[Paper Review] Superstatistical turbulence models

Christian Beck|ArXiv.org|Jun 14, 2005

Statistical Mechanics and Entropy3 citations

TL;DR

This paper proposes a superstatistical extension of the Sawford model to explain non-Gaussian statistics in Lagrangian turbulence, using a fluctuating energy dissipation rate modeled via lognormal statistics. The model successfully reproduces experimental data on acceleration and velocity difference distributions with a single fitting parameter, $ s^2 = 3.0 $, demonstrating excellent agreement with measurements from tracer particle experiments.

ABSTRACT

Recently there has been some progress in modeling the statistical properties of turbulent flows using simple superstatistical models. Here we briefly review the concept of superstatistics in turbulence. In particular, we discuss a superstatistical extension of the Sawford model and compare with experimental data.

Motivation & Objective

• To address the non-Gaussian behavior of acceleration and velocity differences in Lagrangian turbulence, which standard Gaussian models fail to capture.
• To develop a superstatistical extension of the Sawford model that incorporates fluctuations in the energy dissipation rate.
• To reproduce experimental data on acceleration and velocity difference probability densities using a single fitting parameter.
• To establish a connection between the observed statistics and the underlying fluctuating parameter $ \beta $, linked to energy dissipation via $ \beta \propto \epsilon^{-3/2} $.
• To validate the model using experimental data from Taylor-Couette and other turbulence setups, confirming the scaling of $ s^2 $ with scale.

Proposed method

• Introduce a superstatistical framework by replacing the constant energy dissipation $ \bar{\epsilon} $ with a fluctuating $ \epsilon(t) $, leading to a fluctuating parameter $ \beta \propto \epsilon^{-3/2} $.
• Use a lognormal distribution for $ f(\beta) $, motivated by Kolmogorov's theory of lognormally distributed energy dissipation in turbulence.
• Derive the stationary probability density $ p(a) = \int_0^\infty \sqrt{\frac{\beta}{2\pi}} f(\beta) e^{-\frac{1}{2}\beta a^2} d\beta $, which generates fat-tailed distributions.
• Apply the model to the Sawford Langevin equation $ \dot{a} = -\gamma a + \sigma L(t) $, with $ \gamma $ and $ \sigma $ depending on $ \epsilon $, and extend it to superstatistics via $ \beta = 2\gamma / \sigma^2 $.
• Use experimental data from Bodenschatz and Pinton groups to fit $ s^2 = 3.0 $ for acceleration, and $ s^2 = 0.28 $ for velocity differences at a given scale.
• Confirm the scaling of $ s^2 $ with the separation scale $ r $, enabling calculation of structure function scaling exponents $ \zeta_m $.

Experimental results

Research questions

• RQ1 Can a superstatistical extension of the Sawford model reproduce the non-Gaussian fat-tailed distributions of acceleration observed in Lagrangian turbulence experiments?
• RQ2 How does the choice of the fluctuating parameter distribution $ f(\beta) $, particularly lognormal, affect the agreement with experimental data?
• RQ3 What is the relationship between the superstatistical parameter $ s^2 $ and the scale $ r $ of velocity differences in turbulent flows?
• RQ4 Can the joint statistics of acceleration and velocity, including correlations, be consistently modeled within the superstatistical framework?
• RQ5 Is the fluctuating energy dissipation rate $ \epsilon $ directly measurable, and can it be consistently linked to the superstatistical parameter $ \beta $?

Key findings

• The superstatistical model with lognormal $ f(\beta) $ and $ s^2 = 3.0 $ perfectly fits the experimental probability density of acceleration components, as shown in Fig. 1.
• The model successfully reproduces the non-Gaussian, fat-tailed statistics of velocity differences $ u $ in Taylor-Couette experiments, with $ s^2 = 0.28 $ at a given scale.
• The parameter $ s^2 $ varies with the separation scale $ r $, enabling the calculation of scaling exponents $ \zeta_m $ for structure functions via the superstatistical formalism.
• The joint statistics of acceleration and velocity do not factorize, and the model captures these correlations through the underlying fluctuating $ \beta $.
• The fluctuating parameter $ \beta $, related to energy dissipation via $ \beta \propto \epsilon^{-3/2} $, is experimentally measurable and consistent with the observed $ f(\beta) $.
• The model extends beyond Lagrangian turbulence, showing excellent agreement with Eulerian turbulence data, confirming its broad applicability.