Showing papers by "E. A. Kuznetsov published in 2017"

Asymptotic solution for high-vorticity regions in incompressible three-dimensional Euler equations

Abstract: Incompressible three-dimensional Euler equations develop high vorticity in very thin pancake-like regions from generic large-scale initial conditions. In this work, we propose an exact solution of the Euler equations for the asymptotic pancake evolution. This solution combines a shear flow aligned with an asymmetric straining flow, and is characterized by a single asymmetry parameter and an arbitrary transversal vorticity profile. The analysis is based on detailed comparison with numerical simulations performed using a pseudospectral method in anisotropic grids of up to .

Fermi–Pasta–Ulam recurrence and modulation instability

Abstract: We give a qualitative conceptual explanation of the Fermi–Pasta–Ulam (FPU) like recurrence in the onedimensional focusing nonlinear Schrodinger equation (NLSE). The recurrence can be considered as a result of the nonlinear development of the modulation instability. All known exact localized solitary wave solutions describing propagation on the background of the modulationally unstable condensate show the recurrence to the condensate state after its interaction with solitons. The condensate state locally recovers its original form with the same amplitude but a different phase after soliton leave its initial region. Based on the integrability of the NLSE, we demonstrate that the FPU recurrence takes place not only for condensate, but also for a more general solution in the form of the cnoidal wave. This solution is periodic in space and can be represented as a solitonic lattice. That lattice reduces to isolated soliton solution in the limit of large distance between solitons. The lattice transforms into the condensate in the opposite limit of dense soliton packing. The cnoidal wave is also modulationally unstable due to soliton overlapping. The recurrence happens at the nonlinear stage of the modulation instability. Due to generic nature of the underlying mathematical model, the proposed concept can be applied across disciplines and nonlinear systems, ranging from optical communications to hydrodynamics.

Development of high vorticity in incompressible 3D Euler equations: influence of initial conditions

Abstract: The incompressible three-dimensional ideal flows develop very thin pancake-like regions of increasing vorticity. These regions evolve with the scaling $\omega_{\max}(t)\propto\ell(t)^{-2/3}$ between the vorticity maximum and pancake thickness, and provide the leading contribution to the energy spectrum, where the gradual formation of the Kolmogorov interval $E_{k}\propto k^{-5/3}$ is observed for some initial flows [Agafontsev et. al, Phys. Fluids 27, 085102 (2015)]. With the massive numerical simulations, in the present paper we study the influence of initial conditions on the processes of pancake formation and the Kolmogorov energy spectrum development.

Isotropization of two-dimensional hydrodynamic turbulence in the direct cascade

Abstract: We present results of numerical simulation of the direct cascade in two-dimensional hydrodynamic turbulence (with spatial resolution up to ). If at the earlier stage (at the time of order of the inverse pumping growth rate τ-Γmax −1), the turbulence develops according to the same scenario as in the case of a freely decaying turbulence [1, 2]: quasi-singular distribution of di-vorticity are formed, which in k-space correspond to jets, leading to a strong turbulence anisotropy, then for times of the order of 10τ turbulence becomes almost isotropic. In particular, at these times any significant anisotropy in the angular fluctuations for the energy spectrum (for a fixed k) is not visible, while the probability distribution function of vorticity for large arguments has the exponential tail with the exponent linearly dependent on vorticity, in the agreement with the theoretical prediction [3].

Development of high vorticity structures and geometrical properties of the vortex line representation

Abstract: The incompressible three-dimensional Euler equations develop very thin pancake-like regions of increasing vorticity. These regions evolve with the scaling $\omega_{max}\sim\ell^{-2/3}$ between the vorticity maximum and the pancake thickness, as was observed in the recent numerical experiments [D.S. Agafontsev et al, Phys. Fluids 27, 085102 (2015)]. We study the process of pancakes' development in terms of the vortex line representation (VLR), which represents a partial integration of the Euler equations with respect to conservation of the Cauchy invariants and describes compressible dynamics of continuously distributed vortex lines. We present, for the first time, the numerical simulations of the VLR equations with high accuracy, which we perform in adaptive anisotropic grids of up to $1536^3$ nodes. With these simulations, we show that the vorticity growth is connected with the compressibility of the vortex lines and find geometric properties responsible for the observed scaling $\omega_{max}\sim\ell^{-2/3}$.

Isotropization of two-dimensional hydrodynamic turbulence in the direct cascade

Abstract: We present results of numerical simulation of the direct cascade in two-dimensional hydrodynamic turbulence (with spatial resolution up to $16384 \times 16384$). If at the earlier stage (at the time of order of the inverse pumping growth rate $\tau\sim\Gamma_{max}^{-1}$), the turbulence develops according to the same scenario as in the case of a freely decaying turbulence \cite{KNNR-07, KKS}: quasi-singular distributions of di-vorticity are formed, which in $k$-space correspond to jets, leading to a strong turbulence anisotropy, then for times of the order of $10\tau$ turbulence becomes almost isotropic. In particular, at these times any significant anisotropy in the angular fluctuations for the energy spectrum (for a fixed $k$) is not visible, while the probability distribution function of vorticity for large arguments has the exponential tail with the exponent linearly dependent on vorticity, in the agreement with the theoretical prediction \cite{FalkovichLebedev2011}.

Universal scaling of extreme vorticity regions and the structure of the vortex lines representation

Abstract: The incompressible three-dimensional ideal flows develop very thin pancake-like regions of increasing vorticity, which evolve with the scaling \omega_{\max}\propto\ell^{-2/3} between the vorticity maximum and the pancake thickness. We study this process from the point of view of the vortex lines representation (VLR), which describes the associated dynamics of the compressible "flow" of continuously distributed vortex lines. Based on numerical simulations of the VLR equations in adaptive anisotropic grids of up to 1536^3 nodes for two initial flows, we show that the vorticity growth is connected with the compressibility of the vortex lines and find the link between the scaling law \omega_{\max}\propto\ell^{-2/3} and the geometric properties of the VLR.