Theoretical investigation on laminar-turbulent transition of flows through a rectangular duct

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(1)Linear and nonlinear stability analyses on a convective flow in a channel produced by a homogeneously distributed heat source are carried out by imposing a periodic boundary condition on the side walls in the spanwise direction. We expect that the transition mechanism from laminar to turbulent flows in this system would not change much from the one in a rectangular duct case. From the linear analysis we find that there are two kinds of infinitesimal disturbances, transverse-roll type and longitudinal-roll type, which destabilise the flow in the case when the channel is placed almost vertically. Also we are able to obtain finite amplitude solutions resulting from the nonlinear interaction of these two kinds of disturbances.(2)A linear stability on the flow through a duct produced by a homogeneously distributed heat source is examined numerically. We obtain the following results. (i)There are, unexpectedly, several inflection points and reverse-flow points in the basic flow profile. They play important roles for the stability, of the flow. (ii)The flow through a duct at any aspect ratio can become unstable. (iii)The region in the parameter space where the flow becomes unstable coincides with the region where the basic flow profile is inflectional. (iv)The eigenfunction of the unstable mode propagates in the streamwise direction with the phase velocity equal to the average velocity of the basic flow.(3)As a nonlinear extension of the case (2)we derive higher order terms resulting from the nonlinear interaction of unstable eigenmodes, and the modification terms for the mean flow and the mean temperature. By taking into account the particular spatial symmetry in this system we identify the spatial structure of the finite amplitude solution that may bifurcate as a result of instability. The validity of the numerical code which incorporates the above has been checked partially.