The nonlinear interaction of Tollmien–Schlichting waves and Taylor-Görtler vortices in curved channel flows
The nonlinear interaction of Tollmien–Schlichting waves and Taylor-Görtler vortices in curved channel flows
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弯曲河道流中 Tollmien-Schlichting 波与 Taylor-Görtler 涡流的非线性相互作用
DOI：
10.1098/rspa.1988.0060
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发表时间：
1987
期刊：
影响因子：
--
通讯作者：
F. Smith
中科院分区：
文献类型：
--
作者：
P. Hall;F. Smith
已知具有弯曲流线的粘性流体流可以支持Tollmien-Schlichting和Taylor-Görtler不稳定性。吉布森和库克（Q. Jl Mech.appl.Math.27，149（1974））。然而，在两种模式都可能基于线性理论的情况下，必须使用非线性理论来确定不稳定性相互作用的影响。这种相互作用的细节具有实际的重要性，因为它可能对用于层流控制的机制产生灾难性的影响。在这里，这种相互作用的背景下，充分发展的流动在弯曲的渠道。除了与附面层增长有关的技术上的差别外，这种流动中的不稳定性结构与实际上更重要的外部附面层情况中的不稳定性结构非常相似。示出的相互作用有两个不同的阶段，取决于输入干扰的大小。在非常低的振幅下，两个倾斜的Tollmien-Schlichting波与Görtler涡相互作用，使得标度振幅在有限时间内变为无穷大。这种类型的相互作用是由二次非线性常微分振幅方程。一个更强的类型的相互作用发生在较大的输入扰动振幅，并导致一个更复杂的类型的演化方程。现在这些方程的解主要取决于Tollmien-Schlichting波的波前到Görtler涡的方向。因此，如果涡旋和波的方向之间的夹角大于41.6°，这种较强的相互作用又在有限时间内终止于奇点;否则，崩溃是指数的，需要无限的时间。此外，更强的相互作用可以发生在没有曲率的情况下，在这种情况下，纵向涡完全由Tollmien-Schlichting波驱动。
It is known that a viscous fluid flow with curved streamlines can support both Tollmien-Schlichting and Taylor-Görtler instabilities. The question of which linear mode is dominant at finite values of the Reynolds numbers was discussed by Gibson & Cooke (Q. Jl Mech. appl. Math. 27, 149 (1974)). In a situation where both modes are possible on the basis of linear theory a nonlinear theory must be used, however, to determine the effect of the interaction of the instabilities. The details of this interaction are of practical importance because of its possible catastrophic effects on mechanisms used for laminar flow control. Here this interaction is studied in the context of fully developed flows in curved channels. Apart from technical differences associated with boundary-layer growth the structures of the instabilities in this flow can be very similar to those in the practically more important external boundary-layer situation. The interaction is shown to have two distinct phases depending on the size of the input disturbances. At very low amplitudes two oblique Tollmien–Schlichting waves interact with a Görtler vortex in such a manner that the scaled amplitudes become infinite at a finite time. This type of interaction is described by ordinary differential amplitude equations with quadratic nonlinearities. A stronger type of interaction occurs at larger input disturbance amplitudes and leads to a more complicated type of evolution equation. The solution of these equations now depends critically on the orientation of the wavefronts of the Tollmien–Schlichting waves to the Görtler vortex. Thus, if the angle between the directions of the vortex and the waves is greater than 41.6° this stronger interaction again terminates in a singularity at a finite time; otherwise the breakdown is exponential, taking an infinite time. Moreover, the stronger interaction can take place in the absence of curvature, in which case the longitudinal vortex is entirely driven by the Tollmien–Schlichting waves.