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).
Highlights
Достаточно, чтобы был известен закон изменения этой циркуляции во времени
Б. Второе интегральное обобщение инварианта Крокко для 3D-течений за отошедшим головным скачком // Вестник Самарского государственного технического университета
Summary
Следуя [11,12,13,14,15,16,17], для формулировки утверждений будем пользоваться представлением о движении внутри жидкости воображаемых частиц, предложенным в [11]. В области G рассмотрим также течение воображаемой жидкости, частицы которой движутся со скоростью U(x, y, z, t). Частицы воображаемой жидкости не взаимодействуют с частицами реальной жидкости и не влияют на ее движение. Пусть в области G в течение интервала времени (t1, t2) завихренность реальной жидкости Ω и скорость воображаемой жидкости U связаны уравнением. В этом случае из критерия Зоравского [18,19], который также называется теоремой Фридмана [20], следует, что в интервале (t1, t2) отрезки вихревых линий и вихревых трубок перемещаются вместе с частицами воображаемой среды, движущимися со скоростью U, а интенсивность вихревых трубок (циркуляция Γ скорости V по контуру, один раз опоясывающего трубку) сохраняется (до тех пор, пока эти частицы находятся внутри области G).
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