scholarly journals Localization of flow structures using -norm optimization

2013 ◽  
Vol 729 ◽  
pp. 672-701 ◽  
Author(s):  
D. P. G. Foures ◽  
C. P. Caulfield ◽  
P. J. Schmid
Keyword(s):  
Channel Flow ◽  
Energy Production ◽  
Worst Case Analysis ◽  
Two Dimensional ◽  
Time Intervals ◽  
Worst Case ◽  
Optimal Perturbation ◽  
Energy Growth ◽  
Perturbation Problem ◽  
Production Zone

AbstractStability 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 } )\int \nolimits _{\Omega } e(\boldsymbol{x}, t)\hspace{0.167em} \mathrm{d} \Omega $ (where $e(\boldsymbol{x}, t)= \vert \boldsymbol{u}{\vert }^{2} / 2$) 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 $p\gg 1$, 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 $\infty $-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 $\infty $-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.

scholarly journals Optimal perturbation growth on a breaking internal gravity wave

2021 ◽  
Vol 925 ◽  
Author(s):  
J.P. Parker ◽  
C.J. Howland ◽  
C.P. Caulfield ◽  
R.R. Kerswell
Keyword(s):  
Shear Flow ◽  
Gravity Wave ◽  
Three Dimensional ◽  
Internal Gravity Wave ◽  
Internal Gravity Waves ◽  
Mixing Efficiency ◽  
Two Dimensional ◽  
Systematic Analysis ◽  
Optimal Perturbation ◽  
Energy Growth

The breaking of internal gravity waves in the abyssal ocean is thought to be responsible for much of the mixing necessary to close oceanic buoyancy budgets. The exact mechanism by which these waves break down into turbulence remains an active area of research and can have significant implications on the mixing efficiency. Recent evidence has suggested that both shear instabilities and convective instabilities play a significant role in the breaking of an internal gravity wave in a high Richardson number mean shear flow. We perform a systematic analysis of the stability of a configuration of an internal gravity wave superimposed on a background shear flow first considered by Howland et al. (J. Fluid Mech., vol. 921, 2021, A24), using direct–adjoint looping to find the perturbation giving maximal energy growth on this evolving flow. We find that three-dimensional, convective mechanisms produce greater energy growth than their two-dimensional counterparts. In particular, we find close agreement with the direct numerical simulations of Howland et al. (J. Fluid Mech., 2021, in press), which demonstrated a clear three-dimensional mechanism causing breakdown to turbulence. The results are shown to hold at realistic Prandtl numbers. At low mean Richardson numbers, two-dimensional, shear-driven mechanisms produce greater energy growth.

scholarly journals Variational Nonlinear Optimization in Fluid Dynamics: The Case of a Channel Flow with Superhydrophobic Walls

Mathematics ◽  
2020 ◽  
Vol 9 (1) ◽  
pp. 53
Author(s):  
Stefania Cherubini ◽  
Francesco Picella ◽  
Jean-Christophe Robinet
Keyword(s):  
Channel Flow ◽  
Degrees Of Freedom ◽  
Slip Length ◽  
Superhydrophobic Surfaces ◽  
Transition To Turbulence ◽  
Growth Mechanisms ◽  
Boundary Slip ◽  
Optimal Perturbation ◽  
Energy Growth ◽  
Optimal Energy

Variational optimization has been recently applied to nonlinear systems with many degrees of freedom such as shear flows undergoing transition to turbulence. This technique has unveiled powerful energy growth mechanisms able to produce typical coherent structures currently observed in transition and turbulence. However, it is still not clear the extent to which these nonlinear optimal energy growth mechanisms are robust with respect to external disturbances or wall imperfections. Within this framework, this work aims at investigating how nano-roughnesses such as those of superhydrophobic surfaces affect optimal energy growth mechanisms relying on nonlinearity. Nonlinear optimizations have been carried out in a channel flow with no-slip and slippery boundaries, mimicking the presence of superhydrophobic surfaces. For increasing slip length, the energy threshold for obtaining hairpin-like nonlinear optimal perturbations slightly rises, scaling approximately with Re−2.36 no matter the slip length. The corresponding energy gain increases with Re with a slope that reduces with the slip length, being almost halved for the largest slip and Reynolds number considered. This suggests a strong effect of boundary slip on the energy growth of these perturbations. While energy is considerably decreased, the shape of the optimal perturbation barely changes, indicating the robustness of optimal perturbations with respect to wall slip.

scholarly journals On the longitudinal optimal perturbations to inviscid plane shear flow: formal solution and asymptotic approximation

2013 ◽  
Vol 737 ◽  
pp. 387-411 ◽  
Author(s):  
C. Arratia ◽  
J.-M. Chomaz
Keyword(s):  
Shear Flow ◽  
Initial Conditions ◽  
Formal Solution ◽  
Base Flow ◽  
Flow Profile ◽  
Plane Shear ◽  
Optimal Perturbation ◽  
Energy Growth ◽  
Perturbation Problem ◽  
New Formulation

AbstractWe study the longitudinal linear optimal perturbations (which maximize the energy gain up to a prescribed time $T$) to inviscid parallel shear flow, which present unbounded energy growth due to the lift-up mechanism. Using the phase invariance with respect to time, we show that for an arbitrary base flow profile and optimization time, the computation of the optimal longitudinal perturbation reduces to the resolution of a single one-dimensional eigenvalue problem valid for all times. The optimal perturbation and its amplification are then derived from the lowest eigenvalue and its associated eigenfunction, while the remainder of the infinite set of eigenfunctions provides an orthogonal base for decomposing the evolution of arbitrary perturbations. With this new formulation we obtain, asymptotically for large spanwise wavenumber ${k}_{z} , $ a prediction of the optimal gain and the localization of inviscid optimal perturbations for the two main classes of parallel flows: free shear flow with an inflectional velocity profile, and wall-bounded flow with maximum shear at the wall. We show that the inviscid optimal perturbations are localized around the point of maximum shear in a region with a width scaling like ${ k}_{z}^{- 1/ 2} $ for free shear flow, and like ${ k}_{z}^{- 2/ 3} $ for wall-bounded shear flows. This new derivation uses the stationarity of the base flow to transform the optimization of initial conditions in phase space into the optimization of a temporal phase along each trajectory, and an optimization among all trajectories labelled by their intersection with a codimension-1 subspace. The optimization of the time phase directly imposes that the initial and final energy growth rates of the optimal perturbation should be equal. This result requires only time invariance of the base flow, and is therefore valid for any linear optimal perturbation problem with stationary base flow.

scholarly journals Degeneracy of turbulent states in two-dimensional channel flow

2021 ◽  
Vol 917 ◽  
Author(s):  
Vilda K. Markeviciute ◽  
Rich R. Kerswell
Keyword(s):  
Channel Flow ◽  
Two Dimensional

Abstract

Hierarchical Programming for Worst-Case Analysis of Power Grids

2018 ◽  
Author(s):  
Hatim Djelassi ◽  
Stephane Fliscounakis ◽  
Alexander Mitsos ◽  
Patrick Panciatici
Keyword(s):  
Case Analysis ◽  
Power Grids ◽  
Worst Case Analysis ◽  
Worst Case

Formal Worst-Case Analysis of Crosstalk Noise in Mesh-Based Optical Networks-on-Chip

2013 ◽  
Vol 21 (10) ◽  
pp. 1823-1836 ◽  
Author(s):  
Yiyuan Xie ◽  
Mahdi Nikdast ◽  
Jiang Xu ◽  
Xiaowen Wu ◽  
Wei Zhang ◽  
...  
Keyword(s):  
Optical Networks ◽  
Case Analysis ◽  
Worst Case Analysis ◽  
Crosstalk Noise ◽  
Worst Case ◽  
Networks On Chip ◽  
On Chip
Sign in / Sign up

Export Citation Format

Share Document