An Experimentally Validated Cavitation Model for Hydrodynamic Bearings Using Non-Condensable Gas

Abstract

Despite great research effort in recent decades, cavitation in hydrodynamic journal bearings is still a not completely understood phenomenon. In particular, it is unclear which proportions of different cavitation types are present in a bearing. Novel experimental results show a clear deviation from the predictions of hydrodynamic lubrication theory. This article presents a new approach for modeling cavitation in hydrodynamic bearings by using computational fluid dynamics with the volume of fluid method and a phase of non-condensable gas in the lubrication oil. The validation of the model is achieved through the simulation of a large Offset-Halves Bearing and a subsequent comparison of the results with various experimental data, including the fractional film content. In the results, cavitation also occurs in the convergent gap due to a pressure drop caused by inertia forces. The findings indicate that the cavitation effects in oil-lubricated hydrodynamic bearings are caused by a special form of gaseous cavitation, designated as pseudo-cavitation. The presented model with non-condensable gas is able to reproduce the observed phenomena excellently.

1. Introduction

Cavitation is the formation of a gas or vapor phase as a result of a pressure drop. Cavitation is an effect that forms in many slide bearings, particularly in the area of the opening gap, which is why it occurs primarily in fixed-lobe bearings. Cavitation can impair operational safety. The state of knowledge about cavitation in hydrodynamic lubrication was reviewed by Braun and Hannon [1]. Three different types of cavitation can be differentiated:

• Vaporous cavitation occurs when the pressure falls below the vapor pressure of the liquid phase and boiling begins. Depending on the temperature, Ref. [1] gives values for the vapor pressure $p_{\mathrm{v}}$ of typical oil mixture components between $2.1·{10}^{-4}\mathrm{Pa}$ and $52.4\mathrm{kPa}$ that are close to the vacuum.
• Gaseous cavitation is caused by dissolved gas that will be desorbed when pressure drops below saturation pressure. Braun and Hannon provide values for the temperature-dependent saturation pressure $p_{\mathrm{s}}$ of between $34.48\mathrm{kPa}$ and $101.37\mathrm{kPa}$.
• Pseudo-cavitation is a special case of gaseous cavitation when the gas is completely undissolved. This type of cavitation is caused by the expansion of dispersed gas bubbles due to depressurization. The processes of molecular dissolution and absorption do not occur, and the total amount of free gas is constant ($R_{\mathrm{des}}=R_{\mathrm{abs}}=0$). The gas can enter the system either via the lubricant or directly from the environment.

In Figure 1, the mass transfer between the single phases is illustrated for the general case of cavitation. In the event that the pressure falls below the saturation pressure, the dissolved gas desorbs from the oil at a desorption rate $R_{\mathrm{des}}$. Conversely, if the saturation pressure is exceeded, the gas is absorbed at an absorption rate $R_{\mathrm{abs}}$. Similarly, if the pressure falls below the vapor pressure, the oil vaporizes at a vaporization rate $R_{\mathrm{vap}}$. Conversely, if the vapor pressure is exceeded, the vapor condenses at an absorption rate $R_{\mathrm{con}}$.

The most common mathematical description of cavitation in hydrodynamic lubrication is based on the Reynolds equation with Elrod’s cavitation algorithm [2]. Elrod extends the Reynolds equation by an additional variable $\theta$, denoted as fractional film content or density ratio, and $(1-\theta )$ is named the void fraction.

$$\nabla ·\left({\displaystyle \frac{\rho \theta h^3}{12\eta }}\nabla p\right)=\nabla ·\left(\rho \theta \overrightarrow{U}h\right)+{\displaystyle \frac{\partial }{\partial t}}\left(\rho \theta h\right)$$

(1)

For closure, he introduces the complementary condition

$$\begin{array}{cc}\hfill p>p_{\mathrm{cav}}& \iff \theta =1+{\displaystyle \frac{1}{\beta }}(p-p_{\mathrm{cav}}),\hfill \\ \hfill p=p_{\mathrm{cav}}& \iff 0\le \theta <1,\hfill \end{array}$$

(2)

where $\beta$ is the compressibility factor resulting in a value of $\theta$ slightly greater than one. The influence of a gas phase on the density ratio is neglected in this consideration. This algorithm is mass conservative and gives a pressure distribution that aligns closely with experimental results but does not differentiate between the cavitation types and does not consider the real physical processes in the cavitation zone. However, it assumes a specific cavitation formation with gas regions and liquid streamers. The lubricant is advected through the streamers by the Couette flow, and the Poiseuille flow is assumed to be zero. Various modifications and different types of implementations have been made to this algorithm.

Models for vaporous cavitation have been derived by simplifying the Rayleigh–Plesset equation, for example, by Schnerr and Sauer [3], Singhal [4], and Zwart et al. [5]. These models are typically used in combination with a numerical solution of the Navier–Stokes equation and are available in several computational fluid dynamic software packages. The Singhal model is also able to take non-condensable gas into account.

Peeken and Benner [6] were the first to describe a gaseous cavitation model based on a saturation equilibrium. Another equilibrium model was developed by a research group at Tsinghua University in Beijing: Li et al. [7] present a gaseous cavitation model based on a saturation equilibrium, which they apply to a journal bearing in combination with the Reynolds equation. Hao et al. [8] extended the model of Li et al. for a non-saturation equilibrium and applied it in combination with the Reynolds equation to a thrust bearing. Song et al. [9] used a similar gaseous cavitation model to Li et al., which they applied to various slide bearings in combination with the Reynolds equation. The outcomes of these studies demonstrate analogous results to those observed in the context of ordinary cavitation boundary conditions. Ding et al. [10,11] also present a similar gaseous cavitation model and apply it to a journal tilting pad bearing in a computational fluid dynamics simulation, finding a better agreement with experimental data when using the non-equilibrium model than the equilibrium model.

Ibata et al. [12] present an empirical gaseous cavitation model considering the effects of dynamic stimulation on the saturation pressure. Osterland et al. [13] present a gaseous cavitation model that they adapt to hydraulic oil together with the vapor cavitation model by Zwart et al. using experimental data. They set the absorption rate to zero. Reinke et al. [14] investigated gaseous cavitation with high-speed digital photography in a Couette–Poiseuille flow for kerosene oil, finding that the dissolved gas accumulates in the oil over time, resulting in increasing pseudo-cavitation until all the gas is undissolved. They adapted their results to an empirical model and set the absorption rate to zero. In a previous publication [15], the authors of this paper presented a high-speed journal bearing with reduced pad size. A novel methodology for quantifying the fractional film content in a large journal bearing with local resolution was proposed and applied for the first time in [16], where it was tested on this bearing. The results have shown clear deviations to the predictions made by the classical lubrication theory. Comparing Figure 17 in [15] with Figures 13–18 in [16], the experimental results reveal the presence of large partially flooded areas, especially behind the oil feed and at the sides of the pads. In contrast, the lubrication theory, using the Reynolds equation and Elrod’s cavitation algorithm, predict an almost fully-flooded gap. Only at the end of the loaded pad, where the gap narrows, is cavitation predicted in agreement with the measurement. The visualization of cavitation zones had already been presented qualitatively by previous authors using optical methods, for example, in [17], which exhibits cavitation formations similar to those observed in [16], such as the formation of a large fillament in the divergent region. The present study demonstrates that this unpredicted cavitation effect is the result of a combination of inertia forces and undissolved gases. For this purpose, a pseudo-cavitation model and its validation by comparing numerical results with comprehensive experimental data are presented below.

2. Mathematical Model

2.1. Governing Equations

For describing the flow of the given problem, a computational fluid dynamics model that is based on Eulerian treatment by the three-dimensional Navier–Stokes equation is employed. The volume of fluid (VOF) approach is applied to describe the multiphase nature of the flow. With this approach, each phase $\alpha$ within a specified volume V takes up a partial volume $V_{\alpha }$. Based on this assumption, the volume fraction $r_{\alpha }=V_{\alpha }/V$ is defined as a continuous field. Overall, the volume conservation equation

$$\sum _{\alpha }{r}_{\alpha }=1$$

(3)

must be fulfilled in the entire solution domain. To maintain mass conservation, a continuity equation is balanced separately for each phase.

$${\displaystyle \frac{\partial \left(r_{\alpha }{\rho }_{\alpha }\right)}{\partial t}}+\nabla ·\left(r_{\alpha }{\rho }_{\alpha }\overrightarrow{v}\right)=0.$$

(4)

Moreover, the conservation of momentum and energy is balanced for all phases as a whole, which is known as a homogeneous model. The momentum conservation equation reads

$${\displaystyle \frac{\partial \left(\rho \overrightarrow{v}\right)}{\partial t}}+\nabla ·\left(\rho \overrightarrow{v}\otimes \overrightarrow{v}\right)=\nabla ·\mathit{\tau }-\nabla p.$$

(5)

The thermal energy equation is

$${\displaystyle \frac{\partial \left(\rho h\right)}{\partial t}}+\nabla ·\left(\rho \overrightarrow{v}h\right)=\nabla ·\left(\lambda \nabla T\right)+\Phi ,$$

(6)

whereby h is the static enthalpy with $\mathrm{d}h=c_p\mathrm{d}T$ and $\Phi =\mathit{\tau }:\nabla \overrightarrow{v}$ is the dissipation term. In the homogenous model, the mixture fluid properties are assumed to be volume-weighted averages, accordingly $\rho ={\sum }_{\alpha }{r}_{\alpha }{\rho }_{\alpha }$ for the density, $\eta ={\sum }_{\alpha }{r}_{\alpha }{\eta }_{\alpha }$ for the dynamic viscosity, and $\lambda ={\sum }_{\alpha }{r}_{\alpha }{\lambda }_{\alpha }$ for the specific heat conductivity. Additional turbulence and transition effects are considered by using the Shear Stress Transport (SST) turbulence model [18] together with the Gamma-Theta transition model [19]. For the solid domain, the energy equation simplifies in a way that the dissipation term is zero and the solid velocity is specified. Solid deformation is considered using a Lagrangian treatment by a coupled structural mechanic simulation that is based on the principle of virtual displacements in

$${\int }_{\mathrm{\Omega }}\mathit{\sigma }:\delta \mathit{\epsilon }\mathrm{dV}={\oint }_{\mathit{\partial }\mathrm{\Omega }}\overrightarrow{t}·\delta \overrightarrow{u}\mathrm{dA}+{\int }_{\mathrm{\Omega }}\rho \overrightarrow{k}·\delta \overrightarrow{u}\mathrm{dV}$$

(7)

with the external stress $\overrightarrow{t}$ and volumetric forces $\rho \overrightarrow{k}$.

2.2. Material Laws

When modeling gaseous cavitation with non-condensable gases, the compressibility of the gas phase emerges as the pivotal property. Therefore, the ideal gas law is used as the equation of state to determine the density as

$$\rho ={\displaystyle \frac{RT}{p}}.$$

(8)

The viscous stress tensor related to a Newtonian fluid is

$$\mathit{\tau }=\eta \left[\nabla \overrightarrow{v}+{\left(\nabla \overrightarrow{v}\right)}^T-{\displaystyle \frac{2}{3}}\left(\nabla ·\overrightarrow{v}\right)\mathbf{1}\right],$$

(9)

where $\mathbf{1}$ is the unity tensor. The formula by Falz [20] is used to describe a temperature-dependent viscosity with the reference viscosity ${\eta }_0$, the reference temperature $T_0$, and the Falz exponent l.

$$\eta ={\eta }_0{\left({\displaystyle \frac{T}{T_0}}\right)}^{-l}$$

(10)

The density and specific heat of the fluid are described as linear-dependent on the temperature, see (13) and (14). For the Cauchy stress tensor, a linear isotropic model is used.

$$\mathit{\sigma }={\displaystyle \frac{E}{1+\nu }}{\mathit{\epsilon }}^{\mathrm{D}}+{\displaystyle \frac{E}{3(1-2\nu )}}\left(\mathrm{Tr}\left(\mathit{\epsilon }\right)+3a\Delta T\right)\mathbf{1},$$

(11)

wherein $\mathrm{Tr}\left(\mathit{\epsilon }\right)$ means the trace and ${\mathit{\epsilon }}^{\mathrm{D}}$ the deviator of the deformation tensor.

2.3. Numerical Solution

The governing equations are approximated using methods of weighted residuals, the finite element method is employed for the structural model, and the element-based finite volume method is used for the fluid model and the solid heat transfer. A hybrid structured/unstructured technique employing 8-node hexahedron and 6-node prism elements is applied for the mesh generation. The solution of the fluid model and the conjugated heat transfer, as implemented in the Ansys CFX solver, is coupled with an external structural mechanics solver in a custom framework. The fluid model is solved for a few hundreds of iterations, with a time step size between ${10}^{-5}$ and ${10}^{-4}$ s. Subsequently, the temperature and pressure fields are utilized as a load for the structural mechanics model. The mounted bearing is elastically supported in a steel housing, which is modeled as a hollow cylinder with a large outer diameter, the mesh of which is not shown here. The rotor position is determined in order to reach a specified operation point using an algebraic model, whereby tolerances of $\pm 0.05\mathrm{MPa}$ for the specific load and $\pm 0.5^{\circ }$ for the load angle is targeted in this study. Consequently, the structural model generates a mesh displacement, which is then utilized to update the meshes for the fluid model. The submodel’s rotor displacement, fluid model, and solid model are solved repeatedly to obtain a stationary converged solution. The solution can be initialized with that of a previous operating point. The computation time depends on the operating point. A typical time is about two days using 4 partitions.

3. Test Case

3.1. Geometry

As a test case (see Figure 2), an Offset-Halves Bearing is used, which was previously investigated in [15,16]. More details about the bearing and the experimental setup can be found in these references. The bearing has a nominal diameter of $500\mathrm{mm}$, an outer diameter of $800\mathrm{mm}$, and a width of $350\mathrm{mm}$. To reduce the power loss, the width of the upper sliding surface is smaller. Both sliding surfaces are separated by two deep pockets with $142\mathrm{mm}$ length. The oil supply is applied by two leading edge grooves (LEG) with a chord length of $30\mathrm{mm}$ with common ring channel feeding. A hydrostatic jacking pocket that is closed by a check valve during static operation with a chord length of $84\mathrm{mm}$, a width of $80\mathrm{mm}$, and a radius of $90\mathrm{mm}$ is located adjacent to the load direction. The geometric parameters are listed in

Table 1. Geometric parameters of the test bearing.

Parameter	Nominal	Measured	Unit
Rel. bearing clearance	$1.{20}^{+0.09}$	$1.19$	‰
Preload unloaded pad	$1.{75}_{-0.12}^{+0.07}$	$1.60$
Preload loaded pad	$1.{75}_{-0.12}^{+0.07}$	$1.73$
Angle of curvature center of unloaded pad	${235}^{\circ }$	${247}^{\circ }$
Angle of curvature center of loaded pad	${55}^{\circ }$	${58}^{\circ }$
Bore diameter	$500.{470}^{+0.044}$		$\mathrm{mm}$
Pad radius	$250.{460}^{+0.022}$		$\mathrm{mm}$
Outer diameter	800		$\mathrm{mm}$
Width unloaded pad	350		$\mathrm{mm}$
Width loaded pad	210		$\mathrm{mm}$
Shaft outer diameter	$499.870$		$\mathrm{mm}$
Shaft inner diameter	150		$\mathrm{mm}$

.

The related computational grids are displayed in Figure 3. The lubrication gap is meshed with 20 nodes across its height. The model utilizes a total of $2·{10}^5$ fluid nodes and $6·{10}^4$ solid nodes. Mesh Independency is verified in Appendix A.

3.2. Material Parameters

The shaft and the bearing liner are machined from carbon steel. A $4\mathrm{mm}$ thick coating of white metal is applied to act as a sliding surface. The corresponding material parameters are listed in

Table 2. Material parameters of the bearing.

Parameter	Steel	White Metal	Unit
Density $\rho$	7870	7400	$\mathrm{kg}/{\mathrm{m}}^3$
Young’s modulus E	$205·{10}^9$	$56.5·{10}^9$	$\mathrm{N}/{\mathrm{m}}^2$
Poisson’s ratio $\nu$	0.28	0.33	1
Thermal expansion rate a	$11·{10}^{-6}$	$23.4·{10}^{-6}$	1
Specific heat $c_p$	440	230	$\mathrm{J}/\left(\mathrm{kg}\mathrm{K}\right)$
Heat conductivity $\lambda$	$51.9$	35	$\mathrm{W}/\left(\mathrm{m}\mathrm{K}\right)$

.

Table 3. Material parameter of the lubricant.

Parameter	Value							Unit
Temperature T	20	40	60	80	100	120	140	$°\mathrm{C}$
Density $\rho$	$856.1$	$843.5$	$830.8$	$818.0$	$805.1$	$792.1$		$\mathrm{kg}/{\mathrm{m}}^3$
Dynamic viscosity $\eta$	$89.14$	$35.15$	$17.52$	$10.19$	$6.62$	$4.65$		$\mathrm{mPa}\mathrm{s}$
Specific heat $c_p$	$2032.7$			$2247.7$	$2319.3$	$2391.0$	$2462.7$	$\mathrm{J}/\left(\mathrm{kg}\mathrm{K}\right)$
Heat conductivity $\lambda$	$0.134$	$0.134$	$0.134$	$0.134$	$0.134$	$0.134$	$0.134$	$\mathrm{W}/\left(\mathrm{m}\mathrm{K}\right)$

contains the material properties of the used ISO VG 32 turbine oil. applied for lubrication in this study.

The density and the specific heat capacity of the lubricant can be described in close approximation as linearly dependent on temperature. A fit to tabular values given in Table 3 yields the following empirical correlations:

$$\rho =837.1{\displaystyle \frac{\mathrm{kg}}{{\mathrm{m}}^3}}-0.64{\displaystyle \frac{\mathrm{kg}}{{\mathrm{m}}^3\mathrm{K}}}(T-50 °\mathrm{C})$$

(12)

$$c_p=2140.2{\displaystyle \frac{\mathrm{J}}{\mathrm{kg}\mathrm{K}}}+3.5832{\displaystyle \frac{\mathrm{J}}{\mathrm{kg}{\mathrm{K}}^2}}(T-50 °\mathrm{C}).$$

(13)

An adjustment of Equation (10) to these data results in ${\eta }_0=35.28\mathrm{mPa}\mathrm{s}$ and $l=1.79$. The reference temperature used is $T_0=40$ °C. The density of air at a reference state $T=25$ °C, $p=101,325\mathrm{Pa}$ is assumed by $\rho =1.185\mathrm{kg}/{\mathrm{m}}^3$; the specific heat at constant pressure of air is $c_p=1004.4\mathrm{J}/\left(\mathrm{kg}\mathrm{K}\right)$; and the viscosity and the heat conductivity are set to be $\eta =1.83·{10}^{-5}\mathrm{Pa}\mathrm{s}$ and $\lambda =0.0261\mathrm{W}/\left(\mathrm{m}\mathrm{K}\right)$.

3.3. Boundary Conditions

At the inlet on the outside of the ring channel, an oil volume flow rate of $450\mathrm{L}/\min$ is fed with an inlet temperature of $50 °\mathrm{C}$. The volume fraction of the gas phase at the inlet is determined by

$$r_g={\displaystyle \frac{{\alpha }_{\mathrm{B}}}{{\alpha }_{\mathrm{B}}+\frac{p}{p_0}\frac{T_0}{T}}}.$$

(14)

with the Bunsen solubility coefficient ${\alpha }_{\mathrm{B}}=0.09$. At the outer fluid boundary, atmospheric pressure and pure air ($r_{\mathrm{g}}=1$) are specified. The specific load is varied in the range of $\overline{p}=1\mathrm{MPa}$ to $\overline{p}=5\mathrm{MPa}$ and directed to the center of the jacking pocket defined at $\phi ={180}^{\circ }$. The rotational speed of the rotor is varied from $n=1000\mathrm{rpm}$ to $n=3600\mathrm{rpm}$ in the positive z-direction. For the investigated operation points, mirror symmetry is assumed at $z=0$. The remaining boundaries are treated as adiabatic. These are the free surface of the fluid in the ring channel, the outer surface and face of the bearing, and the inner surface and face of the rotor.

4. Numerical Results

4.1. Pressure

Figure 4 depicts a comparison between measured and predicted hydrodynamic film pressure. The simulated pressure shows some irregularities at the end of the hydrostatic jacking pocket. These might be a numerical issue due to the sharp gradients at this location, although it could be physical and not captured by the sensor. Both the experiment and the simulation demonstrate that the pressure drops below the atmospheric pressure shortly behind the lubrication pocket. This is presented in detail in Figure 5 for experimental and simulation data. Here, it should be noted that the pressure sensor provides a smoothed signal due to its size and delay. This pressure drop can be explained by the acceleration of the fluid and the resulting inertia forces. This is illustrated in the section in Figure 5b.

4.2. Volume Fraction and Fractional Film Content

Figure 6a shows the liquid volume fraction $r_{\mathrm{l}}$ of the simulation in the entire fluid region for the maximum investigated rotor speed and $4\mathrm{MPa}$ specific load. In Figure 6b, the liquid volume fraction at the bearing center is depicted across the circumferential angle $\phi$ and the distance y from the rotor surface across the gap. It can be seen that the volume fraction in the cavitation regions behind the oil supply varies across the lubrication gap in such a way that the oil phase separates partially at the sliding surface, while the volume fraction in the diverging gap zone is almost constant. Such a combination of streamers and adhered layers is typical for stationary loaded bearings.

The fractional film content for the simulation is evaluated as the average volume fraction of the lubricant across the gap height.

$$\theta ={\displaystyle \frac{1}{h}}{\int }_0^h{r}_{l}dy$$

(15)

In Figure 7, the fractional film content is compared with the experimental results for different operation points. It can be seen that the cavities behind the leading edge grooves ➀ and ➁ occur at speeds above $2000\mathrm{rpm}$. At lower speeds, only ambient air entering the gap from sides ➂ and ➃ is visible in the results.

This also indicates that the lubricating film pressure in this area is lower than the atmospheric pressure. With increasing speed, the pressure behind the oil feed pockets becomes lower and the expansion of the dispersed gas in the oil becomes more pronounced. With increasing load, these partially flooded and unflooded regions become smaller at the loaded pad and larger at the unloaded pad because of the changed film thickness. At high loads, the ordinary cavitation caused by a diverging gap at the end of the loaded pad can be observed ➄. The simulated and measured oil distributions are in overall agreement.

An alternative application of the gaseous cavitation model from Osterland et al. [13] resulted in an insufficient desorption rate. This supports the assumption that desorbed gas accumulates in the oil during operation and, consequently, pseudo cavitation occurs at stationary operation. Describing this shift from gaseous to pseudo cavitation would require the transient modeling of the full oil circuit, which would necessitate a considerable computational effort. Furthermore, simpler modeling approaches using a single phase of an oil–gas mixture with fixed composition (specified mass fractions) based on the Navier–Stokes as well as the Reynolds equation led to similar results to Elrod’s cavitation algorithm. In the investigated bearing, the mass fractions are not constant at all, especially in the areas that ingest enviromental air through the lateral boundary of the lubrication gap. It is essential to consider the multiphase nature of this cavitation effect by solving a single transport equation for each phase to ensure accurate modeling. Inertia forces must also be taken into account in order to describe the pressure drop.

4.3. Film Thickness

Figure 8 illustrates the film thickness in the bearing center along the circumferencial angle and along the axial direction at the position of the minimum film thickness. It can be seen in Figure 8b that the film thickness varies along the axial direction. This results from the thermal deformation of the rotor and bearing.

4.4. Temperature

The temperature within the bearing was measured by 123 type J (Fe-CuNi) thermocouples (see Figure 2). Figure 9 compares the measured temperature of the sensors in the bearing center with the predictions. These sensors are applied approximately $9\mathrm{mm}$ below the sliding surface. The accuracy is good at low to medium speed. At higher speeds, however, the predicted temperature is too low. This currently represents a limitation.

4.5. Power Loss

Figure 10 shows a satisfactory agreement between calculated and calorimetrically measured frictional power loss. However, at high speed some variance exists, which potentially originates from turbulence modeling. Prediction is particularly difficult at speeds of 3000 rpm and 3600 rpm at low load, as a large area of the lubricating film is in the laminar-turbulent transition.

4.6. Effect of the Bunsen Solubility Coefficient

The Bunsen solubility coefficient specifies the volume of dissolved gas per volume of liquid. For mineral oil, a range of ${\alpha }_{\mathrm{B}}=0.08\dots 0.1$ is given in the literature. A larger amount of air may be present if free air is induced into the oil during operation and if the oil is contaminated or aged. In order to validate the chosen value of ${\alpha }_{\mathrm{B}}$ for the present model, it was varied and compared in terms of agreement of the fractional film content with the measurements. The results are shown in Figure 11. It can be seen that the higher the ${\alpha }_{\mathrm{B}}$ chosen, the narrower the area of incoming air on the sides, especially of the unloaded sliding surfaces due to the higher compressibility. Therefore, the correspondence is better for the investigated operating point, which indicates the presence of additional free gas.

5. Conclusions

In this paper, a thermal-elastohydrodynamic model of a large hydrodynamic Offset-Halves Bearing using non-condensable gases for describing cavitation is presented and validated with comprehensive experimental data.

By the utilization of the presented computational fluid dynamics numerical model, it is shown that cavitation for the given test bearing can be sufficiently described by non-condensable gases and that the cavitation type is pseudo cavitation. This is because undissolved gas accumulates in the lubricating oil over time due to the low resorption rate. As a result, common gaseous cavitation only occurs at the beginning of operation and only pseudo-cavitation is observed during stationary operation. The cavities occur wherever the pressure drops below atmospheric pressure, but not only in the divergent gap. In contrast, the described cavitation effect cannot be explained by the use of the classical hydrodynamic lubrication theory. The pressure drop that occurs subsequent to the oil feed is attributable to an inertia effect, which arises from the acceleration of the lubricant. A similar effect is known at the leading edge of tilting pad bearings where the lubricant is decelerated and a ram pressure occurs [21]. The pressure drop results in an expansion of the gas phase that separates from the liquid phase. These cavities as well as the pressure drop, which are not predicted by the thin film theory, are reproduced very accurately by our model. This shows that cavitation in hydrodynamic bearings is more complex than previously assumed and is not fully understood. In particular, the influence of inertia forces on the dynamic pressure must be taken into account. The partially flooded areas resulting from gaseous cavitation can potentially compromise the operational safety especially, at high speed and low load operation. Further investigation is necessary to elucidate the effects on rotor dynamics.