a Study of Finite Amplitude Bifurcations in Plane Poiseuille Flow.
Abstract
Plane Poiseuille flow is known to be linearly unstable at a Reynolds number of 5772.22 (Drazin and Reid, 1981). In experiments, however, transition to turbulent flow is seen to occur at a Reynolds number of 1000 (Nishioka and Asai, 1985). In an attempt at resolving this conflict, we search for 2D and 3D nonlinear bifurcations at low Reynolds number. Because we wish to study secondary bifurcations, we compute the 2D waves which bifurcate from plane Poiseuille flow. These waves were first computed by Zahn, et al., (1975), and the critical Reynolds number, based on constant pressure, was found to be approximately 2900. To find 2D bifurcations, we study the 2D superharmonic stability of the 2D waves. The stability picture is found to change when switching from a constant flux to constant average pressure gradient boundary condition. For both boundary conditions, we find several Hopf bifurcations on the upper branch of the 2D waves. We calculate the periodic orbits which emanate from these bifurcations and find that no branch extends below the critical 2D wave Reynolds number. We also confirm the results of Jimenez (1988) who detected one of the branches we calculate with a time dependent formulation. To find 3D bifurcations, we study the 3D stability of the 2D waves. Several branches of 3D waves are calculated. In particular, we study 3D bifurcations at a spanwise wave number of 2. No bifurcations are found to branches which extend to low Reynolds numbers. This result conflicts with those found by Rozhdestvensky and Simakin (1984) with a time dependent formulation. In addition, we study 3D oblique waves and 3D standingtravelling waves (standing in the streamwise direction) which bifurcate from plane Poiseuille flow. In particular, we study the bifurcation at spanwise wave numbers greater than.365. Contrary to Bridges' (1988) hypothesis, we find that no branches extend to low Reynolds numbers.
 Publication:

Ph.D. Thesis
 Pub Date:
 1989
 Bibcode:
 1989PhDT.......148S
 Keywords:

 Physics: Fluid and Plasma