New Lagrangian view of vorticity evolution in two-dimensional flows of liquid and gas

Abstract

Purpose of the study is to obtain formulas for such a speed of imaginary particles that the circulation of the speed of a (real) fluid along any circuit consisting of these imaginary particles changes (in the process of motion of imaginary particles) according to a given time law. (Until now, only those speeds of imaginary particles were known, at which the mentioned circulation during the motion remained unchanged). Method. Without implementation of asymptotic, numerical and other approximate methods, a rigorous analysis of the dynamic equation of motion (flow) of any continuous fluid medium, from an ideal liquid to a viscous gas, is carried out. Plane-parallel and nonswirling axisymmetric flows are considered. The concept of motion of imaginary particles is used, based on the K. Zoravsky criterion (which is also called A. A. Fridman’s theorem). Results. Formulas for the speed of imaginary particles are proposed. These formulas include the parameters of the (real) flow, their spatial derivatives and the function of time, which determines the law of the change in time of the (real fluid) velocity circulation along the contours moving together with the imaginary particles. In addition, it turned out that for a given function of time (and, as a consequence, for a given law of change in circulation with respect to time), the speed of imaginary particles is determined ambiguously. As a result, a method is proposed to change the speed and direction of motion of imaginary particles in a certain range (while maintaining the selected law of changes in circulation in time). For a viscous incompressible fluid, formulas are proposed that do not include pressure and its derivatives. Conclusion. A new Lagrangian point of view on the vorticity evolution in two-dimensional flows of fluids of all types is proposed. Formulas are obtained for the velocity of such movement of contours, at which the real fluid velocity circulation along any contour changes according to a given time law. This theoretical result can be used in computational fluid dynamics to limit the number of domains when using a gridless method for calculating flows of a viscous incompressible fluid (the method of viscous vortex domains).

Keywords

• velocity of contours motion
• circulation of velocity
• velocity of imaginary particles
• zoravsky criterion
• friedmann’s theorem
• method of viscous vortex domains