Journal of Fluid Mechanics

Nonlinear oscillatory convection

R. M.  Clever a1 and F. H.  Busse a1
a1 Institut für Physik, Universität Bayreuth, 8580 Bayreuth, West-Germany

Article author query
clever rm   [Google Scholar] 
busse fh   [Google Scholar] 


A numerical analysis has been performed of three-dimensional time-dependent solutions which bifurcate supercritically from two-dimensional convection-roll solutions at the onset of the oscillatory instability. The bifurcating solutions describe a periodic shifting forward and backward of the convection rolls and lead to a strong deformation of the rolls as the Rayleigh number increases. Since the bifurcating solution is stable in the form of a travelling wave, the computational expense can be reduced by assuming a moving coordinate. Travelling-wave solutions have been computed in the case of rigid boundaries as a function of the Prandtl number and of the two basic wavenumbers αx, αy of the problem. The onset of oscillations reduces the heat transport in comparison with that of two-dimensional rolls because the occupation of a new degree of freedom of motion by the oscillation reduces the energy of the heat-transporting component of convection. A limited stability analysis of finite-amplitude travelling waves has been performed and the onset of an asymmetric mode of oscillations is determined as a function of the parameters of the problem. This mode appears to be identical with a mode that was observed in the numerical simulations of Lipps (1976) and McLaughlin & Orszag (1982).

(Published Online April 21 2006)
(Received March 6 1986)
(Revised August 14 1986)