Effect of Prandtl number on wavy rolls in Rayleigh–Bénard convection

We study the effect of the Prandtl number on wavy rolls in three-dimensional (3D) Rayleigh-Benard convection (RBC) with free-slip boundary conditions. Direct numerical simulation (DNS) of the 3D RBC with free-slip boundary conditions is performed in a rectangular box of size L-x x L-y x 1 for this purpose. Dynamics of the wavy rolls are investigated for different values of the horizontal aspect ratio Gamma = L-y/L-x in the range 1 <= Gamma <= 4. For Gamma = 1, DNS shows interesting time-dependent wavy rolls near the onset of convection. A low-dimensional model is constructed from DNS data to understand the origin of different wavy rolls in low Prandtl number (Pr) fluids including quasiperiodic and chaotic ones. The investigation of the model using the tools of dynamical systems reveals that periodic wavy rolls are generated via supercritical Hopf bifurcation of the stationary two-dimensional rolls for 0 < Pr <= 0.4. These periodic wavy rolls remain stable up to 25% above the onset of convection for fluids with Pr in the range of 0.035 <= Pr <= 0.4. However, for lower Prandtl numbers (Pr < 0.035) the periodic wavy rolls undergo a pair of Neimark-Sacker (NS) bifurcations of which one is forward and the other is backward. These NS bifurcations are found to be responsible for the apperance of quasiperiodic wavy rolls from a periodic one and vice versa near the onset of convection. Chaotic wavy rolls are also found to appear in between quasiperiodic wavy rolls for very low Prandtl number fluids. These results of the model are consistent with the DNS results for Gamma = 1. For larger values of Gamma, the bifurcation structures are found to be modified significantly.