The paper discusses the description of vortex flows of an ideal incompressible fluid based on the formalism of Lagrangian mechanics. Using the displacement field of liquid particles as a generalized coordinate, we write out the Lagrangian describing the dynamics of small perturbations (Kopiev, Chernyshev, 2018). The corresponding Lagrange equations are the equation for the displacement field (Drazim, Reid, 1981): This equation is equivalent to the Helmholtz equation for vorticity perturbations. The displacement field is defined as the difference in the positions of liquid particles on trajectories in disturbed and undisturbed flows. Although this definition is given in terms of Lagrangian variables associated with liquid particles, the displacement field itself is an Euler variable, expressed through velocity and vorticity perturbations. An example of using Lagrangian to solve the problem of conservation of the quadrupole moment of a vortex flow is considered. Using the Noether theorem, conditions on a stationary flow are obtained, under which the quadrupole moment of small perturbations of this flow is an integral of motion (Kopiev, Chernyshev, 2018). It is shown that these conditions are satisfied for the jet flows uniform along the longitudinal coordinate. The result obtained is important in aeroacoustics due to the fact that the quadrupole moment of the vortex flow represents the main term of the decomposition of a compact acoustic source in Machnumber (Lighthill, 1952; Crow, 1970; Kopiev, Chernyshev, 1995).
The generalization of these results to the nonlinear case is considered. The Lagrangian is obtained for an arbitrary nonlinear displacement field: nowhere Gis Green’s function of the Laplace equation. The corresponding Lagrange equations coincide with the differential equations describing the nonlinear dynamics of the displacement field (Drazin, Reid, 1981). Expansion of the Lagrangian in small perturbations to quadratic terms gives the Lagrangian of the linear system. The question of the relationship of the proposed approach to the description of the dynamics of an incompressible fluid and known approaches based on the formalism of Lagrangian mechanics with the coordinates of liquid particles as generalized coordinates (Chapman, 1978; Goncharov, Pavlov, 2008; Kuznetsov, Ruban, 1998) is considered. It is shown that the transformation of the Lagrangian obtained in (Kuznetsov, Ruban, 1998) to the Lagrangian can be carried out by transforming Lagrangian variables (coordinates of liquid particles) to Eulerian variables (displacement field).
This study was supported by the Russian Science Foundation, project No. 17-11-01271.
LAGRANGIAN FORMALISM IN PROBLEMS OF SMALL OSCILLATIONS OF VORTEX FLOWS AND ITS CONNECTION WITH THE VARIATIONAL PRINCIPLE FOR IDEAL INCOMPRESSIBLE HYDRODYNAMICS OF VORTEX LINES
