Dissipation of coherent structures in confined two-dimensional turbulence

Abstract

Since the seminal article “Inertial ranges in two-dimensional turbulence” by Kraichnan in 1967, our understanding of the dynamics and transport properties of two-dimensional turbulence is largely built on the assumption of homogeneity and isotropy of statistically steady or decaying turbulence. In the last two decades, more attention has been paid to the presence of lateral walls, either with stress-free or no-slip boundary conditions, and also considering a variety of geometries such as square, rectangular, or circular domains. The impact of confining boundaries on the dynamics of two-dimensional turbulence is important. This is in sharp contrast with three-dimensional turbulence, where homogeneity and isotropy are locally restored due to the cascade process in the inertial range. The impact of confining boundaries is therefore limited in three-dimensional turbulence. The presence of an inverse energy cascade in two-dimensional turbulence, however, will continuously generate large-scale energy-containing eddies, and their vigorous interaction with, in particular, no-slip walls generates large amounts of vorticity and contributes significantly to the dissipation of kinetic energy of the flow. The dissipation is even strongly enhanced compared with the unbounded case. In this review, we will focus on one of the elementary structures observed in two-dimensional turbulent flows: the dipolar vortex. With its self-induced velocity, it propagates easily through the domain and is hence likely to interact with domain boundaries. Standard vortex generation mechanisms allow to create well-defined dipoles and to investigate the collision of such structures with rigid domain boundaries in detail. Relevant aspects of the collision process concern the dynamics and stability of the generated boundary layers, the vorticity, and vorticity gradients contained in these boundary layers, and the dissipation of kinetic energy when dipoles collide with walls. Some of these aspects will be discussed in this review. Moreover, we are interested in the Reynolds-number dependence of these processes, including exploration of the vanishing viscosity limit.