Summary: We study the large-time behavior of boundary- and pressure-gradient-driven incompressible fluid flows in elongated two-dimensional channels with emphasis on estimates for their degrees of freedom, i.e., the dimension of the attractor for the solutions of the Navier-Stokes equations. For boundary-driven shear flows and flux-driven channel flows we present upper bounds for the degrees of freedom of the form \(c\alpha\text{ Re}^{3/2}\), where \(c\) is a universal constant, \(\alpha\) denotes the aspect ratio of the channel (length/width), and Re is the Reynolds number based on the channel width and the imposed “outer” velocity scale. For fixed pressure-gradient-driven channel flows we obtain an upper bound of the form \(c'\alpha\text{ Re}^2\), where \(c'\) is another universal positive constant, and the Reynolds number is based on a velocity defined by the infimum, over all possible trajectories, of the time averaged mass flux per unit channel width. We discuss these results in terms of physical arguments based on small length scales in turbulent flows.