A new method, enabling the computation of steady solutions of the Navier-Stokes equations in globally unstable configurations, is presented. We show that it is possible to reach a steady state by damping the unstable (temporal) frequencies. This is achieved by adding a dissipative relaxation term proportional to the high-frequency content of the velocity fluctuations. Results are presented for cavity-driven boundary-layer separation and a separation bubble induced by an external pressure gradient. © 2006 American Institute of Physics

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. © 2011 Cambridge University Press

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. © 2013 Cambridge University Press

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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 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 Grtler type. © 2009 Cambridge University Press

© 2015 AIP Publishing LLC.The control of flow around a canonical airfoil-like geometry with laminar separation bubble is analyzed using linear stability theory. The theoretical predictions are compared to data from Navier-Stokes simulations [Kotapati et al., "Nonlinear dynamics and synthetic-jet-based control of a canonical separated flow," J. Fluid Mech. 654, 65-97 (2010)], in which the flow was controlled through a zero-net-mass-flux actuator. Very good agreement between the two approaches is found for a range of frequencies from low to high relative to the most dominant frequency for convective instability. The uncontrolled case exhibits periodic vortex shedding from the separation bubble due to an absolute instability. Linear modes with intermediate frequencies are found to exhibit strongest convective amplification, and forcing at these frequencies is most effective in order to reduce the size and extent of the separation bubble. The corresponding physical mechanism relies on a Kelvin-Helmholtz instability of the separated shear layer in conjunction with the non-linear effect of the mean flow deformation. For low frequencies, the front part of the bubble still diminishes due to the interaction of a vortex that starts from the actuator with the wall. This vortex transiently amplifies downstream due to the Orr mechanism. Actuation at high frequencies leads to visible, amplified instability waves in the shear layer, but is not effective in reducing the size of the bubble