Related papers: Unraveling Self-Similar Energy Transfer Dynamics: …
We demonstrate that numerical solutions of Burgers' equation can be obtained by a scale-totality algorithm for fluids of small viscosity (down to one billionth). Two sets of initial data, modelling simple shears and wall boundary layers,…
Similarity solutions play an important role in many fields of science: we consider here similarity in stochastic dynamics. Important issues are not only the existence of stochastic similarity, but also whether a similarity solution is…
In this paper we introduce a new PDE model in frequency space for the inertial energy cascade that reproduces the classical scaling laws of Kolmogorov's theory of turbulence. Our point of view is based upon studying the energy flux through…
We study the statistical properties of solutions to Burgers' equation, $v_t + vv_x = \nu v_{xx}$, for large times, when the initial velocity and its potential are stationary Gaussian processes. The initial power spectral density at small…
The present work is devoted to the evolution of random solutions of the unforced Burgers and KPZ equations in d-dimensions in the limit of vanishing viscosity. We consider a cellular model and as initial condition assign a value for the…
A physical model of a three-dimensional flow of a viscous bubbly fluid in an intermediate regime between bubble formation and breakage is presented. The model is based on mechanics and thermodynamics of a single bubble coupled to the…
The solution of self-similar shock dynamics satisfying the inviscid Burgers equation are provided in closed form for planar, cylindrical and spherical problems. The approach follows Lee's method for obtaining self-similar solutions for the…
We present a study of the Burgers equation in one and two dimensions $d=1,2$ following the analytic approach indicated in the previous paper I. For the problem of the initial conditions decay we consider two classes of initial condition…
Identifying self-similarity is key to understanding and modelling a plethora of phenomena in fluid mechanics. Unfortunately, this is not always possible to perform formally in highly complex flows. We propose a methodology to extract the…
This work is a review with proofs of a group of results on the stochastic Burgers equation with small viscosity, obtained during the last two decades. These results jointly show that the equation makes a surprisingly good model of…
This work is devoted to the study of the decay of multiscale deterministic solutions of the unforced Burgers' equation in the limit of vanishing viscosity. A deterministic model of turbulence-like evolution is considered. We con- struct the…
This paper continue earlier investigations on the decay of Burgers turbulence in one dimension from Gaussian random initial conditions of the power-law spectral type $E_0(k)\sim|k|^n$. Depending on the power $n$, different characteristic…
This work is devoted to investigating stochastic turbulence for the fluid flow in one-dimensional viscous Burgers equation perturbed by L\'evy space-time white noise with the periodic boundary condition. We rigorously discuss the regularity…
The spectrally truncated, or finite dimensional, versions of several equations of inviscid flows display transient solutions which match their viscous counterparts, but which eventually lead to thermalized states in which energy is in…
Simulating turbulence to stationarity is a major bottleneck in many engineering problems of practical importance. The problem can be cast as a multiscale problem involving energy redistribution processes that take place on the long large…
This paper is an introduction to the theory of 1d stochastic Burgers equation under periodic boundary conditions and with a stochastic force, sufficiently smooth in the space variable. We prove the classical results on the existence and…
Data-driven turbulence modeling is experiencing a surge in interest following algorithmic and hardware developments in the data sciences. We discuss an approach using the differentiable physics paradigm that combines known physics with…
The goal of this study is to analyze the fine structure of nonlinear modal interactions in different 1D Burgers and 3D Navier-Stokes flows. This analysis is focused on preferential alignments characterizing the phases of Fourier modes…
First, we discuss the non-Gaussian type of self-similar solutions to the Navier-Stokes equations. We revisit a class of self-similar solutions which was studied in Canonne-Planchon (1996). In order to shed some light on it, we study…
We construct a discrete shell-model for two-dimensional turbulence that takes into account local and nonlocal interactions between velocity modes in Fourier space. In real space, its continuous limit is described by the one-dimensional…