Abstract: An adjoint-based method is presented for determining manufacturing tolerances for aerodynamic surfaces with natural laminar flow subjected to wavy excrescences. The growth of convective unstable disturbances is computed by solving Euler, boundary layer, and parabolized stability equations. The gradient of the kinetic energy of disturbances in the boundary layer (E) with respect to surface grid points is calculated by solving adjoints of the governing equations. The accuracy of approximations of ΔE, using gradients obtained from adjoint, is investigated for several waviness heights. It is also shown how second-order derivatives increase the accuracy of approximations of ΔE when surface deformations are large. Then, for specific flight conditions, using the steepest ascent and the sequential least squares programming methodologies, the waviness profile with minimum L2-norm that causes a specific increase in the maximum value of N- factor, ΔN, is found. Finally, numerical tests are performed using the NLF(2)-0415 airfoil to specify tolerance levels for ΔN up to 2.0 for different flight conditions. Most simulations are carried out for a Mach number and angle of attack equal to 0.5 and 1.25^∘, respectively, and with Reynolds numbers between 9×10⁶ and 15×10⁶ and for waviness profiles with different ranges of wavelengths. Finally, some additional studies are presented for different angles of attack and Mach numbers to show their effects on the computed tolerances. Graphic abstract: (Figure presented.).

The instability characteristics and laminar-Turbulent transition of a series of laminar separation bubbles (LSBs) formed due to a single sinusoidal surface waviness are investigated in the absence of external disturbances or forcing. A scaling based on the geometrical parameters of the waviness and flow Reynolds number is found that enables the prediction of flow separation on the wall leeward side. The analysis of three-dimensional instabilities of two-dimensional base flows reveals a relation between the number of changes in the curvature sign of the recirculating streamlines and the number of unstable centrifugal modes that coexist for the same flow. When multiple curvature changes occur, in addition to the usual steady mode reported for two-dimensional recirculation bubbles, a new self-excited mode with a higher growth rate emerges, localised near the highest streamline curvature, close to the reattachment point. A detailed analysis of the mode growth and saturation using DNS reveals that the localised mode only disturbs the LSB locally, while the usual one leads to a global distortion of the bubble in the spanwise direction; this has a distinctive impact on the self-excited secondary instabilities. Then, the complete transition scenario is studied for two selected LSB cases. The first one only presents an unstable eigenmode, namely the usual centrifugal mode in recirculating flows. The second case presents three unstable eigenmodes: Two centrifugal eigenmodes (the usual and the localised ones) and a two-dimensional eigenmode associated with the self-sustained Kelvin-Helmholtz waves. These results show how completely different transition scenarios can emerge from subtle changes in the LSB characteristics.

Linear global stability analysis is performed on a laminar separation bubble formed dueto surface waviness. The eigenspectrum shows a globally unstable mode, responsiblefor the three-dimensionalisation of the bubble, and a family of low-frequency globallystable modes. An adjoint sensitivity analysis shows high sensitivity of the stable modesupstream of the bubble’s reattachment point. Direct numerical simulation (DNS)alongside with a linear impulse response analysis are performed. DNS shows that, whentransition occurs due to self-excited mechanisms, low-frequency upstream propagatingwaves form inside the bubble; this is not the case in linear impulse analysis. It isconjectured that these upstream propagating waves correspond to the low-frequencystable modes in the spectrum which become active through nonlinearity when transitionoccurs.

The effect of a smooth surface hump on laminar–turbulent transition over a swept-wing is investigated using direct numerical simulation (DNS), and results are compared with wind tunnel measurements. When the amplitude of incoming crossflow (CF) perturbation is relatively low, transition in the reference (without hump) case occurs near 53% chord, triggered by the breakdown of type I secondary instability. Under the same conditions, no transition is observed in the hump case within the DNS domain, which extends to 69% chord. The analysis reveals a reversal in the CF velocity component downstream of the hump’s apex. Within this region, the structure and orientation of CF perturbations are linearly altered, particularly near the wall. These perturbations gradually recover their original state further downstream. During this recovery phase, the lift-up mechanism is weakened, reducing linear production, which stabilises the stationary CF perturbations and weakens spanwise gradients. Consequently, the neutral point of high-frequency secondary CFI modes shifts downstream relative to the reference case, leading to laminar-turbulent transition delay in the presence of the surface hump. In contrast, when the amplitude of the incoming CF perturbation is relatively high, a pair of stationary counter-rotating vortices forms downstream of the hump due to strong destabilisation of higher harmonics of the primary CF instability mode within the CF reversal region. These vortices locally deform the boundary layer and generate regionsof elevated spanwise shear. The growth of secondary instabilities in these high-shear regions leads to rapid advancement of transition toward the hump, in agreement with experimental observations.

This work demonstrates that linear stability analysis can be used to study the effect of a smooth hump on secondary instabilities in incompressible swept-wing boundary layers. Two-dimensional Local Stability Theory (LST-2D) and three-dimensional Parabolised Stability Equations (PSE-3D) are employed to investigate how secondary crossflow instabilities are affected by the presence of a hump, compared to the reference (without hump) case. Comparisons between PSE-3D and DNS results, for the dominant instability induced by the hump, show good agreement. These findings confirm the capability of PSE-3D as an efficient tool for analysing secondary instabilities in boundary layers affected by smooth surface humps.

This study investigates the interactions between stationary crossflow (CF) vortices and smooth surface hump geometries on a swept wing using direct numerical simulations(DNS) and linear stability analysis, under flow conditions similar to those in the experimental study by Rius-Vidales et al. (J. Fluid Mech., vol. 1014, 2025, A35). To analyse the secondary instabilities of the resulting three-dimensional boundary layers, two-dimensional linear stability theory and three-dimensional parabolised stability equations, formulated in a generalised non-orthogonal coordinate system, are employed. It is shown that the induced adverse pressure gradient (APG) downstream of the hump scales almost linearly with the hump slope at its inflection point. Increasing APG, either by narrowing the hump or by increasing its height, progressively weakens, and may eventually reverse the CF velocity component downstream of the hump’s apex. It is further shown that increasing the APG leads to a local amplification of stationary primary disturbances immediately downstream of the hump, which, in the absence of unsteady perturbations, can be followed by a more pronounced stabilisation of primary CF modes further downstream. However, stationary counter-rotating vortex pairs may form downstream of the humps, locally enhancing spanwise shear. Secondary stability analyses reveal that a mode originating within the CF reversal region can undergo rapid amplification over a short distance, with its growth strongly dependent on the local modulation of the spanwise shear induced by the counter-rotating vortex pair. Finally,it is shown that properly designed double-hump configurations can effectively suppress both primary and secondary CF instabilities, offering a promising passive control strategy for delaying laminar-turbulent transition in swept-wing boundary layers.

This study examines the transition to turbulence downstream of fluttering and nonfluttering bioprosthetic aortic valves using global linear stability theory. During systole, increasing inflow velocities result in temporally evolving flow profiles downstream of the valve which are highly influenced by the leaflet kinematics. These profiles are time-averaged at the sinotubular junction over successive windows and used as boundary conditions to obtain base flows for stability analysis. Three-dimensional global modes are computed for one design of each valve type across multiple time windows, revealing several unstable modes whose frequencies and growth rates increase over time. Notably, the non-fluttering valve exhibits higher growth rates than the fluttering valve. The resulting eigenspectra show that, for each case, the most unstable eigenvalues align along two distinct parabolic branches in the complex plane. For each valve case, the modes within each branch are found to have similar group velocities, suggesting that the unstable modes along a branch constitute a coherent structure. Motivated by this, a transient growth analysis is conducted to identify the optimal initial perturbations that maximize energy gain for a given time horizon. When superimposed onto the base flow, these perturbations generate vortical structures that closely resemble thoseo bserved in fully coupled nonlinear fluid–structure interaction simulations for a similar time-scale as the one used to obtain the optimal perturbations. These results suggest that the optimal perturbations may initiate the shear-layer instabilities responsible for transition to turbulence, providing valuable insight into the underlying mechanisms in the flow fields downstream of bioprosthetic valve designs.