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

（1）通过在展向侧壁上施加周期性边界条件，对均匀分布热源引起的通道内对流进行了线性和非线性稳定性分析。我们预计，从层流到湍流的过渡机制在这个系统中不会改变太多的矩形管道的情况下。从线性分析中，我们发现，有两种类型的无穷小扰动，横向滚动型和纵向滚动型，这是不稳定的情况下，几乎垂直放置的通道的流动。我们也能够得到这两种扰动的非线性相互作用的有限振幅解。(2)A本文用数值方法研究了均匀分布热源引起的管道内流动的线性稳定性。我们得到以下结果。（i）在基本流动剖面中，出乎意料地存在几个拐点和逆流点。它们对水流的稳定性起着重要的作用。（ii）在任何纵横比下通过管道的气流都可能变得不稳定。（iii）参数空间中流动变得不稳定的区域与基本流动剖面是弯曲的区域相一致。（iv）不稳定模态的本征函数沿流向传播，相速度等于基本流的平均速度。(3)As对情形（2）作了非线性推广，导出了由不稳定本征模的非线性相互作用引起的高阶项，以及平均流量和平均温度的修正项。考虑到在这个系统中的特定空间对称性，我们确定的空间结构的有限振幅的解决方案，可能会分叉的不稳定性的结果。已对包含上述内容的数字代码的有效性进行了部分检查。

(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.