Unraveling self-similar energy transfer dynamics: A case study for the one-dimensional Burgers system

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 problem in terms of partial differential equation (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 problem are obtained with a state-of-the-art adjoint-based gradient method. Two distinct families of solutions, termed and , 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 behavior 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, two-dimensional turbulence, and, ultimately, three-dimensional turbulence.