Stationary and travelling crossflow vortices in three-dimensional boundary layers: nonlinear interactions within a common critical layer
Creators: name: Amos, Alex John
Publisher: name: Imperial College London
Year: 2017

description: Three-dimensional boundary layers, such as that over a swept wing, are known to exhibit a crossflow instability, which manifests itself in the form of stationary vortices and travelling-wave vortices. Despite their smaller linear growth rates, the former tend to be the dominant cause of transition to turbulence when free-stream turbulence is relatively low. Travelling-wave vortices, which have higher growth rates, dominate at elevated levels of free-stream turbulence. Recent experiments found that the development of stationary vortices may be affected by free-stream disturbance levels representative of flight conditions, suggesting that travelling-wave vortices may play a role. It was also observed that travelling-wave vortices may be affected by stationary modes. Prompted by these observations, we carry out a theoretical study of nonlinear mutual interactions between stationary and travelling-wave vortices. In order to fix the idea, the base flow is taken to be a Falkner-Skan-Cooke boundary layer. The eigenvalue problem, consisting of the Rayleigh equation and homogeneous boundary conditions, is solved. It was found that there exists a pair of stationary and travelling vortices that share a common critical level, where the base flow velocity projected to the directions of the wave vectors is equal to the phase speeds of the vortices. This pair is particularly significant because effective nonlinear interactions take place in the critical layer, a thin region surrounding the common critical level. The mutual and self nonlinear interactions are analysed by employing the nonlinear non-equilibrium critical-layer approach. The governing equations and solutions for the disturbances are expanded asymptotically both in the main boundary layer and inside the critical layer. The analysis of the outer expansion determines the eigenmodes at leading-order, and at the next order leads to solvability conditions involving the velocity jumps across the critical layer. The analysis of the inner expansions, in particular of the inter-modal interactions, provides jumps, which are combined with the solvability conditions to obtain the integro-differential amplitude equations. For the Falkner-Skan-Cooke baseflow under consideration, the stationary and travelling vortices interact at the quadratic level to force the sum mode. The latter interacts with the stationary mode to regenerate the travelling vortices. The resulting nonlinear effects cause the travelling vortices to amplify rapidly in the form of a super-exponential growth. It is also noted that alternative forms of interactions, involving the difference mode, may be possible in general, and for other base flows, travelling vortices may cause stationary vortices to amplify super-exponentially instead. The analysis is then extended to include the nonlinear self-interactions of the stationary and travelling vortices, and the resulting integro-differential amplitude equations contain derivatives of the amplitudes within the integrals. The amplitude equations are solved numerically. We also investigated the interactions of travelling vortices with the distortion induced by distributed surface roughness, which is modelled by a wavy wall with its height being constant or spatially modulated. Of importance is the distortion sharing the same critical level of the travelling vortices. The form of the interaction is similar to that found for the stationary and travelling eigenmodes, however the presence of a viscous wall layer and an associated `blowing velocity' alters the interaction. A more realistic case, roughness consisting of a continuum of wavenumbers, which has previously not been considered, is also investigated. The contribution of the interaction is found to be a superposition of the Fourier modes. These amplitude equations are also solved numerically, and the solutions indicate that roughness elements of moderate height can have significant effects on the instability.