Heat flux at the transition from harmonic to chaotic flow in thermal convection

Numerical simulations of the fully compressible Navier-Stokes equations are used to study the transition from simple-periodic "harmonic" thermal convection to chaotic thermal convection as the Rayleigh number Ra is increased. The simulations suggest that a sharp discontinuity in the relationship between the Nusselt number Nu (the ratio of the total heat flux to the Fourier heat flux) and the Rayleigh number is associated with this transition in flow morphology. This drop in the Nusselt number is also seen in the data reported in independent experiments involving the convection of two characteristically different fluids-liquid mercury [Phys. Rev. E 56, R1302 (1997)] (a nearly incompressible fluid with Prandtl number Pr=0.024) and gaseous helium [Phys. Rev. A 36, 5870 (1987)] (a compressible fluid with unit Pr). The harmonic flow generates a dual-maximum (quasiharmonic) temperature histogram, while the chaotic flow generates a single-maximum histogram at the center point in the simulated cell. This is consistent with the temperature distributions reported for the convecting mercury before and after the drop in Nu. Our simulations also suggest a hysteresis in the Nu-Ra curve linking the two distinctly different flow morphologies, heat fluxes, and temperature-fluctuation histograms at the same Rayleigh number.