Competition of the multiple Gortler modes in hypersonic boundary layer flows

The boundary layer stability analysis with a concave wall is termed the Görtler instability. It has been shown in this study that, apart from the conventional wall-layer mode (mode W) there also exists the trapped wall layer (mode T) when the Mach number is large, e.g. Ma≥4. Competition of the multiple Görtler modes in hypersonic boundary layer flows are thus investigated.

The present study illustrates, for the hypersonic flows, through the local and marching analysis, the crossover of the mode W and the mode T at O(1) wavenumber and large Görtler number regime. In fact, it is at this wavenumber regime that the instability is most likely to occur. The two approaches are expected to deliver similar results and the marching analysis helps to express the details of the crossover and confirm the result of the local analysis.

In fact the study of Görtler instability goes back to the date of the 1940s. Since Görtler's pioneering investigation on the boundary layer instabilities subjected to the negative curvature in 1940 highlighting the existence of the streamwise-oriented counter-rotating vortices, extensive studies have been carried out on this subject especially in the incompressible flows. These vortices are caused by the imbalance between the centrifugal force and the normal pressure gradient near a concave surface and exhibit a quasi-constant spanwise wavelength.

The most distinct difference between the incompressible counterparts is the existence of the trapped-layer mode (mode T) apart from the conventional wall layer mode (mode W) observed in the incompressible cases. The velocity disturbances of the multiple Görtler modes are given below in Figure 1. It is evident that the mode T has its disturbances detached from the wall. As a result of this, the nonlinear development of Görtler vortices in hypersonic boundary layers shows considerable differences. Figure 2 shows the development of the streamwise velocity contours of the Görtler flow up to a fully saturated states. The Mach numbers are 1.5, 3.0, 4.5 and 6.0 respectively. The famous mushroom structures (subsonic and moderate supersonic, e.g., Ma=1.5 and 3.0) are replaced by the bell shapes (hypersonic, e.g., Ma=4.5 and 6.0). This is because the mode T is the most dangerous modal shape in such flows.

To conclude, when considering the flow transition induced by the Görtler instability in hypersonic flows, the mode T highlighted in this article must be considered. Also, with appropriate parameters the mode T and mode W have crossovers. This study is a guidance for future studies of the secondary instability of Görtler vortices and flow transition in hypersonic boundary flows. The readers are also recommended to read the future articles by the authors.

More information:
Jie Ren and Song Fu, "Competition of the multiple Görtler modes in hypersonic boundary layer flows" SCIENCE CHINA Physics, Mechanics & Astronomy 2014（6）, 1178-1193 DOI: 10.1007/s11433-014-5454-9

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