Numerical study of hopf bifurcations in the two-dimensional plane poiseuille flow

Abstract

In this work we try to analyse the dynamics of the Navier-Stokes equations in a problem without domain complexities as is the case of the plane Poiseuille flow. The Poiseuille problem is described as the flow of a viscous incompressible fluid, in a channel between two infinite parallel plates. We have considered it in two dimensions for the most common boundary conditions used to drive the fluid: mean constant pressure gradient or constant flux through the channel. We also specify the relation between this two formulations.<br/><br/>We give the details of the direct numerical solution of the full two-dimensional, time-dependent, incompressible Navier-Stokes equations, formulated by means of spectral methods on the spatial variables and finite differences for time. Unlike other authors we have considered the classical formulation in terms of primitive variables for velocity and pressure. We also describe the approach adopted to eliminate the pressure and the cross-stream component of the velocity, obtaining thus a reduced system of ordinary differential equations from an original system of differential-algebraic equations. <br/><br/>This is translated to a reduction of two thirds in the dimension of the original system and, in addition, it allows us to study the stability of fixed points by means of the analytical Jacobian matrix.<br/><br/>We reproduce previous calculations on travelling waves (which are time-periodic orbits) and its stability to superharmonic disturbances. These solutions are observed as stationary in a Galilean reference in the streamwise direction. We begin by reviewing some results of the Orr-Sommerfeld equation which serve as a starting point to obtain the bifurcating solutions of time-periodic flows for several values of the periodic length in the streamwise direction. In turn, we also calculate several Hopf bifurcations that appear on the branch of periodic flows, for both cases of imposed constant flux and pressure.<br/><br/>Likewise, for each unstable periodic flow, we study the connection of its unstable manifold to other attracting solutions.<br/><br/>Starting at the Hopf bifurcations found for periodic flows, we analyse the bifurcating branches of quasi-periodic solutions at the two first Hopf bifurcations for the case of imposed constant pressure and the first one for constant flux. Those solutions are found as fixed points of an appropriate Poincaré map since, by the symmetry of the channel, they may be viewed as periodic flows in an appropriate moving frame of reference. We also study their stability by analysing the linear part of the Poincaré map. In the case of constant flux we have found a branch of quasi-periodic solutions which, on increasing the Reynolds number, changes from stable to unstable, giving rise to an attracting family of quasi-periodic flows with 3 frequencies. The results referring to the first Hopf bifurcation for constant pressure, are not in qualitative agreement with those of Soibelman & Meiron (1991),<br/>which yield a different bifurcation picture and stability properties for the obtained quasi-periodic flows. From the computed unstable flows we follow their unstable invariant manifold and describe what new attracting solution they are conducted to.