A parabolized set of linear equations is derived, which, in combination with the proposed solution procedure, allows for the study of both non-modal and modal disturbance growth in three-dimensional boundary layers. The method is applicable to disturbance waves whose lines of constant phase are closely aligned with the external streamline. Moreover, strongly growing disturbances may fall outside the scope of application. These equations are used in conjunction with a variational approach to compute optimal disturbances in Falkner Skan Cooke boundary layers subject to adverse and favourable pressure gradients. The disturbances associated with maximum energy growth initially take the form of streamwise vortices which are tilted against the mean crossflow shear. While travelling downstream these vortical structures rise into an upright position and evolve into bent streaks. The physical mechanism responsible for non-modal growth in three-dimensional boundary layers is therefore identified as a combination of the lift-up effect and the Orr mechanism. Optimal disturbances smoothly evolve into crossflow modes when entering the supercritical domain of the flow. Non-modal growth is thus found to initiate modal instabilities in three-dimensional boundary layers. Optimal growth is first studied for stationary disturbances. Influences of parameters such as sweep angle, spanwise wavenumber and position of inception are studied, and the initial optimal amplification of stationary crossflow modes because of non-modal growth is investigated. Finally, general disturbances are considered, and envelopes yielding the maximum growth at each position are computed. In general, substantial growth is already found upstream of the first neutral point. The computations show that at supercritical conditions, maximum growth of optimal disturbances in accelerated boundary layers can exceed the growth predicted for modal instabilities by several orders of magnitude.

This paper represents a continuation of the work by Tempelmann et al. (JFM 2010) on spatial optimal growth in incompressible boundary layers over swept flat plates . We present an extension of the methodology to compressible flow. Also we account for curvature effects. Spatial optimal growth is studied for boundary layers over both flat and curved swept plates with adiabatic and cooled walls. We find that optimal growth increases for higher Mach numbers. In general, extensive non-modal growth is observed for all boundary layer cases even in subcritical regions, i.e. where the flow is stable with respect to modal crossflow disturbances. Wall cooling, though stabilising crossflow modes, destabilises disturbances of non-modal nature. Curvature acts similarly on modal as well as non-modal disturbances. Convex walls have a stabilising effect on the boundary layer whereas concave walls act destabilising. The physical mechanisms of optimal growth in all studied boundary layers are found to be similar to those identified for incompressible flat-plate boundary layers.

The receptivity to localised surface roughness of a swept-wing boundary layer is studied by direct numerical simulation (DNS) and computations using the parabolised stability equations (PSE). The DNS is laid out to reproduce wind tunnel experiments performed by Saric & coworkers, where micron-sized cylinders were used to trigger steady crossflow modes. The amplitudes of the roughness-induced fundamental crossflow wave and its superharmonics obtained from nonlinear PSE solutions agree excellently with the DNS results. A receptivity model using the direct and adjoint PSE is shown to provide reliable predictions of the receptivity to roughness cylinders of different heights and chordwise locations. Being robust and computationally efficient, the model is well suited as a predictive tool of receptivity in flows of practical interest. The crossflow mode amplitudes obtained based on both DNS and PSE are 40% of those measured in the experiments.Additional comparisons between experimental and PSE data for various disturbance wavelengths reveal that the measured disturbance amplitudes are consistently larger than those predicted by the PSE-based receptivity model by a nearly constant factor. Supplementary DNS and PSE computations suggest that possible natural leading-edge roughness and free-stream turbulence in the experiments are unlikely to account for this discrepancy. It is more likely that experimental uncertainties in the streamwise location of the roughness array and cylinder height are responsible for the additional receptivity observed in the experiments.

Adjoint solutions of the linearized incompressible Navier-Stokes equations are presented for a cross-flow-dominated swept-wing boundary layer. For the first time these have been computed in the region upstream of the swept leading edge and may therefore be used to predict receptivity to any disturbances of the incoming free stream as well as to surface roughness. In this paper we present worst-case scenarios, i.e. those external disturbances yielding maximum receptivity amplitudes of a steady cross-flow disturbance. In the free stream, such an 'optimal' disturbance takes the form of a streak which, while being convected downstream, penetrates the boundary layer and smoothly turns into a growing cross-flow mode. The 'worst-case' surface roughness has a wavy shape and is distributed in the chordwise direction. It is shown that, under such optimal conditions, the boundary layer is more receptive to surface roughness than to incoming free stream disturbances.