КВАЗИ-ЛАГРАНЖЕВОЕ ИНТЕГРИРОВАНИЕ ДВУМЕРНЫХ УРАВНЕНИЙ ПРАНДТЛЯ И БУССИНЕСКА В НЕВЯЗКОМ ПРЕДЕЛЕ

Abstract

For the hydrodynamic type systems locally describing incompressible twodimensional fluid flows without dissipation we suggest quasi-Lagrangian approach for their integration. This method is based on application of incomplete Legendre transformation when the independent variables become a Lagrangian invariant (i.e. the constant quantity along the fluid particle trajectory), instead of one of the spatial coordinate, and the rest ones which are another spatial coordinate and time. Thus, this method is based on the inverse transform of one of the spatial coordinate and by this reason differs from the the complete Legendre transformation. The classical example of the complete Legendre transformation is the Hodograph transformation applying to solve the equations for one-dimensional isoentropycal gas flows. In this paper it is shown that equation for the Lagrangian invariant after applying the incomplete Legendre transformation and introducing the stream function transforms into the linear eqation can be resolved by means of the generating function introduction. This method is turned to be effective for solving the inviscid twodimensional Prandtl equation (this equation describes the boundary layer behavior) that allows one to integrate this equation completely. In the case of the constant pressure along the boundary the parallel velocity component represents the Lagrangian invariant. The obtained solution is written through the initial data and satisfies the non-penetrate boundary condition. Analysis of this solution shows the formation of the singularity for the velocity gradient on the wall. This singularity appears as the result of breaking. At the breaking point the velocity gradient tends to infinity according to the power ~(t0–t)-1 where t0 is the singular time. This solution describes the appearance of the folding type singularity. It is shown also that the Prandtl equation admits complete integration for arbitrary dependence of pressure on the longitudinal coordinate. The simplest solution is written for the case of the constant pressure gradient. For the Boussinesq system the equation for density can be resolved by this method that reduces to one equation for the generating function. This work was supported by the RAS Presidium Program «Nonlinear dynamics: fundamental problems and applications».