The temporal growth of Gortler vortices and the associated secondary instability mechanisms are investigated numerically in the case of an asymptotic suction boundary layer on a curved plate. Highly inflectional velocity profiles are generated in both the spanwise and vertical directions. The inflectional velocity profile develops earlier in the spanwise direction. There exist two distinct modes of instability that eventually lead to the breakdown of Gortler vortices: the sinuous mode and the varicose mode. The temporal secondary instability analysis of the three-dimensional inflectional velocity profile reveals that the sinuous mode becomes unstable earlier than the varicose mode. The sinuous mode is shown to be primarily related to shear in the spanwise direction, aU/az, and the varicose mode to shear in the vertical direction, a ulay.

Primary and secondary instabilities of the asymptotic suction boundary layer on a curved plate

This paper investigates streaks and Gortler vortices in a boundary layer over a flat or concave wall in a contracting or expanding stream, which provides a favourable or adverse pressure gradient, respectively. We consider first the excitation of streaks and Gortler vortices by free stream vortical disturbances (FSVD), and their nonlinear evolution. The focus is on FSVD with sufficiently long wavelength, to which the boundary layer is most receptive. The formulation is directed at the general case where the Gortler number increases. Raising FSVD intensity reduces the effects of the pressure gradient and curvature. At a high FSVD level of 14 %, the curvature has no impact on the vortices, while the pressure gradient only influences the saturation intensity. The unsteadiness of FSVD is found to reduce the boundary layer response significantly at FSVD levels, but that effect weakens as the turbulence level increases. A secondary instability analysis of the vortices is performed for moderate adverse and favourable pressure gradients. Three families of unstable modes have been identified, which may become dominant depending on the frequency and streamwise location. In the presence of an adverse pressure gradient, the secondary instability occurs earlier, but the unstable modes appear in a smaller band of frequency, and have smaller growth rates. The opposite is true for a favourable pressure gradient. The present theoretical framework, which accounts for the influence of the curvature, turbulence level and pressure gradient, allows for a detailed and integrated description of the key transition processes, and represents a useful step towards predicting the pretransitional flow and transition itself of the boundary layer over a blade in turbomachinery.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112020004383

Görtler vortices and streaks in boundary layer subject to pressure gradient: excitation by free stream vortical disturbances, nonlinear evolution and secondary instability

Boundary layer flows over concave wall can be unstable to disturbances giving rise to streamwise counter-rotating vortices known as Gortler vortices. These vortices in its nonlinear form are responsible for a strong distortion of the streamwise velocity profiles in the wall-normal and spanwise directions. The resulting inflectional velocity profiles are unstable to unsteady disturbances. These disturbances are called secondary instabilities and can develop into horseshoe vortices or a sinuous motion of the Gortler vortices. These types of secondary instabilities are known as even (varicose) and odd (sinuous) modes, respectively. Although many studies focused this subject, it has not been stated which mode dominates the transition process. In the present study the secondary instability of Gortler flow is investigated using high-order spatial numerical simulation. Multi-frequency unsteady disturbances are introduced with the same spanwise wavelength as the Gortler vortices, but different spanwise phases. Three different spanwise phases are used and the effect on the secondary instability is analyzed. Both, even and odd secondary instabilities are observed, according to the relative spanwise position of the unsteady disturbances. The growth analysis for each secondary crossplane instability mode is made using a temporal Fourier analysis and the physics is explored with the aid of the flow structures visualization. The results introducing disturbances that give rise to odd and even modes simultaneously show that, for the spanwise wavelength analyzed, the odd modes grow first and dominate the transition process.

On the odd and even secondary instabilities of Görtler vortices

Subsonic wind tunnel experiments were conducted to study the effect of tripped boundary layer on the pressure distribution in the contraction region of the tunnel. Measurements were performed by installing trip strip at two different positions in the concave portion of the contraction. The results show that installation of the trip strips, have significant effects on both turbulence and pressure distribution. The reduction in the free stream turbulence and reduction of the wall static pressure distribution deferred signified with the location of the trip strip. Keywords—Contraction, pressure distribution, trip strip, turbulence intensity.

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Control of Pressure Gradient in the Contraction of a Wind Tunnel

/pdf/establishing-very-low-speed-disturbance-free-flow-for-h547j8p90s.pdf

T. Sonoda

Papers

Effects of pressure gradient on the secondary instability of Goertler vortices