Optimal mixing in three-dimensional plane Poiseuille flow at high Peclet number

We consider a passive zero-mean scalar field organised into two layers of different concentrations in a three-dimensional plane channel flow subjected to a constant along-stream pressure gradient. We employ a nonlinear direct-adjoint-looping method to identify the optimal initial perturbation of the velocity field with given initial energy which yields `maximal' mixing by a target time horizon, where maximal mixing is defined here as the minimisation of the spatially-integrated variance of the concentration field. We verify in three-dimensional flows the conjecture by Foures et al. (J. Fluid Mech., vol. 748, 2014, pp. 241-277) that the initial perturbation which maximizes the time-averaged energy gain of the flow leads to relatively weak mixing, and is qualitatively different from the optimal initial `mixing' perturbation which exploits classical Taylor dispersion. We carry out the analysis for two different Reynolds numbers ($Re=U_m h/\nu= 500$, and $Re = 3000$, where $U_m$ is the maximum flow speed of the unperturbed flow, $h$ is the channel half-depth and $\nu$ is the kinematic viscosity of the fluid) demonstrating that this key finding is robust with respect to the transition to turbulence. We also identify the initial perturbations that minimise, at chosen target times, the `mix-norm' of the concentration eld, i.e. a Sobolev norm of negative index in the class introduced by Mathew et al. (Physica D, vol. 211, pp. 23-46, 2005). We show that the `true' variance-based mixing strategy can be successfully and practically approximated by the mix-norm minimisation since we f ind that the mix-norm optimal initial perturbations are far less sensitive to changes in the target time horizon than their optimal variance-minimising counterparts.This work was supported by the UK Engineering and Physical Sciences Research Council (EPSRC) grant EP/H023348/1 for the University of Cambridge Centre for Doctoral Training, the Cambridge Centre for Analysis. The research activity of C.P.C. is supported by EPSRC Programme Grant EP/K034529/1 entitled Mathematical Underpinnings of Strati ed Turbulence. This re- search was also supported in part by the National Science Foundation under Grant No. NSF PHY17-48958

Optimal mixing in two-dimensional stratified plane Poiseuille flow at finite Peclet and Richardson numbers

We consider the nonlinear optimisation of irreversible mixing induced by an initial finite amplitude perturbation of a statically stable density-stratified fluid with kinematic viscosity $\nu$ and density diffusivity $\kappa$. The initial diffusive error function density distribution varies continuously so that $\rho \in [\bar{\rho} - (1/2)\rho_0, \bar{\rho} + (1/2) \rho_0]$. A constant pressure gradient is imposed in a plane two-dimensional channel of depth $2h$. We consider flows with a finite P\'eclet number $Pe= U_m h /\kappa=500$ and Prandtl number $Pr=\nu/\kappa=1$, and a range of bulk Richardson numbers $Ri_b= g \rho_0 h /(\bar{\rho} U^2) \in [0,1]$ where $U_m$ is the maximum flow speed of the laminar parallel flow, and $g$ is the gravitational acceleration. We use the constrained variational direct-adjoint-looping (DAL) method to solve two optimization problems, extending the optimal mixing results of Foures, Caulfield \& Schmid (2014) to stratified flows, where the irreversible mixing of the active scalar density leads to a conversion of kinetic energy into potential energy. We identify initial perturbations of fixed finite kinetic energy which maximize the time-averaged perturbation kinetic energy developed by the perturbations over a finite time interval, and initial perturbations that minimise the value (at a target time, chosen to be $T=10$) of a `mix-norm' as first introduced by Mathew, Mezic \& Petzold (2005), further discussed by Thi eault (2012) and shown by Foures et al. (2014) to be a computationally efficient and robust proxy for identifying perturbations that minimise the long-time variance of a scalar distribution. We demonstrate, for all bulk Richardson numbers considered, that the time-averaged-kinetic-energy-maximising perturbations are significantly suboptimal at mixing compared to the mix-norm-minimising perturbations, and also that minimising the mix-norm remains (for density-stratified flows) a good proxy for identifying perturbations which minimise the variance at long times. Although increasing stratification reduces the mixing in general, mix-norm-minimising optimal perturbations can still trigger substantial mixing for $Ri_b \lesssim 0.3$. By considering the time evolution of the kinetic energy and potential energy reservoirs, we find that such perturbations lead to a flow which, through Taylor dispersion, very effectively converts perturbation kinetic energy into `available potential energy', which in turn leads rapidly and irreversibly to thorough and efficient mixing, with little energy returned to the kinetic energy reservoirs

Fourest, Georges

Notice bio-bibliographique sur Georges Foures

A tale of two airfoils: resolvent-based modelling of an oscillator vs. an amplifier from an experimental mean

The flows around a NACA 0018 airfoil at a Reynolds number of 10250 and angles of attack of alpha = 0 (A0) and alpha = 10 (A10) are modelled using resolvent analysis and limited experimental measurements obtained from particle image velocimetry. The experimental mean velocity profiles are data-assimilated so that they are solutions of the incompressible Reynolds-averaged Navier-Stokes equations forced by Reynolds stress terms which are derived from experimental data. Spectral proper orthogonal decompositions (SPOD) of the velocity fluctuations and nonlinear forcing find low-rank behaviour at the shedding frequency and its higher harmonics for the A0 case. In the A10 case, low-rank behaviour is observed for the velocity fluctuations in two bands of frequencies. Resolvent analysis of the data-assimilated means identifies low-rank behaviour only in the vicinity of the shedding frequency for A0 and none of its harmonics. The resolvent operator for the A10 case, on the other hand, identifies two linear mechanisms whose frequencies are a close match with those identified by SPOD. It is also shown that the second linear mechanism, corresponding to the Kelvin-Helmholtz instability in the shear layer, cannot be identified just by considering the time-averaged experimental measurements as a mean flow due to the fact that experimental data are missing near the leading edge. The A0 case is classified as an oscillator where the flow is organised around an intrinsic instability while the A10 case behaves like an amplifier whose forcing is unstructured. For both cases, resolvent modes resemble those from SPOD when the operator is low-rank. To model the higher harmonics where this is not the case, we add parasitic resolvent modes, as opposed to classical resolvent modes which are the most amplified, by approximating the nonlinear forcing from limited triadic interactions of known resolvent modes.Comment: 32 pages, 23 figure

Optimal mixing in two-dimensional plane Poiseuille flow at finite Peclet number

International audienceWe consider the nonlinear optimisation of the mixing of a passive scalar, initially arranged in two layers, in a two-dimensional plane Poiseuille flow at finite Reynolds and Péclet numbers, below the linear instability threshold. We use a nonlinear-adjoint-looping approach to identify optimal perturbations leading to maximum time-averaged energy as well as maximum mixing in a freely evolving flow, measured through the minimisation of either the passive scalar variance or the so-called mix-norm, as defined by Mathew, Mezić & Petzold (Physica D, vol. 211, 2005, pp. 23-46). We show that energy optimisation appears to lead to very weak mixing of the scalar field whereas the optimal mixing initial perturbations, despite being less energetic, are able to homogenise the scalar field very effectively. For sufficiently long time horizons, minimising the mix-norm identifies optimal initial perturbations which are very similar to those which minimise scalar variance, demonstrating that minimisation of the mix-norm is an excellent proxy for effective mixing in this finite-Péclet-number bounded flow. By analysing the time evolution from initial perturbations of several optimal mixing solutions, we demonstrate that our optimisation method can identify the dominant underlying mixing mechanism, which appears to be classical Taylor dispersion, i.e. shear-augmented diffusion. The optimal mixing proceeds in three stages. First, the optimal mixing perturbation, energised through transient amplitude growth, transports the scalar field across the channel width. In a second stage, the mean flow shear acts to disperse the scalar distribution leading to enhanced diffusion. In a final third stage, linear relaxation diffusion is observed. We also demonstrate the usefulness of the developed variational framework in a more realistic control case: mixing optimisation by prescribed streamwise velocity boundary conditions

Localization of flow structures using infinity-norm optimization

International audienceStability theory based on a variational principle and finite-time direct-adjoint optimization commonly relies on the kinetic perturbation energy density E-1(t ) = (1/V-Omega) integral(Omega) e(x, t) d Omega (where e(x, t) = vertical bar u vertical bar(2)/2) as a measure of disturbance size. This type of optimization typically yields optimal perturbations that are global in the fluid domain Omega of volume V-Omega. This paper explores the use of p-norms in determining optimal perturbations for 'energy' growth over prescribed time intervals of length T. For p = 1 the traditional energy-based stability analysis is recovered, while for large p >> 1, localization of the optimal perturbations is observed which identifies confined regions, or 'hotspots', in the domain where significant energy growth can be expected. In addition, the p-norm optimization yields insight into the role and significance of various regions of the flow regarding the overall energy dynamics. As a canonical example, we choose to solve the infinity-norm optimal perturbation problem for the simple case of two-dimensional channel flow. For such a configuration, several solutions branches emerge, each of them identifying a different energy production zone in the flow: either the centre or the walls of the domain. We study several scenarios (involving centre or wall perturbations) leading to localized energy production for different optimization time intervals. Our investigation reveals that even for this simple two-dimensional channel flow, the mechanism for the production of a highly energetic and localized perturbation is not unique in time. We show that wall perturbations are optimal (with respect to the infinity-norm) for relatively short and long times, while the centre perturbations are preferred for very short and intermediate times. The developed p-norm framework is intended to facilitate worst-case analysis of shear flows and to identify localized regions supporting dominant energy growth