Investigation of viscous fluid flow in an eccentrically deposited annulus using CFD methods

The theory of fluid flow in an eccentrically deposited annulus has of great importance especially in the design of sliding bearings (axial, radial). If the geometry is more complex or shaft is deposited eccentrically, then a suitable alternative for design hydrostatic bearing is using ANSYS Fluent, which solves the general three-dimensional viscous fluid flow also in complex geometry. The problem of flow solves in the narrow gap between the cylinders in this paper, when the inner cylinder is stored with a defined eccentricity. The movement of the inner cylinder is composed of two motions (rotation, precession), i.e. rotation around its own axis and move along the circle whose radius is the size of the eccentricity. Addition the pressure gradient is considered in the axial direction. In the introductory section describes the methodology for defining of motions (rotation and precession of the inner cylinder) when the user function (UDF) is created that defines the rotation and move along the circle in C++. The above described methodology of the solution was then applied to the 3D model with a defined pressure drop when the problem was solved as a time-dependent with a time value corresponding to two turns of the internal shaft.


Introduction
Sliding bearings surface represent two planar or cylindrical surfaces which are each other moving and engineering designs are based on the theory of flow in the gap between these surfaces [6].If the geometry is complicated or there is eccentric shaft, then a suitable variant of hydrostatic bearings design the spatial viscous fluid flow solutions applicable in complex geometry using ANSYS Fluent is utilized.Thus, it can be very effectively to apply the design for different types of hydrostatic bearings (axial, radial, linear bearing, hydrostatic matrix) using the aforementioned program.The results can be used in the design of the seal in many applications, e.g. for pumps.
The challenge is to define a mathematical model of single-phase flow in a narrow gap between the cylinders with rotating of the inner one.This cylinder also held precession (circular motion).The inner cylinder is placed in the basic position with defined eccentricity.In the direction of the axis the pressure gradient is considered.The mathematical model of the flow is given by the equation of continuity and Navier -Stokes equations.The model is then solved numerically by finite volume method in ANSYS Fluent13.0[4].To create a geometry DesignModeler program was used and to create a computational grid ANSYS meshing.
Rotation and precession of the inner cylinder is given by user defined function (UDF) in C + +.Subsequently thus generated function is implemented in the program environment ANSYS Fluent.The user function is specified equations of inner cylinder center motion through equation of a circle, as it is also related to the subsequent definition of the moving computational grid (moving mesh).The above described solution methodology was tuned on a 2D model and then applied to the 3D model.In 3D model the role of the pressure gradient was also considered and the problem was solved as time-dependent.Time value corresponded to the two inner shaft speed.Numerical solutions were used to evaluate the size effect of rotation and precession in the radial and tangential forces acting on the shaft.The angular velocity of rotation and precession was entered in the range from 600 rev/min to 3000 rev/min.For individual variations of numerical simulations the radial and tangential force components acting on the shaft, depending on the rotation were evaluated.Consequently, at the conclusion from the forces process the mean values are plotted in the graph, depending on the angular velocity of precession rotation at a constant size.The results show polynomial dependence of the radial force and the linear dependence of tangential forces on the rotation at precession constant.

Geometry of eccentric narrow gap
Solved geometry is presented on figure 1 and figure 2. Region consists of outer and inner cylinder, which is placed eccentrically and input and output pocket is added.
The main dimensions are shown on figure 2. Input of flowing medium (water) in the narrow gap is illustrated in figure 2 (blue) at a defined pressure.The inner cylinder rotates around its own axis and also held precession (moving along the circle with a given eccentricity).Detailed analysis of the movement is described in Chapter 4. Due to the nature of the geometry the general threedimensional space is assumed.Type of flow is determined from the geometry and operating parameters and subsequently identified dimensionless numbers such as Reynolds number, the size of which determines the type of flow (laminar vs. turbulent) and Taylor number, which determines whether Taylor vortices arise in this gap between the cylinders at zero pressure gradient.The pressure gradient suppresses the existence of these vortices.The definition of these parameters in the case of flow between the cylinders with the rotation is: Reynolds number: where () is the angular velocity, (r 1 ) is the inner radius (r 2 ) is the outer radius, () is the kinematic viscosity.The critical value of the transition from laminar to turbulent flow is in the interval 1100 to 1400.Taylor number : In Table 1 there are tabulated values of the Reynolds and Taylor numbers for given speed and geometry, and the existence of vortices is defined here for the zero pressure gradient (p = 0 Pa) and without eccentricity.The flow is supposed therefore in the region of laminar and transition region between laminar and turbulent flow.
Based on experience with modeling of Taylor vortices it is usefulto use in those cases rather laminar flow mathematical model, since turbulence models too deformed the vortices due to artificial flow viscosity [5].
In test solution of 2D tasks turbulent mathematical model the turbulent viscosity was evaluated.That have been for water in the range from 0.01 to 0.025 Pa • s, therefore

Creation of computational grid
The problem is complicated in that it is expected to move the boundaries area that the outer cylinder does not rotate, but the inner rotating angular speed ω, and moreover, moves in a circle with a given eccentricity (held precession angular velocity Ω).Thus, it is necessary to consider a change of grid in each iteration step.Therefore, selected from various alternatives from slipping grid where each cylinder was surrounded by a layer of cells moving with them and between these layers were created annulus, which deforms and suitably also defines the interface between the layers of cells, see

Variants of numerical simulation, boundary physical properties
Two variants of geometry have been defined for numerical simulation in ANSYS Fluent.The first option was a 2D model without considering the pressure drop (geometry corresponds to figure 4) and a second variant of the 3D model which is described in Chapter 2 (figure 1, figure 2) with a defined pressure drop.

Numerical solutions in 2D geometry without considering the pressure drop
Variant 2D computational domain was used mainly to debug functionality mutual rotation and precession of the rotor.Computational domain is schematically illustrated in figure 5.

Boundary conditions for numerical solution:
- of rotation  and precession  of rotor).The tasks were solved as a time-dependent with a defined time step.The total computing time corresponded to two speed of rotor about the axis of the computational domain (two precession movements).The aim of numerical solution was to evaluate the distribution of radial force F r and tangential force F t on the rotor, depending on the time or turn the rotor.In the software ANSYS Fluent13.0can evaluate the force components F x and F y .Subsequently, based on the conversion relationships can be determined forces F r , F t acting on the rotor.
Size of the forces F x , F y , F r and F t were evaluated with the frequency of rotation of the rotor /4 for two turns.Subsequently from distribution of the radial and tangential forces depending on the rotation were determined mean values of F r F t [1], [2], [3].Thus defined procedure of evaluation forces was carried out for all combinations of rotation and precession.

Numerical solution of the 3D geometry with consideration of pressure drop
Debugging methodology for solving of flow in the 2D narrow gap was subsequently applied to the 3D model.
Evaluation of the radial and tangential forces was carried out in the same manner as described in the previous chapter.In terms of boundary conditions are also considered pressure drop along its longwise.

Boundary conditions for numerical solution:
- Flowing medium is water, which is defined by a constant value of the density ρ = 1000 kg • m -3 and a kinematic viscosity ν = 10 -6 m 2 • s -1 , thus defined incompressible flow.

Evaluation of numerical simulation of 2D and 3D model of the flow in the narrow gap 6.1 Evaluation of forces F x , F y, F r a F t for 3D model of flow
For each variant set of boundary conditions (rotation and precession rotor) was evaluated distribution of F x , F y , F r and F t depending on the rotor position for two turns.From the graphical dependence was evident that can evaluate the mean value of F r and F t .In the next step, the evaluation was drawn graph depending of the radial and tangential forces at a constant rotation  rotor according to the precession of the rotor .For clarity, the boundary conditions are given including parameters setting numerical simulations for one variant, see Table 2.    From the evaluated functional dependence (figure 11) were set polynomial regression dependencies:    The figure 18 shows the distribution of the static pressure on the wall of the rotor.Again, it is evident from the picture input pressure and a gradual decrease in pressure.

Fig. 17
Course of the static pressure in a longitudinal section Fig. 18 The distribution of static pressure on the wall of the rotor

Conclusions
The article defines the problem of mathematical modeling of flow in the narrow gap between the two cylinders, when the inner cylinder is eccentrically placed and held simultaneously rotating and precession motion in a circle.From the result of numerical simulation using ANSYS Fluent software are determined the radial and tangential forces and other dependencies.As defined above approach can be applied for example in the case of a hydrostatic bearing lubrication, flow modeling sealing gaps pumps or turbines, and many other technical applications.

3 .
where (s) is s = r 2 -r 1 .Critical value characterizing the formation of stationary vortices is 41.6 and critical value of periodic wave modes of vortices is up to one hundred times of the critical Taylor number, see figure 3. -vortex flow, fluid flows in the tangential direction B) stationary vortices C) periodical non-stationary vortices Fig. Characteristics of flow between concentric cylinders with the inner cylinder rotation low.This means that the viscosity at molecular level again indicates the transition area.Due to the character of task the problem was solved as time-dependent.Thus, the mathematical model of the flow is given by the equation of continuity and Navier -Stokes equations, when laminar flow is considered.The mathematical model is then solved in ANSYS Fluent13.0[4].Equation of continuity:

figure 4 .
Movement of the inner cylinder (rotation and precession in a circle) is defined by custom function in C + + (UDF function).Number of cells along the height of the individual layers is at least the10.

Fig. 4 .Fig. 5 .
Fig. 4. Description of the problem of rotation, the computational domain Then it can be determine that the total cell number was 1292500.User functions of rotation and precession was tested on a simple 2D area with respect to debugging functionality.Computing grid of individual areas 2D

Table 3 .
speed and precession.The resulting distribution of radial force F r depending on the angle of rotation  for constant rotation ω = 62.8319 rad/s and variable precession ( = 62.8319 rad/s, 125.6637 rad/s, 188.4956 rad/s, 251.3274 rad/s, 303.6873 rad/s , 314.1593 rad/s) are shown in "Fig 9 ".Evaluation of the magnitude of forces F x , F y, F r and F t for variant ( = 62.8319 rad/s,  = 62.8319 rad/s) [] = ° [t] = s [F x ] = N [F y ] = N [F r ] = N [F t ]

Fig. 9 .Fig. 10 . 6 . 2 Fig. 11 .
Fig. 9. Distribution of radial force F r depending on the angle of rotation  for constant rotation ω and variable precession ΩIn the same way the courses were evaluated tangential forces F t , see figure10.A similar procedure was then applied for other values of rotation ω.

Table 4 .Fig. 13 6 . 3
Fig. 12 Course of the coefficient A in depending on the rotation [] = rad/s

Table 1 .
The values of the Reynolds and Taylor numbers for given revolutions and geometry r 1 = 0.0197 m, s = 0.0003 m a pro (p = 0Pa)[Ω] = rad/s

Table 2 .
Boundary conditions, setting parameters x , F y, F r and F t for variant rotation  = 62.8319 rad/s, and the precession  = 62.8319 rad/s rotor including mean values of forces F r and F t are listed in Table3.In a similar manner were evaluated forces for different variants of combinations of 01115-p.4EPJWeb of Conferences