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

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-Benard 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 similar or equal to 80 and r similar or equal to 500 respectively (where r is the normalized Rayleigh number). The system becomes chaotic at r similar or equal to 750, and the size of the chaotic attractor increases at r similar or equal to 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, vertical bar W-101 vertical bar similar to (r-1)(0.62) and vertical bar theta(101)vertical bar similar to (r-1)(-0.34) (velocity and temperature Fourier coefficient for (1, 0, 1) mode). However the Nusselt number scales as (r-1)(0.33).