  Due to ample applications from medical services to industrial activities, the
study of flow and heat transfer through a curved duct has attracted
considerable attention to the researchers. In this paper, a comprehensive
numerical study is presented for the fully developed two-dimensional flow of
viscous incompressible fluid through a curved square duct for various
curvatures. The spectral method is used as a basic tool to solve the system of
nonlinear partial differential equations. Numerical calculations are carried
out over a wide range of the Dean number, $0<D_n\le 5000$, for curvature ratio
$\delta =0.001$, $0.1$, and $0.5$. A temperature difference is applied across
the horizontal walls for the Grashof number $Gr = 1000$, where the bottom wall
is heated while cooling from the ceiling, the outer and inner walls being
thermally insulated. First, the bifurcation structure of steady solutions is
investigated. As a result, two branches of steady solutions consisting of two-
to eight-vortex solutions are obtained for $\delta =0.001$ and $0.1$ while
three branches for $\delta =0.5$. Then we performed time evolution calculation
to investigate unsteady flow characteristics, and it is found that the unsteady
flow undergoes through various flow instabilities, if $D_n$ is increased. Flow
transitions are well determined by obtaining phase space of the time evolution
results. Typical contours of streamlines and isotherms are obtained at several
values of $D_n$ and it is found that the unsteady flow consists of
two-to-eight-vortex solutions. 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 as the number
of secondary vortices increases, that occurs for the chaotic solution, heat
transfer is boosted substantially.

Fluid Dynamics;Numerical Analysis;Numerical Analysis