Hydrothermal Behavior of Fluid Flow and Heat Transfer through a Curved Square Duct with Curvature Effects

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.