Examination of large-scale structures in a turbulent plane mixing layer. Part 2. Dynamical systems model

Abstract

The temporal dynamics of large-scale structures in a plane turbulent mixing layer are studied through the development of a low-order dynamical system of ordinary differential equations (ODEs). This model is derived by projecting Navier–Stokes equations onto an empirical basis set from the proper orthogonal decomposition (POD) using a Galerkin method. To obtain this low-dimensional set of equations, a truncation is performed that only includes the first POD mode for selected streamwise/spanwise (k₁/k₃) modes. The initial truncations are for k₃ = 0; however, once these truncations are evaluated, non-zero spanwise wavenumbers are added. These truncated systems of equations are then examined in the pseudo-Fourier space in which they are solved and by reconstructing the velocity field. Two different methods for closing the mean streamwise velocity are evaluated that show the importance of introducing, into the low-order dynamical system, a term allowing feedback between the turbulent and mean flows. The results of the numerical simulations show a strongly periodic flow indicative of the spanwise vorticity. The simulated flow had the correct energy distributions in the cross-stream direction. These models also indicated that the events associated with the centre of the mixing layer lead the temporal dynamics. For truncations involving both spanwise and streamwise wavenumbers, the reconstructed velocity field exhibits the main spanwise and streamwise vortical structures known to exist in this flow. The streamwise aligned vorticity is shown to connect spanwise vortex tubes.