Stability of Inviscid flows through a given domain

Our main results are:
1. The perturbations of an inviscid flow in the form of point vortices are considered. It leads to the striking discovery of the trapping of a vortex: a point vortex can never leave a channel. Further asymptotic and numerical considerations show that the trapping also takes place for finite vortex patches, where only a part of vorticity can be trapped, while its other part leaves the domain. We believe that the trapping of vorticity represents a universal phenomenon for any outlet boundary conditions; it essentially determines the dynamics of vorticity in a gap. In particular, it makes the formation of recirculation zones or the self-oscillating flow regimes for the cases of large enough initial perturbations unavoidable.
2. The studies of smooth perturbations of an inviscid flow are performed with the use of Arnold’s functional. For two broad classes of flows they provide the decreasing of Arnold's functional that immediately leads to the general criterion of stability and to the nonlinear stability of these flows. Those results represent the generalizations of Rayleigh's inflection point theorem (and related stability theorems by Fjortoft and Arnold) on the channel flows with given inlet and outlet. Hence, they allow us to use Arnold’s stability theory for the essentially new classes of flows that are important in applications.
3. We have proven that for the finite smooth perturbations with small enstrophy (enstrophy=mean-square vorticity) the phenomenon of `washing-off' (i.e. the asymptotic stability) of a basic inviscid flow takes place. Here we have calculated the asymptotic stability threshold (more exactly—its low bound) with the non-dimensional enstrophy of perturbations of order 0.01. This step represents a keystone in our theory: it is the first known result of this kind and its value can be improved in future.
4. The high-Reynolds viscous flows in a duct (oscillating and non-oscillating) have been described with the use of the Vishik-Lyusternik method. The use of this powerful method leads to the global and uniformly valid solution in the whole flow domain. In particular, for steady flows it gives the complete description of the boundary layers at the inlet and outlet that are not accessible for the conventional boundary layer theory. Simultaneously it produces the flows outside the boundary layers. For the oscillating boundary conditions this procedure produces both vibrational boundary layers and the proper generalization of the steady streaming phenomena. An important advantage of our theory is: it does not produce any secular terms in a velocity field on the contrary to the known theory based on the conventional boundary layer equations. The inviscid oscillating flows are considered. Here we give the general theory of an inviscid oscillating flow through a duct including the derivation of the averaged equations of motion and their solutions. We have solved the generalized Bjorknes problem where a solid of arbitrary shape moves in the velocity field of a point source that possesses both constant and oscillating components.
5. We have developed and tested the effective numerical method for the description of the inviscid flows in a duct. It represents a version of the method of particles (or vortex method) adapted to the cases when material particles enter and leave the flow domain.
6. The decreasing of Arnold's functional allows us to establish the analogy with finite-dimensional dissipative systems. This analogy suggests that one can expect three qualitatively different scenarios (regimes) of the flow developments: (i) the evolution to a trivial equilibrium (i.e. the asymptotic stability or the `washing-off’ vorticity), (ii) the forming of steady structures (i.e. the trapping of vorticity with the forming of various recirculation zones), and (iii) self-oscillatory flows. We obtain several numerical solutions that explicitly show examples of all three listed scenarios.