Different methods to estimate the Einstein-Markov coherence length in turbulence

Area of Science:

• Fluid Dynamics
• Statistical Physics
• Turbulence Research

Background:

• Previous research identified the Markov property in turbulent velocity increments for specific nesting structures above the Einstein-Markov coherence length (l(EM)).
• The Einstein-Markov coherence length (l(EM)) is approximately the magnitude of the Taylor microscale (λ).

Purpose of the Study:

• To investigate the Markov property of experimental velocity data in homogeneous isotropic turbulent flows.
• To determine if the Markov property of velocity increments is independent of nesting structure.
• To analyze the Markovian characteristics of the velocity process as a function of spatial position.

Main Methods:

• Analysis of the stochastic "cascade" process of nested velocity increments ξ(r) = u(x+r) - u(x).
• Application of a defined effective step size for analyzing the velocity increment process.
• Statistical testing for the Markov property on the velocity u(x) as a function of spatial position x.
• Utilizing a transition probability matrix method.

Main Results:

• The Markov property for velocity increments holds independently of nesting structure when an effective step size is used.
• A characteristic length scale, l(u(x)), approximately equal to l(EM), is identified for the velocity process u(x).
• The non-Markovian character of u(x) has implications for turbulence's statistical properties.

Conclusions:

• The Markov property in turbulent velocity increments is robust across different nesting structures under appropriate step size definitions.
• A significant length scale related to the Einstein-Markov coherence length characterizes the spatial variation of velocity in turbulent flows.
• Further examination of the non-Markovian nature of u(x) is crucial for understanding turbulence statistics.