A domain decomposition matrix-free method for global linear stability

This work is dedicated to the presentation of a matrix-free method for global linear stability analysis in geometries composed of multi-connected rectangular subdomains. An Arnoldi technique using snapshots in subdomains of the entire geometry combined with a multidomain linearized Direct Numerical Finite difference simulations based on an influence matrix for partitioning are adopted. The method is illustrated by three benchmark problems: the lid-driven cavity, the square cylinder and the open cavity flow. The efficiency of the method to extract large-scale structures in a multidomain framework is emphasized. The possibility to use subset of the full domain to recover the perturbation associated with the entire flow field is also highlighted. Such a method appears thus a promising tool to deal with large computational domains and three-dimensionality within a parallel architecture

Invariant solutions in a channel flow using a minimal restricted nonlinear model

Simulations using a Restricted Nonlinear (RNL) system, where mean flow distortion resulting from Reynolds stress feedback regenerates rolls, is applied in a channel flow under subcritical conditions. This quasi-linear restriction of the dynamics is used to study invariant solutions located in the bulk of the flow found recently by Rawat et al. (2016) [14]. It is shown that the RNL system truncated to a single streamwise mode for the perturbation supports invariant solutions that are found to bifurcate from a relative periodic orbit into a travelling wave solution when the spanwise size is increasing. In particular, the travelling wave solution exhibits a spanwise localized structure that remains unchanged for large values of the spanwise extent as the invariant solution lying on the lower branch found by Rawat et al. (2016) [14]. In addition, travelling wave solutions provided by this minimal RNL system are self-similar with respect to the Reynolds number based on the centreline velocity, and the half-channel height varying from 2000 to 5000

Linear stability of optimal streaks in the log-layer of turbulent channel flows

The importance of secondary instability of streaks for the generation of vortical struc-tures attached to the wall in the logarithmic region of turbulent channels is studied. Thestreaks and their linear instability are computed by solving equations associated withthe organized motion that include an eddy-viscosity modeling the effect of incoherentfluctuations. Three friction Reynolds numbers,Reτ=2000,3000, and 5000, areinvestigated. For all flow cases, optimal streamwise vortices (i.e., having the highestpotential for linear transient energy amplification) are used as initial conditions. Dueto the lift-up mechanism, these optimal perturbations lead to the nonlinear growthof streaks. Based on a Floquet theory along the spanwise direction, we observe theonset of streak secondary instability for a wide range of spanwise wavelengths whenthe streak amplitude exceeds a critical value. Under neutral conditions, it is shown thatstreak instability modes have their energy mainly concentrated in the overlap layer andpropagate with a phase velocity equal to the mean streamwise velocity of the log-layer.These neutral log-layer modes exhibit a sinuous pattern and have characteristic sizesthat are proportional to the wall distance in both streamwise and spanwise directions, inagreement with the Townsend’s attached eddy hypothesis (A. Townsend, the structureof turbulent shear flow, Cambridge university press, 1976 2nd edition). In particular,for a distance from the wall varying fromy+≈100 (in wall units) toy≈0.3h, wherehis half the height of the channel, the neutral log-layer modes are self-similar with aspanwise width ofλz≈y/0.3 and a streamwise length ofλx≈3λz, independently ofthe Reynolds number. Based on this observation, it is suggested that compact vorticalstructures attached to the wall can be ascribed to streak secondary instabilities. Inaddition, spatial distributions of fluctuating vorticity components show that the onsetof secondary instability is associated with the roll-up of the shear layer at the edgeof the low-speed streak, similarly to a three-dimensional mixing layer

Global and Koopman modes analysis of sound generation in mixing layers

It is now well established that linear and nonlinear instability waves play a significant role in the noise generation process for a wide variety of shear flows such as jets or mixing layers. In that context, the problem of acoustic radiation generated by spatially growing instability waves of two-dimensional subsonic and supersonic mixing layers are revisited in a global point of view, i.e., without any assumption about the base flow, in both a linear and a nonlinear framework by using global and Koopman mode decompositions. In that respect, a timestepping technique based on disturbance equations is employed to extract the most dynamically relevant coherent structures for both linear and nonlinear regimes. The present analysis proposes thus a general strategy for analysing the near-field coherent structures which are responsible for the acoustic noise in these configurations. In particular, we illustrate the failure of linear global modes to describe the noise generation mechanism associated with the vortex pairing for the subsonic regime whereas they appropriately explain the Mach wave radiation of instability waves in the supersonic regime. By contrast, the Dynamic Mode Decomposition (DMD) analysis captures both the near-field dynamics and the far-field acoustics with a few number of modes for both configurations. In addition, the combination of DMD and linear global modes analyses provides new insight about the influence on the radiated noise of nonlinear interactions and saturation of instability waves as well as their interaction with the mean flow

A domain decomposition matrix-free method for global linear stability

This work is dedicated to the presentation of a matrix-free method for global linear stability analysis in geometries composed of multi-connected rectangular subdomains. An Arnoldi technique using snapshots in subdomains of the entire geometry combined with a multidomain linearized Direct Numerical Finite difference simulations based on an influence matrix for partitioning are adopted. The method is illustrated by three benchmark problems: the lid-driven cavity, the square cylinder and the open cavity flow. The efficiency of the method to extract large-scale structures in a multidomain framework is emphasized. The possibility to use subset of the full domain to recover the perturbation associated with the entire flow field is also highlighted. Such a method appears thus a promising tool to deal with large computational domains and three-dimensionality within a parallel architecture

Restricted nonlinear model for high- and low-drag events in plane channel flow

A restricted nonlinear (RNL) model, obtained by partitioning the state variables into streamwise-averaged quantities and superimposed perturbations, is used in order to track the exact coherent state in plane channel flow investigated by Toh & Itano (J. Fluid Mech., vol. 481, 2003, pp. 67–76). When restricting nonlinearities to quadratic interaction of the fluctuating part into the streamwise-averaged component, it is shown that the coherent structure and its dynamics closely match results from direct numerical simulation (DNS), even if only a single streamwise Fourier mode is retained. In particular, both solutions exhibit long quiescent phases, spanwise shifts and bursting events. It is also shown that the dynamical trajectory passes close to equilibria that exhibit either low- or high-drag states. When statistics are collected at times where the friction velocity peaks, the mean flow and root-mean-square profiles show the essential features of wall turbulence obtained by DNS for the same friction Reynolds number. For low-drag events, the mean flow profiles are related to a universal asymptotic state called maximum drag reduction (Xi & Graham, Phys. Rev. Lett., vol. 108, 2012, 028301). Hence, the intermittent nature of self-sustaining processes in the buffer layer is contained in the dynamics of the RNL model, organized in two exact coherent states plus an asymptotic turbulent-like attractor. We also address how closely turbulent dynamics approaches these equilibria by exploiting a DNS database associated with a larger domain

Analyse de stabilité globale d'une couche limite décollée.

Une analyse de stabilité globale pour une perturbation tridimensionnelle d'un écoulement modèle basé sur les équations de Falkner-Skan a été menée. Plusieurs scenarii de déstabilisation d'un décollement non révélés par les approches locales classiques ont été mis en évidence. En particuliers, l'analyse des modes globaux fait apparaître deux modes stationnaires: un mode intrinsèque instable et un mode convectif similaire à une instabilité de type Görtler

Instabilities in oblique shock wave/laminar boundary-layer interactions

The interaction of an oblique shock wave and a laminar boundary layer developing over a flat plate is investigated by means of numerical simulation and global linear-stability analysis. Under the selected flow conditions (free-stream Mach numbers, Reynolds numbers and shock-wave angles), the incoming boundary layer undergoes separation due to the adverse pressure gradient. For a wide range of flow parameters, the oblique shock wave/boundary-layer interaction (OSWBLI) is seen to be globally stable. We show that the onset of two-dimensional large-scale structures is generated by selective noise amplification that is described for each frequency, in a linear framework, by wave-packet trains composed of several global modes. A detailed analysis of both the eigenspectrum and eigenfunctions gives some insight into the relationship between spatial scales (shape and localization) and frequencies. In particular, OSWBLI exhibits a universal behaviour. The lowest frequencies correspond to structures mainly located near the separated shock that emit radiation in the form of Mach waves and are scaled by the interaction length. The medium frequencies are associated with structures mainly localized in the shear layer and are scaled by the displacement thickness at the impact. The linear process by which OSWBLI selects frequencies is analysed by means of the global resolvent. It shows that unsteadiness are mainly associated with instabilities arising from the shear layer. For the lower frequency range, there is no particular selectivity in a linear framework. Two-dimensional numerical simulations show that the linear behaviour is modified for moderate forcing amplitudes by nonlinear mechanisms leading to a significant amplification of low frequencies. Finally, based on the present results, we draw some hypotheses concerning the onset of unsteadiness observed in shock wave/turbulent boundary-layer interactions

Global Instability in Shock Wave Laminar Boundary-Layer Interaction

The linear global stability of an interaction between an oblique shock wave and a laminar boundary layer is carried out for various oblique shock angles. It is illustrated that such a flow acts as a noise amplifier. The least temporally damped global modes are classified into three main categories, low, medium and high frequencies. The high frequencies are localized into the attached boundary layer, the medium frequencies are associated with Kelvin–Helmholtz like structures along the shear layer and convective waves in the separated flow downstream whereas the low frequencies are driven by the interaction zone. In particular, a low frequency mode emerges which is scaled by the interaction length and the freestream velocity