Maximum spatial growth of Görtler vortices

We use a numerical optimization technique to determine the optimal forcing, applied at a particular upstream location xo called the inlet, that will maximize the amplitude of Goertler vortices at some downstream location x in a Blasius boundary layer on a concave wall. The spatial evolution of Goertler vortices is ruled by a parabolic system of equations that is numerically solved using the downstream marching technique proposed by Hall. For this parabolic system the perturbation velocity profiles at a given station x is related to the perturbation velocity profiles enforced at the inlet xo trough an operator that we call the spatial propagator of the system. We build a numerical approximation of the spatial propagator by post-processing numerical simulations obtained with a set of linearly independent inlet perturbations. To quantify spatial gain of Goertler vortices we used an Euclidean norm on the state variables of our systems (u,v), we then numerically computed the optimal inlet perturbations and the optimal gain by evaluating the Euclidean norm of the discretized spatial propagator. Recently Luchini & Bottaro analyzed the far downstream receptivity of Goertler vortices to freestream disturbances and wall roughness by backward-integrating the adjoint of the parabolic system governing the spatial evolution. Their technique is able to find optimal perturbations that maximize the gain far enough downstream, when all spatial transients extinguished and the most amplified mode has emerged. The technique we used extends these results also to short range optimizations including spatial transient effects. A similar "all-range" analysis has been very recently developed by Andersson et al., and Luchini on a different probelm, the Blasius boundary layer on a flat plate, using a different technique to evaluate the optimal perturbations based on a series of coupled downstream marching integrations of the parabolic operator and backward integration of the adjoint. The main advantage of the technique we used, compared to the cited ones, is that, being not based on marching integrations in x, it can be extended to the analysis of non-parabolic systems such as the complete Navier Stokes equations. Concerning the streamwise evolution of the velocity field our results confirm those of Day et al.: the growth rates and the perturbation profiles, obtained from different inlet forcing, converge as they evolve sufficiently downstream. It is however found that the total gain experienced by a particular perturbation strongly depends on the chosen inlet forcing so that transition predictions based on critical gains, like the widely used eN method, could be very sensitive to the chosen inlet forcing. The optimal upstream forcings we found strongly differ from both the most amplified mode, that emerges far downstream, and from all upstream forcings previously used in the literature. The use of optimal upstream forcing could substantially move upstream the point where critical amplifications are reached and lead to improved theoretical and numerical transition predictions using eN methods. It is recognized that non-parallel effects play an essential role in the Goertler instability and indeed the non-parallelism of the base flow must be taken into account to correctly describe the propagator of the system. The non-parallelism, however, is not necessary to generate a similar dynamics and we propose an alternative interpretation based on the non-normality of the spatial propagator which exploits the possibility of strong and long transient amplifications. These non-normal effects have been pointed out on a straightforward 2 x 2 model system. The gain curves of that system bear similarities to the one computed for Goertler instability. The toy model is also able to mimic both the strong sensibility, yet pointed out by Hall for the Goertler problem, of the gain curve to the particular inlet perturbation used, and the lack of similarity between optimal upstream forcing and the downstream most amplified modes due to the strong difference bewteen the most amplified mode of the most amplified adjoint mode of the spatial propagator.