This paper reports on a numerical study of forced two-dimensional turbulence in a periodic channel with flat no-slip walls. Since corners or curved domain boundaries, met in the standard rectangular, square or circular geometries, are absent in this geometry, the (statistical) analysis of the flow is substantially simplified. Moreover, the use of a standard Fourier-Chebyshev pseudospectral algorithm enables high integral-scale Reynolds number simulations. The paper focuses on (i) the influence of the aspect ratio of the channel and (ii) the integral-scale Reynolds number on the large-scale self-organization of the flow. It is shown that for small aspect ratios a unidirectional flow emerges spontaneously, notably in the absence of a pressure gradient in the longitudinal direction. For larger aspect ratios the flow tends to organize into an array of counter-rotating vortical structures. The computed energy and enstrophy spectra provide further evidence that the injection of small-scale vorticity at the no-slip walls modify the inertial-range scaling. Additionally, the quasi-stationary final state of decaying turbulence is interpreted in terms of the Stokes modes of a viscous channel flow. Finally, the transport of passive tracer material is studied with emphasis on the role of the large-scale flow on the dispersion and the spectral properties of the tracer variance in the presence of no-slip boundaries.