In this paper we describe the results of various numerical simulations of sideways or horizontal convection. Specifically, a two-dimensional Boussinesq fluid is both heated and cooled from its upper surface, but the walls and the bottom of the tank are insulating and have no flux of heat through them. We perform experiments with a range of Rayleigh numbers up to 10(11), obtained by systematically reducing the diffusivity. We also explore the effects of a nonlinear equation of state and of a mechanical force imposed on the top surface at a fixed Rayleigh number. We find that, when there is no mechanical forcing, both the energy dissipation and the strength of the circulation itself monotonically fall with decreasing diffusivity. At Rayleigh numbers greater than 10(10) the flow is unsteady; however, the eddying flow is still much weaker than the steady flow at smaller Rayleigh numbers. At high Rayleigh numbers, the stratification and the mean circulation are increasingly confined to a thin layer at the upper surface, with the layer thickness decreasing according to Ra-1/5. There is no evidence that the flow ever enters a regime that is independent of Rayleigh number. Using a nonlinear equation of state makes little difference to the flow phenomenology at a moderate Rayleigh number. The addition of an imposed stress at the upper surface makes a significant difference in the flow. A strong, energy-dissipating circulation can be maintained even at Ra = 10(9), and the stratification extends more deeply into the fluid than in the unstressed case. Overall, our results are consistent with the notion that in the absence of mechanical forcing a fluid that is heated and cooled from above cannot maintain a deep stratification or a strong sustained flow at high Rayleigh numbers, even if the interior flow is unsteady.