Unraveling Self-Similar Energy Transfer Dynamics: a Case Study for 1D Burgers System

TL;DR

This study constructs initial conditions for the 1D Burgers equation to produce self-similar energy cascades, using PDE-constrained optimization, revealing solutions that mimic turbulence energy transfer.

Contribution

It introduces a PDE optimization framework to identify self-similar solutions in the Burgers system, advancing understanding of turbulence-like energy transfer mechanisms.

Findings

Identified viscous and inertial families of solutions with distinct enstrophy behaviors.

Inertial solutions exhibit self-similarity only at low viscosity levels.

Methodology can be extended to more complex turbulence models.

Abstract

In this work we consider the problem of constructing initial conditions for a flow model such that the resulting flow evolution leads to a self-similar energy cascade consistent with Kolmogorov's statistical theory of turbulence. As a first step in this direction, we focus on the one-dimensional viscous Burgers equation as a toy model. Its solutions exhibiting self-similar behavior, in a precisely-defined sense, are found by framing this problems in terms of PDE-constrained optimization. The main physical parameters are the time window over which self-similar behavior is sought (equal to approximately one eddy turnover time), viscosity (inversely proportional to the ``Reynolds number") and an integer parameter characterizing the distance in the Fourier space over which self-similar interactions occur. Local solutions to this nonconvex PDE optimization problems are obtained with a state-of-the-art adjoint-based gradient method. Two distinct families of solutions, termed viscous and inertial, are identified and are distinguished primarily by the behavior of enstrophy which, respectively, uniformly decays and grows in the two cases. The physically meaningful and appropriately self-similar inertial solutions are found only when a sufficiently small viscosity is considered. These flows achieve the self-similar behaviour by a uniform steepening of the wave fronts present in the solutions. The results obtained demonstrate that the proposed methodology may be used to search for self-similar behavior in more complex flow models, including shell models, 2D turbulence and, ultimately, 3D turbulence.