1 Introduction

Water-based lubrication is widely used in the maritime industry, notably in stern tube bearings for propeller shafts in ships, and other types of machinery which have regular contact with water. Due to the improved design features of water lubricated bearings, they possess significant potential to replace a large number of oil-based bearings in machineries operating at low to medium speeds. With the advent of polymer-based bearing linings, manufacturers have been able to generate improved efficiencies even at higher speeds. Also, with the increased usage of water lubricated bearings, the oil releases during regular maintenance can be lowered, thereby improving the ocean ecosystem and reducing pollution [1]. Water-based bearings are also simpler and cheaper to produce and maintain, thereby making fiscal sense for both producers and consumers. The most popular bearing lining in use today is nitrile butadiene rubber (NBR). Along with NBR, there are also polymer-based bearings currently in use in maritime industries. Rubber has natural resilience with excellent shock, vibration, and noise absorption properties. Due to the elastic nature of rubber material, it manages well in case of shaft misalignment. Experimental studies on water lubricated rubber journal bearings by Carbera et al. [2] observed the complex pressure variations developed over the loaded rubber staves. Pressure distributions were found to be highly influenced from the elastic bearing surface deflections. However, rubber deflections were not significant enough to modify the lubricant viscosity. Similar variations in pressure and load capacity were also noted from the numerical analysis of straight fluted rubber bearing structure with cavity [3].

A major limitation of water lubricant is its low viscosity properties, which results in the low load-carrying capacity of the water lubricated journal bearing. To compensate for such limitations, water lubricated bearing of different design configurations are developed over the years. Laser surface texturing with different dimple patterns at contacting stave regions has proved to influence the friction reducing effect of water lubricated silicon nitride material [4]. Similar surface modifications in loaded regions can significantly improve the surface properties and performance parameters of water lubricated bearings. Various theoretical and experimental studies have been carried out over the years on water lubricated bearings with multiple grooves. These studies have positively impacted the bearing market with manufacturers producing water lubricated bearings of different materials, sizes and groove patterns. However, limited studies are carried out solely focusing on improving the load capacity of the water lubricated bearing. Gao et al. [5] carried out a numerical analysis using Computational Fluid Dynamics (CFD) technique to study the hydrodynamic performance of journal bearing operated with water lubricant. Effect of varying eccentricity ratio on the film pressure distribution and load capacity was analyzed under different rotational speeds. Improved efficiency was observed for eccentricity ratio of 0.6 and 0.7. Based on the numerical analysis, a design reference for water lubricated bearing was developed using numerical fitting method. He further extended the studies by including a bearing bush with a transition arc design for further improvement in bearing load capacity [6]. Using both CFD and Fluid Structure Interaction (FSI) techniques, Wang et al. [7] analyzed the lubrication behavior of Polytetrafluoroethylene (PTFE)-based water lubricated bearings by considering the cavitation effects. Effect of both elasticity modulus and Poisson’s ratio on load capacity and elastic deformation are studied. Reference data were generated for varied loads and rotational speeds to identify whether elastic deformation need to be considered for composite journal bearings. Datasheets generated for future analysis also need to consider the comparison of rubber journal bearings and polymer bearings.

Earlier, Majumdar et al. [8] conducted theoretical studies on the steady-state and dynamic characteristics of a three-axial groove water lubricated journal bearing. The effect of number of grooves and groove angles on its load capacity, stiffness and damping characteristics and whirl instability were analyzed in detail. Finite difference method and linear perturbation technique were employed for the theoretical analysis. For groove angles such as 360 and 180, a constant pressure profile and a linear pressure drop in the grooves are set as boundary conditions. Results indicated an improvement in load capacity and whirl stability by considering smaller groove angles. Pai et al. [9] employed CFD method to simulate fluid flow in a water lubricated bearing with three equally spaced axial grooves. For varying load, speed, groove geometry, and pressure supply conditions, experimental studies were also conducted on a bearing test rig to analyze the axial and circumferential pressure profiles along and around the bearing. The findings highlighted from CFD study include that the maximum pressure zone in bearing has shifted toward the outlet. Such CFD and FSI studies were even extended to analyze the temperature variation and cavitation effects of three lobed and plain hydrodynamic journal bearings [10, 11]. These studies were limited to minimal number of grooves and land regions. Additional increase in groove number, size and shape can potentially influence the bearing performance under dynamically loaded conditions. Harishkumar [12] presented a detailed review concerning the effects of cavitation and temperature on the load bearing capacity of journal bearings using CFD and FSI techniques. A detailed comparison was presented to highlight the variation in performance parameters such as load capacity, friction force, flow rate, wear and deformation of bearing lining material. Thermal models developed using CFD technique was simulated and observed a notable decline in load capacity with the rise in temperature.

Zhang et al. [13] proposed an efficient method to determine the stiffness coefficients of water lubricated plain journal bearings by accounting the cavitation effects. In the CFD analysis, a cavitation model of Zwart-Gerber-Belamri was applied. Load-stiffness relationship for bearings of varied diameters, clearance spaces, L/D ratios and rotational speeds was analyzed in detail. At low eccentricity ratios, the stiffness coefficients were minimal, and a notable increase was observed with increase in load. Due to low stiffness values, designers will have to select suitable bearing diameter, length and clearance values according to working conditions to make certain larger eccentricity ratios are generated. To analyze the influence of perturbation amplitudes on stiffness coefficients, Liang et al. [14] conducted a three-dimensional CFD-FSI simulation on three types of WLJB having rubber, polymer and steel lining materials to overcome the existing limitation of CFD models of water lubricated bearings. The deformation of lining material selected is found to influence the journal attitude angle and stiffness of water lubricant. Study conducted clearly demonstrated the significance of lining deformation and perturbation amplitudes in determining the bearing stiffness at larger journal eccentricity ratios. Li et al. [15] developed a nonlinear transient hydrodynamic force model by creating a structured mesh movement algorithm. Since the mesh did not experience any distortion or numerical failure, the proposed mesh movement algorithm was found to be suitable for transient flow analysis of WLJB’s. By considering linear velocity perturbation, an effective method was developed and advocated to calculate the hydrodynamic forces and dynamic coefficients, which is based on the computation of 3D transient flow field. Even at nominal whirling radius (R = 0.5 µm), the nonlinear and linear hydrodynamic forces generated indicated a substantial difference. Additionally, when the size of the orbit increased, linear hydrodynamic force amplitude demonstrated significant growth more than the nonlinear hydrodynamic forces. Transient loads have a notable impact on stability of WLJB’s. Simulation results for an increase in transient loads up to 3000 N generated peak hydrodynamic pressures and exhibited a notable reduction in the effective bearing area of water film pressure distribution [16].

By taking account of slip surfaces in bearing models, a significant deviation in performance characteristics can be observed in comparison with models neglecting slip conditions. Lin et al. [17] presented the effect of slip surfaces on the performance of hybrid and hydrodynamic journal bearings. Both journal bearings exhibited monotonic effects of slip intensity on their tribological performance. Load-carrying capacity and pressure variation were studied for different slip conditions mainly in three different zones: cavitation zone, pressure rising zone and pressure drop zone. The boundary slip exhibits a positive effect for a slip surface in the pressure rising zone. Under such conditions, the load-carrying capacity is enhanced, and a further improvement was noted for an increase in the size/area of the slip surface. The slip surface close to the bearing center in the axial direction significantly impacts the increase in load-carrying capacity. At the bearing outlet, the slip surface will adversely affect the load-carrying capacity. In conclusion, improper slip surface design has negative effects and would lead to a reduction in load-carrying capacity. Further by considering both inertial force and wall slip effects, the variation in the hydrodynamic characteristics and friction coefficient of journal bearing with water film was analyzed by Xie et al. [18, 19]. Fluctuation in friction coefficient is found be influenced by the transition of lubrication states and film thickness ratio. Such analytical results are of guiding significance for further design and optimization of water lubricated bearing structure. Further studies even focused on analyzing the influence of turbulence and convective inertia using conventional theoretical approach and CFD simulation in a tilting pad journal bearing with water film as lubricant [20]. In the study, turbulence accounted for around 50% of overall load capacity, highlighting the importance of precise characterization of turbulence effects in water lubricated bearings due to the usage of low viscosity lubricants.

Gu et al. [21] further considered the slip velocity effects at the surface of a PTFE coated water lubricated bearing. The effects of varying slip length on the static performance envelope were determined by considering the three-dimensional slip velocity boundary conditions. Analysis was performed by embedding the 3D slip velocity formulation into the Fluent and simulating the internal flow field in WLJB. Taking account of slip conditions, a notable rise in frictional power and load capacity was observed for an increase in attitude angle. By considering partial wall slip in micro-grooved water lubricated bearings, Feng et al. [22] observed improved load capacity for all combinations of microgroove distributions. Bearing with herringbone-groove yielded higher load capacity followed by spiral and straight grooved bearings. Parametric analysis carried out demonstrates that the load capacity of partial microgroove WLJB decline at larger eccentricity ratios. In water lubricated thrust bearings with herringbone grooves, effect of design parameters such as spiral angle, groove width ratio and operational speeds on the cavitation area was also analyzed in [23, 24]. The effect of cavitation in herringbone grooves under varying viscosity and temperature conditions, and its load capacity and sealing performance are studied in detail. The optimal structural parameters proposed ensure higher stability for herringbone spiral groove bearing structure.

Xie et al. [25] utilized CFD approach to simulate and analyze the micro-interface lubrication mechanisms in a water lubricated bearing. Streamlines, velocity vectors, pressure variations, eddy viscosities and kinetic energy variations were explored in the micro-cavities. Further simulation studies were explored by using different lubrication models in a circuit loop system of nuclear power plant [26]. Optimum eccentricity ratios are identified at which load bearing capacity and frictional characteristics attain a balance point. The outcomes of this study have guiding significance on optimizing and developing new bearing designs structures in the circuit loop system. Zhu et al. [27] further studied the effect of axial vibration on the dynamic characteristics of stern tube bearings of ships under varied rotational speed and misalignment conditions. Bearing stiffness properties was mainly influenced by axial displacement of journal and damping properties are affected by variation in journal velocity. Both parameters are found to be positively correlating with the misaligned angle.

In comparison with numerical and theoretical analysis, limited experimental research is conducted on different geometries of WLJB’s. Litwin [28] developed an experimental test setup to study the effects of surface roughness topography on the performance behavior of WLJB’s. Studies carried out for a variety of polymer materials indicated a strong influence of roughness height on the lubrication performance of bearings. Movement resistance, bearing pressures and journal orbits were measured and simulated. Rubber bearing with groove design is found to provide improved bearing elasticity, good vibration and damping properties and was less susceptible to shaft misalignment. Results proved that the classic bush structure is inappropriate for hydrodynamic lubrication using water [29, 30]. For vertical shaft bearings, study concluded that multiple grooves need to be located along the entire bearing circumference for lightly loaded conditions. However, the location, shape and size of grooves in horizontal shaft bearings has further scope of research considering the durability while lubricating with water containing dirt particles. Litwin et al. [31] also investigated the wear properties of sliding bearings operated in contaminated water lubricant. Different types of bearing materials and bearing sleeve structure are experimentally tested under demanding conditions. Test results indicated that contaminated water lubricant has significant influence over the wear of bearing lining materials. However, higher water velocity is found to minimize the wear on stainless steel shafts. To measure the circumferential pressure variation in WLJB’s, a non-destructive pressure measurement system using wireless technology was proposed by [32, 33]. By analyzing the real-time pressure variation, load bearing performance can be studied in-depth and the factors influencing wireless communication can be utilized for other monitoring requirements of rotary machineries. A time domain averaging algorithm was utilized to minimize the noises produced during pressure signal generation. Liu and Li [34] developed an advanced experimental setup to assess the lubrication behavior of WLJB’s at high rotational speeds. In the test rig, the micro-pressure sensors and signal conditioners are mounted inside a hollow shaft to limit the signal interference. Multiple triaxial force sensors are embedded to determine the simultaneous bearing load in different directions.

Multi-objective optimization techniques are useful tools to identify the optimized performance parameters for hydrodynamic journal bearings. Chen et al. [35] investigated the performance behavior of miniature journal bearings with asymmetrical herringbone grooves using Taguchi analysis. Optimum groove design parameters generating higher load capacity and lower side leakage was identified from the analysis. For an experimental analysis on surface textured journal bearing, Bhasker et al. [36] utilized a fuzzy-based Taguchi optimization method to identify the optimal input parameters for accurate assessment of temperature and pressure distributions. Multi-objective optimization method indicated that both load and speed attained an optimal multi-performance characteristic index (MPCI) value of 0.85 for accurate simulation. Using artificial neural network approach, studies also focused on analyzing the influences of different process parameters such as variance ratios, journal diameter, clearance, lubricant viscosity, and surface roughness features on the performance of journal bearing [37, 38]. Using ANN approach, maximization of minimum film thickness, critical mass parameter and minimization of friction torque is analyzed for multiple bearings with isotropic, transverse and longitudinal kind of roughness patterns. Study conducted validated the suitability of ANN technique for the design optimization of WLJB. Despite the above research studies conducted, optimization of the design parameters influencing the performance of water lubricated journal bearing systems have not been comprehensively studied till date. Even, limited CFD studies have focused on analyzing the effect of slip on the lubrication performance of water lubricated bearings.

The currently existing water lubricated stern tube bearings made of NBR, polymer material and different metal linings has high probability to experience partial slip at the fluid solid boundaries of the grooved surfaces. In this context, the present study aims to analyze the performance behavior of WLJB’s with partial slip considered at the multiple groove regions. For multi-grooved water lubricated bearing (MGWLB) with engineered slip surfaces, design parameters such as groove angle, groove size, attitude angle and number of grooves are varied and analyzed using the CFD approach. CFD model and analysis of MGWLB helps to simulate and explore the interaction between the water lubricant, stave regions, grooves and shaft surfaces. Using Taguchi technique, optimal design parameters influencing the pressure and load-carrying capacity of MGWLB’s are identified and a design reference is proposed. The focus of the present study on improving the load bearing performance of water lubricated bearings by optimizing the groove design parameters can help to develop different structural designs of MGWLJB’s. CFD analysis considering engineered slip surfaces can also help to identify alternate bearing lining materials. Axially located groove configurations along the bearing circumference play a significant role specifically under contaminated water lubricated conditions. Hence, groove design and selection of bearing lining material has a major effect on the bearing performance. Integration of optimization techniques and CFD approach effectively generates reliable set of data applicable to stern tube bearings used in ships. The simulation data generated can be a guiding reference for designing different structures of water lubricated journal bearings with varying number of axial grooves and different kinds of groove configurations.

2 Methodology

2.1 Design of Experiments

Use of statistical tools such as design of experiments is important to gather and evaluate vast volumes of data to obtain complete information on the critical factors being studied. In the present study, the Taguchi method is used to evaluate the influence of various design parameters on the performance of water lubricated grooved bearings. Using Taguchi method, number of simulation models to be developed and analyzed can be significantly minimized using the L9 orthogonal array. Also, the effects of uncontrollable factors on the bearing performance can be reduced. Taguchi method uses a loss function to compute the variation between desired and analyzed values. Table 1 shows the parameter combinations for L9 orthogonal array. The major bearing design parameters considered for the Taguchi analysis are groove height, groove angle, number of grooves and attitude angle. Figure 1 shows the schematic representation of a 4 axial grooved water lubricated bearing with detail of different design parameters considered. Taguchi method is useful in identifying the influential and optimal parameters of the WLJB. CFD simulations are done for nine different combination of design parameter values detailed in the L9 orthogonal array of Table 1. For each of the CFD simulation runs, the pressure variation and load-carrying capacity of water lubricated journal bearing is noted. MINITAB software is used for the Taguchi analysis.

Table 1 L9 orthogonal array of parameter combinations
Full size table
Fig. 1
figure 1

Schematic representation of a water lubricated journal bearing model with 4 axial grooves

Full size image

2.2 Shape Optimization

After identifying the influential parameters using the Taguchi analysis, the optimal parameter values for each factor from the levels selected need to be determined. The response table for mean in Table 4 is used, where the mean value for each factor at each of their levels is obtained using the MINITAB software. Detailed illustration for the main effect plot for means is given in Fig. 11. For each factor, the level at which the mean value is maximum is selected as the optimal value for the factor. Hence, the optimum value for each factor is the level with the maximum mean value. Mean effect plots give an insight to the parameters having highest influence on the load-carrying capacity.

2.3 CFD Model

In this study, all the computational models presented are based on the physical models of axially grooved water lubricated journal bearing systems. For the CFD analysis, multiple simulation models of varying groove angle, groove size, attitude angle and number of grooves are developed based on the optimization studies. Figure 2 represents the meshed fluid domain model of WLJB having four number of grooves. Similar CFD models are developed based on the different levels and factors mentioned in Table 1. The design models of grooved WLJB are developed considering the eccentricity ratio and attitude angle obtained for different load and speed conditions. The design parameters considered for the water lubricated bearings is detailed in Table 2. The fluid model is considered as an incompressible, steady and iso-thermal fluid. These models are built using ANSYS Design Modeler software.

Fig. 2
figure 2

Mesh of computational fluid domain of 4- four-axial groove WLJB with boundary conditions

Full size image
Table 2 Design parameters of water lubricated journal bearing
Full size table

Table 3 details the characteristic parameters of the water film mesh. For all the cases given in Table 1, meshing and simulation is carried out using ANSYS FLUENT. The 3D models of water film in clearance spaces are discretized with hexahedral elements. Further grid refinement on both sides of wall is performed to obtain accurate film pressure values and minimize the simulation errors. In the meshed models, ten divisions are made across the film thickness excluding the groove portion and further applied multizone mesh method and body sizing of 0.1 mm. The quality of mesh is evaluated by analyzing the major parameters affecting the water film mesh such as element quality (optimum value = 1), average aspect ratio (optimum value = 1) and average skewness (optimum value = 0). In the present analysis, element quality obtained is 0.9875, average aspect ratio is 1.02618 and the average skewness is 9.014831 × 10–3, which proves that the developed mesh is of good quality. Improving the mesh quality is required to enhance the simulation efficiency and for accelerating the convergence rates. The grid and boundary conditions applied to the water film mesh models are illustrated in Fig. 3.

Table 3 Characteristics features of fluid mesh model
Full size table
Fig. 3
figure 3

Three-dimensional fluid domain model of an axial groove water lubricated journal bearing a 4-Groove b 3-Groove c 2-Groove

Full size image

2.4 Boundary Conditions

As shown in Fig. 3, one side of the water film is used as inlet and the other side as outlet. Both the inlet and outlet boundary gauge pressures are set to 0 Pa. The inner surface of the water film is modeled as a moving wall with a rotating speed equal to the rotation speed of the journal, and the outer surface is defined as a stationary wall. Initially, both the walls are subjected to a no-slip condition. The numerical analysis was performed using a pressure-based solver. The SIMPLEC algorithm is used to treat the pressure–velocity coupling. Momentum equations are solved using second order upwind discretization scheme. An output parameter load is created through integration of pressure on moving wall. The iterations are continued till the convergence of the load value reaches till 1 × 10–3. For validation, the above-mentioned procedure is applied to a plain journal bearing model with z = L/2 and the resulting dimensionless pressure distributions are compared with the results of Zhang et al. [13]. The pressure values around the journal surface are obtained using CFD Post and converted to non-dimensional form using Eq. (1).

$$\overline{P }= p\psi 2/(\mu \omega )$$
(1)

For mesh independence analysis, an attitude angle of 400 and eccentricity ratio of 0.6 is considered. The variation in load capacity with respect to number of elements is represented in Fig. 4. In the simulation, sweep method is used to get the hexahedral meshes and a body sizing of 0.1 mm is applied to the entire model. With the increase in number of elements, minimal variation in load capacity is noted for varying body sizing values. As the number of elements was increased to 3,44,800, the load capacity exhibited minimal variation and for analyzing the calculation accuracy, the number of elements was increased to 26,94,400. However, by considering the computational time required and calculation accuracy, the number of elements was finalized at 3,44,800.

Fig. 4
figure 4

Mesh independence study

Full size image

2.5 Slip Velocity Model

In order to predict the boundary slip behavior, different numerical models such as slip length model, limiting shear stress model and slip intensity model are often considered. The slip length model describes that slip velocity, Us is directly in relation with surface shear rate, ∂u/∂y and slip length, b as defined in Eq. (2). Slip length, b is the distance below the solid region where the velocity linearly varies to 0 as illustrated in Fig. 5.

Fig. 5
figure 5

Slip length model [17]

Full size image
$${U}_{s}=b \frac{\partial u}{\partial y}$$
(2)

Both slip length and limiting shear stress slip models describes the relation between slip velocity, Us and fluid velocity, U at regions having close proximity to fluid–solid interface. This relationship is defined using a slip-intensity factor, γ given in Eq. (3), which has a value ranging from 0 to 1. The slip-intensity factor, γ is useful to calculate the slip intensity and determine its effect on the performance envelope of water lubricated bearings. The slip intensity model derived from limiting shear stress slip model indicate that the boundary slip happens only in tangential direction. When slip-intensity factor γ = 1, a perfect slip condition occurs through the slip model. Whereas for γ = 0, no-slip boundary condition occurs with no velocity at the stationary wall [39]. The simulation is run for three values of γ, i.e., 1%, 2% and 3%. The values of slip-intensity factor ‘γ’ considered in this study for water lubricated journal bearings are adopted from Amar and Raghunandana [39], where a detailed theoretical analysis was conducted for a non-circular bearing with slip between the sliding surfaces. The fluid velocity is the same as the tangential velocity at the journal surface.

$${U}_{s}=\upgamma \cdot U$$
(3)

Tangential velocity

$$U=\frac{\mathrm{\pi D N}}{60}= \frac{\pi \mathrm{x }0.08\mathrm{ x }3000}{60}= 12.57\mathrm{ m}/\mathrm{s}$$
(4)

Shear Stress

$$\tau = \mu \frac{\mathrm{d}u}{\mathrm{d}y}$$
(5)
$$\tau = \mu \left(\frac{U-{U}_{s}}{0.04 \mathrm{x }{10}^{-3}}\right)$$
(6)

where dy is the radial clearance.

For γ = 1%, Us = 0.1257 m/s

$$\tau = 0.001003 \left(\frac{12.57-0.1257}{0.04 \mathrm{x }{10}^{-3}}\right) = 312.041\mathrm{ Pa}$$
(7)

For γ = 2% and 3% the shear stress values are 308.89 Pa and 305.74 Pa, respectively. Initially, no-slip condition is considered at the bearing surface and resulting pressure variation and load capacity is monitored. Later, partial slip is applied at the multiple groove surfaces by defining slip conditions (i.e., shear stress value) as the boundary conditions. The film pressures and load capacity is monitored for the above three different values of shear stress.

2.6 Method Validation

To validate the CFD solution approach adopted in the present simulation of water lubricated bearings, the solution configuration is applied to a plain journal bearing model. The simulation results of plain journal bearing are compared with the published data of Zhang et al. [13] as a reference by considering similar input parameters and operating conditions. Figure 6 presents the comparison of the dimensionless pressure distribution for ε = 0.6 and L/D = 1. Based on the comparative data, maximum pressure generation is noted in the convergent fluid film region of plain bearing model. In conclusion, the present simulation results are comparable with those of Zhang et al. [13], indicating that the applied CFD solution methodology is validated. All further simulation on axially grooved water lubricated bearing will be based on the validated solution configuration. Further, the accuracy of the computational method is also compared with the published experimental results of Litwin and Olszewski [40]. In this validation, a shaft of 100 mm diameter is considered to be operating at a speed of 7 rev/s. A detailed comparison of computed film pressures and experimentally measured pressure values are illustrated in Fig. 7. An estimated maximum error of 9.91% is observed, which proves that the present computational method is in close agreement with the experimental studies and can be effectively used to study the load bearing performance of water lubricated bearing.

Fig. 6
figure 6

Dimensionless Pressure Distribution at z = L/2 for ε = 0.6, L/D = 1

Full size image
Fig. 7
figure 7

Comparison of calculated film pressures with the experimentally measured values of water lubricated bearing [40]

Full size image

3 Results and Discussions

3.1 Effect of Bearing Design Parameters

For all the combination level of factors mentioned in Table 1, detailed CFD simulation is carried out on axially grooved water lubricated bearing models. Multiple CFD models are developed by considering varying groove angle, groove height, number of grooves and attitude angle. The lubricant is fed along the axial grooves and a lubricant film is developed in the land/stave regions. In the clearance spaces, the fluid flow occurs in circumferential and axial direction. The water film pressure distribution developed in the clearance spaces of WLJB models are illustrated in the form of pressure contours in Fig. 8. Peak hydrodynamic pressures developed in water film are also depicted using graphical representations. Film pressures at the axial grooves are considered as atmospheric pressures due to the larger surface area of the groove structure in comparison with the clearance spaces at land regions. In commercially available WLJB’s, due to side leakage during bearing operation, maximum pressures generated will be concentrated near the central position of the bearing.

Fig. 8
figure 8figure 8figure 8

Pressure contours and dimensionless pressure distribution for different combinations of parameters in axially grooved water lubricated bearing

Full size image

As shown in Fig. 8a, for Model 1 with four number of grooves having 360 groove angle, positive pressures are generated in the loaded stave/land regions of bearing model. Whereas negative pressures are noted in the unloaded stave regions due to the presence of larger film thickness. From the pressure contours generated, higher hydrodynamic pressures developed are found to be distributed over two lands (i.e., Land 2 and Land 3) of water lubricated bearings. Such pressure distribution is mainly influenced by smaller land area, load direction and eccentric journal positions under bearing operating conditions. The positive pressures generated due to wedge action in land regions helps to support the load acting on journal. Diverging film regions (i.e., negative pressures) influenced from journal position and loading conditions are noted in Land 3 and Land 4 of the bearing model. In such divergent regions, cavitation effects will be prominent under starved lubrication conditions. In the graphical representation, two positive pressure peaks of approximately the same magnitudes are noted in regions of Land 2 and Land 3. At halfway across Land 3, pressures generated get transitioned to negative pressure zone. Positive pressures of small magnitude are also noted in Land 1 of bearing model. Such pressure distributions over different land regions are influenced by the presence of axial grooves and will further supplement the load bearing capacity of water lubricated bearing. For Model 2 in Fig. 8b, reduction in groove height and number of grooves of same groove angle has a positive impact on the hydrodynamic pressures developed in clearance spaces. In the entire region of Land 1, a nominal increase in film pressures is noted. Whereas a significant rise in the magnitude of hydrodynamic positive pressures is noted in Land 2. Such increase in film pressures helps to improve the load-carrying capacity of the proposed bearing configuration as illustrated in Fig. 9. Similar pressure variation is also noted in the graphical representation given in Fig. 8b. No pressure variation is observed in the groove regions due to the applied boundary conditions. In actual water lubricated bearings, major amount of lubricant flows through the axial grooves than compared with the clearance spaces of land/staves. Such lubricant flow conditions and larger film thickness are the major reasons for the boundary conditions applied in the present computational study. In normal situations, the pressure drop in the grooves is assumed to be linear from entrance to the exit region of bearing. In comparison with Model 1, 3-groove bearing model is found to generate lower magnitudes of negative film pressures, which can be partially attributed to the reduced groove height. In Fig. 8c, a further increase in positive hydrodynamic pressures is noted in the land regions of a 2-groove bearing model. Reduction in number of grooves and increased attitude angle has significantly influenced the film pressures. Increased attitude angle modifies the journal equilibrium position and loading conditions. Such variation in attitude angle affects the clearance spaces and minimum film thickness regions of the bearing model resulting in the increase in film pressures. From Fig. 9, a significant increase in bearing load capacity is noted for a 2-groove bearing model than compared with 4 and 3-groove WLJB models. However, a nominal increment in groove height (i.e., h = 5 mm) is not found to have a predominant effect on the bearing pressure distribution. The low magnitude negative pressures noted in both the land regions of bearing model is comparatively minimal than observed in earlier bearing configurations.

Fig. 9
figure 9

Load-carrying capacity for different combinations of parameters in axially grooved water lubricated bearing

Full size image

Figure 8d to f presents the pressure contours generated for WLJB models of 18° groove angles with varying groove height, attitude angle and number of grooves. From Fig. 8d, a notable increase in film pressures developed in clearance spaces are observed than compared with the pressure variation of a 4-groove bearing model depicted in Fig. 8a. In the groove design configuration, reduction in groove angle and groove height has effectively contributed to the rise in film pressures. Fluid pressures in the smaller axial groove configurations are found to positively increase the film pressures at the land regions. For ϕ = 60°, the journal gets positioned at a larger eccentric distance away from the bearing center. Such modifications in the journal equilibrium positions have a positive influence on the film thickness developed. Larger attitude angle combined with smaller groove height reduces the minimum film thickness and helps to build the film pressures at the convergent zone. By considering a 3-groove model with 180 groove angles shown in Fig. 8e, a notable change in pressure distribution can be observed. Maximum pressures are generated only in Land 2 of the bearing model. Due to the increase in pressure amplitudes, higher load-carrying capacity is generated by Model 5 as shown in Fig. 9. Increased land area and lesser number of grooves is found to influence the bearing load capacity. For such groove configurations, the effect on friction variable and side leakage needs to be further assessed. A further increase in pressure magnitudes can be observed by decreasing the groove height and considering larger attitude angle conditions. In Fig. 8f, a 2-groove bearing model is found to generate higher hydrodynamic pressures at the convergent region than observed in the previous models. A further rise in peak pressure can be obtained by considering grooves of smaller surface area. From Fig. 9, lesser number of grooves exhibit significant potential in improving bearing load capacity at high rotational speeds.

A reduction in groove angle to 90 further causes a sharp rise in pressure peaks generated for WLJB models having varying number of grooves and groove height. Even at lower groove angles, maximum hydrodynamic pressures are found to be distributed over two lands of a 4-groove bearing model illustrated in Fig. 8g. Two higher pressure peaks developed are also depicted in the graphical representation. Load orientation also plays a major role in distributing the pressures over the two land regions. Such variations can get further exaggerated under turbulence and journal misaligned conditions, which is future part of this study. Further, by considering 3 grooves over the bearing surface, a substantial rise in film pressures is noted in the minimum film thickness region. Even though higher hydrodynamic pressures are developed with smaller groove angles, further enhanced pressure variation can be attained by designing the models with smaller groove height. As shown in Fig. 8f, for a 2-groove bearing model with 90 groove angle, maximum water film pressures are noted than compared with all the previously simulated WLJB models. Such an increase in film pressures is primarily influenced by the smaller groove size and lesser number of grooves in the bearing models. Hence, 2-groove WLJB model of 90 groove angle and smaller groove height can provide higher load-carrying capacity at varied journal eccentricity ratios and higher rotational speeds. Thus, it proves that modifications in groove design configuration can effectively improve the load bearing performance of water lubricated bearings. As shown in Fig. 9, from the present simulation, maximum load-carrying capacity is obtained for Model 9 with varied design and operational parameters than compared with all other WLJB models.

Further statistical analysis is carried out to identify the accurate parameters having higher percentage influence on the load bearing performance. Such design reference data generated is useful for the design and development of an optimized water lubricated journal bearing with improved performance characteristics. The percentage influence of different design and operating parameters including groove angle, groove height, number of grooves and attitude angle on load-carrying capacity is illustrated in Fig. 10. The percentage influence for each factor is determined by taking their sum of squares of means and then dividing with a summation of the sum of squares of means of each factor. Based on the statistical analysis, number of grooves is found to have the highest percentage influence of 89%. Whereas variation in groove angle has a nominal effect with a percentage influence of about 10%. The other two parameters such as attitude angle and groove height have a minimal influence in comparison with the number of grooves and groove angle factors. From the present study, it is noted that higher number of grooves has a negative impact on the water film pressures since the fluid pressures at the grooved regions are considered to be at atmospheric pressures. Also, the axial grooves have larger surface area than compared with the clearance spaces between land and journal surfaces. Hence, reducing the number of grooves helps to increase the hydrodynamic film pressures and thereby provide improved load-carrying capacity. Due to the maritime applications of WLJB’s, the number of grooves is not only to improve the bearing performance, but also to ease the bearing lubrication under difficult operating conditions involving contaminant or dirt particles. Currently, the commercially available water lubricated bearings usually have five or eight circumferentially located axial grooves which also helps in flushing out the contaminant and dirt particles from the bearing. In the present study, optimization of the number of grooves is primarily determined based on the pressure magnitudes generated and load bearing capacity of the WLJB’s. However, the effect of side leakage and fluid friction is not taken into account in the present analysis.

Fig. 10
figure 10

Representation of percentage of influence of each design parameter on load capacity

Full size image

Table 4 lists the factors with highest influence using the response table for means. The factors with highest delta value are identified as the factor with highest influence on load-carrying capacity. These influencing factors are studied using the combinations generated through the L9 orthogonal array in the MINITAB 16 software. Static Taguchi analysis has been adopted to find the optimum combination of influencing parameters. The optimal values of each parameter is obtained by determining the highest value of mean among the three levels. The highest mean value for attitude angle is 60° at level 3. As a result, the optimum values of groove angle, groove height and number of grooves obtained are 9°, 7 mm and 2, respectively. Figure 11 depicts the mean effect plot for means of each parameter. Parameters with larger slope tend to have higher influence on the load capacity for a small variation in the parameter values. It can be observed that the number of grooves has the highest slope in comparison with other combination of factors. It implies that for a small change in the number of grooves, the load-carrying capacity will alter significantly. The groove angle has the second highest slope followed by groove height and attitude angle. Depending upon the bearing lining material, groove angle has a significant role in flushing out the contaminant particles from the bearing lubricant zone. Hence, reference data generated for optimal groove angle will help the designers in developing an efficient water lubricated bearing with higher load capacity. The groove height and the attitude angle are found to have a very small slope and variation in their values will not significantly affect the load-carrying capacity of the water lubricated bearing models.

Table 4 Response table for means
Full size table
Fig. 11
figure 11

Main effect plots for means

Full size image

From the mean effect plots, the optimal value of each parameter identified is detailed in Table 5. The pressure distribution obtained for the optimal parameter values is illustrated in Fig. 12. Higher hydrodynamic positive pressures are generated in the land region of 2-groove bearing model. The pressure contours generated are similar to the water film pressure variation noted for a 2-groove bearing model in Fig. 8. Maximum pressures are developed at the central region of the convergent fluid film zone. Side leakage plays a major role in generating such water film pressure distributions. Increased water film pressures will effectively contribute to improving the bearing load-carrying capacity. By analyzing the peak pressure values developed for an optimal water lubricated bearing model, higher magnitude of pressures are noted in comparison with the pressure values reported for a 3 groove bearing model in Pai et al. [9]. Even at higher loads, around 2.88 times increase in peak pressures are generated from the optimal bearing model reported in the current study.

Table 5 Optimal parameters of water lubricated bearing models
Full size table
Fig. 12
figure 12

Pressure contour and dimensionless pressure distribution for optimal combination of water lubricated bearing design parameters

Full size image

3.2 Effect of Partial Slip on Load Capacity

In traditional lubrication theory, the fluid nearer to the solid boundaries is considered to have zero velocity with respect to the solid surfaces. In many of the applications, no-slip condition is a valid model to predict the lubrication mechanisms. However, for water lubricated stern tube bearings with engineered surfaces, the assumption of no-slip boundary conditions is not valid. Currently, the commercially available water lubricated bearings utilize rubber lining material and different surface coatings resulting in slip at the fluid–solid interface. Considering the practical implications, WLJB models with partial slip conditions is simulated in the present study.

In the present study, partial slip condition is applied only at the groove regions and the resulting pressure distribution and load-carrying capacity are recorded. A 2-groove WLJB model with optimal design parameters mentioned in Table 5 is considered for the CFD simulation with partial slip. The water film pressure distribution generated for three different slip intensity conditions at groove surfaces is presented in Fig. 13. The variation in film pressures is identical to all the three slip intensities. However, a small rise in pressure peak is noted for a reduction in slip intensity. At the minimum film thickness region, the presence of increased film pressures is due to the decrease in fluid velocity as the fluid moves from the slip surface to a non-slip surface, i.e., land region. As a result, the kinetic energy of the fluid gets converted to pressure energy. The maximum and minimum dimensionless pressures are 2.58 and − 0.315, respectively, for slip intensity of 1% as shown in the graphical representation of Fig. 13a. For a further increase in slip intensity, a small drop in pressure is noted in the pressure contours illustrated in Fig. 13b. For a slip intensity of 3%, in Fig. 13c, the maximum and minimum dimensionless pressures recorded are 2.578 and − 0.315, respectively.

Fig. 13
figure 13

Dimensionless Pressure Distribution for different slip intensities

Full size image

From the results, a small decline in pressure peaks is noted for an increase in slip values. However, the peak pressures generated for the bearing model with partial slip is notably higher than the film pressures recorded for no-slip 2-groove model presented in Fig. 12. From the film pressures generated, the load-carrying capacity determined for an optimal 2-groove bearing model under varying slip intensities is listed in Table 6. Based on the simulation, the introduction of slip at the groove surfaces has a nominal effect on the load-carrying capacity. In comparison, higher load capacity is found to be attained by applying low magnitude partial slip conditions at the grooved regions than observed for a simulated model with no-slip condition. Further analysis by applying slip at the land/stave regions can have a major effect on the load bearing performance of water lubricated bearings, which is a future part of this work.

Table 6 Load-carrying capacity for different shear values
Full size table

4 Conclusions

Through CFD simulation, the paper presented a detailed study identifying the optimum design and operating parameters influencing the load capacity of an axial groove water lubricated journal bearing. The influence of varying slip intensities at the grooved regions on bearing load capacity was also studied. Based on the CFD analysis.

  • Multiple number of grooves is found to be the most influential parameter affecting the load bearing capacity followed by groove angle, attitude angle and groove depth.

  • Using Taguchi approach, the optimal values identified for attitude angle, groove angle, groove height and number of grooves are 60°, 9o, 7 mm and 2, respectively.

  • An optimal value of 7 mm groove depth is counterintuitive as increasing groove depth should decrease the load-carrying capacity. This is due to the interaction of multiple design parameters involved in the analysis of CFD bearing model.

  • Peak hydrodynamic pressures and load capacity generated for the 2-groove optimal bearing model is significantly higher than observed for other configurations of bearing structure.

  • The optimized bearing model was further simulated by applying partial slip conditions at the bearing groove surfaces. A small improvement in the load-bearing capacity was noted with the addition of partial slip patterns at selective locations.

  • To avoid the complexity of slip boundary on water lubricated journal bearings, the current study considers the lubricant to be at an isothermal state and ignore the possible thermal distortions of the bearing structure.

  • Further analysis will be conducted and addressed in future by considering the viscosity–temperature relationship and heat transfer to attain an accurate lubrication condition for bearing performance analysis.