Unsteady flow of droplet liquid in hydraulic systems of aircrafts and helicopters: models and analytical solutions

Abstract

The subject of this study is the unsteady flow of liquid in pipelines, which are part of the design of airplanes and helicopters. This name means, first of all, the phenomenon of a sharp increase in pressure in the pipeline, which is known as a hydraulic shock. Although we have already learned to deal with this phenomenon in some parts of the systems, in many structural elements (flexible pipelines), inside which the working pressure reaches several hundred atmospheres, this phenomenon is still quite dangerous. As you know, the best way to deal with an unwanted phenomenon is through theoretical study. To date, there has been a huge amount of work in the direction of hydraulic shock research. This article does not fully cover these studies. It is limited to references to reviews and relevant works. Because the phenomenon of hydraulic shock has a significantly nonlinear character, analytical solutions of systems of equations corresponding to the simplest models were unknown until recently. This work presents, as an overview, already known analytical solutions describing the process of shock wave propagation. Most importantly, new achievements are given, both for the inviscid approximation and for considering internal viscous friction. It is shown that the internal friction within the considered model is negligible almost everywhere, except for the thin shock layer. The asymptotic is proportional to the tangent function and inversely proportional to the square root of the product of the Reynolds number and the dimensionless parameter characterizing the convection effect. Convection of the velocity field significantly affects the distribution of characteristics in hydraulic shock. If the self-similar solutions that were obtained earlier have a power-law character for the velocity distribution in the shock wave, then the simultaneous consideration in the model of convection and friction on the pipeline walls (according to the Weisbach-Darcy model) made it possible to obtain a distribution in the form of an exponential function that decays with increasing distance from the shock wave front. In addition, the work includes an original approach to solving a nonlinear system of differential equations that describes the propagation of a shock wave without considering the friction on the walls. Analytical solutions were obtained in the form of a function of pressure versus the velocity of shock wave propagation. Research methods. This work uses purely theoretical approaches based on the use of well-known fluid flow models, methods of analytical solution of differential equations and their systems, asymptotic methods, derivation of self-model equations, and finding their solutions. Conclusions. Analytical solutions of systems of differential equations were obtained, which describe models of hydraulic shock without considering viscous effects. A comparison of the obtained results with the results of other studies is given.