On the receptivity problem for Görtler vortices: vortex motions induced by wall roughness
On the receptivity problem for Görtler vortices: vortex motions induced by wall roughness
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DOI：
10.1098/rsta.1991.0036
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发表时间：
1991-04
期刊：
影响因子：
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通讯作者：
J. Denier;P. Hall;S. Seddougui
中科院分区：
文献类型：
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作者：
J. Denier;P. Hall;S. Seddougui
研究了壁面粗糙度引起的Görtler涡的感受性问题。粗糙度是通过对流动发生的弯曲壁的小幅度扰动来模拟的。这些扰动的幅度被认为是足够小的诱导Görtler涡被描述的线性理论。假定粗糙度在边界层长度尺度上沿展向变化，而在流动方向上相应的变化在壁面曲率变化的长度尺度上。事实上，后一条件可以放宽，以允许更快的流向粗糙度变化，只要这种变化不像展向变化那样快。描述粗糙度的函数假定为展向和流向相关性可以分离;这使我们能够在适当的地方进行傅立叶变换或拉普拉斯变换。研究了孤立粗糙元和分布粗糙元的情况，并在小波长范围内发现了与强迫振幅和诱导涡振幅有关的耦合系数。结果表明，这个系数是指数小，在后者的限制，使它是不太可能的，这种模式可以直接刺激壁粗糙度。在O（1）波长的情况是完全不同的，这是不同的强迫函数的数值研究。结果发现，一个孤立的粗糙元素诱导的涡流场的增长在一个有限的距离下游的单元楔形。然而，在障碍物的下游，由元件产生的扰动流的振幅衰减。在较大的Görtler数适合相对较大的壁曲率的感受性问题进行了详细讨论。结果表明，Görtler不稳定性方程中增长最快的线性模的波数与Görtler数的五分之一次方成正比。该模式可以与无粘扰动和干扰适当的右手分支的中性曲线的Görtler涡。这个增长最快的涡旋和强迫函数之间的耦合系数以封闭形式存在。
The receptivity problem for Görtler vortices induced by wall roughness is investigated. The roughness is modelled by small amplitude perturbations to the curved wall over which the flow takes place. The amplitude of these perturbations is taken to be sufficiently small for the induced Görtler vortices to be described by linear theory. The roughness is assumed to vary in the spanwise direction on the boundary-layer lengthscale, whilst in the flow direction the corresponding variation is on the lengthscale over which the wall curvature varies. In fact the latter condition can be relaxed to allow for a faster streamwise roughness variation so long as the variation does not become as fast as that in the spanwise direction. The function that describes the roughness is assumed to be such that its spanwise and streamwise dependences can be separated; this enables us to make progress by taking Fourier or Laplace transforms where appropriate. The cases of isolated and distributed roughness elements are investigated and the coupling coefficient which relates the amplitude of the forcing and the induced vortex amplitude is found asymptotically in the small wavelength limit. It is shown that this coefficient is exponentially small in the latter limit so that it is unlikely that this mode can be stimulated directly by wall roughness. The situation at O(1) wavelengths is quite different and this is investigated numerically for different forcing functions. It is found that an isolated roughness element induces a vortex field which grows within a wedge at a finite distance downstream of the element. However, immediately downstream of the obstacle the disturbed flow produced by the element decays in amplitude. The receptivity problem at larger Görtler numbers appropriate to relatively large wall curvature is discussed in detail. It is found that the fastest growing linear mode of the Görtler instability equations has wavenumber proportional to the one-fifth power of the Gortler number. The mode can be related to both inviscid disturbances and the disturbances appropriate to the right-hand branch of the neutral curve for Görtler vortices. The coupling coefficient between this, the fastest growing vortex, and the forcing function is found in closed form.