From stationary to complex time-dependent flows at moderate Rayleigh numbers in two-dimensional annular thermal convection

Abstract

Two-dimensional nonlinear thermal convection in a cylindrical annulus is numerically analyzed for a Boussinesq fluid of low Prandtl number ϭ0.025. For a fixed value of the radius ratio, ϭ0.3, different types of steady columnar patterns are found. The stability of these convection patterns and the spatial interaction between them, which result in the formation of mixed modes, are investigated by considering the full nonlinear set of Navier-Stokes equations. Special attention is paid to the strong spatial interaction of the initially unstable modes with wavenumbers nϭ2 and nϭ4, which leads, through global bifurcations, to multiple stable quasi-periodic states of the system. A detailed picture of the nonlinear dynamics until temporal chaotic patterns set in is presented and understood in terms of local and global symmetry-breaking bifurcations of the O(2)-symmetric system.

Newton–Krylov continuation of periodic orbits for Navier–Stokes flows

“…In the results shown, L Â N ¼ 32 Â 192 for a total of 11,582 unknowns. By using only the time integration code and comparing with resolutions 32 Â 256 and 48 Â 192, we have found differences below 0.2% for the worst resolved frequency of the quasi-periodic solutions obtained above Ra ¼ 18; 000 (see Table 1 in [27]). With the same truncations, we have also estimated differences below 0.5% in the Rayleigh numbers of the bifurcation points between stable solutions up to Ra ¼ 21; 000 used as a test.…”

“…Its physical interest arises from the study of large-scale motions generated by radial temperature gradients in geophysical and astrophysical processes (see [27,26] for details). The nonlinear dynamics of the two-dimensional vortices also provides a simple fluid dynamical system highly attractive from the point of view of bifurcation theory, because it is large enough to provide a rich spatio-temporal dynamics.…”

“…It shows the time evolution of a S-cycle obtained at Ra ¼ 17; 761, for which h 0 % p=2. See also [27] for shadowgraph space-time plots.…”

Amplitude equations close to a triple-(+1) bifurcation point of D4-symmetric periodic orbits in O(2)-equivariant systems

“…2 and 3 (see [12] for further details on the sequence of bifurcations), and is labeled with n = 4 because u s keeps the 7r/2-rotational symmetry of the steady state. The bifurcation breaks the (purely spatial) reflection symmetry of the steady solutions, but preserves a spatio-temporal reflection symmetry.…”

“…The free thermal convection of a Boussinesq fluid in a circular annulus, subject to a constant inward radial gravity and heated from the inside is considered. This problem provides the simplest two-dimensional model for natural convection in the equatorial plañe of atmospheres and planetary interiors (see [9], and the introduction of [12]). Since the problem is independent of the axial coordí-nate z, only roll-like structures, which are exact solutions for a cylindrical annulus subject to stress-free and perfectly conducting top and bottom boundaries [1], and perturbations keeping the two-dimensionality are studied.…”