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

(1)对均匀分布热源产生的通道内的对流流动，采用沿展向施加周期边界条件的方法，进行了线性和非线性稳定性分析。我们预计，该系统从层流到湍流的转变机制与矩形管道情况下的机制不会有太大变化。由线性分析可知，在近乎垂直的情况下，存在横向滚动型和纵向滚动型两种微小扰动，这两种扰动会使水流不稳定。数值研究了均匀分布热源作用下管道流动的线性稳定性。我们得到了以下结果。(I)在基本流动剖面中，意外地出现了几个拐点和倒流点。它们对流动的稳定性起着重要的作用。(Ii)在任何纵横比下，流经管道的流动都可能变得不稳定。(Iii)参数空间中流动变得不稳定的区域与基本流型为拐点的区域重合。(4)不稳定模的本征函数沿流向传播，相速度等于基流的平均速度。(3)作为情况(2)的非线性推广，我们导出了由不稳定本征模的非线性相互作用引起的高阶项，以及平均流和平均温度的修正项。通过考虑该系统特殊的空间对称性，我们确定了有限幅值解的空间结构，该结构可能由于不稳定性而分叉。已部分检查了包含上述内容的数字代码的有效性。

(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.