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Lagrangian and Eulerian velocity structure functions in hydrodynamic turbulence. Article from K. P. Zybin and V. A. Sirota Enrico Dammers & Christel Sanders Course 3T220 Chaos. Content. Goal Euler vs. lagrangian Background Theory from earlier articles Structure functions

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lagrangian and eulerian velocity structure functions in hydrodynamic turbulence

LagrangianandEulerianvelocitystructurefunctionsinhydrodynamicturbulenceLagrangianandEulerianvelocitystructurefunctionsinhydrodynamicturbulence

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

Enrico Dammers & Christel Sanders

Course 3T220 Chaos

content
Content
  • Goal
  • Euler vs. lagrangian
  • Background
  • Theory from earlier articles
  • Structure functions
  • Bridge relations
  • Results
  • Conclusions
goal of the article
Goal of the article
  • Showing eulerian and lagrangian structure formulas are obeying scaling relations
  • Determine the scaling constants analytical without dimensional analyses
euler vs lagrangian
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
structure functions
Structure Functions
  • Kolmogorov:
  • She-Leveque:
background
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

theory of earlier articles navier stokes on vortex filament
Theory of earlier Articles:Navier-stokes on vortex filament
  • Dot product with
    • relation pressure en velocity
  • Change to Lagrange Frame:
    • Lagrange:
    • ,
    • , at r=
theory of earlier articles navier stokes on vortex filament1
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
theory of earlier articles navier stokes on vortex filament2
Theory of earlier Articles:Navier-stokes on vortex filament
  • Combining all terms
    • =
      • 15 different values
      • 10 equations
      • 5 undefined functions
theory of earlier articles navier stokes on vortex filament3
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:
theory of earlier articles eigenfunctions
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
eulerian structure function
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
eulerian structure function1
Eulerian structure function
  • This results in the following condition:
      • : Eddy Turn over time
      • : Eddy size
      • for
    • Gives:
eulerian structure function2
Eulerian structure function
  • The eulerian structure function now becomes:
    • With
lagrangian structure function
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:
lagrangian structure function1
Lagrangian structure Function
  • The lagrangian structure function now becomes:
    • With
bridge relation
Bridge relation
  • Now we have
  • Combination of ’s gives relation:
    • (n-)=2(n-
results
Results
  • Compare with numerical simulation
conclusions
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-