=Paper=
{{Paper
|id=Vol-2837/short5
|storemode=property
|title=Numerical solution of the one-sided compressor thrust bearing dynamics equation (short paper)
|pdfUrl=https://ceur-ws.org/Vol-2837/short5.pdf
|volume=Vol-2837
|authors=Pavel E. Fedotov
}}
==Numerical solution of the one-sided compressor thrust bearing dynamics equation (short paper)==
https://ceur-ws.org/Vol-2837/short5.pdf
Numerical solution of the one-sided compressor thrust bearing
dynamics equation
Pavel E. Fedotova
a
Kazan Federal University, 18 Kremlevskaya Street, Kazan, 420008, Russia
Abstract
The aim of this work is to construct grid algorithms for solving nonstationary second-order
partial differential equations that arise when modeling problems of the hydrodynamic theory
of lubrication of thrust bearings. In constructing the grid schemes in the parts of the bearing,
the finite element method and the version of discontinuous Galerkin method were used. To
solve the pressure equation, the method of adder identities is used. To obtain a solution in a
thrust bearing, a domain decomposition method is built based on the Lions method.
Numerical experiments were performed demonstrating the convergence of the grid scheme of
the Galerkin method on a sequence of condensing grids. A set of programs was built with the
help of which it is possible to study the behavior of the bearing at various geometric and
physical parameters. Determine lubricant consumption and bearing capacity over time.
Keywords 1
boundary value problem, thrust bearing, partial differential equations, heat equation,
discontinuous Galerkin method, domain decomposition,
1. Introduction
Thrust plain bearings are an important design element for centrifugal and screw compressors. They
are designed to take the axial load acting on the rotor, transfer it to the stator, and also to fix the rotor
relative to the housing in the axial direction.
The lubricant flow in the lubricant layer of thrust bearing is mathematically described by a system
of nonlinear differential equations. In this work, we use models of lubricant flow in bearings,
proposed by Kazan mathematicians Sokolov, Khadiev and Maximov [1, 2].
Thrust bearings used in compressors have fixed pads and a rotating collar, between which
lubricant flows. The surface of the pad is profiled and experiences thermal deformations. Therefore,
the thickness of the gap of the lubricating layer is variable. Between the pads are channels through
which lubricant is supplied. In the collar and bearing pads, it is assumed that the linear heat equation
is satisfied. In the lubricating layer, the pressure distribution is described by the two-dimensional
Reynolds equation, and heat transfer is described by the three-dimensional energy equation with
dominant convection. In this case, there is no thermal conductivity in the radial direction in the
lubricating layer.
The temperature distribution in the collar and pads is described by the linear heat conduction
equation. In the lubricating layer, the pressure distribution is described by the two-dimensional
Reynolds equation, and heat transfer is a three-dimensional nonlinear energy equation with dominant
convection.
When modeling bearing dynamics, it is convenient to use a cylindrical coordinate system. We
denote by the coordinate axes corresponding to the radius, angular coordinate, and thickness of the
lubricant layer.
Far Eastern Workshop on Computational Technologies and Intelligent Systems, March 2–3, 2021, Khabarovsk, Russia
EMAIL: paulfedotov@mail.ru (P.E. Fedotov)
© 2021 Copyright for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR Workshop Proceedings (CEUR-WS.org)
2. Issue statement
The two-dimensional equation determining the pressure distribution has the form
p p
r rf0 + f 0 = f , x , (1)
r r r
where p — pressure function, p and f 0 , f — coefficients depending on the temperature of
lubricant layer. At the boundaries of the region, the pressure is set.
The energy equation in the lubricating layer, collar and pad has the form
( t )
b + div(Vt − K t ) = f , x , (2)
where the velocity vector and the thermal conductivity in the lubricating layer has the form
Vr 0 0 0
V = V , K = 0 K K y .
V 0 K K yy
y y
The speed in the pad area is assumed to be zero, V = (0,V ,0)T in the collar area and K —
diagonal matrix.
We supplement equation (2) with boundary conditions. By coordinate in the collar and
lubricant layer we assume that the periodicity condition is fulfilled. In the y direction, at the
boundaries between the lubricating layer and the solid bearing elements, we set the condition for
equal temperatures and heat fluxes. Within the inter-air channel, at the conditional boundary
y = h(r , ) of the lubricating layer, the temperature of the inflowing lubricant is set. At other
boundaries of solid elements, a condition of the third kind of heat exchange with the environment is
set. The equation in the lubricating layer has the following feature. By the variable r , there is only a
convective term. It is assumed that the velocity Vr determines the outflowing flows on the external
and internal radii of the region [1]. Therefore, boundary conditions are not set there. The temperature
of the inflowing grease is considered set.
3. Solution methods
The thickness of the lubricating layer h is a variable in connection with the profiling of the pad, as
well as thermal deformations. Therefore, we first make a change of variables
r = r , = , y = h(r , ) y , that translates the original computational domain into a rectangular one. A
change of variables was performed, preserving the divergent form of the equation.
To numerically solve the obtained boundary value problem in the lubricating layer, a scheme of
the discontinuous Galerkin method with rectangular elements is constructed. The choice is determined
by the local conservatism of the grid schemes built on its basis, low circuit viscosity, and also their
stability for a wide class of problems with dominant convection in a wide range of grid
parameters [3]. In the constructed scheme, piecewise constant inside the computational domain h
and piecewise linear near the boundary y of the space of approximating functions were used. The
choice of this type of approximating functions makes it possible to significantly reduce the amount of
resources required for the calculation, without a tangible loss in the quality of the solution. A method
of constructing schemes of this kind is given in [4].
The grid scheme allows us to reduce issue to a grid equation, which in operator notation has the
form
( uh )
B + ( Av + Aq + A )uh = F + Fg , (3)
where uh is a grid approximation of the temperature function t .
The operators in equation (3) are defined by the following forms
Av u wh = Av uwh dx = ( −uV wh )dx + \ y [u+ p (V p ) − u− p (V p ) ]( wh , + p − wh , − p ) dx,
− +
K h K
Aq u wh = ( qh wh )dx + \ y ( wh ,+ p − wh ,− p )qh ,+ p pdx,
K h K
A u wh = uwh dx, Bu wh = buwh dx,
y Kh K
F wh = fwh dx, Fg wh = gwh dx
K h K y
for any wh from the space of approximating functions. Here h is the set of finite elements of the
region , K h — finite element; p — unit normal vector to the boundaries of the elements of the
partition area, oriented so that e p 0 , e = (1,1,1) / 3 , w — positive or negative part of the
function w . The spaces of approximating functions contain discontinuous functions. Here, the
symbols w p denote the limit values of the functions of the partitioning elements adjacent to the
boundary from the side p . Equation (3) is approximated by an implicit scheme for which an
iterative process is constructed with lowering to the lower time layer.
For the numerical solution of equations in solids, the finite element method was used. To
approximate equation (1), the method of adder identities is used. For solution obtained by the grid
method, the upper relaxation method is used.
In order to consider heat transfer between regions, a method based on the Lyons method of
decomposition of regions is constructed [5, p. 59]. Iterations of the decomposition method for two
regions, with different speeds and thermal conductivity coefficients, are given by the equations:
div(V1u1k +1 − K1u1k +1 ) = f , div(V2u2k +1 − K 2u2k +1 ) = f ,
−(V1u1k +1 − K1u1k +1 ) n1 + u u1k +1 = 1k , −(V2u2k +1 − K 2u2k +1 ) n2 + u u2k +1 = 2k , (4)
1k +1 = −2k + 2u u2k +1 , 2k +1 = −1k + 2u u1k +1 ,
where n1 , n2 — normal vectors to the boundary of the first and second regions, Vi , K i , ui — velocity
vector, thermal conductivity coefficient and temperature of the i-th region.
4. Results
To solve the constructed grid circuits, a set of C ++ programs was created using the Eigen 3.0 class
library [6, 7]. With its help, numerical experiments were carried out, demonstrating the convergence
of schemes on condensing grids. In numerical experiments, the approximate solution uh was
compared with the exact solution t of the issue test model. Figure 1 shows a graph of the solution
error depending on the grid step and number of points of partition of space and time n. The figure
shows that there is a convergence of the constructed method with linear velocity with increasing n .
Figure 1: Graph of the solution error depending on the number of points n and grid step h .
The sought quantities in the problem under consideration are temperature and pressure. To find
them with real parameters, the methods and program complexes described above are used. Figure 2
shows a graph of isobars, and Figure 3 shows a graph of isotherms in the lubricating layer of a thrust
bearing when considering the stationary version of the problem.
Figure 2:Graph of isobars.
Figure 3:Graph of isotherms.
The decomposition method (4) is used to solve the problem. Figure 4 shows a graph of the
temperature distribution at fixed r and at the centers of the calculated areas of the pad, collar and
lubricant layer.
Figure 4: Graph of the temperature distribution at fixed r and .
During bearing operation, the disc moves in a vertical direction. A set of programs allows to study
the behavior of the bearing at various geometric and physical parameters over time. The Figure 5
shows graphs of changes in bearing characteristics during sinusoidal collar motion. The dots mark the
characteristics under consideration, and the line shows the thickness of the lubricating layer during the
movement of the disk.
Figure 5: Graphs of changes in bearing characteristics during collar motion.
5. Acknowledgements
This paper has been supported by the Kazan Federal University Strategic Academic Leadership
Program.
6. References
[1] N. Sokolov, M. Khadiev, T. Maksimov, E. Fedotov, P. Fedotov, Mathematical modeling of
dynamic processes of lubricating layers thrust bearing turbochargers. J. Phys.: Conf. 1158 (2019)
[2] N. Sokolov, M. Khadiev, T. Maximov, E. Fedotov, P. Fedotov, Mathematical modeling of a
dynamically loaded thrust bearing of a centrifugal compressor, Actual problems of marine
energy: materials of the eighth international scientific and technical conference (2019) 307–311
[3] B. Cockburn, C. Shu, The local discontinuous Galerkin method for time-dependent convection-
diffusion systems, Hampton (VA): NASA Langley Research Center. Institute for Computer
Applications in Science and Engineering. (1997)
[4] E. Fedotov, Limit Galerkin-Petrov schemes for the nonlinear convection-diffusion equation,
Differential equations 46 (2010) 1033–1043.
[5] V. Dolean, P. Jolivet, F. Nataf, An Introduction to Domain Decomposition Methods: algorithms,
theory and parallel implementation, Master, France, 2015.
[6] Sparse linear algebra. Eigen., 2020. URL:
http://eigen.tuxfamily.org/dox/groupsparse/chapter.html.
[7] P. Fedotov, E. Fedotov, N. Sokolov, M. Khadiev, Sm2px3tx𝜏 - dynamically loaded thrust plain
bearing when setting a direct problem., 2020. Certificate of the state registration of a computer
program No. 2020615227.