Lagrangian and Eulerian velocity structure functions in hydrodynamic turbulence

Article from K. P. Zybin and V. A. Sirota

Enrico Dammers & Christel SandersCourse 3T220 Chaos

Content

Goal Euler vs. lagrangian Background Theory from earlier articles Structure functions Bridge relations Results Conclusions

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Goal of the article

Showing eulerian and lagrangian structure formulas are obeying scaling relations

Determine the scaling constants analytical without dimensional analyses

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Euler vs. Lagrangian

LAGRANGIAN EULER

Measured between t and t+τ Along streamline

Structure function

Measured between r and r+l Between fixed points

Structure function

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Structure Functions

Kolmogorov: She-Leveque:

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Background

Turbulent flow, Assumptions:

Stationary Isotropic Eddies , which are characterized by velocity scales and time scales(turnover time)

Model: Vortex Filaments Thin bended tubes with vorticity, ω. Assumption:

Straight Tubes Regions with high vorticity make the main contribution to structure functions

ω

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Theory of earlier Articles:Navier-stokes on vortex filament Dot product with

relation pressure en velocity

Change to Lagrange Frame: Lagrange: ,

, at r=

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Theory of earlier Articles:Navier-stokes on vortex filament Taylor expansion of v’ and P around r=

, Splitting in sum of symmetric and anti-

symmetric term

Vorticity

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Theory of earlier Articles:Navier-stokes on vortex filament Combining all terms

= 15 different values 10 equations 5 undefined functions

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Theory of earlier Articles:Navier-stokes on vortex filament Assumption:

are random functions, stationary With:

Where is a function depending on profile

When For Simplicity:

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Theory of earlier Articles:Eigenfunctions Small n, value of order , non-linear

function In real systems for large n:

assumption of article Where is maximum possible rate of vorticity

growth

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Eulerian structure function

Assume circular orbit of particle in a filament:

Average over all point pairs:

l must be smaller then R:

This restriction gives a maximum to t for the filament

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Eulerian structure function

This results in the following condition:

: Eddy Turn over time : Eddy size for

Gives:

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Eulerian structure function

The eulerian structure function now becomes:

With

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Lagrangian structure function

For the lagrangian function:

: curvature radius of the trajectory Assume which is the same restriction as

in the euler case,

Same steps as with the eulerian function gives:

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Lagrangian structure Function

The lagrangian structure function now becomes:

With

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Bridge relation

Now we have Combination of ’s gives relation:

(n-)=2(n-

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Results

Compare with numerical simulation

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Conclusions

Showing eulerian and lagrangian structure formulas are obeying scaling relations

Determine the scaling constants analytical without dimensional analyses Using Eigen functions:

(n-)=2(n-

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Questions?

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Results

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Results

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