### Butymova L. Модорский В.Я.

## Development and application of a unified algorithm for solving the interdisciplinary problem of modeling aeroelastic processes in the labyrinth seal of centrifugal compressors

### Reporter: Butymova L.

To ensure a contactless connection between a rotating rotor and a stationary body in aircraft engines [16], high-pressure pumps [13, 14], etc., labyrinth seals (LS) are used. In labyrinth seals, the working medium is sealed by throttling it when moving through successively located contractions and extensions.

The main task of the LS is to ensure the tightness of the rotor, therefore the flow expansion and contraction of the flow in the LS are usually considered in the direction parallel to the axis of the rotor. However, in order to ensure aerovibration resistance, it is necessary to take into account the processes of the working medium motion that take place in the peripheral direction of the LS under the rotor vibrations. It should be noted that sequencing of the constrictions and expansions affects the oscillation amplitude in the gas-dynamic cavity between the LS and the rotor increasing the flow non-uniformity. Consequently, the refusal to take these elements into account in the aeroelastic calculation [15, 21] can give an additional margin from the point of view of reducing oscillations in the LS and, which is important for solving the related problems of mechanics of continuous media [19], reduce the complexity and time of calculation.

Based on the above said, the calculation of the LS is replaced by calculating the gap seal, equivalent (with margin) to the labyrinth seal, if we consider processes occurring in the circumferential direction of the LS.

IT is known that LS operation occurs at high temperatures and high rotation speeds. Under critical operating conditions the LS is influenced by significant loads from the gas-dynamic flow as well as the LS influences the gas-dynamic flow. The impact of this process is ambiguous and requires more detailed research. Publications related to vibrational processes in LS, consider the influence of precession [11], geometric characteristics of LS [12] and do not take into account the influence of gas-dynamic forces.

Vibrational gas-dynamic processes occurring in the LS with the rotor vibration caused, in turn, for example, by technological imbalances may lag behind the rotor vibrations in the phase. It is necessary to analyze the possibility of amplifying or weakening the rotor vibrations and the dependence of these processes on the LS characteristics [9-10].

The classical formulation of the vibration problem allows to take into account the influence of structural [20], physical-mechanical and technological parameters on vibrations, but does not allow for the influence of gas dynamic loads.

When considering the problem of the vibration effect on gas-dynamic processes in the labyrinth seals of a centrifugal model compressor of the gas transmittal unit in a dynamic related formulation it is possible to take into account the gas-dynamic factors [13, 17]. In addition, it becomes possible to calculate the oscillation parameters of gas-dynamic forces acting on the rotor [4-8].

To solve the related dynamic problem mathematical model is suggested based on using conservation equations of mass, momentum and energy in the differential form for gas and structure and closed by the state equation of compressible gas, ratios for displacements, geometric Cauchy ratios, generalized Hooke's law, as well as by the initial and boundary conditions [21].

Let us describe the problem solution steps. First, a finite-difference approximation of the original system of differential equations is carried out. The subsequent algorithm for solving the resulting system of finite-difference relations is determined by the solution method. As the basic one the method of large particles is chosen [22]. The main idea of the method consists in splitting the initial non-stationary system of Euler equations in physical processes, written in the form of conservation laws. The medium is modeled by a system of particles that coincide at the given moment with a cell of the Euler mesh. The stationary solution of the problem, if it exists, is obtained as a result of the establishment, and therefore the whole process of computation consists of repetition of the steps.

Each computational cycle, in turn, is divided into seven stages. The first three stages are designed to solve the gas dynamic problem, the next four steps are to evaluate the parameters of the dynamic SSS of the structure.

In the first stage (Eulerian), all the effects associated with the elementary cell movement are neglected (there is no mass flow across the cell boundaries), and we take into account the acceleration effects of the material only due to pressures. Here, for a large particle, the intermediate values of the sought flow parameters are determined;

In the second stage (Lagrangian), mass flows through the boundaries of Eulerian cells are calculated;

In the third stage (final), the final values of the flow parameters for each cell and the whole system at a fixed calculation mesh are determined at a new time instant.

The obtained parameters of the gas-dynamic flow are the initial data for the subsequent step in time and go into the calculation of the boundary conditions of the problem of the SSS structure estimating.

In the fourth stage (Eulerian) we ignore all the effects associated with the movement of the unit cell (there is no mass flow across the cell boundaries), and we take into account the acceleration effects of the material only due to stresses. Here, for a large particle, the intermediate values of the sought flow parameters are determined;

In the fifth stage (Lagrangian), mass flows through the boundaries of Eulerian cells are calculated;

At the sixth stage (final), the final values of the flow parameters for each cell and the whole system are determined at a new time instant on a fixed calculation mesh.

The seventh stage is new from the standpoint of traditional representations of the large particle method and includes algorithms for determining the movements, strains and stresses at each time step.

This concludes the computational cycle of one time step. The results of the calculation at this time step are the initial data for the subsequent step.

A special feature of the dynamic problems of the elasticity theory is a low level of the displacement velocity, which can lead to solution oscillations. To ensure the computation stability, in general, it is necessary to choose the right finite-difference scheme in determining the SSS parameters. Sometimes it is advisable to use asymmetrical difference schemes when determining stresses and strains. In the paper, schemes of the first order of accuracy are used, both in space and in time. It is required to satisfy the conditions of stability by Courant Friedrichs Levy.

Based on the results of the work done, the following conclusions can be drawn:

1. With the increase of 3 times compression wave speed arising from the convergence of the rotor with the LS surface under vibration the amplitude of gasdynamic forces increases by 2 times. The frequency does not change and is equal to 400 Hz. As the speed increases, you can expect a vibration increase in the LS.

2. When increasing the shaft diameter by 4 times a change in the maximum amplitude of the gas-dynamic force by 11 times is observed. The shaft diameters corresponding to the minimum and maximum values of the amplitudes of the gas-dynamic forces are obtained. It can be seen that as the rotor diameter increases, the nominal values of the gas dynamic force increase. This is due to the increase in the rotor area at a constant nominal pressure. In this case the maximum amplitudes of gasdynamic forces are observed when the oscillation frequency f of the rotor equal to the first natural frequency of the pressure circumferential fluctuations of the gas-dynamic cavity in the gap. Amplitudes of oscillations of gasdynamic forces are lower at the rotor frequency of oscillation f equal to the second natural frequency of the gas-dynamic pressure oscillations of the gas-dynamic cavity in the gap. The gas-dynamic forces oscillation amplitude at frequency f of the rotor oscillations equal to the natural frequency of the fourth circumferential gas-dynamic cavity pressure fluctuations in the gap are even lower. The amplitude of oscillations of the gas dynamic force at the frequency of natural oscillations of the gas-dynamic cavity at a non-multiple rotor frequency was also low.

3. Weak influence of the working fluid (air, methane) on vibration in LS can be noted.

4. With increasing elastic modulus of the LS material by 4 times from 50 GPa to 200 GPa the amplitude of pressure oscillations is reduced by 5 times from 0.97 MPa to 0.19 MPa, pressure oscillation frequency is increased by almost 2 times, from 134 kHz to 256 kHz.

5. Changing the rotor diameter, you can reduce vibration. Thus, it is possible to reduce the designated gaps in LS and reduce leakage, thereby increasing pump efficiency.

The study was performed by a grant from the Russian Science Foundation (project №14-19-00877).

REFERENCES

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