Unsteady flow of bubble liquid in hydraulic systems of aircraft and helicopters

Abstract

The subject of this work is the phenomenon of a water hammer in a liquid that contains a small volume of gas bubbles. Historically, this phenomenon began to be studied as the dynamics of gas bubbles (Rayleigh-Pleset equation). Today, thanks to progress in computer technology, this phenomenon is studied at the level of bubble deformation during hydraulic shock. Another approach is to consider the dynamics of a multiphase (two-phase) medium in the form of a bubbly liquid. After several assumptions, the main one being a relatively small gas content in the liquid, the model consists of two differential equations with respect to the shock wave propagation speed and the resulting pressure perturbations. The specified system of equations differs from the corresponding classical water hammer equations: they consider the convection of the velocity field. In addition, the friction of the liquid against the wall according to the Weisbach-Darcy model is considered. Because of the small content of gas bubbles, the Weissbach-Darcy friction is approximated in the same way as in a homogeneous liquid, i.e., in a certain sense, greater than the real friction. Maybe that is why more or less physical results are obtained only for small values of the dimensionless parameter responsible for the friction of the liquid against the wall. It concerns the non-contradiction of the assumptions and the results obtained on their basis. Thus, in the front region of the shock pulse, where the pressure increases, the radial velocity of the bubbles is negative; however, for relatively large values of the friction parameter, the maximum pressure disturbance moves from the center of the shock pulse. This contradicts the assumption about compression: after passing the maximum pressure, gas bubbles expand due to a decrease in pressure. The graphical dependence obtained in this study are compared with the results related to a homogeneous liquid. They agree, but the shock pulse in a bubbly liquid is not as concentrated in space as that in a homogeneous liquid. Its length is 10-12 times greater than the corresponding value in a homogeneous liquid. Research methods are purely theoretical. The well-known bubble liquid model is used as a single-speed model continuum. Differential equations are solved analytically, approximately (series expansion), and numerically. In addition, the original approach of obtaining an analytical solution of an autonomous system is used-finding the function of pressure disturbances from the velocity of propagation of the shock pulse (and vice versa). Conclusions. A simple one-dimensional hydraulic model of shock wave (impulse) propagation in a bubbly liquid is proposed. In contrast to classical ideas (solutions) about a water hammer, which consists of two waves of opposite directions of propagation, a shock pulse is a region of pressure disturbances in which the speed of motion of fluid particles is also variable – from the maximum value to almost zero.