BIFURCATION ANALYSIS OF THE FLOW PATTERNS IN TWO-DIMENSIONAL RAYLEIGH–BÉNARD CONVECTION

Abstract

We investigate the origin of various convective patterns for Prandtl number P = 6.8 (for water at room temperature) using bifurcation diagrams that are constructed using direct numerical simulations (DNS) of Rayleigh–Bénard convection (RBC). Several complex flow patterns resulting from normal bifurcations as well as various instances of "crises" have been observed in the DNS. "Crises" play vital roles in determining various convective flow patterns. After a transition of conduction state to convective roll states, we observe time-periodic and quasiperiodic rolls through Hopf and Neimark–Sacker bifurcations at r ≃ 80 and r ≃ 500 respectively (where r is the normalized Rayleigh number). The system becomes chaotic at r ≃ 750, and the size of the chaotic attractor increases at r ≃ 840 through an "attractor-merging crisis" which results in traveling chaotic rolls. For 846 ≤ r ≤ 849, stable fixed points and a chaotic attractor coexist as a result of an inverse subcritical Hopf bifurcation. Subsequently the chaotic attractor disappears through a "boundary crisis" and only stable fixed points remain. These fixed points later become periodic and chaotic through another set of bifurcations which ultimately leads to turbulence. As a function of Rayleigh number, |W101| ~ (r - 1)0.62 and |θ101| ~ (r - 1)-0.34 (velocity and temperature Fourier coefficient for (1, 0, 1) mode). However the Nusselt number scales as (r - 1)0.33.