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.

We investigate the hydrodynamic stability of compressible boundary layers over adiabatic walls with fluids at supercritical pressure in the proximity of the Widom line (also known as the pseudo-critical line). Depending on the free-stream temperature and the Eckert number that determines the viscous heating, the boundary-layer temperature profile can be either sub-, trans-or supercritical with respect to the pseudo-critical temperature, T pc . When transitioning from sub-to supercritical temperatures, a seemingly continuous phase change from a compressible liquid to a dense vapour occurs, accompanied by highly non-ideal changes in thermophysical properties. Using linear stability theory (LST) and direct numerical simulations (DNS), several key features are observed. In the sub-and supercritical temperature regimes, the boundary layer is substantially stabilized the closer the free-stream temperature is to T pc and the higher the Eckert number. In the transcritical case, when the temperature profile crosses T pc , the flow is significantly destabilized and a co-existence of dual unstable modes (Mode II in addition to Mode I) is found. For high Eckert numbers, the growth rate of Mode II is one order of magnitude larger than Mode I. An inviscid analysis shows that the newly observed Mode II cannot be attributed to Mack's second mode (trapped acoustic waves), which is characteristic in high-speed boundary-layer flows with ideal gases. Furthermore, the generalized Rayleigh criterion (also applicable for non-ideal gases) unveils that, in contrast to the trans-and supercritical regimes, the subcritical regime does not contain an inviscid instability mechanism.

The objective of this work is to investigate linear modal and algebraic instability in Poiseuille flows with fluids close to their vapour-liquid critical point. Close to this critical point, the ideal gas assumption does not hold and large non-ideal fluid behaviours occur. As a representative non-ideal fluid, we consider supercritical carbon dioxide (CO 2 ) at pressure of 80 bar, which is above its critical pressure of 73.9 bar. The Poiseuille flow is characterized by the Reynolds number (Re = ρ * w u * r h * /µ * w ), the product of Prandtl (Pr = µ * w C * pw /κ * w ) and Eckert number (Ec = u * 2 r /C * pw T * w ), and the wall temperature that in addition to pressure determines the thermodynamic reference condition. For low Eckert numbers, the flow is essentially isothermal and no difference with the well-known stability behaviour of incompressible flows is observed. However, if the Eckert number increases, the viscous heating causes gradients of thermodynamic and transport properties, and non-ideal gas effects become significant. Three regimes of the laminar base flow can be considered, subcritical (temperature in the channel is entirely below its pseudo-critical value), transcritical, and supercritical temperature regime. If compared to the linear stability of an ideal gas Poiseuille flow, we show that the base flow is more unstable in the subcritical regime, inviscid unstable in the transcritical regime, while significantly more stable in the supercritical regime. Following the corresponding states principle, we expect that qualitatively similar results will be obtained for other fluids at equivalent thermodynamic states.

Abstract

No abstract

No abstract

The linear and nonlinear development of the Görtler vortices in hypersonic boundary layer flows with the consideration of multi-Görtler-modes are investigated using the normal mode analysis and the parabolized stability equations. The linear local analysis predicted the crossover of the multiple Görtler modes (between the trapped-layer mode and the primary wall-layer mode) when the Görtler number and Mach number are large enough, e.g. Ma≥4, G>2000. The Ma=4 case is verified with the linear marching analysis. Five particular regions can be distinguished depending on the behavior of the multi modes. The nonlinear analysis is performed for a Ma=6 case proposed by Li et al. 1 The saturation of the disturbances is reached and the mushroom structures are generated. The multi-modecrossover is also observed with nonlinear effects.

This work aims to investigate the anisotropic fracture and energy dissipation characteristics of marbles cored along an angle of 0°, 30°, 60° and 90° with respect to interbed planes and subjected to multilevel cyclic loading conditions. Rock fatigue deformation, strength, lifetime and dissipated energy first decrease and then increase with increasing interbed orientation, and they get to the minimum for the sample at a 30° interbed orientation. Rock stiffness degradation is significant with the increase of cyclic level, and the stiffness evolution is affected by the interbed structure. The incremental rate of dissipated energy becomes faster with increase of cyclic loading level, and it presents an abrupt increasing trend at the last cyclic loading level. A damage evolution model was first established based on dissipated energy to describe the two‐phase damage accumulation characteristics. It suggests that the proposed model fits to the testing data well and favourably represents the non‐linear characteristics of damage accumulation.