Recall that the first aim of this thesis was to assess whether we can derive bi-orthogonality relations demonstrating that the adjoint LST eigenvectors are orthogonal to the direct eigenvectors, for both the compressible and incompressible LST equations; thereby resolving the non-orthogonality complication. It is to that end that we derived the adjoint operators for the incompressible and compressible LST equations using both the discrete and the continous approach in chapter 3. Subsequently, we derived, implemented and assessed bi-orthogonality relations using the adjoint eigenvectors resulting from the discrete and continous approaches in chapter 3. It was found that theoretically relations can be derived for which a weight matrix ensures bi-orthogonal sets of direct and adjoint eigenvectors. Numerical implementation of these biorthogonality relations for both the incompressible and compressible LST yielded diagonal biorthogonality coefficient matrices for the discrete case. Results for the continous approach have shown that the coefficient matrices are diagonally dominant, but do contain substantial offdiagonal terms. As per the theoretical bi-orthogonality derivation, the complex conjugates of the adjoint eigenvalues were shown to match the direct eigenvalues in the physically interesting range, for both the incompressible and compressible LST spectra. The discrete and continous adjoint eigenmodes were found to be similar in shape and location, but the modes did not coincide. A potential explanation for the discrepancy might be the use of the Chebyshev collocation nodes in the continous adjoint EVP. The adjoint EVP may require different stretching of the mesh. Alternatively, closing the continuous adjoint system using the ’adjoint wall-normal equation’ as a compatiblity condition for the adjoint pressure amplitude might possibly adversely affect the continous adjoint eigenmodes. An interesting observation was made when considering the spatial structure defining the overlap between direct eigenmodes and discrete adjoint eigenmodes. For both a wall mode, and a ’middle region’ mode, the location of the maximum amplitude of the direct eigenmode was found to differ spatially from that of the maximum amplitude of the discrete adjoint eigenmode.

Micro-ramps are passive flow control devices used to delay flow separation. Their use is widespread due to their reduced drag and structural robustness. We reproduce with Direct Numerical Simulations (DNS) recent Particle Image Velocimetry (PIV) experiments of the micro-ramp flow performed at TU Delft to study the wake of a micro-ramp immersed in a laminar and incompressible boundary layer. The micro-ramp is a vortex generator which induces a pair of streamwise counter-rotating vortices. The current literature identifies this structure as the main flow feature contributing to the increase of the near-wall momentum. The micro-ramp is also a surface roughness element which can trigger laminar-turbulent transition. The action of the induced vortices introduces a strong detached shear layer into the flow field, susceptible to Kelvin-Helmholtz (K-H) instability. We analyse the micro-ramp flow dynamics and the transitional mechanisms which develop in the micro-ramp wake. Furthermore, we intend to contribute to the discussion on the micro-ramp working principle, which has been put into question by other authors. We show the importance of the transitional perturbation development in the micro-ramp functionality. %In the past decade, numerous computational and experimental studies performed inquired into the topology of the flow behind a micro-ramp and its relation to the functionality of the device.

Downstream-travelling streamwise vortices and transitional disturbances serve to the same purpose of increasing the momentum close to the surface. To examine their relative contribution in this regard, we numerically decompose the micro-ramp flow field into a laminar steady state and a time-dependant perturbation field. To achieve that, we apply Selective Frequency Damping (SFD), a numerical technique used to compute the steady solutions of globally unstable dynamical systems. SFD is a popular method nowadays and the preferred approach for aerospace applications. However, it has two case-dependant model parameters which are key to the method's effectivity and efficiency, and whose selection remains a challenge in the literature. Not every combination of the model parameters guarantees the success of SFD, and even if so, the required computational time may be so large that the approach is impractical. We provide the first rigorous analysis of the influence of these parameters to the functionality of SFD, leading to simple expressions and procedures for choosing them optimally. Furthermore, we prove that, under certain conditions, SFD is always able to stabilise a globally unstable flow configuration.