Numerical evidence of anomalous energy dissipation in incompressible Euler flows: towards grid-converged results for the inviscid Taylor–Green problem

Abstract

The well-known energy dissipation anomaly in the inviscid limit, related to velocity singularities according to Onsager, still needs to be demonstrated by numerical experiments. The present work contributes to this topic through high-resolution numerical simulations of the inviscid three-dimensional Taylor–Green vortex problem using a novel high-order discontinuous Galerkin discretisation approach for the incompressible Euler equations. The main methodological ingredient is the use of a discretisation scheme with inbuilt dissipation mechanisms, as opposed to discretely energy-conserving schemes, which – by construction – rule out the occurrence of anomalous dissipation. We investigate effective spatial resolution up to $8192^3$ (defined based on the $2{\rm \pi}$ -periodic box) and make the interesting phenomenological observation that the kinetic energy evolution does not tend towards exact energy conservation for increasing spatial resolution of the numerical scheme, but that the sequence of discrete solutions seemingly converges to a solution with non-zero kinetic energy dissipation rate. Taking the fine-resolution simulation as a reference, we measure grid-convergence with a relative $L^2$ -error of $0.27\,\%$ for the temporal evolution of the kinetic energy and $3.52\,\%$ for the kinetic energy dissipation rate against the dissipative fine-resolution simulation. The present work raises the question of whether such results can be seen as a numerical confirmation of the famous energy dissipation anomaly. Due to the relation between anomalous energy dissipation and the occurrence of singularities for the incompressible Euler equations according to Onsager's conjecture, we elaborate on an indirect approach for the identification of finite-time singularities that relies on energy arguments.

From anomalous dissipation through Euler singularities to stabilized finite element methods for turbulent flows

“…in a regime where asymptotic rates of convergence cannot be reached due to the broad spectrum of spatial scales. The above theoretical estimate could explain the numerical results in [53] for the 3D inviscid Taylor-…”

“…The work [63] reported numerical evidence of a finite-time singularity, results supported very recently by mathematical proofs in [64]. To not make the present contribution exhaustive, we want to refer to the study by Fehn et al [53] and references therein for a more detailed overview of techniques and numerical studies. Here, we want to focus in particular on techniques exploiting the connection between singular behavior and anomalous dissipation according to Onsager's conjecture.…”

“…In this section we present numerical results for the inviscid Taylor-Green vortex problem. The recent study [53] has simulated the three-dimensional inviscid Taylor-Green problem using the stabilized L 2conforming method presented in Section 4.7. A numerical convergence study has been performed, increasing the spatial resolution from 8 3 up to 8192 3 degrees of freedom per velocity component (defined w.r.t.…”

“…There seems to be an inherent conflict to construct numerical methods for dissipative hyperbolic dynamics that are energy-conserving and at the same time free of thermalization. In our opinion, the one-dimensional Burgers' equation with formation of shocks appears to be most illustrative to convince oneself regarding these conflicting goals of both energy-conserving (or dissipation-free) and thermalization-free numerics [53,56].…”

“…The chosen approach is thus in line with the philosophy of implicit turbulence modeling. According to the numerical studies by [52] and [53], stabilized (dis-)continuous Galerkin methods might indeed be suitable candidates to predict the phenomenon of anomalous dissipation numerically and, thereby, qualify as physics-compatible numerical methods. Let us note that this discretization approach differs in its philosophy rather fundamentally from the goals persued e.g.…”