Selfsimilarity of Developed Turbulence

  • H. Effinger
  • S. Grossmann
Conference paper
Part of the Springer Series in Synergetics book series (SSSYN, volume 37)

Abstract

One aspect of fluid flow with large Reynolds number and energy input is to understand its chaotic structure, the Lyapunov spectrum, the K-entropy, the invariant measure, etc., starting from the properties of the Navier-Stockes equations1. This would justify a statistical mechanics of turbulence. The question of which measurable statistical properties emanate from that have found even longer lasting interest2. A quantity of particular interest is the structure function
$$ D(r) = \ll {\left| {\overrightarrow u (\overrightarrow x + \overrightarrow r ) - \overrightarrow u (\overrightarrow x )} \right|^2} \gg $$
(1)
in stationary, homogeneous, isotropic turbulence. \( \overrightarrow u (\overrightarrow x, t) \) is the Eulerian velocity field, ≪...≫ denotes the ensemble average (which then is independent of \( \overrightarrow x \) and t). \( \overrightarrow u \) is solenoidal and solves the Navier-Stockes equations (NS-E)
$$ {\partial_t}{u_i}(\overrightarrow x, t) = - {u_j}(\overrightarrow x, t) - {\partial_{{{x_i}}}}p(\overrightarrow x, t) + \nu {\Delta_x}{u_i}(\overrightarrow x, t) + {f_i}(\overrightarrow x, t) $$
(2)

Keywords

Advection 

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Copyright information

© Springer-Verlag Berlin Heidelberg 1987

Authors and Affiliations

  • H. Effinger
    • 1
  • S. Grossmann
    • 1
  1. 1.Fachbereich Physik der Philipps-UniversitätMarburgFed. Rep. of Germany

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