Markov Processes linking Thermodynamics and Turbulence

Abstract

This PhD thesis deals with the Markov picture of developed turbulence from the theoretical point of view. The thesis consists of two parts. The first part introduces stochastic thermodynamics, the second part aims at transferring the concepts of stochastic thermodynamics to developed turbulence. / Central in stochastic thermodynamics are Markov processes. An elementary example is Brownian motion. In contrast to macroscopic thermodynamics, the work done and the entropy produced for single trajectories of the Brownian particles are random quantities. Statistical properties of such fluctuating quantities are central in the field of stochastic thermodynamics. Prominent results are so-called fluctuation theorems which express the balance between production and consumption of entropy and generalise the second law. / Turbulent cascades of eddies are assumed to be the predominant mechanism of turbulence generation and fix the statistical properties of developed turbulent flows. An intriguing phenomenon of developed turbulence, known as small-scale intermittency, are violent small-scale fluctuations in flow velocity that exceed any Gaussian prediction. / In analogy to Brownian motion, it is demonstrated in the thesis how the assumption of the Markov property leads to a Markov process for the turbulent cascade that is equivalent to the seminal K62 model. In addition to the K62 model, it is demonstrated how many other models of turbulence can be written as a Markov process, including scaling laws, multiplicative cascades, multifractal models and field-theoretic approaches. Based on the various Markov processes, the production of entropy along the cascade and the corresponding fluctuation theorems is discussed. In particular, experimental data indicates that entropy consumption is linked to small-scale intermittency, and a connection between entropy consumption and an inverse cascade is suggestive.