Multi-scale invariant solutions in plane Couette flow: a reduced-order model approach

Plane Couette flow at Re=1200 (based on the channel half-height and half the velocity difference between the top and bottom plates) is investigated in the minimal multi-scale flow unit (i.e. a flow unit with only two spanwise integral length scales), a system for which the computation of invariant solutions that are physically representative of the turbulent state has been understood to be challenging. To address this challenge, our approach is to employ an accurate reduced-order model with 600 degrees of freedom (Cavalieri & Nogueira, Phys. Rev. Fluids, vol. 7, 2022, L102601). Using the two-scale energy budget and the temporal cross-correlation of key observables, it is first demonstrated that the model contains most of the multi-scale physical processes identified recently (Doohan et al., J. Fluid Mech., vol. 913, 2021, A8): i.e. the large- and small-scale self-sustaining processes, the energy cascade for turbulent dissipation, and an energy-cascade mediated small-scale production mechanism. Invariant solutions of the reduced-order model are subsequently computed, including 96 equilibria and 43 periodic orbits. It is found that all the computed equilibrium solutions are not able to reproduce sound energy balance associated with the multi-scale dynamics of turbulent state. Incorporation of unsteadiness into invariant solutions is seen to be essential for a sensible description of the multi-scale turbulent dynamics and the related energetics, at least in this type of flow, as periodic orbits with a sufficiently long period are mainly able to describe the complex spatiotemporal dynamics associated with the known multi-scale phenomena

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

Jet-edge interaction tones

Motivated by the problem of jet-flap interaction noise, we study the tonal dynamics that occur when a sharp edge is placed in the hydrodynamic nearfield of an isothermal turbulent jet. We perform hydrodynamic and acoustic pressure measurements in order to characterise the tones as a function of Mach number and streamwise edge position. The distribution of spectral peaks observed, as a function of Mach number, cannot be explained using the usual edge-tone scenario, in which resonance is underpinned by coupling between downstream-travelling Kelvin-Helmholtz wavepackets and upstream-travelling sound waves. We show, rather, that the strongest tones are due to coupling between the former and upstream-travelling jet modes recently studied by Towne et al. (2017) and Schmidt et al. (2017). We also study the band-limited nature of the resonance, showing a high-frequency cut-off to be due to the frequency dependence of the upstream-travelling waves. At high Mach number these become evanescent above a certain frequency, whereas at low Mach number they become progressively trapped with increasing frequency, a consequence of which is their not being reflected in the nozzle plane. Additionally, a weaker, low-frequency, forced-resonance regime is identified that involves the same upstream travelling jet modes but that couple, in this instance, with downstream-travelling sound waves. It is suggested that the existence of two resonance regimes may be due to the non-modal nature of wavepacket dynamics at low-frequency.Comment: 21 pages, 15 figure

Wavepacket eduction in turbulent jets based on eigenmode decomposition of PIV data

The dynamics of large scale structures in unforced turbulent jets at subsonic speeds have been related to the generation of the peak noise radiated the aft direction. The utility of instability wavepackets computed by linear stability theory or parabolised stability equations (PSE) have been demonstrated for the modeling of the near-field pressure fluctuations associated with the coherent structures. In this paper, we investigate whether the velocity field corresponding to the wavepackets also represents adequately that of the coherent structures. Previous research showed remarkable agreement in the velocity field up to the end of the potential core, but the agreement is lost gradually downstream. Locally-parallel linear stability theory (LST) of jet velocity profiles is revisited to further study the evolution of the wavepackets and the manner in which PSE models them. An adjoint-based eigenmode decomposition technique is used to project cross-sectional velocity profiles measured using time-resolved particle image velocimetry (PIV) on the Kelvin-Helmholtz eigenmode responsible for the wavepacket amplification. The instability wave thus extracted is then compared, both in amplification and shape, to the PSE wavepacket and to the dominant coherent structures obtained from the proper orthogonal decomposition of the PIV measurements. The comparisons between PSE models and POD-filtered fluctuations define three spatial regions along the streamwise direction that are explained in terms of changes in the LST eigenspectrum

Widest scales in turbulent channels

The widest spanwise scales in turbulent channel flows are studied through the use of three periodic channel-flow simulations at friction Reynolds number $\mathrm{Re}_{\tau}=550$. The length and height of the channels are the same in all cases ($L_x/h=8\pi$ and $L_y/h=2$ respectively), while the width is progressively doubled: $L_z/h = \{4\pi, 8\pi, 16\pi\}$. The effects of increasing the domain can not be determined with statistical significance in our simulations, since the difference in the statistics between the simulations is of the same order as the errors of convergence. A channel flow similar to the smaller one ($\textit{J. Fluid Mech.}$, vol. 500, 2004, pp. 135--144), which was averaged over a very long time, was used as a reference. The one-dimensional spanwise spectrum of the streamwise velocity is computed with the aim of assessing the domain-size effect on the widest scales. Our results indicate that $90\%$ of the total streamwise energetic fluctuations is recovered without a significant influence of the size of the domain. The remaining $10\%$ of the energy reflects that the widest scales in the outer layer are the ones most significantly affected by the spanwise length of the domain. The power-spectral density for $k_z = 0$ remains constant even if the size of the domain in the spanwise direction is increased up to 4 times the standard spanwise length, indicating that wide, spanwise coherent structures are not an artifact of domain truncation

Lift-up, Kelvin-Helmholtz and Orr mechanisms in turbulent jets

Three amplification mechanisms present in turbulent jets, namely lift-up, Kelvin–Helmholtz and Orr, are characterized via global resolvent analysis and spectral proper orthogonal decomposition (SPOD) over a range of Mach numbers. The lift-up mechanism was recently identified in turbulent jets via local analysis by Nogueira et al. (J. Fluid Mech., vol. 873, 2019, pp. 211–237) at low Strouhal number ( St ) and non-zero azimuthal wavenumbers ( m ). In these limits, a global SPOD analysis of data from high-fidelity simulations reveals streamwise vortices and streaks similar to those found in turbulent wall-bounded flows. These structures are in qualitative agreement with the global resolvent analysis, which shows that they are a response to upstream forcing of streamwise vorticity near the nozzle exit. Analysis of mode shapes, component-wise amplitudes and sensitivity analysis distinguishes the three mechanisms and the regions of frequency–wavenumber space where each dominates, finding lift-up to be dominant as St/m→0 . Finally, SPOD and resolvent analyses of localized regions show that the lift-up mechanism is present throughout the jet, with a dominant azimuthal wavenumber inversely proportional to streamwise distance from the nozzle, with streaks of azimuthal wavenumber exceeding five near the nozzle, and wavenumbers one and two most energetic far downstream of the potential core

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