Dissipation element analysis of turbulence

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Dissipation element analysis of turbulence
Lipo Wang, Norbert Peters Institut für
Technische Verbrennung RWTH-Aachen Germany TMBW-0
7 21.08.2007 Trieste, Italy
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Content

• 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

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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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Parametric description
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.
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The joint PDF
The typical joint PDF from DNS
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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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The PDF of length scale

• 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
  Reconnection
• and one drift term in the evolution equation.

( L.Wang and N.Peters, JFM (2006), vol.554,
pp.457-475)
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The PDF of length scale
The PDF of length scale comparison of model
with DNS
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Model of the joint PDF

• 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 lt??nlgt and its
  scaling exponents
• Then 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 ??.

PDF decomposition
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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
K1, C1.5
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Results anomalous scaling exponents
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Results conditional means
Conditional first moment with different defect
factor c
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Results the marginal PDF of ?
The marginal PDF of ??
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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.
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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 ).
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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.