity νT needs to be specified (modeled)

Reynolds-stress transport models

¯ Du′

iu′ j

¯ Dt = . . .

◮ various unknown terms

(cf. lecture 5)

Non-linear turbulent viscosity models

u′

iu′ j = non-linear-function (ui,j, k, ε, . . .)

(cf. lecture 7)

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RANS modeling The turbulent viscosity assumption Generalities Algebraic TVMs One-equation models

Assumptions behind Boussinesq’s hypothesis

u′

iu′ j − 2

3k δij = −2νT ¯ Sij

Reynolds stress assumed proportional to local mean strain rate

• 1. mechanisms generating Reynolds stress are assumed local

→ transport effects neglected

• 2. turbulent stress and mean strain are assumed aligned

→ this stems from the linearity of the relation assumptions in general not true!

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RANS modeling The turbulent viscosity assumption Generalities Algebraic TVMs One-equation models

The locality assumption: example of failure

Experiments demonstrate:

◮ importance of history

effects

◮ contraction with ¯

Sij =cst but: increasing anisotropy

◮ ¯

Sij =0 in straight section but: non-zero stress

Turbulent viscosity models will not work in this case!

• Straight section

Axisymmetric contraction Straight section Turbulence generating grid − − 1 2 − Sij = 0 Sij = 0 − S11

k

S22 = S33 = −  −

k

x1

0.0 0.5 1.0

• 0.30
• 0.20
• 0.10

0.00 0.10 0.20

bij Sλt

0.0 0.2 0.4 0.6

Contraction Straight Section tε/k b11 b22

• exp. Tucker (1970)
• exp. Warhaft (1980)

(from Pope “Turbulent flows”) 11 / 24

RANS modeling The turbulent viscosity assumption Generalities Algebraic TVMs One-equation models

Assumption of stress/strain alignment

Boussinesq: bij = −νT k ¯ Sij

But, data shows:

◮ even in simple equilibrium flows

→ anisotropy NOT aligned with mean strain rate

◮ example: plane channel flow ◮ problem worse in more complex

flows

bij

DNS data for channel flow

400 800 1200 1600 2000 0.2 0.4 0.6 0.8 1 −0.4 −0.2 0.2 0.4

b11 b33 b22 b12

y/h

(Jimenez et al., Reτ = 2000) 12 / 24

RANS modeling The turbulent viscosity assumption Generalities Algebraic TVMs One-equation models

The analogy: Newtonian stress/turbulent viscosity

Kinetic theory for ideal gases → Newtonian stress law

−σij/ρ − p/ρδij = −2νSij with: ν ≈ 1

2 ¯

Cλ

◮ ¯

C mean molecular speed, λ mean free path

◮ time scale ratio in shear flow: λ ¯ C S = O(10−10)

Eddy viscosity hypothesis for turbulent flow

u′

iu′ j − 2

3k δij = −2νT ¯ Sij

◮ typical time scale ratio: k εS = O(1) ◮ local equilibrium assumption in general NOT valid!

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RANS modeling The turbulent viscosity assumption Generalities Algebraic TVMs One-equation models

Linear turbulent viscosity models

How can the turbulent viscosity νT be determined?

◮ uniform turbulent viscosity

(cf. lecture on jet flow)

◮ algebraic expressions (mixing-length etc.) ◮ one-equation models (k-model, Spalart-Allmaras) ◮ two-equation models (k-ε, k-ω)

(cf. lecture 4)

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RANS modeling The turbulent viscosity assumption Generalities Algebraic TVMs One-equation models

Mixing-length model (Prandtl 1925)

Consider two-dimensional shear flow (channel or BL)

◮ dimensionally: νT = u∗ · ℓm ◮ fluid “lump” travels δy = ℓm ◮ maintains original u(y) ◮ for constant shear S:

u′ = −S · ℓm

◮ Prandtl’s approximation:

u∗ ≈ ℓm

• du

dy

• ⇒

νT = ℓ2

m

• du

dy

• u(y),v’(y)

u’=u(y)−u(y+lm) lm x y 15 / 24

RANS modeling The turbulent viscosity assumption Generalities Algebraic TVMs One-equation models

Mixing-length coefficients for different flows

Self-similar free shear flows

◮ mixing length: ℓm = α · r1/2 α plane wake 0.180 mixing layer 0.071 plane jet 0.098 round jet 0.080 (from Wilcox 2006)

Fully-developed wall-bounded shear flows

◮ van Driest function for buffer and log-region:

ℓm = κy (1 − exp(−y+/A+)) A+ = 26

◮ simple cut-off for the outer region:

max(ℓm) = 0.09 δ

◮ more elaborate models for boundary layers:

Cebeci & Smith (1967), Baldwin & Lomax (1978)

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RANS modeling The turbulent viscosity assumption Generalities Algebraic TVMs One-equation models

Assessment of mixing-length models

Advantage

◮ numerically efficient:

• nly solve averaged Navier-Stokes + algebraic expressions

Drawbacks

◮ turbulent velocity scale entirely determined by mean flow ◮ incompleteness: flow-dependent mixing length

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RANS modeling The turbulent viscosity assumption Generalities Algebraic TVMs One-equation models

Turbulent kinetic energy model

u′

iu′ j − 2

3k δij = −2νT ¯ Sij νT = u∗ · ℓ∗

Determine characteristic velocity u∗ from TKE

◮ u∗ often not given by mean flow

e.g. decaying grid turbulence

◮ Kolmogorov (1942), Prandtl (1945):

u∗ = c √ k with: c = 0.55, and: ℓ∗ = ℓm ⇒ determine k from transport equation ℓm still needs to be provided flow by flow

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RANS modeling The turbulent viscosity assumption Generalities Algebraic TVMs One-equation models

Turbulent kinetic energy model: closure

The TKE transport equation

¯ Dk ¯ Dt − P = −     1 2u′

iu′ iu′ j + u′ jp′/ρ − νk,j

• ˜

T′

   

,j

− ˜ ε

◮ production term closed through Boussinesq hypothesis ◮ model for dissipation from high-Re assumption:

˜ ε = CD k3/2/ℓm with: CD = c3 (from log-law)

◮ model for flux term from gradient-transport hypothesis:

˜ T′ = −

• ν + νT

σk

• ∇k

with: σk = 1

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RANS modeling The turbulent viscosity assumption Generalities Algebraic TVMs One-equation models

Prediction of the individual model terms (1)

Algebraic dissipation model

◮ ˜

ε = CD k3/2/ℓm

◮ consider plane channel flow ◮ with adapted constant:

CD = 0.125

◮ 2-layer mixing length:

ℓ(1)

m =κy (1−exp(−y+/A+))

ℓ(2)

m = 0.09 δ ◮ reasonable in outer region

strong discrepancies near the wall (y+ < 40)

˜ ε+

— —DNS Hoyas & Jimenez Reτ = 2000

0.2 0.4 0.6 0.8 1 0.005 0.01 0.015 0.02 0.025 0.03

ℓ(1)

m

ℓ(2)

m

y/h

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RANS modeling The turbulent viscosity assumption Generalities Algebraic TVMs One-equation models

Prediction of the individual model terms (2)

Model for the energy flux

◮ ˜

T′ = −

• ν + νT

σk

• ∇k

◮ plane channel flow ◮ usual value: σk = 1 ◮ reasonable model

some discrepancies in buffer layer (10 ≤ y+ ≤ 20)

˜ T ′+

y,y

˜ T ′+

y,y — —DNS Hoyas & Jimenez Reτ = 2000

10 20 30 40 50 60 70 80 90 100 −0.2 −0.1 0.1 0.2 0.3

– – – – model predictions

500 1000 1500 2000 −4 −2 2 4 x 10

−3

y+

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RANS modeling The turbulent viscosity assumption Generalities Algebraic TVMs One-equation models

Incompleteness of the TKE model

Problem of the one-equation model based on TKE

the lenght scale ℓ∗ needs to be specified ⇒ incompleteness

Is there a “complete” one-equation model?

⇒ models with transport equation for turbulent viscosity νT

◮ Nee & Kovasznay (1969) ◮ Baldwin & Barth (1990) ◮ Spalart & Allmaras (1992) ◮ Menter (1994) 22 / 24

RANS modeling The turbulent viscosity assumption Generalities Algebraic TVMs One-equation models

The Spalart-Allmaras model for turbulent viscosity

¯ DνT ¯ Dt = ∇ · νT σν ∇νT

• + Sν(ν, νT , ¯

Ω, |∇νT |, ℓw)

◮ convection-diffusion equation + source term ◮ source includes various mechanisms of generation/destruction

◮ mean flow rotation ¯

Ω

◮ near-wall behavior through wall-distance ℓw ◮ destruction term (|∇νT |2), . . .

◮ basic model: 8 closure coefficients, 3 closure functions ◮ calibrated for aerodynamical applications

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RANS modeling The turbulent viscosity assumption Generalities Algebraic TVMs One-equation models

Assessment of the Spalart-Allmaras model

Spreading rate of free shear flows

SA model measured plane wake 0.341 0.32-0.40 mixing layer 0.109 0.103-0.120 plane jet 0.157 0.10-0.11 round jet 0.248 0.086-0.096

Skin friction of boundary layers

pressure gradient SA model error favorable 1% mild adverse 10% moderate adverse 10% strong adverse 33%

(from Wilcox 2006)

not satisfactory in some free shear flows

◮ reasonable predictions for attached boundary layers

discrepancies in separated flows

⇒ Need a more universal model for general flows

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Conclusion Outlook Further reading

Summary

Main issues of the present lecture

◮ How can the Reynolds-averaged equations be closed? ◮ What are the different types of models commonly used?

◮ Boussinesq’s turbulent viscosity hypothesis ◮ algebraic models ◮ transport equations for one or two turbulent scales ◮ transport equations for the Reynolds stress

◮ Do simple eddy viscosity models allow for acceptable

predictions?

◮ mixing-length type models are not complete ◮ one-equations models offer modest advantages

both types lack universality

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Conclusion Outlook Further reading

Outlook on next lecture: k–ε and other eddy viscosity models

How can the turbulent viscosity be completely determined from field equations? Does this improve the predictive capability?

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Conclusion Outlook Further reading

Further reading

◮ S. Pope, Turbulent flows, 2000

→ chapter 8 & 10

◮ P.A. Durbin and B.A. Pettersson Reif, Statistical theory and

modeling for turbulent flows, 2003 → chapter 6

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

→ chapter 2, 3 & 4

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