Modal growth and non-modal growth in a stretched shear layer

The evolution of two-dimensional linear perturbations in a uniform shear layer stretched along the streamwise direction is considered in this work. The velocity field of the basic flow is assumed to be given by the following exact solution of Navier-Stokes equations U=(x+(1/S(t))erf(y/a(t)),-y,0) where erf is the error function, a(t) and S(t) are time-varying functions. The solution is governed by two parameters: the Reynolds number Re and the stretching rate (non-dimensionalized by the initial maximum vorticity) which is assumed to be a positive constant. Using a direct-adjoint technique, perturbations which maximize the energy gain during a time interval (0,tf) are computed for various tf, and Re. For each case, the results are compared with those obtained by considering a single local normal mode (WKBJ approach). For small tf (tf20), instability takes over transients: the WKBJ approximation is shown to provide a good estimate of the maximum gain whatever the Reynolds number (>10) and the stretching rate (