Subharmonic instabilities of Tollmien-Schlichting waves in two-dimensional Poiseuille flow

Abstract

The stability of the upper branch of shear traveling waves in two-dimensional Poiseuille flow, when the total flux through the channel is held constant, is considered. Taking into account the length of the periodic channel, perturbations of the same wave number (superharmonic), and different wave number (subharmonic) of the uniform wave trains are imposed. We mainly consider channels long enough to contain M=4 and M=8 basic wavelengths. In these cases, subharmonic bifurcations are found to be dominant except in a small region of parameters. From this type of bifurcation, we show that if the wave number is decreased, the periodic train of finite amplitude waves evolves continuously towards the stable localized wave packets obtained in long channels by other authors and whose existence has been associated to the vicinity of an inverted Hopf bifurcation. Depending on the basic wave number of the periodic train destabilized, different types of solutions for a given length of the channel can be obtained. Furthermore, for moderate Reynolds numbers, configurations of linearly stable wave trains exist, provided that their basic wave number is alpha approximately 1.5.

A mechanism for streamwise localisation of nonlinear waves in shear flows

“…As a matter of fact, previous subharmonic stability analyses of travelling waves in PPF (Drissi et al 1999) reveal oscillatory Hopf bifurcations from which emanate branches of streamwise modulated wave trains (Jimenez 1990) that are time-periodic when observed from a reference frame moving at the average streamwise speed of the waves.…”

“…We adopt the streamfunction formulation of Drissi et al (1999), which naturally satisfies mass conservation and the no slip boundary condition at the walls, and at the same time preserves a constant massflow. The nondimensional streamfunction Ψ (x, y; t) is related to the nondimensional velocity field through u = ∂ y Ψ and v = −∂ x Ψ .…”

“…Although continuation of fully three-dimensional localised solutions has been shown possible (Chantry et al 2014), the refined multiparametric study required here is unaffordable for three-dimensional systems such as pipe or channel flows, where the initial steps towards explaining the phenomenon have been undertaken, but becomes feasible for the simplest flow exhibiting archetypal streamwise localisation, ie 2D plane-Poiseuille flow (PPF). This problem gathers all the required ingredients: subcritical bifurcation of upper and lower branch streamwise-periodic solutions (Tollmien-Schlichting waves, Chen & Joseph (1973); Zahn et al (1974)), subharmonic bifurcation of localised wave-packet solutions (Jimenez 1990;Drissi et al 1999) and the advantage of two-dimensionality. In this Rapids we exploit this fact to resolve the problem of streamwise localisation in shear flows.…”

Continuation of travelling-wave solutions of the Navier–Stokes equations

“…The solution of the discretized problem satisÿes both the boundary conditions and the condition of incompressibility. We had previously used this method to calculate numerically the travelling waves arising in two-dimensional planar Poiseuille ow [1] and in Rayleigh-BÃ enard convection [7], and to obtain thermal Rossby waves [8].…”

“…In addition, both stable and unstable travelling-wave solutions can be obtained. This approach has been used in the study of Rayleigh-BÃ enard convection [7] and in the analysis of Poiseuille ow [1], among other problems.…”