Hagen-Poiseuille flow linear instability

Abstract

In the suggested here linear theory of hydrodynamic instability of the Hagen Poiseuille flow it is counted the possibility of quasi periodic longitudinal variations, when there is no separation of the longitudinal and radial variables in the description of the disturbances field. It is proposed to use the energetic method and the Galerkin approximation method that takes into account existence of different values of longitudinal variability periods for different radial modes corresponding to the equation of evolution of extremely small axially symmetric velocity field tangential component disturbances and boundary condition on the tube surface and axis. We found that even for two linearly interacting radial modes the HP flow may have linear instability, when Re > Re-th (p) and the value Re-th (p) very sensitively depends on the ratio p of two longitudinal periods each of which describes longitudinal variability for its own radial mode only. Obtained from energetic method for the HP flow linear instability realization minimal value Re-thmin approximate to 704 ( for N=600 radial modes) and from Galerkin approximation Re-th min 448 min ( for N=2 modes with p=1.516) which quantitatively agrees with the Tolmin- Shlihting waves in the boundary layer arising, where also the threshold value Re-th = 420 is obtained. We state also the agreement of the phase velocity values of the considered in our theory vortex disturbances with the experimental data on the fore and rear fronts of the turbulent puffs spreading along the pipe axis.