Numerical study of bifurcations for the 2–D Poiseuille problem

We study the dynamics of two-dimensional Poiseuille flow. Firstly we check our calculations with previous results concerning the laminar solution and the minimum Reynolds number such that it becomes unstable. Next we studied time periodic solutions which, because of the imposed periodicity in the stream-wise direction, are rotating waves, what allow us to treat them as stationary flows in a moving system of reference. We use this fact to obtain also unstable time periodic flows and bifurcation branches from the laminar solution for different values of the wave number. Finally we introduce the case of bifurcation to quasi-periodic solutions. Poiseuille flow We consider the flow of a viscous incompressible fluid, in a channel between two parallel walls at y = 1, driven by a stream-wise pressure gradient and governed by the dimensionless Navier–Stokes equations @u @t + u@u @x + v @u @y = @p @x + 1 Re @2u @x2 + @2u @y2 +G @v @t + u@v @x + v @v @y = @p @y + 1 Re @2v @x2 + @2v @y2 @u @x + @v @y = 0; (1) where (u; v) are the components of the velocity, p is the pressure, Re is the Reynolds number and G = 2=Re is the constant pressure gradient. The Reynolds number is defined in terms of the channel half-length and the center velocity of a stationary solution, known as laminar flow. In dimensionless coordinates this laminar solution is written as u(y) = 1 y2; v = 0; r p = 0: As boundary conditions we suppose no-slip on the channel walls and period L = 2 = ( is the wave number) in the stream direction x, i.e. u(x; 1; t) = v(x; 1; t) = 0 (u; v; p)(x+ L; y; t) = (u; v; p)(x; y; t) x 2 IR; y 2 [ 1; 1]; t 0: The initial profile of velocities is simply subjected to the incompressibility condition. We are concerned with the dynamics of Poiseuille flow in varying the parameters Re and . The stability of the laminar solution has been studied extensively through the literature. For instance, Orszag (see [3]), solved numerically the linearization around the laminar flow of Navier–Stokes equations expressed in terms of the vorticity, what is known as the Orr– Sommerfeld equation. This is an eigenvalue problem for each value of the parameters Re and , in such a way that if c = cr+ ici is a complex eigenvalue with ci > 0, then the laminar flow is unstable to small disturbances according to linear theory. The neutral stability curve is presented in figure 1. 0.5 0.6 0.7 0.8 0.9 1 1.1 Recr=5772.22 10000 5000