PSI - Issue 32

R. Tsvetkov et al. / Procedia Structural Integrity 32 (2021) 209–215

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R. Tsvetkov et al./ Structural Integrity Procedia 00 (2021) 000 – 000

measured by sensors. The HLS can be equipped with any necessary number of sensors-vessels and it is not required to provide a line of sight between the sensors. In the practical use of such systems, the questions always arise of what is the optimal position of the sensors, what should be the dimensions of the sensors and connecting pipes, which liquid should be chosen. The properties of the liquid should be as little dependent on temperature and time as possible. There is no ideal liquid for leveling. Its choice is based on a compromise, due to the operating conditions of the system and the required parameters, such as the dimensions of the system, the operating temperature range, etc. Important parameter of the system is the interval required for liquid equilibration. It determines the maximum possible frequency of measurements. The system does not have the ability to register processes with higher speeds. The paper proposes a mathematical model of transient processes in the hydrostatic leveling system associated with mechanical impact on some of its elements. Analysis of the obtained numerical solutions allows one to choose the optimal parameters of the system (liquid properties, radii of hoses and vessels, etc).

Nomenclature h 0

average height of the liquid column in measuring vessel

the height of the liquid column in the i -th measuring vessel, normalized by h 0

h̃ i R R̃

radius of horizontal connecting segment radius of horizontal segment, normalized by h 0 radius of measuring vessel, normalized by h 0 length of the i -th horizontal connecting segment radius of measuring vessel

r r̃

L i L̃ i

length of the i -th horizontal connecting segment, normalized by h 0

character time

τ

dimensionless velocity in the i -th connecting segment dimensionless velocity in the i -th measuring vessel pressure drop between i and i- 1 connecting segment

Ṽ i ṽ i P i H̃ i

dimensionless pressure

2. The hydraulic model of HLS The description of dynamic processes in a liquid is generally based on the Navier-Stockes equation. The position of liquid free surface in measuring vessels may be calculated by the Volume of Fluid Method described in Hirt (1981). However, numerical simulation of fluid flow in very long small-diameter tubes with a large number of free surfaces requires very large computational resources. The simplified hydraulic models are often used to describe transient processes in such systems. Analytical formulas for evaluating dynamic processes in a tiltmeter (representing a 2-point HLS) wer e used by d’Oreye (2005) and Boudin (2008). A similar approach was proposed by Manukin (2014) when modeling long base hydrostatic leveling system. Dimensionless version of hydraulic model of the multipoint hydrostatic leveling system was developed by Tsvetkov (2020). In this model, HLS includes N vertical measuring vessels and N horizontal connecting segments (hoses). One of the vertical vessels is significantly larger than the others. This vessel is used as a storage tank. It is necessary to maintain the required liquid level in the measuring vessels and to mix the liquid in the system. Fig. 1 shows a fragment of HLS, which is a vertical measuring vessel with two adjacent horizontal connecting hoses.