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Dissipation element analysis of turbulence

Lipo Wang, Norbert Peters

Institut für Technische Verbrennung

RWTH-Aachen

Germany

TMBW-07

21.08.2007 Trieste, Italy

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Content

• Introduction

• The concept of dissipation element

• PDF of the length scale of dissipation elements

• Joint PDF of the length scale and scalar difference

• Modelling

• Conditional moments

• Relation of the joint PDF to Intermittency

• Summary

Content

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Introduction

Previous geometrical studies of turbulence

• Vortex tubes (Townsend, 1952, She et al., 1990)

-not space-filling

• Critical points (Gibson, 1968; Perry and Chong, 1987; Vassilicos, 2003)

-analysis only in the vicinity of those points

The objective of dissipation element analysis is to decompose the entire field into small units to better understand turbulence.

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Construction of dissipation element by gradient trajectories

Starting from each material point in a flow field in ascending directions along scalar gradients, each trajectory will inevitably reach the maximal and minimal points. The ensemble of material points sharing the same pair ending points is named a dissipation element. This decomposition is space-filling and non-arbitrary.

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Introduction of dissipation element

For illustration: Dissipation elements in 2D turbulence

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3D-DNS calculation

Various simulations of homogenous shear flow in a 2 cubic box

Here we focus only on the passive scalar field.

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Interaction with vortex tubes

Interaction between dissipation elements and vortex tubes

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Among the many parameters to describe the statistical properties of dissipation elements, we have chosen l and ∆’, which are defined as the straight line connecting the two extremal points and the scalar difference at these points, respectively.

Parametric description

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The typical joint PDF from DNS

The joint PDF

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Results: conditional mean

The compensated conditional mean from DNS joint PDF

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A model for the length scale PDF

Fast (jump) processes:

1. The Poisson process of random cutting of a line into small segments.

This gives an exponential distribution.

2. Add a reconnection mechanism by molecular diffusion.

This removes the small elements

Slow process (drift term)

3. Continuous change of length by connection and diffusion of end points.

This enforces the .

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There are four terms describing fast processes• GC: Generation (of small elements) by Cutting• GR: Generation (of large elements) by

Reconnection• RC: Removal (of large elements) by Cutting• RR: Removal (of small elements) by Reconnectionand one drift term in the evolution equation.

The PDF of length scale

( L.Wang and N.Peters, JFM (2006), vol.554, pp.457-475)

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The PDF of length scale: comparison of model with DNS

The PDF of length scale

0 1 2 3 4 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Re =98.0Re =125.0Re =170.0model

P (l/lm)

l/lm

0 1 2 3 4 510-3

10-2

10-1

100

P (l/lm)

l/lm

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Model of the joint PDF

PDF decomposition:

The joint PDF contains the foremost information for the modeling of scalar mixing in turbulence:• the marginal PDF of the length of elements• all conditional moments <∆’n|l> and its scaling exponentsThen there is a need to go further and model the joint PDF.

Once an element is cut or reconnected, the ∆’ of the new element(s) will be forced to change. Therefore the fast processes changing the length of an element will determine the change of ∆’.

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A compensation-defect model

For scalar difference, there is a compensation in the cutting process and defect in the reconnection process, respectively.

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Model of the joint PDF

The resulting joint PDF equation

and c are modeling constants.

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Model of the joint PDF

x0 1 2 3 4 50.0

1.0

2.0

3.0

'

K=1, C=1.5

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Results: anomalous scaling exponents

0 2 4 6 8 100.0

0.5

1.0

1.5

2.0

DE analysis from DNSjoint PDF model from DE analysisExperiment, Antonia et al.

nmoment

K41

scal

ing

exponen

tq

19x10-2 10-1 10010-1

100

101

c =1.0c =1.2c =1.5

x

Conditional first moment with different defect factor c

Results: conditional means

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The marginal PDF of ∆’

Results: the marginal PDF of

0 1 2 3 40

0.2

0.4

0.6

0.8

DNSmodel

'

PDF

0 1 2 3 410-3

10-2

10-1

100

PDF

'

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Intermittency in the conventional representation

The occurence of strong events during the cascade process makes turbulent flows inhomogeneous and intermittent. Therefore the PDFs at different scales are not self-similar.

-1 -0.5 0 0.5 110-3

10-2

10-1

100

101

PDF

'

small distance(r/ =3.83)

large distance(r/ =127.7)

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Intermittency in the context of the joint PDF

At large scales, the conditional PDF is Gaussian around the conditional mean. At small scales, the conditional PDF is skewed toward large scalar difference, which implies the occurrence cliff structure (large and ).

0

1

2

3

'

P( l)

small l (0.3) large l (2.0)

DNSmodel

'

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Summary

1. A given (diffusive) scalar field can be decomposed into dissipation elements, which are space-filling and non-arbitrary.

2. The length scale PDF from the cutting-reconnection model agrees well with the results from DNS.

3. By setting appropriate parameters in the joint PDF equation, also a fair agreement with DNS is obtained.

4. The conditional moments from the joint PDF reproduce the inertial range scaling exponents.

5. Intermittency and cliff structure in scalar fields can be related with and explained from the joint PDF.