Bifurcation structure and unsteady solutions through a curved square duct with bottom wall heating and cooling from the ceiling

Abstract

The present paper investigates a spectral-based numerical study for fully developed two-dimensional flow of viscous incompressible fluid through a curved square duct for the constant curvature δ = 0.1. The bottom wall of the duct is heated while cooling from the ceiling; the inner and outer sidewalls being thermally insulated. Flow characteristics are investigated over a wide range of the Dean number 0 < Dn ≤ 5000 for the Grashof number Gr = 100. First, we investigated solution structure of the steady solutions by using Newton-Raphson iteration method. As a result, four branches of steady solutions are obtained with a bifurcating relationship among the branches. Then, we studied unsteady solutions by time evolution calculations justified by their phase spaces, and it is found that the unsteady flow undergoes in the scenario “steady-state→periodic (multi-periodic)→chaotic”, if Dn is increased. The present study demonstrates the role of secondary vortices on convective heat transfer and it is found that convective heat transfer is significantly enhanced by the secondary flow; and the chaotic flow, which occurs at large Dn’s, enhances heat transfer more effectively than the steady-state or periodic solutions.