Investigation of weakly-nonlinear development of unsteady Görtler vortices

The nonlinear development of stationary Görtler vortices leads to a highly distorted mean flow field where the streamwise velocity depends strongly not only on the wall-normal but also on the spanwise coordinates. In this paper, the inviscid instability of this flow field is analysed by solving the two-dimensional eigenvalue problem associated with the governing partial differential equation. It is found that the flow field is subject to the fundamental odd and even (with respect to the Görtler vortex) unstable modes. The odd mode, which was also found by Hall & Horseman (1991), is initially more unstable. However, there exists an even mode which has higher growth rate further downstream. It is shown that the relative significance of these two modes depends upon the Görtler vortex wavelength such that the even mode is stronger for large wavelengths while the odd mode is stronger for short wavelengths. Our analysis also shows the existence of new subharmonic (both odd and even) modes of secondary instability. The nonlinear development of the fundamental secondary instability modes is studied by solving the (viscous) partial differential equations under a parabolizing approximation. The odd mode leads to the well-known sinuous mode of break down while the even mode leads to the horseshoe-type vortex structure. This helps explain experimental observations that Görtler vortices break down sometimes by sinuous motion and sometimes by developing a horseshoe vortex structure. The details of these break down mechanisms are presented.

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The fully nonlinear development of Görtler vortices in growing boundary layers

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

10.1098/rspa.1988.0021 ◽

1988 ◽

Vol 415 (1849) ◽

pp. 421-444 ◽

Cited By ~ 46

Keyword(s):

Boundary Layer ◽

Evolution Equations ◽

Mean Flow ◽

Initial Development ◽

Fully Nonlinear ◽

Nonlinear Development ◽

Görtler Vortices ◽

Starting Point ◽

The Mean ◽

Gortler Vortices

The fully nonlinear development of small-wavelength Görtler vortices in a growing boundary layer is investigated by a combination of asymptotic and numerical methods. The starting point for the analysis is the weakly nonlinear theory of Hall ( J. Inst. Math. Applies 29, 173 (1982)) who discussed the initial development of large-wavenumber small-amplitude vortices in a neighbourhood of the location where they first become linearly unstable. That development is unusual in the context of nonlinear stability theory in that it is not described by the Stuart-Watson approach. In fact, the development is governed by a pair of coupled nonlinear partial differential evolution equations for the vortex flow and the mean flow correction. Here the further development of this interaction is considered for vortices so large that the mean flow correction driven by them is as large as the basic state. Surprisingly it is found that such a nonlinear interaction can still be described by asymptotic means. It is shown that the vortices spread out across the boundary layer and effectively drive the boundary layer. In fact, the system obtained by the equations for the fundamental component of the vortex generates a differential equation for the basic state. Thus the mean flow adjusts so as to make these large amplitude vortices locally neutral. Moreover in the region where the vortices exist the mean flow has a ‘square-root’ profile and the vortex velocity field can be written down in closed form. The upper and lower boundaries of the region of vortex activity are determined by a free-boundary problem involving the boundary-layer equations. In general it is found that this region ultimately includes almost all of the original boundary layer and much of the free stream. In this situation the mean flow has essentially no relation to the flow that exists in the absence of the vortices.

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The nonlinear development of Görtler vortices in growing boundary layers

Journal of Fluid Mechanics ◽

10.1017/s0022112088002137 ◽

1988 ◽

Vol 193 (-1) ◽

pp. 243 ◽

Cited By ~ 85

Author(s):

Philip Hall

Keyword(s):

Boundary Layers ◽

Nonlinear Development ◽

Görtler Vortices ◽

Gortler Vortices

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The excitation of Görtler vortices by free stream coherent structures

Journal of Fluid Mechanics ◽

10.1017/jfm.2017.380 ◽

2017 ◽

Vol 826 ◽

pp. 60-96 ◽

Cited By ~ 7

Author(s):

L. J. Dempsey ◽

P. Hall ◽

K. Deguchi

Keyword(s):

Boundary Layer ◽

Coherent Structures ◽

Free Stream ◽

Nonlinear Regime ◽

Weakly Nonlinear ◽

Stream Structure ◽

Görtler Vortices ◽

Gortler Vortices ◽

Asymptotic Suction ◽

Imperfect Bifurcation

The effect of free stream coherent structures in the asymptotic suction boundary layer on the initiation of Görtler vortices is considered from both the ‘imperfect’ bifurcation and receptivity viewpoints. Firstly a weakly nonlinear and a full numerical approach are used to describe Görtler vortices in the asymptotic suction boundary layer in the absence of forcing from the free stream. It is found that interactions between different spanwise harmonics occur and lead to multiple secondary bifurcations in the fully nonlinear regime. Furthermore it is shown that centrifugal instabilities of the asymptotic suction boundary layer behave quite differently than their counterparts in either fully developed flows such as Couette flow or growing boundary layers. A significant result is that the most dangerous disturbance is found to bifurcate subcritically from the unperturbed state. Within the weakly nonlinear regime the receptivity of Görtler vortices to the free stream exact coherent structures discovered by Deguchi & Hall (J. Fluid Mech., vol. 752, 2014, pp. 602–625; J. Fluid Mech., vol. 778, 2015, pp. 451–484) is considered. The presence of free stream structures results in a resonant excitation of Görtler vortices in the main boundary layer. This leads to imperfect bifurcations reminiscent of those found by Daniels (Proc. R. Soc. Lond. A, vol. 358, 1977, pp. 173–197) and Hall & Walton (Proc. R. Soc. Lond. A, vol. 358, 1977, pp. 199–221; J. Fluid Mech., vol. 90, 1979, pp. 377–395) in the context of transition to finite amplitude Bénard convection in a bounded region. In order to understand the receptivity problem for the given flow the spatial initial value problem for this interaction is also considered when the free stream structure begins at a fixed position along the wall. Remarkably, it will be shown that free stream structures are incredibly efficient generators of Görtler vortices; indeed the induced vortices are found to be larger than the free stream structure which provokes them! The relationship between the imperfect bifurcation approach and receptivity theory is described.

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