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Three‐dimensional numerical simulations of flow through a channel with spatially periodic temperature boundary conditions in the streamwise direction have been carried out. A spectral method employing Fourier series expansion in the streamwise and spanwise directions and Chebyshev expansion in the vertical direction is used. A supercritical bifurcation to three‐dimensional flow is found to occur before the onset of unsteadiness. The bifurcation to unsteady three‐dimensional flow is also found to be supercritical. The unsteady three‐dimensional flow oscillates at a slightly lower frequency than the two‐dimensional flow, and introduces a slightly higher incommensurate frequency causing quasiperiodic flow and a slow spanwise wave motion.

Two‐dimensional numerical simulations of flow through a channel with spatially periodic temperature boundary conditions at the lower wall have been carried out. A spectral method with a Fourier series expansion in the streamwise direction and a Chebyshev expansion in the vertical direction is employed. A bifurcation to a limit cycle occurs at Reynolds numbers as low as Re=4 and Rayleigh numbers as low as Ra=14 500. The frequency of oscillation decreases with Re and is essentially independent of Ra. Maps of flow regimes for varying periodicities have been created. Regimes with periodicities longer than the imposed temperature boundary condition have been found.