Non-linear and non-stationary modes of the lower branch of the incompressible boundary layer flow due to a rotating disk
Non-linear and non-stationary modes of the lower branch of the incompressible boundary layer flow due to a rotating disk
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旋转盘引起的不可压缩边界层流下支的非线性和非平稳模态
DOI：
10.1090/s0033-569x-07-01050-x
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发表时间：
2007
影响因子：
0.8
通讯作者：
M. Turkyilmazoglu
中科院分区：
文献类型：
--
作者：
M. Turkyilmazoglu
本文从理论上研究了旋转圆盘作用于不可压VonKarman边界层流动的三维小扰动下分支中性稳定性模态的结构。特别注意的是在足够高的雷诺数与合理的小尺度频率的短波长非线性非定常横流涡模式。密切关注渐近框架介绍了[Proc。Soc.伦敦系列A 406（1986），93-106]和[Proc. Roy. Soc.伦敦Ser.A413（1987），497-513]对于定常线性和非线性模式，这里揭示了具有足够长时间尺度的非定常模式也可以由基于三层理论的渐近展开过程来描述。利用这种方法，它考虑了非线性和非平行的影响，非平稳模式的渐近结构被证明是由粘性和科里奥利力之间的平衡调整，并导致在磁盘表面的剪切应力消失的事实。作为相邻区域中的解的匹配的结果，发现在线性情况下，波数和下分支模式的取向由本征关系控制，这类似于先前在[Proc. Roy. Soc.伦敦系列A 406（1986），93-106]中描述。渐近理论表明，非平行性有一个不稳定的影响。在无限大雷诺数的限制下，通过弱非线性分析，也得到了调制涡振幅的Landau型方程，其系数通常很难从有限雷诺数计算中得到。还发现非线性对于正频率波和负频率波都是不稳定的，尽管具有接近中性位置的正频率的扰动的有限振幅增长更有效。
In this paper a theoretical study is undertaken to investigate the structure of the lower branch neutral stability modes of three-dimensional small disturbances imposed on the incompressible Von Karman's boundary layer flow due to a rotating disk. Particular attention is given to the short-wavelength non-linear non-stationary crossflow vortex modes at sufficiently high Reynolds numbers with reasonably small scaled frequencies. Following closely the asymptotic frameworks introduced in [Proc. Roy. Soc. London Ser. A 406 (1986), 93-106] and [Proc. Roy. Soc. London Ser. A 413 (1987), 497-513] for the stationary linear and non-linear modes, it is revealed here that the non-stationary modes with sufficiently long time scale can also be described by an asymptotic expansion procedure based on the triple-deck theory. Making use of this approach, which takes into account the non-linear and non-parallel effects, the asymptotic structure of the non-stationary modes is shown to be adjusted by a balance between viscous and Coriolis forces, and resulted from the fact of vanishing shear stress at the disk surface. As a consequence of the matching of the solutions in adjacent regions it is found that in the linear case the wavenumber and the orientation of the lower branch modes are governed by an eigenrelation, which is akin to the one obtained previously in [Proc. Roy. Soc. London Ser. A 406 (1986), 93-106] for the stationary modes. The asymptotic theory shows that the non-parallelism has a destabilizing effect. A Landau-type equation for the modulated vortex amplitude with coefficients that are often difficult to get from finite Reynolds number computations has also been obtained from a weakly non-linear analysis in the limit of infinitely large Reynolds numbers. The non-linearity has also been found to be destabilizing for both positive and negative frequency waves, though finite amplitude growth of a disturbance having positive frequency close to the neutral location is more effective.