Laminar-turbulent transition behavior of fully developed air flow in a heated horizontal tube

I have experimentally studied the influence of the buoyancy force and inlet flow conditions on the laminar-turbulent transition process of fully developed air flow in a heated horizontal tube with uniform wall heat flux at modified Rayleigh number 3.1×10⁶. Eight time-series of the air temperature were simultaneously obtained using eight thermocouples positioned within the tube along a vertical line passing through the tube's axis. I have studied the time and space dependence of the transition behavior by analyzing these instantaneous time-series. By calculating a set of Lyapunov exponents and the correlation dimension of the time-series of a single thermocouple, these transitional flows are found to be chaotic.