Numerical prediction of non-isothermal flow through a curved square duct

A numerical study is presented for the solution structure, stability and transitions of non-isothermal flow through a curved square duct by using a spectral method and covering a wide range of the Dean number, Dn, 0 ≤ Dn ≤ 6000 and the curvature, δ, 0 < δ ≤ 0.5. A temperature difference is applied across the vertical sidewalls for the Grashof number Gr Combining double low line 500, where the outer wall is heated and the inner one cooled. First, steady solutions are obtained by the Newton-Raphson iteration method. As a result, two branches of asymmetric steady solutions are obtained. Linear stability of the steady solutions is then investigated. It is found that only the first branch is linearly stable in a couple of interval of Dn for small δ; for large δ, however, the same branch is linearly stable in a single but wide interval of Dn though the branching pattern of the bifurcation diagram is unchanged. When there is no stable steady solution, time evolution calculations as well as their spectral analysis show that typical transition occurs from steady flow to chaos through various flow instabilities, if Dn is increased. It is also found that the transition to periodic or the chaotic state is delayed if the curvature is increased.