From Anomalous Dissipation Through Euler Singularities to Stabilized Finite Element Methods for Turbulent Flows

It is well-known that kinetic energy produced artificially by an inadequate numerical discretization of nonlinear transport terms may lead to a blow-up of the numerical solution in simulations of fluid dynamical problems such as incompressible turbulent flows. However, the community seems to be divided whether this problem should be resolved by the use of discretely energy-preserving or dissipative discretization schemes. The rationale for discretely energy-preserving schemes is often based on the expectation of exact conservation of kinetic energy in the inviscid limit, which mathematically relies on the assumption of sufficient regularity of the solution. There is the (contradictory) phenomenological observation in turbulence that flows dissipate energy in the limit of vanishing molecular viscosity, an “anomalous” phenomenon termed dissipation anomaly or the zeroth law of turbulence. As already conjectured by Onsager, the Euler equations may dissipate kinetic energy through the formation of singularities of the velocity field. With the proof of Onsager’s conjecture in recent years, a consequence for designing numerical methods for turbulent flows is that the smoothness assumption behind conservation of energy in the inviscid limit becomes indeed critical for turbulent flows. The velocity field rather has to be expected to show singular behavior towards the inviscid limit, supporting the dissipation of kinetic energy. Our main argument is that designing numerical methods against the background of this physical behavior is a strong rationale for the construction of dissipative (or dissipation-aware) numerical schemes for convective terms. From that perspective, numerical dissipation does not appear artificial, but as an important ingredient to overcome problems introduced by energy-conserving numerical methods such as the inability to represent anomalous dissipation as well as the accumulation of energy in small scales, which is known as thermalization. This work discusses stabilized \(H^1\), \(L^2\), and \(H(\mathrm{div})\)-conforming finite element methods for incompressible flows with a focus on the energy-stability of the numerical method and its dissipation mechanisms to predict inertial dissipation. Finally, we discuss the achievable convergence rate for the kinetic energy in under-resolved turbulent flow simulations.