Reduced-order Galerkin models of plane Couette flow

Reduced-order models were derived for plane Couette flow using Galerkin projection, with orthonormal basis functions taken as the leading controllability modes of the linearised Navier-Stokes system for a few low wavenumbers. Resulting Galerkin systems comprise ordinary differential equations, with a number of degrees of freedom ranging from 144 to 600, which may be integrated to large times without sign of numerical instability. The reduced-order models so obtained are also found to match statistics of direct numerical simulations at Reynolds number 500 and 1200 with reasonable accuracy, despite a truncation of orders of magnitude in the degrees of freedom of the system. The present models offer thus an interesting compromise between simplicity and accuracy in a canonical wall-bounded flow, with relatively few modes representing coherent structures in the flow and their dominant dynamics.Comment: 10 pages, 4 figure

Transition to chaos in a reduced-order model of a shear layer

The present work studies the non-linear dynamics of a shear layer, driven by a body force and confined between parallel walls, a simplified setting to study transitional and turbulent shear layers. It was introduced by Nogueira \& Cavalieri (J. Fluid Mech. 907, A32, 2021), and is here studied using a reduced-order model based on a Galerkin projection of the Navier-Stokes system. By considering a confined shear layer with free-slip boundary conditions on the walls, periodic boundary conditions in streamwise and spanwise directions may be used, simplifying the system and enabling the use of methods of dynamical systems theory. A basis of eight modes is used in the Galerkin projection, representing the mean flow, Kelvin-Helmholtz vortices, rolls, streaks and oblique waves, structures observed in the cited work, and also present in shear layers and jets. A dynamical system is obtained, and its transition to chaos is studied. Increasing Reynolds number $Re$ leads to pitchfork and Hopf bifurcations, and the latter leads to a limit cycle with amplitude modulation of vortices, as in the DNS by Nogueira \& Cavalieri. Further increase of $Re$ leads to the appearance of a chaotic saddle, followed by the emergence of quasi-periodic and chaotic attractors. The chaotic attractors suffer a merging crisis for higher $Re$, leading to chaotic dynamics with amplitude modulation and phase jumps of vortices. This is reminiscent of observations of coherent structures in turbulent jets, suggesting that the model represents dynamics consistent with features of shear layers and jets.Comment: 28 pages, 18 figure

Resolvent-based analysis of streaks in turbulent jets

Large scale, elongated structures, similar those ones widely studied in wall-bounded flows, are also present in turbulent jets. Several characteristics of these streaks can be identified via reduced order models such as resolvent analysis. The present work involves a resolvent-based study of these structures in turbulent jets. We focus on obtaining the optimal forcing that generates these energetic coherent structures. Results are compared with experimental data post-processed using spectral proper orthogonal decomposition, allowing us to draw conclusions about the nature of the non-linear forcing, since the two analyses should provide equivalent results if this term is modelled as spatially white. By identifying streaks in a global framework, we expect to better understand the mechanism by which they are generated

The colour of forcing statistics in resolvent analyses of turbulent channel flows

QC 20200511</p

The colour of forcing statistics in resolvent analyses of turbulent channel flows

QC 20200511</p