PSI - Issue 32

R. Tsvetkov et al. / Procedia Structural Integrity 32 (2021) 209–215 R. Tsvekov et al. / Structural Integrity Procedia 00 (2021) 000 – 000

211

3

Fig. 1. (a) Drawing of the HLS part; (b) Drawing of the HLS part with dimensionless parameters.

The system of equations describing the motion of a fluid in HLS is written in dimensionless form. It includes the following equations: a pressure balance equation for measuring modules (1) and connecting tubes (2); a kinematic equation (3) for determining the relationship between the fluid velocity in the vessel and the rate of change in the height of the liquid column; a continuity equation (a mass balance of a fluid) that balances fluid flow rates at the junction between the vessel and adjacent tubes (4): Dimensionless variables are introduced as follows: geometric characteristics are related to the height of the liquid column in the measuring vessel in a stable state ( h 0 ), the time is divided by the characteristic time τ , determined by the ratio of inertial and viscous forces in the fluid flow, acceleration is related to gravitational acceleration g , velocity – to τg , pressure – to ρgh 0 .

dv

0 ,

i dt − + − = i i i h H h

(1)

dV

(2)

0 ,

i + − + − − L

1 i i i H H L K V i i

=

i

dt

2

dh g

h  =

(3)

,

v

i

i

dt

0

( )

( )

2

2

( )

2

1 i i V R V R v r − − + − = . 0 i i

(4)

Character time τ and dimensionless pressure H̃ depends on the viscosity μ and density ρ and gravitational acceleration g :

2 / 8 R    = ,

(5)

0 i i H P gh  = . /

(6)

Equations (1)-(4) are written under the assumption that the fluid is ideal, and the fluid velocity is constant in the cross-sectional areas of all tubes and vessels. These assumptions are valid because a low-viscosity liquid is usually used to fill levels, and the length of the connecting tube is much less than its radius L>>R . However, the viscous properties of the fluid are important in transient processes. We made the assumption that viscous friction must be taken into account when the fluid moves in the connecting tubes, but it can be neglected for the measuring vessels because the total length of the tubes is significantly greater than that of the vessels. Adding the last term to equation (2) makes it possible to take this fact into account. The friction coefficient K depends on the viscosity μ and density ρ and the oscillation frequency ω :