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.

Direct numerical simulation data of bypass transition in flat-plate boundary layers are analysed to examine the characteristics of turbulence in the transitional regime. When intermittency is 50 % or less, the flow features a juxtaposition of turbulence spots surrounded by streaky laminar regions. Conditionally averaged turbulence statistics are evaluated within the spots, and are compared to standard time averaging in both the transition region and in fully turbulent boundary layers. The turbulent-conditioned root-mean-square levels of the streamwise velocity perturbations are notably elevated in the early transitional boundary layer, while the wall-normal and spanwise components are closer to the levels typical for fully turbulent flow. The analysis is also extended to include ensemble averaging of the spots. When the patches of turbulence are sufficiently large, they develop a core region with similar statistics to fully turbulent boundary layers. Within the tip and the wings of the spots, however, the Reynolds stresses and terms in the turbulence kinetic energy budget are elevated. The enhanced turbulence production in the transition zone, which exceeds the levels from fully turbulent boundary layers, contributes to the higher skin-friction coefficient in that region. Qualitatively, the same observations hold for different spot sizes and levels of free-stream turbulence, except for young spots which do not yet have a core region of developed turbulence.

Flows induced by centrifugal buoyancy occur in rotating systems in which the centrifugal force is large when compared to other body forces and are of interest for geophysicists and also in engineering problems involving rapid rotation and unstable temperature gradients. In this numerical study we analyse the onset of centrifugal buoyancy in a rotating cylindrical cavity bounded by two plane, insulated disks, adopting a geometrical configuration relevant to fundamental studies of buoyancy-induced flows occurring in gas turbine’s internal air systems. Using linear stability analysis, we obtain critical values of the centrifugal Rayleigh number $Ra$ and corresponding critical azimuthal wavenumbers for the onset of convection for different radius ratios. Using direct numerical simulation, we integrate the solutions starting from a motionless state to which small sinusoidal perturbations are added, and show that nonlinear triadic interactions occur before energy saturation takes place. At the lowest Rayleigh number considered, the final state is a limit-cycle oscillation affected by the presence of the disks, having a spectrum dominated by a certain mode and its harmonics. We show that, for this case, the limit-cycle oscillations only develop when no-slip end walls are present. For the largest $Ra$ considered chaotic motion occurs, but the critical wavenumber obtained from the linear analysis eventually becomes the most energetic even in the turbulent regime.

The paper describes a numerical investigation of linear and nonlinear instability in high-speed boundary layers. Both a frozen gas and a finite-rate chemically reacting gas are considered. The weakly nonlinear instability in the presence of a large-amplitude two-dimensional wave is investigated for the case of fundamental resonance. Depending on the amplitude of this two-dimensional primary wave, strong growth of oblique secondary perturbations occurs for favourable relative phase differences between the two. For essentially the same primary amplitude, secondary amplification is almost identical for a reacting and a frozen gas. Therefore, chemical reactions do not directly affect the growth of secondary perturbations, but only indirectly through the change of linear instability and hence amplitude of the primary wave. When the secondary disturbances reach a sufficiently large amplitude, strongly nonlinear effects stabilize both primary and secondary perturbations.

Nonlinear instability in a supersonic boundary layer at Mach 4.8 with two-dimensional roughness is investigated by means of spatial direct numerical simulations (DNS). It was previously found that an important effect of a two-dimensional roughness is to increase significantly the amplitude of two-dimensional waves downstream of the roughness in a certain frequency band through enhanced instability and transient growth, while waves outside this band are damped. Here, we investigate the nonlinear secondary instability induced by a large-amplitude two-dimensional wave, which has received a significant boost in amplitude from this additional roughness-induced amplification. Both subharmonic and fundamental secondary excitation of the oblique secondary waves are considered. We found that even though the growth rate of the secondary perturbations increases compared to their linear amplification, only in some of the cases was a fully resonant state attained by the streamwise end of the domain. A parametric investigation of the amplitude of the primary wave, the phase difference between the primary and the secondary waves, and the spanwise wavenumber has also been performed. The transient growth experienced by the primary wave was found to not influence the secondary instability for most parameter combinations. For unfavourable phase relations between the primary and the secondary waves, the phase speed of the secondary wave decreases significantly, and this hampers its growth. Finally, we also investigated the strongly nonlinear stage, for which both the primary and the subharmonic secondary waves had a comparable, finite amplitude. In this case, the growth of the primary waves was found to vanish downstream of the transient growth region, resulting in a lower amplitude than in the absence of the large-amplitude secondary wave. This feedback also decreases the amplification rate of the secondary wave.

The convective primary amplification of a forced two-dimensional perturbation initiates the formation of essentially two-dimensional large-scale vortices in a laminar separation bubble. These vortices are then shed from the bubble with the forcing frequency. Immediately downstream of their formation, the vortices get distorted in the spanwise direction and quickly disintegrate into small-scale turbulence. The laminar–turbulent transition in a forced laminar separation bubble is dominated by this vortex formation and breakup process. Using numerical and experimental data, we give an in-depth characterization of this process in physical space as well as in Fourier space, exploiting the largely periodic character of the flow in time as well as in the spanwise direction. We present evidence that a combination of more than one secondary instability mechanism is active during this process. The first instability mechanism is the elliptic instability of vortex cores, leading to a spanwise deformation of the cores with a spanwise wavelength of the order of the size of the vortex. Another mechanism, potentially an instability of flow in between two consecutive vortices, is responsible for three-dimensionality in the braid region. The corresponding disturbances possess a much smaller spanwise wavelength as compared to those amplified through elliptic instability. The secondary instability mechanisms occur for both fundamental and subharmonic frequency, respectively, even in the absence of continuous forcing, indicative of temporal amplification in the region of vortex formation.

The evolution of two- and three-dimensional small-amplitude disturbances in the laminar part of a laminar separation bubble is investigated in detail. We apply a combination of local linear stability theory, results from different experimental measurement campaigns and direct numerical simulations to identify two different discrete eigenmodes in the laminar part of the bubble. A stable eigenmode, the outer mode, governs unsteady oscillations in the upstream part of the bubble. However, this perturbation is quickly overtaken by an unstable eigenmode, the inner mode, which eventually leads to transition of the detached shear layer. Such a behaviour is observed due to an acceleration region with a favourable pressure gradient preceding the adverse-pressure-gradient region. The flow is stable in the acceleration region, in which the outer mode is only moderately damped, while the inner mode is strongly damped. At the onset of instability for the unstable eigenmode upstream of separation, both viscous Tollmien–Schlichting and inviscid Kelvin–Helmholtz instability mechanisms contribute to amplification, while deeper inside the bubble only the inviscid mechanism is active. If the explicit forcing is moved to a region downstream of the favourable pressure gradient, only the unstable eigenmode appears. The same behaviour is found for two-dimensional and weakly oblique waves.

Short laminar separation bubbles can develop on a flat plate due to an externally imposed pressure gradient. Here, these bubbles are computed by means of direct numerical simulations. Laminar–turbulent transition occurs in the bubble, triggered by small disturbance input with fixed frequency, but varying amplitude, to keep the bubbles short. The forcing amplitudes span a range of two orders of magnitude. All resulting bubbles differ with respect to their mean flow, linear-stability characteristics and distance between transition and mean reattachment locations. Mechanisms responsible for these differences are analysed in detail. Switching off the disturbance input or reducing it below a certain, very small threshold causes the short bubble to grow continuously. Eventually, it no longer exhibits typical characteristics of a short laminar separation bubble. Instead, it is argued that bursting has occurred and the bubble displays characteristics of a long-bubble state, even though this state was not a statistically steady state. This hypothesis is backed by a comparison of numerical results with measurements. For long bubbles, the transition to turbulence is not able to reattach the flow immediately. This effect can lead to the bursting of a short bubble, which remains short only when sufficiently large disturbances are convected into the bubble. Large-scale spanwise-oriented vortices at transition are observed for short but not for long bubbles. The failure of the transition process to reattach the flow in the long-bubble case is ascribed to this difference in transitional vortical structures.

The mutual interaction of laminar–turbulent transition and mean flow evolution is studied in a pressure-induced laminar separation bubble on a flat plate. The flat-plate boundary layer is subjected to a sufficiently strong adverse pressure gradient that a separation bubble develops. Upstream of the bubble a small-amplitude disturbance is introduced which causes transition. Downstream of transition, the mean flow strongly changes and, due to viscous–inviscid interaction, the overall pressure distribution is changed as well. As a consequence, the mean flow also changes upstream of the transition location. The difference in the mean flow between the forced and the unforced flows is denoted the mean flow deformation. Two different effects are caused by the mean flow deformation in the upstream, laminar part: a reduction of the size of the separation region and a stabilization of the flow with respect to small, linear perturbations. By carrying out numerical simulations based on the original base flow and the time-averaged deformed base flow, we are able to distinguish between direct and indirect nonlinear effects. Direct effects are caused by the quadratic nonlinearity of the Navier–Stokes equations, are associated with the generation of higher harmonics and are predominantly local. In contrast, the stabilization of the flow is an indirect effect, because it is independent of the Reynolds stress terms in the laminar region and is solely governed by the non-local alteration of the mean flow via the pressure.

A novel flow configuration devised for investigation of active control of separated airfoil flows using synthetic jets is presented. The configuration consists of a flat plate, with an elliptic leading edge and a blunt trailing edge, at zero incidence in a free stream. Flow separation is induced on the upper surface of the airfoil at the aft-chord location by applying suction and blowing on the top boundary of the computational domain. Typical separated airfoil flows are generally characterized by at least three distinct frequency scales corresponding to the shear layer instability, the unsteadiness of the separated region and the vortex shedding in the wake, and all these features are present in the current flow. Two-dimensional Navier–Stokes simulations of this flow at a chord Reynolds number of 6 × 104 have been carried out to examine the nonlinear dynamics in this flow and its implications for synthetic-jet-based separation control. The results show that there is a strong nonlinear coupling between the various features of the flow, and that the uncontrolled as well as the forced flow is characterized by a variety of ‘lock-on’ states that result from this nonlinear coupling. The most effective separation control is found to occur at the highest forcing frequency for which both the shear layer and the separated region lock on to the forcing frequency. The effects of the Reynolds number on the scaling of the characteristic frequencies of the separated flow and its subsequent control are studied by repeating some of the simulations at a higher Reynolds number of 1 × 105.

A numerical investigation of the disturbance amplification in a Mach 4.8 flat-plate boundary layer with a localized two-dimensional roughness element is presented. The height of the roughness is varied and reaches up to approximately 70% of the boundary-layer thickness. Simulations are based on a time-accurate integration of the compressible Navier–Stokes equations, with a small disturbance of fixed frequency being triggered via blowing and suction upstream of the roughness element. The roughness element considerably alters the instability of the boundary layer, leading to increased amplification or damping of a modal wave depending on the frequency range. The roughness is also the source of an additional perturbation. Even though this additional mode is stable, the interaction with the unstable mode in the form of constructive and destructive interference behind the roughness element leads to a beating and therefore transiently increased disturbance amplitude. Far downstream of the roughness, the amplification rate of a flat-plate boundary layer is recovered. Overall, the two-dimensional roughness element behaves as disturbance amplifier with a limited bandwidth capable of filtering a range of frequencies and strongly amplifying only a selected range.

Steady linear three-dimensional disturbances are investigated in a two-dimensional laminar boundary layer. The boundary layer is subject to a streamwise favourable-to-adverse pressure gradient and eventually undergoes separation. The separating flow corresponds to the first part of a pressure-induced laminar-separation bubble on a flat plate. Streamwise disturbance development in such a flow is studied by means of direct numerical simulation, a water-tunnel experiment and an adjoint-based parabolic theory suited to study spatial optimal growth. A complete overview of the disturbance evolution in various areas of the favourable-to-adverse pressure gradient laminar boundary layer is given. Results from all investigation methods show overall good agreement with respect to disturbance growth and shape within the entire domain. In the favourable pressure-gradient region and, again, slightly downstream of separation, transient growth caused by the lift-up effect dominates disturbance behaviour. In the adverse pressure-gradient region, a modal instability is observed. Evidence is presented that this instability is of Görtler type.