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Numerical study of Dean vortices and unsteady solutions through a curved square duct flow. (English) Zbl 1432.76105

Summary: Flow instability in a curved duct with square cross section is studied numerically by using a spectral method, and covering a wide range of the Dean number \(0\leq Dn\leq 5000\) for the curvature \(\delta=0.1\). A temperature difference is applied across the vertical sidewalls for the Grashof number \(Gr=100\), where the outer wall is heated and the inner wall cooled. After a comprehensive survey over the parametric ranges, two branches of asymmetric steady solutions with two- and four-vortex solutions are obtained by the Newton-Raphson iteration method. 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 \(Dn\), while the other branch is linearly unstable. Steady values of the Nusselt numbers, \(Nu\), are also calculated for two differentially heated vertical sidewalls. When there is no stable steady solution, time evolutions of \(Nu\) as well as their phase spaces are obtained, and it is found that in the unstable region the flow undergoes in the scenario “steady \(\to\) periodic \(\to\) multiperiodic \(\to\) chaotic”, if the Dean number is increased.

MSC:

76E30 Nonlinear effects in hydrodynamic stability
76D17 Viscous vortex flows
76M22 Spectral methods applied to problems in fluid mechanics
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
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