Theory and predictions for finite‐amplitude waves in two‐dimensional plane Poiseuille flow

1990 ◽  
Vol 2 (6) ◽  
pp. 955-970 ◽  
Author(s):  
Dwight Barkley
Keyword(s):  
Poiseuille Flow ◽  
Finite Amplitude ◽  
Two Dimensional ◽  
Plane Poiseuille Flow ◽  
Dimensional Plane ◽  
Finite Amplitude Waves

A systems theory approach to the feedback stabilization of infinitesimal and finite-amplitude disturbances in plane Poiseuille flow

1997 ◽  
Vol 332 ◽  
pp. 157-184 ◽  
Author(s):  
Sanjay S. Joshi ◽  
Jason L. Speyer ◽  
John Kim
Keyword(s):  
Feedback Control ◽  
Systems Theory ◽  
Poiseuille Flow ◽  
Finite Amplitude ◽  
Feedback Stabilization ◽  
Bypass Transition ◽  
Theory Approach ◽  
Two Dimensional ◽  
Plane Poiseuille Flow ◽  
Fluid System

A systems theory framework is presented for the linear stabilization of two-dimensional laminar plane Poiseuille flow. The governing linearized Navier-Stokes equations are converted to control-theoretic models using a numerical discretization scheme. Fluid system poles, which are closely related to Orr-Sommerfeld eigenvalues, and fluid system zeros are computed using the control-theoretic models. It is shown that the location of system zeros, in addition to the well-studied system eigenvalues, are important in linear stability control. The location of system zeros determines the effect of feedback control on both stable and unstable eigenvalues. In addition, system zeros can be used to determine sensor locations that lead to simple feedback control schemes. Feedback controllers are designed that make a new fluid-actuator-sensorcontroller system linearly stable. Feedback control is shown to be robust to a wide range of Reynolds numbers. The systems theory concepts of modal controllability and observability are used to show that feedback control can lead to short periods of highamplitude transients that are unseen at the output. These transients may invalidate the linear model, stimulate nonlinear effects, and/or form a path of ‘bypass’ transition in a controlled system. Numerical simulations are presented to validate the stabilization of both single-wavenumber and multiple-wavenumber instabilities. Finally, it is shown that a controller designed upon linear theory also has a strong stabilizing effect on two-dimensional finite-amplitude disturbances. As a result, secondary instabilities due to infinitesimal three-dimensional disturbances in the presence of a finite-amplitude two-dimensional disturbance cease to exist.

scholarly journals Optimal mixing in two-dimensional plane Poiseuille flow at finite Péclet number

2014 ◽  
Vol 748 ◽  
pp. 241-277 ◽  
Author(s):  
D. P. G. Foures ◽  
C. P. Caulfield ◽  
P. J. Schmid
Keyword(s):  
Scalar Field ◽  
Poiseuille Flow ◽  
Peclet Number ◽  
Passive Scalar ◽  
Two Dimensional ◽  
Plane Poiseuille Flow ◽  
Péclet Number ◽  
Dimensional Plane ◽  
Scalar Variance ◽  
Optimal Mixing

AbstractWe 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.

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