On the stationarity of linearly forced turbulence in finite domains

TL;DR

This paper investigates the theoretical underpinnings of how linearly forced turbulence in finite domains evolves towards a stationary state, emphasizing symmetry considerations, domain effects, and the emergence of isotropy.

Contribution

It provides a symmetry-based theoretical explanation for the stationarity of linearly forced turbulence in finite domains, linking domain size and symmetry breaking to turbulence behavior.

Findings

System evolves to a stationary state due to symmetry breaking.

Finite domain size influences turbulence isotropy and stationarity.

Stationary state can be modeled as a stable fixed point in self-preserving turbulence models.

Abstract

A simple scheme of forcing turbulence away from decay was introduced by Lundgren some time ago, the `linear forcing', which amounts to a force term linear in the velocity field with a constant coefficient. The evolution of linearly forced turbulence towards a stationary final state, as indicated by direct numerical simulations (DNS), is examined from a theoretical point of view based on symmetry arguments. In order to follow closely the DNS the flow is assumed to live in a cubic domain with periodic boundary conditions. The simplicity of the linear forcing scheme allows one to re-write the problem as one of decaying turbulence with a decreasing viscosity. Scaling symmetry considerations suggest that the system evolves to a stationary state, evolution that may be understood as the gradual breaking of a larger approximate symmetry to a smaller exact symmetry. The same arguments show that the finiteness of the domain is intimately related to the evolution of the system to a stationary state at late times, as well as the consistency of this state with a high degree of isotropy imposed by the symmetries of the domain itself. The fluctuations observed in the DNS for all quantities in the stationary state can be associated with deviations from isotropy. Indeed, self-preserving isotropic turbulence models are used to study evolution from a direct dynamical point of view, emphasizing the naturalness of the Taylor microscale as a self-similarity scale in this system. In this context the stationary state emerges as a stable fixed point. Self-preservation seems to be the reason behind a noted similarity of the third order structure function between the linearly forced and freely decaying turbulence, where again the finiteness of the domain plays an significant role.