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 p th > and the value ) ( Re p th 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 704 min
Highlights
Fundamental and applied problem of defining of the turbulence arising mechanism for the Hagen-Poiseuille (HP)1 flow more than century is left mysterious because of the linear stability paradox of the flow with respect to extremely small by amplitude disturbances for any Reynolds number valueRe = Vmax R ν [1,2,3,4]
We show that possibility of linear absolute instability of the HP flow is defined by the value of complementary to the Reynolds number Re control parameter p, which characterizes frequency-wave features of the disturbances and determines the value of the threshold Reynolds number Reth( p) independently from the amplitude of the initial disturbances
Obvious correspondence of the results following from Fig. 1a) allows us making the conclusion about similarity of the linear dissipative instability mechanisms realized for the HP flow (when meeting the condition (14) or (A.3)), as well as for Tolmin-Shlihting waves excitation in a boundary layer
Summary
Fundamental and applied problem of defining of the turbulence arising mechanism for the Hagen-. Obvious contradiction with experiments corresponding to the paradox now is used to be coped with based on an assumption of permissibility of the HP flow instability with respect to disturbances having sufficiently large finite amplitude strict non-linear mechanism only [5,6,7,8,9,10] The basis for such the assumption (see [3, 4]) gives one side interpretation of experiments [11] in which many-fold increase of the threshold Reynolds number value Reth up to 100000 is achieved due to the increase of the level of smoothness of the streamlined pipe surface. This confirms expected above similarity of their viscous dissipative realization mechanisms
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