MATHEMATICAL MODEL OF FLUID FREIGHT BEHAVIOR DURING TRANSPORTATION
UDC 519.87:621.6
RUZHYTSKYI I.,
candidate of physics and mathematics sciences, senior lecturerof Kyiv National University of Trade and Economics
MATHEMATICAL MODEL OF FLUID FREIGHT BEHAVIOR DURING TRANSPORTATION
Background. Nowadays the problem of joint movement of systems that consist of reservoirs of spherical and ellipsoidal shape partially filled with liquid is not studied enough.
The aim of investigation is the numerical implementation for the model system consisting of a reservoir partially filled with fluid which will be suitable for use in the study of processes in axisymmetric non cylindrycal cavities.
Material and methods. The research is based on the combined use of analytical methods of nonlinear dynamics and variational methods of mathematical physics to the problem in variational formulation. Description of the problem is performed using non-Cartesian parameterization for the domain occupied by liquid. The method of an auxiliary domain was used to satisfy conditions of solvability of the problem and for construction of the coordinate functions that satisfy the condition of not overflow not only for the level of unperturbed state of fluid on the solid boundaries, but also on tank walls, where crests of waves can reach. Discrete model of minimal dimension which is derived for ellipsoidal forms of reservoir from the combined use of classical variational principle and the method of modal decomposition is obtained based on the analytical exclusion of all kinematic ties before solving nonlinear variational problem.
Results. The steps that are performed to solve the problem are described. The choice of analytical approaches of use for modal decomposition methods for modeling nonlinear vibrations of fluid with free surface in the case of ellipsoidal shape reservoir and subsequent numerical implementation upon receipt of a second-order differential equations for the model is justified. Linearity of the system relative to the second derivatives of unknown quantities allowed to organize the computational process in which every step of the numerical integration of systems of differential equations, using a computer, converted to Cauchy normal form, and then using standard Runge-Kutt method, numerical integration over time is performed. The case when the system consists of a spherical reservoir and fluid that partially fills it, based on the model using five coordinate functions that made it possible to draw conclusions about the compliance of the obtained coefficients, with coefficients of equations in a form that is presented in the works of Academician of NAS of Ukraine I. O. Lukovskii.
Conclusion. It is concluded that the developed methods and algorithms, and the obtained results of numerical simulations, can be used to analyze the dynamics applications at aviation and space systems, and systems associated with the transportation of liquid cargo. The obtained results of the study will help to: improve the development of control algorithms and analysis of the development of processes of compliant motion of fluid and reservoir in nonstationary regimes; explain and predict complex processes of wave generation and interactions of components in the system, "ellipsoidal reservoirfluid with free surface".
Key words: mathematical model, fluctuations of fluid, reservoir, ellipsoidal shape.
MATHEMATICAL MODEL OF FLUID FREIGHT BEHAVIOR DURING TRANSPORTATION
RUZHYTSKYI I.,
candidate of physics and mathematics sciences, senior lecturerof Kyiv National University of Trade and Economics
MATHEMATICAL MODEL OF FLUID FREIGHT BEHAVIOR DURING TRANSPORTATION
Background. Nowadays the problem of joint movement of systems that consist of reservoirs of spherical and ellipsoidal shape partially filled with liquid is not studied enough.
The aim of investigation is the numerical implementation for the model system consisting of a reservoir partially filled with fluid which will be suitable for use in the study of processes in axisymmetric non cylindrycal cavities.
Material and methods. The research is based on the combined use of analytical methods of nonlinear dynamics and variational methods of mathematical physics to the problem in variational formulation. Description of the problem is performed using non-Cartesian parameterization for the domain occupied by liquid. The method of an auxiliary domain was used to satisfy conditions of solvability of the problem and for construction of the coordinate functions that satisfy the condition of not overflow not only for the level of unperturbed state of fluid on the solid boundaries, but also on tank walls, where crests of waves can reach. Discrete model of minimal dimension which is derived for ellipsoidal forms of reservoir from the combined use of classical variational principle and the method of modal decomposition is obtained based on the analytical exclusion of all kinematic ties before solving nonlinear variational problem.
Results. The steps that are performed to solve the problem are described. The choice of analytical approaches of use for modal decomposition methods for modeling nonlinear vibrations of fluid with free surface in the case of ellipsoidal shape reservoir and subsequent numerical implementation upon receipt of a second-order differential equations for the model is justified. Linearity of the system relative to the second derivatives of unknown quantities allowed to organize the computational process in which every step of the numerical integration of systems of differential equations, using a computer, converted to Cauchy normal form, and then using standard Runge-Kutt method, numerical integration over time is performed. The case when the system consists of a spherical reservoir and fluid that partially fills it, based on the model using five coordinate functions that made it possible to draw conclusions about the compliance of the obtained coefficients, with coefficients of equations in a form that is presented in the works of Academician of NAS of Ukraine I. O. Lukovskii.
Conclusion. It is concluded that the developed methods and algorithms, and the obtained results of numerical simulations, can be used to analyze the dynamics applications at aviation and space systems, and systems associated with the transportation of liquid cargo. The obtained results of the study will help to: improve the development of control algorithms and analysis of the development of processes of compliant motion of fluid and reservoir in nonstationary regimes; explain and predict complex processes of wave generation and interactions of components in the system, "ellipsoidal reservoirfluid with free surface".
Key words: mathematical model, fluctuations of fluid, reservoir, ellipsoidal shape.
MATHEMATICAL MODEL OF FLUID FREIGHT BEHAVIOR DURING TRANSPORTATION