Study on the Performance of Liquid Metal Lubricated V-Groove Bearing Considering Turbulence

applied sciences

Article Study on the Performance of Liquid Metal Lubricated V-Groove Bearing Considering Turbulence

Mingyu Xu and Wei Chen *

Key Laboratory of Education Ministry for Modern Design and Roter-Bearing System, Xi’an Jiaotong University, Xi’an 710049, China; [email protected] * Correspondence: [email protected]

Abstract: To study the application of liquid metal (LM) in the field of practical lubrication, the viscosity of gallium based liquid metal (GBLM) was measured initially, and the relationship between viscosity and temperature was fitted to obtain the viscosity under high temperature. Under the simulated high temperature and vacuum working environment of computed tomography tube (CTT) machines, considering the influence of turbulence, the changes, with eccentricity, of bearing capacity, discharge, friction power consumption, temperature rise, stiffness, and critical mass of the GBLM lubricated V-groove bearing (V-g B) were analyzed. Due to the special structure of V-g B, the coordinate transformation was carried out and the turbulent Reynolds equation was solved by using the finite difference method and the local integral method. The bearing film thickness and pressure distribution under the two coordinate systems were analyzed and compared and the pressure distribution of V-g B under small eccentricity and large eccentricity was studied, respectively. The performance of GBLM lubricated V-g B was studied, which provides theoretical guidance and an analytical method for LM bearing of high-performance CT equipment.

Keywords: liquid metal; V-groove bearing; turbulence; lubrication; static and dynamic performance

Citation: Xu, M.; Chen, W. Study on 1. Introduction the Performance of Liquid Metal Lubricated V-Groove Bearing Computed tomography (CT) equipment can find the internal problems of the object Considering Turbulence. Appl. Sci. before the implementation of measures. It can reduce the cost and risk, and plays an 2021, 11, 940. https://doi.org/ increasingly important role in medical diagnosis, equipment diagnosis, and other aspects. 10.3390/app11030940 As the high-value and consumable parts of CT machines, one or two CTT are generally needed to be replaced once a year per CT equipment. Therefore, to medical giants in Received: 10 December 2020 various countries, CTT is the core component in the development and production process Accepted: 12 January 2021 of CT machines [1] and has a great market value. Published: 21 January 2021 In order to avoid the local damage of a fixed anode due to heat accumulation in the CTT, a mature technology is to use a rotating anode instead of a fixed one, so that the heat Publisher’s Note: MDPI stays neu- will be dispersed into an arc area, and not a point, preventing materials from local damage. tral with regard to jurisdictional clai- Considering the actual working conditions of CTT, it is required that the lubricant used in ms in published maps and institutio- the rotating anode should not pollute the vacuum tube. The lubricant should also be able to nal affiliations. withstand high temperature and conduct electricity. Meanwhile, it is better to conduct heat. Gallium based liquid metal (GBLM) has excellent electric and thermal conductivity as well as fluidity. These qualities make it the best choice to lubricate the rotating anode. In the Copyright: © 2021 by the authors. Li- Netherlands, the Philips Medical Institution has produced the IMRC CTT [1]. This CTT censee MDPI, Basel, Switzerland. changed the anode applied to medical diagnosis from fixed to rotating, and used a spiral This article is an open access article groove anode bearing with LM as the lubricant. This technology has not only improved distributed under the terms and con- the performance of CTT, but also broadened the research field. Next, the technology was ditions of the Creative Commons At- developed, comparatively well-established in Europe and America [2–5]. Professor Chen tribution (CC BY) license (https:// Wei of Xi’an Jiaotong University in China [6] provided a detailed design and research on creativecommons.org/licenses/by/ this type of LM lubricated spiral groove bearing. 4.0/).

Appl. Sci. 2021, 11, 940. https://doi.org/10.3390/app11030940 https://www.mdpi.com/journal/applsci Appl. Sci. 2021, 11, 940 2 of 14

GBLM has the advantages of low vapor pressure, high thermal conductivity, and good fluidity, but is also non-toxic and stable in thermo-physical properties. The research on GBLM is increasing, especially its lubrication performance, and its application is growing extensively [7]. For example, Hughes [8] used LM as a high temperature conductive lubricating fluid in magnetohydrodynamic bearings for nuclear power, and found that the bearings’ capacity had been greatly improved; Burton [9] used gallium–indium–tin liquid metal (G–I–T LM) to lubricate and found that the resistance and calorific value of the brush with high current density decreased; Cerkema [10] discovered that GBLM could lubricate deep groove ball bearings in an oxygen free environment, at the same time, the surface tension and adhesion of GBLM had a great influence on the lubrication performance; Kezik [11] studied the lubrication properties of LM at different temperatures; Tian Yu’s research [12] showed that GBLM could effectively prevent the welding of sliding interface under extreme pressure and high load; and Liu Weimin’s team [13–16] has conducted many friction and wear tests, showing that a GBLM oxide film has a certain lubricating effect. The research of spiral groove bearing has a history of more than 90 years [17]. Due to the spiral groove bearing’s self-sealing performance, this kind of bearing does not need external oil supply equipment. The hydrodynamic sliding bearing realizes load carrying through the dynamic pressure oil film. Due to the special structure of the spiral groove bearing, this kind of bearing forms an oil film by the pumping pressure effect and periodic step effect of the spiral groove. Compared with the traditional ball bearing used in CTT, the spiral groove bearing has many advantages such as compact structure, good stability, low vibration noise, low friction power consumption, low wear, and long service life [17]. Therefore, it can better meet the needs of high-speed and stable rotation of the anode target. At present, the research on the lubrication performance of GBLM is mainly focused on the effective anti-friction and load-bearing performances through the friction and wear tests. Meanwhile the theoretical research on the application of GBLM in hydrodynamic lubrication is almost blank. The theoretical lubrication research of liquid metal bearing can help to better understand its working principle, which is of great significance in product design. In this paper, the viscosity of GBLM at different temperatures was measured experimentally, and the viscosity of it at high temperature was obtained by fitting data; as the rotating speed of CTT is high and the GBLM’s viscosity at high temperature is low, the turbulent lubrication model was established, and considering the special structure of V-g B, the lubrication performance of liquid metal bearing was calculated and studied by coordinate transformation.

2. Viscosity Measurement of Gallium Based Liquid Metal Viscosity is a basic property of fluid. It is the measurement of internal friction force between fluid molecules when the fluid is displaced by external force. It indicates the fluidity of fluid, generally varying with different fluids. The smaller the viscosity, the better the fluidity. As an important index to evaluate the quality of lubricants, the viscosity reflects the internal friction force of fluid and is directly related to the performance, wear degree, and service life of the bearing [18]. In addition, in the theoretical analysis of lubrication, it is necessary to know the viscosity of the lubricant, judge its flow form according to the working conditions, and then establish the theoretical lubrication model for analysis. Therefore, in order to carry out the theoretical analysis of the bearing lubrication performance, it is very important to measure the viscosity of gallium based liquid metal. First, we prepared the GBLM, which in this paper was a gallium–indium–tin alloy with a mass ratio of 68.5:21.5:10. According to the configuration method described by Liu Jing’s team in [19], the high-purity metal was selected according to the mass ratio and mixed in the container, which should be cleaned with deionized water. Then, the mixed raw materials were heated to a certain temperature until the metal melted, stirred slightly and cooled to room temperature to obtain GBLM. As GBLM is very easy to oxidize in air, gallium oxide will be produced, which will affect the viscosity and surface tension [20,21]. Therefore, impurity removal steps should be carried out. As is well known, NaOH can Appl. Sci. 2021, 11, x FOR PEER REVIEW 3 of 15

First, we prepared the GBLM, which in this paper was a gallium–indium–tin alloy with a mass ratio of 68.5:21.5:10. According to the configuration method described by Liu Jing’s team in [19], the high-purity metal was selected according to the mass ratio and mixed in the container, which should be cleaned with deionized water. Then, the mixed raw materials were heated to a certain temperature until the metal melted, stirred slightly and cooled to room temperature to obtain GBLM. As GBLM is very easy to oxidize in air, gallium oxide will be produced, which will affect the viscosity and surface tension [20,21]. Therefore, impurity removal steps should be carried out. As is well known, NaOH can Appl. Sci. 2021, 11, 940 3 of 14 react with oxides to remove oxide impurities. We mixed a certain concentration of NaOH solution and the prepared GBLM alloy, both incompatible with each other proportionally; in addition, the density of the liquid metal was larger, so it will be layered after standing; reactthen the with LM oxides after to impurity remove oxideremoval impurities. can be ob Wetained mixed by a taking certain the concentration lower layer of solution NaOH solutionwith a rubber and the tip prepared dropper. GBLM alloy, both incompatible with each other proportionally; in addition,Considering the density that the of thecapillary liquid viscometer metal was larger,is easy so to it operate will be and layered the aftererror standing; is small, thenthis paper the LM used after this impurity method removal to measure can th bee obtainedviscosity byof GBLM. taking theThe lower experimental layer solution setup with(Model a rubber SYD-265E tip dropper. kinematic viscometer, China, SANUO) is shown in Figure 1a. Given that theConsidering actual working that the temperature capillaryviscometer can reach 300 iseasy °C, it to was operate difficult and for the the error test istemper- small, thisature paper to reach. used Therefore, this method in toorder measure to obtain the viscositythe working of GBLM. viscosity, The first, experimental the viscosity setup of (ModelGBLM at SYD-265E different kinematic temperatures viscometer, was measured China, SANUO) by changing is shown the in test Figure temperature,1a. Given thatand ◦ thethen actual the viscosity–temperature working temperature relationship can reach 300 wasC, fitted it was by difficult using the for least the testsquares temperature method. toThe reach. results Therefore, are shown in orderin Figure to obtain 1b, and the the working viscosity–temperature viscosity, first, the equation viscosity can of GBLMbe ob- attained different as follows: temperatures was measured by changing the test temperature, and then the viscosity–temperature relationship was fitted by using the least squares method. The results are shown in Figure1b, and theln𝜈 viscosity–temperature = −0.004638 × equation 𝑇 − 0.616012 can be obtained as follows:(1) where 𝜈 is the kinematic viscosity and 𝑇 is the temperature in centigrade. ln ν = −0.004638 × T − 0.616012 (1) With Equation (1), the viscosity of GBLM under actual working conditions can be wherepredicted:ν is thewhen kinematic the actual viscosity working and temperatureT is the temperature reached 300 in centigrade. °C, the viscosity of GBLM was 0.134 mm/s, converted into a dynamic viscosity of 8.63 × 10 Pa ∙ s.

(a) (b)

Figure 1. (a) The experimental setupsetup forfor thethe capillarycapillary viscometer viscometer method. method. ( b(b)) The The ﬁtting fitting curve curve of of the viscosity–temperaturethe viscosity–temperature relationship. relationship.

With Equation (1), the viscosity of GBLM under actual working conditions can be By measuring the viscosity of GBLM, it was found that its◦ viscosity was low under predicted:the actual working when the condition, actual working and the temperature rotating speed reached of the 300 anodeC, target the viscosity in CTT ofwas GBLM very 2 × −4 · washigh 0.134(up to mm about/s, converted10,000 r/min into). aBefore dynamic theviscosity establishment of 8.63 of the10 theoreticalPa s. model, the ReynoldsBy measuring number thewas viscosity calculated, of GBLM, which it was was about found 2500. that its This viscosity exceeded was lowthe underReynolds the actual working condition, and the rotating speed of the anode target in CTT was very high number limit of laminar flow, which is 2000. Therefore, the turbulent model should be (up to about 10, 000 r/min). Before the establishment of the theoretical model, the Reynolds used in the theoretical analysis of lubrication. number was calculated, which was about 2500. This exceeded the Reynolds number limit of laminar ﬂow, which is 2000. Therefore, the turbulent model should be used in the 3. Numerical Model and Simulation theoretical analysis of lubrication.

3. Numerical Model and Simulation The spiral groove bearing in the CT tube analyzed in this paper was a V-g B. Its essence is to machine a circle of V-groove on the surface of traditional smooth circular bearing. The structure is shown in Figure2, in which, h is the oil ﬁlm thickness; e is the eccentric distance; θ is the deﬂection angle; b is the bearing width; d is the bearing diameter; α is the included angle of herringbone groove; Cg is the trench depth; Lr the ridge width; and Lg is the groove width. Appl. Sci. 2021, 11, x FOR PEER REVIEW 4 of 15

Appl. Sci. 2021, 11, x FOR PEER REVIEW 4 of 15

The spiral groove bearing in the CT tube analyzed in this paper was a V-g B. Its es- senceThe is to spiral machine groove a circle bearing of V-groove in the CT on tube the surfaceanalyzed of intraditional this paper smooth was a circularV-g B. Its bear- es- h e senceing. The is to structure machine is a showncircle of in V-groove Figure 2, on in thwhich,e surface is of the traditional oil film thickness; smooth circular is the bear- ec- centric distance; θ is the deflection angle; b is the bearing width; d is the bearing diam- Appl. Sci. 2021, 11, 940 ing. The structure is shown in Figure 2, in which, h is the oil film thickness; e is the4 of ec- 14 α Cg Lr centriceter; distance; is the included θ is the angle deflection of herringbone angle; b groove; is the bearing is thewidth; trench d isdepth; the bearing the diam- ridge Lg eter;width; α and is the included is the groove angle width.of herringbone groove; Cg is the trench depth; Lr the ridge width; and Lg is the groove width.

FigureFigure 2. 2.The The structuralstructural diagramdiagram ofof V-gV-g B. B. Figure 2. The structural diagram of V-g B. 3.1.3.1. EstablishmentEstablishment ofof ReynoldsReynolds EquationEquation forfor V-GV-G BB 3.1. EstablishmentConsideringConsidering thatofthat Reynolds thethe Ng–PanNg–Pan Equation turbulentturbulent for V-G B modelmodel basedbased onon thethe lawlaw ofof thethe wallwall isisthe the most widely used turbulent model. Meanwhile, its turbulent factor is linear, which makes mostConsidering widely used thatturbulent the Ng–Pan model. turbulentMeanwhile, model its turbulent based on factor the lawis linear, of the which wall makesis the the calculation more simple and suitable for light load hydrodynamic sliding bearing. mostthe calculation widely used more turbulent simple model. and suitable Meanwhile, for light its turbulent load hydrodynamic factor is linear, sliding which bearing. makes Therefore, by choosing this model, the dimensionless Reynolds equation is as follows: theTherefore, calculation by choosing more simple this model, and suitable the dimens for ionlesslight load Reynolds hydrodynamic equation slidingis as follows: bearing. Therefore, by choosing∂ this H model,∂3 ∂𝐻P ∂𝑃the dimensd 𝑑2 ∂ionless∂ H𝐻3 ∂ ∂𝑃ReynoldsP 1 ∂𝐻∂ Hequation is as follows: ++ == (2) (2) ∂ϕ ∂𝜑k∂x ∂ϕ𝐻𝑘 ∂𝑃∂𝜑 b 𝑑𝑏 ∂λ∂𝜆∂ k𝐻z𝑘∂λ∂𝑃∂𝜆 14∂𝐻∂𝜑∂ϕ + = (2) ∂𝜑 𝑘 ∂𝜑 𝑏 ∂𝜆 𝑘 ∂𝜆 4 ∂𝜑 wherewherek 𝑘is is the the turbulent turbulent factor factor and and the the value value is is where 𝑘 is the turbulent factor and𝑘 the= 12value + 0.0136 is (𝑅𝑒). ( 0.9 k = 12 + 0.0136(Re) . (3) x𝑘 = 12 + 0.0043(𝑅𝑒). 𝑘 = 12 + 0.0136(𝑅𝑒0.98) (3) k = 12 + 0.0043(Re) z . (3) It can be seen from Figure 2 that𝑘 = the 12 V-gr + 0.0043oove’s(𝑅𝑒 structure) is special because its trend is inclinedItIt cancan bebein seenaseen straight fromfrom Figureline.Figure The2 2that rectangularthat the the V-groove’s V-gr gridoove’s under structure structure the Cartesian is is special special coordinate because because its its system trend trend isis(CCS) inclinedinclined will inincreasein aa straight straight the line. difficulline. TheThety rectangular inrectangular dealing with grid grid the under under flow the the discontinuity Cartesian Cartesian coordinate coordinate (i.e., at the system systemgroove (CCS)(CCS)step). will Therefore,will increaseincrease transforming the the difﬁculty difficul tycoordinate, in in dealing dealing with thewith calculation, the the ﬂow flow discontinuity discontinuity and analysis (i.e., (i.e., were at at the carriedthe groove groove out step).step).in the Therefore, Therefore,oblique coordinate transformingtransforming system coordinate,coordinate, (OCS) consiste thethe calculation,calculation,nt with the and andgroove analysisanalysis direction. werewere Figure carriedcarried 3 out outis a inindiagram thethe obliqueoblique of the coordinatecoordinate coordinate system system transformation. (OCS) (OCS) consistent consiste nt withwith thethe groovegroove direction. direction. FigureFigure3 3is is a a diagramdiagram ofof thethe coordinatecoordinate transformation.transformation.

Figure 3. Diagram of coordinate transformation. Figure 3. Diagram of coordinate transformation. FigureAs 3. Diagramshown in of Figure coordinate 3, with transformation. the inclination angle of half of the included angle of the V-groove,AsAs shownshown the CCS in Figure was transformed 33,, withwith the into inclination the OC S.angle angle The ofcorresponding of half half of of the the included includedcalculation angle angle relation- of ofthe theV-groove,ship V-groove, is the CCS the CCSwas transformed was transformed into the into OC theS. The OCS. corresponding The corresponding calculation calculation relation- relationshipship is is  λ α η = ϕ + ϕ = η − ξ cos  tan( α ) 2 ⇔ 2 (4) = α λ λ ξ sin 2  ξ = α sin( 2 ) Appl. Sci. 2021, 11, 940 5 of 14

The expression of pressure partial derivative is obtained as followed:

 ∂P ∂P ∂η ∂P ∂ξ ∂P  ∂ϕ = ∂η ∂ϕ + ∂ξ ∂ϕ = ∂η ∂P = ∂P ∂η + ∂P ∂ξ = 1 + 1 (5)  ∂ξ ∂η ∂λ ∂ξ ∂λ α ∂P α ∂P tan 2 ∂η sin 2 ∂ξ

Then, bring Equation (5) into Equation (2), and the Reynolds equation in OCS is as follows: ∂ H3 ∂P + d2 ∂ H3 ∂P + d2 ∂ H3 ∂P ∂η kx ∂η b2 tan2( α ) ∂η kz ∂η b2 tan( α ) sin( α ) ∂η kz ∂ξ 2 2 2 (6) + d2 ∂ H3 ∂P + d2 ∂ H3 ∂P = 1 ∂H 2 α α ∂ξ kz ∂η 2 2 α ∂ξ kz ∂ξ 4 ∂η b tan( 2 ) sin( 2 ) b sin ( 2 )

At this time, the oil ﬁlm thickness equation is: ridge: α H = 1 + εcosϕ = 1 + εcos η − ξcos − θ (7) 2 groove: α H = 1 + εcosϕ + γ = 1 + εcos η − ξcos − θ + γ (8) g 2 g cg e where γg = c is the groove depth ratio; ε = c is the eccentricity; cg is the groove depth; and c is the radius gap.

3.2. Equation Solving 3.2.1. The Discretization of Equations In general, the calculation in CCS can directly carry out differential processing of the partial derivative of the Reynolds equation. However, the equations calculated in OCS have cross partial derivatives, and the direct differential processing will make pressure values at non nodes. Therefore, the ﬁve-point difference method was not applicable here, and the local integration method was chosen to deal with this problem. When the bearing is meshed along the direction of the V-groove and its boundary, there Appl. Sci. 2021, 11, x FOR PEER REVIEWwill appear continuous and discontinuous areas of oil ﬁlm. As an example, the process6 ofof 15 the local integration method can be explained by taking the node falling at the continuous area. The integral area is shown in Figure4, recorded as marked area A.

FigureFigure 4. Diagram 4. Diagram of continuousof continuous oil oil ﬁlm film integration integration area area A. A.

BothBoth sides sides of Equationof Equation (6) (6) were were conducted conducted on on a surface a surface integral integral in thein the region region shown shown belowbelow in Figurein Figure5: 5:

" ∂ 𝐻 ∂𝑃 𝑑 ∂ 𝐻 ∂𝑃# 1 ∂𝐻 ∂ H3 ∂P d 2 ∂ H3 ∂P 1 ∂H + + d𝜑d𝜆= = d𝜑d𝜆 (9) x ∂𝜑 𝑘 ∂𝜑 𝑏 ∂𝜆 𝑘 ∂𝜆dϕdλ x4 ∂𝜑 dϕdλ (9) ∂ϕ kx ∂ϕ b ∂λ kz ∂λ 4 ∂ϕ A A Using the Green formula, the surface integral was changed into a curve integral as follows:

𝐻 ∂𝑃 𝑑 𝐻 ∂𝑃 1 d𝜆 − d𝜑 = 𝐻d𝜆 (10) 𝑘 ∂𝜑 𝑏 𝑘 ∂𝜆 4 Taking Equations (4) and (5) into Equation (10), the integration was carried out according to the curve direction shown in Figure 5, and the five-point difference formula in oblique coordinate was obtained:

𝐴,𝑃, +𝐵,𝑃, +𝐶,𝑃, +𝐷,𝑃, −𝐹, 𝑃, = (11) 𝐸, where

𝛼 cos Δ𝜉 𝛼 H., 𝑑 −𝐻,. +𝐻,. Δ𝜉 2 𝐻., 𝐴, = sin − 𝛼 − 𝛼 (12) Δ𝜂 2 𝑘 𝑏 2𝑘 tan Δ𝜂 tan 𝑘 2 2

𝛼 cos Δ𝜉 𝛼 H., 𝑑 𝐻,. +𝐻,. Δ𝜉 2 𝐻., 𝐵, = sin − 𝛼 − 𝛼 (13) Δ𝜂 2 𝑘 𝑏 2𝑘 tan Δ𝜂 tan 𝑘 2 2

𝑑 Δη 1 𝐻,. α −𝐻., +𝐻., 𝐶, =− − 𝛼 − cos 𝛼 (14) 𝑏 Δξ sin 𝑘 2 2𝑘 sin 2 2

𝑑 Δη 1 𝐻,. α 𝐻., −𝐻., 𝐷, =− − 𝛼 −cos 𝛼 (15) 𝑏 Δξ sin 𝑘 2 2𝑘 sin 2 2

𝐸, = 𝐴, +𝐵, +𝐶, +𝐷, (16)

1 𝛼 𝐹 = sin Δ𝜉𝐻 −𝐻 (17) , 4 2 ., ., For the discontinuous areas of oil film (i.e., the nodes falling at the junction of ridge and groove), it is only necessary to locally integrate the left and right parts of Figure 4, Appl. Sci. 2021, 11, x FOR PEER REVIEW 7 of 15

respectively. For the dynamic Reynolds equation, the small perturbation method is directly used to deal with this as the discrete form is similar to that of the static equation.

Appl. Sci. 2021, 11, 940 3.2.2. Boundary Conditions 6 of 14 Due to the symmetry of the V-groove structure, half of it was selected for research, as shown in Figure 5.

Figure 5. Diagram of the bearing boundary.

TheUsing boundary the Green conditions formula, of the static surface Reynolds integral equation was changed for V-g intoB are a as curve follows: integral as follows: Z 𝐸𝑛𝑑3 𝑑𝑖𝑠𝑐ℎ𝑎𝑟𝑔𝑒 2 Z 3 𝑏𝑜𝑢𝑛𝑑𝑎𝑟𝑦:𝑃| Z =0 ⎧H ∂P d H ∂P ± 1 dλ − dϕ = Hdλ (10) ⎪kx ∂ϕ b kz ∂λ 4 L 𝑃𝑒𝑟𝑖𝑜𝑑𝑖𝑐L 𝑏𝑜𝑢𝑛𝑑𝑎𝑟𝑦: 𝑃 =𝑃 L (18) Taking Equations⎨ (4) and (5) into Equation (10), the𝜕𝑃 integration was carried out ⎪𝑅𝑒𝑦𝑛𝑜𝑙𝑑𝑠 𝑏𝑜𝑢𝑛𝑑𝑎𝑟𝑦: 𝑃 =0; =0 according to the curve direction shown in Figure5, and the𝜕𝜂 ﬁve-point difference formula in ⎩ oblique coordinate was obtained: For the calculation of the symmetric boundary, just assign the pressure symmetrically. Ai,jPi+1,j + Bi,jPi−1,j + Ci,jPi,j+1 + Di,jPi,j−1 − Fi,j Pi,j = (11) The boundary condition of the dynamic ReynoldsEi,j equation of V-g B is that the dis- turbance pressure was zero around the complete oil film area determined by the above staticwhere Reynolds boundary.

3 2 3 3 α 3 ! ∆ξ α Hi+0.5,j d −Hi,j−0.5 + Hi,j+0.5 ∆ξ cos Hi+0.5,j 3.2.3.A Iterative= Calculationsin − − 2 (12) i,j ∆η 2 k b α ∆η α k The successive over-relaxationx method2k Z(SORtan method)2 is easytan to2 applyz with a wide application range, and its programming is simple. Therefore, the SOR method was used H3 2 H3 + H3 α H3 ! to solve the static∆ξ andα dynai−0.5,jmic differenced equations.i,j−0.5 i,j+0.5 ∆ξ cos 2 i−0.5,j Bi,j = sin − α − α (13) In order to∆ ηjudge2 whetherkx the bresult of 2eachkZ tan iteration2 has∆ reachedη tan 2 enoughkz accuracy, and then decide whether the iteration process can be stopped, the following convergence 2 H3 −H3 + H3 ! criteria were set here: d ∆η 1 i,j+0.5 α i+0.5,j i−0.5,j Ci,j = − − α − cos α (14) b ∆ξ sin kz 2 2k sin m n 2 m n z 2 2 Pk -Pk-13 / Pk 3 ≤δ 3 ! (19) d ∆η 1i,j i,jHi,j−0.5 α Hi,j i+0.5,j − Hi−0.5,j D = − − − cos (15) i,j j=2 ξi=2 α j=2 i=2 α b ∆ sin 2 kz 2 2kz sin 2 where the allowable relative error δ is generally about 10. Ei,j = Ai,j + Bi,j + Ci,j + Di,j (16) Figure 6 shows the iterative flow chart of the V-g B performance calculation. 1 α F = sin ∆ξ H + − H − (17) i,j 4 2 i 0.5,j i 0.5,j For the discontinuous areas of oil ﬁlm (i.e., the nodes falling at the junction of ridge and groove), it is only necessary to locally integrate the left and right parts of Figure4, respectively. For the dynamic Reynolds equation, the small perturbation method is directly used to deal with this as the discrete form is similar to that of the static equation.

3.2.2. Boundary Conditions Due to the symmetry of the V-groove structure, half of it was selected for research, as shown in Figure5. Appl. Sci. 2021, 11, 940 7 of 14

The boundary conditions of static Reynolds equation for V-g B are as follows:  End discharge boundary : P| =± 1 = 0  ξ ( α )  sin 2 Periodic boundary : P = = P = (18) η 0 η 2π  Reynolds boundary P = ∂P =  : η≥η1 0; ∂η 0 η≥η1

For the calculation of the symmetric boundary, just assign the pressure symmetrically. The boundary condition of the dynamic Reynolds equation of V-g B is that the distur- bance pressure was zero around the complete oil ﬁlm area determined by the above static Reynolds boundary.

3.2.3. Iterative Calculation The successive over-relaxation method (SOR method) is easy to apply with a wide application range, and its programming is simple. Therefore, the SOR method was used to solve the static and dynamic difference equations. In order to judge whether the result of each iteration has reached enough accuracy, and then decide whether the iteration process can be stopped, the following convergence criteria were set here: ! ! m n m n k − k−1 k ≤ δ ∑ ∑ Pi,j Pi,j / ∑ ∑ Pi,j (19) j=2 i=2 j=2 i=2 Appl. Sci. 2021, 11, x FOR PEER REVIEW 8 of 15 where the allowable relative error δ is generally about 10−3.

Figure6 shows the iterative ﬂow chart of the V-g B performance calculation.

FigureFigure 6. 6.Flowchart Flowchart of of the the numerical numerical procedure. procedure.

3.2.4.3.2.4. Performance Performance Calculation Calculation InIn order order to to reduce reduce the the friction friction resistance resistance of of parts parts in in operation operation and and improve improve the the bear- bear- inging capacity capacity of of lubricating lubricating ﬁlm, film, and and then then improve improve the the stability stability of of bearing bearing performance, performance, thethe pressure pressure distribution, distribution, supporting supporting force, force, friction friction resistance resistance and and stiffness stiffness of of lubricating lubricating ﬁlmfilm should should be be calculated calculated by by using using lubrication lubrication theory. theory. TheThe following following is is the the calculation calculation formula formula for for the the bearing bearing performance performance parameters: parameters: (1) Bearing capacity:

⎧ Total bearing capacity:𝐹 = 𝐹 +𝐹 ⎪ ⎪ Horizontal bearing capacity: 𝐹 = 𝑃sin𝜑d𝜑d𝜆 (20) ⎨ ⎪ Vertical bearing capacity:𝐹 = 𝑃cos𝜑dφd𝜆 ⎩

(2) Deflection angle: The deflection angle is solved by iteration. In the beginning, when the value of the bearing deflection angle is not known, an initial deflection angle should be given, and then the load angle can be calculated after calculating the oil film bearing capacity:

𝐹 𝛼 =arctg (21) 𝐹 The deflection angle is corrected according to the calculated load angle:

𝜃=𝜃 −𝛼 (22) After the new deflection angle is obtained, the real offset angle can be obtained by returning to the program for iterative solution again until the deflection angle correction converges.

(3) Discharge:

𝑟 𝐻 𝜕𝑃 𝑟 𝐻 𝜕𝑃 𝑄 =−4𝑐𝑈𝑏 − dφ (23) 𝑏 𝑘 𝜕𝜆 𝑏 𝑘 𝜕𝜆 Appl. Sci. 2021, 11, 940 8 of 14

(1) Bearing capacity:  q  = 2 + 2  Total bearing capacity : F Fx Fy R 1 R 2π Horizontal bearing capacity : Fx = P sin ϕdϕdλ (20)  −1 0  R 1 R 2π Vertical bearing capacity : Fy = −1 0 P cos ϕdϕdλ

(2) Deflection angle: The deflection angle is solved by iteration. In the beginning, when the value of the bearing deflection angle is not known, an initial deflection angle should be given, and then the load angle can be calculated after calculating the oil film bearing capacity: Fξ αF = arctg (21) Fη

The deﬂection angle is corrected according to the calculated load angle:

θ = θ0 − αF (22)

After the new deﬂection angle is obtained, the real offset angle can be obtained by returning to the program for iterative solution again until the deﬂection angle correction converges. (3) Discharge:

Z r2 H3 ∂P r2 H3 ∂P = − − QL 4cUb 2 2 dϕ (23) b kz ∂λ λ=1 b kz ∂λ λ=−1

(4) Friction resistance, friction power consumption, and temperature rise: Friction resistance:

ωr2µb 1 12H ∂P Ft = x τc − dϕdλ (24) 2c H kx ∂ϕ

where τc is the Kutta shear stress; τc = −1 in laminar ﬂow; and τc is a function of Reynolds number in turbulent ﬂow, which is obtained from the experimental data in [22]:

0.94 τc = 1 + 0.0012(Re) (25)

Friction power consumption: Nt = Ft·U (26) Temperature rise: F ·U ∆T = t (27) QLρcv (5) Four dimensionless stiffness of bearing:

 K sinφ  xε = − R 2π R 1 P dλ dφ  K 0 −1 ε cosφ yε (28) K sinφ  xθ − R 2π R 1  0 −1 Pθdλ dφ Kyθ cosφ Appl. Sci. 2021, 11, 940 9 of 14

(6) Four dimensionless damping of bearing:

 D sinφ  xx = − R 2π R 1 P dλ dφ  D 0 −1 Vξ cosφ yx (29) D sinφ  xy − R 2π R 1  0 −1 PVηdλ dφ Dyy cosφ

(7) Equivalent stiffness: Keq = Kxx Dyy + KyyDxx − KxyDyx − Kyx Dxy / Dxx + Dyy (30)

(8) Limit eddy ratio square:

2 γst = Keq − Kxx Keq − Kyy − KxyKyx / Dxx Dyy − DxyDyx (31)

(9) Dimensionless critical mass: 2 M = Keq/γst (32)

3.3. Model Validation There has been validation between the above numerical model and references. One ◦ Appl. Sci. 2021, 11, x FOR PEER REVIEWwas the experimental study on the turbulent lubrication performance of the 360 sliding10 of 15 bearing performed by Smith and Fuller [23], and another was the simulation result of

bearing capacity in [24]. The results are shown in Figure7.

Figure 7. Verification of the bearing capacity of the circular bearing. Figure 7. Veriﬁcation of the bearing capacity of the circular bearing.

AsAs shown shown in in Figure Figure7, the7, the numerical numerical model model was was in in good good agreement agreement with with the the literature literature andand experimental experimental results. results.

4.4. Results Results and and Discussion Discussion TheThe parameters parameters of of V-g V-g B B and and the the lubricant lubricant used used in in the the following following analysis analysis are are listed listed in in TablesTables1 and 1 and2. 2.

TableTable 1. 1.The The parameters parameters of of V-g V-g B. B.

ParameterParameter NumericalNumerical Value Value BearingBearing diameter diameter d/mm d/mm 30 30 RadiusRadius gap c/mm gap c/mm 0.02 0.02 Number of slots n 10 Number of slots n 10 Groove-ridge ratio rg 1 Width-diameterGroove-ridge ratio L/d ratio r 1 1 HelixWidth-diameter angle α/◦ ratio L/d 60 1 Rotation rateHelix N/ r ·anglemin−1 α/° 10,800 60 GrooveRotation depth rate cg/mm N/(r∙min ) 0.02 10,800

Groove depth c/mm 0.02

Table 2. The parameters of the lubricant.

Parameter Numerical Value Density 6.44 g/cm Viscosity 1.8 × 10 Pa ∙ s Specific heat capacity 365.6 J/kg ∙ ℃

4.1. Comparison of Cartesian and Oblique Coordinates With the eccentricity of 0.4, Figure 8 shows the oil film thickness and pressure distributions of liquid metal bearing, where (a) and (b) are calculated in CCS, and (c) and (d) are calculated in OCS. Appl. Sci. 2021, 11, 940 10 of 14

Table 2. The parameters of the lubricant.

Parameter Numerical Value Density 6.44 g/cm3 Viscosity 1.8 × 10−3 Pa·s Speciﬁc heat capacity 365.6 J/kg·°C

4.1. Comparison of Cartesian and Oblique Coordinates

Appl. Sci. 2021, 11, x FOR PEER REVIEW With the eccentricity of 0.4, Figure8 shows the oil ﬁlm thickness11 and of 15 pressure distributions of liquid metal bearing, where (a) and (b) are calculated in CCS, and (c) and (d) are calculated in OCS.

(a) (b)

Figure 8. OilFigure ﬁlm thickness 8. Oil film and thickness pressure and distributions pressure distributi in differentons coordinatein different systems: coordinate (a,b systems:) are calculated (a,b) are in CCS; (c,d) are calculated incalculated OCS. in CCS; (c,d) are calculated in OCS.

As shown in FigureAs shown 8, it can in Figure be seen8, itthat, can in be different seen that, coordinates, in different the coordinates, film thickness the film thickness and pressure distributionsand pressure were distributions consistent, were respectively, consistent, respectively,and the pressure and the value pressure reached value reached the the maximum maximumnear the minimum near the minimum film thicknes films. thickness. Compared Compared with the withdistribution the distribution calcu- calculated in lated in CCS, CCS,the thefilm film thickness thickness and and pressure pressure distribution calculatedcalculated in in OCS OCS were were smoother, and the smoother, anddirection the direction of iteration of iteration was morewas mo consistentre consistent with thewith shape the shape boundary boundary of the V-shapedof groove. the V-shaped groove.It was more It was consistent more consistent with the supportingwith the supporting principle principle of V-shaped of V-shaped spiral groove bearing as spiral groove bearingwell as closeras well to as the closer flow to direction the flow of direction the lubricating of the lubricating medium. medium. The film thicknessThe filmdistribution thickness of distribution the circumferential of the circumferential was similar to was that similar of the tocircu- that of the circular lar bearing. However,bearing. as However, the V-groove as the was V-groove uniformly was uniformly distributed distributed on the bearing on the surface, bearing surface, it pre- it presented a stepsented distribution. a step distribution. Due to the Due ex toistence the existence of wedge of wedgeeffect, the effect, pressure the pressure distri- distribution in bution in the circumferentialthe circumferential direction direction was wassimilar similar to that to thatof the of circular the circular bearing. bearing. In this paper, Inthe this spiral paper, groove the spiral was grooveset on wasthe setshaft on and the shaftthe corresponding and the corresponding shaft shaft sleeve sleeve cavity wascavity smooth, was smooth, the shaft the rotated shaft rotatedat a certain at a speed, certain and speed, the andliquid the on liquid one side on one side of the of the sleeve was pumped. This phenomenon is called the pumping pressure effect. There- fore, because of the existence of the V-groove’s pumping pressure effect, the pressure at different positions in the circumferential direction increases rapidly at the tip of the V- groove center, forming a local peak. Moreover, as a result of the existence of both effects, almost every V-groove corresponded to a local peak.

4.2. Comparative Analysis of Small Eccentricity and Large Eccentricity Setting the eccentricity to 0.1 and 0.6, respectively, the bearing pressure distribution was calculated by using the model described in the previous section. Figure 9 shows the calculated bearing pressure distribution, where (a) is calculated with the eccentricity of 0.1, and (b) is calculated with the eccentricity of 0.6. Appl. Sci. 2021, 11, 940 11 of 14

sleeve was pumped. This phenomenon is called the pumping pressure effect. Therefore, because of the existence of the V-groove’s pumping pressure effect, the pressure at different positions in the circumferential direction increases rapidly at the tip of the V-groove center, forming a local peak. Moreover, as a result of the existence of both effects, almost every V-groove corresponded to a local peak.

4.2. Comparative Analysis of Small Eccentricity and Large Eccentricity Setting the eccentricity to 0.1 and 0.6, respectively, the bearing pressure distribution was calculated by using the model described in the previous section. Appl. Sci. 2021, 11, x FOR PEER REVIEW 12 of 15 Figure9 shows the calculated bearing pressure distribution, where (a) is calculated

with the eccentricity of 0.1, and (b) is calculated with the eccentricity of 0.6.

(a) (b)

Figure 9. PressureFigure distributions9. Pressure distributions under different under eccentricities: different eccentricities: (a) Dimensionless (a) Dimensionless pressure distributable pressure 0.1; distrib- (b) Dimensionless pressure distributionutable 0.1; calculated (b) Dimensionless with the eccentricitypressure distribution of 0.6. calculated with the eccentricity of 0.6.

From the analysisFrom of theFigure analysis 9, the of pressure Figure9 tends, the pressure to be gentle tends along to be the gentle circumference along the circumference with the smallwith eccentricity, the small and eccentricity, the pressure and thepeak pressure value peakis prominent value is prominentand changes and vio- changes violently lently when thewhen eccentricity the eccentricity is large. In is the large. case In of the small case eccentricity, of small eccentricity, the pressure the is pressure mainly is mainly pro- provided by thevided V-groove’s by the V-groove’s pumping pumpingpressure pressureeffect and effect relatively and relatively scattered, scattered, which is which is mainly mainly distributeddistributed in the groove in the groove rather ratherthan at than the ridge; at the ridge;meanwhile, meanwhile, each V-groove each V-groove cor- corresponds responds to a localto a localpressure pressure peak peak point point with with a small a small value. value. In the In case the caseof large of large eccentricity, eccentricity, the overall wedge-shaped effect of bearing is more obvious, the pressure distribution is more concen- the overall wedge-shaped effect of bearing is more obvious, the pressure distribution is trated with a larger peak value, which is mainly distributed in the ridge, while at the same more concentrated with a larger peak value, which is mainly distributed in the ridge, time, the pump pressure effect is evident near the minimum ﬁlm thickness. while at the same time, the pump pressure effect is evident near the minimum film thick- Regardless of eccentricity, compared with ordinary smooth bearing, V-g B had a more ness. dispersed pressure distribution with a wider range, more peak points, and more stable Regardless of eccentricity, compared with ordinary smooth bearing, V-g B had a pressure change. more dispersed pressure distribution with a wider range, more peak points, and more stable pressure4.3. change. Static and Dynamic Performance Analysis of Bearing The static and dynamic performances of hydrodynamic V-g B was analyzed by chang- 4.3. Static and Dynamic Performance Analysis of Bearing ing eccentricity from 0.1 to 0.9 while other parameters remained unchanged. The static andFigure dynamic 10 shows performances the curve of thehydrodynamic static and dynamic V-g B performancewas analyzed parameters by of V-g B changing eccentricitywith eccentricity. from 0.1 to 0.9 while other parameters remained unchanged. Figure 10 shows the curve of the static and dynamic performance parameters of V-g B with eccentricity.

(a) (b) Appl. Sci. 2021, 11, x FOR PEER REVIEW 12 of 15

(a) (b)

Figure 9. Pressure distributions under different eccentricities: (a) Dimensionless pressure distributable 0.1; (b) Dimensionless pressure distribution calculated with the eccentricity of 0.6.

From the analysis of Figure 9, the pressure tends to be gentle along the circumference with the small eccentricity, and the pressure peak value is prominent and changes violently when the eccentricity is large. In the case of small eccentricity, the pressure is mainly provided by the V-groove’s pumping pressure effect and relatively scattered, which is mainly distributed in the groove rather than at the ridge; meanwhile, each V-groove corresponds to a local pressure peak point with a small value. In the case of large eccentricity, the overall wedge-shaped effect of bearing is more obvious, the pressure distribution is more concentrated with a larger peak value, which is mainly distributed in the ridge, while at the same time, the pump pressure effect is evident near the minimum film thickness. Regardless of eccentricity, compared with ordinary smooth bearing, V-g B had a more dispersed pressure distribution with a wider range, more peak points, and more stable pressure change.

4.3. Static and Dynamic Performance Analysis of Bearing The static and dynamic performances of hydrodynamic V-g B was analyzed by Appl. Sci. 2021, 11, 940 12 of 14 changing eccentricity from 0.1 to 0.9 while other parameters remained unchanged. Figure 10 shows the curve of the static and dynamic performance parameters of V-g B with eccentricity.

Appl. Sci. 2021, 11, x FOR PEER REVIEW 13 of 15

(a) (b)

(e) (f)

Figure 10. CurveFigure of the10. performanceCurve of the parametersperformance of parameters V-g B with eccentricity: of V-g B with (a) eccentricity: Dimensionless (a) bearingDimensionless capacity; (b) Discharge; (c) Dimensionlessbearing power capacity; consumption; (b) Discharge; (d) Temperature(c) Dimensionless rise; (powee) Dimensionlessr consumption; equivalent (d) Temperature stiffness; rise; (f) Dimensionless(e) critical mass.Dimensionless equivalent stiffness; (f) Dimensionless critical mass.

Based on the analysis,Based on it the can analysis, be found it that can bewith found the increase that with in the eccentricity, increase in the eccentricity, clear- the clear- ance between ancethe bearing between and the bearing bearing bush and decreases, bearing bush leading decreases, to the leadingincrease to of the the increase wedge of the wedge effect, then theeffect, oil film then pressure the oil increases film pressure as well increases as the bearing as well ascapacity. the bearing At the capacity. same time, At the same time, the dischargethe increases discharge linearly increases with linearly the increase with theof eccentricity. increase of eccentricity. When the eccentricity When the eccentricity in- increased to aboutcreased 0.6, to the about oil 0.6,film the became oil film thinner, became causing thinner, a causing significant a significant rise in the rise shear in the shear force, force, then thethen friction the frictionpower consumption power consumption increased increased rapidly. rapidly.Therefore, Therefore, due to the due com- to the combined bined influenceinfluence of friction of friction power power consumption consumption and anddischarge, discharge, the thetemperature temperature rise rise in- increases with creases with thethe trend trend of of eccent eccentricity,ricity, which which rose rose rapidly rapidly when when the the eccentricity eccentricity climbed climbed to to 0.6, and the ◦ 0.6, and the totaltotal temperature temperature rise rise was was within within 6.4 6.4 °C.C. Simultaneously, Simultaneously, the stiffness and critical mass critical mass keptkept enhancing enhancing with with the the increase increase of of eccentricity. eccentricity.

5. Conclusions In this paper, the viscosity of G-I-T LM was measured, and the relationship curve between viscosity and temperature was fitted. After considering the influence of turbulence, the film thickness and pressure distribution of the bearing calculated in CCS and OCS were analyzed and compared by using the Ng–Pan turbulent model; the difference of bearing pressure distribution under small and large eccentricity was studied, respectively, and the static and dynamic performances of GBLM lubricated V-g B such as bearing capacity, discharge, friction power consumption, temperature rise, stiffness and critical mass, with change of eccentricity were analyzed. The conclusions are as follows: Appl. Sci. 2021, 11, 940 13 of 14

5. Conclusions In this paper, the viscosity of G-I-T LM was measured, and the relationship curve between viscosity and temperature was fitted. After considering the influence of turbulence, the film thickness and pressure distribution of the bearing calculated in CCS and OCS were analyzed and compared by using the Ng–Pan turbulent model; the difference of bearing pressure distribution under small and large eccentricity was studied, respectively, and the static and dynamic performances of GBLM lubricated V-g B such as bearing capacity, discharge, friction power consumption, temperature rise, stiffness and critical mass, with change of eccentricity were analyzed. The conclusions are as follows: 1. The viscosity of G–I–T LM will decrease with the increase of temperature, so under the working condition of high speed and high temperature, the flow form of G–I–T LM is turbulent flow. 2. The spiral groove bearing’s carrying capacity is provided by the pump pressure effect and wedge effect together: in the case of small eccentricity, the pump pressure effect of the herringbone groove mainly provides pressure, and the overall wedge effect of bearing is more obvious under large eccentricity. 3. The static and dynamic performance parameters of bearing increase with the rise in eccentricity.

Author Contributions: Methodology, M.X.; investigation, M.X.; writing-original draft, M.X.; super- vision, W.C.; writing—review and editing, W.C. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by the National Science and Technology Major Project of China (grant number 2017YFC0111500). Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: The data presented in this study are available on request from the corresponding author. Acknowledgments: We acknowledge the support of the National Science and Technology Major Project of China and Xi’an Jiaotong University. Conﬂicts of Interest: The authors declare that there is no conﬂict of interests regarding the publication of this article.

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