Growth and wall-transpiration control of nonlinear unsteady Gortler vortices forced by free-stream vortical disturbances

The generation, nonlinear evolution, and wall-transpiration control of unsteady Gortler vortices in ¨ an incompressible boundary layer over a concave plate is studied theoretically and numerically. Gortler rolls are initiated and driven by free-stream vortical perturbations of which only the low- ¨ frequency components are considered because they penetrate the most into the boundary layer. The formation and development of the disturbances are governed by the nonlinear unsteady boundaryregion equations with the centrifugal force included. These equations are subject to appropriate initial and outer boundary conditions, which account for the influence of the upstream and free-stream forcing in a rigorous and mutually consistent manner. Numerical solutions show that the stabilizing effect on nonlinearity, which also occurs in flat-plate boundary layers, is significantly enhanced in the presence of centrifugal forces. Sufficiently downstream, the nonlinear vortices excited at different free-stream turbulence intensities Tu saturate at the same level, proving that the initial amplitude of the forcing becomes unimportant. At low Tu, the disturbance exhibits a quasi-exponential growth with the growth rate being intensified for more curved plates and for lower frequencies. At higher Tu, in the typical range of turbomachinery applications, the Gortler vortices do not undergo a modal stage ¨ as nonlinearity saturates rapidly, and the wall curvature does not affect the boundary-layer response. Good quantitative agreement with data from direct numerical simulations and experiments is obtained. Steady spanwise-uniform and spanwise-modulated zero-mass-flow-rate wall transpiration is shown to attenuate the growth of the Gortler vortices significantly. A novel modified version of the Fukagata- ¨ Iwamoto-Kasagi identity, used for the first time to study a transitional flow, reveals which terms in the streamwise momentum balance are mostly affected by the wall transpiration, thus offering insight into the increased nonlinear growth of the wall-shear stress.