Three approaches were used to verify the obtained results - analytical, experimental and CFD. An integrated process was also followed to create and analyse the results using these three methods, as shown in Fig. 2. Based on the analytical calculation, experimental conditions were determined. Afterwards, boundary conditions of the CFD analysis were set according to the resulting analytical results and experimental conditions. Subsequently, all results were compared and discussed.
2.1 Analytical approach
Navier-Stokes equations play a significant role in the field of fluid mechanics. Nonetheless, due to the complexity of the Navier-Stokes equations, it is almost impossible to obtain analytical expressions to calculate required flow characteristics. Therefore, certain assumptions can be considered to help to obtain a simpler form of equations. The assumptions are as follows:
-
Film thickness is constant and small compared to its size in other directions.
-
The fluid is Newtonian, incompressible and isoviscous.
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Inertia terms are negligible compared to the viscous forces.
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No squeeze nor sliding exists, and the bearing surfaces are stationary.
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The pressure of lubricant is constant in the direction of the film thickness \(\left(\partial p/\partial z=0\right)\).
- There Is A Continuous Supply Of Lubricant, And The Flow Is Laminar.
Then the Reynolds equation for computation of the pressure distribution in the simplified form is as follows:
$$\frac{\partial }{\partial x}\left(\frac{{h}^{3}}{12\mu }\frac{\partial p}{\partial x}\right)+\frac{\partial }{\partial y}\left(\frac{{h}^{3}}{12\mu }\frac{\partial p}{\partial y}\right)=0$$
1
After obtaining the pressure distribution, the load-carrying capacity (or acting force) can be expressed as:
$$W={P}_{r}\bullet {A}_{eff}$$
2
where Ps is recess pressure and Aeff is effective area of the pad, which can be expressed from effective pad area proportion β (β < 1) and total area of the pad Atot :
$${A}_{eff}=\beta \bullet {A}_{tot}$$
3
The equations (1–3) can be further customized and utilized to express required flow supply to the bearing as follows:
$$Q={Q}_{f}\frac{W}{{A}_{tot}}\frac{{h}^{3}}{12\bullet \mu }$$
4
where h is film thickness, µ is dynamic viscosity of lubricant, and Qf is flow factor, dependent on the geometry and the effective area of the bearing.
Loeb [17] introduced a new methodology for estimating optimal proportions of the bearing pad and recess based on electric analog field plotter. Generalized analytical equations could be used to calculate the bearing performance using read values obtained from graphs, presented in Loeb and Rippel [16]. Similarly, the performance factors (Eq. 5–7), can be determined based on known values of bearing characteristics.
a) pressure factor | \({P}_{f}=\frac{{P}_{r}\bullet {A}_{tot}}{W}\) | (5) |
b) flow factor | \({Q}_{f}=\frac{12\bullet \mu \bullet Q\bullet {A}_{tot}}{W\bullet {h}^{3}}\) | (6) |
c) power loss factor | \({H}_{f}=\frac{{P}_{f}\bullet {Q}_{f}}{\eta }\) | (7) |
The equations (5,6) can be transformed to obtain the required variable from the expression. An example of designing a HS bearing pad proportions using one-parameter optimization is shown in Fig. 3. The ratio (a/l) represents the characteristic dimension of the recess to the characteristic dimension of the pad, that is, the diameter of the recess to the pad edge length. One primary parameter is kept constant (usually recess size to retain the required lifting area), while the other is optimized. The aim of the pad geometry optimization is generally to achieve the best possible performance with minimal pumping power loss, in this case, the power loss factor – Hf. And for the (a/l) ratio of the minimum power loss factor, the values of optimal pressure (Pf) and flow (Qf) factors are obtained, respectively.
In the basic approach, only one of the two parameters: a or l, are used, while the recess position (t = l), is assumed. The optimization approach using performance curves in a 2D graph utilizes (l = 4a). However, we defined the new variable parameter (t) to describe the position of the recess. Thus, the proposed method uses two-parameter criteria, recess position (t), and recess area to total pad area ratio (Ar/Atot), for performance optimization. The area and the position of the recess were chosen as the primary parameters because the recess shapes may be chosen to better accommodate the pad shape and thus improve the bearing performance. The length of the edge of the pad (2l) remained constant, so the second variable assumed in this study was the recess radius – a. Attention should be paid to prevent exceeding the recess area out of the pad geometry, i.e., in case the pad has rounded corners, etc.
Table 1
Performance factors for experimental pad proportions read from Fig. 4 of [16]
Parameter | Label | Value |
Pressure factor | Pf | 1.52 |
Flow factor | Qf | 27.4 |
Power loss factor | Hf | 41.65 |
2.2 Experimental validation
2.2.1 Lubricant viscosity
To obtain all the inputs required to build a relevant mathematical model, the dynamic viscosity measurement was performed on a rotational viscometer HAAKE RotoVisco® 1 (PSL Systemtechnik, Germany) rotational viscometer. Measured data was filtered with 99.88% reliability using a Bingham rheological model. The dynamic viscosity of the used ISO VG 46 grade hydraulic oil was fitted using the Vogel-Fulcher equation with R2 = 99.89%. The experiments were carried in static conditions, with stable temperature 23°C recorded using temperature sensors in the recess. The results shown very low temperature difference throughout the measurements, only ± 1°C. Thus, constant viscosity of 0.104 Pa∙s was considered in all simulations. It has been previously observed by Schmelzer et al. [44] that the pressure-viscosity dependence of liquids of constant composition is extremely small (around 2%) in the investigated pressure range (0–1 MPa). Therefore, the pressure-viscosity dependence was neglected in this study.
2.2.2 Two-pad HS experimental bearing (2-PAD)
The experimental device used to obtain the measured data for evaluation is schematically shown in Fig. 4. It consists of two main parts, the bearing, and the hydraulic circuit, which supplies pressurized lubricant with a pump powered by an electromotor. The hydraulic circuit is equipped with safety features such as a check valve and a pressure relief valve. The hydraulic accumulator minimizes the pressure spikes generated by the pump. The flowmeter measures the total supplied flow to the whole bearing. Flow restrictors (throttle valves), secure bearing stability in the event of asymmetrical load or misalignment. A pressure sensor of 0.25 bar precision is mounted on each of the inlets to the recess. The lubricant that flows out of the contact area to the collector is returned to the oil tank.
Three contact potentiometer proximity sensors are mounted on each of the 2-PAD pads for obtaining film thickness with precision of 0.01 mm and range of 3 mm. To measure the actual temperature in the recess area, a thermometer is fitted in each of the recesses for calculation of the viscosity value of the lubricant. The load is created using four set screws, each mounted with a load cell with ± 5 N precision, which provides a real-time measurement of the loading force. The experimental measurements were carried out three times after the temperature in the recess area stabilized at 23°C (corresponding to a dynamic viscosity of 0.104 Pa⸱s). Since the hydraulic circuit has relatively large tank (100 liter), no additional cooling was necessary. The experimental conditions were based on the previous test bearing comparison with analytical equations in ref. [45]. The total supplied flow into both of the pads was set to 8.5 l/min. A comparison of predicted and measured film thickness with variable load was presented in reference [45], where the highest precision was achieved at 16 kN load. Therefore, for all measurements, a total load of 16 kN was applied in this study. The resulting average film thickness was 0.197 mm. The pad size used in experiments was 140x140 mm with recess diameters of 35 mm and positions of t = 35 mm. Inlet hole diameter was 12 mm, and recess depth was 5 mm.
2.3 Numerical approach
CFD analysis has become a favoured method for the analysis of fluid dynamics. It provides a detailed understanding of the lubricant flow in hydrostatic bearings. In this study, the CFD analysis was done using commercial software ANSYS Fluent 2021 R2 based on the Finite Volume Method with Cell-Centered formulation. The code solves the conservation equations for mass (Eq. 8) and momentum (Eq. 9), respectively.
\(\rho \left(v\bullet \nabla \right)v=-\varDelta p+\mu {\nabla }^{2}v\) | (9) |
In this case, pressure-based steady-state incompressible flow with absolute velocity formulation was investigated without considering thermal effects and gravity. A uniform lubricant film was considered in all simulations. Because relatively high film thickness and low pressure were assumed for very short experimental tests isothermal conditions were assumed – constant viscosity (0.104 Pa⸱s) and density (875 kg/m3) were established for all simulation design points. As indicated in the previous research papers, turbulent models were used in HS bearing simulations to improve the obtained results with considering the influence of vortices emerging in the recess area. Moreover, the previous research [35–37], as well as preliminary simulation results showed better agreement of the predicted results for turbulent model k-ω SST than for the laminar viscous model. Therefore, k-ω SST viscous model with 5% turbulent intensity and turbulent viscosity ratio of 10 was used for all simulations. The operating conditions included an atmospheric pressure of 101.3 kPa to reflect the real conditions. The 3D geometry of the fluid domain was parametrized in 3D modelling software of the program. The variable parameters, recess position and diameter, are listed in Table 2. The simulation of all design points was conducted using the setting described below.
Table 2
Combination table of variable parameters for simulation design points
Area ratio Ar/Atot (-) | 0.05 | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.75 |
Recess diameter D (mm) | 17.66 | 24.98 | 35.32 | 43.26 | 49.96 | 55.85 | 61.18 | 66.08 | 68.40 |
Recess position t (mm) | 8.83 | 12.48 | 17.66 | 21.63 | 24.97 | 27.92 | 30.59 | 33.04 | 34.20 |
19.29 | 21.49 | 24.59 | 26.97 | 28.98 | 30.75 | 32.35 | 33.82 | 34.52 |
29.76 | 30.49 | 31.53 | 32.32 | 32.99 | 33.58 | 34.12 | 34.61 | 34.84 |
40.23 | 39.50 | 38.46 | 37.67 | 37.01 | 36.41 | 35.88 | 35.39 | 35.16 |
50.70 | 48.50 | 45.40 | 43.02 | 41.01 | 39.24 | 37.64 | 36.17 | 35.48 |
The fluid domain of one quarter of the pad was discretized into a polyhedral mesh in built-in ANSYS Fluent Meshing (Fig. 6). The target mesh size in the upper plane and the lubricating film regions was set to 0.128 µm to enhance the results of the forces acting in the upper plane. A mesh sensitivity analysis was performed with the same boundaries as in the final analysis to determine the mesh quality and stability (Fig. 7). An approximate number of polyhedral elements, 1e + 07, was used in all simulations with average maximum skewness 0.4 and minimal orthogonality of 0.35. Six boundary layers were added to the walls in the recess region, and four in the inlet region, to enhance near wall fluid behaviour. The number of boundary layers was established according to preliminary simulations. The resulting wall Y + value along the top surface was in the range from 0.12 to 0.18. However, the precision of the model was primarily judged according to the analytically predicted and experimentally obtained results. There is no limitation for the film thickness range, but lower film thicknesses may require longer computational times to secure smooth mesh element transition from the recess area to the film region.
Hydrostatic bearings are usually modelled only with the part of the inlet region directly flowing to the recess. However, the initial comparison of the fluid domain only with straight vertical pipe showed a bigger difference to experimental data. Therefore, the fluid domain was modelled with the inlet region to the location of the pressure sensor location on the experimental rig. For all simulations, a uniform film thickness of 0.197 mm, as obtained from experimental measurements, was used for all design points. The boundary conditions can be seen in Fig. 8. Starting with the mass flow inlet (Q) set at 7.096E-5 m3/s, symmetry conditions were employed on the symmetry planes of the quarter model and pressure outlet with atmospheric pressure operating condition (Patm). Output parameters were the resulting force on the top plane of the bearing and average pressure over the recess area.
The residuals of continuity and momentum equations condition were set below 10− 4 to achieve convergence. Underrelaxation factors for both velocity and pressure, were set at 0.5 and 0.5, respectively, without any further need for adjustment. We used the coupled pressure-velocity coupling scheme, which should provide robust and high performance for steady-state flows compared to the segregated schemes available in ANSYS Fluent. The gradient least squares-based spatial discretization method was used. The first-choice schemes – second order, and second order upwind were kept for pressure and momentum, respectively. Hybrid initialization was conducted to improve the convergence. One case of the calculation took approximately 2 hours with the used computer specifications: 12 physical cores of 3,47 GHz processor, 96 GB of RAM, and 1 TB SSD to store retained case data.