Multiplicity and stability of convection in curved ducts: Review and progress

The present review starts with a literature review of multiplicity and stability of convection in curved ducts, a phenomenon widely observed from piping systems for various fluids to blood flows in the human arterial system. As a result of this review, several key unresolved issues are identified: (i) multiple steady-state solutions up to a high value of Dean number; (ii) conclusive determination of linear stability of multiple solutions and their dynamic responses to finite random disturbances; (iii) gain and loss of flow stability along solution branches without passing limit/bifurcation points; (iv) development of temporal oscillations in the range of dynamic parameters where no stable steady fully developed flows exist; and (v) phenomena related to the transition to the turbulence at high Dean numbers such as temporal oscillation, period doubling, intermittency, and chaotic oscillation. An attempt to address these issues has led to our recent numerical study on the fully-developed bifurcation structure and stability of forced convection in a curved duct of square cross-section. In addition to the extension of three known solution branches to the high Dean number region, three new asymmetric solution branches are found from three symmetry-breaking bifurcation points on the isolated symmetric branch. The flows on these new branches are either an asymmetric 2-cell state or an asymmetric 7-cell structure. The linear stability of multiple solutions is conclusively determined by solving the eigenvalue system for all eigenvalues. Only 2-cell flows on the primary symmetric branch and on the part of isolated symmetric branch are linearly stable. The symmetric 6-cell flow is also linearly unstable under asymmetric disturbances although it was ascertained to be stable under symmetric disturbances in the literature. The linear stability is observed to change along some solution branches even without passing any bifurcation or limit points. Furthermore, dynamic responses of the multiple solutions to finite random disturbances are also examined by the direct transient computation. It is found that physically realizable fully-developed flows possibly evolve, as the Dean number increases, from a stable steady 2-cell state at a lower Dean number to a temporal periodic oscillation state, another stable steady 2-cell state, a temporal oscillation with the intermittency, and a chaotic temporal oscillation. Among them, three temporal oscillation states have not been reported in the literature. A temporal periodic oscillation between symmetric/asymmetric 2-cell flows and symmetric/asymmetric 4-cell flows are found in the range where there are no stable steady fully-developed solutions. The symmetry-breaking point on the primary solution branch is determined to be a subcritical Hopf point by the transient computation.