Competition of the multiple Görtler modes in hypersonic boundary layer flows

2015

J. Fluid Mech.

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Görtler vortices developed in laminar boundary layer experience remarkable changes when the flow is subjected to compressibility effects. In the present study, five Ma numbers, covering incompressible to hypersonic flows, at 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 (Ma = 0.015, 1.5 and 3.0) are governed by the conventional wall-layer mode (mode W). In hypersonic flows (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 × 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 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.

Section: Formulation Of the Problemmentioning

confidence: 99%

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

2015

J. Fluid Mech.

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“…The actual eigen-equation is omitted here as is very well known. See [35] for more details of the formulation. In the present work, the spatial problem is considered with the angular frequency ω being prescribed and α solved as the eigenvalue with the Arnoldi method.…”

Section: Lst Approachmentioning

confidence: 99%

“…The trapped-layer mode (mode T) has its disturbances detached from the wall in the upper region of the boundary layer -the temperature adjustment layer. Crossover of the growth rate of the two modes can take place at certain flow conditions [34,35]. The unsteady Görtler mode together with the high turbulence intensity is responsible for the transition in turbomachinery flows, e.g., over the turbine blades [29,36].…”

Section: Introductionmentioning

confidence: 99%

Study of the Discrete Spectrum in a Mach 4.5 Görtler Flow

Flow Turbulence Combust

2014

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Evolution of the discrete spectrum in the Ma = 4.5 boundary layer is studied with LST and PSE approaches. Both two-dimensional (2-D) and three-dimensional (3-D) disturbances are considered with streamwise curvature effects. The concave curvature shows a destabilizing effect on the 2-D second/third mode when the fast mode (mode F (1) , mode F (2) ...) synchronizes with the slow mode (mode S). The spectrum branching in the synchronization between the mode F (2) and mode S is also observed. The increase in the spanwise wavenumber(3-D disturbances), on the other hand, suppresses the synchronization between mode F and mode S and reduces the growth rate of the unstable mode. With regard to the 3-D disturbances subjecting to the concave curvature, the mode S originating from the slow acoustic wave amounts to the unsteady Görtler mode while the quasi-steady Görtler mode emanates from the continuous spectrum of the vorticity/entropy wave.

“…In addition to the cases B1, B2 and B3, the growth rate of another As stated in Section I, the mode T has the largest growth rate in hypersonic cases except for very large Re numbers. 14 Fig. 3 shows the modal profiles u ′ ρ ′ and T ′ of the disturbances for case M1-B2, M2-B2, M3-B2 and M4-B2 at Re = 400 and 1000.…”

Section: Mathematical Formulationmentioning

confidence: 99%

“…The readers may refer to Dando & Seddougui for the asymptotic analysis 13 and Ren & Fu for the local & marching analysis. 14 The secondary instability of Görtler vortices in hypersonic flows has received much fewer attentions. Fu & Hall 15 showed the destabilizing effects of Görtler vortices, wall cooling and gas dissociation on the Rayleigh secondary instability in the hypersonic boundary layer.…”

Section: Introductionmentioning

confidence: 99%

The role of Görtler vortices in the hypersonic boundary layer transition

Liu

44th AIAA Fluid Dynamics Conference

2014

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The Görtler vortices which manifest as the counter-rotating streamwise vortices in highspeed boundary layers are studied. Four groups of M a numbers (M a = 1.5, 3.0, 4.5 and 6.0) and three groups of wavenumbers (B = 0.5, 1.0 and 2.0×10 −3 ) are specified to investigate the parametric effect of M a numbers and the spanwise wavenumbers on the spatial development and the interrelated secondary instability of Görtler vortices. The Görtler vortices in moderate supersonic flows (Ma=1.5, 3.0) are governed by the conventional wall-layer mode (mode W). In hypersonic flows (Ma=4.5, 6.0), it is the trapped-layer mode (mode T) that becomes dominating. The linear and nonlinear development of Görtler vortices are studied using the LST and the direct marching of the nonlinear parabolic equations. The secondary instabilities of Görtler vortices are then analyzed on the saturated state of the Görtler vortices with Floquet theory. The conclusion that "the dominating secondary modes (odd or even) depend on the spanwise wavenumbers of the Görtler vortices 1 " is shown to be unapplicable in compressible cases. The relationship between the primary Görtler instability and the corresponding secondary instability are clarified. In the Görtler-dominated transition process, the most dangerous wavelength of the primary Görtler instability give rise to the largest growth rate of the secondary instability. The role of the Görtler vortices on the blunt compression/flared cone 2 is shown to be minor compared with the Mack mode. For the blunt compression wedge, it is the Görtler mode that becomes responsible for the flow transition as the Mack mode is considerably suppressed.