Global Linear Stability Analysis of Weakly Non-Parallel Shear Flows

The global linear stability of incompressible, two-dimensional shear flows is investigated under the assumptions that far-field pressure feedback between distant points in the flow field is negligible and that the basic flow is only weakly nonparallel, i.e. that its streamwise development is slow on the scale of a typical instability wavelength. This implies the general study of the temporal evolution of global modes, which are time-harmonic solutions of the linear disturbance equations, subject to homogeneous boundary conditions in all space directions. Flow domains of both doubly infinite and semi-infinite streamwise extent are considered and complete solutions are obtained within the framework of asymptotically matched WKBJ approximations. In both cases the global eigenfrequency is given, to leading order in the WKBJ parameter, by the absolute frequency w_o(X^t) at the dominant turning point X^tof the WKBJ approximation, while its quantization is provided by the connection of solutions across X^t. Within the context of the present analysis, global modes can therefore only become time-amplified or self-excited if the basic flow contains a region of absolute instability.