Effects of Geometries on the Nonlinearity of Thermal Fluids in Curved Ducts of Heat Exchangers
Authors: Liu Fang 
Published: March 2016

The present work is on comparison of bifurcation and stability of fully-developed forced convection in a curved duct with various aspect ratios and various curvature ratios. In this study, water was used as the fluid assuming the properties are constant. Boundary conditions are non-slip, impermeability and uniform peripheral temperature. The governing differential equations from the conservation laws are discretized by the finite volume method and then solved for parameter-dependence of flow and temperature fields by the Euler–Newton continuation. The Dk number and the local variable are used as the control parameters in tracing the branches. The Dk number is the ratio of the square root of the product of inertial and centrifugal forces to the viscous force. The test function and branch switching technique are used to detect the bifurcation points and switch the branch respectively. The flow stability on various branches is determined by direct transient computation on dynamic responses of the multiple solutions.
For the curved ducts with of aspect ratio 1 and curvature ratio 5 × 10−6, ten solution branches (either symmetric or asymmetric) are found with eight symmetry-breaking bifurcation points and thirty-one limit points. Thus a rich solution structure exists with the co-existence of various flow states over certain ranges of governing parameters. Dynamic responses of the multiple steady flows to finite random disturbances are examined by the direct transient computation. It is found that possible physically realizable fully developed flows under the effect of unknown disturbances evolve, as the Dean number increases, from a stable steady 2-cell state at lower Dean number to a temporal periodic oscillation, another stable steady 2-cell state, a temporal intermittent oscillation, and a chaotic temporal oscillation.
There exist no stable steady fully-developed flows in some ranges of governing parameters. For the curved ducts with of aspect ratio 1 and curvature ratio 0.5, ten solution branches, two symmetric and eight asymmetric, are found. Among them, one symmetric branch and seven asymmetric branches have not been reported in the literature. On these new branches, the flow has a structural 2-, 4-, 5-, 6-, 7- or 8-cell. The mean friction factor and Nusselt number are different on various solution branches. In tightly curved ducts, the secondary flow enhances the heat transfer more significantly than the friction increase.
For the curved ducts with of aspect ratio 10 and curvature ratio 0.5, seven symmetric and four asymmetric solution branches were found. As Dean number increases, finite random disturbances lead the flows from a stable steady state to another stable steady state, a periodic oscillation, an intermittent oscillation, another periodic oscillation and a chaotic oscillation. The mean friction factor and the mean Nusselt number are obtained for all physically-realizable flows.
Heat transfer enhancement potential of the flow and the evolution of stability as Dk increases in curved ducts with different aspect ratio and curvature ratio are compared. It is found that a significant enhancement of heat transfer can be achieved at the expense of a slight increase of flow friction, especially for the square curved ducts.