Direct Numerical Simulation of Square Vortex Flows

Abstract

Atmospheric and oceanic flows are often modeled as two-dimensional due to their large planar extents (∼ 1000 km) and small vertical extent (∼ 10 km). Most phenomena in these systems are governed by the theory of two-dimensional turbulence. Rather than truncating the third dimension, its effect is modeled using additional friction term in the 2D Navier-Stokes equations. These equations are said to be applied to a class of flows called quasi-two-dimensional (Q2D) flows, which are essentially three-dimensional but almost planar flows. The approximation of Q2D-ness is widely applied for shallow fluid layers in laboratory experiments. We study Q2D flows generated by electromagnetically driving a shallow electrolyte layer. The specific form of forcing mimics that of a chessboard, where the laminar flow is composed of counter rotating square vortices arranged in a chessboard like array. The forcing function is carefully chosen to mimic experimental measurements of a laboratory realization, which is not purely sinusoidal as in the well-known Taylor Green vortex flow. The steadily forced flow leads to a statistically stationary state which is analyzed over a wide Reynolds number range. We analyze how the dynamics and flow structures change with increasing Reynolds number. We illustrate methods to discuss when and how the Q2D model ceases to accurately represent experiments.