article from k. p. zybin and v. a. sirota enrico dammers & christel sanders course 3t220 chaos
TRANSCRIPT
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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