arXiv subjects

Dmitriy Zhigunov

Publications and source records attributed to Dmitriy Zhigunov.

4 recordsLinked to original sources

arXiv2026

Exact coherent structures as building blocks of turbulence on large domains

Exact unstable solutions of the Navier-Stokes equations are thought to underpin the dynamics of turbulence, but are usually computed in minimal computational domains. Here, we extend this dynamical systems approach to spatially extended turbulent flows featuring multiple interacting 'substructures', and show how new simple invariant solutions can be constructed by spatial tiling of exact solutions from small-box calculations. Candidate solutions are found via gradient-based optimization of a scalar loss function which targets autorecurrence in spatially-masked regions of the flow. We apply these ideas to a vertically-extended Kolmogorov flow, where we first identify large numbers of relative periodic orbits (RPOs) which are combinations of high-dissipation, small-box solutions with laminar patches. We then show that vertically-stacked combinations of pairs of distinct small-box RPOs can form robust guesses for dynamically-relevant two-tori in the larger domain. Finally, we show how our optimization procedure can identify 'turbulent' trajectories which locally shadow a small-box RPO for multiple periods in a subdomain. These small-box combinations are possible as the flow spends prolonged periods in a regime where it can be effectively considered as a pair of weakly-coupled small-box systems, due to shielding effects associated with higher-dissipation flow structures.

physics.flu-dyn↗

arXiv2024

Self-similarity and the direct (enstrophy) cascade in two-dimensional fluid turbulence

A widely used statistical theory of 2D turbulence developed by Kraichnan, Leith, and Batchelor (KLB) predicts a power-law scaling for the energy, $E(k)\propto k^α$ with an integral exponent $α={-3}$, in the inertial range associated with the direct cascade. In the presence of large-scale coherent structures, a power-law scaling is observed, but the exponent often differs substantially from the value predicted by the KLB theory. Here we present a dynamical theory which describes the key physical mechanism behind the direct cascade and sheds new light on the relationship between the structure of the large-scale flow and the scaling of the small-scale structures in the inertial range. This theory also goes a step beyond KLB, to predict the upper and lower bounds of the inertial range as well as the energy scaling in the dissipation range.

physics.flu-dyn↗

arXiv2023

Exact Coherent Structures in Fully Developed Two-Dimensional Turbulence

This paper reports several new classes of weakly unstable recurrent solutions of the 2+1-dimensional Euler equation on a square domain with periodic boundary conditions. These solutions have a number of remarkable properties which distinguish them from analogous solutions of the Navier-Stokes equation describing transitional flows. First of all, they come in high-dimensional continuous families. Second, solutions of different types are connected, e.g., an equilibrium can be smoothly continued to a traveling wave or a time-periodic state. Third, and most important, many of these solutions are dynamically relevant for turbulent flow at high Reynolds numbers. Specifically, we find that turbulence in numerical simulations exhibits large-scale coherent structures resembling some of our time-periodic solutions both frequently and over long temporal intervals. Such solutions are analogous to exact coherent structures originally introduced in the context of transitional flows.

physics.flu-dyn↗

arXiv2015

Hawking Radiation and Classical Tunneling

Acoustic waves in fluids undergoing the transition from sub- to supersonic flow satisfy governing equations similar to those for light waves in the immediate vicinity of a black hole event horizon. This acoustic analogy has been used by Unruh and others as a conceptual model for `Hawking radiation.' Here we use variational methods, originally introduced by Brizard for the study of linearized MHD, and ray phase space methods, to analyze linearized acoustics in the presence of background flows. The variational formulation endows the evolution equations with natural Hermitian and symplectic structures that prove useful for later analysis. We derive a $2\times 2$ normal form governing the wave evolution in the vicinity of the `event horizon.' This shows that the acoustic model can be reduced locally (in ray phase space) to a standard (scalar) tunneling process weakly coupled to a unidirectional non-dispersive wave (the `incoming wave'). Given the normal form, the Hawking `thermal spectrum' can be derived by invoking standard tunneling theory, but only by ignoring the coupling to the incoming wave. Deriving the normal form requires a novel extension of the modular ray-based theory used previously to study tunneling and mode conversion in plasmas. We also discuss how ray phase space methods can be used to change representation, which brings the problem into a form where the wave functions are less singular than in the usual formulation, a fact that might prove useful in numerical studies.

physics.flu-dyn↗