Secondary instability of stationary Görtler vortices originating from first/second Mack mode

2020 ◽  
Vol 32 (3) ◽  
pp. 034109 ◽  
Keyword(s):  
Secondary Instability ◽  
Görtler Vortices ◽  
Gortler Vortices

Fundamental and subharmonic secondary instabilities of Görtler vortices

1995 ◽  
Vol 297 ◽  
pp. 77-100 ◽  
Author(s):  
Fei Li ◽  
Mujeeb R. Malik
Keyword(s):  
Flow Field ◽  
Vortex Structure ◽  
Streamwise Velocity ◽  
Secondary Instability ◽  
Mean Flow ◽  
Partial Differential ◽  
Relative Significance ◽  
Nonlinear Development ◽  
Görtler Vortices ◽  
Gortler 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.

Nonlinear evolution and secondary instability of steady and unsteady Görtler vortices induced by free-stream vortical disturbances

2017 ◽  
Vol 829 ◽  
pp. 681-730 ◽  
Author(s):  
Dongdong Xu ◽  
Yongming Zhang ◽  
Xuesong Wu
Keyword(s):  
Boundary Layer ◽  
Free Stream ◽  
Nonlinear Evolution ◽  
Low Frequency ◽  
Secondary Instability ◽  
Moderate Intensity ◽  
Mean Flow ◽  
Görtler Vortices ◽  
Gortler Vortices ◽  
The Impact

We study the nonlinear development and secondary instability of steady and unsteady Görtler vortices which are excited by free-stream vortical disturbances (FSVD) in a boundary layer over a concave wall. The focus is on low-frequency (long-wavelength) components of FSVD, to which the boundary layer is most receptive. For simplification, FSVD are modelled by a pair of oblique modes with opposite spanwise wavenumbers $\pm k_{3}$, and their intensity is strong enough (but still of low level) that the excitation and evolution of Görtler vortices are nonlinear. For the general case that the Görtler number $G_{\unicode[STIX]{x1D6EC}}$ (based on the spanwise wavelength $\unicode[STIX]{x1D6EC}$ of the disturbances) is $O(1)$, the formation and evolution of Görtler vortices are governed by the nonlinear unsteady boundary-region equations, supplemented by appropriate upstream and far-field boundary conditions, which characterize the impact of FSVD on the boundary layer. This initial-boundary-value problem is solved numerically. FSVD excite steady and unsteady Görtler vortices, which undergo non-modal growth, modal growth and nonlinear saturation for FSVD of moderate intensity. However, for sufficiently strong FSVD the modal stage is bypassed. Nonlinear interactions cause Görtler vortices to saturate, with the saturated amplitude being independent of FSVD intensity when $G_{\unicode[STIX]{x1D6EC}}\neq 0$. The predicted modified mean-flow profiles and structure of Görtler vortices are in excellent agreement with several steady experimental measurements. As the frequency increases, the nonlinearly generated harmonic component $(0,2)$ (which has zero frequency and wavenumber $2k_{3}$) becomes larger, and as a result the Görtler vortices appear almost steady. The secondary instability analysis indicates that Görtler vortices become inviscidly unstable in the presence of FSVD with a high enough intensity. Three types of inviscid unstable modes, referred to as sinuous (odd) modes I, II and varicose (even) modes I, are identified, and their relevance is delineated. The characteristics of dominant unstable modes, including their frequency ranges and eigenfunctions, are in good agreement with experiments. The secondary instability is intermittent when FSVD are unsteady and of low frequency. However, the intermittence diminishes as the frequency increases. The present theoretical framework, which allows for a detailed and integrated description of the key transition processes, from generation, through linear and nonlinear evolution, to the onset of secondary instability, represents a useful step towards predicting the pre-transitional flow and transition itself of the boundary layer over a blade in turbomachinery.

scholarly journals Visualization of Secondary Instability Flows by Forced Wavelength Gortler Vortices

2009 ◽  
Vol 29-1 (2) ◽  
pp. 1175-1175
Author(s):  
Yoichi YAHAGI ◽  
Kazuyuki HATAKEYAMA ◽  
Akira ITO
Keyword(s):  
Secondary Instability ◽  
Görtler Vortices ◽  
Gortler Vortices
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