Conceptual dynamical models for turbulence

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.Understanding the complexity of anisotropic turbulence processes over a wide range of spatiotemporal scales in engineering shear turbulence (1–3) as well as climate atmosphere ocean science (4–6) is a grand challenge of contemporary science. This is especially important from a practical viewpoint because energy often flows intermittently from the smaller scales to affect the largest scales in such anisotropic turbulent flows. The typical features of such anisotropic turbulent flows are the following (2–4):

• (A)The large-scale mean flow is usually chaotic but more predictable than the smaller-scale fluctuations. The overall single point probability distribution function (pdf) of the flow field is nearly Gaussian whereas the mean flow pdf is sub-Gaussian, in other words, with less extreme variability than a Gaussian random variable.
• (B)There are nontrivial nonlinear interactions between the large-scale mean flow and the smaller-scale fluctuations which conserve energy.
• (C)There is a wide range of spatial scales for the fluctuations with features where the large-scale components of the fluctuations contain more energy than the smaller-scale components. Furthermore, these large-scale fluctuating components decorrelate faster in time than the mean-flow fluctuations on the largest scales, whereas the smaller-scale fluctuating components decorrelate faster in time than the larger-scale fluctuating components.
• (D)The pdfs of the larger-scale fluctuating components of the turbulent field are nearly Gaussian, whereas the smaller-scale fluctuating components are intermittent and have fat-tailed pdfs, in other words, a much higher probability of extreme events than a Gaussian distribution (see figures 8.4 and 8.5 from ref. 3 for such experimental features in a turbulent jet).

The goal here is to develop the simplest conceptual dynamical model for anisotropic turbulence that captures all of the features in (A)–(D) in a transparent qualitative fashion. In contrast with deterministic models of turbulence which are derived by Galerkin truncation of the Navier–Stokes equation (7) and do not display all of the features in (A)–(D), the conceptual models developed here are low-dimensional stochastic dynamical systems; the nonlinear interactions between the large-scale mean-flow component and the smaller-scale fluctuating components are completely deterministic but the potential direct nonlinear interactions between the smaller-scale fluctuating components are modeled stochastically by damping and stochastic forcing (6, 8). The conceptual models developed here are not derived quantitatively from the Navier–Stokes equations but are developed to capture the key features in anisotropic turbulent flows listed in (A)–(D) by mimicking key physical processes. Besides aiding the understanding of anisotropic turbulent flows, such conceptual models are useful for designing and testing numerical algorithms for prediction and data assimilation in such complex turbulent systems.