A NONLINEAR-ANALYSIS OF THE GORTLER INSTABILITY

A model system of ordinary differential equations is used as an approximation to the partial differential system governing a small disturbance with spanwise periodicity superimposed on a two-dimensional laminar boundary layer along a concave wall. The linearized version formulates a simple modification of the well-known Gortler equations, including an additional term associated with non-parallelism of the basic flow. The additional term plays an important role outside the boundary layer and the modified system yields a neutral stability curve possessing a critical Gortler number at a finite value of spanwise wavenumber. The formulation is extended to the weakly nonlinear theory to evaluate the Landau coefficients. A new type of singularity is found, which is due to a resonance between the fundamental Fourier component and the second harmonic of disturbances in a supercritical region slightly above the linear critical point. Weakly nonlinear analysis of the resonance indicates very strong growth of the two Fourier components. This fact offers a plausible explanation to the wavenumber selection of disturbances observed in experiments.