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

Abstract

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 =(γx+(1/S(t)) erf(y/a(t)),−γy,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 γ (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, t f ) are computed for various t f , γ and Re. For each case, the results are compared with those obtained by considering a single local normal mode (WKBJ approach). For small t f ( t f <10) and large Reynolds numbers, transient effects associated with non-modal growth are clearly visible: they favor large wavenumbers which are locally stable. However, they are found not to provide important energy gains. Moreover, transient growth is shown not to be significantly affected by stretching and to diminish with viscosity. For larger t f ( t f >20), 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 (<0.025). However, differences concerning the most amplified wavenumbers remain visible and increase with γ. For very large times, stretching moves the local wavenumber toward zero. A non-viscous asymptotic study performed for small k shows that although the perturbation energy ultimately diminishes, it decreases less rapidly than the basic flow energy density. Stretching therefore never stabilizes the shear layer for large Reynolds numbers. The results obtained in the WKBJ framework are also extended to more general configurations including three-dimensional perturbations and triaxial stretching fields.