Evolution of two-dimensional disturbances in the Rayleigh–Bénard problem and their preferred wavenumbers

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Abstract
A nonlinear non-stationary problem of the development of two-dimensional convective motions in a plane horizontal fluid layer bounded by free surfaces and heated from below is studied. Horizontal dependences of the velocity and temperature are not assumed to be periodic or almost periodic; they have continuous wavenumber spectra and are represented by Fourier integrals. Vertical dependence of each variable is represented by several Fourier harmonics. Spectrum evolution is studied by means of the numerical integration of an initial-value problem. Initial disturbances of two qualitatively different classes are considered; viz those localized in horizontal extent within a narrow part of the layer as well as having the form of a roll set throughout a rather wide region. In both cases convective flows tend to evolve towards the arrays of well-established rolls with the same horizontal wavenumber ap, which apparently seems to be the physically optimal (preferred) one for two-dimensional convection at given Rayleigh number R and Prandtl number P. We see no indications that the deviation of the initial roll-disturbance wavenumber from ap must exceed some threshold value for giving rise to the flow readjustment in wavenumber. At sufficiently small P a decrease in ap with increasing R is observed which agrees with experiments. A comparison is made of various theoretical models and various experimental circumstances, whence it is seen that the less stable the flow (i.e. the easier it can readjust), the better the preferred wavenumber manifests itself. In particular, roll flows periodic in a horizontal direction all over the infinite layer are highly stable, and when only such flows are considered, as has most often been the case, the preferred wavenumber is not revealed.