Three-dimensional wavelike equilibrium states in plane Poiseuille flow.

*(English)*Zbl 0724.76036Summary: In the quest for a physically more realistic transition criterion, the prechaotic bifurcation behaviour of plane Poiseuille flow is studied. Various classes of nonlinear time-periodic equilibrium solutions are computed via Keller’s pseudo-arclength continuation method [H. B. Keller, Application of bifurcation theory, Proc. Adv. Semin., Madison/Wis. 1976, 359-384 (1977; Zbl 0581.65043)]. In particular, attention is focused on three-dimensional nonlinear travelling-wave type secondary bifurcation branches. These saturated equilibrium states originate on the nonlinear primary bifurcation surface from neutral, phase-locked secondary instability modes. Taking advantage of symmetries, only those nonlinear secondary branches which correspond to symmetric and antisymmetric linear secondary instability modes are investigated.

It appears that a new family of secondary bifurcation solutions which contains only even spanwise Fourier modes is particularly important. Dominated largely by the spanwise (0,2) mode and discovered by investigating bicritical secondary bifurcations, the mean quantities of these solutions show a certain resemblance to those observed in transitional flow during the ‘spike’ stage. The friction factor of this new solution branch is in the experimentally observed range and the critical Reynolds number, defined with the mean flow velocity, is reduced to about 1000 in general agreement with experiments.

It appears that a new family of secondary bifurcation solutions which contains only even spanwise Fourier modes is particularly important. Dominated largely by the spanwise (0,2) mode and discovered by investigating bicritical secondary bifurcations, the mean quantities of these solutions show a certain resemblance to those observed in transitional flow during the ‘spike’ stage. The friction factor of this new solution branch is in the experimentally observed range and the critical Reynolds number, defined with the mean flow velocity, is reduced to about 1000 in general agreement with experiments.

##### MSC:

76F20 | Dynamical systems approach to turbulence |