Conceptual dynamical models for turbulence

Abstract

Understanding the complexity of anisotropic turbulent processes in engineering and environmental fluid flows is a formidable challenge with practical significance because energy often flows intermittently from the smaller scales to impact the largest scales in these flows. Conceptual dynamical models for anisotropic turbulence are introduced and developed here which, despite their simplicity, capture key features of vastly more complicated turbulent systems. These conceptual models involve a large-scale mean flow and turbulent fluctuations on a variety of spatial scales with energy-conserving wave-mean-flow interactions as well as stochastic forcing of the fluctuations. Numerical experiments with a six-dimensional conceptual dynamical model confirm that these models capture key statistical features of vastly more complex anisotropic turbulent systems in a qualitative fashion. These features include chaotic statistical behavior of the mean flow with a sub-Gaussian probability distribution function (pdf) for its fluctuations whereas the turbulent fluctuations have decreasing energy and correlation times at smaller scales, with nearly Gaussian pdfs for the large-scale fluctuations and fat-tailed non-Gaussian pdfs for the smaller-scale fluctuations. This last feature is a manifestation of intermittency of the small-scale fluctuations where turbulent modes with small variance have relatively frequent extreme events which directly impact the mean flow. The dynamical models introduced here potentially provide a useful test bed for algorithms for prediction, uncertainty quantification, and data assimilation for anisotropic turbulent systems.

Keywords: stochastic model; wave–mean interaction.

Fig. 1.

Negative large-scale damping: time series (Left) and pdfs (Right) of the turbulent signal u, and with . Note the logarithmic scale of pdfs in the y axis. Dashed lines are Gaussian distributions with the same mean and variance.

Fig. 2.

Positive large-scale damping: time series (Left) and pdfs (Right) of the turbulent signal u, and with . Note the logarithmic scale of pdfs in the y axis. Dashed lines are Gaussian distributions with the same mean and variance.