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

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Introduction Filtering More on filtering

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 8

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Introduction Filtering More on filtering

LECTURE 8 Introduction to Large Eddy Simulation & Spatial Filtering

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Introduction Filtering More on filtering

Questions to be answered in the present lecture

What are the basic elements of the LES approach? How can (explicit) spatial filtering be realized?

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Introduction Filtering More on filtering

Basic idea of Large-Eddy Simulation

◮ consider the time-dependent Navier-Stokes equations ◮ resolve only the large scales of motion numerically ◮ replace the action of the small scales by a model

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Introduction Filtering More on filtering

Conceptual steps involved in an LES analysis

Intermediate steps

1. define a spatial filter
2. derive the filtered Navier-Stokes equations
3. choose a model for the unclosed subgrid-stress term
4. solve the closed equations numerically

Result

→ approximation of the large-scale motions (single flow realization)

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Introduction Filtering More on filtering Generalities Fourier space Discrete data

Resolving the large scales in LES

physical space DNS LES

E(κ)

spectral space

10

−2

10 10

−8

10

−6

10

−4

10

−2

10

LES DNS

∆x−1

LES

∆x−1

DNS

wavenumber κη

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Introduction Filtering More on filtering Generalities Fourier space Discrete data

Two different viewpoints of LES & filtering

I Explicit filtering (through a filter operation)

◮ spatial filtering, residual stress modelling and numerical

solution of the model equations are treated separately

II Implicit filtering (by the numerical grid)

◮ numerical discretization errors are deliberately treated as part

f the model

→ we will first consider explicit filtering

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Introduction Filtering More on filtering Generalities Fourier space Discrete data

Definition of the filter operation

General definition

u(x, t) =

G(r, x)

filter fct. u(x − r, t)

signal

dr (volume integral)

◮ normalization condition:

G(r, x) dr = 1

∀ x

◮ decomposition:

u(x, t) = u(x, t) filtered + u′(x, t) residual → appears analogous to Reynolds decomposition

◮ BUT: u(x, t) is a random field! ◮ AND: u′ = 0

ALSO: u = u

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Introduction Filtering More on filtering Generalities Fourier space Discrete data

Filtering in one spatial dimension

◮ 1D case, consider homogeneous filter functions G(r)

u(x) = ∞

−∞

G(r) u(x − r) dr

Example

◮ artificial signal u(x) ◮ “box filter” with width

∆ = 0.35 → filtered signal u is smoother → filtered residual u′ is not zero → repeated filtering increases smoothness

from Pope (2000) 2 4 6 8 10

1

1 2 3 4

U u’ x ∆

u u u′ u′ x

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Introduction Filtering More on filtering Generalities Fourier space Discrete data

Some common filter functions in physical space

G(r) box

1 ∆ H( 1 2∆ − |r|)

Gauss

6

π∆2

1/2 exp

− 6r2

∆2

sharp

sin(π r/∆)/(π r) spectral

G ∆

from Pope (2000)

4
2

2 4 0.0 1.0

∆

box – – – – Gauss — — sharp spectral — · —

r/∆

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Introduction Filtering More on filtering Generalities Fourier space Discrete data

Identities: filtering, derivatives and statist. averaging

Filtering and temporal derivative commute.

∂u ∂t = ∂u ∂t

Filtering and spatial differentation do not generally commute.

∂ui ∂xj = ∂ui ∂xj

+ . . .

. . . except when the filter function is spatially homogeneous.

Filtering and statistical averaging commute.

u = (u)

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Introduction Filtering More on filtering Generalities Fourier space Discrete data

Fourier space view: transfer function

◮ writing the Fourier transform of u(x) as:

ˆ u(κ) = F {u(x)}

◮ one obtains from the convolution theorem:

ˆ u = F {u(x)} = ˆ G(κ) ˆ u(κ) → where ˆ G is the transfer function: ˆ G(κ) = 2πF {G(r)}

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Introduction Filtering More on filtering Generalities Fourier space Discrete data

Transfer functions for common filters

ˆ G(κ) box sin(κ∆/2)/(κ∆/2) Gauss exp

− κ2∆2

24

sharp

H(κc − |κ|) spectral κc = π/∆

ˆ G

from Pope (2000)

10
5

5 10 0.0 0.5 1.0

κ) κ/κ

box – – – – Gauss — — sharp spectral — · —

κ/κc

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Introduction Filtering More on filtering Generalities Fourier space Discrete data

Further properties of filtering (Fourier space)

◮ filtered field:

ˆ u(κ) = ˆ G(κ) “low-pass” ˆ u(κ)

◮ residual field:

u′(κ) =
1 − ˆ

G(κ)

“high-pass”

ˆ u(κ)

◮ double-filtered field:

ˆ u(κ) =

ˆ

G(κ) 2 ˆ u(κ) ⇒ u = u requires ˆ G2 = ˆ G (in general not true)

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Introduction Filtering More on filtering Generalities Fourier space Discrete data

Discrete point of view of filtering

discrete grid: xi = (i − 1)∆x

xi i i + 1 i − 1 i − 2 i + 2 ∆x ∆x

Discrete filter function w

◮ vector w with length 2n + 1, normalization: n

q=−n

wq = 1

◮ example: box filter, n = 1:

w(n=1)

box

= 1 3, 1 3, 1 3

Discrete filter operation

◮ ui = n

q=−n

wq ui+q ∀ i = 1 . . . N

◮ example: box filter , n = 1:

ui = 1 3 (ui−1 + ui + ui+1)

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Introduction Filtering More on filtering Generalities Fourier space Discrete data

Discrete point of view of filtering (2)

Discrete filter operation – applied twice

◮ example: box filter , n = 1: ui = 1

3 (ui−1 + ui + ui+1) = 1 9 (ui−2 + 2ui−1 + 3ui + 2ui+1 + ui+2)

Conclusion

◮ doubly-filtered signal u = u ◮ applying box filter twice ⇒ triangle-shaped filter ◮ applying filter twice ⇒ widening filter support

→ more examples in Matlab exercise

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Introduction Filtering More on filtering Non-uniform grid & Boundaries 3D filtering Filtered energy spectrum

Filtering on non-uniform grid & near domain boundaries

Filter G is in general a function of position x

◮ u(x) =

∞

−∞

G(r, x) u(x − r) dr

Non-uniform grid:

⇒ define filter in mapped (uniform) space

(Vasilyev, Lund & Moin, 1998)

Boundaries:

⇒ filter becomes asymmetric near boundary

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Introduction Filtering More on filtering Non-uniform grid & Boundaries 3D filtering Filtered energy spectrum

Filtering in three spatial dimensions (homogeneous case)

1D filtering

◮ u(x) =

∞

−∞

G(r) u(x − r) dr

3D isotropic filter

◮ u(x) =

∞

−∞

G(|r|) u(x − r) dr

3D anisotropic filter (“rectangular grid filter”)

◮ u(x) =

∞

−∞

G(r1, ∆1)G(r2, ∆2)G(r3, ∆3) u(x − r) dr

More complex multi-dimensional filters may be used . . .

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Introduction Filtering More on filtering Non-uniform grid & Boundaries 3D filtering Filtered energy spectrum

Filtered energy spectrum – in homogeneous turbulence

One-dimensional energy spectrum

◮ definition:

E11(κ) = 1 π ∞

−∞

R(r) exp(−Iκr) dr

◮ autocovariance of filtered field:

R(r) = u(x + r) u(x) ⇒ energy spectrum of the filtered field: E11(κ) =

ˆ

G(κ)

2

attenuation factor E11(κ)

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Introduction Filtering More on filtering Non-uniform grid & Boundaries 3D filtering Filtered energy spectrum

Filtered energy spectrum – attenuation factors

◮ different localization

properties of filters in Fourier space

| ˆ G|2

from Pope (2000) 10-1 100 101 102 10-6 10-5 10-4 10-3 10-2 10-1 100 10-1 100 101 102 10-6 10-5 10-4 10-3 10-2 10-1 100 10-1 100 101 102 10-6 10-5 10-4 10-3 10-2 10-1 100

κ/κ κ)2

box – – – – Gauss — — sharp spectral — · —

κ/κc

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Introduction Filtering More on filtering Non-uniform grid & Boundaries 3D filtering Filtered energy spectrum

Example of filtered spectrum

◮ Pope’s model spectrum ◮ Reλ = 500 ◮ Gaussian filter ◮ ∆ = L11/6 E11/(u′u′L11)

from Pope (2000) 10-1 100 101 102 103 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 11

κ=κc L κ

E11 E11

L11κ

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Conclusion References Further reading

Summary of today’s lecture

What are the basic elements of the LES approach?

◮ definition of a spatial filter ◮ derivation of the filtered Navier-Stokes equations ◮ choice of a model for the unclosed subgrid-stress term ◮ numerical solution of the closed equations

How can (explicit) spatial filtering be realized?

◮ filtering = convolution of signal with filter function ◮ some common filter functions & properties

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Conclusion References Further reading