Selective Frequency Damping (SFD) is a popular method for the computation of globally unstable steady-state solutions in fluid dynamics. The approach has two model parameters whose selection is generally unclear. In this article, a detailed analysis of the influence of these parameters is presented, answering several open questions with regard to the effectiveness, optimum efficiency and limitations of the method. In particular, we show that SFD is always capable of stabilising a globally unstable systems ruled by one unsteady unstable eigenmode and derive analytical formulas for optimum parameter values. We show that the numerical feasibility of the approach depends on the complex phase angle of the most unstable eigenvalue. A numerical technique for characterising the pertinent eigenmodes is presented. In combination with analytical expressions, this technique allows finding optimal parameters that minimise the spectral radius of a simulation, without having to perform an independent stability analysis. An extension to multiple unstable eigenmodes is derived. As computational example, a two-dimensional cylinder flow case is optimally stabilised using this method. We provide a physical interpretation of the stabilisation mechanism based on, but not limited to, this Navier-Stokes example.

We study the interaction between a stationary crossflow instability and forward-facing steps in a swept-wing boundary layer using Direct Numerical Simulations (DNS). The stationary primary crossflow mode is imposed at the inflow. Steps of several heights are modeled. Particular emphasis is placed on ensuring a fully stationary solution, in order to isolate the modulation of the primary instability at the step and the generation of stationary secondary perturbations. The main features of the base flow are step-induced flow reversal and secondary inflection points. When approaching the step, the incoming crossflow instability lifts up and passes over it. Additional perturbation streaks arise, which are accompanied by secondary stationary vortices that have the same spanwise wavenumber as the fundamental crossflow disturbance. For sufficiently high steps, secondary perturbation structures attain amplitude values comparable in magnitude to primary instability in the vicinity of the step. We propose metrics to quantify the impact of the step on the development of the crossflow disturbance upon interaction. Considering energy-based criteria, we find that the crossflow perturbation is amplified upstream of the step. Interestingly, the incoming primary crossflow instability undergoes strong stabilization immediately downstream of the largest step. The mechanisms revealed in this work provide a first insight into the possible causes of transition on swept wings due to forward-facing steps.

Micro-ramps are popular passive flow control devices which can delay flow separation by re-energising the lower portion of the boundary layer. We compute the laminar base flow, the instantaneous transitional flow, and the mean flow around a micro-ramp immersed in a quasi-incompressible boundary layer at supercritical roughness Reynolds number. Results of our Direct Numerical Simulations (DNS) are compared with results of BiLocal stability analysis on the DNS base flow and independent tomographic Particle Image Velocimetry (tomo-PIV) experiments. We analyse relevant flow structures developing in the micro-ramp wake and assess their role in the micro-ramp functionality, i.e., in increasing the near-wall momentum. The main flow feature of the base flow is a pair of streamwise counter-rotating vortices induced by the micro-ramp, the so-called primary vortex pair. In the instantaneous transitional flow, the primary vortex pair breaks up into large-scale hairpin vortices, which arise due to linear varicose instability of the base flow, and unsteady secondary vortices develop. Instantaneous vortical structures obtained by DNS and experiments are in good agreement. Matching linear disturbance growth rates from DNS and linear stability analysis are obtained until eight micro-ramp heights downstream of the micro-ramp. For the setup considered in this article, we show that the working principle of the micro-ramp is different from that of classical vortex generators; we find that transitional perturbations are more efficient in increasing the near-wall momentum in the mean flow than the laminar primary vortices in the base flow.

The evolution of secondary instabilities in a three-dimensional stationary-crossflow-dominated boundary layer is investigated by means of Direct Numerical Simulations (DNS) and linear spanwise BiGlobal stability analysis. Single-frequency unsteady disturbances and a critical stationary crossflow mode are considered. Unsteady perturbation content at 1 kHz manifests in the form of the type-III instability mechanism in the lower portion of the boundary layer in both the DNS and the stability approach. Considering disturbances at 6 kHz, the results from the stability analysis reveal the existence of largely amplified type-I and type-II secondary instability mechanisms. Strong growth displayed by the former is measured in the DNS, which potentially overshadows manifestations of the type-II mechanism. Laminar-turbulent transition primarily induced by the growth of type-I disturbances is captured in the 6 kHz case. Overall, we report good agreement between DNS and stability analysis in terms of perturbation organization and growth rate for all cases studied.

The interaction between forward-facing steps of several heights and a pre-existing critical stationary crossflow instability of a swept-wing boundary layer is analysed. Direct numerical simulations (DNS) are performed of the incompressible three-dimensional laminar base flow and the stationary distorted flow that arise from the interaction between an imposed primary stationary crossflow perturbation and the steps. These DNS are complemented with solutions of the linear and the nonlinear parabolised stability equations, used towards identifying the influence of linearity and non-parallelism near the step. A fully stationary solution of the Navier–Stokes equations is enforced numerically, in order to isolate the mechanisms pertaining to the interaction of the stationary disturbance with the step. Results provide insight into the salient modifications of the base laminar boundary layer due to the step, and the response of the incoming crossflow instability to these changes. The fundamental spanwise Fourier mode of the disturbance field gradually lifts up as it approaches the step and passes over it. The flow environment around the step is characterised by a sudden spanwise modulation of the base-flow streamlines. Additional stationary perturbation structures are induced at the step, which manifest in the form of spanwise-aligned velocity streaks near the wall. Shortly downstream of the step, the fundamental component of the crossflow perturbation maintains a rather constant amplification for the smallest steps studied. For the largest step, however, the fundamental crossflow perturbation is stabilised significantly shortly downstream of the largest step. This surprising result is ascribed to a modulation of the kinetic energy transfer between the base flow and the fundamental perturbation field, which is brought forward as a new step interaction mechanism. Possible non-modal growth effects at the step are discussed. Furthermore, the results from DNS indicate significant amplification of the high-order harmonic crossflow components downstream of the step.