Nonlinear mode selection in a model of trailing line vortices

Abstract

Nonlinear mode selection, from initial random Gaussian field perturbations, in a model of trailing line vortices (swirling jets), in the breakdown regime, is addressed by direct numerical simulations with a Reynolds number equal to 1000. A new concept of mode activity in the nonlinear evolution is introduced. The selected modes, according to their activities, are reported and related to strain eigenvectors (with maximum eigenvalues) of the basic flow corresponding to the trailing line vortex under consideration. The selected modes are also related to results from the linear eigenmode (exponential growth) instability theory using the concept of dispersion relation envelope. It is found that the global mode hypothesis of the linear eigenmode theory is violated near the flow axis when the swirl number increases. However, far from the flow axis the linear eigenmode theory is in good agreement with the nonlinear evolution in the breakdown regime. The discrepancy between the nonlinear evolution and the linear eigenmode theory is related to the transient growth of optimal perturbations resulting from the non-normality of the linearized Navier–Stokes equations about shear flows. A clear distinction between an eigenmode, an optimal perturbation (non-modal) and a direct numerical simulation (DNS) mode is made. It is shown that the algebraic (transient) growth contributions from the inviscid continuous spectrum could trigger nonlinearities near the flow axis. The DNS mode selected in the nonlinear regime coincides with the long-wave eigenmode benefiting from the algebraic growth in the linear regime. This eigenmode is different from the short-wave eigenmode with the absolute maximum exponential growth. Although it is promoted by transients, in the linear regime, the long-wave component is selected nonlinearly.