Lagrangian and Eulerian Representations of Fluid Flow: Kinematics
Lagrangian and Eulerian velocity structure functions in hydrodynamic turbulence
-
Upload
charlotte-schroeder -
Category
Documents
-
view
33 -
download
2
description
Transcript of Lagrangian and Eulerian velocity structure functions in hydrodynamic turbulence
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
2
Goal of the article
Showing eulerian and lagrangian structure formulas are obeying scaling relations
Determine the scaling constants analytical without dimensional analyses
3
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
4
Structure Functions
Kolmogorov: She-Leveque:
5
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
6
Theory of earlier Articles:Navier-stokes on vortex filament Dot product with
relation pressure en velocity
Change to Lagrange Frame: Lagrange: ,
, at r=
7
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
8
Theory of earlier Articles:Navier-stokes on vortex filament Combining all terms
= 15 different values 10 equations 5 undefined functions
9
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:
10
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
11
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
12
Eulerian structure function
This results in the following condition:
: Eddy Turn over time : Eddy size for
Gives:
13
Eulerian structure function
The eulerian structure function now becomes:
With
14
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:
15
Lagrangian structure Function
The lagrangian structure function now becomes:
With
16
Bridge relation
Now we have Combination of ’s gives relation:
(n-)=2(n-
17
Results
Compare with numerical simulation
18
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-
19
Questions?
20
Results
21
Results
22