Two-dimensional Plane Poiseuille Flow Research Articles

Subcritical transitional flow in two-dimensional plane Poiseuille flow

Recently, subcritical transition to turbulence in the quasi-two-dimensional (quasi-2-D) shear flow with strong linear friction (Camobreco et al., J. Fluid Mech., vol. 963, 2023, R2) has been demonstrated by the 2-D mechanism at $Re = 71\,211$ , and the nonlinear Tollmien–Schlichting (TS) waves related to the edge state were approached independently of initial optimal disturbances. For 2-D plane Poiseuille flow, transition to the fully developed turbulence requires that the Reynolds number is several times larger than the critical Reynolds number $Re_c$ (Markeviciute & Kerswell, J. Fluid Mech., vol. 917, 2021, A57). In this paper, we observed the subcritical transitional flow in 2-D plane Poiseuille flow driven by the nonlinear TS waves by both linear and nonlinear optimal disturbances ( $Re < Re_c$ ) with different quantitative edge states. The nonlinear optimal disturbances could trigger the sustained subcritical transitional flow for $Re \geqslant 2400$ . The initial energy for nonlinear optimal disturbance is more efficient than the linear optimal disturbance in reaching the subcritical transitional flow for $2400 \leqslant Re \leqslant 5000$ . Moreover, the initial energy of linear optimal disturbance is larger than the energy of its edge state. The nonlinear TS waves along the edge state are formed by the nonlinear optimal disturbances to trigger transitional flow, which agrees well with the main conclusions of Camobreco et al. (J. Fluid Mech., vol. 963, 2023, R2), while the required $Re$ of 2-D plane Poiseuille flow is much smaller.

On the Stability and Control of Two-Dimensional Channel Flow

Abstract A direct numerical simulation is used to solve the two-dimensional plane Poiseuille flow for three different stability cases; specifically, they are a supercritical case at a Reynolds number of 10,000, a critical-stable case at a Reynolds number of 5772.22, and subcritical case at a Reynolds number of 1000. The perturbations are developed using the eigenfunctions of Orr–Sommerfeld equation's solution. In many applications, the flow was required to be fully laminar. As a result, a model-based controller is developed for stabilizing the flow field using unsteady strong suction and blowing technique, through distributed slots on the two walls of the channel. This technique is introduced to cancel the propagating wave in the boundary-layers near the two walls. Promising results are achieved for the unstable and critical-stable cases. The stable case is tested. Slight improvements in its growth rate are recorded. In turn, a faster response for the flow field is achieved.

Pattern preservation during the decay and growth of localized wave packet in two-dimensional channel flow

In this paper, the decay and growth of localized wave packet (LWP) in a two-dimensional plane-Poiseuille flow are studied numerically and theoretically. When the Reynolds number (Re) is less than a critical value Rec, the disturbance kinetic energy Ek of LWP decreases monotonically with time and experiences three decay periods, i.e., the initial and the final steep descent periods and the middle plateau period. Higher initial Ek of a decaying LWP corresponds to longer lifetime. According to the simulations, the lifetime scales as (Rec−Re)−1/2, indicating a divergence of lifetime as Re approaches Rec, a phenomenon known as “critical slowing-down.” By proposing a pattern preservation approximation, i.e., the integral kinematic properties (e.g., the disturbance enstrophy) of an evolving LWP are independent of Re and single valued functions of Ek, the disturbance kinetic energy equation can be transformed into the classical differential equation for saddle-node bifurcation, by which the lifetimes of decaying LWPs can be derived, supporting the −1/2 scaling law. Furthermore, by applying the pattern preservation approximation and the integral kinematic properties obtained as Re<Rec, the Reynolds number and the corresponding Ek of the whole lower branch, the turning point, and the upper-branch LWPs with Ek<0.15 are predicted successfully with the disturbance kinetic energy equation, indicating that the pattern preservation is an intrinsic feature of this localized transitional structure.

Self-sustaining and propagating mechanism of localized wave packet in plane-Poiseuille flow

In this Letter, it is revealed that the convection velocity of a localized wave packet in two-dimensional plane-Poiseuille flow is determined by the solitary wave at the centerline of a vortex dipole, which is evinced by subtracting the base flow from the mean flow. The fluctuation component propagates obeying the local dispersion relation of the mean flow and oscillates with a global frequency selected by the upstream marginal absolute instability and, hence, is a traveling wave mode. The vortex dipole provides an unstable region for the fluctuation waves to grow up, and the Reynolds stress of the fluctuation waves leads to streaming components enhancing the vortex dipole. By applying localized initial disturbances, a nonzero wave-packet density is achieved at the threshold state, suggesting a first-order transition.

Symmetry-breaking waves and space-time modulation mechanisms in two-dimensional plane Poiseuille flow

Two distinct scenarios of spatial modulation in two-dimensional plane Poiseuille flow have been studied. The first one is based on the identification of a new family of asymmetric Tollmien-Schlichting waves (TSW) breaking the reflectional symmetry about the channel midplane. The second follows the fate of a branch of time-periodic space-modulated waves that exclusively bridge upper-branch TSW-trains of different number of replicas by means of a codimension-2 bifurcation point. These modulated waves may therefore play a relevant role in the strange saddle governing domain-filling turbulent dynamics at high Reynolds numbers.

A mechanism for streamwise localisation of nonlinear waves in shear flows

We present the complete unfolding of streamwise localisation in a paradigm of extended shear flows, namely two-dimensional plane Poiseuille flow. Exact solutions of the Navier–Stokes equations are computed numerically and tracked in the streamwise wavenumber–Reynolds number parameter space to identify and describe the fundamental mechanism behind streamwise localisation, a ubiquitous feature of shear flow turbulence. Unlike shear flow spanwise localisation, streamwise localisation does not follow the snaking mechanism demonstrated for plane Couette flow.

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

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.

Parameterization of travelling waves in plane Poiseuille flow

The first finite-dimensional parameterization of a subset of the phase s pace of the Navier‐Stokes equations is presented. Travelling waves in two-dimensional plane Poiseuille flow are shown numerically to approximate maximum-entrop y configurations. In a coordinate system moving with the phase velocity, the e nclosed body of the flow exhibits a hyperbolic sinusoidal relationship betwee n the vorticity and stream function. The phase velocity and two amplitude parameters describe the stable manifold on the slow viscous time scale. This original parameterization provides a valuable visualization of this subset of the phase space of the Navier‐Stokes equations. These new results provide physical insight into an important intermediate stage in the instability process of plane Poiseuille flow . Parameterization, plane Poiseuille flow, maximum-entropy configurations

Hopf bifurcations to quasi-periodic solutions for the two-dimensional plane Poiseuille flow

This paper studies various Hopf bifurcations in the two-dimensional plane Poiseuille problem. For several values of the wavenumber α, we obtain the branch of periodic flows which are born at the Hopf bifurcation of the laminar flow. It is known that, taking α≈1, the branch of periodic solutions has several Hopf bifurcations to quasi-periodic orbits. For the first bifurcation, calculations from other authors seem to indicate that the bifurcating quasi-periodic flows are stable and subcritical with respect to the Reynolds number, Re. By improving the precision of previous works we find that the bifurcating flows are unstable and supercritical with respect to Re. We have also analysed the second Hopf bifurcation of periodic orbits for several α, to find again quasi-periodic solutions with increasing Re. In this case the bifurcated solutions are stable to superharmonic disturbances for Re up to another new Hopf bifurcation to a family of stable 3-tori. The proposed numerical scheme is based on a full numerical integration of the Navier–Stokes equations, together with a division by 3 of their total dimension, and the use of a pseudo-Newton method on suitable Poincaré sections. The most intensive part of the computations has been performed in parallel. We believe that this methodology can also be applied to similar problems.

Unstable manifold computations for the two-dimensional plane Poiseuille flow

We follow the unstable manifold of periodic and quasi-periodic solutions in time for the Poiseuille problem, using two formulations: holding a constant flux or mean pressure gradient. By means of a numerical integrator of the Navier–Stokes equations, we let the fluid evolve from an initially perturbed unstable solution until the fluid reaches an attracting state. Thus, we detect several connections among different configurations of the flow such as laminar, periodic, quasi-periodic with two or three basic frequencies, and more complex sets that we have not been able to classify. These connections make possible the location of new families of solutions, usually hard to find by means of numerical continuation of curves, and show the richness of the dynamics of the Poiseuille flow.

Control of boundary layer flow transition via distributed

A reduced-order linear feedback controller, which is used to control the linear disturbance in two-dimensional plane Poiseuille flow, is applied to a boundary layer flow for stability control. Using model reduction and linear-quadratic-Gaussian/loop-transfer-recovery control synthesis, a distributed controller is designed from the linearized two-dimensional Navier-Stokes equations. This reduced-order controller, requiring only the wall-shear information, is shown to effectively suppress the linear disturbance in boundary layer flow under the uncertainty of Reynolds number. The controller also suppresses the nonlinear disturbance in the boundary layer flow, which would lead to unstable flow regime without control. The flow is relaminarized in the long run. Other effects of the controller on the flow are also discussed.

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

It has been found by Pugh and Saffman [J. Fluid Mech. 194, 295 (1988)] that in a two-dimensional channel, the stability of finite-amplitude steady waves to modulated waves can depend on the boundary conditions imposed on the flow. In particular, near the limit point in Reynolds number, stability can depend on whether the flux or the mean pressure gradient is prescribed for the flow. Here a continuous range of intermediate boundary conditions is defined and studied using bifurcation theory. Based only on previous numerical solutions to the Navier–Stokes equations at constant mean flux and constant mean pressure gradient, it is shown that the finite-amplitude steady waves must have a double-zero eigenvalue at some intermediate boundary condition. From this a unifying picture emerges for the dynamics near the limit point in Reynolds number and specific predictions are made for finite-amplitude solutions to the Navier–Stokes equations. These predictions include the existence of a homoclinic orbit and a degenerate Hopf bifurcation.

The convective nature of instability in plane Poiseuille flow

By numerical solution of the Orr–Sommerfeld equation for complex frequency and complex wavenumber for a wide range of Reynolds numbers R and by asymptotic analysis for large R, it is shown that there is no absolute instability in a two-dimensional plane Poiseuille flow for any R and that the flow is convectively unstable for Rc <R<∞, where Rc is the critical Reynolds number.