Gortler vortices: a backward-in-time approach to the receptivity problem

A quantitative definition of the receptivity of the Gortler instability is given by the Green's functions that external disturbances must be scalarly multiplied by in order to yield the amplitude of the most amplified instability mode, defined sufficiently far downstream of the plate's leading edge. These Green's functions tone for each kind of external disturbance, either coming from the free stream or from the wall) are here displayed for the first time. Calculating such functions from a numerical solution of the instability equations would require repeating the calculation for each of a complete set df different initial and boundary conditions; although numerical simulations of the Gortler instability abound in the literature, such a systematic screening has never been attempted. Here, instead, we calculate the Green's functions directly from a numerical solution of the adjoint of the linearized boundary-layer equations, which exploits the fact that the direct and adjoint parabolic problems have opposite directions of stable time-like evolution. The Green's functions can thus be obtained by marching backward in time at the same computational cost as a single forward-in-time integration of the direct problem. The backward-in-time technique is not limited to the Gortler problem; quantitative receptivity calculations for other types of instability can easily be envisioned.