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

Introduction Constructing Reynolds-stress models Performance

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 5

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Introduction Constructing Reynolds-stress models Performance

LECTURE 5 Reynolds-stress transport models

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Introduction Constructing Reynolds-stress models Performance

Questions to be answered in the present lecture

How can the equations be closed at the second-moment level?

◮ why resort to Reynolds-stress models? ◮ how to derive the u′ iu′ j transport equation? ◮ how to model the principal unknown terms?

How do Reynolds-stress models perform?

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Introduction Constructing Reynolds-stress models Performance

Why use Reynolds-stress transport models?

Fundamental deficiency of turbulent viscosity models:

◮ Reynolds stress is assumed local function of mean strain-rate

→ transport/history effects are neglected (e.g. failure in relaxation from mean strain – cf. lecture 3)

Attractive features of Reynolds-stress transport models:

◮ avoid any turbulent viscosity hypothesis ◮ transport & production terms are in closed form

→ transport effects “built-in” → stress production model-free (important in complex strain)

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Introduction Constructing Reynolds-stress models Performance Deriving the stress transport equation Modelling the pressure-strain correlation Inhomogeneous flows

Deriving the transport equation for the Reynolds stress

Steps in deriving the exact equation from Navier-Stokes

note that ∂tu′

iu′ j = u′ j∂tu′ i + u′ i∂tu′ j

• 1. write transport equation for fluctuating velocity u′
• 2. multiply ith-component with u′

j

• 3. multiply jth-component with u′

i

• 4. add results from 2. and 3.
• 5. take average of result from 4.

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Introduction Constructing Reynolds-stress models Performance Deriving the stress transport equation Modelling the pressure-strain correlation Inhomogeneous flows

The exact transport equation for the Reynolds stress

¯ Du′

iu′ j

¯ Dt +        u′

iu′ ju′ k + 1

ρ p′u′

jδik + 1

ρ p′u′

iδjk − νu′ iu′ j,k

• turbulent transport Tkij

      

,k

= −ui,ku′

ku′ j − uj,ku′ ku′ i

• production Pij

+ 1 ρ

• p′u′

j,i + p′u′ i,j

• pressure-strain Rij

− 2νu′

i,ku′ j,k

• dissipation tensor εij

◮ pressure–rate-of-strain Rij and dissipation εij are unclosed ◮ first three terms of turbulent transport Tkij are unclosed ◮ half the trace of this equation yields TKE equation ◮ pressure–rate-of-strain is absent in TKE equation

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Introduction Constructing Reynolds-stress models Performance Deriving the stress transport equation Modelling the pressure-strain correlation Inhomogeneous flows

Importance of the terms in a boundary layer flow

0 = −

¯

Du′

iu′ j

¯

Dt −u′

iu′ ju′ k,k

+νu′

iu′ j,kk

+Pij +Πij −εij

(1) (2) (3) (4) (5) (6)

Budget of streamwise normal stress u′u′

◮ principal production: P11 ≈ −2u′v′u,y ◮ mainly balanced by: dissipation & pressure–rate-of-strain note: Πij ≡ Rij −

1 ρ

• p′u′

jδik + p′u′ iδjk

• ,k

(DNS Spalart 1988, Reθ = 1410) 7 / 28

Introduction Constructing Reynolds-stress models Performance Deriving the stress transport equation Modelling the pressure-strain correlation Inhomogeneous flows

Importance of the terms in a boundary layer flow

0 = −

¯

Du′

iu′ j

¯

Dt −u′

iu′ ju′ k,k

+νu′

iu′ j,kk

+Pij +Πij −εij

(1) (2) (3) (4) (5) (6)

Budget of wall-normal stress v′v′

◮ no production: P22 ≈ 0 ◮ gain from pressure–rate-of-strain ◮ approximately balanced by dissipation

(DNS Spalart 1988, Reθ = 1410) 8 / 28

Introduction Constructing Reynolds-stress models Performance Deriving the stress transport equation Modelling the pressure-strain correlation Inhomogeneous flows

Nature of the pressure–rate-of-strain correlations

Observation from flow data:

◮ pressure terms are of significant magnitude ◮ pressure–rate-of-strain correlation has redistributive character ◮ due to incompressibility, term has zero trace:

Rij ≡ 1

ρ

• p′u′

j,i + p′u′ i,j

• →

Rii = 0 ⇒ no contribution to turbulent kinetic energy

Pressure–rate-of-strain is main challenge for modelling!

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Introduction Constructing Reynolds-stress models Performance Deriving the stress transport equation Modelling the pressure-strain correlation Inhomogeneous flows

Dissipation tensor components in boundary layer flow

Observations:

◮ for high Reynolds number:

dissipation tensor is approximately isotropic

◮ low Reynolds in DNS:

→ some residual anisotropy but: significant anisotropy near wall (cf. lecture 6)

0.0 0.2 0.4 0.6 0.8 1.0

• 0.5

0.0 0.5 1.0 1.5 2.0 2.5

ε11 ε12 ε22 ε33 ε ~

2 3

εij y/δ

(DNS Spalart 1988, Reθ = 1410)

⇒ dissipation tensor often modelled as isotropic: εij = 2 3 ˜ ε δij

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Introduction Constructing Reynolds-stress models Performance Deriving the stress transport equation Modelling the pressure-strain correlation Inhomogeneous flows

Unclosed terms in the Reynolds-stress transport equation

Reynolds-stress equation, assuming isotropic dissipation:

¯ Du′

iu′ j

¯ Dt +          u′

iu′ ju′ k

• T (u)

kij

+ 1 ρ p′u′

jδik + 1

ρ p′u′

iδjk

• T (p)

kij

+νu′

iu′ j,k

        

,k

= Pij + Rij − 2

3 ˜

εδij

Models need to be prescribed for the following terms:

◮ triple correlation T (u) kij and pressure transport T (p) kij ◮ the scalar (pseudo) dissipation rate ˜

ε → similar to k-ε model

◮ the pressure–rate-of-strain correlation Rij

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Introduction Constructing Reynolds-stress models Performance Deriving the stress transport equation Modelling the pressure-strain correlation Inhomogeneous flows

The Poisson equation for pressure fluctuations

Fluctuating pressure: given by linear equation

1 ρ∇2p′ = −2ui,j u′

j,i −

• u′

iu′ j − u′ iu′ j

• ,ij

◮ pressure can be decomposed into 3 contributions

p′ = p(h) + p(r) + p(s)

◮ homogeneous pressure p(h):

∇2p(h) = 0

◮ rapid pressure p(r):

∇2p(r) = −2ρui,j u′

j,i ◮ slow pressure p(s):

∇2p(s) = −ρ

• u′

iu′ j − u′ iu′ j

• ,ij

⇒ 3 different contributions to pressure-strain correlation

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Introduction Constructing Reynolds-stress models Performance Deriving the stress transport equation Modelling the pressure-strain correlation Inhomogeneous flows

Contributions to pressure-strain correlation

Homogeneous pressure → R(h)

ij ≡ p(h)(u′ i,j + u′ j,i)/ρ ◮ influenced by boundary conditions only ◮ R(h) ij

vanishes in homogeneous turbulence

◮ contribution important near walls

(details in lecture 6)

Rapid pressure

◮ reacts instantly to mean velocity gradients ◮ dominant contribution for large strain rate Sk/ε

Slow pressure

◮ determined by self-interaction of turbulent field ◮ principal mechanism for return to isotropy without strain

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Introduction Constructing Reynolds-stress models Performance Deriving the stress transport equation Modelling the pressure-strain correlation Inhomogeneous flows

Modelling the slow part of pressure-strain

Homogeneous turbulence without mean velocity gradients:

∂tu′

iu′ j = Rij − 2

3 ˜ εδij

◮ no mean velocity gradients → Rij = R(s) ij ◮ modelling ansatz:

R(s)

ij = ˜

ε F(s)

ij (bij) ◮ the most general tensor function is:

F(s)

ij

= C1bij + C2

• b2

ij − 1

3b2

kk δij

• ◮ C1, C2 are scalar functions

◮ Rotta’s linear model:

C1 = −2CR, C2 = 0 ⇒ linear return-to-isotropy: dtbij = −(CR − 1) ˜

ε k bij

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Introduction Constructing Reynolds-stress models Performance Deriving the stress transport equation Modelling the pressure-strain correlation Inhomogeneous flows

Slow pressure-strain models: comparison with experiment

Return-to-isotropy after distorting duct

• 1=3

• =

◮ mean strain is imposed in a

distorting duct

◮ then: in straight section,

turbulence relaxes to isotropy

◮ simple linear model works in

this case

evolution of bij in straight duct section

b33 b22 b11

distance x

(×, ◦, • experiment LePenven et al. 1985) (— — Rotta model, CR = 1.5) 15 / 28

Introduction Constructing Reynolds-stress models Performance Deriving the stress transport equation Modelling the pressure-strain correlation Inhomogeneous flows

But: return-to-isotropy depends on complete state of b

◮ process can be described by

two invariants of tensor b: IIb = − 1

2bijbji

IIIb = 1

3bijbjkbki ◮ isotropy: IIb = IIIb = 0 ◮ experiments: return rate

depends on invariant IIIb !

◮ linear model:

dtIIb = −2(CR − 1) ˜

ε kIIb

model should be non-linear ⇒ Shih/Lumley 1985: C1(II, III) ⇒ Speziale et al 1991: C2 = 0

IIb

case A

• 1=3

• =

case B

• 1=3

• =

2nd invariant of b

1 2 3 4 5 −0.03 −0.02 −0.01

case A (IIIb > 0) case B (IIIb < 0) distance x 16 / 28

Introduction Constructing Reynolds-stress models Performance Deriving the stress transport equation Modelling the pressure-strain correlation Inhomogeneous flows

Return to isotropy in Lumley’s triangle

◮ two invariants fully

determine tensor bij

◮ Pope proposes η, ξ:

6η2 = −2 IIb 6ξ3 = 3 IIIb

◮ “triangular” region,

bounded by: axisymmetric & 2-component states

◮ isotropy at origin

η

−1/6 1/6 1/3 1/6 1/3 2C 1C axi−2p axi−1p iso

ξ

data from exp. of LePenven et al. (1985),

• case A (IIIb > 0), • case B (IIIb < 0)

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Introduction Constructing Reynolds-stress models Performance Deriving the stress transport equation Modelling the pressure-strain correlation Inhomogeneous flows

Aside: channel flow in Lumley’s triangle

η

data from DNS of Jim´ enez et al. (2006) −1/6 1/6 1/3 1/6 1/3 2C 1C axi−1p axi−2p iso

ξ

• y+ = 1, • y+ = 5, • y+ = 50, • y/h = 0.1, • y/h = 1

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Introduction Constructing Reynolds-stress models Performance Deriving the stress transport equation Modelling the pressure-strain correlation Inhomogeneous flows

Modelling the fast part of pressure-strain

Homogeneous turbulence with mean velocity gradients:

∂tu′

iu′ j = Pij + Rij − 2

3 ˜ εδij

◮ here: Pij = 0 and Rij = R(r) ij + R(s) ij ◮ exact expression: R(r) ij = 2ul,k (Mkjil + Mikjl)

where Mijkl is an integral of two-point correlations

◮ in single-point closures: Mijkl modelled as function of (b, k) ◮ symmetry, tensorial considerations & realizability constraints

→ lead to functional form: R(r)

ij = k 8 n=3 f(n) T (n) ij

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Introduction Constructing Reynolds-stress models Performance Deriving the stress transport equation Modelling the pressure-strain correlation Inhomogeneous flows

Some common rapid pressure-strain models:

R(r)

ij

= k

8

• n=3

f(n) T (n)

ij

n T (n)

ij

f(n) : LRR SSG SL

(Launder et al. (Speziale et al. (Shih & Lumley 1975) 1991) 1985)

3 ¯ Sij

4 5 4 5 + 1.3√IIb 4 5

4 ¯ Sikbkj + bik ¯ Skj − 2

3 ¯

Sklblkδij

6 11 (2 + 3C2) 5 4

12C2 5 ¯ Ωikbkj − bik ¯ Ωkj

2 11 (10 − 7C2) 2 5 4 3 (2 − 7C2)

6 ¯ Sikb2

kj + b2 ik ¯

Skj − 2

3 ¯

Sklb2

lkδij 4 5

7 ¯ Ωikb2

kj − b2 ik ¯

Ωkj

4 5

8 bik ¯ Sklblj − 1

3 ¯

Sklb2

lkδij

− 8

5 C2 = 0.4 C2 =

1 10 (1+ 4 5 g(IIb, IIIb))

satisfies realizability: no yes yes where: ¯ Sij ≡ 1

2 (ui,j + uj,i), ¯

Ωij ≡ 1

2 (ui,j − uj,i) 20 / 28

Introduction Constructing Reynolds-stress models Performance Deriving the stress transport equation Modelling the pressure-strain correlation Inhomogeneous flows

Results for pressure-strain models in homogeneous shear

Comparison with experimental measurements

◮ note: this test involves rapid and slow parts R(r) ij + R(s) ij ◮ equilibrium results: experiment LRR SSG SL Tavoularis & Karnik (1989) b11 0.155 0.219 0.135 0.18 b22

• 0.122
• 0.146
• 0.136
• 0.11

b33 0.033 0.073

• 0.001

0.07 b12

• 0.188
• 0.164
• 0.108
• 0.16

◮ LRR and SSG models provide reasonable values ◮ Shih/Lumley model yields too weak tangential component b12

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Introduction Constructing Reynolds-stress models Performance Deriving the stress transport equation Modelling the pressure-strain correlation Inhomogeneous flows

Reynolds-stress transport in non-homogeneous flow

¯ Du′

iu′ j

¯ Dt +          u′

iu′ ju′ k

• T (u)

kij

+ 1 ρ p′u′

jδik + 1

ρ p′u′

iδjk

• T (p)

kij

+νu′

iu′ j,k

        

,k

= Pij + Rij − 2

3 ˜

εδij

Additional modeling for non-homogeneous flows:

◮ use local pressure–strain model Rij as in homogeneous flow

(except for wall corrections – cf. lecture 6)

◮ models for triple correlation T (u) kij and pressure transport T (p) kij ◮ equation for the dissipation rate, involving transport ˜

ε

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Introduction Constructing Reynolds-stress models Performance Deriving the stress transport equation Modelling the pressure-strain correlation Inhomogeneous flows

Modeling combined turbulent transport

Gradient diffusion models T (t)

kij = T (u) kij + T (p) kij ◮ isotropic eddy diffusivity:

T (t)

kij = − Cs

k2 ˜ ε

νT ∂u′

iu′ j

∂xk

with constant Cs = 0.09 as in k-ε model

◮ eddy diffusivity tensor model:

T (t)

kij = − Cs

k ˜ ε u′

ku′ l

• (νT )kl

∂u′

iu′ j

∂xl

with constant Cs = 0.22

◮ more general models, symmetric w.r.t. indices i, j, k

(Mellor & Herring, 1973; Hanjalic & Launder, 1972)

◮ models based on transport equation for u′ iu′ ju′ k

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Introduction Constructing Reynolds-stress models Performance Deriving the stress transport equation Modelling the pressure-strain correlation Inhomogeneous flows

Experimental data for triple correlation

Self-similar mixing layer

◮ triple term dominates

transport

◮ compare T (u) kij with

isotropic eddy diffusivity model: u′v′v′ = −Cs k2

˜ ε u′v′,y

⇒ reasonable predictions

y r1/2

u′v′v′/U3

measurement Bell & Mehta (1990)

— — eddy diffusivity model

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Introduction Constructing Reynolds-stress models Performance Deriving the stress transport equation Modelling the pressure-strain correlation Inhomogeneous flows

Scalar dissipation rate equation for RSM

Similar model equation as in k-ε model:

¯ D˜ ε ¯ Dt =

• ν · δlk + (νT )kl

σε

• ˜

ε,k

• ,l

+ Cε1 P˜ ε k − Cε2 ˜ ε2 k

Here: minor differences w.r.t. k-ε model:

◮ production is model-free:

P = −u′

iu′ jui,j ◮ turbulent diffusion term has anisotropic eddy viscosity:

(νT )kl = Cε k

εu′ ku′ l

with: Cε/σε = 0.15

◮ other constants take standard values: Cε1 = 1.44, Cε2 = 1.92

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Introduction Constructing Reynolds-stress models Performance Free shear flow Secondary flow in square duct BL on curved walls

Predictions of complete model for mixing layer flow

Self-similar mixing layer

◮ spreading rate dδ/dx: exp. LRR SSG k-ε 0.019 0.019 0.018 0.016

→ LRR, SSG yield good predictions

◮ turbulence structure similar

to homogeneous shear flow ⇒ performance as calibrated (pressure-strain)

y δ

• ✁✄✂✆☎
• ✁

• ❏■▲❑

−b12

(experiment of Bell & Mehta, 1990) 26 / 28

Introduction Constructing Reynolds-stress models Performance Free shear flow Secondary flow in square duct BL on curved walls

Flow through a straight square duct

secondary flow vectors

(from Durbin and Pettersson Reif, 2001) v∂yωx + w∂zωx = ∂yz

• v′v′ − w′w′
• +
• ∂yy − ∂zz
• v′w′ + ν
• ∂yy + ∂zz
• ωx

◮ Reynolds-stress models can predict: secondary shear v′w′

& normal-stress anisotropy v′v′ − w′w′

◮ this example: secondary flow strength underpredicted

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Introduction Constructing Reynolds-stress models Performance Free shear flow Secondary flow in square duct BL on curved walls

Boundary layer on curved walls

Concave wall Convex wall

y δ99

−u′v′

y δ99

−u′v′

• experiments, —

— Reynolds-stress model (Durbin 1993)

◮ increased production

→ enhanced turbulence

◮ reduced production

→ damped turbulence ⇒ effect captured by Reynolds stress transport model

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Conclusion Outlook Further reading

Summary

Why resort to Reynolds-stress models?

◮ convective transport and production mechanisms are

model-free

Pressure-strain correlation is principal unknown

◮ splitting in slow and rapid part ◮ modelling for homogeneous flow (tensor fct., realizability)

Performance of Reynolds-stress models:

◮ account for complex straining fields, normal stress anisotropy

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Conclusion Outlook Further reading

Outlook: Boundary conditions and wall treatment

How can RANS models be applied in wall-bounded flows?

◮ the wall-function approach ◮ specific model modifications for the wall region ◮ elliptic relaxation models for the pressure-strain correlation

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Conclusion Outlook Further reading

Further reading

◮ S. Pope, Turbulent flows, 2000

→ chapter 11

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

Statistical theory and modeling for turbulent flows, 2003 → chapter 7

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