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

TL;DR

The paper addresses how to engineer fluid motions that exhibit self-similar energy transfer consistent with turbulence theory by solving PDE-constrained optimization for the 1D Burgers equation on a periodic domain. It introduces a self-similarity constraint in Fourier space, defines an optimization functional $\mathcal{J}_{\nu,\lambda}$ on a constrained initial-data manifold, and computes gradients via a forward Burgers solver coupled to an adjoint equation. The main findings reveal two solution families—viscous and inertial—with inertial flows achieving self-similar spectral transfer through uniform front steepening, realizable only at sufficiently high Reynolds numbers (small $\nu$) and over a finite time window, while viscous flows dissipate and do not transfer energy across scales. The work demonstrates the viability of PDE-constrained optimization to uncover time-dependent self-similar dynamics and suggests a path to extend these ideas to more complex models, including shell models and 2D/3D turbulence, potentially offering new insights into energy cascades at high Reynolds numbers.

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.

Figures (5)

• Figure 1: (a) Decrease of the objective functional \ref{['eq:J']} with iterations $n$ and (b) the corresponding distributions $f(k;\varphi_{\nu, \lambda})$ obtained at the minima when solving \ref{['pb:1']} with $\nu = 2.5 \times 10^{-5}$ and (blue solid line) $\lambda = 2$, (red dashed line) $\lambda = 4$, and (yellow dot-dashed line) $\lambda = 6$.
• Figure 2: (a) Decrease of the objective functional \ref{['eq:J']} with iterations $n$ after refinement of the numerical resolution, (b) dependence of the objective functional $\mathcal{J_{\nu, \lambda}}(\phi^{\eta})$ on the noise magnitude $\eta$ in the perturbed initial condition \ref{['eq:phieta']}, and (c) the error function $g(t;\varphi_{\nu, \lambda})$ defined over a sliding time window for $t \in [T,2T]$ for $\nu = 2.5 \times 10^{-5}$ and (blue solid line) $\lambda = 2$, (red dashed line) $\lambda = 4$, and (yellow dot-dashed line) $\lambda = 6$.
• Figure 3: Comparison of the properties of (left column) the viscous solutions and (right column) the inertial solutions obtained by solving \ref{['pb:1']} with $\nu = 2.5 \times 10^{-3}$ and $\lambda=2$: (a,b) optimal initial conditions $\varphi_{\nu, \lambda}(x)$ (blue, left axis), the magnitude of their gradients $\left|d\varphi_{\nu, \lambda}(x)/d x \right|$ (black, right axis) and the corresponding solutions $u(T,x;\varphi_{\nu, \lambda})$ at the final time (red, left axis); (c,d) the solutions at the final time $u(T,x;\varphi_{\nu, \lambda})$ (red, left axis) and their gradients $\left|\partial u(T,x;\varphi_{\nu, \lambda}) / \partial x \right|$ (black, right axis); (e,f) the Fourier spectra $\widehat{\varphi_{\nu, \lambda}}(k)$ (blue) and $\widehat{u}(T,k;\varphi_{\nu, \lambda})$ (red) of the optimal initial conditions and of the corresponding final states; (g, h) the functions $f(k;\varphi_{\nu, \lambda})$ vs. $k$ and $\mathcal{E}(u(t, \cdot;\varphi_{\nu, \lambda}))$ vs. $t$, cf. \ref{['eq:J']} and \ref{['eq:E']}, respectively, at the minimum for the viscous solution (purple) and the inertial solution (green).
• Figure 4: Solutions at the final time $u(T,x;\varphi_{\nu, \lambda})$ (red, left axis) and their gradients $\left|\partial u(T, x;\varphi_{\nu, \lambda}) / \partial x \right|$ (black, right axis)) obtained by solving \ref{['pb:1']} with $\nu = 2.5 \times 10^{-5}$ and (a) $\lambda = 2$, (b) $\lambda = 3$, (c) $\lambda = 4$, (d) $\lambda = 5$, (e) $\lambda = 6$, and (f) $\lambda = 7$.
• Figure 5: Dependence of $|\kappa(\epsilon) - 1|$ on $\epsilon$ for (a) $\lambda = 2$ and (b) $\lambda = 7$. Expression \ref{['eq:kappa']} was evaluated using the temporal discretizations (empty symbols) $\Delta t = 1 \times 10^{-3}$ and (filled symbols) $\Delta t = 2 \times 10^{-4}$, and the perturbations (blue circles) $\phi' = \sin(-x)$ and (red triangles) $\phi' = \frac{1}{2+\sin(x)}$.

Theorems & Definitions (1)

• Definition 1