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

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Introduction Filtering the equations More

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 9

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Introduction Filtering the equations More

LECTURE 9 LES equations

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Introduction Filtering the equations More

Questions to be answered in the present lecture

Which unknowns are generated by filtering the equations? How can the residual stresses be decomposed? What does the kinetic energy balance in LES involve?

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Introduction Filtering the equations More

Roadmap

The basic elements of the LES approach

1. definition of a spatial filter
2. derivation of the filtered Navier-Stokes equations
3. choice of a model for the unclosed subgrid-stress term
4. numerical solution of the closed equations

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Introduction Filtering the equations More Continuity & momentum Kinetic energy

Filtering the Navier-Stokes equations

Navier-Stokes equations for incompressible flow

◮ instantaneous velocity u(x, t), pressure p(x, t) ◮ Cartesian coordinates, index notation, u = (u1, u2, u3)T

∂ui ∂t + ∂(ujui) ∂xj + 1 ρ ∂p ∂xi = ν ∂2u ∂xj∂xj ∂uj ∂xj =

◮ recall: Reynolds decomposition

u(x, t) = u + u′

◮ here: apply spatial filtering

u(x, t) = u + u′′

(notation u′′ to distinguish from statist. fluctuations)

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Introduction Filtering the equations More Continuity & momentum Kinetic energy

Filtering the Navier-Stokes equations (2)

Applying a spatial filter to the equations

◮ consider a homogeneous filter → filter & derivative commute

Filtered continuity equation

∂uj ∂xj

= ∂uj

∂xj = 0

◮ filtered field u is divergence-free

∂u′′

j

∂xj = ∂ ∂xj (uj − uj) = 0

◮ residual field u′′ is also divergence-free

→ analogous to continuity in the RANS context

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Introduction Filtering the equations More Continuity & momentum Kinetic energy

Filtering the Navier-Stokes equations (3)

Filtered momentum equation

∂ui ∂t + ∂ujui ∂xj + 1 ρ ∂p ∂xi = ν ∂2u ∂xj∂xj = 2ν ∂Sij ∂xj

◮ since ujui = uj ui we have:

∂ui ∂t + ∂ujui ∂xj + 1 ρ ∂p ∂xi = 2ν ∂Sij ∂xj + ∂ujui ∂xj − ∂ujui ∂xj

◮ from which:

Dui Dt + 1 ρ ∂p ∂xi = 2ν ∂Sij ∂xj − ∂τ R

ij

∂xj with τ R

ij ≡ ujui − uj ui

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Introduction Filtering the equations More Continuity & momentum Kinetic energy

Filtering the Navier-Stokes equations (4)

Residual stress tensor τ R

ij ≡ ujui − uj ui ◮ alternatively called sub-grid scale (SGS) stress tensor ◮ analogy to RANS context:

τ RANS

ij

≡ ujui − ujui = u′

iu′ j

Modified filtered equations

◮ residual kinetic energy:

kr ≡ 1

2τ R kk ◮ anisotropic residual stress tensor:

τ r

ij ≡ τ R ij − 2 3krδij ◮ define modified filtered pressure:

˜ p ≡ p + 2

3krδij

Dui Dt + 1 ρ ∂˜ p ∂xi = 2ν ∂Sij ∂xj − ∂τ r

ij

∂xj

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Introduction Filtering the equations More Continuity & momentum Kinetic energy

Comments on the filtered equations

Modified filtered equations

Dui Dt + 1 ρ ∂˜ p ∂xi = 2ν ∂Sij ∂xj − ∂τ r

ij

∂xj ∂uj ∂xj =

◮ equations are unclosed ◮ residual stress tensor τ r ij needs modeling (cf. next lecture) ◮ the fields u and ˜

p are three-dimensional, unsteady, random

◮ residual stress tensor depends on type and parameters of filter

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Introduction Filtering the equations More Continuity & momentum Kinetic energy

Decomposition of the residual stresses

filtered convective term contains: ujui = uj ui + τ r

ij + 2

3δijkr

Leonard decomposition (1974)

τ R

ij = uj ui − uj ui

≡Lij

+ uju′′

i + uiu′′ j

≡Cij

+ u′′

i u′′ j

≡Rij

◮ Lij are termed “Leonard stresses” ◮ Cij are the “cross stresses” ◮ Rij are the “SGS Reynolds stresses”

→ all terms non-zero in general case! BUT: the stresses Lij and Cij are not Galilean invariant (when considering shifted velocity → decomposition changes)

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Introduction Filtering the equations More Continuity & momentum Kinetic energy

Galilean invariant decomposition of the residual stresses

Germano decomposition (1986)

τ R

ij = uj ui − uj ui

≡Lo

ij

+ uju′′

i + uiu′′ j − uj u′′ i − ui u′′ j

≡Co

ij

+ u′′

i u′′ j − u′′ i · u′′ j

≡Ro

ij

◮ Lo ij are termed “objective Leonard stresses” ◮ Co ij are the “objective cross stresses” ◮ Ro ij are the “objective SGS Reynolds stresses”

→ terms Lo

ij, Co ij, Ro ij are invariant under arbitrary shifts

⇒ this is the preferred decomposition

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Introduction Filtering the equations More Continuity & momentum Kinetic energy

Conservation of energy

Instantaneous kinetic energy Ek(x, t) ≡ 1

2u · u = 1 2uiui

Filtered kinetic energy

Ek = 1 2ui ui = 1 2ui ui

≡Ef

+ 1 2ui ui − 1 2ui ui

≡kr= 1

2 τ R ii

Filtered kinetic energy equation

DEf Dt + 1 ρui˜ pδij − 2νuiSij + uiτ r

ij

,j

= −Pr − εf

◮ rate of production of residual kinetic energy:

Pr = −τ r

ijSij ◮ viscous dissipation rate due to filtered field:

εf = 2νSijSij

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Introduction Filtering the equations More Continuity & momentum Kinetic energy

Conservation of energy – comparison with RANS

Reynolds decomposition: Ek = ¯ E + k

∂t ¯ E +

uj ¯

E + uiu′

iu′ j + ujp/ρ − 2νui ¯

Sij

,j

= −P − ¯ ε ∂tk +

ujk + 1

2u′

iu′ iu′ j + u′ jp′/ρ − 2νu′ iS′ ij

,j

= +P − ε

LES decomposition: Ek = Ef + kr

∂tEf +

ujEf + uiτ r

ij + uj ˜

p/ρ − 2νuiSij

,j

= −Pr − εf & similar equation for kr

note formal analogy between RANS and LES

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Introduction Filtering the equations More Continuity & momentum Kinetic energy

Amount of residual kinetic energy & filter width

Residual kinetic energy kr as function of filter width ∆

◮ suppose high Reynolds number, Kolmogorov spectrum ◮ choose sharp spectral filter, cut-off κc = π/∆

→ kr = 3

2Ckol (ε∆/π)2/3

Choice of filter width ∆ to resolve fraction of kr

◮ use length scale L = k3/2/ε and L ≈ ℓEI(6/0.43)

→

∆ ℓEI =

2

3 Ckol kr k

3/2 6π/0.43

◮ e.g. with 80% resolved TKE (kr = 0.2) → ∆/ℓEI = 1.16

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Introduction Filtering the equations More Output of LES

What results are provided by an LES computation?

Output of LES code: filtered field u(x, t) Desired results

1. statistically averaged mean flow field

u(x, t)

but: applying average yields u = u = u ! → in practice: difference often neglected u ≈ u

2. Reynolds stress components

u′

iu′ j(x, t)

◮ available in LES: (ui − ui)(uj − uj) = ui ui − uiuj ◮ defining u∗

i = ui − ui one obtains:

u′

iu′ j = u∗ i u∗ j

LES result

+ τ R

ij

model

+ u′′

i uj + u′′ j ui + u′′ i u′′ j

≈ 0

→ LES result corresponds to resolved part of Reynolds stress (diagonal elements smaller than true value)

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Conclusion References Further reading

Summary of today’s lecture

Derivation of the filtered flow equations

◮ analogy to RANS; physical meaning of terms different

How can the residual stresses be decomposed?

◮ Leonard decompositiopn; Germano decomposition; Galilean

invariance

What does the kinetic energy balance in LES involve?

◮ kinetic energy of filtered field: similar equation as RANS

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Conclusion References Further reading