Secondary instabilities of Görtler vortices in high-speed boundary layer flows

2015 ◽  
Vol 781 ◽  
pp. 388-421 ◽  
Author(s):  
Jie Ren ◽  
Song Fu
Keyword(s):  
Boundary Layer ◽  
High Speed ◽  
Incompressible Flows ◽  
Hypersonic Flows ◽  
Type I ◽  
Görtler Vortices ◽  
Layer Mode ◽  
Threshold Amplitude ◽  
Gortler Vortices ◽  
Secondary Instabilities

Görtler vortices developed in laminar boundary layer experience remarkable changes when the flow is subjected to compressibility effects. In the present study, five $\mathit{Ma}$ numbers, covering incompressible to hypersonic flows, at $\mathit{Ma}=0.015$, 1.5, 3.0, 4.5 and 6.0 are specified to illustrate these effects. Görtler vortices in subsonic and moderate supersonic flows ($\mathit{Ma}=0.015$, 1.5 and 3.0) are governed by the conventional wall-layer mode (mode W). In hypersonic flows ($\mathit{Ma}=4.5$, 6.0), the trapped-layer mode (mode T) becomes dominant. This difference is maintained and intensifies downstream leading to different scenarios of secondary instabilities. The linear and nonlinear development of Görtler vortices which are governed by dominant modal disturbances are investigated with direct marching of the nonlinear parabolic equations. The secondary instabilities of Görtler vortices set in when the resulting streaks are adequately developed. They are studied with Floquet theory at multiple streamwise locations. The secondary perturbations become unstable downstream following the sequence of sinuous mode type I, varicose mode and sinuous mode type II, indicating an increasing threshold amplitude. Onset conditions are determined for these modes. The above three modes can each have the largest growth rate under the right conditions. In the hypersonic cases, the threshold amplitude $A(u)$ is dramatically reduced, showing the significant impact of the thermal streaks. To investigate the parametric effect of the spanwise wavenumber, three global wavenumbers ($B=0.5$, 1.0 and $2.0\times 10^{-3}$) are specified. The relationship between the dominant mode (sinuous or varicose) and the spanwise wavenumber of Görtler vortices found in incompressible flows (Li & Malik, J. Fluid Mech., vol. 297, 1995, pp. 77–100) is shown to be not fully applicable in high-speed cases. The sinuous mode becomes the most dangerous, regardless of the spanwise wavelength when $\mathit{Ma}>3.0$. The subharmonic type can be the most dangerous mode while the detuned type can be neglected, although some of the sub-dominant secondary modes reach their peak growth rates under detuned states.

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

1995 ◽  
Vol 283 ◽  
pp. 249-272 ◽  
Author(s):  
Daniel S. Park ◽  
Patrick Huerre
Keyword(s):  
Boundary Layer ◽  
Velocity Profile ◽  
Three Dimensional ◽  
Vertical Direction ◽  
Secondary Instability ◽  
Spanwise Direction ◽  
Görtler Vortices ◽  
Gortler Vortices ◽  
Secondary Instabilities ◽  
Asymptotic Suction

The temporal growth of Görtler 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 Görtler 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, ∂U/∂z, and the varicose mode to shear in the vertical direction, ∂U/∂y.

Measurements in a Transitional Boundary Layer With Görtler Vortices

1996 ◽  
Author(s):  
Ralph J. Volino ◽  
Terrence W. Simon
Keyword(s):  
Boundary Layer ◽  
Turbulence Intensity ◽  
Free Stream ◽  
Streamwise Velocity ◽  
Mean Velocity ◽  
Free Stream Turbulence ◽  
Görtler Vortices ◽  
Concave Wall ◽  
The Mean ◽  
Gortler Vortices

The laminar-turbulent transition process has been documented in a concave-wall boundary layer subject to low (0.6%) free-stream turbulence intensity. Transition began at a Reynolds number, Rex (based on distance from the leading edge of the test wall), of 3.5×105 and was completed by 4.7×105. The transition was strongly influenced by the presence of stationary, streamwise, Görtler vortices. Transition under similar conditions has been documented in previous studies, but because concave-wall transition tends to be rapid, measurements within the transition zone were sparse. In this study, emphasis is on measurements within the zone of intermittent flow. Twenty-five profiles of mean streamwise velocity, fluctuating streamwise velocity, and intermittency have been acquired at five values of Rex, and five spanwise locations relative to a Görtler vortex. The mean velocity profiles acquired near the vortex downwash sites exhibit inflection points and local minima. These minima, located in the outer part of the boundary layer, provide evidence of a “tilting” of the vortices in the spanwise direction. Profiles of fluctuating velocity and intermittency exhibit peaks near the locations of the minima in the mean velocity profiles. These peaks indicate that turbulence is generated in regions of high shear, which are relatively far from the wall. The transition mechanism in this flow is different from that on flat walls, where turbulence is produced in the near-wall region. The peak intermittency values in the profiles increase with Rex, but do not follow the “universal” distribution observed in most flat-wall, transitional boundary layers. The results have applications whenever strong concave curvature may result in the formation of Görtler vortices in otherwise 2-D flows. Because these cases were run with a low value of free-stream turbulence intensity, the flow is not a replication of a gas turbine flow. However, the results do provide a base case for further work on transition on the pressure side of gas turbine airfoils, where concave curvature effects are combined with the effects of high free-stream turbulence and strong streamwise pressure gradients, for they show the effects of embedded streamwise vorticity in a flow that is free of high-turbulence effects.

Visualization of transition

1969 ◽  
Vol 38 (3) ◽  
pp. 473-480 ◽  
Author(s):  
F. X. Wortmann
Keyword(s):  
Boundary Layer ◽  
Experimental Study ◽  
Flow Visualization ◽  
Three Dimensional ◽  
Vortex Pair ◽  
Instability Mode ◽  
Vortex Pattern ◽  
Görtler Vortices ◽  
Instability Modes ◽  
Gortler Vortices

In an experimental study the development of transition downstream of Görtler vortices was investigated. With the tellurium method it was possible to distinguish beyond the Görtler vortices to successive instability modes. The first deforms the vortex pattern in a steady way and produces between each vortex pair boundary-layer profiles with two points of inflexion. When this has been established another instability mode starts, consisting of regular three-dimensional oscillations. By detailed flow visualization a nearly complete picture of the different flow patterns can be obtained.

Crossflow effects on the growth rate of inviscid Görtler vortices in a hypersonic boundary layer

1994 ◽  
Vol 276 ◽  
pp. 343-367 ◽  
Author(s):  
Yibin Fu ◽  
Philip Hall
Keyword(s):  
Boundary Layer ◽  
Growth Rate ◽  
Exact Solution ◽  
Eigenvalue Problem ◽  
Hypersonic Boundary Layer ◽  
Minimum Order ◽  
Görtler Vortices ◽  
Neutral Mode ◽  
Gortler Vortices ◽  
The Relationship

The effects of crossflow on the growth rate of inviscid Görtler vortices in a hypersonic boundary layer with pressure gradient are studied in this paper. Attention is focused on the inviscid mode trapped in the temperature adjustment layer; this mode has greater growth rate than any other mode at the minimum order of the Görtler number at which Görtler vortices may exist. The eigenvalue problem which governs the relationship between the growth rate, the crossflow amplitude and the wavenumber is solved numerically, and the results are then used to clarify the effects of crossflow on the growth rate of inviscid Görtler vortices. It is shown that crossflow effects stabilize Görtler vortices in different manners for incompressible and hypersonic flows. The neutral mode eigenvalue problem is found to have an exact solution, and as a byproduct, we have also found the exact solution to a neutral mode eigenvalue problem which was formulated, but unsolved before, by Bassom & Hall (1991).

Hypersonic boundary layer transition on a concave wall: stationary Görtler vortices

2019 ◽  
Vol 865 ◽  
pp. 1-40 ◽  
Author(s):  
X. Chen ◽  
G. L. Huang ◽  
C. B. Lee
Keyword(s):  
Boundary Layer ◽  
Numerical Simulations ◽  
Stability Theory ◽  
Direct Numerical Simulations ◽  
Boundary Layer Transition ◽  
Hypersonic Boundary Layer ◽  
Mode Instability ◽  
Görtler Vortices ◽  
Gortler Vortices ◽  
The Stability

This study investigates the stability and transition of Görtler vortices in a hypersonic boundary layer using linear stability theory and direct numerical simulations. In the simulations, Görtler vortices are separately excited by wall blowing and suction with spanwise wavelengths of 3, 6 and 9 mm. In addition to primary streaks with the same wavelength as the blowing and suction, secondary streaks with half the wavelength also emerge in the 6 and 9 mm cases. The streaks develop into mushroom structures before breaking down. The breakdown processes of the three cases are dominated by a sinuous-mode instability, a varicose-mode instability and a combination of the two, respectively. Both fundamental and subharmonic instabilities are relevant in all cases. Multiple modes are identified in the secondary-instability stage, some of which originate from the primary instabilities (first and second Mack modes). We demonstrate that the first Mack mode can be destabilized to either a varicose-mode or sinuous-mode streak instability depending on its frequency and wavelength, whereas the second Mack mode undergoes a stabilizing stage before turning into a varicose mode in the 6 and 9 mm cases. An energy analysis reveals the stabilizing and destabilizing mechanisms of the primary instabilities under the influence of Görtler vortices, highlighting the role played by the spanwise production based on the spanwise gradient of the streamwise velocity in both varicose and sinuous modes. The effects introduced by the secondary streaks are examined by filtering the secondary streaks in two new simulations with nominally identical conditions to those of the 6 and 9 mm cases. Remarkably, the secondary streaks can destabilize the Görtler vortices, therefore advancing the transition. The stability theory results are in good agreement with those from direct numerical simulations.

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