Characterization of stability limits of Ledinegg instability and density wave oscillations for two-phase flow in natural circulation loops

Abstract

A detailed study of excursive (or Ledinegg) instability and density wave oscillations (DWOs) is carried out for two-phase flow in a natural circulation loop. The maps in the parametric space have been obtained, which indicate excursive instability (Ledinegg instability) as well as density wave oscillations (DWOs). The dynamic (DWOs) and static (Ledinegg) stability boundary on these maps have been drawn. These diagrams dealing with the interaction of Ledinegg instability and DWOs have rarely been reported, and detailed mathematical analysis of these maps is lacking. In the present work, a detailed study of such interactions is carried out. It is also noted that the Ledinegg instability is a manifestation of saddle node bifurcation or limit point (LP) for the dynamical system of natural circulation loop. Density wave oscillations in the dynamical system of natural circulation loop can be observed with the Hopf bifurcation. The detailed bifurcation analysis of these instabilities shows that stability maps can be divided into three broad regions, the first region can only have DWOs (Hopf bifurcation), and the second region can have Ledinegg as well as DWOs (Hopf bifurcation and saddle node bifurcation co-exists), while the third region can have only Ledinegg instability (saddle node bifurcation). The first and second region are separated by a Cusp point (CP) and it is found that between CP and Bogdanov-Takens (BT) bifurcation point both Hopf and LP exist, due to presence of both DWOs and Ledinegg instability. The region beyond BT bifurcation point only has Ledinegg or excursive instability. Moreover, DWOs in the first region have two types of Hopf bifurcation depending on the nature of limit cycles. These subcritical and supercritical Hopf bifurcations, are separated by the Generalized Hopf (GH) bifurcation. Abovementioned three regions are observed only for the Type II DWOs stability boundary, whereas Type I DWOs have only subcritical and supercritical Hopf bifurcations. The impact of design parameters on the saddle-node curve along with the shifting of interaction point (Bogdanov-Takens bifurcation) have been investigated as well.