The effects of no-slip boundaries and external force torque on two-dimensional turbulence in a square domain

Abstract

We study two-dimensional turbulence in a square no-slip domain without bottom drag using direct numerical simulations. The dynamics are shown to depend strongly on the torque $M$ of the external forcing. When $M$ is relatively large, a long-lived coherent vortex forms at the domain center, establishing a persistent angular momentum. At lower torques, the angular momentum undergoes random sign reversals due to spontaneous switching of the central vortex circulation, though it predominantly aligns with the torque direction. We investigate the transition between these regimes by smoothly varying $M$, observing that the time-averaged angular momentum of the system follows $\langle \mathcal{L} \rangle \propto M^{1/3}$. A significant part of the energy dissipates near the domain boundaries, requiring a revision of scaling laws for homogeneous systems. New scaling relations are proposed and they enable velocity profiles in the boundary layers to collapse onto a universal curve for simulations with varying fluid viscosities and forcing amplitudes. The velocity profiles feature a linear viscous sublayer followed by a short logarithmic region. The boundary layer thickness $\delta$ scales with the large-scale Reynolds number $Re$ as $\delta \sim L \cdot Re^{-1/2}$, indicating that the friction at no-slip walls is not sufficient to halt the inverse energy cascade before it reaches the system size $L$. The results highlight how no-slip boundaries and forcing asymmetry govern the dynamics of large-scale coherent structures in confined two-dimensional turbulence.