Flow instability and development of dean vortices in a bending rectangular channel with bottom wall heating and cooling from the ceiling

Investigated here are flow instability and the development of Dean vortices with convective heat transfer in a curved rectangular channel of aspect ratio 2. A temperature difference is maintained between the horizontal walls with bottom-wall heating and top-wall cooling, while the vertical sidewalls are thermally insulated. Using spectral methods, numerical simulations were performed for a curvature of 0.001, a Grashof number of 500, and a wide-ranging pressure gradient parameter, i.e., the Dean number ( 0 < D n ≤ 1500 ). Newton–Raphson iteration and path continuation were used to obtain the branching structure of the steady solutions. Consequently, two branches of steady solutions were obtained comprising symmetric 2- to 12-vortex solutions on the first and second branch. Linear stability analysis of the branches showed that the first branch is linearly stable for 168 ≤ D n ≤ 349 whereas the other branch is always unstable. The transition of the flow state and its nonlinear behavior were investigated in terms of time-evolution calculations, and various flow instabilities were examined over the chosen range of Dean number. Chaotic flow was observed the most (66.73 %). The phase trajectories and power spectral density were plotted to clarify the periodic and chaotic behavior, where 2- to 6-vortex flow patterns were found in the unsteady cases. Finally, the temperature gradients were studied with the Nusselt number as an index of heat transfer. The results show that the centrifugal force influences the convection originating from the secondary flow and magnifies the heat transfer typically from the heated wall to the fluid. The rate of heat transfer is found to be significant if the flow approaches the periodic or chaotic state.